Nonlinear Internal Model Control With Automotive Applications

Nonlinear Internal Model Control

with Automotive Applications

Vom Promotionsausschuss derFakultät für Elektrotechnik und Informationstechnik

derRuhr-Universität Bochum

zur Erlangung des akademischen GradesDoktor-Ingenieur

genehmigte Dissertation von

Dieter Schwarzmann

aus Temeschburg, Rumänien

Stuttgart, 2007

1st Reviewer: Prof. Dr.-Ing. Jan LunzeRuhr-Universität Bochum, Germany

2nd Reviewer: Prof.(i.R.) Dr.-Ing.Dr.h.c. Michael ZeitzUniversität Stuttgart, Germany

Thesis submitted on: 01.05.2007Date of examination: 24.07.2007

Acknowledgement

This dissertation is the result of four years of research as a student at theRuhr-Universität Bochum in collaboration with Robert-Bosch GmbH.

The present achievement was made possible thanks to the excellentsupervision of Prof. Lunze, who encouraged new ideas and approachesand constructively removed the bad ones. His work ethic, personal com-mitment and structured approach towards solving problems are amongstmany values I chose to adopt.

A special thanks goes to Dr. Rainer Nitsche without whom I probablywould have never started this Odyssee. His commitment and attitude to-wards advising PhD students allowed this research to be undisturbed bythe daily routine encountered at the company, Robert Bosch GmbH. Ofthat company, I would like to thank Dr. Thomas Bleile and Dr. MartinRauscher whose insight into function development and turbocharged en-gines was invaluable.

The largest contribution to the results of this thesis comes from all thestudents who have worked with me to solve the many problems. I am veryfortunate, having had so many capable minds helping me. Especially thededication of Andreas Schanz and Marco Schmidt will stay in my memoryas we have spent insanely many nights in front of a white board tryingto wrap our heads around the two-stage turbocharged air system. Manyof the ideas presented in this thesis stem from one of those evenings andmany more still need to be written down.

I would like to thank my colleagues at the Universität Bochum and mycolleagues at Robert Bosch GmbH. I am indebted to Tobias Kreuzinger,Jochen Assfalg, Dr. Rainer Nitsche, Dr. Olivier Cois and Dr. MatthiasBitzer, who, together with Jonas Hanschke and Prof. Zeitz from the Uni-versity of Stuttgart, sacri�ced their time to allow me to bounce ideas o�of them.

On a personal level, I would like to thank my parents and my friendsfor their patience and forgiveness during the time I committed myself tothis work. I am deeply grateful for Lisa Nicole Henson, who accompaniedme with her kindness and love through all the rough patches.

Thank you.

Stuttgart, January 2008 Dieter Schwarzmann

iv

�So many new ideas are at �rst strange and horrible, though

ultimately valuable that a very heavy responsibility rests upon those

who would prevent their dissemination.�

J. B. S. Haldane(1892-1964, British geneticist and scientist )

�Eccentricity is not, as dull people would have us believe, a form

of madness. It is often a kind of innocent pride, and the man of ge-

nius and the aristocrat are frequently regarded as eccentrics because

genius and aristocrat are entirely unafraid of and unin�uenced by

the opinions and vagaries of the crowd.�

Dame Edith Sitwell(1887-1964, English poet, critic and biographer;

Sister of Sir Osbert Sitwell)

�The engineer is the key �gure in the material progress of the

world. It is his engineering that makes a reality of the potential

value of science by translating scienti�c knowledge into tools, re-

sources, energy and labour to bring them into the service of man

. . .To make contributions of this kind the engineer requires the imag-

ination to visualize the needs of society and to appreciate what is

possible as well as the technological and broad social age understand-

ing to bring his vision to reality. �

Sir Eric Ashby(1904-1992, British botanist and administrator)

vi

CONTENTS

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Deutsche Kurzfassung (German summary) . . . . . . . . . . xiii

Nomenclature and Abbreviations . . . . . . . . . . . . . . . . xxiii

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . 11.1 Thesis Objective . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . 21.3 Controller Design in the Automotive Industry . . . . . . . 5

1.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . 51.3.2 Automotive Control Requirements . . . . . . . . . 61.3.3 Phenomenological Controller Design . . . . . . . . 81.3.4 Model-Based Controller Design . . . . . . . . . . . 13

1.4 Outline of the results. . . . . . . . . . . . . . . . . . . . . 15

Part I INTERNAL MODEL CONTROL 17

2. Internal Model Control of Linear SISO Systems . . . . . 192.1 Structure and Properties of Internal Model Control (IMC) 19

2.1.1 Considered System Class . . . . . . . . . . . . . . 192.1.2 IMC Structure . . . . . . . . . . . . . . . . . . . . 202.1.3 IMC Properties . . . . . . . . . . . . . . . . . . . . 22

2.2 Classical IMC Design . . . . . . . . . . . . . . . . . . . . . 262.2.1 IMC Design of Minimum Phase Systems . . . . . . 262.2.2 IMC Design of Non-Minimum Phase Systems . . . 302.2.3 Input Constraints . . . . . . . . . . . . . . . . . . 35

2.3 Feasibility of IMC as Automotive Controller . . . . . . . . 362.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Contents

3. Internal Model Control of Nonlinear SISO Systems . . 393.1 Mathematical Preliminaries . . . . . . . . . . . . . . . . . 393.2 Output-Function Space . . . . . . . . . . . . . . . . . . . 443.3 Right Inverse of the Model . . . . . . . . . . . . . . . . . . 493.4 Structure and Properties of Nonlinear SISO IMC . . . . . 533.5 IMC Design Procedure for Minimum Phase Models with

Well-De�ned Relative Degree . . . . . . . . . . . . . . . . 563.5.1 Control Goal and IMC Design Procedure . . . . . 573.5.2 Substitution of the Internal Model . . . . . . . . . 633.5.3 Robust Stability for Unstructured Uncertainties . . 643.5.4 Application to Linear Plants . . . . . . . . . . . . 66

3.6 Feasibility of Nonlinear IMC as Automotive Controller . . 683.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4. Internal Model Control Design for Input-A�ne SISO Sys-tems and Flat SISO Systems . . . . . . . . . . . . . . . . . 694.1 IMC of Flat SISO Systems . . . . . . . . . . . . . . . . . . 694.2 IMC of SISO Systems in Input/Output Normal Form . . 754.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5. Extensions of the Basic Principle . . . . . . . . . . . . . . 855.1 Input Constraints . . . . . . . . . . . . . . . . . . . . . . . 855.2 IMC for Systems with Ill-De�ned Relative Degree . . . . . 965.3 Non-Minimum Phase IMC Design using a Perfect Inverse 99

5.3.1 Background . . . . . . . . . . . . . . . . . . . . . . 1005.3.2 Introducing the Non-Minimum Phase IMC Design

Using Linear Systems . . . . . . . . . . . . . . . . 1035.3.3 Non-Minimum Phase IMC Design for Nonlinear Sys-

tems . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.3.4 Construction of the Non-Minimum Phase IMC Filter 115

5.4 Treating Measured Disturbances . . . . . . . . . . . . . . 1165.5 IMC of Simple Quadratic Nonlinear MIMO Systems . . . 1195.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

viii

Contents

Part II INTERNAL MODEL CONTROL OFTURBOCHARGED ENGINES 123

6. Control of a One-Stage Turbocharged Diesel Engine . . 1256.1 Function of a One-Stage Turbocharged Diesel Engine . . . 1256.2 Control Problem . . . . . . . . . . . . . . . . . . . . . . . 1266.3 Nonlinear IMC of the Air-System . . . . . . . . . . . . . . 128

6.3.1 Model Inverse . . . . . . . . . . . . . . . . . . . . . 1286.3.2 IMC Filter . . . . . . . . . . . . . . . . . . . . . . 1306.3.3 Complete IMC Law . . . . . . . . . . . . . . . . . 130

6.4 Test Bed Results . . . . . . . . . . . . . . . . . . . . . . . 1316.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7. Control of a Two-Stage Turbocharged Diesel Engine . . 1377.1 Function of a Two-Stage Turbocharged Diesel Engine . . . 1377.2 Control Problem . . . . . . . . . . . . . . . . . . . . . . . 1397.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . 1417.4 Nonlinear IMC of the Air-System . . . . . . . . . . . . . . 143

7.4.1 Model Inverse . . . . . . . . . . . . . . . . . . . . . 1437.4.2 Singularity of the Model Inverse of a Two-Stage Tur-

bocharged Engine . . . . . . . . . . . . . . . . . . 1447.4.3 IMC Filter . . . . . . . . . . . . . . . . . . . . . . 1467.4.4 Complete IMC Law . . . . . . . . . . . . . . . . . 1487.4.5 Robust Stability Analysis . . . . . . . . . . . . . . 149

7.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . 1537.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

8. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . 159

ix

Contents

Appendix 163

A. Compositional Model Library for Turbocharged Diesel En-gines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165A.1 Modelling Goal and Assumptions . . . . . . . . . . . . . . 166A.2 Connecting Individual Component Models . . . . . . . . . 167A.3 Component Models . . . . . . . . . . . . . . . . . . . . . . 170

A.3.1 Storage Models . . . . . . . . . . . . . . . . . . . . 170A.3.2 Coupling Models . . . . . . . . . . . . . . . . . . . 171

A.4 Parameter Estimation and Model Veri�cation . . . . . . . 180A.5 Model Simpli�cation . . . . . . . . . . . . . . . . . . . . . 183

A.5.1 Simpli�ed Chamber Model . . . . . . . . . . . . . 183A.5.2 Model Simpli�cation Procedure . . . . . . . . . . . 184

A.6 Control Design Models of One- and Two-Stage TurbochargedDiesel Engines . . . . . . . . . . . . . . . . . . . . . . . . 186A.6.1 Simpli�ed Model of a One-Stage Turbocharged Diesel

Engine . . . . . . . . . . . . . . . . . . . . . . . . . 186A.6.2 Simpli�ed Model of a Two-Stage Turbocharged Diesel

Engine . . . . . . . . . . . . . . . . . . . . . . . . . 187A.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

B. Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 199B.1 Two-Degrees-of-Freedom IMC . . . . . . . . . . . . . . . . 199

B.1.1 Linear Case . . . . . . . . . . . . . . . . . . . . . . 199B.1.2 Nonlinear Case . . . . . . . . . . . . . . . . . . . . 200

B.2 Ad Hoc Non-Minimum Phase IMC Design for Flat Systemsusing a Minimum Phase Model . . . . . . . . . . . . . . . 201

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

x

Abstract

This work develops an internal model control (IMC) design method fornonlinear plants and employs this method to design pressure controllersfor a one-stage and a two-stage turbocharged diesel engine.

The main focus lies on developing an applicable controller designmethod for automotive control problems. Automotive applications arecharacterised by a combination of the limited computational power of thecar's on-board control unit and the nonlinear character of the systems tobe controlled. Moreover, a required step in the development of series pro-duction controllers is the manual adaptation (calibration) of the controllerparameters after the actual design. For this reason, the controller shouldprovide tunable parameters. IMC is proposed as the control structure andthe parameters of the internal model are chosen as tunable parameters.

The contribution of this thesis is two-fold. First, this work presents anIMC design procedure for nonlinear single-input, single-output systems.The nonlinear IMC, as proposed here, is based on the IMC structureknown from linear systems and is based on a nonlinear feedforward controldesign. It is inversion-based and uses a low-pass state-variable �lter whichconnects to the right inverse of the plant model to obtain a realisable IMCcontroller. Basic system properties, such as relative degree and internaldynamics, are exploited to extend the system class to stable and invertibleplants. Input constraints and model singularities are taken into accountby using a nonlinear low-pass �lter that is made aware of the possibleinput/output behaviour of the model. This awareness is introduced by amodel-dependent constraint of the �lter's highest output derivative. Thenonlinear IMC provides robust stability and robust tracking of the closed-loop system.

Second, the feasibility of this control scheme is presented. A single-input, single-output boost-pressure IMC controller is designed for a one-stage turbocharged diesel engine. The controlled plant was tested at thetest bed and showed good results, surpassing the performance of the pro-duction PID-type controller. Two-stage turbocharging recently producedinterest among car manufacturers and poses a challenging control problemdue to the nonlinearity of the MIMO plant and a singularity of its inverse.This thesis presents the �rst model-based solution to this control prob-lem. A multi-input, multi-output nonlinear IMC controller is designedand tested in simulations, showing good performance and robustness.

Contents

xii

DEUTSCHE KURZFASSUNG

In der vorliegenden Arbeit wird eine Methode zum Entwurf eines nicht-linearen �Internal Model Control� (IMC) Reglers entwickelt. Diese Meth-ode wird verwendet, um Ladedruckregler für einen einstu�g sowie einenzweistu�g aufgeladenen Motor zu entwickeln. In dieser Zusammenfassungwird beispielhaft auf die anwendung der IMC-Regelung bei dem zweistu�gaufgeladenen Dieselmotor eingegangen.

Entwurf eines nichtlinearen SISO IMC-Reglers

IMC-Struktur

Abbildung 1 zeigt die IMC-Struktur mit IMC-Regler Q, Regelstrecke Σund Modell der Regelstrecke Σ. Die Arbeitsweise des IMC ist die Folgende:Ist das Modell Σ exakt (Σ = Σ) und es treten keine Störungen auf (d =0), so verschwindet das Rückführsignal (y(t) − y(t) = 0) und der IMC-Regler Q agiert als Vorsteuerung (s. Abb. 1b). Bei Modellfehlern undStörungen wirkt die Rückführung diesen E�ekten automatisch entgegen.Für Systeme mit kleinen Modellunsicherheiten und Störungen ist es alsowünschenswert, den IMC-Regler Q als Vorsteuerung zu entwerfen. Mit

w uK

Q

Σ~

Σdy

y~

w~−

−

(a) IMC-Struktur

w Q u yΣ=Σ ~

(b) IMC für Σ = Σ undd = 0

Abb. 1: IMC-Struktur.

w =w − y + y

y =Σ ◦Qw(1)

�ndet man folgende strukturelle Eigenschaften der IMC-Struktur, welcheauch für nichtlineare Systeme gelten [28]:

Eigenschaft 1 (Stabilität). Wenn das Modell exakt ist (Σ = Σ), ist dergeschlossene Regelkreis in Abb. 1 stabil, wenn der IMC-Regler Q und dieRegelstrecke Σ stabil sind.

Eigenschaft 2 (Perfekte Regelung). Der Ausgang y(t) folgt der Führungs-gröÿe perfekt (y(t) = w(t), ∀t > 0) für beliebige Störungen d(t), wenn der

IMC-Regler Q die Inverse des Modells ist (Σ◦Q = 1) und der geschlosseneRegelkreis in Abb. 1 stabil ist.

Eigenschaft 3 (Verschwindende Regelabweichung). Es entsteht keinebleibende Regelabweichung bei konstanten Führungsgröÿen (limt→∞ y(t) =limt→∞ w(t) = w) und bei konstanten Störungen limt→∞ d(t) = d, wenndie stationäre Verstärkung Qss des IMC-Reglers invers zu der stationärenVerstärkung des Modell (Σss ◦Qss = 1) und der Regelkreis in Abb. 1 stabilist.

Eigenschaften 2 und 3 gelten ohne explizite Forderung an die Modell-güte. Eigenschaft 3 besagt, dass eine verschwindende Regelabweichungerreicht wird, wenn die stationären Verstärkungen von Modell Σ undIMC-Regler Q invers zueinander sind. Es ist also nicht notwendig, einenexpliziten Integralteil in den Regler einzuführen.

IMC-Entwurf für nichtlineare Systeme

Im Folgenden wird eine IMC Entwurfsmethode für nichtlineare Systemevorgeschlagen. Hier wird auf Eingröÿensysteme mit wohl de�niertem rel-ativen Grad und vorerst ohne Eingangsbeschränkungen eingegangen. Dienichtlineare Regelstrecke Σ, das nichtlineare Modell Σ der Regelstreckeund der IMC Regler Q werden als nichtlineare Operatoren betrachtet. DieRechtsinverse Σr des Modells Σ wird als Operator mit der Eigenschaft

Σ ◦ Σr ◦ y(t) = y(t) (2)

de�niert. Die Rechtsinverse Σr generiert die Eingangstrajektorie u(t), sodass der Modellausgang y(t) der gegebenen Trajektorie yd(t) exakt folgt.

xiv

Deutsche Kurzfassung

Ein nichtlinearer IMC-Regler Q kann nun angegeben werden als Verknüp-fung eines linearen Tiefpass�lters F und der Rechtsinversen Σr

Q = Σr ◦ F . (3)

Bestimmung der Rechtsinversen. Die Rechtsinverse kann zum Beispiel(angelehnt an [35]) wie folgt bestimmt werden. Gegeben sei das ModellΣ

Σ : x = f(x, u), x(0) = x0 (4a)

y = h(x) (4b)

mit relativem Grad r, welcher durch

r = arg minr

{∂

∂uLrfh(x, u) 6= 0

}de�niert ist. Hierbei stellt Lf die Lie-Ableitung entlang des Vektorfeldsf dar. Mittels der Zustandstransformation

[y, ˙y, . . . , y(r−1),η]T = φ(x) mit (5a)

y(i) = Lifh(x) = φi+1, i = 0, . . . , r − 1 (5b)

η = φη(x) ∈ Rn−r (5c)

kann das System Σ in die nichtlineare Ein-/Ausgangs-Normalform

y(r) = α(y, ˙y, . . . , y(r−1),η, u

)(6a)

η = β(η, y, ˙y, . . . , y(r−1), u

)(6b)

mit α(·) = Lrfh ◦ φ−1 und β(·) = Lfφη,i ◦ φ−1 transformiert werden. Eswird angenommen, dass der relative Grad r wohl de�niert ist und damit∂α∂u 6= 0 zumindest lokal gilt.

Aus der Ein-/Ausgangs-Normalform (6a) wird die Rechtsinverse Σr als

Σr : u = α−1(yd, ˙yd, . . . , y

(r)d ,η

)(7a)

η = β(η, yd, ˙yd, . . . , y

(r−1)d , u

)(7b)

bestimmt.

xv

F

d~y

r~Σ

)(d~ ry

−

−w w~ Σ

Σ~

u y

y~

Q

M

Abb. 2: IMC-Struktur nichtlinearer Systeme Σ.

−w~ ∫ ∫ ∫

+

+

1−rk

2−rk

)1(d~ −ry

)2(d~ −ry

d~yL

O

rk/1

M

+

)(d~ ry

M

F

Abb. 3: IMC-Filter F .

IMC-Filter. Die Rechtsinverse Σr benötigt denWert der Solltrajektorieyd sowie dessen erste r Zeitableitungen ˙yd, . . . , y

(r)d . Diese müssen vom

IMC-Filter F berechnet und der Rechtsinversen übergeben werden.Die resultierende IMC-Struktur wird in Abb. 2 gezeigt. Damit das

IMC-Filter F die Trajektorie yd sowie die Information der Ableitungen˙yd bis y(r)

d übergeben kann, wird vorgeschlagen, das IMC-Filter F alsZustandsvariablen�lter (siehe Abb. 3) mit der Übertragungsfunktion

F (s) =yd(s)w(s)

1krsr + kr−1sr−1 + · · ·+ 1

!=1

(s/λ+ 1)r(8)

zu implementieren. Es ergibt sich somit als Di�erentialgleichung für denFilterausgang yd

kry(r)d + kr−1y

(r−1)d + . . .+ yd = w. (9)

Zusammenfassung des IMC-Entwurfs. Der IMC-Entwurf für nicht-lineare Systeme kann in den folgenden Schritten zusammengefasst werden:

1. Berechne die Rechtsinverse Σr des Modells Σ der Regelstrecke Σ.

2. Entwerfe das IMC-Filter F wie in Gl. (8).

xvi


3. Implementiere das IMC-Filter als Zustandsvariablen�lter wie inAbb. 3.

4. Die resultierende Reglerstruktur ist in Abb. 2 dargestellt.

Der IMC-Regler Q für nichtlineare Systeme (4), welche in die Ein-/Ausgangs-Normalform transformiert werden können, setzt sich ausden Gleichungen (7) für die Rechtsinverse und der Gleichung (9) desFilters zusammen.

Die Existenz der Rechtsinversen ist eine notwendige Voraussetzungfür den IMC-Entwurf. Die Berechnung der Rechtsinversen eines Modellsist abhängig von der Systemklasse des Modells. Die Bestimmung vonRechtsinversen für �ache und Systeme in Ein-/Ausgangsnormalform wer-den in dieser Arbeit gezeigt.

Der dargestellte Entwurf eines nichtlinearen IMC ist eine direkte Er-weiterung des linearen IMC-Entwurfs. Der resultierende Regelkreis basiertauf einer Ausgangsrückführung, ist nominell stabil und hat eine verschwind-ende bleibende Regelabweichung.

Berücksichtigen von Stellgröÿenbeschränkungen. Sind die Stell-gröÿen u ∈ [umin, umax] beschränkt, so kann das Filter geändert werden,dass es nur eine solche Wunschtrajektorie yd produziert, welche mit derbeschränkten Stellgröÿe vom Modell exakt erreicht werden kann (yd = y).

Die dafür notwendige Änderung erkennt man mittels Gl. (6a). Sindnämlich das Vektorfeld f sowie die Ausgangsfunktion h des Modells (4)analytische Funktionen von ihren Argumenten x und u, so folgt, dassdie Funktion α auch analytisch (und damit stetig) in x und u ist. Somitbildet α bei gegebenen yd, . . . , y

(r−1)d das zusammenhängende Intervall der

zulässigen Stellgröÿe u ∈ [umin, umax] in ein zusammenhängendes Intervallfür die zulässige höchste Ableitung yd ∈ [y(r−1)

d,min , y(r−1)d,max] ab. Dies geschieht

mittels

y(r)d,min = min

u∈[umin,umax]

{α(yd, ˙yd, . . . , y

(r−1)d ,η, u

)}(10a)

y(r)d,max = max

u∈[umin,umax]

{α(yd, ˙yd, . . . , y

(r−1)d ,η, u

)}. (10b)

Wird das IMC-Filter wie in Abb. 4 verändert, so wird durch die Sätti-gungskennlinie gesichtert, dass nur solche Trajektorien yd generiert wer-den, deren höchste Ableitungen y(r)

d in dem erlaubten Invervall sind.

xvii

−w~

)(max,d

~ ry)(min,d

~ ry

∫ ∫ ∫

+

+

1−rk

2−rk

L

O

rk/1

M

+

M

F )1(d~ −ry

)2(d~ −ry

d~y

)(d~ ry

Abb. 4: Nichtlineares IMC-Filter mit Sättigungskennlinie zur Real-isierung von Eingangsbeschränkungen.

Dadurch wird die Rechtsinverse nie eine Stellgröÿe u produzieren, welchedie Stellgröÿenbeschränkungen verletzt.

IMC Druckregelung eines zweistu�gturboaufgeladenen Motors

HD Laufwerk

NDK

NDT

HDT

HDK

Motor

HDK Bypass

HDT BypassLPT

Wastegate

Ladeluftkühler

Abgasnach-behandlung

ND Laufwerk

NDTA

HDTA

HDKA

1V

2V

3V

4V

5V

5V 6V

0V

En

fm&

Abb. 5: Zweistu�g aufgeladener Motor.

Ein zweistu�g turboaufgeladener Motor mit Drehzahl nE und Mengeeingespritztem Kraftsto�s mf (siehe Abb. 5) hat zwei sequentiell angeord-

xviii


nete Turbolader. Jeder Turbolader wird als Stufe bezeichnet, wobei dieStufe am Motor die Hochdruckstufe (HD) und die Stufe an der Umgebungdie Niederdruckstufe (ND) genannt wird. Die eingesaugte Luft wird zuerstvom Niederdruckkompressor (NDK) und dann vom Hochdruckkompres-sor (HDK) komprimiert. In derselben Weise wird der Abgasstrom zuerstüber die Hochdruckturbine (HDT) und dann über die Niederdruckturbine(NDT) geleitet. Der Fluidstrom kann mittels Bypässen um den HDK alsauch um beide Turbinen geleitet werden. Hierfür müssen die Ö�nungs-querschnitte AHDK, AHDT und ANDT entsprechend gestellt werden.

Es gilt, einen Regler zu �nden, so dass die folgendenden Bedingungenerfüllt werden:

• Die gemessenen Drücke y = [p2, p3, p4]T , welche in den KammernV1, V2 und V3 auftreten, folgen der Führungsgröÿe w ohne bleibendeRegelabweichung.

• Die Stellgröÿen sind die Querschnitts�ächen der Bypässeu = [AHDK, AHDT, ANDT]T

• Die Stellgröÿen u sind beschränkt durch 0 ≤ ui ≤ uimax mit i =1, 2, 3.

Die Eigenschaften der Regelstrecke erschweren die Regelungsaufgabeaus folgenden Gründen.

• Die Motordrehzahl nE und die eingespritzte Kraftsto�menge mf kön-nen für die Regelungsaufgabe als messbare Störungen interpretiertwerden. Deren Wirkung muss vom Regler kompensiert werden.

• Es handelt es sich um eine Mehrgröÿenregelung, so dass der zuvorvorgestellte IMC-Entwurf auf den Mehrgröÿenfall angewendet wer-den muss.

• Die Regelstrecke verliert die Eigenschaft der Invertierbarkeit währenddes Betriebs, wenn der Ladedruck p2 gleich dem Druck p1 zwischenden Kompressoren ist, denn dann verliert der Eingang u1 = AHDK

seine Wirkung auf das System.

Die Regelstrecke wird durch ein stabiles nichtlineares eingangsa�nesMehrgröÿensystem repräsentiert. Der nichtlineare IMC nutzt die Ein-/Ausgangsnormalform für eingangsa�ne Systeme, um die Rechtsinversezu bestimmen. Die Stellgröÿenbeschränkung wird mittels des limitierten

xix

IMC-Filter F nach Abb. 3 realisiert, welches die Trajektorie yd des Ein-gangs der Rechtsinversen Σr so berechnet, dass diese ohne Verletzen derStellgröÿenbeschränkung erreicht werden kann. Die Singularität der Regel-strecke wird behandelt, indem der Regler das System genau so durch dieSingularität steuert, dass diese Bewegung mittels vorhandener Stellgröÿenerreicht werden kann. Dadurch ist die Rechtsinverse stets de�niert.

Abbildung 6 zeigt das Simulationsergebnis des Ladedrucks p2, des Ab-gasgegendrucks p3 und des Drucks zwischen den Turbinen p4 im Vergleichzu ihren Sollwerten w1, w2 und w3. Ferner ist die Motordrehzahl nE sowieder eingespritzte Kraftsto� mf als auch die aus dem IMC-Regler berech-neten Bypass�ächen u1, u2 und u3 dargestellt. Die Simulation wurde sogewählt, dass sie realistische Betriebsbedingungen wiedergibt. Auÿerdemwerden Modellfehler in der Simulation angenommen.

Die Simulationsergebnisse zeigen eine hohe Regelgüte als auch Robus-theit gegenüber Modellfehlern.

xx


0 25 50 75Drehzahl [100 U/min]Kraftstoff [mg/hub]

n

Em

f

0 1 2 3

Ladedruck [bar]

w

2y1

=p

2p

1

1 2 3

Abgasgegendruck undZwischenturb. Druck

[bar]

w

2y2

=p

3w

3y3

=p

4

05

1020

2530

0 5 10

Zeit [s]

Bypassfläche [cm2]

u

1u

2u

3

Abb. 6: Simulationsergebnis für einen realistisch parametriertenzweistu�g aufgeladenen Dieselmotor mit IMC-Regelung.

xxi

xxii

NOMENCLATURE AND ABBREVIATIONS

A list of symbols is not given for the complete thesis, as some variableschange their meaning in dependence on the context. All variables areproperly labelled when used. An example is ended by the symbol �, ade�nition is ended by the symbol �, and a proof is ended by �.

The commonly seen format for variables, vectors, and matrices areused (see e. g., [63] for a description). Additionally, the entries of a vector,say x, are understood to be labelled as x1, x2, . . ., and so forth. Thus,when introducing the variables x1, x2, . . . it can be assumed, without anexplicit introduction, that they belong to a vector x.

Signals that are given in Laplace domain are denoted by small letterstogether with their dependency on the independent variable s, for exam-ple, u(s) denotes the Laplace transform of the input signal u(t). When itis desired to emphasise the linearity of a system, then this system is pre-sented in its Laplace transform, for example, G(s) means the linear andLaplace transformed system. The usual abuse of notation, namely notto use units in transfer functions or their poles and zeros, see e. g., [87],is employed. More precisely, within this text time is understood toalways have the unit of seconds and amplitude is dimensionless,unless noted di�erently. Therewith, the independent variable s in Laplacedomain has units of 1/seconds. However, in this text transfer functionsand their poles and zeros do not convey those units.

Finally, the following table lists the frequently used abbreviations(acronyms) and their meaning.

List of abbreviations.

HPC High-pressure compressorI/O Input-to-output, or input/outputIMC Internal model controlISE Integral square error (norm)LPC Low-pressure compressorMIMO Multi-input multi-outputMP Minimum phaseNMP Non-minimum phaseOCU On-board control unitPID Proportional Integral Derivative (Control)RHP Right-half (of the complex) planeSISO Single-input single-outputSVF State-variable �lterVNT Variable-nozzle turbine

xxiv

1. INTRODUCTION

1.1 Thesis Objective

This thesis is motivated by control problems in the automotive industry.The objective is to develop a controller design method which is particu-larly suited for automotive control problems. Automotive control prob-lems are characterised by a combination of the limited computationalpower of the car's on-board control unit and the nonlinear character ofthe systems to be controlled. The demands on closed-loop behaviour in-clude performance criteria in time domain, such as rise time, settling timeand overshoot. Moreover, a required step in the development of seriesproduction controllers is the manual adaptation (calibration) of the con-troller parameters. For this reason, the controller should provide tunableparameters.

The predominant controller design for automotive applications is atedious process which relies heavily on trial-and-error. It is proposedto substitute this design method with a model-based controller designmethodology by employing internal model control (IMC).

The classical IMC design concerns the control of linear systems. Al-though concepts to use IMC to control nonlinear systems exist, they do notshare the IMC design philosophy and are not based on the IMC structure;as a consequence, they do not retain the typical IMC properties. This the-sis proposes a nonlinear IMC design methodology which is based on thetypical IMC design philosophy and employs the classical IMC structure.As a result of this, the nonlinear IMC inherits all closed-loop properties ofthe classical IMC, including robust stability and zero steady-state o�set.

Two automotive control problems are solved using the proposed non-linear IMC design method. They are the boost pressure control problemof a one-stage turbocharged diesel engine and a multi-input, multi-outputpressure control problem of a two-stage turbocharged engine. In the caseof the two-stage turbocharged engine, the proposed IMC presents the �rstpublished control solution to this problem which is fully able to utilise the

1. INTRODUCTION

behaviour of the plant.

1.2 Literature Review

Evaluation of relevant existing control design methods. The fol-lowing brie�y evaluates some controller design methods that are related tothe IMC design as presented here. Therefore, some nonlinear control de-signs (often referred to as �constructive nonlinear control�) [57, 83], suchas construction of a Lyapunov function, backstepping, etc. are not dis-cussed. Concerning the extensions proposed in Chapter 5, a topic-speci�cliterature review is given when the individual extensions are introduced.

Internal model control. The internal model control structure is thebasis for this thesis. It is presented in detail in Chapter 2. The mainidea of IMC is that its structure degenerates to a feedforward control loopin the case of an exact model. The IMC design focuses on feedforwardcontrol design and relies on model inversion.

Concepts similar to IMC have been used since the late 1950s [50, 73]to design optimal feedback controllers. However, in 1974, it was �rst pro-posed [31] to use the typical IMC feedback structure to control processes.In 1982 [72], the IMC structure was employed in the �eld of robust control.IMC is at the core of the Smith Predictor [88] as well as some predictivecontrol strategies [32]. It has been extended by an adaptive control strat-egy in [21] and is reviewed in some standard textbooks on control engineer-ing like [63, 87]. Today, [72] is widely accepted as the standard referenceof IMC. In 2002 [9], an IMC design procedure was introduced that guaran-tees robust performance of linear systems and presents an advanced linearIMC design method.

One �nds that IMC meets all demands on an automotive controller,with the exception that its classical design method is limited to linearsystems.

Nonlinear internal model control. The classical IMC design was ex-tended to nonlinear systems (see e. g., [28, 31, 44, 101]). It was establishedalready in the original work on IMC [31] in 1974 that the IMC structure isfeasible to control nonlinear systems. The publications [28, 31] introduceda nonlinear IMC which relies on on-line numerical inversion of the model,based on iterative methods. Since this approach to nonlinear IMC uses a

2

1.2. LITERATURE REVIEW

computational intense operation, it is likely that it cannot be executed inreal-time by a car's on-board control unit (OCU).

In [44, 101], the goal is an exact I/O linearisation around which theIMC structure is placed. The contribution of the IMC structure to anI/O linearisation is that the internal model works as a feedforward stateobserver and that zero steady-state o�set is guaranteed. However, thisnonlinear IMC does not employ the typical IMC feedback structure, whichdoes not have state feedback from the model. These nonlinear IMC ap-proaches share the limitations of I/O linearisations as well as the limi-tations of IMC: The system class is limited to input a�ne, stable, andminimum phase1 models. Thus, this approach to nonlinear IMC is un-able to reproduce the classical IMC which, for example, is able to controlnon-minimum phase plants.

In summary, current extensions of IMC to nonlinear systems either usenumerical inversion methods or do not employ the classical IMC structureor its design philosophy.

The main contribution of this work is the development of a generaldesign concept for a nonlinear IMC which does not necessitate the feed-back linearisation of the plant or plant model. It is based on some ideaspresented in [28] as it also relies on the concept of the right inverse ofthe plant model. Using concepts from exact I/O linearisation and �at sys-tems, the right inverse is obtained analytically and not numerically. Asthis inverse is, in general, not realisable since it depends on di�erentia-tions of its input signal, it is proposed to combine it with a low-pass �lter.The composition of the �lter with the inverse results in a non-anticipativesystem, which is realisable. The functional analytic interpretation of theclosed-loop behaviour allows to extend the system class to which the non-linear IMC is applicable. That is, it can treat input constraints, modelsingularities, and unstable model inverses. None of these can be treatedby available approaches to nonlinear IMC. Moreover, the resulting controlstructure is identical to the classical IMC structure and can be designedand interpreted as the classical IMC.

The result is a nonlinear (IMC) controller which is robustly stable,guarantees zero steady-state o�set, is straightforward to calibrate, re-spects input constraints, can handle model singularities, and is not re-stricted to minimum phase models.

1 The property of (non-) minimum phase will, later on in this work, be de�ned as(in-)stability of the model inverse.

3

1. INTRODUCTION

Exact Input-Output Linearisation Techniques The idea behind exactinput-output (I/O) linearisation is to design a controller for a nonlinearplant such that the behaviour from the input of the controller to plantoutput is linear. The proposed nonlinear IMC uses some of the mathemat-ical tools and concepts (in particular the notion of relative degree and theI/O normal form) that are used also in the context of I/O linearisation.

The majority of the literature on this topic (see e. g., [4, 5, 20, 41�43, 51, 54, 58, 60, 61]) focuses on input-a�ne nonlinear systems. An I/Olinearisation uses a nonlinear model of the plant which is typically derivedfrom physical laws and has physical parameters. Hence, a calibration of anI/O linearising controller consists of changing the physical model parame-ters which simpli�es controller calibration. The computational burden isacceptable for an OCU since no on-line optimisation or similar computa-tionally intense operations are needed. However, major drawbacks are thenecessity of state feedback and the limitation to minimum phase plants.Moreover, robust stability of the closed-loop is di�cult to prove.

If a system possesses the property of di�erential �atness (see e. g., [6,29, 30, 39, 81]), it can elegantly be exploited for nonlinear controllerdesign. The work of [39] addresses robustness issues of �atness-basedcontrollers. The major advantage of �at systems is the simplicity withwhich a nonlinear feedforward control law can be designed [40, 102, 103].

In a two-degrees-of-freedom structure, a �atness-based feedforwardcontroller can also be used with a linear PID feedback controller and,thus, also only necessitates output feedback.

Control of turbocharged engines.

One-stage turbocharged engine. There are numerous publications con-cerning the control of this plant. Work on control solutions of one-stageturbocharged engines includes [52], where a multi-input, multi-output lin-ear parameter variant controller for both boost pressure and exhaust gasrecirculation is presented. In [82], linear models for each operating pointare computed through system identi�cation based on test bed measure-ments. For each operating point, a linear model predictive controlleris designed. The resulting controller is obtained by scheduling betweenthe individual operating point-speci�c controllers. However, nominal sta-bility is not guaranteed. In [71, 74] several linear control concepts forboost pressure and exhaust gas recirculation are introduced and evalu-ated. Of these, an H∞ approach yields nominal and robust stability.

4

1.3. CONTROLLER DESIGN IN THE AUTOMOTIVE INDUSTRY

However, no dedicated calibration parameters are o�ered and one designiteration of an H∞ controller can take several minutes on a standard PCand does not address time-domain behaviour directly; thus, calibrationis time consuming. Some other solutions use nonlinear model predictivecontrol [33, 45, 78, 95].

In conclusion, there is no current solution which o�ers calibration pa-rameters that can be selected online at the test bed, is guaranteed to benominally stable, and is inexpensive enough to be implemented in today'sor future OCUs.

Two-stage turbocharged diesel engine. For the two-stage turbocharg-ed diesel engine there exists no solution capable of using the full potentialof the plant. However, there are two published approaches [90, 96] thatdeal with the control of a two-stage turbocharged engine. Both designsare essentially based on trial-and-error and control the MIMO plant of atwo-stage turbocharged engine by a single-input, single-output switchingcontroller which switches not only its parameterization but also its con-trolled variable. More detail on these publications is given in Section 7.3since their assessment requires some introduction to the plant.

1.3 Controller Design in the Automotive Industry

1.3.1 Introduction

Today's automobiles are complex machines and perform a variety of tasks,beyond mere transportation. These tasks range from safety-related pro-cesses like anti-lock braking or deployment of an airbag; comfort-relatedfunctions like power steering, cruise control and air conditioning; andenvironmental-oriented functions like injection timing or catalytic conver-sion of exhaust gas. These tasks are controlled by programs (controllers)that run on the on-board control units (OCUs). Modern vehicles havenearly one hundred OCUs, each executing several control tasks.

Today, approximately (see [11])

• 90% of all innovations in automobiles are driven by electronics andsoftware,

• 50%-70% of an OCU's development cost is determined by software,and

5

1. INTRODUCTION

• 40% of a car's production cost stems from electronics and softwaredevelopment.

In addition, a car's OCUs collectively

• contain about ten million lines of code and

• run about 2500 features/tasks on

• four di�erent bus systems concurrently.

From an engineering perspective, the automotive industry can be splitinto car suppliers and manufacturers. Most components of a car such asengine, power windows, drive train, electric motors, etc. are common tomany cars and pose common control problems. The controllers are of-ten developed by the supplier who delivers the respective OCU to severalcar manufacturers and therefore has to solve many control problems forcomponents that share the same functionality (but not necessarily size orshape). Thus, the majority of control problems are not new but rather avariant of an older one. For such plants, a controller is obtained by man-ually adapting (calibrating) the available controller. As a result, today'scontrollers o�er dedicated calibration parameters with which the closed-loop behaviour can be adjusted by a calibration engineer, with less e�ortthan a complete new controller design. In order to do this e�ciently, thefunction of the controller should be easy to comprehend. Today, thereis an organisational separation between controller design and controllercalibration. Hence, either the two are done by di�erent departments in asingle organisation, or they are done by di�erent companies. Thus, if amodern control design method is proposed, it should still o�er calibrationparameters, since the process of calibration has become an important stepin the development of the �nal product.

Since this thesis is concerned with methods for controller design, therequirements of the automotive industry must be considered. These re-quirements are discussed in detail in the following section.

1.3.2 Automotive Control Requirements

The goal of control design is to obtain a controller K which meets someprede�ned requirements on the closed-loop behaviour. Figure 1.1 showsa general control loop with controller K, plant Σ, control input u, distur-bance d, reference signal w and plant output y. In this thesis, the plant Σis assumed to be asymptotically stable. The closed-loop behaviour must

6


K Σuw y

d

Fig. 1.1: General control-loop structure.

meet given speci�cations, despite some disturbance d that a�ects the plantin an undesirable way. Figure 1.2 shows the standard control loop withcontroller C, which can be used as a speci�c realisation of the general con-

d

w C u−

y

Fig. 1.2: Standard control loop.

trol loop shown in Fig. 1.1. The reference value w is in�uenced directly bythe driver. Thus, future reference values of automotive control loops areunknown, which stands in contrast, for example, to the process industryin which the desired plant behaviour is de�ned by a given recipe.

In general, automotive control problems share the following propertiesand demands:

D1: Demands are posed in the time domain. Performance criteria con-cern properties like settling time, overshoot, and decay ratio of theclosed-loop step response.

D2: The closed-loop behaviour should be robustly stable and obtain zerosteady-state o�set, despite part production tolerances and a con-stant disturbance.

The following are automotive-speci�c restrictions on the control law.

D3: The control algorithm has to be executed in real-time on the car'sOCU2.

D4: The controller must have adjustable parameters with which proper-ties of the closed-loop system can be changed (calibrated), preferably

2 Due to cost constraints, OCUs are computationally weak and o�er little memoryin comparison to a standard PC.

7

1. INTRODUCTION

on-line, during a test bed run or during a test drive in an experimen-tal vehicle. By manually calibrating the controller parameters, thecontroller should be able to control all structurally identical plants.

D5: Since parameters of the controller have to be calibrated, its func-tionality should be easily understood by the calibration engineerand the customer. Hence, the simplicity of a controller function isan important point to be considered.

Demands D1 and D2 are typically encountered in control engineering ofmany �elds. Since the behaviour in time-domain is important, a controldesign, which can directly incorporate these demands in the design pro-cess, is preferred. A direct result of Demand D3 is that the controllershould not rely on computationally intensive on-line optimisation proce-dures. Demands D4 and D5 are unusual in control engineering and stemfrom the unique requirements of the automotive industry.

Remark 1.1 (Additional requirements). The implementation of a con-troller in an OCU requires two additional demands to be met. These arethat the controller has to be represented by a sampling control algorithmand that, as of today, mainly �xed point arithmetic must be used.

Although both of these are important demands, they are not treated inthis thesis explicitly. Concerning the necessity for a sampling control,it is assumed that the engineer is able to implement a continuous timecontroller in a sampling control algorithm. Since future OCUs will havethe possibility to process �oating point calculations, this requirementseemed less important and its treatment is assumed beyond the scopeof this work.

The following chapter investigates and evaluates today's predominantcontroller design procedure.

1.3.3 Phenomenological Controller Design

The term �phenomenological� is used to qualify a process that relies mainlyon repetitive observation and adaptation, in such that every change in thecontroller is motivated by an observed phenomenon of the plant. Thus, aphenomenological approach can be described as �trial-and-error�.

Today's predominant controller design is done in a phenomenologicalprocess. This process consists of two phases. In the �rst phase, the control

8


structure is chosen. In the second phase, the controller parameters arecalibrated until the closed-loop behaviour satis�es the speci�cations.

Predominantly gain-scheduled PID controllers are used because of theclear implications of changes of the gains KP, KI, and KD and the ex-istence of tuning rules (e.g. Ziegler-Nichols) which provide closed-loopstability and some robustness (see [86] for a detailed investigation ofthe most popular tuning rules and [1] for an overview on gain schedul-ing). The transfer function C(s) of a PID-controller, which is imple-mented in the standard control loop shown in Fig. 1.2, is given by C(s) =KP + KI

s + KDs1+T1s

.

Control structure. Figure 1.3 shows the typical control structure. It

Setpoints

Adaptation

Feedforward

+-

Driver

Parameters

Environment

u yw Σ

C

d

k

Feedforward

PID

Fig. 1.3: Typical closed-loop control used to control automotiveplants.

consists of a feedforward and a feedback path. However, the feedforwardcontroller is not designed to improve performance but rather to compen-sate static nonlinearities: If the plant was linearised around the state xss

and the input uss, the feedforward controller is a static block with re-inforcement uss. Therefore, its degree-of-freedom is not used since it isdetermined by the steady-state behaviour of the plant Σ. Setpoints ware de�ned through the driver's demand and the current output y of thevehicle.

The gains, KP,KI,KD are adapted in dependence upon the current op-eration mode and therein also in dependence upon the current operatingpoint. The main idea, presented in the following in detail, is to partitionthe range of the measurement signals and then to �nd controller parame-

9

1. INTRODUCTION

ters in dependence of each partition. The following de�nes the notion ofoperation mode and operating point, as it is typically understood in theautomotive industry, from a system theoretical perspective:

Operating point: Suppose that the chosen sensor signals are writtenin a vector k = [k1, . . . , kp]T , where e. g., k1 might be the current ambienttemperature and k2 could be the engine speed. Each entry ki lies in arange [kimin, kimax] with i = 1, 2, . . . , p. This range is discretised intomi ∈ N data points. Denote a discretised sensor signal as ki. The currentvalue of the vector of discretised sensor signals k = [k1, . . . , kp]T is theoperating point. Note that the mi data points of each discretised vectorki serve as nodes for look-up tables.

Operation mode: De�ne a set K, here called the measurement space

K = {k, ki ∈ [kimin, kimax]} , i = 1, 2, . . . , p. (1.1)

The set K consists of all possible combinations of measurement values. Inorder to de�ne l operation modes, this set is divided into l distinct subsetsKξ with ξ = 1, . . . , l

Kξ ⊂ K, where Ki ∩ Kj = {} for i 6= j

and⋃ξ

Kξ = K. (1.2)

When the current measurement lies in the ξ-th subset (k(t) ∈ Kξ), onesays that the plant is in operation mode ξ. Operation modes are used tovary the structure of the adaption algorithm. Figure 1.4 gives an exam-ple of two sensed signals k1, k2 which are divided to obtain �ve subsetsK1, . . . ,K5, resembling �ve operation modes. The discretisation of k1 andk2 is indicated by the dotted lines.

The function of the adaptation algorithm can now be given as fol-lows: Set the PID gains KP,KI,KD by using an operation mode-speci�cfunction. For example, at the operation mode ξ one would write for theproportional gain KP

KP = LP, ξ(k(t)), (1.3)

where the value of the function LP, ξ is obtained from a look-up table independence upon the current measurement vector k(t). The nodes of thelook-up table are given by the discretised vector k.

10


1k

2k Operating Point

K1

K2

K3

K4

K5

K

Fig. 1.4: Example of a measurement space K formed by p = 2 sensedsignals divided into ξ = 5 subsets. The dotted lines indi-cate the discretisation of the measurement signals.

Hence, the design of the adaptation algorithm for a speci�c controlproblem includes the following steps:

1. Select an appropriate set of p available measurement signals ki andtheir maximum and minimum values. This de�nes the measurementspace K.

2. Choose a discretisation ki of the measurement signals. This givesthe nodes of the look-up tables. The intersection of the nodes givesthe operating points.

3. Create an appropriate number of ξ operation modes and their bound-aries.

4. Specify the structure of the adaptation functions LP,ξ, LI,ξ, LD,ξ.

All of the above is done phenomenologically, i. e., by trial-and-error di-rectly on a sample of the real plant. Thus, such a control design is atedious process. From a system theoretical point of view, the aforemen-tioned design method results in a switching controller, which is known [85]to not ensure performance, robustness or stability in neither the transitionfrom one operating point to another nor in switching between operationmodes.

Calibration. Before a controller can be employed on the �nal product,it needs to be calibrated. This means adjusting the scheduling tables'values (e.g. the parameters of the functions LP, ξ, LI, ξ, LD, ξ), associated

11

1. INTRODUCTION

to each operating point, until a desired objective is met. Calibration isdone manually in an iterative way for each entry in each table and forall operating points. Calibration is time-consuming and, therefore, anexpensive procedure.

The current control structure ful�ls the automotive-speci�c require-ments D3-D5:

about D3: It uses few resources: The adaptation algorithm uses only basicarithmetic operations and the PID controller only has an order oftwo.

about D4: It provides calibration parameters: All values in the look-uptables can be adjusted.

about D5: It is easily comprehensible: The PID controller is quickly un-derstood and the adaptation algorithm is developed using the expe-rience gained by trial-and-error.

The following are bene�ts of the current development process.

• The exact behaviour of the plant at the test bed is considered, sincethe engineer develops the controller directly on the �nal product.

• Initial results are obtained quickly since, in each iteration, the con-troller is tested immediately on the real plant.

• The process is straightforward. The design of the adaptation algo-rithm does not require control engineering knowledge and calibrat-ion can be performed with basic understanding of the function of aPID controller.

Thus, this procedure seems to be an attractive method for automotiveapplications.

Some demerits of this control design method result from plant nonlin-earities and from the heuristic design procedure. Both points contributeto the large adaptation complexity. Therefore, the de�ciencies are thefollowing:

• The plant complexity is not fully considered a priori by the con-trol structure. Due to the phenomenological approach, even sim-ple linear single-input, single-output plants (e. g., electronic throt-tle plates) are typically controlled using PID controllers with much

12


more complex adaptation algorithms than necessary: Even if a sin-gle PID controller could control the plant, it is unlikely that thiscontroller is found during the phenomenological design process.

• The major design e�ort lies in calibration. Several weeks are nec-essary to manually calibrate a controller for a moderately complexplant.

• Expertise, which is gained by individual engineers on a certain plant,cannot be preserved easily because it is intuitive, rather than instruc-tive.

• There are no guarantees with respect to either stability or robustness.Although the �nal product goes through a series of tests on regularand extreme samples of the plant, failures are still encountered inthe �eld.

In conclusion, today's phenomenological development procedure is be-coming increasingly expensive for modern components. The following pro-poses a di�erent approach to control design which promises to be moretimely and, thus, less costly.

1.3.4 Model-Based Controller Design

The term �model-based control design� means that a (mathematical) modelis used during development. However, it does not mean that the resultingcontroller includes this model. If a controller does include the model ofthe plant then it is called a �model-based controller�.

Figure 1.5 shows the model-based control design procedure. It is as-sumed that an appropriate plant model already exists3, and that the con-trol problem is given. The control engineer then proceeds to select a con-trol method (which may or may not result in a model-based controller)and a performance criterion with which this controller can be computed.The performance criterion is chosen to re�ect the speci�cations of theclosed- loop behaviour. The controller is then tested in simulations usingthe model of the plant. If it fails to pass the speci�cations, the perfor-mance criterion is altered. Once the controller ful�ls the speci�cations,it can be tested at the test bed. The control design needs to provide

3 The choice or development of a plant model is part of model-based control de-sign. However, modelling is not a main topic of this thesis. For more information onmodelling, the reader is referred to e. g., [53, 55] and the references therein.

13

1. INTRODUCTION

Control Problem

Performance Criterion

Controller computation

Controller test using the model

Controller

Specifications ok?

yes

Controller test at testbed

Specifications ok?

yes

Change of the performance criterion

Change of selected parameters

no

no

Plant model

Result: Calibrated Controller

Design

Calibration

Fig. 1.5: Methodological controller development process.

some parameters with which the closed-loop behaviour can be calibrated.Thus, calibration is performed by adjusting only these parameters at thetest bed.

The main advantage of the model-based design (Fig. 1.5) over the phe-nomenological design is a reduction in the number of iterations (trial-and-error) and a reduction in the number of calibration parameters. In thecase of a good model and a good choice of the control method, there willbe no iterations, whatsoever. The main advantages of a methodologicaldesign procedure are the following:

• The calibration procedure is reduced to few, possibly physicallymeaningful parameters, as well as distinct calibration parameters.

• Many in�uences (external and internal) do not have to be treatedphenomenologically, since they are modelled and, thus, are automat-

14

1.4. OUTLINE OF THE RESULTS.

ically treated by the controller.

• Guarantees about the resulting closed-loop behaviour and its robust-ness are possible for some control methods. Therefore, productiontolerances for various plant components can be accounted for di-rectly.

• The knowledge about the plant (i. e., its model) and the controlconcept is available explicitly and can be documented.

In conclusion, a methodological control design is well suited for auto-motive applications. However, not all control methods ful�l the demandsD3-D5. In particular, a control method should be found to conclude themethodological control design process. This method should provide cali-bration parameters, is able to control nonlinear systems, addresses time-domain speci�cations directly, and does not use online-optimisations

1.4 Outline of the results.

This dissertation consists of two parts, preceded by this introduction.

Part I (Internal Model Control). Part I begins with a review of theclassical IMC concept. The design of internal model control focuses on�nding a controller that achieves a desired closed-loop input-to-output be-haviour. It is based on feedforward control design and relies on model in-version. The IMC structure is simple and plausible and provides valuableproperties such as nominal and robust stability as well as zero steady-stateo�set.

However, the classical IMC is limited to linear systems. The theoreticalcontribution of this thesis is the design methodology of a novel nonlinearIMC: This thesis proposes a nonlinear IMC controller to be designed asa feedforward controller by using the right inverse of the plant modeltogether with a low-pass �lter, called IMC �lter. The connection of thetwo results in a realisable, non-predictive nonlinear feedforward controller.The requirement for this method is a stable and invertible plant model. Inthe IMC structure, this results in a nonlinear output feedback controllerthat is robustly stable and ensures zero steady-state o�set. A functional-analytic view together with basic concepts of geometric control allows toextend the system class of the proposed nonlinear internal model controldesign method. The extensions include

15

1. INTRODUCTION

• incorporating input constraints into the control design,

• guaranteeing �nite inputs in the presence of singularities of themodel inverse, and

• a method to control nonlinear systems with unstable inverses (non-minimum phase models).

The IMC design is shown in detail for �at systems and systems that canbe transformed into the I/O normal form.

Part II (Internal Model Control of Turbocharged Engines). The modelderivation of turbocharged engines as well as the reduction to control-oriented models are given in Appendix A. Part II exclusively deals withcontrol design. The control-oriented model of the one-stage turbochargedair-system is �at, thus, a �atness-based IMC is developed. The IMC �lteris chosen with respect to input constraints and the resulting controllerwas tested on a real engine at a test bed. The �atness-based IMC com-pares favourably to the current series production PID controller in termsof performance and calibration e�ort.

Finally, the IMC controller of a two-stage turbocharged air-system isdeveloped. It provides tracking control of the boost pressure, the exhaustback pressure, and the pressure between the turbines of a two-stage tur-bocharged engine. Since the model is input-a�ne, the model inversionexploits the I/O normal form to develop the IMC controller. It is shownthat invertibility of the plant model is lost if the pressure between thecompressors equals the boost pressure. This implies a singularity of themodel inverse. Due to the proposed singularity handling of the novel IMC,the controlled system never loses its relative degree: The setpoints are al-tered by the IMC �lter automatically such that the resulting trajectoriesfor the pressures can be achieved with the available inputs. A stabilityanalysis shows that the closed-loop is robustly stable.

In conclusion, a novel nonlinear IMC controller is proposed and it isapplied to two automotive control problems.

16

Part I

INTERNAL MODEL CONTROL

This part presents the theoretical contribution of the thesis.First, the control design method of internal model control(IMC) for linear SISO systems is reviewed and it is shownthat it ful�ls the demands of the automotive industry. Then,the IMC design method for nonlinear SISO systems is de-veloped, with its speci�c application for �at and input-a�nesystems.

2. INTERNAL MODEL CONTROL OF LINEAR

SISO SYSTEMS

The linear IMC (also referred to as classical IMC in this thesis) is intro-duced in detail in this chapter to establish a common basis for its extensionto nonlinear systems. It is shown that IMC leads to a controller whichmeets all demands of the automotive industry.

2.1 Structure and Properties of Internal ModelControl (IMC)

2.1.1 Considered System Class

In this chapter, the system class under consideration includes all lineartime-invariant systems which can be described by transfer functions. Allresults also hold in the multi-input, multi-output (MIMO) case. However,to demonstrate the concept and design of IMC, only single-input, single-output (SISO) systems are regarded here. For a detailed discussion onthe design of a MIMO IMC, the reader is referred to e. g., [72].

In the IMC design philosophy, plant Σ and plant model Σ are notconsidered equal. The IMC design uses only the model Σ. Unlike mostother control methods, IMC understands the plant Σ as part of the con-trol loop but not as part of the control design. The plant Σ is interpretedas the real machine to be controlled and, as such, cannot be representedmathematically. The plant model Σ is used for control design. With suchan interpretation, some model properties may be designed to intention-ally di�er from plant properties (see [9], where the model is not selectedfor plant match but for closed-loop robust performance). Therefore, themajority of this thesis is concerned with �nding a controller based on theknowledge of the plant model Σ.

2. INTERNAL MODEL CONTROL OF LINEAR SISO SYSTEMS

The plant model Σ(s) is described by the transfer function

Σ(s) =b0 + b1s+ . . .+ bqs

q

a0 + a1s+ a2s2 + . . .+ ansn, q ≤ n (2.1)

and consists of n poles pi (i = 1, . . . , n) and q zeros zj (j = 1, . . . , q).

De�nition 2.1 (Relative degree of a transfer function, [63]). Therelative degree r of a transfer function Σ(s) is the number of excessivepoles

r = n− q. (2.2)

�

Systems with a relative degree equal to zero (r = 0) are said to havea direct feedthrough (i. e., the system input u a�ects the system output ywithout delay).

2.1.2 IMC Structure

w uK

Q

Σ~

Σdy

y~

w~−

−

Fig. 2.1: IMC structure.

The IMC structure with IMC controller Q(s) is shown in Fig. 2.1. Themain idea of IMC is to include the model Σ(s) of the plant Σ(s) into thecontroller K(s), which quali�es IMC as a model-based controller. Notethat the IMC structure is a speci�c realisation of the general control loopshown in Fig. 1.1 on page 7.

In the nominal case1, one �nds from Fig. 2.1 that the feedback signalvanishes (y(t) − y(t) = 0, ∀ t) and, consequentially, the IMC structuredegenerates to a feedforward control where the IMC controller Q(s) actsas a feedforward controller for the plant model Σ(s), as shown in Fig. 2.2.In the presence of a modelling error and the e�ect of a disturbance, the

1 In the nominal case, there is no disturbance (d = 0) and the plant model Σ(s) is

an exact representation of the plant (Σ(s) = Σ(s)).

20

2.1. STRUCTURE AND PROPERTIES OF INTERNAL MODEL CONTROL(IMC)

w Q u yΣ=Σ ~

Fig. 2.2: Resulting IMC structure in the case of an exact model andno disturbance, i. e., the nominal case.

w Q u−

yΣ−Σ ~w~

d

Fig. 2.3: Realisation of the IMC structure to visualise that Q actson modelling errors (Σ− Σ) and the disturbance d.

IMC structure from Fig. 2.1 is equal to the control structure shown inFig. 2.3 which can be interpreted as a standard control loop (cf. Fig. 1.2)in which the feedback controller Q(s) deals only with the modelling errorΣ(s)− Σ(s) and the disturbance d(t).

From Fig. 2.2 and 2.3, one concludes that an IMC controller Q(s)essentially is a feedforward controller which is also used in feedback toattenuate the e�ect of the disturbance d(t) and the modelling error (Σ(s)−Σ(s)).

The IMC structure in Fig. 2.1 can also be implemented in a standardcontrol loop (cf. Fig. 1.2), as shown in Fig. 2.4. One �nds the follow-

w Q u

+−y

Σ

Σ~

dC

Fig. 2.4: IMC structure implemented in the standard control loop.

ing relationships between the IMC controller Q and a standard feedbackcontroller C

C(s) =(

1−Q(s)Σ(s))−1

Q(s) (2.3)

Q(s) = C(s)(

1 + Σ(s)C(s))−1

. (2.4)

Equations (2.3) and (2.4) show that every standard controller C(s) canbe realised by an IMC controller Q(s) and vice versa. It is argued that

21


an IMC controller Q(s) can be designed in a straightforward fashion sincethe closed-loop speci�cations can be addressed directly. Moreover, it o�ersinherent structural closed-loop properties that are more di�cult to obtainwhen directly designing a standard feedback controller C(s).

2.1.3 IMC Properties

This section reviews general properties of IMC which result from the struc-ture of the feedback loop shown in Fig. 2.1 and apply independently of thedesign method used to get the IMC controller Q(s). They are indepen-dent in the sense that they only require Q(s) to have basic properties, likestability and a unity steady-state gain.

The following three properties can be derived [72]:

Property 2.1 (Stability). Assume the model to be exact (Σ(s) = Σ(s)).Then, the closed-loop system in Fig. 2.1 is internally stable if and only ifthe controller Q(s) and the plant Σ(s) are stable.

The proof closely follows that of [72] and is presented here for the sakeof completeness.

Proof. In order to show internal stability, all transfer functions betweenthe possible inputs and the possible outputs have to be stable. Figure 2.5

w uQ

Σ~

Σ1d

y

y~

w~2d

−

−

Fig. 2.5: IMC realisation used for internal stability conditions.

shows three independent inputs w, d1, d2 and one �nds with Σ(s) = Σ(s)thaty(s)

u(s)y(s)

=

Σ(s)Q(s) Σ(s) (I − Σ(s)Q(s))Σ(s)Q(s) 0 −Σ(s)Q(s)

Σ(s)Q(s) Σ(s) −Σ(s)2Q(s)

w(s)d1(s)d2(s)

(2.5)

holds. The transfer functions in Eq. (2.5) are stable if and only if bothQ(s) and Σ(s) are stable.

22


This result is to be expected, since the IMC structure degenerates to afeedforward control loop (Fig. 2.2) in the case of an exact model. An openloop control structure is only internally stable if each transfer function isstable. From this result, one concludes the following:

1. Only stable plants can be controlled with the IMC structure shownin Fig. 2.1: It is well known that an unstable plant can only bestabilised by feedback. Since the feedback signal of IMC may van-ish, one would have to intentionally introduce modelling errors toestablish a feedback signal which could then be used by a stabilisingcontroller. Such a procedure, however, would defeat the purpose ofa plant model.

2. Nominal stability of an IMC loop is trivially guaranteed for all stableplants Σ(s) simply by chosing any stable transfer function Q(s).This is a structural property of the IMC loop and it does not holdfor a standard control loop (cf. Fig. 1.2) where stability of the closed-loop has to be be shown for each controller C(s), even in the nominalcase.

Property 2.2 (Perfect Control). Assume that the IMC controller is

equal to the model inverse (Q(s) = Σ−1(s)) and that the closed-loop sys-tem in Fig. 2.1 is stable. Then, the plant output y(t) follows the referencesignal w(t) perfectly y(t) = w(t), ∀ t for an arbitrary disturbance d(t).

Proof. An analysis of the block diagram in Fig. 2.1 gives

y(s) = Σ(s)(I +Q(s)

(Σ(s)− Σ(s)

))−1

Q(s) · (w(s)− d(s)) + d(s),

(2.6)

which yields y(s) = w(s) for Q(s) = Σ−1(s) for arbitrary disturbancesd(s).

The IMC property of perfect control holds in the presence of modellingerrors and allows an interesting interpretation of the IMC structure: Fromthe IMC loop in Fig. 2.1 one also �nds

u(s)w(s)

= Q(s)(I + Σ(s)Q(s)−Q(s)Σ(s)

)−1

, (d = 0), (2.7)

23


which yields for Q(s) = Σ−1(s) the closed-loop setpoint-to-input relation-ship

u(s)w(s)

= Σ−1(s). (2.8)

Thus, by chosing the IMC controller as the model inverse (and assum-ing closed-loop stability) the IMC structure essentially inverts the plantbehaviour even though the plant behaviour is not exactly known.

However, perfect control cannot be realised in practise due to thefollowing reasons:

1. Consider an IMC implemented in a standard control loop (cf. Fig. 2.4).One �nds from Eq. (2.3) that if the IMC controller is a perfect in-verse of the plant model (Q(s) = Σ−1(s)) the equivalent feedbackcontroller C(s) is mathematically not de�ned due to a division byzero.

2. If the model Σ(s) is non-minimum phase, its inverse is unstable and,thus, leads to an internally unstable closed-loop.

3. The model inverse is only realisable (proper) if the model has thesame number of poles and zeros (i. e., if it has direct feedthrough).In such a case, however, the IMC structure would lead to a controllercontaining an algebraic loop which cannot be implemented, either.

Later on, the proposed IMC design will be interpreted as a low-frequencyapproximation of the perfect controller from above.

Property 2.3 (Zero O�set). Assume that the steady-state controller

gain is equal to the inverse of the steady-state model gain (Q(0) = Σ(0)−1)and that the closed-loop system in Fig. 2.1 is stable. Then, for any asymp-totically constant reference signal limt→∞ w(t) = wss and disturbancelimt→∞ d(t) = dss, no steady-state control error occurs (limt→∞ y(t) =wss).

Proof. Inserting Q(0) = Σ(0)−1 into Eq. (2.6) one �nds w(s) = y(s) fors = 0 .

Property 2.3 implies that if the steady-state gain of the IMC controlleris the inverted steady-state gain of the model, the IMC structure has im-plicit integral action. Thus, it is not necessary (and rather unreasonable)

24


to add an explicit integrator to the IMC controller to make the steady-state error vanish for constant w(t) and d(t). Unlike perfect control, whichis not realisable, the property of zero steady-state o�set is of practical use.

In the presence of model uncertainties, stability can be an issue. Inorder to show robust stability as a structural property, the Small-GainTheorem [64, 99] is used. Consider a stable multiplicative output un-certainty2 ∆(s) of the plant Σ(s), as shown in Fig. 2.6. One �nds therepresentation of the plant as

Σ(s) = (1 + ∆(s))Σ(s) (2.9)

with the output uncertainty ∆(s). It is assumed that ∆(s)Σ(s) is proper.

Σ~∆

+Σ

u y

Fig. 2.6: Multiplicative output uncertainty ∆.

Property 2.4 (Robust Stability). Assume that Σ(s) and Q(s) arestable transfer functions, for which the IMC structure of Fig. 2.1 is stablefor an exact model Σ(s) = Σ(s). Then, the IMC structure remains stablefor multiplicative output uncertainties ∆(s) if

‖∆(s)Σ(s)Q(s)‖∞ < 1 (2.10)

holds, with ‖·‖∞ denoting the H∞-norm3.

Proof. The open-loop transfer function from the signal w to the feedbacksignal y − y is

Σ0(s) ,y(s)− y(s)

w(s)=(

Σ(s)− Σ(s))Q(s). (2.11)

2 A multiplicative output uncertainty is a typical uncertainty description of the �eldof robust control. It is chosen here, since robust stability of an IMC loop can be shownin a straightforward fashion using the well-known Small-Gain Theorem.

3 The H∞-norm is de�ned as ‖G(s)‖∞ = supω

σmax[G(jω)] where σmax(·) denotes

the maximum singular value.

25


Inserting Eq. (2.9) into Eq. (2.11) results in

Σ0(s) = ∆(s)Σ(s)Q(s).

The Small-Gain Theorem says that if the gain |Σ0(jω)| of the open-looptransfer function is smaller than one for all frequencies ω

‖Σ0(s)‖∞ < 1

then closing the loop yields a stable feedback system.

Note that for a given upper boundary ∆(ω) of the amplitude |∆(jω)|of the model uncertainty

|∆(jω)| ≤ ∆(ω), ∀ ω, (2.12)

the system is robustly stable if

∆(ω) <1

|Σ(jω)Q(jω)|, ∀ ω (2.13)

holds.In conclusion, the IMC controller Q(s) works as a feedforward con-

troller which is also used to attenuate modelling errors and disturbances.It has important properties like nominal and robust stability as well aso�set-free control which makes it attractive for controller design. Thefollowing introduces a design method for internal model control for linearsystems.

2.2 Classical IMC Design

The IMC design will be presented �rst for stable minimum phase (MP)systems. Then, it will be extended to non-minimum phase (NMP) sys-tems.

The design presented here presents the standard design method as, forexample, given in [87]. Note that this proposed design works well on manyplants but still its performance can be improved by a more sophisticatedapproach [9, 72].

2.2.1 IMC Design of Minimum Phase Systems

Consider the model Σ(s) with relative degree r from Eq. (2.1) to be asymp-totically stable and minimum phase. Thus, the n poles pi (i = 1, . . . , n)and the q zeros zj (j = 1, . . . , q) of the model Σ(s) are on the left half ofthe complex plane (Re (pi) , Re (zj) < 0, ∀ i, j).

26

2.2. CLASSICAL IMC DESIGN

Design. A feasible IMC controller Q(s) for an NMP model Σ(s) withrelative degree r is designed by

Q(s) = Σ−1(s)F (s) , (2.14a)

with the IMC �lter F (s)

F (s) =1(

sλ + 1

)r with λ > 0. (2.14b)

The IMC �lter F (s) has an r-fold pole at −λ. The variable λ is the onlyparameter to be chosen by the designer. Thus, with a given model Σ, anIMC design focuses on choosing the parameter λ of the IMC �lter. Here,it is proposed to place λ such that the desired closed-loop bandwidth4

ωB and high-frequency noise ampli�cation coincides with that of the �lterF (s).

Remark 2.1. Any stable transfer function F (s) with the property F (0) =1 and a relative degree of at least r would be feasible as an IMC �lter.The presented IMC �lter F (s) is convenient since the value of only oneparameter is to be chosen.

The resulting IMC structure is shown in Fig. 2.7.

w u

Q

Σ~

Σdy

y~

w~ F 1~−Σ−

−

Fig. 2.7: IMC structure with IMC controller Q from Eq. (2.14a).

Explanation. An inverted model Σ−1(s) is a perfect feedforward con-troller for Σ(s) and in the IMC structure, perfect control would follow (cf.IMC Property 2.2). However, since

Σ−1(s) =a0 + a1s+ a2s

2 + . . .+ ansn

b0 + b1s+ . . .+ bqsq(2.15)

4 Let the bandwidth ωB of a low-pass �lter F (s) with |F (0)| = 0dB be de�ned as|F (jωB)| = −3dB.

27


holds, the model inverse is not realisable if the model's relative degreeis greater than zero (r = n − q > 0). In order to obtain a realisabletransfer function Q(s), the model inverse Σ−1(s) is padded with the IMC�lter F (s), which compensates for the r excess zeros in Eq. (2.15) byintroducing r additional poles. Those r additional poles are placed at thesame location, namely at −λ, for convenience. The steady-state gain ofthe IMC �lter is one (F (0) = 1) and it follows from Eq. (2.14a) that thesteady-stage gains of the IMC controller is inverse to the steady-stage gainof the model (i. e., Q(0) = Σ−1(0)). Thus, an IMC loop with the IMCcontroller from Eq. (2.14a) will not have a steady-state o�set (cf. IMCProperty 2.3), even in the presence of model uncertainties.

Since the model Σ(s) is minimum phase, its inverse Σ−1(s) is stableand nominal stability of the IMC loop with IMC controller Q(s) fromEq. (2.14a) follows (cf. IMC Property 2.1). Considering robust stability,one �nds with the IMC controller Q from Eq. (2.14a) and the stabilitycriterion (2.10) with Σ0 = ∆(s)F (s) that if

‖∆(s)F (s)‖∞ < 1 (2.16)

holds, then the closed loop is robustly stable with this IMC controllerfrom which the condition

|F (jω)| < 1∆(ω)

, ∀ω. (2.17)

for robust stability can be derived. As the �lter parameter can be chosenarbitrarily and ‖F (s)‖∞ = supω|F (jω)| = |F (0)| = 1 holds, Eq. (2.17)shows a considerable robustness of the nonlinear IMC loop, because sta-bility is ensured for sup(∆(ω)) < 1.

Using the IMC controllerQ(s) from Eq. (2.14a) and assuming the plantto be represented using output uncertainties from Eq. (2.9), one �nds withEq. (2.6) that

y(s)w(s)

= T (s) =(I + ∆(s))F (s)I −∆(s)F (s)

(2.18)

holds with this controller. Thus, the closed-loop behaviour T (s) tends tothe behaviour of the IMC �lter F (s) for small model uncertainties ∆(s)

lim∆(s)→0

T (s) = F (s), ∀s. (2.19)

28


Interpretation. The IMC controller given in Eq. (2.14a) can be inter-preted in several ways:

• The IMC controllerQ(s) is designed as feedforward controller for themodel Σ(s): A perfect feedforward controller is the inverse Σ−1(s)of the model. In order to be able to realise the IMC controllerQ(s), it is designed as an approximation of the model inverse Σ−1(s).The bandwidth ωB of the IMC �lter F (s) governs which complexfrequencies s of the model inverse Σ−1(s) are approximated well byQ(s). Well below the IMC �lter bandwidth the IMC controller Q(s)given in Eq. (2.14a) is a good approximation of the model inverseΣ−1(s). This does not hold true for frequencies above the bandwidthof the IMC �lter.

• The design of the IMC controller Q(s) is equivalent to model refer-ence control design: Model reference control [98] deals with �ndinga controller such that the closed-loop behaviour T (s) is equal toa given stable reference model R(s). Changing the IMC design inEq. (2.14a) by setting the IMC �lter F (s) equal to the referencemodel R(s)

F (s) = R(s), (2.20)

it follows from Eq. (2.19) that the presented IMC design is equivalentto model reference control for stable minimum phase plants.

Example 2.1 (IMC control of a linear minimum phase process):Consider a plant model

Σ(s) =1

s2 + s+ 1(2.21)

with no zeros and two complex conjugate poles at p1/2 = − 12±√

32j. From

Eq. (2.14a) one �nds the IMC controller

Q(s) =s2 + s+ 1

1︸︷︷︸Σ−1(s)

1(sλ

+ 1)2︸︷︷︸

F (s)

(2.22)

where λ > 0 is chosen by the designer to establish the desired closed-loopbandwidth. Now, consider a distorted plant

Σ(s) =10

3s2 + 5s+ 7(2.23)

29


from which a multiplicative output uncertainty ∆(s) (cf. Eq. (2.9))

∆(s) = Σ(s)Σ−1(s)− 1

=7s2 + 5s+ 3

3s2 + 5s+ 7

(2.24)

can be derived. Figure 2.8 shows bode magnitude plots of the IMC �lterF (s) with the choice λ = 1, the uncertainty ∆(s) and the open loop transferfunction Σ0 = ∆(s)F (s). It shows that the closed IMC loop with this

10−2

10−1

100

101

102

−80

−60

−40

−20

0

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

F (s)

∆(s)

∆(s)F (s)

Fig. 2.8: Bode magnitude plot of IMC �lter F (s), uncertainty ∆(s)and open loop Σ0(s) = ∆(s)F (s), given in Example 2.1.

uncertainty ∆(s) and IMC �lter F (s) is stable since ‖∆(s)F (s)‖∞ < 0dB =1 holds. The high frequency magnitude of the uncertainty ∆(s) is above one(limω→∞|∆(jω)| = 7

3), but the IMC �lter attenuates these high frequency

gains. Thus, the choice of the IMC �lter pole λ determines not only theclosed-loop bandwidth but also robust stability (see [63, 72] for an in-depthdiscussion on this matter). Finally, Fig. 2.9 shows the step response yd(t) ofthe nominal case and the case with the modelling errors y(t) assuming theplant from Eq. (2.24). It is assumed that the shape of both step responsesare acceptable since they are non-oscillatory. As expected, zero steady-stateo�set is achieved even for the distorted plant. �

2.2.2 IMC Design of Non-Minimum Phase Systems

Consider the model Σ(s) with relative degree r from Eq. (2.1) to be stableand NMP. Thus, it either has a time delay or zeros in the right half plane.Time delays are not considered in this work. The basic idea of dealingwith an NMP system is the following:

30


0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

time (s)

Am

plitu

de

w(t)

yd(t)

y(t)

Fig. 2.9: Response of the nominal output yd and actual output y ofExample 2.1, to a reference step in w.

1. Split the model Σ(s) into two parts where one part ΣMP(s) is MPand the other part ΣNMP(s) contains the NMP behaviour.

2. Design the IMC controller Q(s) as described in Eq. (2.14a) but onlyfor the MP part ΣMP(s).

3. Implement the original NMP model Σ(s) as internal model in theIMC structure.

By inverting only the MP part ΣMP(s) of the model Σ(s), the IMC con-troller Q(s) will be stable (as required) and implementing the full modelΣ(s) as internal model yields the best model-plant match. However, sincethis approach does not invert the NMP part ΣNMP(s) of the model Σ(s),the closed-loop IMC behaviour T (s) will still exhibit this NMP behaviour.

Design. Consider the q zeros of the model Σ(s) to be sorted such thatthe �rst ζ zeros z1, . . . , zζ are in the right half of the complex plane(Re(z1), . . . ,Re(zζ) > 0) and the other q− ζ zeros to be in the left half ofthe complex plane.

The following introduces two speci�c possibilities to split an NMPmodel Σ(s) into an MP ΣMP(s) and an NMP part ΣNMP(s):

Σ(s) = ΣMP(s) · ΣNMP(s) with either

31


Case I: ΣNMP(s) =

(sz1

+ 1)· . . . ·

(szζ

+ 1)

(s−z1 + 1

)· . . . ·

(s−zζ + 1

) , ΣA(s), or

(2.25)

Case II: ΣNMP(s) =

(sz1

+ 1)· . . . ·

(szζ

+ 1)

(sλ + 1

)ζ (2.26)

Both cases share the properties that the steady-state gain of the NMPpart equals to one (ΣNMP(0) = 1) and that the relative degree of the MPpart ΣMP(s) is equal to the relative degree r of the original model Σ(s).

The design of an IMC controller Q(s) holds for both cases and isanalogous to the one portrayed for MP systems as introduced in Eq. (2.10)

Q(s) = Σ−1MP(s)F (s) (2.27)

with the IMC �lter F (s) from Eq. (2.10) and is shown as a block diagramin Fig. 2.10.

w u

Q

Σ~

Σdy

y~

w~ F1~−ΣMP−

−

Fig. 2.10: IMC structure with IMC controller Q from Eq. (2.27).

Explanation. The resulting IMC controller Q(s) is stable since it isobtained by inverting the MP part ΣMP(s) of the model. Thus, nominalstability follows. Considering robust stability, one gets with the multiplica-tive output uncertainty ∆(s) from Eq. (2.9) with Property 2.4 on page 25that if

‖Σ0(s)‖∞ = ‖∆(s)ΣNMP(s)F (s)‖∞ < 1 (2.28)

holds then the closed IMC loop is stable. Thus, given an uncertain plant,the choice of the NMP part ΣNMP(s) of the model will in�uence robuststability.

32


Since the IMC �lter F (s) has a steady-state gain of one (|F (0)| = 1),it follows from Q(0) = ΣMP(0)−1 = Σ(0)−1 and Property 2.3 that theclosed IMC loop yields zero steady-state o�set.

Considering the closed-loop behaviour, one �nds from Fig. 2.10 that

T (s) =y(s)w(s)

=F (s)ΣNMP(s) (I −∆(s))

I + F (s)ΣNMP(s)∆(s)(2.29)

holds. Hence, even for small model uncertainties ∆(s) the closed-loop willretain the NMP behaviour ΣNMP(s) of the plant Σ(s):

lim∆(s)→0

T (s) = F (s)ΣNMP(s), ∀s. (2.30)

from which one �nds

y(s) = F (s)ΣNMP(s)w(s) +(

1− F (s)ΣNMP(s))d(s) (2.31)

as the behaviour of the output signal y(s). This, however, is to be expectedsince (with the demand of internal stability) an NMP behaviour of a plantcannot be in�uenced by feedback or feedforward by any control algorithm[63, 87].

Interpretation. Case I: The model Σ(s) is split into an MP part ΣMP(s)and a stable all-pass part ΣA(s). The all-pass part ΣA(s) has a gain ofone over all frequencies |ΣNMP(jω)| = 1, ∀ω and contains all right halfplane zeros. Thus, the amplitude-over-frequency shape of the MP partΣMP(s) is equal to that of the original model Σ(s):

|Σ(jω)| = |ΣMP(jω)|, ∀ω. (2.32)

It is important to appreciate that a number of ζ zeros are introduced tothe remaining MP part ΣMP(s) in Eq. (2.25) at the mirror image of theNMP zeros (namely at −z1, . . . ,−zζ). Thus, the system ΣMP(s) will havethe same number of zeros and the same relative degree as the original nonminimum-phase model Σ(s).

It can be shown [63, 72], that by splitting the model according to caseI and choosing an in�nitely fast �lter pole λ → ∞, the resulting IMCcontroller Q(s) from Eq. (2.10) will minimise the integral square error(ISE) norm, i. e.,

minQ(s)

∫ ∞0

|w(t)− y(t)|2dt. (2.33)

33


Thus, Q(s) from Eqns. (2.10) and (2.25) (case I) is ISE-optimal for λ→∞.Case II: Case II di�ers from case I in the choice of the ζ zeros that

are introduced to the MP part ΣMP(s). The property of Eq. (2.33) isinteresting but not necessary for IMC design. Thus, since ζ zeros need tobe introduced to the MP part ΣMP(s) it is convenient to place them atthe same location as the IMC �lter pole, namely at −λ.

The ISE-optimal IMC controller Q(s) is interesting from a mathemat-ical perspective but not necessarily a good choice for practical appli-cations.

Aside from the unrealistic necessity of using an in�nitely fast IMC �lterpole (one could use an arbitrary but �nite �fast� pole), such a designmight lead to excessive noise ampli�cation. Foremost, however, an integralcriterion does not account for the shape of a control response since itmerely represents its norm. Thus, ISE-optimal controllers may lead tooscillating control inputs u(t) and oscillating plant outputs y(t) whichresult in high wear and tear of the actuator. Moreover, a control responseis judged intuitively by an engineer, who tries to avoid oscillating inputand output signals as they are not considered good responses. Therefore,the author encourages the designer to choose case II since it o�ers morefreedom in designing the controller.

The following example demonstrates the di�erence between an IMCdesign for non-minimum phase systems using either Case I or Case II.

Example 2.2 (IMC control of a linear non-minimum phase pro-cess):This example has been adapted from [9]. Consider the non-minimum phaseexact model

Σ(s) = Σ(s) =0.225s2 − 0.0964s+ 1

0.137s3 + 1.274s2 + 2.137s+ 1(2.34)

with complex conjugate zeros at z1/2 = 0.21 ± j2.1, a two-fold pole atp1 = p2 = −1, and a single pole at p3 = −7.3. Hence, the plant is stableand NMP. With case I, one �nds for an ISE-optimal controller Qise(s) withEqns. (2.14b),(2.25) and (2.27)

Qise(s) =0.137s3 + 1.274s2 + 2.137s+ 1

0.225s2 + 0.0964s+ 1· 1

s/λ+ 1. (2.35)

With case II (cf. Eq. (2.26)), one �nds with Eqns. (2.14b) and (2.27):

Qλ(s) =0.137s3 + 1.274s2 + 2.137s+ 1

(s/λ+ 1)2· 1

s/λ+ 1. (2.36)

34


0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

time (s)

Am

plitu

de

w(t)

yd(t)

yise(t)

yλ(t)

Fig. 2.11: Step response of Example 2.2.

Both controllers were designed using λ = 3. As the �lter parameter λ isnot chosen to be in�nite, Qise will be referred to as ISE-oriented, hence-forth in the example. Figure 2.11 shows the step response of the plantΣ(s) controlled with both the ISE-oriented IMC controller Qise(s) (outputyise(t)) and IMC controller Qλ(s) (output yλ(t)). The ISE-oriented con-troller Qise(s) shows an oscillating output yise(t), since the NMP zeros wereunderdamped which leads to underdamped poles in the controller. In anyautomotive control system, such an output trajectory would be unacceptabledue to the oscillations and the slow settling time of about 20 seconds. Thisstep response yields an ISE (cf. Eq. (2.33)) of

∫∞0|w(t)−yise(t)|2dt ≈ 0.3595.

The output yλ(t) of the closed-loop with the IMC controller Qλ(s), wherethe additional MP zeros were placed at the IMC �lter pole −λ, shows consid-erably more damping and a settling time of about four seconds. To an engi-neer, the output yλ(t) is signi�cantly more appealing than the ISE-oriented,although its ISE norm

∫∞0|w(t)−yλ(t)|2dt ≈ 0.6214 is almost twice as large.

�

In conclusion, IMC for NMP systems can be designed to be ISE-optimal,however, a low value of the ISE-norm usually does not stand in anyrelationship with the desired closed-loop behaviour. Therefore, an ISE-optimal control design is not of any interest in an automotive application.

2.2.3 Input Constraints

The most basic form of incorporating input constraints into IMC is brie�yreviewed since this topic is of high importance in virtually all practicalcontrol applications.

35


Input constraints are given as

umin ≤ u(t) ≤ umax. (2.37)

Since this is a property of the plant, one possibility of treating this limita-tion is to include it also in the model Σ. Figure 2.12 shows the resulting

dw C u

−y

w Q u−

yΣ−Σ ~w~

d

w uK

Q

Σ~

Σ

dy

y~

w~

w Q u

+−

yΣ

Σ~

dC

w Q u yΣ=Σ ~

w Q u− yΣ=Σ ~w~

d

w uQ

Σ~

Σ

1d

y

y~

w~2

d

Σ~

∆+

Σ

u y

w u

Q

Σ~

Σd

y

y~

w~ F 1~−Σ

w u

Q

Σ~

Σd

y

y~

w~ F1~−ΣMP

w u

Σ~

Σ

dy

y~

Q

FBQ

Strecke

Modell-

-IMC-Regler

G~

GQu y

y~

d+

u

w w uQ

Σ~

Σ

dy

y~

w~

Fig. 2.12: Handling input constraints by including the limitation inthe model Σ.

IMC loop where the model is enhanced to respect input constraints.This approach does not change robust stability since the modelling

error is not a�ected. However, since the IMC controller Q(s) does nothave any information on limited inputs, performance will decrease. Otherapproaches to limit inputs with IMC are, for example, given in [9, 84, 91,104]. These approaches yield better performance than the one portrayedhere, but are mathematically more involved, which spoil the attractivesimplicity of IMC design. To this end, Section 5.1 develops an e�ectiveand straightforward method of respecting input constraints with IMC.

2.3 Feasibility of IMC as Automotive Controller

IMC is a feasible control concept for automotive applications. In order fora control concept to be of interest to the automotive industry, it shouldcomply with the demands D1-D5 (listed in Section 1.3.2) and it should bea model-based design procedure (described in Section 1.3.4 on pages 13�).Since IMC is a model-based controller, it naturally �ts in the model-baseddesign procedure. The following shows that an IMC controller meets therequirements on automotive controllers.

Comparing the automotive requirements D1-D5 to the IMC de-sign method.

D1: Performance criteria are given in time-domain. Time domain criteriaare incorporated into an IMC design in a straightforward fashion

36

2.3. FEASIBILITY OF IMC AS AUTOMOTIVE CONTROLLER

by choosing an IMC �lter F (s) which meets these criteria. Con-sidering minimum phase plants, the closed-loop behaviour T (s) isidentical to the IMC �lter F (s) (see Eq. (2.18)). Considering NMPplants, the closed-loop behaviour is governed by the IMC �lter F (s)together with the (unavoidable) NMP behaviour ΣNMP(s) of theplant (see Eq. (2.29)). Hence, performance criteria in time-domainare accounted for by choosing the IMC �lter F (s), according to thedesired closed-loop behaviour (and, if applicable, appreciating theunavoidable NMP behaviour of plants).

D2: Robust stability and zero steady-state o�set are required. These con-ditions are basic properties of the IMC structure and are ful�led au-tomatically (see Section 2.1.3) if the design procedure, as introducedabove, is followed.

D3: An IMC must be implemented in a car's OCU. An IMC does not relyon numerically intense calculations such as on-line matrix inversionsor on-line optimisations. It can be assumed, that an IMC can beimplemented in a standard OCU with a computational burden ofthe same order of magnitude as today's PID oriented automotivecontrollers.

From the design as introduced in the preceding sections, it followsthat, since the model is part of the �nal controllerK(s) (see Fig. 2.1),a controller K(s) implemented in the IMC structure will have anorder of 2n, where n is the order of the model. However, if anIMC is implemented in a standard control loop (see Fig. 2.4 andEq. (2.3)), it follows that the minimal realisation of the controllerC(s) has an order of n. Hence, it is proposed to always implementan IMC controller as a standard controller C(s).

D4: A controller must be straightforward to calibrate. Calibration of anIMC is signi�cantly easier than calibration of a PID controller. AnIMC can be calibrated by adjusting the parameters b0, . . . , bq anda0, . . . , an (cf. Eq. (2.1)) of the internal model. The advantage of ad-justing model parameters over adjusting controller parameters, likePID gains, is appealing to an engineer, especially if the model param-eters have a physical interpretation. Moreover, model parameterscan be calibrated on-line at a test bed without a time-intensive newdesign.

If desired, the �lter parameter λ can also be calibrated. However,

37


since this parameter de�nes the nominal closed-loop bandwidth andnoise ampli�cation, it can be determined from the closed-loop spec-i�cations, alone.

D5: A controller should be easy to understand. The IMC design as pre-sented here relies on the basic properties of the IMC structure (seeSection 2.1.3) which can be comprehended quickly.

In conclusion, the IMC concept ful�ls all requirements of the automo-tive industry.

2.4 Summary

The design of internal model control focuses on �nding a controller suchthat a given closed-loop I/O behaviour is achieved. It is based on feed-forward control design and relies on model inversion. The IMC structureis simple and plausible and provides valuable properties such as nominaland robust stability as well as zero steady-state o�set.

The attractiveness of IMC to industry comes from the internal modeland the simple design law. Through this model the IMC controller Q(s)is de�ned to a great extent. Once an IMC controller Q(s) is determinedfor a speci�c plant Σ(s), it can be adapted to control similar plants bycalibrating the internal model parameters and � if desired � the tuningparameter λ. By changing the model parameters, the IMC controller pa-rameters are changed accordingly. This enables non-control engineers tocalibrate an existing IMC controller, since knowledge of the plant su�cesto determine the model parameters. IMC does not use on-line optimisa-tion procedures. Hence, it can be implemented in a real-time environmentlike an OCU.

In order to use the IMC structure to control nonlinear systems a non-linear IMC controller Q for stable nonlinear systems is developed in thefollowing.

38

3. INTERNAL MODEL CONTROL OF

NONLINEAR SISO SYSTEMS

This chapter develops the theoretical contribution of this thesis, namelythe extension of the SISO IMC principle to nonlinear SISO systems. TheIMC design philosophy, including the IMC structure of the linear case asintroduced in Chapter 2, is retained. Hence, a nonlinear IMC is designedas a feedforward controller for the nonlinear system by employing a modelinverse in connection with a low-pass �lter.

This chapter uses basic system theoretical properties, like the relativedegree, and basic functional analytic ideas as a right inverse, operatorsand signal norms to propose a control design method. The appeal ofthe proposed design method is that it is applicable to industrial controlproblems, its application is straightforward, and, despite its simplicity,o�ers important properties such as nominal and robust stability.

3.1 Mathematical Preliminaries

For simplicity, the concept of nonlinear IMC is presented for the SISOcase, only. This chapter uses a functional analytic view on dynamicalsystems, where the input/output (I/O) behaviour of the system Σ is givenby understanding Σ as an operator which maps a signal u contained in afunction space U into a signal y, contained in the function space Y, i. e.,

Σ : U → Y. (3.1)

It is written as y = Σu, for a u ∈ U . The usual composition symbol�◦� is omitted in this text, except for instances where the composition isemphasised. In its state space representation, the model Σ is given as

Σ : x(t) =f (x(t), u(t)) , x(0) = x0, x ∈ X , (3.2a)

y(t) =h (x(t), u(t)) , u ∈ U , y ∈ Y. (3.2b)

3. INTERNAL MODEL CONTROL OF NONLINEAR SISO SYSTEMS

The signals u(t) ∈ R1, x(t) ∈ Rn and y(t) ∈ R1 denote values at aspeci�c time point t. The trajectory of a signal is denoted by omittingthe dependence of t (e. g., x, y) and refers to the input u ∈ U , state x ∈ Xor output y ∈ Y as the whole time function. Thus, instead of using theexpression of, for example, x(·) to denote a trajectory, this work omits�(·)�. The time t is de�ned on the set t ∈ T = [0,∞). Moreover, the modelΣ is assumed to be time-invariant and asymptotically1 stable.

The vector �eld f and the function h are analytic2 in their argumentsx(t) and u(t) for all x ∈ X and u ∈ U . Thus, the solution x of Eq. (3.2a)exists and is unique [46]. Moreover, analycity implies that the functionsf and h are non-singular with respect to all possible arguments x(t) andu(t). The measurement map h : Rn ×R1 → R1 maps the current valuesx(t) and u(t) of the states and input signal into the current value y(t)of the output signal. Note that the function h in Eq. (3.2b) may or maynot be directly dependent on u. If the input u appears in the mapping hexplicitly, the model is said to have a direct feedthrough.

The initial state x0 of the system is assumed to be given and �xed,which simpli�es the following notation in such that a system can be repre-sented by a single operator Σ. If the initial condition x0 would be allowedto vary, one would have to introduce a di�erent operator for each initialcondition or use the concept of relations instead of operators [99].

The output-function space

Y ={y ∈ Y : y = Σu, u ∈ U

}(3.3)

contains exclusively all signals which can be produced by the model Σunder the given initial state x0 and permissible controls u. Hence, theI/O map Σ is surjective on this set.

In the following, the identity operator I will be used extensively andis de�ned as

u = Iu for any trajectory u. (3.4)

Thus, the identity operator preserves its input signal.In order to de�ne the system gain, signal norms are introduced. The

1 Asymptotic stability (see e. g., [54, 83]) is a property of the behaviour of the statesof a system. Since this thesis is mainly concerned with I/O behaviour, stability condi-tions in dependence of the I/O behaviour are given in the following.

2 An analytic function (see e. g., [16, 56] for its properties) is also referred to as�holomorphic function.�

40

3.1. MATHEMATICAL PRELIMINARIES

norm Lp of a signal u is de�ned as

‖u‖Lp =(∫

t∈T|u|pdt

) 1p

(3.5)

with 1 ≤ p <∞. A signal u for which the norm ‖u‖Lp exists, is denotedby u ∈ Lp. For p =∞ the L∞-norm is

‖u‖L∞ = supt∈T|u(t)| <∞. (3.6)

In the following, a signal norm is denoted by ‖·‖ whenever the dependenceon p is not �xed.

Remark 3.1. The following expressions are used synonymously:

• u ∈ L∞• u is bounded piecewise continuous (cf. [54, 99])

• u is non-explosive (cf. [99])

• u is measurable

The above all mean that the signal u(t) is de�ned at each instance intime, i. e., ∃u(t), ∀t ∈ T .

De�nition 3.1 (Finite-Gain Stability [54]). An I/O map Σ is calledto be �nite-gain Lp-stable if there exist non-negative constants γ and βsuch that

‖Σu‖ ≤ γ‖u‖+ β (3.7)

holds for all u ∈ Lp.�

In this work, stability means �nite-gain stability unless noted di�er-ently.

De�nition 3.2 (System gain, [54]). For a �nite-gain Lp-stable system,the smallest value γ, for which inequality (3.7) is satis�ed, is called thegain of the system, denoted by g(Σ) = γ. �

41


Remark 3.2. If the operator Σ has the property

0 = Σ 0 (3.8)

where 0 is a trajectory u(t) = 0, ∀t then the system gain can also beexpressed as [99]

g(Σ) = sup‖(Σu)‖‖u‖ (3.9)

where the supremum is taken over all u ∈ U for which u 6= 0.

If the operator Σ does not have the property 0 = Σ 0 it can be enforcedby adding a compensating bias β (cf. Eq. (3.7)) to the output y.

Gains share the properties of norms. In addition [99], a gain ful�ls theinequality

g(A ◦B) ≤ g(A)g(B) (3.10)

for any two stable operators A and B.

Lie-derivative. The following relationships make use of the Lie deriva-tive [51] to simplify notation. The time derivative of h(x) along f(x, u)is denoted by

Lfh(x) =∂h(x)∂x

f(x, u)

and is equal to ˙y = ddth(x). The function Lkfh(x) satis�es the recursion:

Lkfh(x) =∂Lk−1

f h(x)∂x

f(x, u),

with L0fh(x) = h(x).

Remark 3.3. Consider the Lie-derivative of a function h as a timederivative of that function given in dependence upon the system states x.It is an abbreviation for the chain-rule with the subsequent substitutionof x by its de�nition Eq. (3.2a).

In this work, the expressions Lkfh(x, u) and Lkfh(x) are used synony-mously, despite the omitted dependency on u. In the case of the explicit

42

3.1. MATHEMATICAL PRELIMINARIES

dependency on u, the Lie-derivative has to be de�ned by

Lfh(x, u) =∂h(x, u)

∂xf(x, u) +

∂h

∂uu with the recursion

Lkfh(x, u) =∂Lk−1

f h(x, u)

∂xf(x, u) +

k−1∑i=1

∂Lifh(x, u)

∂u(i−1)u(i)

and L0fh(x, u) = h(x, u).

Fortunately, derivations of the output function h which contain deriva-tives of u will never be used in this work. In this light, the de�nitiongiven in this remark is never employed and merely given for complete-ness.

De�nition 3.3 (Relative degree [35, 42]). The relative degree r of asystem Σ is the smallest value r ∈ N0 for which y(r) can be expressed byan algebraic function ϕ which explicitly depends on the input u, i. e.,

y(r) = Lrfh(x, u) = ϕ(x, u) (3.11)

holds with

∂

∂uLifh(x, u) = 0, 0 ≤ i ≤ r − 1

∂

∂uLrfh(x, u) 6= 0.

If the relative degree r does not exist (i. e., an equality (3.11) cannot begiven), it will be referred to as r → ∞ (see e. g., [19, 93], where thisconvention is also used).

The relative degree is well-de�ned if the value of r is constant in thestate-space region of concern (i. e., locally or globally, depending on thecontext). It is called ill-de�ned if it is not well-de�ned. �

Remark 3.4. The relative degree r can also be de�ned more condensedby

r = arg mink

{∂

∂uLkfh(x, u) 6= 0

}.

It is sometimes also called relative order and can be interpreted as thenumber of integrations that the input u (or some algebraic function of it)has to undergo until it a�ects the output y. Therefore, r is an indicationof the sluggishness of the system response [19]. For the relative degreeof r = 0, the system has a direct feedthrough.

43


This de�nition coincides with De�nition 2.1 of the relative degree r of atransfer function: By multiplying the transfer function with sr (which isequivalent to taking r derivatives) one obtains a transfer function withdirect feedthrough.

An excellent, detailed, structural interpretation of the relative degree rand its implications can be found in [19]. The work of [19] is stronglyrecommended to the interested reader, as its results complement the de�-nition and interpretation of the relative degree given here. For the resultsof this thesis, the notion of relative degree is of fundamental importanceand the relationship (3.11) will be used extensively.

3.2 Output-Function Space

This thesis mainly focuses on the input/output behaviour of nonlinearsystems by using the abstract notion of operators, as in Eq. (3.1). Withinthis context, the de�nition of the input space U and the output space Yof the operator (plant model) Σ becomes necessary. In the nonlinear IMCdesign, as it is proposed in the following, the precise knowledge of theinput-space U and, more importantly, the output-space Y is necessary formore than just the sake of completeness of the mathematical concept ofoperators. These spaces, describing the shapes of the input function uand the resulting output-function y, are used for control design and theirknowledge is necessary for e. g., respecting input constraints.

Remark 3.5. Note that, here, the word shape is used to distinguishthe goal of describing the output-function space Y from the usually en-countered de�nition of the output-function space Y by means of a signalnorm. In loose terms, it means that the look of all output functionsy ∈ Y is to be described by their initial behaviour at time t = 0 as wellas how many times they are (at least) continuously di�erentiable in thedevelopment for t > 0. The number of times a function is continuouslydi�erentiable can be interpreted as its smoothness.

Throughout this thesis, the input space U is to be regarded as the spaceof all piecewise continuous functions with constant constraints umin, umax,i. e.,

U : {u| u ∈ L∞ : umin ≤ u(t) ≤ umax} , umin, umax ∈ R. (3.12)

44

3.2. OUTPUT-FUNCTION SPACE

In general, it is not necessary for the constraints umin, umax to be constant.However, this demand su�ciently describes the typical scenario of controlproblems and simpli�es later results.

The set Y, which contains all possible output functions y of an operatorΣ, is dependent on the structure of Σ (in particular upon its relative degreer) and the possible input functions u ∈ U . In the following, the output-function space Y is de�ned in two propositions. Proposition 3.1 addressesthe initial (i. e., at t = 0) behaviour of all functions y ∈ Y and Proposition3.2 de�nes the output function space Y from t > 0 in dependence uponthe models relative degree r and the input function space U .

The results obtained below present a generalisation of the results ob-tained by [19]. It will be shown that, for a nonlinear system (3.2) withrelative degree r, the output trajectory y has a �xed initial shape and itssmoothness can be de�ned by the system's relative degree r.

Proposition 3.1 (Initial shape of all output functions in Y).Let Σ be a model as given by Eq. (3.2) with the initial state x0 andthe relative degree r in some (arbitrarily small) neighbourhood around

x0. The set Y contains only such signals y that have an initial shape(i. e., at time t = 0) given by

y(0) = h(x0)˙y(0) = h(x0) = Lfh(x0)

...

y(r−1)(0) = Lr−1f h(x0).

(3.13)

Proof. Equations (3.13) follow directly from di�erentiations of the outputmap h(x) (cf. Eq. (3.2b)) under initial conditions x0, given in Eq. (3.2a).

Note that the initial shape of the output function y up to its r − 1-thderivative y(r−1) is not in�uenced by the initial control u(0). Rather, it isde�ned completely by the initial state x0. In [69], this initial behaviouris investigated from a di�erential-algebraic perspective and provides addi-tional inside into the matter.

45


Proposition 3.2 (Output-function space of a system with a

well-de�ned relative degree r). Consider a model Σ given inEq. (3.2) with a well-de�ned relative degree r ∈ N.Then, the output function space Y can be de�ned as followsa: At agiven state signal x, the set Y contains only signals y that have theinitial shape as de�ned in Proposition 3.1 where the possible followingshape (i. e., t ≥ 0) is de�ned by

Y ={y | y ⊆ Cr−1 : y(r) ∈ {ϕ(x, u) : u ∈ U}

}. (3.14)

a Note that the expression Ck means the space of all k-times continuously dif-ferentiable functions.

In words, the shape of all output functions y ∈ Y is de�ned via therange of the r-th derivative y(r) = ϕ(x, u) (cf. De�nition 3.3) that isreachable from the input u ∈ U .

The proof extensively uses some of the basic properties of (real-)analyticfunctions. Those are the following (see e. g., [16, 56]):

• Analytic functions are in�nitely often continuously di�erentiable(i. e., they are C∞).

• The Taylor series of an analytic function converges.

• Products, sums and compositions of analytic functions are also ana-lytic.

• A division fg of two analytic functions f and g is also an analytic

function except for the point where the singularity occurs (i. e., atg = 0).

Remark 3.6. The proof of the Proposition above could be brie�y givenas �The proof follows directly from the analycity of f and h�. Such awording is usually encountered in mathematical treatments on similarissues. Here, however, the proof is given in detail as the author believesthat some explanations are helpful in understanding this matter.

Proof. The proof is given in three steps. First, it is shown that if r existsthen ϕ(x, u) exists (in the sense that it has a �nite value at all times t ≥ 0):

46

3.2. OUTPUT-FUNCTION SPACE

By de�nition, both the output map h(x, u) and the vector �eld f(x, u)are analytic in x and u. Therefore, the function ϕ(x, u), introduced inEq. (3.11), is also an analytic function of its arguments x and u since itis obtained only by di�erentiations, products and sums of the analyticfunctions h and f (see De�nition 3.3). As ϕ(x, u) is analytic, it is non-singular for all permissible x and u and, thus, has a �nite value at alltimes t ≥ 0. Therefore, ϕ(x, u) exists if r exists.

As an intermediate step, it is shown that ϕ(x, u) is integrable in time:The function ϕ(x, u) is dependent on x and u. The state trajectory xis an n-vector of continuous functions of time t (xi ∈ C0) and the inputu is, in the most general case, discontinuous in time t (u ∈ L∞). Bothexist at all times. It follows from the chain-rule that the composition ofan analytic function (here ϕ) with a piecewise continuous L∞- (or Ck-)function is also a L∞- (or Ck-) function (with k ∈ N). Therefore, ϕ(x, u)can generally be described as a piecewise continuous function of time tsince u can � in the worst case � be discontinuous in time:

ϕ(x, u) = y(r) ∈ L∞. (3.15)

As ϕ(x, u) is piecewise continuous, it is integrable.Finally, the proof of Eq. (3.14) is completed by showing that the output

signal y is r − 1 times continuously di�erentiable (i. e., y ∈ Cr−1): FromDe�nition 3.3 and Eq. (3.15) one �nds

y(t) =∫T· · ·∫T︸︷︷︸

r

ϕ(x(τ), u(τ))dτ r + c, where c ∈ R. (3.16)

c is some constant that can be obtained from Proposition 3.1. It followsthat y is r-times di�erentiable, since it can be obtained by r integrations.

Interpretation. The implications of Proposition 3.2 are brie�y discussed.Consider linear systems and therein especially low-pass �lters of ordersfrom zero to three, e. g.,

F0(s) =11, F1(s) =

1s+ 1

, F2(s) =1

(s+ 1)2.

Their respective relative degree is 0, 1, 2. Using Proposition 3.2, it ispossible to explain the well-known behaviour of these systems. Systems,

47


t

2=r1=r

0=r

Fig. 3.1: Output shape of systems F0(s), F1(s) and F2(s) with rela-tive degree r under the input discontinuity of a step input

as F0(s), which have a direct feedthrough, will react with a discontinuousoutput trajectory y to a discontinuous input u. The �rst order �lterF1(s), having an integrator between input and output, can only producecontinuous output trajectories y but will produce a discontinuous �rstderivative ˙y upon a control discontinuity. Similarly, F2(s) will alwaysproduce at least continuous �rst derivatives ˙y but discontinuous secondderivatives ÿ following a discontinuous input u. This behaviour is shown inFig. 3.1. Note, that the above considerations do not hold when u is chosenas a Dirac impulse, as this input will �bypass� one integrator. However,such an input is prohibited, as it does not lie in the set of L∞.

As a main result, if follows from Proposition 3.2 that this observa-tion about system behaviour also holds for nonlinear systems with a well-de�ned relative degree r. Thus, the relative degree r describes a struc-tural property of dynamic systems, namely the minimal smoothness ofthe system output. The word �minimal� refers to the case when the inputtrajectory u is chosen to be discontinuous (i. e., u ∈ L∞ and u /∈ C0).The following corollary addresses the question as to how the shape Y ofthe output y changes if u is chosen to have a certain smoothness itself(U ⊆ Ck with k ∈ N).

Corollary 3.1 (System output space Y under smooth inputs

u ∈ Ck and well-de�ned relative degree). Consider a model Σgiven in Eq. (3.2) with a well-de�ned relative degree r ∈ N. Further,the input-function space is U ⊆ Ck with k ∈ N.Then, the output function space Y can be de�ned as follows: At agiven state signal x, the set Y contains only signals y that have theinitial shape as de�ned in Proposition 3.1 where the possible followingshape (i. e., t ≥ 0) is de�ned by

Y ={y | y ⊆ Cr−1+k : y(r) ∈ {ϕ(x, u) : u ∈ U}

}. (3.17)

48

3.3. RIGHT INVERSE OF THE MODEL

In loose terms, the implication of the above corollary is that the smooth-ness of the input u is �added� to the minimal smoothness of the outputy.

Finally, the following addresses the case where the relative degree ris ill-de�ned (i. e., its value may change abruptly in time). The result isobtained from a piecewise application of Proposition 3.2 for time periodswhere the relative degree is well-de�ned. Consider a model Σ given inEq. (3.2) with an ill-de�ned relative degree r(t) ∈ N or r(t) → ∞. Theseries of values r(t) over time can, thus, be constant or non-existent overa time interval. Then, the output function y belongs piecewise to sets Ytas follows: If r(t) is �nite and constant within t1 ≤ t < t2 then y(t) ∈ Ytcan be de�ned within t by applying Proposition 3.2 to obtain Yt.

If the relative degree is non-existent (i. e., r → ∞) then y(t) ∈ Yt isan analytic function within t. y(t) can then be given by its Taylor series

y(τ) =h(x(t1)) + Lfh(x(t1)) · (τ − t1) +L2fh(x(t1))

2!· (τ − t1)2

+ . . .+Lkfh(x(t1))

k!· (τ − t1)k for k →∞.

(3.18)

Letrmin = min

tr(t), ∀t ∈ [0,∞). (3.19)

The set Y can be de�ned as containing signals y with an initial behavioury(0) as de�ned by Proposition 3.1 with the following behaviour (i. e., t ≥ 0)given by

Y ⊆ Crmin−1. (3.20)

The main results of the above are that if the relative degree does not exist(i. e., r → ∞) then the output trajectory y is in�nitely smooth and notin�uenced by the system input u and that the minimal smoothness ofthe system output y depends on the smallest value rmin that the relativedegree r possesses in time.

3.3 Right Inverse of the Model

This section introduces the properties of a right inverse of a dynamicalSISO system. The notion of the right inverse is of high importance inthe design of a nonlinear IMC controller. However, at this point it willnot be discussed how a right inverse of a given dynamical system can

49


be obtained but rather how it is de�ned. In Sections 4.1 and 4.2 rightinverses for di�erent system classes are given.

De�nition 3.4 (Right inverse [28]). The right inverse

Σr : Y → U (3.21)

of the system Σ is a mapping with the property

ΣΣry = y (3.22)

for all y ∈ Y. �

d~~ yy =d

~yr~Σu

Σ~

Fig. 3.2: Right inverse.

Thus, for every given signal yd ∈ Y, the right inverse Σr generatesan input u such that the model output y exactly follows the trajectoryyd (Fig. 3.2). The domain Y of the right inverse Σr is equal to the rangeY of the plant model Σ. According to Eq. (3.21), the right inverse isonly applicable to signals yd(= y) in Y. Thus, it cannot be applied tosignals w /∈ Y that cannot be produced as output from the model underpermissible inputs u ∈ U . Since the function space Y was de�ned toexclusively contain the possible output signals of the model Σ, it is obviousthat the right inverse exists for each trajectory y ∈ Y. This holds truesince Y was de�ned using the shape of all possible outputs y rather thantheir norm.

Remark 3.7 ([80]). Let Σ be a linear system and let Σ(s) denote its

Laplace-transform. Then, its right inverse is given by Σr = Σ−1(s), if

Σ−1(s) exists.

Remark 3.8 (Left Inverse). Although the left inverse does not play anyrole in this thesis, its di�erence to the right inverse is noted for the sakeof completeness: The left inverse Σl is de�ned by ΣlΣu = u and impliesinjectivity of the I/O map Σ while the right inverse implies surjectivity

50

3.3. RIGHT INVERSE OF THE MODEL

[80]. The left inverse reconstructs the speci�c input signal u that hasbeen used to obtain a measured model output y. Thus, the left inversedoes not exist if two di�erent input signals u1 6= u2 lead to the sameoutput, i. e., y = Σu1 = Σu2. In such a case, a left inverse does notexist since it is impossible to detect which input was used to obtain thesensed output y. However, in this case the right inverse does exist sinceit would only have to select any one of the two possible input signals u1

or u2 which yields the requested output y = yd.

Right invertibility. The property of right invertibility is introducedbelow and deserves some words of introduction. In [80], it is establishedthat right invertibility directly relates to surjectivity of the I/O map Σ.Here, the output function space Y (cf. Eq. (3.3)) was chosen such thatthe I/O map Σ is surjective. Thus, right invertibility would follow for theintroduced plant model (3.2). However, with this choice of the outputfunction space Y, a right inverse exists for all systems (3.2) and includessuch systems that are only able to produce a unique output trajectory y.Such systems essentially behave like autonomous systems. Consider thefollowing example.

Example 3.1:Consider the system x = u with an output y = 1 which is unin�uencedby the states and some initial condition x(0) = x0. Clearly, there is onlyone possible output function, namely y = 1, ∀t, regardless of the choiceof the input u. The system output behaves autonomously. Therefore, theset of output trajectories Y only contains the single signal y = 1. Hence,inversion does not make any sense since the system can only generate thissingle signal. �

Similarly, a system only has a single output trajectory y if the set ofinput signals U only contains a single function u. In summary, inversionof systems with an autonomous I/O behaviour is senseless as nothing buttheir autonomous behaviour can be achieved.

Thus, in this thesis, the property of right invertibility will be under-stood as the feasibility of inversion. In other words, a system (3.2) will beconsidered right invertible if it makes sense to design a controller based onan inverse. This is clearly not the case if the system can only produce aunique output signal. Hence, for a given initial condition x0, the output-function space Y must contain more than one signal for inversion to beuseful.

51


De�nition 3.5 (Right Invertibility). A system 3.2 is considered rightinvertible, if the set of its output function space Y contains more thanone signal. �

Remark 3.9 (Interpretation of right invertibility). The property of rightinvertibility is de�ned di�erently than right invertibility as discussed,for example in [80]. In [80], the input function space U as well as the

output function space Y were de�ned as sets that contain all analyticfunctions. Thus, they were de�ned less restrictive than in this thesis. Inthat case, the question about invertibility of a plant appears di�erently.The question becomes as to what property a dynamical system Σ shouldpossess such that it can produce any arbitrary (but analytic) outputfunction y. Or, worded di�erently, what property must a dynamicalsystem possess such that it is surjective if its domain and its rangeconsists of all analytic functions? This, however, is not conducive toan inversion-based control design since then the introduction of inputconstraints would su�ce to render virtually any system non-invertible.

In this thesis the property of invertibility does not have the same im-portance or meaning since surjectivity of the I/O map is established by

selecting a model-dependent output function space Y (see Proposition3.2). Since the problem of right inversion is approached by de�ning the

output function space Y such that the I/O map Σ is surjective on theinput function space U , a right inverse always exists. However, the set ofsignals on which the right inverse may be applied to, may, in the worstcase, only consist of a single signal. An inversion of such a model is tobe avoided.

With the introduction of the right inverse, the term (non-) minimum-phase can be de�ned [60, 61].

De�nition 3.6 (Minimum-phase behaviour). The model Σ is said tobe minimum phase if its right inverse Σr is �nite-gain Lp-stable. Other-wise, the model Σ is said to be non-minimum phase. �

Remark 3.10 (Minimum-phase behaviour). In the de�nition above,non-minimum phase behaviour is given as it is most helpful for thisthesis. Note, however, that the term non-minimum phase was origi-nally de�ned for linear systems with right-half plane zeros or time-delays.Hence, linear systems are non-minimum phase if their phase cannot bededuced by their amplitude-over-frequency behaviour alone. Thus, for

52

3.4. STRUCTURE AND PROPERTIES OF NONLINEAR SISO IMC

linear systems, the property of minimum phase and non-minimum phasebehaviour is de�ned by a property of its transfer function which is anI/O operator. De�nition 3.6 coincides with linear system theory in suchthat (in-) stability of the system inverse directly results from (non-)minimum phase behaviour of the system.

For nonlinear systems, it has then been de�ned by [51] as the instabilityof the zero dynamics. Interestingly, this de�nition uses a property ofthe system states (and not directly its I/O behaviour). As this thesisfocuses on I/O behaviour instead of the behaviour of some states it isnecessary to describe this property as a property of the I/O behaviourof nonlinear systems.

Note that the notion of zero dynamics [51] has not yet been introducedand, thus, (non-) minimum phase behaviour is not de�ned as (in)stabilityof the zero dynamics. In Section 5.3 non-minimum phase behaviour isdiscussed in more detail and the notion of internal dynamics and itsstability is then used to de�ne non-minimum phase behaviour. In thischapter, however, a system is de�ned by its I/O behaviour and thede�nition above has been chosen to accommodate this fact.

3.4 Structure and Properties of Nonlinear SISOIMC

This section gives a survey of the IMC structure and shows that all prop-erties of the linear case (see Section 2.1.3) also hold in the nonlinear casewith the necessary change in notation.

IMC structure. Figure 3.3 shows the IMC structure with nonlinear

w Q u

−

−y

y~Σ

Σ~

w~d

Fig. 3.3: IMC Structure for nonlinear plants.

IMC controller Q, nonlinear plant Σ and nonlinear plant model Σ. Dis-turbances d on the plant Σ include input and output disturbances as wellas internal disturbances which a�ect the model error. The block diagram

53


is identical to the classical IMC as shown in Fig. 2.1 with the exception ofhow the disturbances d enter the control structure.

Also, all observations obtained for the classical IMC given in Chapter2 concerning the function of IMC still hold. This can be observed sincethe Figures 2.2 and 2.3 can also be obtained in the nonlinear case fromwhich one concludes (as in the linear case) the following:

• A nonlinear IMC controller Q can be interpreted as a feedforwardcontroller for the model Σ.

• A nonlinear IMC controller Q is used in feedback to attenuate dis-turbances d and model uncertainties Σ− Σ.

The following shows that all IMC properties as given in Section 2.1.3also hold in the nonlinear case with the necessary change in notation.

IMC properties. The block diagram in Fig. 3.3 yields

w =w − y + y (3.23)

y =ΣQw. (3.24)

From Eqns. (3.23) and (3.24) the following properties can be derived [28].

Property 3.1 (Nominal Stability). Assume an exact model (Σ = Σ)in the absence of disturbances (d = 0). Then, the closed-loop system in

Fig. 3.3 is internally stable if the controller Q and the plant Σ are stable.

Proof. In the case of an exact model (Σ = Σ) and no disturbances (d = 0),the IMC structure degenerates to the feedforward control shown in Fig. 2.2,which is internally stable for the conditions mentioned above.

Property 3.2 (Perfect Control). Assume that the right inverse of the

model Σr exists and that the closed-loop system is stable with controllerQ = Σr. Then, the control will be perfect (y = w) for arbitrary distur-bances d.

Proof. Substituting Eq. (3.24) with Q = Σr into Eq. (3.23) yields w =y.

54

3.4. STRUCTURE AND PROPERTIES OF NONLINEAR SISO IMC

Remark 3.11 (Perfect Control). It is very important to realise that theproperty of perfect control does not necessitate an exact model, nor doesit necessitate the absence of disturbances. In fact, when the IMC loopis stable, the above property states that perfect control is achieved evenif the model has never been designed to resemble the plant, but ratherwas chosen arbitrarily. Also note that it is only the model which needsto be inverted. However, as will be discussed later on, an inverse alone isusually not realisable and thus, this property will remain an interestingbut useless fact. The reader may go back to page 23 where this issue isbrie�y investigated for linear systems.

Property 3.3 (Zero O�set). Assume that, for a steady-state signallimt→∞ w = wss, the IMC controller Q acts as a right inverse in steady-state, i. e.,

limt→∞

(ΣQ wss

)= wss (3.25)

holds, and that the closed-loop system is stable. Then, o�set-free con-trol yss = wss is attained for asymptotically constant reference signalslimt→∞ w = wss and constant disturbances limt→∞ d = dss.

Proof. With Eq. (3.25) inserted into Eq. (3.24) one gets

yss = wss (3.26)

for t → ∞ with limt→∞ w = wss. Substituting Eq. (3.26) into Eq. (3.23)(and taking the limit as t→∞) yields

yss = wss.

Property 3.4 (Robust Stability). Assume that Σ and Q are stable sys-tems, for which the IMC structure in Fig. 3.3 is stable for an exact modelΣ = Σ. Then, the IMC structure remains stable if the model deviationΣ− Σ satis�es the gain inequality (see De�nition 3.2)

g(

(Σ− Σ)Q)< 1. (3.27)

The proof follows directly from the Small-Gain Theorem.Comparing Properties 3.1 to 3.4 of the nonlinear IMC with the Prop-

erties 2.1 to 2.4 one �nds that the structural properties of IMC remainthe same, independently whether linear or nonlinear systems are to becontrolled. Hence, just as in the linear case,

55


• the stability of the plant Σ and the controller Q are necessary con-ditions for closed-loop stability,

• the property of perfect control cannot be realised either as will bediscussed in the following section, and

• the IMC structure guarantees zero steady-state o�set and a certainrobust stability.

From the above, one concludes that with the properties also all inter-pretations of the function of IMC that have been obtained in the linearcase carry over to the nonlinear case with the necessary change in notation.Thus, the same design procedure applies [28].

3.5 IMC Design Procedure for Minimum PhaseModels with Well-De�ned Relative Degree

For the sake of a straightforward introduction of the concept of nonlinearIMC design, the system class is limited as follows: This section dealswith �nite-gain stable (cf. De�nition 3.1), minimum-phase systems (cf.De�nition 3.6) represented by the plant model (3.2). Moreover, the plantmodel should have a well-de�ned relative degree r and it is �rst assumedthat the set U of permissible control signals contains all arbitrary but�nite functions (i. e., u ∈ U = L∞).

The restriction to stable systems is inherent in the IMC structure (seeProperty 3.1) and invertibility of the plant model Σ is a necessary condi-tion for IMC design. These two conditions are not removed throughoutthis thesis. The restriction to minimum-phase systems implies stable in-verses, which allows for a simple controller design procedure since nominalstability is guaranteed (cf. Property 3.1). The restriction to a well-de�nedrelative degree r means that Proposition 3.2 applies. These properties willbe exploited below.

The assumptions of a minimum-phase model and a well-de�ned rel-ative degree r severely limit the system class and exclude many auto-motive plants including the two-state turbocharged diesel engine which isdiscussed in Chapter 7. These excessive assumptions are dropped in Chap-ter 5, where the basic idea of IMC design is extended to stable systemswith an ill-de�ned relative degree r and to non-minimum phase systems.In that same chapter, the permissible set of control signals U will be re-stricted to incorporate input constraints.

56

3.5. IMC DESIGN PROCEDURE FOR MINIMUM PHASE MODELS WITHWELL-DEFINED RELATIVE DEGREE

3.5.1 Control Goal and IMC Design Procedure

Control goal. The goal is to �nd an IMC controller Q such that thefollowing requirements are satis�ed:

• The output y of the nonlinear plant Σ tracks an arbitrary but�nite reference signal w ∈ W = L∞ with zero steady-state o�-set (assuming a constant steady-state value, i. e., yss = wss forwss = limt→∞ w = const and dss = limt→∞ d = const).

• With t as the current time, future reference values w(τ) with τ > tare unknown and the controller can react only in dependence on thecurrent and earlier values. Thus, a non-predictive control scheme isdesired.

Main idea. The main idea is to construct the nonlinear IMC controllerQ as the composition of a linear �lter F and the right inverse Σr of themodel Σ, namely

Q = Σr F . (3.28)

The resulting IMC structure is shown in Fig. 3.4. Figure 3.4 di�ers from

w r~Σ u

−

−y

y~Σ

Σ~

w~d

F

Q

Fig. 3.4: Generalised IMC Structure.

Fig. 2.7 in such that, in the linear case, the IMC controller Q is representedas a single transfer function for implementation. Here, however, the �lterF and the inverse Σr are each implemented individually, as shown by thedashed and solid lines.

Although this approach follows the design procedure for linear systems(see Eq. (2.14a)), it needs to be explained di�erently.

Realisability of the right inverse. According to Property 3.2, thedesired IMC controller is the right inverse Q = Σr of the model Σ be-cause perfect control follows. However, this controller is not realisable:As introduced in Section 3.3, the right inverse Σr maps signals from the

57


output-function space of the model Y into the range U of the permissibleinput signals. In the IMC structure (Fig. 3.3), the signal w ∈ W is theinput signal for Q. Since the reference signal may be arbitrary but lim-ited (i. e., w ∈ L∞) one concludes that also the �lter input w lies in thesame space (w ∈ L∞). Hence, for the input w of the controller Q, therelationship

w ∈ L∞ ⊃ Cr−1 ⊇ Y for r > 0 (3.29)

holds. That is, the signal w is not necessarily in the domain of the rightinverse Σr and, hence, Σr cannot be used as controller Q. Therefore, apure right inverse as IMC controller is not realisable.

Remark 3.12. For r = 0, the IMC controller as well as the model havea direct feedthrough and the IMC structure results in an algebraic loopwhich is an ill-posed feedback loop [83] and, thus, cannot be realised.

Function of the IMC �lter. To circumvent this di�culty, the IMC�lter F is introduced as an operator which ful�ls

F : L∞ → Cr−1.

It produces for every w ∈ L∞ an output yd ∈ Y that is acceptable asinput to the right inverse Σr. The interpretation of this idea is that theIMC �lter acts as a translator from the signal w ∈ W into the set Y ofsignals that the right inverse �understands.� The block diagram in Fig. 3.5clari�es the above change in signals through the feedforward path of theIMC structure.

F Σr Σw

∈L∞yd

∈Cr−1

u

∈L∞y = yd

∈Cr−1

Fig. 3.5: Signal spaces from IMC �lter input w to model output y.

As in the linear case, the IMC controller Q given in Eq. (3.28) can beinterpreted as a feedforward controller for the model Σ as can be seenfrom the equalities

y = ΣQw = Σ (ΣrF ) w = (Σ Σr)F w

= Fw = yd.(3.30)

58


If zero steady-state o�set is desired (Property 3.3), the IMC �lter Fhas to ful�l the condition

limt→∞

Fwss = wss. (3.31)

This property is denoted by Fss = 1 in the following and means theequivalent of a unitary steady-state gain. Note that an ill-posed feedbackloop is avoided if F has a relative degree of r > 0.

The proposed IMC design procedure is summarised in the followingalgorithm:

Algorithm 3.1. Generalised IMC Design Procedure

Given: A stable and invertible minimum phase plant model Σ

Step 1: Compute the model right inverse Σr

Step 2: Find an IMC �lter F : W → Y which

• has a �good� step response from w to y, and

• has a steady-state gain of one.

Step 3: With Q = ΣrF , the IMC control loop is given in Fig. 3.4.

Result: Nonlinear output feedback IMC control loop.

Note that the derivation of the right inverse (Step 1) is given in Chapter4 for �at and input a�ne SISO systems. In the following, the focus lieson the IMC �lter F and its composition with the right inverse Σr.

Composition of the right inverse and IMC �lter. In the linearcase, the implementation of the series of F (s) and Σ(s) is done by mul-tiplying the two transfer functions to obtain Q(s) = Σ(s)F (s) and, as Qis proper, one can implement the complete IMC controller Q(s) as onetransfer function. In the nonlinear case, however, the implementation ofthe IMC controller is mathematically more involved.

The domain and range of any (nonlinear) operator are signal spaces(cf. Section 3.1). These signals (e. g., y) are de�ned over all time (i. e., t ∈(−∞,∞)). Hence, it is possible to exactly compute all existing derivativesof these signals at any instance in time τ as the signal value for t > τ isknown. Therefore, by interpreting the right inverse Σr of the model Σas a SISO operator (Fig. 3.2), there is no problem using any number of

59


derivatives y(i)d (where i = 1, . . . , r) of the input signal yd [48, 80] to

compute the output u:

u = Σryd

u(t) , Σr(yd(t), ˙yd(t), . . . , y(r)

d (t)),

(3.32)

where t is the current time.However, implementation of the right inverse Σr means to provide Σr

with all necessary information, at any instance in time, so that it cancompute the output u. This means that the derivatives y(i)

d of the inputy must be provided to Σr for the following reason: The control problemconcerns reference signals w, which are only known up to the currenttime t. Consequently, the input w to the IMC �lter and its output yd

are also de�ned only up to the current time t. Therefore, Σr represents anon-realisable operator since the derivatives y(i)

d (t) cannot be computedwithout the knowledge of future values of the model output y(t).

Hence, it is proposed to implement the right inverse Σr such that, inaddition to the signal yd, it also takes the information of the derivativesy

(j)d (t) with j = 1, . . . , r from the �lter F (Fig. 3.6). The chain connection

Fw~

d~y

r~Σ u

)(d~ ry

Fig. 3.6: I/O signals of the implementation of the composition of F

and Σr.

of F and Σr as shown in Fig. 3.6 forms a realisable (non-anticipatory)system Q. This implementation of Σr is realisable for unknown futurevalues of w(τ) with τ > t which means that the IMC controller Q isnon-anticipative and, hence, technically realisable.

Implementation of the IMC �lter. The IMC �lter F maps its inputsignals w ∈ W into the domain of the right inverse Y (see Algorithm 3.1)

yd = Fw. (3.33)

The �lter output yd should follow its input signal w �closely�, and havea steady-state gain of Fss = 1. The domain W is assumed to contain alllimited functions (i. e., w ∈ L∞).

60


It is proposed to use a linear IMC �lter F with transfer function

F (s) =yd(s)w(s)

=1

krsr + kr−1sr−1 + · · ·+ 1(3.34)

and it is proposed to choose the ki such that

F (s) =1(

sλ + 1

)r (3.35)

holds where λ can be chosen by the designer. The �lter F has the rel-ative degree r and, thus, (cf. Proposition 3.2) its output yd is r-timesdi�erentiable (yd ∈ Cr−1). Hence, the �lter F maps arbitrary limitedsignals w into signals yd which are in the function space yd = y ∈ Cr−1.Additionally, Eq. (3.31) holds with this IMC �lter, since Fss = 1.

Remark 3.13 (Nonlinear IMC �lters). In some publications on inver-sion based control (e. g., [2, 28]), it is mentioned that the use of non-linear �lters (or nonlinear reference systems) would result in better per-formance or robustness. Although nonlinear IMC �lters will also beproposed later on for special cases, here, essentially a linear IMC �lteris used since it o�ers a simple structure and is straightforward to tune.

Besides the output signal yd, the �lter has to deliver the �rst r deriva-tives of yd. To this end, it is further proposed to implement F as astate-variable �lter [97] as shown in Fig. 3.7. Such an implementation au-

−w~ ∫ ∫ ∫

+

+

1−rk

2−rk

)1(d~ −ry

)2(d~ −ry

d~yL

O

rk/1

M

+

)(d~ ry

M

F

Fig. 3.7: IMC �lter F implemented as an SVF.

tomatically delivers r derivatives of the output yd. The initial states ofthe integrators of the IMC �lter F follow from Eq. (3.13). It only di�ersfrom the classical IMC �lter in the fact that it also gives r derivatives of

61


the output. The implementation of the composition of IMC �lter F andmodel inverse Σr results in the connection signals displayed in Fig. 3.6.

Note that the �lter F (s) with the transfer function (3.34) can also beimplemented in state-space using the control normal form. In state-space,the �lter F consists of the r-state vector xF(t) ∈ Rr, input w and ther + 1 outputs [yd, ˙yd, . . . , y

(r)d ]T :

xF =

0 1 0 · · · 00 0 1 · · · 0...

. . . 00 0 · · · 0 1− 1kr−k1kr · · · −kr−1

kr

xF +

00...01kr

w

yd

˙yd

...

y(r−1)d

y(r)d

=

1 0 0 · · · 00 1 0 · · · 0

0 0. . . 0 0

0 0 · · · 0 1− 1kr−k1kr · · · −kr−1

kr

xF +

00...01kr

w(3.36a)

with the initial conditions from Eq. (3.13) as

xF(0) =[h(x0), Lfh(x0), · · · , Lr−1

f h(x0)]T. (3.36b)

Interpretation of the IMC �lter F . The IMC �lter is chosen to havethe same relative degree r as the plant model Σ. Moreover, the initial con-dition (3.36b) of the IMC �lter is chosen such that its output matches theinitial shape (cf. Eq. (3.13)) of the model output y. Employing Proposi-tion 3.1 and Proposition 3.2 one �nds that the IMC �lter F and the modelΣ have the same output-function space Y. Therewith one �nds that theinverse Σr (cf. De�nition 3.4) is always de�ned as it always operates onthe output function space Y of the model Σ.

It is important to appreciate that the proposed implementation ofIMC �lter and right inverse shown in Fig. 3.6 does not change the factthat, from a functional analytic view, the right inverse Σ as well as the�lter F still are SISO operators.

Remark 3.14. Note that y(r)d could be regarded as the only necessary

information from the IMC �lter F . This requires the integrator chain

62


to be implemented once in the �lter F and once in the inverse Σr withidentical initial conditions. This, however, is a waste of processing powerand memory since the resulting IMC controller Q would have r redun-dant states. Therefore, the integrator chain is only implemented in theIMC �lter.

Moreover, consider inverses of linear systems given in Laplace domain.The order of such an inverse is identical to the number of zeros of themodel to be inverted. Clearly, the order of the inverse is neither therelative degree of the model to be inverted nor the number of its states.In this respect, the procedure proposed above represents an analogy toinversion of linear systems.

3.5.2 Substitution of the Internal Model

For a reduction of the implementation e�ort, the structure shown inFig. 3.4 can be simpli�ed.

Theorem 3.1. Consider the IMC structure as shown in Fig. 3.4. Thecomputation of the model output y can be performed by y = Fw. There-with, the internal model Σ is redundant and can be substituted.

If follows that the IMC structure in Fig. 3.4 is equivalent to the structureshown in Fig. 3.8, where no plant model appears explicitly.

The proof follows directly from Eq. (3.30). This theorem implies a dra-

Fd~y

r~Σ

)(d~ ry

w u

−

−y

Σw~dQ

Fig. 3.8: Generalised IMC controller with no internal model.

matic complexity reduction of the �nal controller. In the original versionshown in Fig. 3.4, the controller consists of the r-th order �lter F , (n− r)-th order inverse Σr and the n-th order plant model and has the overallorder of 2n, whereas in Fig. 3.8 the plant model is omitted, resulting inan overall order n.

This interesting result is achieved since the signal yd between IMC�lter F and right inverse Σr is available in the �nal implementation (seeFig. 3.6). With this result, it is proposed to replace Step 3 of Algorithm3.1 by:

63


Algorithm 3.2. IMC control loop implementation

Algorithm 3.1 with the following replacement:Step 3: Compute the model output y by y = Fw and omit the internalmodel (see Fig. 3.8).

Interestingly, in the literature it has not yet been proposed to substi-tute the internal model in the IMC loop for linear systems. This is obviousfor two reasons:

• In the linear case, the IMC controller Q(s) is always implementedas a single transfer function. Thus, the output signal yd of the �lterF (s) is not available.

• A minimal realisation of a linear IMC loop can be obtained by com-puting the equivalent controller C(s) for a classical control loop fromEq. (2.3). Hence, there is another way to reduce the order of the �nalcontroller.

3.5.3 Robust Stability for Unstructured Uncertainties

Suppose that the plant model Σ has an unstructured multiplicative uncer-tainty ∆ such that (see Fig. 3.9)

Σ = (I + ∆)Σ (3.37)

holds. The uncertainty ∆ can be interpreted as some nonlinear dynamical

Σ~∆

+Σ

u y

Fig. 3.9: Plant Σ represented with multiplicative output uncertainty∆.

system. Then, with Eq. (3.22) and the nonlinear IMC structure fromFig. 3.8 without disturbances (d = 0) one �nds as open loop system Σ0

from �lter input signal w to feedback signal y − y

Σ0 = (Σ− Σ)ΣrF = (ΣΣr − I)F= ∆F (3.38)

which is shown in Fig. 3.10.

64


w−

∆w~ Fd~yy −

d~y

Fig. 3.10: Resulting IMC control loop with multiplicative outputuncertainty ∆.

First, robust stability of the control structure in Fig. 3.10 is shownusing the Small-Gain Theorem: Let ∆ be an upper boundary for theuncertainty ∆, such that

g(∆) ≤ ∆ (3.39)

holds, where g(∆) is the gain of the system operator ∆ (cf. De�nition 3.2).The Small-Gain Theorem [99] says that if ∆ and F are stable systems andif the open-loop gain is smaller than one, i. e.,

g(∆F ) < 1 (3.40)

holds, then closing the loop yields a stable feedback system. Hence, theclosed-loop is stable if

∆ · g(F ) < 1 (3.41)

holds. Since the IMC �lter F was designed as a linear system, well knownmethods from linear control theory can be employed to determine its gain:

g(F ) = ‖F (s)‖∞ = maxω|F (jω)|. (3.42)

The H∞-norm in frequency domain relates to the ‖·‖L2 signal norm [54].Thus, let g(∆), and therewith ∆, be de�ned using the ‖·‖L2 signal norm.The following robustness test follows directly from the Small-Gain Theo-rem:

Theorem 3.2 (Robust Stability of an IMC Loop). An IMC loopis robustly stable for the nonlinear plant given in Eq. (3.37) with theupper uncertainty boundary ∆ if

∆ · ‖F (s)‖∞ < 1 (3.43)

holds.

As the �lter parameters can be chosen arbitrarily and

‖F (s)‖∞ = maxω|F (jω)| = |F (0)| = 1,

65


Eq. (3.43) shows a considerable robustness of the nonlinear IMC loop,because stability is ensured for ∆ < 1.

The above shows a structural robustness of the IMC feedback loop.Therewith, a nonlinear IMC control has an advantage over some otherapproaches, such as exact linearisation, for which no conceptual robust-ness has been shown. However, the robustness boundaries are conservativesince they are only dependent on the gains of F and ∆ and do not accountfor their actual behaviour.

A less conservative robustness test can be given if one further exploitsthe fact that the feedback loop shown in Fig. 3.10 consists of a linear oper-ator, namely F , and a nonlinear operator, namely ∆. For such feedbackloops, specialised robustness criteria can be given. The reader is referredto the literature cited in [54, 99] for an overview. Here, it is proposed toemploy the Circle Theorem as developed in [99]. The Circle Theorem em-ploys a Nyquist-like stability criterion where the location of the Nyquistcurve of F (s) de�nes the boundaries of the uncertainty ∆ as conic regionsin the instantaneous3 I/O behaviour. Although the Circle Theorem is alsonot a necessary condition for stability, it is signi�cantly less conservativethan the Small-Gain Theorem. A thorough review of the Circle Theoremis beyond the scope of this thesis and since it is directly applicable tothe feedback structure of Fig. 3.10 and its introduction is not necessary tofollow the nonlinear IMC design, the reader is referred to [99].

3.5.4 Application to Linear Plants

In this section, the previous results are brie�y compared to the classicalIMC design method for linear systems as discussed in Chapter 2.

Theorem 3.3 (Analogy of the IMC design for nonlinear sys-

tems to the classical IMC design). For linear systems Σ(s), theproposed IMC design for nonlinear systems using the design law (3.28)and the IMC �lter from Eq. (3.34) with the proposed implementationas shown in Fig. 3.6 yields the same IMC controller Q as the classicaldesign law (2.14a) with classical IMC �lter from Eq. (2.14b).

Proof. According to Remark 3.7, the (right) inverse is given by Σr =Σ(s)−1. Therewith, both the IMC design law for linear plants (cf.

3 Instantaneous in the sense that only signal values of the input and output at thecurrent time t are regarded.

66


Eq. (2.14a)) and the proposed IMC design law for nonlinear plants (cf.Eq. (3.28)) come to the same result, namely

Q(s) = Σ(s)−1F (s). (3.44)

The inverse Σ−1(s) is

Σ−1(s) =u(s)yd(s)

=a0 + a1s+ . . .+ an−1s

n−1 + sn

b0 + b1s+ . . .+ bqsq, (3.45)

which is an improper transfer function. It can be rewritten as

u(s) =

Part 1︷︸︸︷1

b0 + b1s+ . . .+ bqsq·

Part 2︷︸︸︷(a0 [yd(s)] + a1 [syd(s)] + . . .+ ar [sryd(s)]) +ar+1s+ · · ·+ ans

q

b0 + b1s+ . . .+ bqsq· [sryd(s)]︸︷︷︸

Part 3

.

(3.46)

The available outputs of the SVF F are in their Laplace-transform yd(s),syd(s), . . . , sryd(s). Every part of Eq. (3.46) is by itself realisable (i. e., itsvalue can be computed at time t) by inserting the output of F : Part 1 ofEq. (3.46) is strictly proper, Part 2 of Eq. (3.46) is a linear combinationof the �lter outputs

[sj yd(s)

](j = 0, . . . , r). Finally, Part 3 of (3.46) is a

proper transfer function multiplied by the last �lter output.Thus, the linear IMC controller Q(s) can be implemented as it is

proposed for the nonlinear IMC controller Q, namely with the connectionsignals shown in Fig. 3.6.

Thus, even with a linear IMC controller Q(s), the IMC �lter F (s) andmodel inverse Σ−1(s) can be implemented independently of each otherusing the connection signals displayed in Fig. 3.6 despite of Σ−1(s) beingimproper. Hence, the proposed nonlinear IMC design is a direct extensionof the classical IMC design to nonlinear systems. In this respect, thisproposed nonlinear IMC design method is unique since other extensions ofIMC to the nonlinear case do not yield the same IMC controller structurefor linear systems.

67


3.6 Feasibility of Nonlinear IMC as AutomotiveController

In Section 2.3, the linear IMC controller was evaluated and it was foundthat a linear IMC is a feasible automotive controller since it ful�ls thedemands listed in Section 1.3.2. The proposed design of a nonlinear IMCis a direct extension of the linear IMC design and all structural proper-ties of the IMC structure hold true for the nonlinear case. Section 2.3explains that a linear IMC controller is a feasible automotive controller.This conclusion is based on the design and the structural properties ofIMC and, hence, completely carries over to the nonlinear case. Thus,one concludes that the proposed nonlinear IMC controller is a feasibleautomotive controller.

3.7 Summary

In this chapter, the IMC controller Q was introduced as the series of theIMC �lter F and the right inverse Σr of the model Σ. The requirement forthis method is a stable and minimum-phase model Σ. It was shown that,for nonlinear systems, such an IMC controller yields a nominally stableclosed loop which produces zero steady-state o�set. If the plant Σ canbe represented by a model with output uncertainties, a closed-loop robuststability analysis is possible if an upper boundary of the gain of the outputuncertainty is given. A considerable robustness concerning stability canbe expected.

The internal model, which is an important part of the internal modelcontrol method, can generally be omitted since its output y is equal to theoutput of the IMC �lter F (i. e., yd = y). Although omitting the internalmodel does not o�er new properties of the closed loop, it will reduce theorder of the overall controller to be implemented.

However, a method to obtain a right inverse has not been discussed.This will be done in the following for the system classes of �at and input-a�ne systems. Once a right inverse of any model is established, it can beimplemented such that the proposed IMC �lter can be used, resulting ina robustly stable nonlinear output feedback control loop.

Note that not before Chapter 5 will the demands on a well-de�nedrelative degree and a minimum-phase model be dropped.

68

4. INTERNAL MODEL CONTROL DESIGN FOR

INPUT-AFFINE SISO SYSTEMS AND FLAT

SISO SYSTEMS

In the preceding chapter, the concept of nonlinear IMC design has beenintroduced by considering mainly the I/O behaviour of dynamical sys-tems. A state-space description of the IMC loop could be avoided sinceall necessary properties of the closed-loop have been shown using I/Oconsiderations alone.

The right inverse has been de�ned by the direction of the mappingbetween the function spaces Y and U . This chapter addresses the problemof �nding a right inverse for a given dynamical system. The system classfor which this is discussed includes �at systems and systems in I/O normalform. For this purpose, however, a state-space approach needs to be takenas a tool to create a right inverse which can be used in the IMC design.Note that the state-space considerations will be limited to the constructionof the IMC controller and do not concern the closed-loop behaviour. Notethat the closed-loop behaviour has been explained in the previous chapterusing I/O considerations alone.

This chapter is divided into Section 4.1, which addresses inverses fordi�erentially �at systems, and Section 4.2, which addresses inverses forinput-a�ne systems. Those two sections are independent of each otherand the reader may skip either one.

4.1 IMC of Flat SISO Systems

This section reviews some basic properties of �at systems and employsthese properties to construct the right inverse of the plant model Σ (Step1 of the design Algorithm 3.2). The right inverse is then used to proposean IMC controller following the design idea presented in the previouschapter.

4. INTERNAL MODEL CONTROL DESIGN FOR INPUT-AFFINE SISOSYSTEMS AND FLAT SISO SYSTEMS

Di�erential �atness of dynamical systems. Consider a nonlinearSISO system Σ in state-space representation (3.2).

De�nition 4.1 (Flatness [30, 81]). The system Σ is called �at if there isa variable z(t) (called the �at output), such that the following conditionsare satis�ed:

1. The �at output z(t) can be represented in terms of the state x(t)

z(t) = Φ (x(t)) . (4.1)

2. The state x(t), the input u(t), and their time derivatives can be rep-resented in terms of z(t) and a �nite number of its time derivativesz, . . . , z(n):

x(t) = ψx

(z(t), z(t), . . . , z(n−1)(t)

)(4.2a)

u(t) = ψu

(z(t), z(t), . . . , z(n)(t)

). (4.2b)

�If the conditions in Eqns. (4.1) and (4.2) are satis�ed then the output

y(t) can be expressed as [40]

y(t) =h(ψx

(z(t), . . . , z(n−1)(t)

), ψu

(z(t), . . . , z(n)(t)

))(4.3a)

,ψy

(z(t), . . . , z(q)(t)

), with q = n− r. (4.3b)

Equation (4.3b) is the output map in dependence upon z. The integerr = n − q is the relative degree which determines the order of the IMC�lter F .

As an intermediate step, assume that requirements on the behaviourof the closed-loop system are given in terms of a desired trajectory zd forthe �at output z. Further, assume that zd is (n − 1)-times continuouslydi�erentiable (i. e., z ∈ Cn−1) and that all derivatives zd

(i) with i =0, . . . , n are known. Then, the control input u(t) can be determined byEq. (4.2b) as

u(t) = ψu

(zd(t), . . . , zd

(n)(t)), (4.4)

which is an algebraic equation whose arguments are the known functionszd, zd, . . . , zd

(n) . For the purpose of this thesis, the main consequenceof the �atness property is a perfect feedforward controller with respect tothe �at output z:

70

4.1. IMC OF FLAT SISO SYSTEMS

udz

)(dnz

dzz =y~Σ~uψ

Eq. (4.4)

Fig. 4.1: Implementation of the �atness-based feedforward controlstructure.

Corollary 4.1 (Perfect Feedforward Controller for the Flat Out-

put). The system Σ follows a given n-times di�erentiable trajectory zd(t) ∈Cn−1 with known derivatives zd

(i)(t) (i = 1, . . . , n) exactly

z(t) = zd(t), (4.5)

if the control input u(t) given by Eq. (4.4) is used and

zd(i)(0) = LifΦ(x(0)), i = 0, . . . , n− 1 (4.6)

holds.

Equation (4.6) ensures that the desired trajectory zd matches the ini-tial state x(0) of the model Σ. Figure 4.1 shows the implementation ofthe resulting �atness-based feedforward control structure.

However, since the model output y and its �at output z are not neces-sarily identical, the following introduces a mapping between them.

Right inverse of �at systems. In the following, the goal is to map thesignal yd (which is supplied by the IMC �lter F ) into the correspondingsignal zd of the �at model output. Then, the mapping from yd intoz (denoted by the operator Fy→z), composed with ψu, yields the rightinverse Σr which is part of the IMC controller.

The relationship between yd(t) and zd(t) is obtained by using Eq. (4.3)

ψy

(zd(t), . . . , zd

(q)(t))

= yd(t), with q = n− r (4.7a)

together with the initial condition given in Eq. (4.6). Equation (4.7a) isa di�erential equation which needs to be solved for zd(t). The solution isstable and can be obtained numerically because the model is assumed tobe minimum-phase1.

1 If Fy→z is an unstable operator then the right inverse Σr would be unstable which

would imply an NMP model Σ (cf. De�nition 3.6).

71


In the sense of feedforward control, the control law (4.4) requires theknowledge of all derivatives up to the n-th order, whereas a numericalsolution of Eq. (4.7a) only yields zd

(q)(t) as highest derivative. To thisend, additional r derivatives of Eq. (4.7a) have to be determined by

d

dt

(ψy

(zd(t), . . . , zd

(q)(t)))

= ˙yd(t) (4.7b)

...

dr

dtr

(ψy

(zd(t), . . . , zd

(q)(t)))

= y(r)d (t). (4.7c)

A numerical solution of the di�erential equation (4.7a) yields the re-quired di�erentiations of the �at output from zd to zd

(q). The remaininghigher derivatives can be obtained by the algebraic relationships (4.7b)to (4.7c) (and all in between) by iteratively inserting the solutions zd tozd

(q) into the equations from �rst to last and each time solving for thehighest derivative of the �at output zd. The equations (4.7b) to (4.7c)are algebraic as each only has one unknown variable (namely its highestderivative for zd) as all lower derivatives of zd are known from previousiterations or the di�erential equation (4.7a).

Remark 4.1 (Order of a �atness-based IMC). From the above one �ndsthat the order of the right inverse (i. e., the number of necessary integra-tors) is q = n − r which results in an order of a �atness-based IMC ofn.

For a given signal y(r)d , solving the di�erential and algebraic equations

(4.7) for zd is denoted by the mapping

Fy→z : Y → Z, (4.8)

whose implementation must also deliver all derivatives of zd up to n-thorder.

The right inverse Σr of a �at system is given by the composition

Σr = ψu ◦ Fy→z (4.9)

which is described by Eqns. (4.7c) and (4.4).

72

4.1. IMC OF FLAT SISO SYSTEMS

IMC controller. The following theorem proposes the �atness-basedinternal model controller.

Theorem 4.1 (Flatness-based IMC). If the plant Σ can be rep-

resented by a stable minimum-phase �at model Σ, a nonlinear IMCcontroller Q is described by

kry(r)d + kr−1y

(r−1)d + . . .+ yd = w (4.10a)

ψy

(zd(t), . . . , zd

(q)(t))

= yd(t), with q = n− r...

dr

dtr

(ψy

(zd(t), . . . , zd

(q)(t)))

= y(r)d (t)

(4.10b)

u = ψu

(zd, . . . , zd

(n)), (4.10c)

with the �lter coe�cients ki (i = 1, . . . , r) and the function ψu resultingdue to the �atness property (4.2b) of the model. The Eqns. (4.10b)represent the operator Fy→z, de�ned in Eq. (4.8). The initial condi-tions for the di�erential equations (4.10a) and (4.10b) are given inEqns. (3.13) or (4.6), respectively.

Equations (4.10a) and (4.10b) represent the dynamics of Q and the alge-braic relationship (4.10c) maps the states of the dynamics into the plantinput u. The di�erential equation (4.10a) follows directly from the trans-fer functions (3.34) and (3.35) of the IMC �lter F . The parameters kiwith i = 1, . . . , r represent the degree-of-freedom of Q. The di�erentialequation (4.10b) together with the output map (4.10c) is the right inverseΣr of the �at model Σ.

The resulting IMC control loop shown in Fig. 4.2 is robustly stable(cf. Theorem 3.2) and if stability is achieved, the closed-loop has zerosteady-state o�set (cf. Property 3.3). The resulting closed-loop behaviourdepends on the choice of the ki and the modelling errors. According toStep 3 of Algorithm 3.2, the IMC control loop can be implemented with-out the explicit model as shown in Fig. 4.2. The important improvementobtained by the method presented in this thesis is the fact that this ap-plication of the IMC principle is not based on feedback linearisation. Thecontroller described by Theorem 4.1 achieves robust tracking performanceand stability directly for the nonlinear plant.

73


Σy

y~

d

w

d~y

(r)d~y

dz

(n)dz u

Qw~

r~Σ

−

−

F zyF → uψ(4.10a) (4.10b) (4.10c)

Fig. 4.2: Flatness-based IMC control structure.

Example 4.1 (Flatness-based IMC):Consider the plant model

Σ : x1 = −x31 + x2

x2 = −x32 − x1x2 + σu

x(0) =[2, 0

]T(4.11a)

y = x1 + x2, (4.11b)

where σ 6= 0 is an uncertain parameter with the nominal value σ = 1.

The �rst time-derivative of the output y depends explicitly on the input u.Thus, the system (4.11), has a relative degree of r = 1. The variable z = x1

is a �at output, since

z = x1 ⇔ x1 = z = ψ11(z) (4.12a)

z = x1 = −x31 + x2 ⇔ x2 = z + z3 = ψ12(z, z) (4.12b)

z = −3x21x1 + x2 = −3x2

1(−x31 + x2)− x3

2 − x1x2 + σu

⇔ σu = z + (z + z3)3 + z(z + 3z2) + z4 (4.12c)

holds. Consequently, for σ = 1, the �at feedforward control law (seeEq. (4.4)) can be determined as:

u = ψu (zd, zd, zd) = zd + (zd + z3d)3 + zd(zd + 3z2

d) + z4d (4.13)

The mapping Fy→z is constructed according to the procedure described inSection 4.1:

zd + z3d + zd = yd (4.14a)

zd + 3z2dzd + zd = ˙yd. (4.14b)

Equation (4.14a) is a nonlinear di�erential equation for z with input yd. Itis stable and, thus, can be solved numerically with the initial conditionszd(0) = x1(0) and zd(0) = x1(0). To obtain the additional derivative zd,Eq. (4.14b) is used. The structure of the required dynamic model Fy→z ispresented in Fig. 4.3. The IMC �lter F has to be implemented as a �rst-

74

4.2. IMC OF SISO SYSTEMS IN INPUT/OUTPUT NORMAL FORM

d~y

d~y&

∫dz&

dz

dz&&

−

×

× 3

− −

Fig. 4.3: Fy→z from Eq. (4.14).

−w~ ∫F

10d~y&

d~y

Fig. 4.4: First-order IMC �lter F .

order �lter (Fig. 4.4). The �lter pole is chosen as λ = −10 and the initialcondition set to yd(0) = x1(0) + x2(0)=2.

Finally, the �at IMC controller Q can be implemented according to thestructure in Fig. 4.2, namely as the interconnection of the �rst-order �lterF (see Fig. 4.4), the mapping Fy→z shown in Fig. 4.3, and the feedforwardcontrol law (4.13).

Figure 4.5 shows the performance of the closed-loop. The reference w isinitially zero and modi�ed by two steps occurring at t = 1s and t = 2s. Itcan be seen that for the nominal case σ = 1, exact tracking of the generatedtrajectory yd is achieved, and that for a changed parameter σ robust controlperformance is provided. �

4.2 IMC of SISO Systems in Input/Output NormalForm

In this section, the I/O normal form of an input-a�ne system is exploitedto obtain a right inverse (Step 1 of Algorithm 3.2).

I/O normal form. A scalar input-a�ne model Σ is de�ned by

Σ : x =f (x) + g (x)u, x(0) = x0

y =h (x) .(4.15)

75


−2

−1

0

1

2

3

Reference wFilter output yd for σ = 1Output y for σ = {1, 0.7, 3}

0 1 2 3−70

−35

0

35

70

time (s)

Control input u for σ = {1, 0.7, 3}

Fig. 4.5: Step responses of the closed-loop system of Example 4.1.

It is assumed that the elements of the vector �elds f and g are analyticfunctions of their arguments. The following reviews the I/O normal formof nonlinear systems.

It is assumed that the model Σ can be transformed with x∗ as the newcoordinates de�ned by (cf. [51])

x∗1 = y = h(x) = φ1(x) (4.16a)

x∗2 = ˙y = x∗1 = Lfh(x) = φ2(x) (4.16b)

......

x∗r = y(r−1) = x∗r−1 = Lr−1f h(x) = φr(x) (4.16c)

x∗r+1 = φr+1(x) (4.16d)

...

x∗n = φn(x). (4.16e)

The functions φi(x) for i = r+ 1, . . . , n can be chosen arbitrarily withthe requirement that the map φ(x) = [φ1, · · · , φn]T is a di�eomorphism2.

2 A di�eomorphism is de�ned as a di�erentiable map between manifolds which has

76


Locally, a di�eomorphism is required to ful�l

Rank(∂φ(x)∂x

)= n. (4.17)

After r di�erentiations of y, the input u appears for the �rst time explicitly

y(r) = Lrfh(x) + LgLr−1f h(x)︸︷︷︸6=0

u. (4.18)

The number r is the relative degree of Σ with respect to its output y. It isassumed that r < n holds. For r = n the output y is a �at output and theresult of the preceding section solves the problem. For r = 0 the outputy explicitly depends on u, resulting in a direct feedthrough (y = h(x, u))for which the following procedure is still applicable.

In the new coordinates x∗, Σ can be written in the I/O normal formx∗1x∗2...

x∗r−1

=

0 1 0 · · · 0

0 0 1. . . 0

.... . . 0

0 · · · 1

(r−1×r)

·

x∗1x∗2...x∗r

(4.19a)

x∗r = a(x∗) + b(x∗)u, b(x∗) 6= 0 (4.19b)[x∗r+1 · · · x∗n

]T = p(x∗) + q(x∗)u (4.19c)

with the initial condition x∗(0) = φ(x(0)) where a(x∗), b(x∗),p(x∗) andq(x∗) are given in new coordinates x∗ by (see e. g., [51, 60, 61])

a(x∗) = Lrfh ◦ φ−1(x∗)

b(x∗) = LgLr−1f h ◦ φ−1(x∗)

pi(x∗) = Lfφi ◦ φ−1(x∗)

qi(x∗) = Lgφi ◦ φ−1(x∗)

for i = r + 1, · · · , n.

Right inverse of I/O-linearisable systems. From the I/O normalform, a right inverse can be obtained: Let the input be de�ned by

u(t) =1

b(x∗(t))(−a(x∗(t)) + ν(t)) , (4.20)

a di�erentiable inverse.

77


where ν(t) is a new input, that changes Eq. (4.19b) into x∗r(t) = ν(t) andfrom Eq. (4.16) it follows that

x∗r(t) = ν(t) = y(r)(t) (4.21)

holds. Demanding y!= yd, Eqns. (4.19b) and (4.19c) can be rewritten as

u =1

b(x∗)

(−a(x∗) + y

(r)d

)(4.22)[

x∗r+1, · · · , x∗n]T = p(x∗) + q(x∗)y(r)

d . (4.23)

Additionally, it is known from Eq. (4.16) that a part of the transformedstate vector x∗ is given by the IMC �lter F

[x∗1, . . . , x∗r ]T = [yd, . . . , y

(r−1)d ]T . (4.24)

In order to use the control law (4.22), the value of the unknown states[x∗r+1, . . . , x

∗n]T needs to be determined. It is proposed to do this by

numerically solving the di�erential equation (4.23), which is possible dueto the restriction to minimum-phase models. The dynamics posed fromEq. (4.23) is called the internal dynamics.

It seems important to clarify the relationship between the internaldynamics and the zero dynamics. Although the notion of zero dynamicsis not directly used for controller design, it is discussed brie�y in thefollowing remark for the sake of completeness.

Remark 4.2 (Zero dynamics [51]). The notion of zero dynamics meansthe case of the trajectory of the internal dynamics that is generatedby the input function u0 = γ(x∗) which keeps the output y at zerofor all time t > 0 (assuming matching initial conditions), which can beexpressed as x∗1(t), . . . , x∗r(t) = 0, ∀t ≥ 0.

The zero dynamics are then de�ned as the dynamics of the states ofthe internal dynamics (4.23) x∗r+1, . . . , x

∗n under the in�uence of u0 and

x∗1, . . . , x∗r = 0. Despite their interesting system theoretical properties,

the zero dynamics are not exploited for an IMC control design. There-fore, the notion of the zero dynamics does not play a direct role in IMCdesign.

In order for the zero dynamics to be important in control design, theoutput y has be de�ned as the di�erence of the output of a referencesystem and the output of the plant. Then, it is desired to maintain thisoutput y constantly at zero. Such a situation is discussed, for example,in [65].

78


With the above, the right inverse Σr for systems which can be trans-formed into the I/O normal form is given by Eqns. (4.22)-(4.24).

The implementation of the right inverse Σr of I/O linearisable systems isdisplayed in Fig. 4.6. Note that the input vector signal [yd, . . . , y

(r)d ]T is

delivered by the IMC �lter F .

u

)(d

** ~)()( ryxqxp + ∫[ ]Tnr

xx **1,,L+

rΣ~

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

)(d

*

*1

)(d

d

d

~~

~

~

rr

ry

x

x

y

y

y

M

M

&

)(

~)(*

)(d

*

x

x

b

ya r+−

Fig. 4.6: Right inverse Σr of a system in I/O normal-form (4.19).Bold arrows indicate vector signals.

IMC controller. The following theorem summarises the results of thissection.

Theorem 4.2 (IMC for Input-A�ne Systems). If the plant Σcan be represented by a stable minimum-phase input-a�ne model Σ,which can be transformed into the I/O normal form (4.19), a nonlinearIMC controller Q is given by

kry(r)d + kr−1y

(r−1)d + . . .+ yd = w

[x∗1, . . . , x∗r ]T = [yd, . . . , y

(r−1)d ]T

(4.25a)

[x∗r+1, · · · , x∗n

]T = p(x∗) + q(x∗)y(r)d (4.25b)

u =1

b(x∗)

(−a(x∗) + y

(r)d

), (4.25c)

with the �lter coe�cients ki (i = 1, . . . , r) and the relationships (4.25b)and (4.25c) which result from the I/O normal form (4.19) of the modeland the introduction of the new input in (4.20). The initial conditionsfor the di�erential equations (4.25a) and (4.25b) are given by x∗(0) =Φ(x(0)).

79


Eqns. (4.25a) and (4.25b) represent the dynamics of Q and the alge-braic relationship (4.25c) maps the states of the dynamics into the plantinput u. The parameters ki with i = 1, . . . , r represent the degree-of-freedom of Q and can be chosen freely with the limitation that the di�er-ential equation (4.25a) must be stable. The di�erential equation (4.25a)follows directly from the transfer function (3.34) of the IMC �lter F .

The resulting IMC control loop (Fig. 3.8) relies on output feedback,is robustly stable (Theorem 3.2), and if stability is achieved, the closed-loop has zero steady-state o�set (Property 3.3). The resulting closed-loopbehaviour depends on the choice of the ki and the modelling error.

Example 4.2 (IMC of an I/O linearisable system):Consider the input-a�ne model

x1 = −σx31 + x2 + u

x2 = −x32 − x1x2 + u

x(0) =[−3.5, 3

]T(4.26a)

y = x1 + x2, (4.26b)

where σ is an uncertain parameter and the IMC controller is designed forthe nominal case σ = 1.

In the I/O normal form, the �rst transformed state x∗1 is equal to the output:

x∗1 = y = x1 + x2 = Φ1(x). (4.27)

Its �rst derivative

x∗1 = ˙y = −x31 + x2 − x3

2 − x1x2 + 2u (4.28)

depends on u, thus the relative degree is r = 1. Employing Eq. (4.22) one�nds

u =1

2

(x3

1 − x2 + x32 + x1x2 + ν

), (4.29)

where ν is the new input. Equation (4.29) is obtained by setting ν to thedi�erentiated IMC �lter output: ν = ˙yd. Choose

x∗2 = x1 − x2 = Φ2(x) (4.30)

as state for the internal dynamics, which consequently is determined as

x∗2 = −x31 + x2 + x3

2 + x1x2. (4.31)

Since the relative degree is r = 1, the �rst-order �lter shown in Fig. 4.4can be used. Equations (4.29) and (4.31) are still expressed in the original

80


u

d~y

−w~ ∫

F

10

Q

∫1x

2x∗2x

∗2x

∗1x

vrΣ~

d~y&

Eq. (4.32) Eq. (4.31)

Eq. (4.29)

Fig. 4.7: Structure of the controller Q.

−1

0

1

2

3

Reference wFilter output yd (for σ = 1)Output y for σ = {1, 0, 0.5, 3}

0 1 2 3 4 5 6 7 8 9−20

−10

0

10

20

time (s)

Control input u for σ = {1, 0, 0.5, 3}

Fig. 4.8: Simulation results of Example 4.2.

coordinates. For implementation, these coordinates can be substituted bythe inverse coordinate transformation, given by

x1 =1

2(x∗1 + x∗2) x2 =

1

2(x∗1 − x∗2). (4.32)

The structure of the complete resulting IMC controller Q is presented inFig. 4.7. Figure 4.8 displays the simulation results for the system (4.26a),(4.26b) controlled by the IMC controller shown in Fig. 4.7. It gives theresponse of the closed-loop system subject to two consecutive steps in w attime t = 2s and t = 5s. The system output y is plotted for the nominal case(σ = 1) as well as for σ = {0, 0.5, 3}. It is shown that despite this signi�cant

81


modelling error, the closed-loop behaviour robustly yields zero steady-stateo�set. �

Example 4.3 (Comparison to Exact I/O Linearisation):The example above is continued to portray the di�erences between an IMCbased on the I/O normal form and an exact I/O linearisation. An I/O lin-earisation for the system (4.26) is performed for the nominal case σ = 1according to [51]: The input law (4.29) remains the same with the excep-tion that the states x1 and x2 have to be measured from the plant (4.26).However, the new input ν is now used to establish the linear error dynamics

ν = ˙yd − c1(y − yd). (4.33)

The pole of the error dynamics is set to the pole of the IMC �lter:

c1 = −λ. (4.34)

An exact I/O linearisation only stabilises the plant around an a priori given

and r-times di�erentiable trajectory yd. In this example, this trajectoryis generated by sending the reference signal w through the �lter F andits outputs ˙yd and yd are then assumed to be the given trajectory. Inconclusion, the exact I/O linearisation requires state feedback and consistsof Eqns. (4.29), (4.33) and (4.34).

For the nominal case (σ = 1), simulation results are identical to the proposednonlinear IMC controller presented in Example 4.2. However, simulationswith varying parameter σ (assuming a given reference signal w and initialcondition x0) show that the exact I/O linearisation becomes unstable forσ ≤ 0.73, while the IMC loop remains stable for all positive values of σ andeven for negative values σ ≥ −0.4.

Figure 4.9 shows a comparison for σ = 0.72 between the nonlinear IMCcontroller obtained in Example 4.2 and the exact I/O linearisation as devel-oped here. The result shows that, here, the exact I/O linearisation is notas robust as IMC. The I/O linearised (but perturbed) plant shows unstablebehaviour between 4s ≤ t ≤ 5s while the output of the IMC controlled plantshows an almost nominal behaviour.

In this example, the nonlinear IMC is to be preferred over the exact I/Olinearisation since it only relies on output feedback instead of state feedback,possesses zero steady-state o�set in the presence of modelling errors andshows larger robustness boundaries. Simulation studies with several otherexamples also indicate that IMC seems generally more robust than exactlinearisation methods. �

82

4.3. SUMMARY

−2

0

2

4

6

8

0 1 2 3 4 5 6 7 8 9−20

0

20

40

time (s)

Reference w

IMC: Output yd (σ = 0.72)I/O Lin.: Output yd (σ = 0.72)

IMC: Input u (σ = 0.72)I/O Lin.: Input u (σ = 0.72)

Fig. 4.9: Simulation results of Example 4.3.

4.3 Summary

The main contributions of this chapter are Theorem 4.1 and Theorem 4.2.The theorems give the necessary equations for designing an IMC controllerQ for �at or input-a�ne systems, respectively. Essentially, right inversesfor these system classes were introduced and combined with the IMC �lteras introduced in Section 3.5.1.

Therefore, this chapter can be interpreted as the link between therather abstract operator-oriented view on nonlinear IMC, as presentedin Chapter 3, and the nonlinear IMC design for systems given in theirstate-space form.

83


84

5. EXTENSIONS OF THE BASIC PRINCIPLE

The main contribution of this chapter is the extension of the system class.In the preceding chapters, the plant model was restrained in such thata well-de�ned relative degree was necessary and arbitrarily large inputsignals were allowed. Moreover, the model was assumed to be MP (i. e., astable inverse was required).

All of these assumptions are dropped in this chapter. In Section 5.1input constraints are introduced to accommodate the fact that virtuallyall real plants only accept a limited range of the inputs u. In Section 5.2,the assumption of a well-de�ned relative degree r is dropped and replacedby the much weaker assumption of model invertibility (cf. De�nition 3.5).As a result, the model may be singular at some points; that is, it maydrop or lose its relative degree completely. As the example of the two-stage turbocharged engine in Chapter 7 shows, real plants may exhibitsuch a behaviour.

The minimum-phase assumption is removed in Section 5.3. A methodis proposed with which a perfect (and therewith unstable) inverse can beemployed by using an appropriate IMC �lter F . It is shown how such anIMC controller can be built to be internally stable.

Finally, Sections 5.4 and 5.5 introduce some necessary amendmentsto the design method which are necessary for the applications in PartII. These amendments are the introduction of measured disturbances andtreating simple quadratic MIMO systems.

5.1 Input Constraints

In all existing plants (e. g., machinery, chemical processes), the accessibleinputs u ∈ U are limited. This is addressed by the input-function space

U = {u | umin ≤ u(t) ≤ umax, ∀t} . (5.1)

The control problem to be solved involves respecting these input con-straints. This means that the output u of the right inverse Σr must ful�l


Eq. (5.1).

Main idea. From the nonlinear IMC design of Chapter 3 one �nds thefollowing: De�nition 3.4 shows that the right inverse Σr is only de�ned forinput signals yd in the output-function space (i. e., yd ∈ Y) of the modelΣ. The IMC �lter F is responsible for mapping arbitrary input signalsw ∈ L∞ into signals that the right inverse �understands� (i. e., yd ∈ Y).

Therefore, input constraints can be considered by assuring that theIMC �lter F only produces signals yd which lie in the range of the modelΣ (i. e., yd ∈ Y) reached from the domain of the limited inputs of Eq. (5.1).Thus, it is proposed to change F to produce only such trajectories yd ∈ Ythat can be obtained with permissible inputs u ∈ U .

Applying Proposition 3.2 shows that the output-function space Y isin�uenced by the input-function space U via the relative degree r. Hence,a limitation on the plant input u directly relates to a limitation on the

highest derivative y(r) of the model output y. With the substitution y!=

yd (cf. (3.30)) one �nds the solution to the problem.

Theorem 5.1 (Input Limitation with IMC). Consider a model

Σ with a relative degree r. If the highest derivative y(r)d of the �lter

output yd is limited to

y(r)d,min(t) ≤ y(r)

d (t) ≤ y(r)d,max(t) (5.2a)

with

y(r)d,min(t) = min

u∈Uϕ(x(t), u)

y(r)d,max(t) = max

u∈Uϕ(x(t), u)

(5.2b)

and ϕ(x(t), u) as given in De�nition 3.3 then the output u of the right

inverse Σr will always respect the limitation (5.1) (i. e., it ful�ls u ∈ U)regardless of the setpoint w, disturbance d and modelling error.

Proof. With a relative degree r and at a state x(t), Eq. (3.11) (given inDe�nition 3.3) the r-th output derivative y(r)(t) satis�es

y(r)(t) = ϕ(x(t), u(t)).

Since ϕ(x, u) is analytic in u (see the proof of Proposition 3.2), the con-tiguous range in u, given in Eq. (5.1), is mapped into a contiguous range

86

5.1. INPUT CONSTRAINTS

)(max~ ry

)(min~ ry

maxuminu

)(~ ry

u

Fig. 5.1: Example of the mapping y(r)(t) = ϕ(x(t), u(t)) assuminga given x(t).

in y(r) .Figure 5.1 illustrates a mapping between u and y(r) at a given x(t).

It shows the above properties of the function ϕ(·, u), namely that a con-tiguous range in u is mapped into a contiguous range in y(r), that y(r)

over u is a smooth function (it is analytic), and that this mapping is notnecessarily invertible.

It follows that for each y(r)(t) in the set [y(r)d,min(t), y(r)

d,min(t)], the alge-braic relationship in Eq. (3.11) can be solved for a feasible input u(t) in thepermissible range [umin, umax]. From Eq. (3.30) it follows that the IMC�lter F achieves y = yd and, thus, Eq. (3.11) is also valid for y = yd.

Implementation. It is proposed to implement the limitation (5.2a) bya saturation block in the SVF structure of the IMC �lter F as shownin Fig. 5.2. This proposed implementation ensures that the �lter F still

−w~

)(max,d

~ ry)(min,d

~ ry

∫ ∫ ∫

+

+

1−rk

2−rk

L

O

rk/1

M

+

M

F )1(d~ −ry

)2(d~ −ry

d~y

)(d~ ry

Fig. 5.2: Nonlinear limited IMC �lter F to be used for input con-straints.

achieves its steady-state value despite arbitrarily many intermitted limi-

87


tations on y(r)d , as long as no limitation occurs for t→∞.

Interpretation. The resulting nonlinear state-variable IMC �lter F ,shown in Fig. 5.2 can be interpreted as a control structure where the ki(i = 1, . . . , r) can be interpreted as the gains of a state feedback controllerfor the plant which consists of the integrator chain. The saturation canthen be interpreted as a modelling error of the integrator chain whosee�ect needs to be attenuated.

The function of the limited IMC �lter F , shown in Fig. 5.2, is thefollowing:

• If the input u is not in saturation then the behaviour of the saturated�lter F (Fig. 5.2) is identical to the behaviour of the unsaturated�lter given by Eq. (3.34). Hence, if not in saturation, the IMC �lteryields the linear I/O behaviour as chosen by the designer.

• If in saturation, the IMC �lter F produces that same output trajec-tory yd = y that the model Σ produces with the saturated input u.Then, the IMC �lter F has a nonlinear I/O behaviour.

Literature. One concludes that this result and the design method ofIMC yield a feedforward controller Q that can respect input constraintson-line. The design of nonlinear feedforward controllers is a topic of cur-rent research (see e. g., [40, 75, 94, 100]). To this end, several algorithmsto design trajectories and their di�erentiations for use with a right inverseare proposed in the literature mentioned above. In [94], the authors pro-pose the use of polynomials to generate yd(t) and its necessary derivatives.In order to respect the input constraints, a numerical iteration is proposedwhere the slope of the polynomial is adjusted iteratively until the outputu of the right inverse is within the acceptable input constraints. Thisapproach does not meet the real-time demands in the automotive indus-try nor can it handle ramps in w(t) nicely. In [40], the authors proposean o�ine trajectory planning algorithm which relies on the solution ofa boundary value problem. This approach relies on the knowledge of afuture setpoint from which the information of the second boundary canbe obtained. It can handle input constraints well, but its application is re-stricted to problems where the setpoints are known a priori. Finally, [100]proposes a similar approach as the one discussed here with the di�erencethat the trajectory yd is not computed by a linear �lter but rather is theoutput of a sliding mode controller. Although this idea is mathematically

88


more involved than limiting an SVF, it does o�er the possibility to respectadditional constraints on, for example the model states.

In conclusion, the limited IMC �lter F o�ers a computationally cheapand straightforward way to generate trajectories for nonlinear feedforwardcontrol that respect input constraints.

Finally, Fig. 5.3 shows the feedforward controller Q which can be usedas an internal model controller. Note that not all states xi in the state

(r)d~y

uw~

)(min,d

~ ry )(max,d

~ ry

d~y

x

Σ~y~

Q

r~ΣF

Eq. (5.2b)

Fig. 5.3: Feedforward control structure using the input constraintsas proposed here.

vector x as depicted in Fig. 5.3 need to be determined by the right inverseΣr. For systems without internal dynamics, the limitations in Eq. (5.2b)can be computed using the information from the �lter F only.

The actual calculation of x(t), which is used to determine the bound-aries y(r)

d,min(t) and y(r)d,max(t) using Eq. (5.2b), depends on the system class

and is discussed in the following.

Input constraint with a �atness-based IMC. In case of a �atness-based IMC, the relationship Eq. (3.11) between the highest derivative y(r)

d

of the �lter output yd and the input u of the model can be obtained byusing the �at output z.

Using Theorem 4.1 on page 73, Eq. (4.10c) can always1 uniquely besolved for the highest derivative zd

(n) of the �at output. Suppose thesolution is written as a function ψ−1

u of the lower derivatives and theinput u

zd(n) = ψ−1

u (zd, zd, . . . , zd(n−1), u). (5.3)

As an intermediate step, the permissible range of the highest derivativezd

(n) is to be limited according to the permissible u ∈ U :

zd,min(n)(t) ≤ zd

(n)(t) ≤ zd,max(n)(t) (5.4a)

1 This is the case because a �atness-based inverse is both a left and a right inverse[81] with respect to the �at output z.

89


with

zd,min(n)(t) = min

u∈U

(ψ−1

u (zd(t), . . . , zd(n−1)(t), u)

)zd,max

(n)(t) = maxu∈U

(ψ−1

u (zd(t), . . . , zd(n−1)(t), u)

) (5.4b)

This limitation can be mapped to a limitation on the highest derivativeof the IMC �lter output y(r)

d using Eq. (4.10b) which gives the desiredpermissible range

y(r)d,min ≤ y

(r)d ≤ y(r)

d,max. (5.5)

With these boundaries, the nonlinear IMC �lter shown in Fig. 5.2 canbe employed. Figure 5.4 shows the structure of a �atness-based IMCenhanced to respect input constraints.

Σy

y~

d

w-

-

Fd~y

(r)d~y

dz

(n)dz

uψ uzyF →

w~

)(min,d

~ ry )(max,d

~ ry(5.4),(4.10b)

Fig. 5.4: Structure of a �atness-based IMC that respects input con-straints.

A �atness-based IMC, as de�ned in Theorem 4.1, can be extended totake input constraints into account.

Corollary 5.1 (Respecting input constraints using a �at-ness-based IMC). The IMC controller Q is de�ned by Eqns. (4.10a)-(4.10c) and a limitation on the �lter output yd


d (t) ≤ y(r)d,max(t),

obtained from Eq. (5.4) inserted into Eq. (4.10b).

The following example employs the proposed handling of input constraints.A linear plant is chosen for the sake of simplicity. The nonlinear case willbe discussed in Part II for both automotive examples.

90


Example 5.1 (Respecting input constraints using a �atness-basedIMC):Consider the problem of designing an IMC controller (feedforward con-troller) Q for the model

Σ(s) =s/5 + 1

s3 + 2s2 + s+ 1. (5.6)

The model can be represented in control normal form in state-space by

x =

0 1 00 0 1−1 −1 −2

x+

001

uy =

[1 1

50]x.

(5.7)

All initial conditions are assumed to be zero. A linear system in controlnormal form always has a �at output z = x1 [ii]:

z = x1 z = x1 = x2

z = x2 = x3...z = x3 = −x1 − x2 − 2x3 + u

= −z − z − 2z + u (5.8)

From Eq. (5.8) one �nds the �atness-based feedforward control law (cf. Eq.(4.10c))

u = ψu(...z d, zd, zd, zd) =

...z d + 2zd + zd + zd. (5.9)

In order to obtain the relationship Fy→z (cf. Eq. (4.10b)) between the desiredmodel output yd and the desired �at output zd the output equation (see

Eq. (5.7)) y = x1 + 15x2 is used with yd

!= y and the relationship (5.8) and

yields

yd = zd + zd/5 ˙yd = z + z/5 ÿd = zd +...z d/5. (5.10)

The di�erential equation (5.10) needs to be solved (note that the signalsyd, ˙yd, ÿd are given from the IMC �lter F ) for zd, zd, zd and

...z d with initial

conditions zd(0) = zd(0) = zd(0) = 0. The operator Fy→z presents thissolution.

The IMC �lter F is designed to exhibit the linear I/O behaviour (when notin saturation)

F (s) =1

(s/λ+ 1)2=

1

s2/λ2 + 2s/λ+ 1(5.11)

and implemented as SVF, as shown in Fig. 5.2. The dynamics of the �lterF are in time domain (cf. (4.10a))

w =1

λ2ÿd +

2

λ˙yd + yd (5.12)

91


with initial conditions yd = ˙yd = 0. At this point in the example, a �atness-based IMC has been designed and it consists of Eq. (5.9), the solution of thedi�erential equation (5.10), and the �lter dynamics (5.12).

Now, input constraints are introduced as in Eq. (5.1). The following demon-

strates how the restrictions on the highest derivative y(r)d of the �lter can

be obtained. From Eq. (5.9) one �nds

...z d,min = min

u∈Uψ−1

u (zd, zd, zd, u) = −2zd − zd − zd + umin

...z d,max = max

u∈Uψ−1

u (zd, zd, zd, u) = −2zd − zd − zd + umax,(5.13)

the permissible range...z d,min ≤ zd ≤

...z d,max of the highest derivative of

the trajectory of the �at output zd. Inserting this result into Eq. (5.10) one�nds the desired limits

ÿd,min = zd +1

5

...z d,min

ÿd,max = zd +1

5

...z d,max.

(5.14)

Hence, the �atness-based IMC will never violate the input constraints ifthe relationships (5.13) and (5.14) are used to limit the highest attainablederivative ÿd of the IMC �lter F , given in the di�erential equation (5.12) byyd,min ≤ ÿd ≤ yd,max.

Figure 5.5 shows the simulation results of the closed-loop system with a �lterdesign using λ = 4 and the input constraints umax = 2 and umin = −2 for astep in the reference signal w(t) occurring at time t = 1s. The dashed linesin Fig. 5.5 show the model output y and input u (assuming an exact model)without any limitations on the input u. It is plotted to show how the rightinverse Σr calculates the input signal u without limitations. The plant modelΣ has two weakly damped poles and, hence, tends to oscillate. During timet = 1s until time t ≈ 1.8s, the dashed input u peaks to about maxt(u(t)) =80 and excites the model. In between time t = 1.8s and t ≈ 2s the unlimitedright inverse Σr decelerates the plant model Σ with mint(u(t)) ≈ −11 suchthat no oscillations occur and the plant output follows the �lter outputexactly y = yd.

If limitations on the input are active and the classical method of respectinginput constraints, as discussed in Section 2.2.3, is used by limiting both plantand model inputs (cf. Fig. 2.12 on page 36), then the result is unsatisfactory

(solid line in Fig. 5.5). In this case, the right inverse Σr and the �lter Fdo not know that the inputs u are limited. Hence, the input trajectory u(solid line) is identical to the unlimited case (dashed line) with the exceptionthat it is cut o� at +2 and −2. Thus, the controller Q is unaware that thedynamics of its output u do not reach the model and the plant. Note that the

92


0

0.5

1

1.5

Out

put

y

0 1 2 3 4 5 6 7 8 9 10−5

0

5

time [s]

Inpu

tu

limited IMC filter

unlimited

classical limitation

Fig. 5.5: Simulation results of Example 5.1, with λ = 4 and umax =2 and umin = −2.

model output y even �rst moves into the negative. Moreover, the model hasbeen excited without actually compensating the weakly damped dynamicsand, thus, oscillations occur and settle not until time t ≈ 40s.

The result using the proposed limited IMC �lter F (dash-dotted lines) isexcellent. In this case, the IMC �lter accounts for the limitation of the inputs.The limited IMC �lter leads to an acceleration of the plant at the upperboundary u(t) = umax for a much longer time period, namely 1s ≤ t ≤ 2.3s.The deceleration occurs much later and also slightly longer than in theunlimited case from 2.3s ≤ t ≤ 3.2s. Hence, the proposed limitation ofthe IMC �lter F does not merely cut o� the input trajectory. Somethingmuch more sophisticated happens: It produces an alternative trajectoryyd (dash-dotted line) which can be achieved by the model (i. e., y = yd)with permissible inputs u ∈ U . Hence, the weakly damped dynamics arecompensated by the right inverse Σr since all of the calculated input ureaches the plant model Σ. Moreover, all calculations are performed on-line since only di�erential equations need to be integrated. Thus, arbitraryreference signals w(t) are handled directly without changing the design. �

This method for treating input constraints produces an appealing re-sult and it does not need to be designed by trial-and-error (i. e., there is noperformance criterion to be chosen or poles to be placed) � it incorporatesthe knowledge of input constraints automatically into the existing design.

93


Input constraint with an IMC for systems in I/O normal form.For systems in I/O normal form, the relationship between the highestderivative y(r)

d and the input u is given in Theorem 4.2 by Eq. (4.25c) and

can be solved for y(r)d (t)

y(r)d = b(x∗)u+ a(x∗).

Now, the desired limits on the highest derivative y(r)d (cf. Eq. (5.2b)) can

be obtained directly by

y(r)d,min(t) = a(x∗(t)) + min (b(x∗(t))umin, b(x∗(t))umax)

y(r)d,max(t) = a(x∗(t)) + max (b(x∗(t))umin, b(x∗(t))umax) .

(5.15)

The necessary transformed state vector x∗ in Eq. (5.15) is given partiallyby the IMC �lter F itself (see Eq. (4.25a)) and the other part is the so-lution of the internal dynamics (see Eq. (4.25b)). Thus, the time-varyingpermissible range on the highest derivative yd(t)(r) in Eq. (5.15) can becomputed.

An IMC for input-a�ne systems, as de�ned in Theorem 4.2 can beextended to respect input constraints Eq. (5.1).

Corollary 5.2 (IMC with input constraints for input-a�nesystems). The IMC controller Q is then de�ned by Eqns. (4.25a)-(4.25c) and a limitation on the �lter output yd


d (t) ≤ y(r)d,max(t),

obtained from Eq. (5.15).

Example 5.2 (IMC with input constraints using for input-a�nesystems):Consider, again, the plant (5.7) from the previous example. Di�erentiatingthe output y with respect to time gives

y = h(x) = x1 +1

5x2 = φ1(x) (5.16a)

˙y = Lfh(x) = x2 +1

5x3 = φ2(x) (5.16b)

ÿ = L2fh(x) =

3

5x3 −

1

5(x1 + x2 + u) (5.16c)

94


a relative degree r = 2. Choosing φ3(x) = x1 gives Rank (∂φ(x)/∂x) = 3and hence φ(x) = [φ1(x) φ2(x) φ2(x)]T is a feasible transformation x∗ =φ(x). The inverse transformation is

x = φ−1(x∗) =

x∗35(x∗1 − x∗3)

5x∗2 − 25(x∗1 − x∗3)

(5.17)

and gives the model Σ in the new coordinates x∗ by x∗ = Lfφ(x) with thesubstitution x 7→ φ−1(x∗)

x∗ =

x∗215(−80x∗1 + 15x∗2 + 79x∗3) + 1

5u

5(x∗1 − x∗3)

y = x∗1.

(5.18)

With x∗1 = yd and x∗2 = ˙yd one �nds from the above

u = 5ÿd − 10 ˙yd + 80yd − 79x∗3 (5.19)

with the internal dynamics

x∗3 = 5(yd − x∗3). (5.20)

Thus, the resulting nonlinear IMC controller Q gets the signals yd, ˙yd and ÿd

from the IMC �lter dynamics (5.12), uses Eq. (5.19) to calculate the controlinput u and, for this purpose, has to solve the di�erential equation (5.20) ofthe internal dynamics for the state x∗3. Again, it is proposed to implementthe �lter F as shown in Fig. 5.2.

At this point in the example, a nonlinear IMC has been developed, based onthe I/O normal form, and does not respect input constraints. Now, considerthe input constraints (5.1). From Eq. (5.19) one �nds

ÿd,min =1

5(+10 ˙yd − 80yd + 79x∗3) +

1

5umin (5.21)

ÿd,max =1

5(+10 ˙yd − 80yd + 79x∗3) +

1

5umax (5.22)

as the desired limitations on the highest derivative of the IMC �lter. Thesimulation results are identical to the ones shown in Example 5.1, since thesame model is controlled. �

95


5.2 IMC for Systems with Ill-De�ned RelativeDegree

The relative degree r is said to be ill-de�ned if it changes its value independence on the current state x(t), say at x(t). A right inverse Σr,designed for the relative degree r0 found at some x(t), becomes singularat the state x(t) (where the relative degree is > r0), in the sense that theoutput u of the inverse Σr is not de�ned (i. e., u /∈ L∞).

The typical approach of dealing with a �nite raise of the relative degreer at the state x(t) has been discussed by e. g., [48, 51]. Their approachincludes generating k further di�erentiations of the output y until the re-lationship between the highest derivative of the input u(k) and the highestderivative of the output y(r0+k) is not singular if solved for u(k). Then,u(k) is chosen as the new input ν = u(k) and its lower derivatives followfrom an integrator chain. This can be interpreted as an extension of thestate-space of the model. In light of Corollary 3.1, such a procedure re-sults in a smoother u as a larger constant relative degree r is enforcedglobally and, thus, a smoother I/O behaviour of the controlled systemfollows.

If this strategy is successful, then this e�ectively yields a non-singularinverse that is valid for the whole state trajectory x. This right inversemaps a given (smoother) trajectory y(r0+k)

d into the new input ν. Thisapproach can also be followed for the proposed design of an IMC. However,if k does not exist, then the approach fails to generate a feasible rightinverse. The following example illustrates this.

Example 5.3 (Model with a Singularity):Consider the system

x1 = x2

x2 = −x2 − 4x1 + (x2 − 2)u

y = x1

(5.23)

with x1(0) = x2(0) = 0, for which one �nds

˙y = x2

ÿ = x2 = −x2 − 4x1 + (x2 − 2)u.(5.24)

This system has a relative degree of r = 2 for x2(t) 6= 2. Since any arbi-trary number of further di�erentiations of the output y does not establish

96

5.2. IMC FOR SYSTEMS WITH ILL-DEFINED RELATIVE DEGREE

a relation to the input u for x2(t) = 2, the system has no relative degree(i. e., r →∞) for x2(t) = 2.

The right inverse Σr is established using the second derivative ÿ as

u =4y + ˙y + ÿ

−2 + ˙y(5.25)

and it becomes singular if the relative degree is lost:

limyd(t)→2

u(t) =∞. (5.26)

�

This section proposes a method which deals with a loss in relative de-gree and, thus, presents an IMC which deals with singular model inverses.

Main idea. As in the previous section, where input constraints werediscussed, it is proposed to alter the �lter F such that it generates atrajectory yd that can be reached by the model output y exactly (i. e., y =yd), with permissible inputs u ∈ U .

Remark 5.1. It is important to realise that a loss in relative degreer → ∞ is a system property and stems from the function of the plant.It results in an autonomous behaviour of the plant model since the inputu has no e�ect on the model output y. Hence, at the singularity x(t), it

is impossible to �nd a right inverse Σr that is de�ned for an arbitrarilydemanded trajectory yd.

While the model is in the singularity, the only possibility to deal withan ill-de�ned relative degree is to alter the IMC �lter F such that itproduces a demanded trajectory yd that coincides with the autonomousoutput y of the model. Then, the right inverse is still not de�ned, but theinput u can be selected arbitrarily (the right inverse becomes a relationwith an in�nite number of possible outputs u(t) ∈ R). Thus, the inputu(t) can be selected to be admissible u ∈ U .Therefore, the goal becomes �nding such an IMC �lter F .

The solution to singular inverses is to include input limitations:

Theorem 5.2 (Handling Singular Model Inverses with IMC).If input constraints using any boundaries umin ≤ umax are treated asproposed in Theorem 5.1, then the output u of the right inverse Σr canalways be selected to be in the permissible range u ∈ U and the modeloutput y is equal to the �lter output (i. e., yd = y).

97


Proof. The function ϕ(x, u) exists for all possible trajectories x (and u),independent of existence or the current value of r (i. e., if r raises its value,then ϕ becomes independent on u, but it is still de�ned for the relativedegree r0 that it was originally designed for.)

The method of respecting input constraints, described in Theorem 5.1,ensures (see (5.2a) and (5.2b)) that, at time t, the highest derivative y(r0)

d

is chosen such that

minu∈U

ϕ(x(t), u) ≤ y(r0)d ≤ max

u∈Uϕ(x(t), u) (5.27)

holds. This holds at every instance in time t ∈ T .The function ϕ(x, u) exists also if the relative degree r → ∞ van-

ishes. Then, ϕ(x, u) is only dependent on x (i. e., ϕ(x, u) = ϕ(x, ·)).Therefore, the limits minu∈U ϕ(x(t), u) and maxu∈U ϕ(x(t), u) producethe same value and the highest derivative of the �lter output y(r0)

d ischosen to be identical to the autonomous behaviour of the model out-put (i. e., y(r0)

d = y(r0) = ϕ(x, ·)). This demand can be met with anyu ∈ U .

Interpretation. Loosely speaking, the above Theorem states that, sincethe input u is limited, it cannot be chosen as in�nite by the right inverse.As a result, it follows that the right inverse produces a feasible outputu ∈ U . However, this method has a substantial demerit: It forces thesystem into an autonomous behaviour even if the relative degree changesto a larger but �nite value. In such a case, the system may not recoverfrom its self-induced autonomy despite the ability of the model and theplant to leave this state.

Literature. Several publications have addressed the issue of singulari-ties that originate from an ill-de�ned relative degree where the relativedegree vanishes completely (e. g., [13, 34, 49, 62, 68]). All publicationsaddress changing the inverse when the plant is close to the singularity.Thus, their approach is inherently di�erent from the one portrayed here:Here, it is not proposed to change the inverse. Rather, the trajectory is(automatically) chosen to avoid the singularities in the sense that a per-missible u ∈ U can be chosen at any instance in time. The inverse itselfis not changed. Hence, while the approach proposed in this thesis guaran-tees exact tracking of an altered trajectory2, the approaches in the cited

2 The altered trajectory is the only possible trajectory in the case of a vanishing r.Thus, the choice of the trajectory is not to be considered arbitrary.

98

5.3. NON-MINIMUM PHASE IMC DESIGN USING A PERFECT INVERSE

literature do not change the trajectory, but allow for imperfect trackingwhile travelling near a singularity.

The advantage of the approach proposed here is that once input con-straints are considered (which they virtually always have to be) singulari-ties are also treated in the sense that the input signal u is non-explosive(i. e., u ∈ L∞). Thus, singularities of model inverses do not necessitate aspecial treatment.

Example 5.4 (Example 5.3 continued):In order to design an IMC controller Q for the model (5.23) presented inthe previous example, input constraints need to be assumed

umin ≤ u(t) ≤ umax (5.28)

with umax = −umin = 1000. The value of 1000 is chosen arbitrarily. FromEq. (5.25) one determines the boundaries for the limited IMC �lter F as

ÿd,min = −4yd − ˙yd + min(( ˙yd − 2)umin, ( ˙yd − 2)umax

)ÿd,max = −4yd − ˙yd + max

(( ˙yd − 2)umin, ( ˙yd − 2)umax

).

(5.29)

Thus, an IMC controllerQ for the plant model (5.23) consists of the equation(5.25) of the right inverse and the IMC �lter dynamics w = ÿd/λ

2 +2 ˙yd/λ+yd (with zero initial conditions) that is restricted by ÿd,min ≤ ÿd ≤ ÿd,max

from Eq. (5.29).

Figure 5.6 shows the simulation results of this example for a reference stepw(t) = 0 for t < 1s and w(t) = 10 for t ≥ 1s. Without any singularityhandling the input u would become singular.

Figure 5.6 shows that the input u does not exceed the lower limit−1000. Thederivative of the output ˙yd does not move into the singularity at ˙yd = x2 = 2since this is prevented by the introduction of the input constraints. Notethat the introduced system is theoretically unable to go past the singularityif u is to be de�ned at all times. �

5.3 Non-Minimum Phase IMC Design using aPerfect Inverse

If the inverse of a model is unstable then the model is NMP (see De�nition3.6). It follows from Property 3.1 that the nonlinear IMC design fromAlgorithm 3.1 (using Eq. (3.28)) leads to an internally unstable closed-loop if the model Σ is NMP.

99


0

5

10y d

0

1

2

˙ y d

0 2 4 6 8 10 12 14 16

−1000

−500

0

u

time (s)

Fig. 5.6: Simulation results with singularity handling of Example5.4. The IMC �lter uses λ = 1.

This section proposes a novel method to design an IMC for an NMPmodel Σ. As a by-product, the proposed method allows to develop a feed-forward controller for nonlinear NMP systems which can be computedon-line. The ability to control a nonlinear NMP system is a major advan-tage of the IMC structure over feedback linearisation methods.

It is proposed to use an exact inverse Σr of the NMP plant modelΣ. In order to obtain a feasible controller Q, two subsequent steps aresuggested: The �rst step requires to introduce the NMP behaviour ofthe model Σ into the IMC �lter F . This leads to an internally unstablezero-pole cancellation, but stable I/O behaviour of Σr ◦ F . In the secondstep the unstable internal dynamics of the inverse Σr are removed andreplaced by the solution of the �lter, resulting in an internally stable IMCcontroller.

Before the method is introduced in detail, it is brie�y discussed whyinverses of NMP stable models become unstable.

5.3.1 Background

In order to explain the internal behaviour of an NMP model, the inputa�ne system (4.15) is used. The internal dynamics of (4.15) becomevisible when the system is transformed into the I/O normal form (4.19).Figure 5.7 is a representation of the I/O normal form. Note that in Fig. 5.7,

100


∫ ∫ ∫L

*rx

*1x

AS

u)()( ** xqxp + ∫BS [ ]Tnr

xx **1,,L+

u uba )()( ** xx +N

Fig. 5.7: Structure of an input-a�ne system Σ in I/O normal form.

the system has been structured into the subsystems, SA and SB. Thesubsystem SA contains the input relationship (4.19b) and the integratorchain of length r. Subsystem SB represents the internal dynamics (4.19c).

The systems SA and SB form a feedback connection. The connec-tion signals are the states x∗1, . . . , x

∗r from SA into SB and the states

x∗r+1, . . . , x∗n of the internal dynamics (4.19c) from SB into SA. Since

only stable systems are considered, this feedback connection is stable byde�nition.

In the case of an NMP system Σ, one or both of the subsystems SA

and SB are open-loop unstable. In this case, the subsystems stabilise eachother3. When constructing the right inverse Σr, the input u is chosen as inEq. (4.22), in order for the subsystem SA to behave like a pure integratorchain of length r. As an undesired result, it cancels the feedback of thestates of the internal dynamics SB on SA. Moreover, a new feedback loopis closed via the input calculation (4.22). This feedback, however, maynot stabilise the internal dynamics.

In summary, instability of an NMP right inverse Σ occurs for one oftwo reasons:

1. The subsystem SB is unstable. Due to cancellation of the stabil-ising feedback signal, SB remains unstable.

2. It is the introduction of the input u by (4.22) which, insertedinto SB, renders the internal dynamics unstable.

3 For example: Assume SA is stable. Then SA acts as a stabilising state feedbackcontroller for the unstable internal dynamics SB.

101


Literature. No literature was found on the topic of NMP nonlinear IMCdesign. However, since the IMC design is essentially based on the con-struction of a feedforward controller, some literature on nonlinear NMPfeedforward control design is presented.

In [59] it is proposed, as in the linear case (cf. Section 2.2.2), to splitthe model Σ into a minimum phase part ΣMP and a nonlinear all-pass partΣNMP = ΣA. Unfortunately, this problem has only been solved for input-a�ne models of up to second order. As in the linear case, the resultingright-inverse leads to an ISE-optimal controller. As it was established inExample 2.2 on page 34, that ISE-optimal controllers are of no concernfor an automotive control engineer, the approach in [59] is not followedhere. However, if a general solution to split a nonlinear model into theabove mentioned parts can be found, it would be of great value, also forIMC design.

The work of [65] discusses how the internal model principle, which alsoholds for nonlinear systems, can be exploited for control design. The mainproblem solved is to follow a reference trajectory generated by a knownexosystem or to attenuate disturbances generated by a known exosystem.As a result, also a non-minimum phase plant can be made to asymptoti-cally follow the trajectory of an exosystem at the cost of some (unknown)transient tracking error.

The contributions [12, 23, 24] establish a feedforward controller fora priori known reference trajectories with compact support. The con-trol scheme does not produce a transient error despite the non-minimumphase characteristic of the plant. This, however, comes at the cost ofa non-causal inverse as the nonlinear and unstable zero dynamics of theplant need to be integrated backwards in time to obtain a feasible initialcondition. Despite the shortcoming of a non-causal controller, the authorshave established that the design of a feedforward controller for an a pri-ori known trajectory can be interpreted as a two-point boundary valueproblem. The solution of this problem yields the necessary control input.This idea was picked up by [36] and generalised to trajectories withoutcompact support. Additionally, [36] was able to put constraints on themodel states and the control input during the transient behaviour of thetrajectory. However, the solution to the trajectory tracking problem needsto be found o�-line, in particular by numerical methods.

In conclusion, the solutions above are all interesting and feasible fortheir respective control problems. However, none solves the control prob-lem discussed in this thesis as they all violate at least one of the following

102


requirements (cf. Section 1.3.2):

• An exosystem is not given. Rather, the reference values w(t) consid-ered here are arbitrary w ∈ L∞ as they are in�uenced directly bythe driver.

• The reference trajectory w is not known before hand, which makesa non-causal inverse or an o�-line solution to trajectory trackingimpossible.

The problem at hand (stated in Chapter 3.5) appears to be much simplerthan the control problems as discussed by the aforementioned publications.However, no solutions on this control problem were found. In the view ofthe available literature one can interpret the problem that, here, it is theexosystem to be designed (namely the IMC �lter F ) such that a causalinverse Σr (the composition of the two is the IMC controller Q) tracks areference signal w without producing an exploding control input u. Thisapproach is persued in this chapter.

5.3.2 Introducing the Non-Minimum Phase IMC DesignUsing Linear Systems

In the following, a method to design an IMC for NMP models is presentedwhich retains the IMC design philosophy in such that the IMC �lter F isdesigned as the demanded I/O behaviour of the nominal closed-loop. Thismethod is �rst introduced by the equivalent procedure for linear systems.Then, the necessary amendments for nonlinear systems are given, followedby a summary in a theorem.

Main idea. It is proposed to design the linear IMC controller Q(s)for linear NMP models Σ as in the classical IMC design procedure (cf.Eq. (2.14a)), for linear minimum phase model, namely by

Q(s) = Σ−1(s)F (s).

Hence, the IMC controller Q(s) is obtained by a perfect inversion of anNMP model. Figure 5.8 shows this open loop structure. Note that Fig. 5.8displays the lower branch of the IMC structure given in Fig. 2.1 or Fig. 3.3.

If the model Σ(s) is NMP, it has at least one zero4 z in the right halfof the complex plane (Re (z) > 0). The feedforward behaviour Σ(s)Q(s)

4 Note that pure time delays are not discussed in this work.

103


w~F 1~−Σ Σ~d~y u

d~~ yy =

Fig. 5.8: Feedforward control structure.

is I/O stable if and only if the IMC �lter F (s) shares all right-half planezeros of the model Σ(s). The zeros of the �lter are used to cancel theunstable poles that result from the inversion Σ−1(s). Note that an exactcancellation is, in principle, possible, since Σ(s) is arti�cial and, thus,perfectly known.

Remark 5.2 (Internal stability). Internal stability requires that all trans-fer functions from every possible input to every possible output must bestable. The expression �possible input� means that an exogenous signalis allowed to alter any signal of the control structure.

Considering the signals in Fig. 5.8, only w can be a�ected by exogenoussignals because all of the operators are arti�cial. This means that allintermediary signals from F to Σ are not subject to alteration by exoge-nous signals, like disturbances. One could, therefore, argue that the feed-forward control structure with controller Q(s) and model Σ(s) is inter-

nally stable because the systems F (s), Σ−1(s)F (s), and Σ(s)Σ−1(s)F (s)are I/O stable.

Although this is true from a mathematical perspective, it does not holdtrue for the actual implementation. Since arti�cial systems are ulti-mately implemented on a digital processor, all signal values are restrictedto a certain precision. Hence, there are numerical errors on all signalswhich render the structure Fig. 5.8 internally unstable because the trans-fer function Σ−1(s) is unstable.

Interpretation. Obtaining the IMC controller Q from an inversion ofan NMP model results in an internally unstable feedforward path. How-ever, it can be used to review some useful facts:

• There exist some input trajectories5 yd = F (s)w to the inverseΣ−1(s) which, despite of the instability of Σ−1(s), do not result inan exploding output signal u.

5 These trajectories yd do not excite the unstable modes of the model Σ.

104


• These input signals yd can be obtained by including the right-halfplane zeros of the model Σ(s) into the IMC �lter F (s).

The main bene�t of this approach is that the IMC philosophy is re-tained for NMP systems. Thus, the IMC �lter F (s) is designed as thedemanded I/O behaviour from w(s) to y(s) of the closed loop and

y(s)w(s)

= F (s) (5.30)

holds in the nominal case for NMP systems. The de�ciency of this ap-proach is that F can no longer be selected as a linear low-pass �lterwithout any zeros, as introduced in Chapter 3.

Internal stability. One can obtain an internally stable feedforward con-trol structure by implementing the minimal realisation of Q(s). Thus, thezero-pole cancellation of Σ−1(s)F (s) is performed before the implementa-tion of Q(s). Comparing this procedure with the results of the linear IMCfor NMP systems in Section 2.2.2 (therein, especially Eq. (2.26), togetherwith Eq. (2.27)), one �nds that they are equivalent.

The following brief example illustrates the considerations above.

Example 5.5 (NMP IMC design for linear systems.):

Consider the model Σ(s) = s−1(s+2)(s+4)

with an NMP zero at z = 1. The

IMC �lter is chosen as F (s) = −(s−1)

(s/λ+1)2and also exhibits the zero at z = 1.

The model inverse is given by Σ−1(s) = (s+2)(s+4)s−1

and, consequentially, hasan unstable pole at p = 1. The IMC controller Q(s) results in

Q(s) = Σ−1(s)F (s)

=(s+ 2)(s+ 4)

s− 1· s− 1

(s/λ+ 1)2

=(s+ 2)(s+ 4)

1· 1

(s/λ+ 1)2

(5.31)

where the last line of the equation above presents the minimal realisation ofQ(s), which clearly is a stable transfer function due to the choice of λ > 0.�

105


5.3.3 Non-Minimum Phase IMC Design for NonlinearSystems

Main idea. It is proposed to extend the above design for NMP linearsystems to nonlinear input-a�ne6 systems. Thus, the idea is to constructan IMC controller Q for nonlinear NMP systems in the same manner asit is designed for MP systems in Eq. (3.28), namely by perfect inversionof the nonlinear NMP model

Q = ΣrF.

Loosely speaking, the idea is to

1. design a stable IMC �lter F , which exhibits the same NMP be-haviour as the model Σ, and

2. �nd an internally stable implementation of Q.

To begin with, the following de�nes the condition under which thetrajectories of the internal dynamics of two systems are identical.

De�nition 5.1 (Identical behaviour of the internal dynamics oftwo systems). Assume two input-a�ne systems A and B, both given inthe I/O normal form as in Eq. (4.19), but with all variables de�ned withthe subscript A or B, respectively. Then, the internal dynamics of A andB are identical if

pA(x∗A) + qA(x∗A)uA − pB(x∗B)− qB(x∗B)uB = 0 (5.32a)

holds with equal initial conditions[x∗A,r+1(0) · · · x∗A,n(0)

]T=[x∗B,r+1(0) · · · x∗B,n(0)

]T. (5.32b)

�De�nition 5.1 establishes equality of two sets of di�erential equations

by requesting identical right hand sides (see Eq. (5.32a)) and identicalinitial conditions (see Eq. (5.32b)). As a result, the trajectories of theinternal dynamics of the two systems A and B are identical. Thus,[

x∗A,r+1(t) · · · x∗A,n(t)]T

=[x∗B,r+1(t) · · · x∗B,n(t)

]T(5.33)

holds for each instance in time t ≥ 0.6 The proposed design is valid for all nonlinear systems that can be implemented in

an I/O normal form but it is introduced using this limited system class for reasons ofclarity.

106


Remark 5.3. De�nition 5.1 does not imply equality of the individualvector-valued functions p(·) or q(·) nor does it imply equality of theinput signals u(·). Hence, in general, the trajectories of the states of theinternal dynamics of A and B may be identical even if

pA(x∗A) 6= pB(x∗B)

qA(x∗A) 6= qB(x∗B)

uA 6= uB

holds. This is exempli�ed here: The two systems xA = uA and xB =−3xB + u2

B have identical solutions xA = xB with identical initial condi-tions xA(0) = xB(0) and the input signal uA = −3xB + u2

B.

Moreover, De�nition 5.1 does not make any implications about the statesx∗·,1, . . . , x

∗·,r which do not belong to the internal dynamics.

Using De�nition 5.1 as a starting point, the following lemma gives thecondition under which the internal dynamics of the IMC �lter F are equalto the internal dynamics of the inverse Σr assuming a series connection ofF and Σr as it appears in an IMC design (cf. Fig. 3.6). To this end, themodel Σ is regarded under the in�uence of input u from Eq. (4.22).

Lemma 5.1 (Identical internal dynamics of IMC �lter F and

model Σ under the in�uence of the right inverse Σr). Assume an

input-a�ne model Σ, given in I/O normal form as de�ned in Eq. (4.19),with input u, and output y = x∗1. Further, assume an input-a�ne systemF which is also given in I/O normal form, as de�ned in Eq. (4.19), butwith all variables de�ned with a subscript F. Hence, the states of F arex∗F,1, . . . , x

∗F,n with input uF = w, the output is yd = yF = x∗F,1 and

y(r)d = xF,r∗ holds. The system F is chosen to have the same number of

states n and the same relative degree r as the system Σ. Finally, let theinput u to the model Σ be de�ned by Eq. (4.22), or, in a more generalsetting, by

u =1

b(x∗)

(−a(x∗) + y

(r)d

), γ

(x∗, y(r)

d

). (5.34)

Note that the input transformation of Eq. (5.34) is part of the right inverseΣr of the model Σ.

With the assumptions above, the trajectory of the states of the internaldynamics of the model Σ under the in�uence of the right inverse Σr are

107


identical to the internal dynamics of the IMC �lter F if and only if

p(x∗) + q(x∗) · γ(x∗, y(r)

d

)− pF(x∗F)− qF(x∗F)w = δ (x∗,x∗F) (5.35a)

holds, where the function δ (x∗,x∗F) has the property

δ (x∗(t),x∗F(t)) = 0 for x∗(t) = x∗F(t). (5.35b)

Proof. Lemma 5.1 meets De�nition 5.1 in such that Eq. (5.35a) corre-sponds to Eq. (5.32a) and, since the initial condition of IMC �lter F andmodel Σ are demanded equal (x∗(0) = x∗F(0)) by the IMC design, thefunction δ (x∗,x∗F) will be zero at t = 0 (which meets Eq. (5.32b)). Dueto Eq. (5.35a), it will remain zero for each instance in time t > 0.

Remark 5.4. Note that, if two systems A and B have identical zero dy-namics, it does not necessarily follow that the behaviour of their internaldynamics are identical in the sense of Lemma 5.1 or De�nition 5.1.

This holds, since the notion of zero dynamics assumes particular initialconditions and a particular output trajectory, which cannot be assumedto be given in the general case.

The following corollary establishes an important intermediary resultand is the conclusion of Lemma 5.1 and the IMC design procedure (cf.Section 3.5). It is the basis for an internally stable implementation of theproposed IMC controller Q = ΣrF .

Corollary 5.3 (Identical state trajectories in feedforward control).Assume the model Σ and the �lter F to be given as in Lemma 5.1. Ifthe conditions of Lemma 5.1 hold true (i. e., the state trajectories of the

internal dynamics of IMC �lter F and right inverse Σr are identical) then

the trajectories of all states of F and Σr are equal. Thus, the equality[x∗1(t), · · · , x∗n(t)

]T =[x∗F,1(t), · · · , x∗F,n(t)

]T(5.36)

applies at each instance in time t ≥ 0.

Since, for IMC design, the initial states of the Filter F and the rightinverse Σr are chosen identical to the model Σ, if follows that

x∗F(0) = x∗(0) (5.37)

108


holds. With the IMC �lter F implemented in an I/O normal form andwith the purpose of the right inverse Σ to obtain the equality yd

...

y(r−1)d

=

x∗F,1...

x∗F,r

=

x∗1...x∗r

(5.38)

(which is achieved using Eq. (4.22)), one �nds that the �rst r states areequal due to the design of the right inverse and the last n − r statesare equal due to the request for identical internal dynamical behaviour(Lemma 5.1 holds true by de�nition in Corollary 5.3).

The result of Corollary 5.3 can now be used to de�ne the requirementson the IMC �lter F that is a feasible choice for NMP models Σ in suchthat it produces a stable I/O behaviour of Q = ΣrF . Moreover, Corollary5.3 can also be used to determine an internally stable implementationof the IMC controller Q. The following theorem summarises the preced-ing results and presents the main contribution of this section; that is, itstates the requirements on F and presents the minimal realisation of thenonlinear IMC controller Q.

109


Theorem 5.3 (IMC design for non-minimum phase nonlinear

systems using an exact model inverse). Assume the model Σand the �lter F to be given as in Lemma 5.1. Additionally, inputconstraints u ∈ U of the model Σ, as given in Eq. (5.1), exist.Assume the IMC �lter F to be designed to have the following proper-ties:

• F is stable (in the sense that all trajectories of the states x∗F ofF are non-explosive).

• F has identical trajectories of the internal dynamics as the modelΣ in the sense of Lemma 5.1 (i. e., under a feedforward connec-tion using the input u from Eq. (5.34)).

• The steady-state gain of F is one (i. e., limt→∞ Fwss =limt→∞ yd(t) = wss for t→∞).

Then, an internally stable IMC controller Q for the model Σ is ob-tained by the IMC �lter F (implemented in the I/O normal form) andthe (algebraic) input calculation

u = γ(x∗F, y

(r)d

), (5.39)

with y(r)d obtained from (cf. Eq. (4.19))

y(r)d = xF,r = aF (x∗F) + bF (x∗F) w (5.40)

but limited by the saturation proposed in Theorem 5.1 with x = x∗F.With this IMC controller Q it follows that the output yd of the IMC�lter F is identical to the output y = x∗1 of the model Σ (yd = y).

Proof. First, stability of the composition Q = ΣrF (i. e., stability of theI/O behaviour of the IMC controller Q) is shown:

By de�nition of F , the behaviour of its internal dynamics is identicalto the behaviour of the internal dynamics of the model Σ. Since a feed-forward control structure depending on the model inverse is discussed,Corollary 5.3 is applicable, and it follows that the state trajectories x∗F ofthe IMC �lter F are identical to the state trajectories x∗ of the model Σ(i. e., x∗F = x∗). Due to the stability de�nition of F , it follows that x∗F,and therewith x∗, is non-explosive. Consequentially, y = yd holds and is

110


also non-explosive since h is analytic.Second, stability of the implementation of the individual operators

Σr and F is shown (i. e., internal stability): Equation (5.39) di�ers fromEq. (5.34) in such that the states of the model inverse x have been replacedby the states x∗F of the IMC �lter F . This is a feasible step, since the tra-jectories of those states are equivalent (i. e., x∗F = x∗). Since the trajectoryxF is non-explosive, and since singularities are avoided by respecting theinput constraints, the input u obtained by the algebraic equation (5.39)is �nite (i. e., u ∈ L∞).

Theorem 5.3 implies an implementation of the IMC �lter F whichdi�ers from the one proposed for minimum phase models (cf. Section 3.5and Fig. 3.6). Here, the IMC �lter F delivers all signals to the right inverseΣ, de�ned by Eq. (5.39). Figure 5.9 displays these needed connectionsignals.

Fw~

*1,Fd

~ xy =

( ))(d

*F

r ~,~ ryxγ=Σ u

*,F

)(d~

rr xy &=

M

*,F nx

*,F

)1(d~

rr xy =−

M

*1,F +rx

Fig. 5.9: Implementation signals of IMC �lter F and right inverseΣr in case of a non-minimum phase model.

Interpretation. Theorem 5.3 summarises the results of this section insuch that it states the demands on the IMC �lter F as well as the internallystable implementation of Q for IMC control of NMP systems.

The application of Theorem 5.3 is given in an algorithm below forreason of comparison to the IMC design for minimum phase systems asdescribed in Algorithm 3.1:

Algorithm 5.1. NMP IMC design using a perfect inverse.

Given: A stable and invertible NMP plant model Σ.

111


Step 1: Compute the model inverse Σr (speci�cally, �nd the necessary re-lationship corresponding to Eq. (5.34)).

Step 2: Find an IMC �lter F : W → Y which

• is stable,

• has a �good� step response from w to y with a steady-state gainof one (i. e., w(t) = yd(t) for t→∞), and

• has identical internal dynamics as the model Σ in the sense ofLemma 5.1.

Step 3: The IMC controller Q can now be implemented by F and Eq. (5.34)).The IMC control loop is given in Fig. 3.4 or, alternatively, by usingAlgorithm 3.2, also by substituting the internal model as in Fig. 3.8.

Result: Nonlinear output feedback IMC control loop.

The main di�erence between the generalised IMC design procedure ofAlgorithm 3.1 and the IMC design for NMP models in Algorithm 5.1 isthe additional requirement on the IMC �lter F to exhibit the identicalinternal dynamics as the model Σ. The following example clari�es thisconcept and uses the results obtained above.

Example 5.6 (IMC of a nonlinear non-minimum phase plant):

Consider the plant model Σ with two states (n = 2) and a relative degreeof r = 1

Σ : x∗ =

x∗2+z(−(x∗1)2u+(x∗1)3+x∗2(z+1))z2

x∗1u−x∗2+z((x∗1)3+x∗2)

z

, x∗(0) = x∗0

y = x∗1

(5.41)

in I/O normal form, with some parameter z ∈ R and z 6= 0. Hence, x∗2 is

the (single) state of the internal dynamics. Computing the right inverse Σr

of the above model means solving the right-hand side of x∗1 with x∗1 7→ ˙yd

for u, which yields

u =−z2 ˙yd + z(x∗1)3 + x∗2(1 + z) + z2x∗2

z(x∗1)2, γ

(x∗, y

(r)d

). (5.42)

112


Inserting Eq. (5.42) into Eq. (5.41) one �nds the composition of right inverse

Σr with the plant model Σ

Σ ◦ Σr : x∗ =

[˙yd

zx∗2 − z ˙yd

], x∗(0) = x∗0

y = x∗1.

(5.43)

The composition has, by design, linear dynamics of its �rst state x∗1 and, bychance, also a linear behaviour of the internal dynamics x∗2. One sees thatthe pole of the internal dynamics is at z. Thus, for z < 0 the model Σ isminimum phase and for z > 0 it is NMP. Since it is desired to treat an NMPsystem, the parameter z is chosen as z > 0.

Consider an IMC �lter F with input w

F : x∗F =

[x∗F,2(z2+2zλ+λ2)

z2+

λ2(−w+x∗F,1)z

−x∗F,2λ(2z+λ)

z+(w − x∗F,1

)λ2

], x∗F(0) = x∗0

yd = x∗F,1

˙yd = x∗F,1.

(5.44)

This IMC �lter F is stable and has a steady-state gain of one.

Using Lemma 5.1, it is shown that the behaviour of the internal dynamicsof the IMC �lter F is identical to Σr. This is done by subtracting the right-hand sides of x∗F,2 (cf. Eq. (5.44)) and x∗2 under inversion (cf. Eq. (5.43))

and the de�nition of ˙yd given in Eq. (5.44). One �nds that

=x∗2︷︸︸︷z(x∗2 − x∗F,2

)− 2x∗F,2λ+

1

z

(zw − zx∗F,1 − x∗F,2

)λ2−

−2x∗F,2λ+1

z

(zw − zx∗F,1 − x∗F,2

)λ2︸︷︷︸

=x∗F,2

= z(x∗2 − x∗F,2

)︸︷︷︸,δ(x∗,x∗F)

(5.45)

holds. It follows from Lemma 5.1 that the internal dynamics of the �lterF and the right inverse Σr are identical and, consequentially, Corollary 5.3states that all states of F and Σr are identical.

Finally, according to Theorem 5.3, the desired IMC controller Q is de�nedby the �lter F given by Eq. (5.44) and the input calculation from Eq. (5.42)but using the states of the �lter F , namely

u = γ(x∗, ˙yd

)with x∗ 7→ x∗F, ˙yd 7→ x∗F,1

=z · (x∗F,1)3 − z2wλ2 + z2x∗F,1λ

2 + x∗F,2(1 + z + z2 + 2z2λ+ zλ2

)z · (x∗F,1)2

(5.46)

113


Thus, the IMC controller Q can be interpreted to be the �lter F with thenew output u given by Eq. (5.46).

0

0.5

1

Out

put

y

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Inpu

tu

time (s)

Fig. 5.10: Closed-loop simulation with a reference step occurringat time t = 1s. Nominal values are λ = 1 and z =2. The nominal response is plotted in the bold dashedline. Responses with only the plant parameter z adjustedbetween 1.3 and 4 are plotted in thin solid lines.

Figure 5.10 shows the simulation results from a reference step occurring att = 1s. The �lter pole is chosen as λ = 1 and in the nominal case z is atz = 2.

The following can be seen from Fig. 5.10: The bold dashed line indicates thenominal plant output y and nominal plant input (IMC controller output) u.In the nominal case, the output yd of the IMC �lter F and the output y ofthe plant are equal (yd = y). The �lter output shows an NMP characteristic(inverse response) due to the zero at z. As expected, the input trajectory uis non-explosive despite the perfect inversion of the NMP nonlinear modelΣ.

In order to explicitly show that this approach does not rely on a perfectinversion (no pole-zero cancellation) of the plant Σ, the plant's NMP be-

haviour is altered, maintaining the model Σ and, therewith, nominal IMCcontroller Q. This is done by changing z in the plant equations between1.4 and 5 in increments of 0.8. Thus, there is a signi�cant modelling errorconcerning the NMP characteristic. Despite this, the closed-loop behaviourof the IMC is good and, as expected, this is achieved with a non-explosive

114


input signal u in all cases. �

5.3.4 Construction of the Non-Minimum Phase IMC Filter

Despite the successful application of Theorem 5.3 in the example above,there is still one signi�cant open problem: How this IMC �lter F is ob-tained. This section addresses this problem by showing that F can beobtained by the solution of a control problem.

The following constructs the structure of F and de�nes the controlproblem. This is done in three steps:

Algorithm 5.2 (Construction of an NMP �lter F ).

Given: An NMP nonlinear model Σ.

Step 1 (Structure of F ): Assume the model Σ to be given in I/O normalform (4.19), shown in Fig. 5.7.

Construct the IMC �lter F with the same number of states n, thesame relative degree r in the same I/O normal form (4.19) but withall variables de�ned with a subscript F, the output y = x∗F,1 and theinput w.

Step 2 (Internal Dynamics of F ): Insert the inverting input (5.34) into

the internal dynamics of the plant model Σ. This results in

[x∗r+1, · · · , x∗n]T = p(x∗) + q(x∗) · γ(x∗, y(r)

d

). (5.47)

Since, according to Corollary 5.3, the states x∗F of the �lter F and

the states of the inverted model Σr are going to be identical, one�nds the states of the internal dynamics of the IMC �lter F fromEq. (5.47) with the substitution

x∗ 7→ x∗F, the input (5.34) and

y(r)d = x∗F,r = aF(x∗F) + bF(x∗F)w

asx∗F,r+1...

x∗F,n

= p(x∗F)+q(x∗F) · −a(x∗F) + aF(x∗F) + bF(x∗F)wb(x∗F)

. (5.48a)

115


Step 3 (F as the solution of a control problem): The remaining states ofF are (cf. Eq. (4.19))

x∗F,i = x∗F,i+1 for i = 1, . . . , r − 1 (5.48b)

x∗F,r = aF(x∗F) + bF(x∗F)w. (5.48c)

The solution of the following control problem gives the desired IMC�lter F with NMP behaviour:

Control Problem (Construction of an NMP IMC �lter F ). Thesystem to be controlled (i. e., the plant in this context) consists ofthe integrator chain (5.48b) and the internal dynamics (5.48a). Thestate feedback controller to this plant is given by (5.48c).

The plant is perfectly known and no disturbances need to be con-sidered since the complete IMC �lter is an arti�cial operator. Theclosed-loop behaviour of the plant (5.48a)-(5.48b) and the controller(5.48c) is the IMC �lter F .

The control problem requires to �nd some functions aF(x∗F) andbF(x∗F) such that

• the closed-loop system is stable, and

• that the steady-state gain of F is one (i. e., w(t) = yd(t) fort→∞).

Result: The result is an IMC �lter F that meets all requirements of The-orem 5.3.

In this thesis, the control problem given above is not solved for the gen-eral case. This is left as an open problem to be solved for each applicationindividually.

5.4 Treating Measured Disturbances

In some cases, a disturbance is measured and, thus, is known. Suchdisturbances are denoted by dm. The e�ect of a measured disturbancecan be attenuated by the IMC controller Q, directly (i. e., the attenuationis not based on the feedback signal but can be initiated in the feedforwardpath), since the disturbance value is available. In some cases, its e�ect canbe removed completely in the feedforward path. The following is basedon [42].

116

5.4. TREATING MEASURED DISTURBANCES

Main idea. It is proposed to model the e�ect of the measured distur-bance dm:

Σ : x(t) =f (x(t), dm(t), u(t)) , x(0) = x0, x ∈ X , (5.49a)

y(t) =h (x(t), dm(t), u(t)) , u ∈ U , y ∈ Y. (5.49b)

The operator of the plant model Σ can now be written as

Σ : U × Dm → Y, (5.50)

and maps the input trajectory u ∈ U and the trajectory of the measureddisturbance dm ∈ Dm into the output trajectory y ∈ Y. The right inverseΣr of this operator is de�ned along the lines of De�nition 3.4 and withthe purpose of the control design in mind as

Σr : Y × Dm → U (5.51a)

and ful�ls

Σ ◦(

Σr ◦ (yd, dm) , dm

)= yd. (5.51b)

Thus, the plant model Σ and its right inverse Σr are de�ned equivalentlyto De�nition 3.4 on page 50 with the di�erence being that the informationof the disturbance dm enters both. According to the De�nition of the rightinverse above, the e�ect of the disturbance will be cancelled completely.Figure 5.11 shows such a feedforward control structure.

uΣ~

yy ~~d

=dy~

rΣ~

md

Fig. 5.11: Right inverse with measured disturbances.

Design. If one extends the notation in the previous section from f(x, u)into f(x, dm, u) then the resulting right inverse will automatically be ob-tained with the methods reviewed in Chapter 4. Hence, the e�ect ofmeasured disturbance will automatically be cancelled by the right in-verses of �at and input-a�ne systems if the dependency of f(x, dm, u)

117


and h(x, dm, u) on the measured disturbance dm is considered appropri-ately. For this reason, the Lie derivative needs to be altered to

Lkfh(x, dm, u) =∂

∂x

{Lk−1f h(x, dm, u)

}f(x, dm, u)

L0fh(x, dm, u) = h(x, dm, u).

(5.52)

The resulting IMC structure, which respects a measured disturbance, isshown in Fig. 5.12.

w

Q

y~

w~ Fu Σ

md

ydy~

rΣ~

d

−

−

Fig. 5.12: IMC structure with a measured disturbance dm and un-measured disturbances d.

However, depending on where the disturbance enters the model, itcannot always be cancelled if it is only known up to the current time t.To clarify this statement, the following introduces a relative degree ρ ofthe measured disturbance dm:

ρ = mink

{∂

∂dmLkfh(x, dm, u) 6= 0

}(5.53)

With the relative degree r of the input u one �nd the following two cases(cf. [42]):

r ≤ ρ: If the relative degree r of the input u is smaller or equal to therelative degree ρ of the measured disturbance dm then the rightinverse Σr will depend on the current measured disturbance dm(t).Additionally, Σr will not depend on di�erentiations of dm(t), likedm(t).

In other words: The input u a�ects the output y more directly thanthe measured disturbance dm.

r > ρ: If the relative degree r of the input u is greater than the relativedegree ρ of the measured disturbance dm then the right inverse Σr

will depend on r − ρ di�erentiations of the measured disturbance.Hence it will need the signal d(r−ρ)

m (t) and its lower derivatives.

118

5.5. IMC OF SIMPLE QUADRATIC NONLINEAR MIMO SYSTEMS

In other words: The measured disturbance dm a�ects the output ymore directly than the input u. Hence, the disturbance cannot becancelled unless its derivatives are known.

In most control problems, derivatives of a measured disturbance are notavailable. Thus, in the case r > ρ, the measured disturbance cannot becancelled perfectly since the necessary signals (namely d(r−ρ)

m (t) and itslower derivatives) are not available. In such a case, however, it is proposedto generate an approximation of d(r−ρ)

m (t) and its lower derivatives byfeeding dm into an SVF of relative order r−ρ. An SVF is especially usefulsince the designer can choose the bandwidth of the SVF as a compromisebetween noise ampli�cation and phase lag.

Fortunately, the control problems discussed in Chapters 6 and 7 bothbelong to the case r > ρ, where no di�erentiations of the measured distur-bances are necessary.

5.5 IMC of Simple Quadratic Nonlinear MIMOSystems

It is not desired to treat MIMO systems extensively. The following merelygives a brief discussion on some structural changes in the IMC design forsimple quadratic systems with p inputs and outputs, without claimingto be an exhaustive examination. For an in-depth treatment on MIMOinverses, the reader is referred to [20, 48, 51, 80, 81] and the referencestherein.

First, some nomenclature for MIMO systems is given. Consider theMIMO model (3.2) with p inputs u = [u1, . . . , up]T and p outputs y =[y1, . . . , yp]T . The relative degree r is a p-vector. Loosely speaking, the

value rj says that the rj-th derivative y(rj)k of the k-th output yk depends

on some input uj (with j ∈ {1, 2, . . . ,m}) which no other output derivativeuses to de�ne its relative degree. Such a relationship is to be established(if possible) for each entry in the u vector for a di�erent output.

Pre-integration. One �nds that, in general, the right inverse Σr of aMIMO nonlinear model depends not only on the demanded trajectory yd

and some of its derivatives, but also on the input vector u and some of itsderivatives u, u, . . .. This is established in e. g., [80] for general nonlinearsystems, can be seen from the I/O normal form of input-a�ne systems(see e. g., [20, 48, 51]), and the de�nition of di�erential �atness of MIMO

119


systems (see e. g., [81]). A typical solution to generate derivatives forsome inputs is pre-integration [80]. Suppose l derivatives of uj are needed.Then choose uj(l) as a new input νj and extend the state-space

xn+1 = xn+2 = uj

...

xn+l−1 = xn+l = uj(l−1)

xn+l = uj(l) = ν.

The relative degree for input uj to output yk changes from rj to rj + l,with respect to the new input νj . In the following, it is assumed thatsuch a pre-integration has been performed and the new input and thenew relative degree are, again, denoted by u or r, respectively. Thus,input derivatives are not considered any further.

IMC �lter. An inversion of a MIMO system Σ results in an I/O decou-pling. This is obvious from the composition Σ ◦ Σr = I, which is impliedby the de�nition (De�nition 3.4) of the right inverse Σr. Therefore, theMIMO IMC �lter F needs to be a diagonal p× p operator

F =

F1 0 0

0. . . 0

0 0 Fp

, (5.54)

which maps the vector signal w into the signal yd. Each entry F1, · · · , Fpis designed and implemented as proposed in Section 3.5.1 for the SISOcase.

In conclusion, the MIMO case does not change the philosophy of howan IMC controller is designed. The only changes lie in the inversion whichmight have to be extended by the concept of pre-integration and that Fis a MIMO diagonal operator.

Input constraints. Input constraints in the SISO case are considered(cf. Section 5.1) by establishing a relationship between the input u andthe highest derivative y(r)

d of the �lter output yd. In the MIMO case, thisrelationship contains more entries of the input vector u. Assuming thatthe necessary pre-integration has been performed, then this relationship

120

5.6. SUMMARY

can, in general, be given as

y(rj)d,k = ϕj(x, u1, . . . , uk, . . . , um) for k = 1, . . . , p. (5.55)

Thus, if not all desired highest derivatives y(rj)d,k of the output vector yd

of the �lter F can be reached with permissible inputs umin ≤ u ≤ umax

then, in general, there is no unique solution as to which derivative (or

combination of) y(rj)d,k to limit. Thus, in this case, the engineer is left

with an in�nite number of possibilities as to which derivatives to limit.Currently, there is no generally applicable solution to this problem.

Let ϕ = [ϕ1, . . . , ϕp]T . Note that a unique solution exists if the condi-tion

Rank∂ϕ(x,u)

∂u= p (5.56)

holds. This is the case if, for example, the ϕj(·) can be sorted such that alower or upper triangular form of the inputs u1, . . . , um can be achieved.For example, the following relationships

y(r1)d,1 = ϕ1(x, u1)

y(r2)d,2 = ϕ2(x, u1, u2)

y(r3)d,3 = ϕ3(x, u1, u2, u3)

present a lower triangular form in the inputs u1, u2, u3 and, thus, a limi-tation in u can be mapped to a unique solution for a limitation in yd.

5.6 Summary

The basic design procedure of IMC, as introduced in Section 3.5, hasa rather limited system class, namely minimum phase models with well-de�ned relative degree and arbitrary but �nite input signals. This chapterextended the system class to

• non-minimum phase models,

• models with an ill-de�ned relative degree,

• models with input constraints, and

• models with measured disturbances.

121


Most extensions exploit Proposition 3.2 . Thus, the IMC design for thisextended system class �ts perfectly into a larger picture and can, therefore,be regarded as a generalisation of the basic IMC design procedure asintroduced in Section 3.5.

122

Part II

INTERNAL MODEL CONTROL OF

TURBOCHARGED ENGINES

This part presents the practical contribution of this thesis.It applies the proposed nonlinear internal model control totwo automotive control problems. Those are the boost pres-sure control of a one-stage turbocharged diesel engine andthe control of three pressures of a two-staged turbochargeddiesel engine. The proposed controller for a two-stage tur-bocharged diesel engine is the �rst solution of its kind.

6. CONTROL OF A ONE-STAGE

TURBOCHARGED DIESEL ENGINE

This chapter develops a SISO IMC boost pressure controller for a turbo-charged diesel engine with a variable-nozzle turbine (VNT) (see Fig. 6.1).For this purpose, the basic IMC design procedure (cf. Section 3.5) is em-ployed where the model inverse is computed using the �at model Σ (cf.Section 4.1). The design is extended to handle limited constraints (cf.Section 5.1) and treats measured disturbances (cf. Section 5.4).

6.1 Function of a One-Stage Turbocharged DieselEngine

The function of an internal combustion engine is described in detail ine. g., [47]. The article [25] focuses on the speci�cs of diesel engines. Thevarious methods of forced air induction are described in [105] and [27].The work of [26] focuses on turbocharging itself.

Shaft

Engine

Chargecooler

Exhaustaftertreatment

Compressor

Turbine

1V

2V

3V

4V 5V

En

fm&

vntu

Fig. 6.1: Air-system of a one-stage turbocharged engine.

6. CONTROL OF A ONE-STAGE TURBOCHARGED DIESEL ENGINE

Figure 6.1 shows the air-system of a common on-stage turbochargedengine, using a VNT [26]. The engine speed is denoted by nE and theinjected fuel mass �ow is denoted by mf .

In the following, the air path through the system is described, startingwith air from the environment V1. In the one-stage turbocharged enginewith a VNT, fresh air is aspirated from the environment V1 and is com-pressed by the compressor. Before the compressed air enters the intakemanifold V2, it is cooled down by the intercooler that simultaneously in-creases the air density. The pressure in the intake manifold V2 is calledthe �boost pressure� p2. The boost pressure p2 forces the air into theengine where it is mixed with fuel and, in turn, leads to combustion. Thehot exhaust gas is pushed into the exhaust manifold V3. Its pressure iscalled the �exhaust back pressure� p3. The exhaust gas is led over theturbine, whose nozzle geometry uvnt a�ects the amount of exhaust energythat is converted to drive the compressor over the shaft. Finally, the ex-haust gas �ows through the exhaust pipe V4, through the aftertreatment(e. g., catalytic converters), and back into the environment V5. The en-gine speed nE and the injected fuel-mass �ow mf deliver energy to theair-system. They can be considered to be exogenous variables, stimuli,or disturbances, depending on the problem to be solved (e. g., parameteridenti�cation or control design).

This method of forced induction is the most widely used for dieselengines. Due to their higher combustion temperatures, spark ignitionengines pose higher demands on materials used for the variable nozzles.Currently, only the Porsche 911 Turbo (type 997) [7] uses this technology.Other turbocharged spark ignition engines use a wastegate turbine insteadof the variable nozzle turbine.

6.2 Control Problem

The aim of turbocharging is to induce a desired amount of oxygen into thecombustion chamber. Since this value is correlated to the measured boostpressure p2, the boost pressure becomes the main control variable. Theinput is the nozzle geometry (u = uvnt). Engine speed nE and injected fuelmass �ow mf are assumed to be exogenous variables and are treated asmeasured disturbances (dm = [nE, mf ]T ). The behaviour of the one-stageair-system can be described using a model Σ which is given by Eq. (6.1).The input and output signals, including the measured disturbances dm

are shown in Fig. 6.2.

126

6.2. CONTROL PROBLEM

Σvntu 2p

md

Fig. 6.2: Input and output signals of a one-stage turbocharged en-gine.

Control Problem (one-stage). The measured variable is the boost pressurey = p2 and the control input is the VNT position u = uvnt. The controllershould ensure that

• the boost pressure p2 tracks a constant reference signal w with zerosteady-state o�set,

• the input constraint 0 = umin ≤ uvnt ≤ umax = 1 of the VNTposition is not violated,

• the e�ect of the measured disturbances dm is compensated, and

• robust stability is guaranteed.

Unfortunately, the precise requirements for the behaviour of the con-trolled system cannot be reviewed here in detail for reasons of con�den-tiality. In general, performance requirements concern mainly a desiredspeed, a demand on overshoot of boost pressure where less overshoot isbetter, and a demand in undershoot of boost pressure where less under-shoot is better. Typically, the permissible overshoot in the boost-pressureis requested to be at most 0.15 bar and the undershoot is requested to beless than 0.03 bar. Note, however, that even the series production con-troller violates these demands frequently, as will be shown later. Hence,those requirements should be interpreted more as a guideline than as asstringent demands.

Model. The derivation of the model equations is given in Appendix A.For the sake of better readability, the equations of the resulting control-oriented model are repeated here.

The input u is a function of the nozzle position of the variable nozzleturbine (VNT). The engine speed nE and the injected fuel mass mf areconsidered to be measured disturbances dm = [nE, mf ]T . The model Σ is

127


given by

Σ : x1 =k1k2

x1(k3(dm) + k6(dm)k4x2)ϕ1(x2)− k1k5ϕ3(x2)

x1ϕ2(x1, x2)

− k1k2

x1(k3(dm) + k6(dm)k4x2)ϕ1(x2)u

x2 =ϕ4(x2)k7

ϕ2(x1, x2)− k9

k7k6x2

(6.1a)

with the initial condition x(0) = x0 and the output equation

y =x2 = p2 (6.1b)

with

ϕ1(x2) =k9

k4+

k17(dm)k14(dm) + k4x2

, T3

ϕ2(x1, x2) = k8k10x

21 − ϕ3(x2)k11x1

, mC,out

ϕ3(x2) = k16

((x2

pamb

)k12− 1

),HC,out + HC,in

mC,out

ϕ4(x2) = k15ϕ3(x2) + k13 , T2,

where the coe�cients ki are system parameters (e. g., diameters, inertiaof the turbocharger, etc.), pamb is the ambient pressure and the ϕj(·) aresome nonlinear functions of the states. The model is stable and minimumphase and, thus, can be used for IMC design.

6.3 Nonlinear IMC of the Air-System

6.3.1 Model Inverse

For the design model (6.1), it is shown that the boost pressure p2 is a �atoutput as well as the system output y

z = y = p2. (6.2)

In order to show the �atness of the system, the state variables x = [ω, p2]T

as well as the input u have to be expressed as a function of z and a �nite

128

6.3. NONLINEAR IMC OF THE AIR-SYSTEM

number of its time derivatives (cf. Eqns. (4.1)-(4.4)). Equation (6.2)relates the �at output z to the �rst state, namely the boost pressure p2.By di�erentiating Eq. (6.2) along the model dynamics (6.1a), one gets

z =ϕ4(x2)k7

ϕ2(x1, x2)− k9

k7k6(dm)x2

=ϕ4(z)k8

(k10x

21 − ϕ3(z)

)k7k11x1

− k9

k7k6(dm)z.

(6.3)

The quadratic equation Eq. (6.3) has the following positive unique solutionfor the second state, the turbocharger speed ω, given that z > pamb, i. e.p2 > pamb holds:

x1 =k7k11z + k9

k7k6(dm)z

2k8k10ϕ4(z)

+

√(k7k11z + k9

k7k6(dm)z)2 + 4k2

8k10ϕ3(z)ϕ4(z)

2k8k10ϕ4(z)

(6.4)

Equation (6.4) expresses the turbocharger speed ω = x1 to the �at outputz and its �rst derivative z. Using the second derivative of Eq. (6.2) andtaking Eq. (6.4) into account, one gets

z = ϕ3(z, z,dm) + ϕ5(z, z,dm)u, (6.5)

which leads to:

u =z − ϕ3(z, z,dm)ϕ5(z, z,dm)︸︷︷︸ψu(z,z,dm)

; ϕ5(z, z,dm) 6= 0. (6.6)

A solution exists for all z, z, if z ≥ pamb. This is always the case, since thelowest attainable boost pressure p2 = y = z is the ambient pressure pamb.The right inverse Σr of the model of a one-stage turbocharged air-systemcan be obtained from Eq. (6.6) by letting z = yd

Σr : u =ÿd − ϕ3(yd, ˙yd,dm)ϕ5(yd, ˙yd,dm)

(6.7)

Since the actual output y is equal to the �at output z (cf. Eq. (6.2))no di�erential equation (i. e., Fy→z = I) needs to be solved to calculateone from the other.

129


6.3.2 IMC Filter

The IMC �lter needs to create the trajectory for the desired output yd(t)and the �rst two derivatives ˙yd(t) and ÿd(t). The transfer function of the�lter F from its input signal w to the output trajectory yd is chosen asproposed in Section 3.5 by

yd(s)w(s)

=1

(s/λ+ 1)2. (6.8)

The input constraints

umin ≤ u ≤ umax. (6.9)

have to be considered. This is done according to Section 5.1.From Eq. (6.5) one �nds maximum and minimum permissible second

derivatives of yd

ÿmax =ϕ3(y, ˙y, n, mf)+

max(ϕ5(y, ˙y, n, mf)umax, ϕ5(y, ˙y, n, mf)umin)ÿmin =ϕ3(y, ˙y, n, mf)+

min(ϕ5(y, ˙y, n, mf)umax, ϕ5(y, ˙y, n, mf)umin).

(6.10)

The limited IMC �lter is shown in Fig. 6.3.

++

max~y

min~y

d~y

d~y

d~y2λ

λ2

w∫ ∫

F

Fig. 6.3: IMC �lter F with saturation to respect input constraints.

6.3.3 Complete IMC Law

The resulting IMC feedback structure is presented in Fig. 6.4 with F fromFig. 6.3 and Σr from Eq. (6.7).

130

6.4. TEST BED RESULTS

u y

y~

dw r~Σ

Q

-

-

F d~y

(n)d~y

Σ

md

Fig. 6.3 Eq. (6.7)

Fig. 6.4: Closed-loop control structure of the �atness-based IMCboost pressure controller. The calculations of the bound-aries in Eq. (6.10) are not shown in this �gure.

The complete �atness-based IMC controller can thus be written by (cf.Theorem 4.1)

Q :1λ2

ÿd +2λ

˙yd + yd = w

with ÿmin ≤ ÿd ≤ ÿmax from Eq. (6.10)(6.11a)

u =ÿd − ϕ3(yd, ˙yd,dm)ϕ5(yd, ˙yd,dm)

(6.11b)

where the di�erential equation (6.11a) presents the dynamics of the IMC�lter F with the conditions to respect input constraints. The initial con-ditions of the �lter are obtained from Eqns. (6.2) and (6.3) as

yd(0) = x1(0)

˙yd(0) =ϕ4(x2(0))

k7ϕ2(x1(0), x2(0))− k9

k7k6(dm(0))x2(0).

(6.11c)

Equation (6.11b) is the right inverse Σr.According to the IMC properties, zero steady-state o�set is to be ex-

pected. The closed-loop algorithm has no iterations and can be imple-mented online. Furthermore, the input constraints Eq. (6.9) will neverbe violated. This holds for arbitrary modelling errors and for arbitrarymeasured and unmeasured disturbances.

6.4 Test Bed Results

The IMC controller was tested on an engine test bed. The results portraythe capabilities of the control concept. The parameter λ of the transferfunction (6.8) of F is the degree-of-freedom of this IMC. It can be selectedto be a compromise between performance and robustness of the closed-loop behaviour. In this case, the calibration of λ was performed manually

131


0.5

1

1.5

2

2.5

Stim

uli

1.2

1.4

1.6

1.8

Boo

stpr

essu

rey

=p

2[b

ar]

0 5 10 15 20 25 30 35 400

0.5

1

time [s]

Inpu

tu

nE [1000rpm]mf [g/s]

wy with IMCy with PID

u with IMCu with PID

Fig. 6.5: Closed-loop performance on an engine test bed.

at the test bed which allowed to consider measurement noise and theperformance criterions on the �nal plant. Most model parameters wereidenti�ed o�-line using manufacturer data and additional measurements.Only the characteristics of the vacuum-controlled diaphragm box, whichdrives the VNT position, was adjusted at the test bed. Calibration of theIMC was completed at the test bed within minutes. The input constraintsare 0 ≤ u(t) ≤ 1.

Figure 6.5 (showing time 0s ≤ t ≤ 40s), Fig. 6.6 (showing time 200s ≤t ≤ 290s), and Fig. 6.7 (showing time 290s ≤ t ≤ 350) portray the closed-loop tracking performance of the boost pressure p2 at the test bed aftercalibration. The time intervals were chosen since they show the mostsigni�cant di�erences between the gain-scheduled PID and the IMC con-troller. At all other times, the two controllers performed nearly identically.The reference value w is generated according to the engine manufacturerspeci�cations in dependence on the current engine speed nE and fuel mass�ow mf , which are portrayed in the �rst subplot of each �gure. The sec-ond subplot shows a comparison of the output y = p2 between the IMC

132

6.4. TEST BED RESULTS

0

2

4

6St

imul

i

1.2

1.4

1.6

1.8

2

Boo

stpr

essu

rey

=p

2[b

ar]

200 210 220 230 240 250 260 270 280 290

0

0.5

1

time [s]

Inpu

tu





controller and the series production controller of that engine, labelled asPID. The series production controller has the structure shown in Fig. 1.3and is a gain-scheduled PID with �nal calibration. Subplot three showsthe comparison of the input signals of the IMC and the gain-scheduledPID controller.

One �nds that the IMC controller is superior to the series produc-tion gain-scheduled PID controller considering its tracking behaviour. Attimes t = 33s, t = 227s, t = 267s and t = 297s the gain-scheduled PIDcontroller shows more overshoot than the IMC controller. Less overshootcan also be observed, for example, at times t = 265s and t = 328s. The≈ 40% overshoot of the gain-scheduled PID controller at time t = 33s isnot due to a change in the reference signal w but rather due to a change inengine speed nE. One concludes that IMC compensates measured distur-bances better (to which the engine speed nE belongs) than the PID-typecontroller, which relies on the scheduling algorithm to deal with such in-�uences.

In conclusion, the IMC tracks the reference signal better in many situ-

133


2

4

6

Stim

uli

1.2

1.4

1.6

1.8

2

2.2

Boo

stpr

essu

rey

=p

2[b

ar]

290 300 310 320 330 340 350

0

0.5

1

time [s]

Inpu

tu





ations. As expected, the IMC controller achieves zero steady-state o�set,despite signi�cant modelling errors and disturbances, including unmea-sured sensor noise. Thus, the IMC controller surpasses the series produc-tion controller in terms of performance with comparable high frequencyinput gain. The implementation e�ort of this nonlinear IMC controller isto be considered small. The controller itself has only two integrators (theIMC �lter) to be implemented and the equations require only few andeasily computed operations.

6.5 Summary

This chapter solves the pressure control problem of a one-stage turbo-charged diesel engine using a variable-nozzle turbine. Since the simpli�edmodel of the air-system is �at, a �atness-based IMC is developed. TheIMC �lter was chosen to respect input constraints and the resulting con-troller was tested on a real engine at a test bed. The �atness-based IMCcompares favourably to the current series production gain-scheduled PID

134

6.5. SUMMARY

controller in terms of performance and calibration e�ort. From an eco-nomical point of view, a nonlinear IMC is to be preferred over the typicalPID controller design used in the automotive industry since it can becalibrated signi�cantly faster, saving personnel cost. Moreover, an IMCuses less memory, since only the value of the physical parameters needto be stored (about thirty parameters) instead of about a dozen look-uptables with hundreds of values each. The computational e�ort is to beconsidered to be about in the same order of magnitude as the productionPID.

135


136

7. CONTROL OF A TWO-STAGE

TURBOCHARGED DIESEL ENGINE

This chapter develops an IMC pressure controller of a two-stage turbo-charged air-system. Figure 7.1 shows the two-stage turbocharged air-system. The controller achieves tracking control of pressures in threepipes, namely the boost pressure, the exhaust back pressure and the in-ner turbines pressure, and is able to drive the plant into any physicallypossible state.

This control solution employs the basic IMC design procedure (cf. Sec-tion 3.5) where the model inverse is developed exploiting the input-a�neplant model (cf. Section 4.2). The design is extended to handle input con-straints (cf. Sections 5.4), to handle the singularity of the inverse of theplant model (cf. Section 5.2) and to incorporate measured disturbances(cf. Section 5.4).

This chapter presents the main practical contribution of this thesis. Itis the �rst published control solution which is able use the full capabilitiesof this plant.

7.1 Function of a Two-Stage Turbocharged DieselEngine

For the next generation of turbocharged diesel engines, current researchis going towards the use of various combinations of turbines and compres-sors. One of the most promising combinations [3, 79, 90] is two-stageturbocharging. The motivation for this system is its higher possible boostpressure, reduced turbo lag1, and lower boost threshold2, in comparisonto one-stage turbocharged engines.

1 Turbo lag denotes the time the air-system needs to build the desired boost pressure.2 Boost threshold denotes the lowest engine speed nE at which the air-system can

build signi�cant boost pressure.

7. CONTROL OF A TWO-STAGE TURBOCHARGED DIESEL ENGINE

The idea of charging an engine in various stages has existed sincethe early 20th century [105]. However, it has recently produced interestamongst car manufacturers for several reasons: The higher achievableboost pressures allow signi�cant downsizing3, have potential for signi�-cantly lower emissions through higher pressure ratios, and result in more�fun to drive�.

HP Shaft

LPC

LPT

HPT

HPC

Engine

HPC Bypass

HPT BypassLPT

Wastegate

Chargecooler

Exhaust aftertreatment

LP Shaft

LPTA

HPTA

HPCA

1V

2V

3V

4V

5V

5V 6V

0V

En

fm&

Fig. 7.1: Two-stage turbocharged engine.

In the concept of two-stage turbocharging (see Fig. 7.1), two turbocharg-ers are placed in sequence. Each turbocharger is called a �stage�. Theturbocharger closest to the engine is the high-pressure (HP) stage andthe low-pressure (LP) stage is positioned closer to the environment.

In the following, the air path through the system is described, startingwith air from the environment V0. The �rst compressor is the low-pressurecompressor (LPC). The air is compressed for a second time in the high-pressure compressor (HPC). The HPC can be bypassed through an ori�cewith variable cross section area AHPC. After combustion, the exhaust is�rst led over the high-pressure turbine (HPT), through the pipe V4, andthen over the low-pressure turbine (LPT). Both turbines can be bypassed

3 The term �downsizing� refers to building engines with smaller displacement thatstill generate a high power output through forced induction.

138

7.2. CONTROL PROBLEM

by setting the cross section areas AHPT and ALPT accordingly. The crosssection areas in�uence the amount of exhaust gas �owing through theturbines, which ultimately drive the compressors via the shafts.

The high pressure turbocharger is smaller and lighter and designedto operate at its optimum when the engine is at low speeds (i. e., under2500rpm) enabling it to quickly deliver signi�cant boost pressure fromabout 1300rpm. At higher engine speeds, the high pressure turbochargerreaches its saturation and is bypassed. In the range from about 2500rpmup to about 6000rpm, the low-pressure turbocharger is designed to deliverthe necessary boost pressure.

Currently, only one manufacturer, namely BMW4, o�ers a two-stageturbocharged engine. It is a three litres inline six cylider engine [18, 90].However, all major car manufacturers are currently developing two-stageturbocharged engines, e. g., [38]. One of the major obstacles in introducingthis new technology is its di�cult control problem [3].

7.2 Control Problem

A two-stage turbocharged air-system (cf. Fig. 7.1) is structurally di�erentfrom a one-stage air-system (cf. Fig. 6.1) in such that it has three inputsinstead of one. Those are the bypass cross sections AHPC, AHPT and ALPT.The three inputs allow to pose requirements in addition to controlling theboost pressure p2. In this work, the freedom given by the inputs is used totrack reference values for the exhaust back pressure p3 and the pressurebetween the turbines p4. Figure 7.2 shows the input and output signals

ΣHPCA 2p

md

4p3p

HPTALPTA

Fig. 7.2: Inputs and output signals of a two-stage turbocharged en-gine.

of a two-stage turbocharged engine.

Control Problem (Two-stage). The measured variable y is

y = [p2, p3, p4]T . (7.1)4 BMW named its two-stage turbocharged engines �twin charged�. Colloquially, two-

stage turbocharging is sometimes called �twin turbocharging�. However, the expression�twin turbocharging� is not properly de�ned. It is used inconsistently for various (other)charging solutions and it is not used in scienti�c literature.

139


The available input consists of the cross section areas of the bypasses

u = [AHPC, AHPT, ALPT]T . (7.2)

Input constraints are as follows.

0 = umin ≤ u ≤ umax ≤ 12.5 (7.3)

The requirements are summarised below.

• The output y should track the reference value w without steady-state o�set.

• The e�ect of the measured disturbances dm should be compensated.

• The closed-loop should be robustly stable.

The reason for the choice of the controlled variables is given in the fol-lowing: The exhaust back pressure p3 determines the force against whichthe exhaust gas needs to be pushed out. A low value results in betterengine e�ciency and a high value increases the engine braking power.Depending on the current driving condition, one is to be favoured overthe other and, thus, this choice o�ers a straightforward in�uence on animportant property of engine operation. Note that in the case of theone-stage turbocharged engine, the value of the exhaust back pressure p3

was fully determined by the choice of the boost pressure p2 (which is thecontrolled variable). The choice of the inner turbines pressure p4 comesfrom the need of a third controlled variable for the control problem tobe determined and since it yields a relative degree to the input u3 of one.Moreover, this value o�ers the engineer a direct in�uence over which stageshould provide how much thermal energy.

The raise in di�culty of controlling a two-stage compared to a one-stage turbocharged air-system is a structural change from a SISO to aMIMO control problem. It will also be shown below, that the two-stageturbocharged air-system becomes singular at some operating conditionswhich also presents an additional control challenge.

Model. The derivation of the model equations is given in Appendix A.For the sake of better readability, the equations of the resulting control-oriented model are repeated here. The six state variables of the reducedmodel of a two-stage turbocharged diesel engine are explained in the Ap-pendix in Tab.A.1 and Tab.A.2 introduces the parameters.

140

7.3. LITERATURE REVIEW

The reduced-order, input-a�ne model Σ, with which the IMC con-troller is developed, is given by

Σ : x = f(x,dm) +G(x,dm)u, x(0) = x0, x ∈ R6 u ∈ R3 (7.4a)

y = h(x) =[x1, x2, x3

]T, y ∈ R3, (7.4b)

with model output y. The vector �eld f is de�ned as

f(x,dm) =

κ2−1V2

(HHPC,out + HE,in

)κ3−1V3

(HE,out + HHPT,in

)κ4−1V4

(HHPT,out + HLPT,in

)mHPT,out + mLPT,in

1JLPωLP

(PLPT − PLPC − dLPω

2LP

)1

JHPωHP

(PHPT − PHPC − dHPω

2HP

)

(7.4c)

and the matrix G has the following structure:

G(x,dm)=

g11(x,dm) 0 0

0 g22(x,dm) 00 g32(x,dm) g33(x,dm)0 g42(x,dm) g43(x,dm)0 0 00 0 0

· 1/m2. (7.4d)

The description of the components of G is given in the Appendix inEq. (A.45e).

7.3 Literature Review

The choice of controlled variables given above, does not re�ect any cur-rent demand from a car manufacturer. Since this charging solution is adramatic structural change (from one to three controlled variables), carmanufacturers and suppliers �nd it di�cult to fully exploit its possibili-ties. Rather, it is desired to maintain the control problem of the one-stageturbocharged engine (control the boost pressure with one input variable)for which a solution has grown over the last two decades.

There are few articles concerning the control of two-stage turbochargedengines, namely [96]5 and [90]6. The control problem solved by existing

5 Developed by the supplier ERFI6 Developed by the car manufacturer BMW

141


solutions consists of only one controlled variable, namely the boost pres-sure p2. This, however, yields an underdetermined control problem, sincea speci�c boost pressure p2 can be obtained by in�nitely many choicesof the three control inputs. This dilemma is evaded by switching betweenthree SISO controllers which all control the boost pressure p2 but withdi�erent input variables. Thus, in dependence upon the current operat-ing point, an overlying logic sets two of the three inputs at some constantvalue (either fully open or fully closed) and uses the remaining input tocontrol the boost pressure p2. Which inputs are kept constant and whichis used to control the boost pressure p2 depends on thresholds of either thecurrent values of engine speed nE and fuel mass �ow mf [90] or the valueof the estimated turbocharger speeds nLP, nHP and the boost pressure p2

[96].

The cited solutions lack the �exibility of a MIMO controller as it is pro-posed in this chapter, since it has the potential to drive the plant throughall its physically possible behaviour simply by changing the reference val-ues. For example, some exhaust gas aftertreatment methods require aboost pressure below ambient pressure. A two-stage turbocharged enginehas the potential to do this. However, the current solutions are not capa-ble of driving the boost pressure to such a low value since the overlyingswitching logic opens and closes two of the three bypasses for normal driv-ing, prohibiting to reach the necessary low boost pressure p2 with only thethird input. Thus, the existing scheduling algorithms would need to beenhanced by additional operation modes (cf. Section 1.3.3), which wouldlead to a more complex switching logic. In the case of a MIMO controller,the necessary pressures can be reached simply by changing the referencevalues.

In the case of [90], the solution was �rst commercially available in the2006 BMW 535d. The control law is essentially a PID controller of thesame structure as shown in Chapter 1.3 with the extension that it canchange the choice of input variable. It works well with a large enoughadaptation algorithm and is not based on a methodological design.

The solution of [96] includes an algebraic feedforward control based onphysical laws and includes a PID feedback controller. Nominal stabilitycannot be guaranteed. This approach can also be put into the typicalstructure of automotive controller shown in Chapter 1.3 with the onlydi�erence of switching the choice of input and computing the static feed-forward control input via physical laws (instead of measuring it at the testbed).

142


In conclusion, the control problem is di�cult and unsolved in the sensethat no control algorithm exists which is able to drive the system into allphysically possible states. The existing solutions are still based on an adhoc approach, are not able to take advantage of the full potentials of theplant, o�er no guarantees concerning nominal stability, and require a highcalibration e�ort.

In the following, a MIMO IMC controller is developed which is ableto drive the plant into any physically possible state, guarantees nominaland robust stability and respects the input constraints.

7.4 Nonlinear IMC of the Air-System

7.4.1 Model Inverse

The model (7.4) is input-a�ne and allows to create an inverse using theI/O normal form as described in Section 4.2. Here, the MIMO case needsto be considered as discussed in Section 5.5.

As in the SISO case, the transformation of the system equations intothe I/O normal form begins with repetitive di�erentiations of the individ-ual outputs yi in Eq. (7.4b) with respect to time and the substitution ofthe elements by the state equations Eq. (7.4a) until an input uj appearsexplicitly. One �nds

˙y1

˙y2

˙y3

=

a(x,dm)︷︸︸︷f1(x,dm)f2(x,dm)f3(x,dm)

+

g11(x,dm) 0 00 g22(x,dm) 00 g32(x,dm) g33(x,dm)

︸︷︷︸

B(x,dm)

u1

u2

u3

,(7.5)

where the �rst derivative of each output component yi already containsan input component uj . Hence, the relative degree with respect to eachinput is one. Each input a�ects a di�erent output with this relative de-gree7. Since only one derivative is necessary, no di�erentiations of themeasured disturbance dm occur (cf. Section 5.4). The matrix B is calledthe decoupling matrix.

7 Note that y3 was chosen such that each input has a relative degree of one.

143


With Eq. (7.5), a transformation Φ(x) of the system Σ into I/O normalform (see e. g., [51]) is obtained by[

yη

]= Φ(x) =

[x1 x2 x3 x4 x5 x6

]T. (7.6)

Thus, the system Σ is already in I/O normal form. The transformed

states are y =[x1 x2 x3

]Tand η =

[x4 x5 x6

]T. The states y will

be controlled directly (since they are also outputs) whereas the states ηare the states of the internal dynamics. With the demand

y!= yd (7.7)

the right inverse Σr is obtained by solving Eq. (7.5) for the input u with(7.6). One gets

u = B−1 (yd,η,dm) ·(yd − a

(yd,η,dm

)). (7.8)

In Section 4.2 it is proposed to obtain the states η of the internal dynamicsby integrating their di�erential equation

η =

f4(yd,η,dm)f5(yd,η,dm)f6(yd,η,dm)

+

0 g42 (yd,η,dm) g43 (yd,η,dm)0 0 00 0 0

uη(0) =

[x4(0) x5(0) x6(0)

]T (7.9)

numerically. Simulations show that the solution of the internal dynamicsis stable.

According to Theorem 4.2, the input u can be calculated by the inverseΣr shown in Fig. 7.3. Note that the measured disturbance dm also entersEqns. (7.9) and (7.8) but is omitted in Fig. 7.3 for clarity.

In short, Eq. (7.8) together with the solution of Eq. (7.9) represent themodel inverse Σr. However, the decoupling matrix B in Eq. (7.5) must beregular, otherwise the inverse would not be de�ned as the relative degreemight be lost.

7.4.2 Singularity of the Model Inverse of a Two-StageTurbocharged Engine

The inputs u are the cross-sections of the bypasses (cf. Fig. 7.1) u =[AHPC, AHPT, ALPT]T . If the pressures before and after a bypass are

144


dy~

dy&~

u

∫η

r~Σ

Eq.(7.8)

Eq.(7.9)

Fig. 7.3: Structure of the right inverse Σr of the model (7.4) of thetwo-stage turbocharged air-system.

equal then the mass �ow through the bypass vanishes independently ofits cross-section. Then, the input u cannot in�uence the system whichresults in a singularity (cf. Section 5.2).

The mathematical equivalence of this problem is that the decouplingmatrix B loses its regularity. The inverse of the matrix B in Eq. (7.8)

B−1 =

1/g11 0 00 1/g22 00 −g32/(g22g33) 1/g33

(7.10)

is only de�ned if all denominators in Eq. (7.10) are non-zero:

g11, g22, g33 6= 0. (7.11)

For the elements g22 and g33 (cf. Eq. (A.45e)) this is guaranteed by thephysical mode of operation: All pressures from the engine into the envi-ronment V6 decrease and, therefore, there would always be positive �uid�ow and therewith positive enthalpy �ow through the turbine bypasses ifthey had unitary cross section areas.

However, the term g11 can become zero during normal driving condi-tions. It can be interpreted as a the part of the enthalpy �ow that isresponsible for building the boost pressure (cf. Eqns. (A.3) and (A.45e)).This happens if the pressure p1 between the compressors is equal to theboost pressure p2.

p1 = p2 ⇔ g11 = 0. (7.12)

This is the case, for example, when the high-pressure turbocharger isbypassed and the required boost pressure is delivered by the low-pressurecompressor. Then, a pressure equalisation between the pipes V1 and V2

(cf. Fig. 7.1) can take place and no �uid �ows through the HPC bypass,independently of its cross section area. Thus, at p1 = p2 the feedforwardcontroller is not de�ned since the plant becomes singular.

145


As discussed in Section 5.2, a loss in relative degree or invertibility canbe accommodated by an IMC �lter which also respects input constraints.Such a �lter is introduced in the following section.

7.4.3 IMC Filter

In Section 5.5 it was established that the IMC �lter for MIMO systems isa diagonal operator. Here, it is proposed to design each diagonal elementFi (with i = 1, 2, 3) as

ydi(s)wi(s)

=1

s/λi + 1. (7.13)

Each has a pole at −λi. The series of the SVFs with the right inverseshown in Fig. 7.3 yields a realisable MIMO feedforward controller.

In Section 5.2 it was established that the introduction of input con-straints also handles model singularities. Here, it will be shown, usingplant-speci�c explanations, that the singularity at p1 = p2 is handled bythe input constraints. Equation (7.5) presents an algebraic relationshipbetween the inputs u and the highest derivative ˙y. This relationship alsoholds true, if some (or all) of the inputs saturate and yields the respectivemaximal and minimal speeds ˙yimax, ˙yimin.

˙y1max = f1 + max (g11u1,max, g11u1,min)˙y1min = f1 + min (g11u1,max, g11u1,min)

(7.14a)

˙y2max = f2 + max (g22u2,max, g22u2,min)˙y2min = f2 + min (g22u2,max, g22u2,min)

(7.14b)

˙y3max = f3 + g32u2 + max(g33u3,max, g33u3,min)˙y3min = f3 + g32u2 + min(g33u3,max, g33u3,min).

(7.14c)

If the IMC �lter ensures that the speeds ˙ydi are restricted to

˙yimin ≤ ˙ydi ≤ ˙yimax (7.15)

then the input u obtained by Eq. (7.8) never violates its limitations (7.3).The limited IMC �lter is shown in Fig. 7.4.

Finally, the following corollary says that the right inverse is de�neddespite moving through the singularity if the above mentioned �lter isused:

146


−iw

∫ iyd

iyd&

iλ

maxiy&

miniy&

iF

,IMC

Fig. 7.4: Nonlinear IMC �lter FIMC,i with saturation.

Corollary 7.1. If the IMC �lter from Fig. 7.4 is used with the limitsdescribed by Eq. (7.14) then a feedforward controller using Eq. (7.8)is globally de�ned despite the singularity of the matrix B−1 fromEq. (7.10).

Proof. As the system approaches the singular point g11 → 0 the input u1

will necessarily saturate at either boundary. Then, using the IMC �lterfrom Fig. 7.4 with Eq. (7.14) one �nds

yd1 = f1 + g11u1max or yd1 = f1 + g11u1min (7.16)

and in general Eq. (7.8) yields for the �rst input u1

u1 =yd1 − f1

g11. (7.17)

Using Eq. (7.16) in Eq. (7.17) results in

u1 =g11

g11u1max or u1 =

g11

g11u1min. (7.18)

Taking the limit of Eq. (7.18) as g11 → 0 (using L'Hospital's rule)

limg11→0

u1 = u1max or limg11→0

u1 = u1min (7.19)

shows that the feedforward control from Eq. (7.8) yields a valid inputwhich will be placed at either boundary .

From a physical point of view, the explanation why the IMC �lterfrom Fig. 7.4 results in a valid input despite the rank de�ciency is thefollowing: Since the HPC bypass has no in�uence at the singular pointp1 = p2, the model can only travel through it in a certain fashion (namelywith the exact rate ˙y1 = f1(x,dm)). The IMC �lter ensures that thetrajectory yd, which is generated from the reference signal w, can beachieved by the system exactly (see Eq. (7.7)) with permissible inputs.Thereby, it automatically leads the system through the singularity withthe prescribed rate.

147


7.4.4 Complete IMC Law

If the non-realisable (but perfect) feedforward controller Σr from Fig. 7.3is �padded� with an IMC �lter FIMCi from Fig. 7.4 for each reference signalwi then a realisable feedforward controller Q results, which also respectsthe input constraints (7.3). The resulting structure of this feedforwardcontroller is shown in Fig. 7.5. The IMC controller Q for a two-stage

Fw dy~

dy~

rΣ u

η

minmax,~y

Q

Fig. 7.4

Eq. (7.14)

Fig. 7.3

Fig. 7.5: IMC controller Q for a two-stage turbocharged air-system.

turbocharged air-system can thus be written as:

1λi

˙ydi + ydi = wi with i = 1, 2, 3

with yimin ≤ ˙ydi ≤ yimax from Eq. (7.14)(7.20a)

η =

f4(yd,η,dm)f5(yd,η,dm)f6(yd,η,dm)

+

0 g42 (yd,η,dm) g43 (yd,η,dm)0 0 00 0 0

uu = B−1 (yd,η,dm) ·

( ˙yd − a(yd,η,dm

)) (7.20b)

where Eq. (7.20a) presents the dynamics of the MIMO �lter F , that re-spects the input constraints. Equation (7.20b) presents the right inverseof the model with internal dynamics and control law. The initial conditionfor the �lter dynamics and the internal dynamics are

yd(0) =[x1(0), x2(0), x3(0)

]T(7.20c)

η(0) =[x4(0), x5(0), x6(0)

]T. (7.20d)

This controller is containable in a standard OCU: It consists of six in-tegrators (three integrators in the MIMO IMC �lter and three integratorsin the internal dynamics). The controller algorithm has to compute somepowers and square roots on top of the usual additions and subtractions.It does not contain any computationally intense on-line operations, likematrix inversions.

148


The calibration of the controller concerns �nding the model parametersand the three poles λi of the MIMO IMC �lter F . The model parameterscan be determined using the algorithm given in Section A.4 in about twohours. The �lter poles can be calibrated within a matter of minutes.

In conclusion, this IMC is a feasible automotive controller, since it isimplementable in a standard OCU and o�ers dedicated calibration param-eters.

7.4.5 Robust Stability Analysis

This section is concerned with showing that the above designed nonlinearIMC controller for the two-stage air-system provides robust stability. Amajor issue in a stability analysis for industrial problems is that an upperboundary ∆ of unstructured uncertainties ∆ is almost never given a priori.Thus, the �rst part of this analysis is focused on determining such an upperboundary ∆ for the two-stage turbocharged engine. In the second part,the actual analysis will take place.

Finding an upper boundary ∆

The procedure as performed in the following should be interpreted as a�hands-on� approach which can be performed in industry and should givethe engineer a rough guess where the uncertainties lie. It should not beregarded as mathematically su�cient.

Main idea. The main idea is based on the observation presented inSection 3.5.3, namely, that the stability analysis of an IMC loop can beperformed on the feedback loop in Fig. 3.10, where F is a linear system.In this analysis, a linear approximation of the uncertainty ∆ is gainedwhich is then used for a linear stability analysis.

The following deals with the su�cient conditions of stability given inEqns. (3.40)- (3.43). Since the uncertainty ∆ is a nonlinear system, itsgain g(∆) cannot be computed even if ∆ was known. To this end, it isproposed to approximate the nonlinear uncertainty ∆ by a linear (andLaplace transformed) approximation ∆(s). Using a linear approximationof the uncertainty ∆ is reasonable since the plant model Σ itself shouldbe chosen to contain the main nonlinearities of the plant.

In the following, a linear approximation (and Laplace transform) ∆(s)of the nonlinear uncertainty ∆ is obtained. The closed-loop robustness

149


analysis is then performed using the linear approximation ∆(s). For thisprocedure to be acceptable, the relationship

g(∆(s)) ≥ g(∆) (7.21)

must be assumed to hold. Hence, this section presents a method to obtaina linear approximation ∆(s) of the nonlinear model uncertainty ∆. Theresult of the following procedure is an upper boundary ∆(ω) which is thenassumed to be given for the robustness analysis.

Procedure. First, a set of models M is generated from the nominalmodel Σ by varying a vector p of signi�cant parameters of Σ within givenboundaries:

M ={

Σp | pmin ≤ p ≤ pmax

}(7.22)

The expression Σp signi�es a model Σ with the parameterization p. Thus,M represents a set of nonlinear models of two-stage turbocharged engines.

The open-loop behaviour Σ0 of the generalised IMC structure in Fig. 3.4is depicted in Fig. 7.6.

ΣF r~Σ

−

Fig. 7.6: Open-loop structure Σ0 = F ◦(

ΣΣr − I)of an IMC con-

troller Q = ΣrF .

It is proposed to linearise Σ0 numerically around all operating pointso while Σ is substituted by all models Σ→ Σ(p) ∈M. This is a requiredintermediate step since it incorporates the e�ect of the model inverse inobtaining the model uncertainty ∆(s). With Eq. (3.38) and

F (s) =

1

s/λ1+1 0 00 1

s/λ2+1 00 0 1

s/λ3+1

(7.23)

each linearisation with model parameterization p and operating point owill give linear MIMO uncertainties ∆p,o(s) by

∆p,o(s) = G0p,o(s)F−1(s). (7.24)

150


Finally, an upper boundary ∆(ω) can be found by

∆(ω) = maxω|∆p,o(jω)|, ∀ p, o (7.25)

where max | · | denotes an element-wise maximum of the magnitude of thetransfer function. Thus, ∆(ω) is a matrix containing the function of theamplitude over frequency of each element. The elements at any frequencyare equal to the largest amplitude of that element at that frequency of allobtained uncertainties ∆p,o(s).

Figure 7.7 shows the result of the open-loop linearisations of the IMCcontroller for the two-stage turbocharged engine with modelling errors inthe loop. The linearisations concern the feedforward control structureas shown in Fig. 7.6. Therewith, a linear upper boundary on the MIMO

-20

0

dB

10-1101103

-20

0

rad/s 10-1101103

to:y

3−y

3to:y

2−y

2to:y

1−y

1

from: w1 from: w2 from: w3

Fig. 7.7: Comparison of the amplitudes of the linearised models|Σ0p,o(jω)| of the open-loop structure (cf. Fig. 7.6).

output uncertainties ∆(ω) is determined which is shown in Fig. 7.8. Anevaluation of the upper boundary ∆(ω) in Fig. 7.8 shows that some of itselements have a signi�cant high frequency gain of about 20.

Stability analysis

It is assumed, thatg(∆) ≤ g(∆p,o(s)), ∀p, o (7.26)

151


−20

0dB

10−1

101

103

−20

0

rad/s 10−1

101

103

to:y

3−y

3to:y

2−y

2to:y

1−y

1

from: y1 from: y2 from: y3

Fig. 7.8: Upper boundary ∆(ω) of the linearised model uncertaintiesderived from parameter variations.

holds; that is: The gain of the actual nonlinear uncertainties ∆ is smalleror equal to the gain of the linearisation obtained with the procedure above.Additionally, the determination of the upper boundary ∆(ω) can in realityonly be performed for a �nite number of parameter variations and �nitenumber of operating points. Nevertheless, it is assumed that such anapproximation will give a representative upper boundary ∆(ω).

Finally, the maximum singular value σmax can be computed for∆(jω)|F (jω)| for all frequencies ω. The result is displayed in Fig. 7.9. Avisual inspection of σmax in Fig. 7.9 shows that it is smaller than one forall frequencies ω. According to [64, 87], this implies robust stability of theclosed-loop. The high frequency gain of the uncertainties is compensatedby the �lter F which can now be interpreted as ensuring robust stabilityat those frequencies.

Since model parameters were varied within manufacturing tolerancesand the model represents the plant well, this presented approach yieldssu�cient con�dence for the control engineer that the nonlinear IMC willsuccessfully control the two-stage turbocharged diesel engine, despite man-ufacturing tolerances and ageing.

However, it is important to note that this approach is conservative.Simulations have shown stability for the closed-loop even for norms well

152

7.5. SIMULATION RESULTS

10−2

100

102

104

10−2

10−1

100

rad/s

σm

ax

Fig. 7.9: Maximum singular value σmax of ∆(ω)|F (jω)| over all fre-quencies ω.

beyond one. In simulations, there was no parameter variation which ledto instability.

7.5 Simulation Results

Simulation results are presented in this section. It can be assumed thatthe simulation results are representative of a test bed experiment, since

• the IMC of the one-stage turbocharged air-system showed that sim-ulation results are a close match to what is observed at the test bedand

• the model quality of the two-stage turbocharged air-system is excel-lent in both transient and steady-state behaviour.

The controlled plant is the original model with twelve states. Thus,modelling errors are present. The results are shown in Fig. 7.10. Enginespeed nE and fuel mass �ow mf change during the simulation run. The en-gine speed nE ramps from 3000rpm to 5500rpm in �ve seconds (from time1s to 6s) under full throttle. At time 6s, acceleration ends and the engineproceeds in constant highway driving. Thus, until time 11s, a mediumfuel �ow and medium boost pressure is required. During this time, theboost pressure is delivered mainly by the LPC. From time 11s until 16s,the capabilities of this controller are displayed by demanding a boost pres-sure below ambient pressure. The controller achieves this by closing theHPC bypass and opening both turbine bypasses. This behaviour cannot

153


0 25 50 75

Eng. speed [100 rpm]Inj. fuel [mg/cyc]

n

Em

f

0 1 2 3

Boost pressure [bar]

w

2y1

=p

2p

1

1 2 3

Exhaust back and inner turbines pressure

[bar]

w

2y2

=p

3w

3y3

=p

4

05

1020

2530

0 5 10

time [s]

Cross section [cm2]

u

1u

2u

3

Fig. 7.10: Simulation results

154

7.5. SIMULATION RESULTS

0 5 10 153.08

3.1

3.12

3.14

3.16

3.18

time [s]

Pres

sure

[bar

]noise

y2

w2

Fig. 7.11: Simulation results of y2 with ≈ 1% measurement noise

be reproduced by any currently available control solution. At time 16s theair-system is set to attain normal driving conditions for nE = 3000rpmand medium fuel �ow. The engine quickly loses speed from time 21s-22s,which resembles a shift in gears. Finally, the engine arrives at 1500rpmin constant driving conditions, for example medium fuel �ow and a boostpressure of 2bar. At this point, the HPC delivers almost the entire boostpressure.

When the air-system travels through the singularity at p1 = p2, theHPC bypass switches between its maximum and minimum values immedi-ately. This happens e. g., at times 16s and 21s. To verify the singularity,the pressure p1 is also plotted. It can be observed that the HPC bypassopens immediately when the LPC is generating more pressure than theHPC. This is an interesting result, since the theoretically developed IMCcontroller acts in full accordance to what experts on turbocharging suggest[10].

Figure 7.11 shows the answer of the controlled system with modellingerrors under the in�uence of measurement noise. Noise attenuation andperformance can be calibrated online at the engine test bed through theIMC �lter poles −λi. As shown here, noise attenuation can be selected tobe satisfactory.

In order to show the behaviour of the controlled plant during age-ing or production tolerances, some parameters of the plant were alteredaccording to Tab. 7.1. The resulting closed-loop behaviour with variouscombinations of the parameter variations of Tab. 7.1 is shown in Fig. 7.12for a random driving scenario. It shows that despite parameter changes,the closed-loop does not become unstable. However, there is a non-zerosteady-state o�set. This o�set comes from the existing input constraints.If the closed-loop IMC controlled two-stage turbocharged air-system doesnot reach its reference value in steady-state then this means that the

155


0 5 10 15 20 251.4

1.6

1.8

2

2.2

time[s]

Boo

stpr

essu

rep

2[b

ar]

Fig. 7.12: Simulation results of y2 under some of the plant parame-ter variations given in Tab. 7.1.

desired reference value is not attainable with the available inputs.In summary, the results show that the theoretically developed nonlin-

ear IMC control approach successfully solves the di�cult control problem.The good tracking performance, the ability of the IMC concept to controlthe system through its singularities, and to lead it into any physicallyachievable condition, shows that the nonlinear internal model controlleris a very good solution to the control problem of a two-stage turbochargeddiesel engine under practical conditions.

7.6 Summary

A nonlinear IMC is developed that controls the boost pressure, the ex-haust back pressure, and the pressure between the turbines of a two-stageturbocharged engine. Since the model is input-a�ne, the model inversionexploited the I/O normal form to develop the IMC controller. However,it is shown, that the inverse becomes singular, if the pressure between thecompressors equals the boost pressure. Due to the proposed IMC �lter,which was limited to respect the input constraints, the controlled systemdrives through the singularity. Thus, the setpoints are altered by the IMC�lter automatically such that the resulting trajectories for the pressurescan be achieved precisely with the available inputs. A stability analysis

Tab. 7.1: Parameter variations

Parameter Nominal value VariationAmbient pressure pamb 0.97 bar 83%-105%Engine fuel heat value 1.52 · 104 kJ

kg 90%-110%Exhaust ori�ce cross section 9.96 cm2 70%-100%

156

7.6. SUMMARY

showed that the closed-loop is robustly stable. In conclusion, this con-trol solution is a feasible choice as controller of this plant under practicalconditions.

157


158

8. CONCLUSION

In this thesis, the focus was on the automotive industry and its inherentrequirements for controllers. It was established that:

• reference signals are in�uenced by the driver and, thus, their futurevalues are unknown;

• the closed-loop performance speci�cations are given by time-domaincriteria (as opposed to integral criteria found in optimal control, ordemands on the amplitude-over-frequency behaviour of some trans-fer functions);

• a controller is calibrated (tuned) by non-control engineers at the testbed or in the experimental vehicle and, therefore, needs dedicatedtuning-parameters to allow for controller calibration;

• computationally intense operations, such as on-line numerical opti-misations, are forbidden due to the weak processing power of OCUs.

There are some control design methods for linear systems which o�erthe above properties (e. g., loop-shaping, IMC, reference system controldesign). However, many automotive plants are nonlinear (e. g., catalyticconverters, air-systems) and future technologies are expected to introducemore nonlinear plants (e. g., Lean NOx Trap). Therefore, a nonlinearcontrol design method which meets the above requirements is desired.

When it comes to nonlinear controller design, the list of available meth-ods is rather sparse, especially if demanding output feedback control whichis a standard requirement on all industrial controllers.

In this thesis, a novel nonlinear control design method was developedby extending the classical IMC design to nonlinear plants. The resultingnonlinear controller ful�lls the above requirements and is applicable onstable plants.

The design of internal model control focuses on �nding a controllersuch that a given closed-loop I/O behaviour is achieved. It is based

8. CONCLUSION

on feedforward control design and relies on model inversion. The IMCstructure is simple and plausible and provides valuable properties such asnominal and robust stability as well as zero steady-state o�set. The attrac-tiveness of IMC to industry comes from the internal model and the simpledesign law. The tuning knob of an IMC, as far as nominal performancegoes, is reduced to a single parameter that sets the nominal closed-loopbandwidth and results in a well damped control response. Once an IMCcontroller is determined for a speci�c plant, it can be calibrated throughthe model parameters. This enables non-control engineers to calibrate anexisting IMC controller, since knowledge of the plant su�ces to determinethe model parameters. IMC does not use on-line optimisation procedures.Hence, it can be implemented in a real-time environment like an OCU.

The main idea to design a nonlinear IMC controller is to employ theright inverse of the nonlinear plant model together with a low-pass IMC�lter. A right inverse is, by itself, a non-realisable operator as it requiresa number of derivatives of its input signal. It is the composition of theright inverse with a low-pass �lter that yields a non-anticipatory operatorwhich can be used as an IMC controller. The connection between thetwo is established by implementing the IMC �lter as a state-variable �lterwhich gives the requested derivatives by its implementation. It was shownthat such an IMC controller yields a nominally stable closed loop whichproduces zero steady-state o�set and has a certain structural robustness.

It was shown that properties of the plant model, like input constraintsand singularity of its inverse, can be addressed by the IMC controller ifthe IMC �lter is altered appropriately. To this end, the notion of relativedegree of nonlinear systems was exploited. It was possible to describe theshape of the output of a dynamical system in dependence of the shape ofthe initial conditions, the input function, and the relative degree. Withthis tool, an algebraic relationship between the input and the highest nec-essary output derivative was established. This relationship was used tocreate an IMC �lter which, in composition with the right inverse, respectsinput constraints and does not lead to in�nite control responses in thepresence of model singularities. Interpreting the IMC �lter as a trajec-tory generator, one �nds that the IMC �lter is altered such that it iscreating only such trajectories that can be a produced by the model withpermissible inputs.

A generalisation of this interpretation of the IMC �lter was used to pro-pose a novel method to treat models with unstable inverses (non-minimumphase models). The basic idea is founded on the equivalent procedure for

160

linear systems where a zero is used to cancel an unstable pole. The min-imum realisation yields an internally stable feedforward controller as thecancellation is done before hand. In the nonlinear case, it was �rst es-tablished under which conditions such a cancellation takes place. Thiscondition was called identical internal dynamics. It was shown that if theIMC �lter shares the internal dynamics of the model then the composi-tion and minimal realisation with a perfect right inverse yields a perfectlyfeasible IMC controller for non-minimum phase nonlinear plants. Theconditions on the IMC �lter were given and it was suggested how such anIMC �lter can be obtained.

With all of the above combined one �nds that the proposed nonlinearIMC controller is applicable to a system class which covers many automo-tive control problems, namely all invertible and stable models, including

• non-minimum phase models,

• models with an ill-de�ned relative degree,

• models with input constraints

• models with measured disturbances, and

• models with singular inverses.

However, the application of the proposed nonlinear IMC also has somedemerits. For good performance, it needs a good control model. With sig-ni�cant modelling errors present, performance is likely to be unsatisfactory(see [22] on the bene�t of feedforward control design used together withfeedback � an issue related to the IMC design as discussed here). Thisholds especially for MIMO plants, as dominant cross-couplings (i. e., nondiagonally dominant plants) may yield a high sensitivity to modelling er-rors if inversion based control is employed. IMC may lead to high ordercontrollers. In its minimal realisation, the order of an IMC controller isequal to the order of the model. Clearly, high order plant models auto-matically lead to a controller of high order. In order to obtain an IMCcontroller of low order, the order of the plant model needs to be reduced�rst. Another drawback is, that this thesis has only considered continu-ous time controllers where, clearly, a discrete time controller needs to beobtained for implementation in an OCU. The task of obtaining a discretetime equivalent is left to the engineer.

Despite these demerits, the presented nonlinear IMC design method isconsidered very attractive for automotive applications and may be evenof interest to other industries.

161

8. CONCLUSION

The nonlinear IMC design method was employed to obtain a controllerfor two automotive problems. First, the pressure control problem of aone-stage turbocharged diesel engine using a variable-nozzle turbine, andsecond, the pressure control of three pressures of a two-stage turbochargeddiesel engine. Other applications can also be found (see e. g., [viii, xix]).

For the control of the one-stage air-system, the respective model wasestablished to be �at. Therefore, a �atness-based IMC was developed.The IMC �lter was chosen to respect input constraints and cancel thee�ect of measured disturbances. The resulting controller was tested on areal engine at a test bed. The �atness-based IMC compared favourablyto the current series production gain-scheduled PID controller in terms ofperformance and calibration e�ort. It showed less overshoot and less un-dershoot in various situations, with comparable closed-loop speed. Hence,it was shown that the proposed control scheme is actually feasible in real-ity.

For the control of the two-stage turbocharged air-system, only simula-tion results were available. A nonlinear IMC was developed that controlsthe boost pressure, the exhaust back pressure, and the pressure betweenthe turbines of a two-stage turbocharged engine. It presents the �rstcontrol solution to this control problem. Moreover it is the �rst solutioncapable to control the plant throughout its physically possible range of op-eration. Since the model is input-a�ne, the model inversion exploited theI/O normal form to develop the IMC controller. However, it was shown,that the inverse becomes singular, if the pressure between the compres-sors equals the boost pressure. Due to the proposed IMC �lter, whichwas limited to respect the input constraints, the controlled system neverloses its relative degree. Thus, the setpoints are altered by the IMC �lterautomatically such that the resulting trajectories for the pressures can beachieved precisely with the available inputs. Moreover, the IMC controllercancelled the e�ect of measured disturbances. The control performancewas very good, even under assumed modelling errors and measurementnoise. A stability analysis showed that the closed-loop is robustly stable.It is assumed that the presented controller would perform well under prac-tical conditions as the model quality is similar to that of the one-stageair-system and the controller was obtained using the same control designmethod.

162

APPENDIX

A. COMPOSITIONAL MODEL LIBRARY FOR

TURBOCHARGED DIESEL ENGINES

This chapter develops a compositional model library for turbocharged air-systems. Models of single turbocharged engines are found, for example,in [14, 52, 70] and follow some di�erent model assumptions and mod-elling goals as the model presented here. They cannot be used to developa model of a two-stage turbocharged engine, mainly due to the lack ofsatisfactory turbine and compressor components. The predominant tools(e. g., [37] or [89]) of modelling engines and air-systems are geared towardscomponent design (as opposed to control system design) and are based oncomputational �uid dynamics. Computational �uid dynamics describe anair path using �nite volumes that are each governed by partial di�erentialequations. The focus is on high-frequency e�ects like pressure waves andit is not useful for controller design due to the resulting model complexityand since di�erential equations cannot be extracted from the resulting�nite volume model.

The model library, as presented here, is described exhaustively on thenext pages. The treatment of the modelling is presented in some detailfor the following reasons:

• It is desired to present the reader a self-contained solution to thecontrol problems of a one- and two-stage turbocharged engine. Inthe opinion of the author, this should include a complete modelderivation; especially, since the model library here di�ers in someaspects from others.

• The library presented here is used to compose the �rst publishedmodel of a two-stage turbocharged air-system that is geared towardscontrol design.

• The turbine and compressor models, as proposed here, are a contri-bution of this thesis for modelling turbochargers for control applica-tions. Unlike given models in the literature, the presented models

A. COMPOSITIONAL MODEL LIBRARY FOR TURBOCHARGED DIESELENGINES

are also valid in atypical operating conditions as they appear intwo-stage turbocharging.

• Finally, there are no publications on identi�cation and model reduc-tion of turbocharged engines, although those are important steps inobtaining models for control design.

A.1 Modelling Goal and Assumptions

Modelling goal. The goal of modelling an air-system of a turbochargeddiesel engine for control design is to obtain

1. a low-frequency model of low order

2. that is composed of re-usable components.

A low-frequency model is important if a model-based controller is to bedesigned. A compositional approach allows re-use of the component mod-els, to model di�erent turbocharging solutions (e. g., parallel turbines andsequential compressors) simply by re-arrangement.

The modelling focus lies in pressures and temperatures in the pipesand the operation of the turbochargers. The diesel engine is an essentialpart of this system, however, torque and emission generation are irrele-vant considering a pressure control problem and, hence, are not modelled.Similarly, the engine speed nE does not need to be modelled since it isnot a control goal and its value is readily available through measurement.

Modelling assumptions. In order to reach the modelling goal, thefollowing assumptions are made:

1. All pipes are assumed to behave like plenum chambers with a uni-form distribution of pressure and temperature. This means thathigh-frequency e�ects, like pressure waves through a pipe, are ne-glected and the system can be considered as a lumped parametersystem.

2. A mean-value engine model is used, which means that the functionof an engine is interpreted as delivering a continuous mass �ow,heated by the injected fuel mass. Thus, high-frequency e�ects, thattypically result from a four stroke gas exchange cycle, are neglected.

3. All gases are assumed to be perfect.

166

A.2. CONNECTING INDIVIDUAL COMPONENT MODELS

Perfect gases follow the law [67]

pV = mRT, (A.1a)

with pressure p in a chamber of volume V , gas mass m, gas constant Rand temperature T . Additionally, speci�c enthalpy h is the product ofconstant pressure speci�c heat cp and temperature T :

h = cpT. (A.1b)

Finally, the following relationships

κ = cp/cv (A.1c)

R = cp − cv (A.1d)

H = mh, (A.1e)

with polytropic exponent κ, constant volume speci�c heat cv, total en-thalpy �ow H and mass �ow m hold.

The following provides the method with which component models areto be connected to one another. More precisely, the physically meaningfulinput and output signals, through which the components communicate,are de�ned.

A.2 Connecting Individual Component Models

The component models are divided into two categories, namely

• models with dynamics (storage models), and

• models without dynamics (coupling models).

As a convention, storage models have �ow-variables as inputs and somefunction of their states as outputs. It is not possible to connect energystorage models directly to one another. They can only communicate overa coupling model which provides the matching inputs and outputs. Thus,coupling models compute the �ow variables from the outputs of the stor-age models.

An analysis of Figure 6.1 and Figure 7.1 on pages 125 and 138 showsthat there are eight distinct components to be modelled. Those are

• an environment (which is essentially an in�nitely large chamber),used to represent V1, V5 and V0, V6 for a one-stage and a two-stageturbocharged air-system, respectively;

167


• a plenum chamber, which is used to model the pipes V2 to V4 andV1 to V5 for a one-stage and a two-stage turbocharged air-system,respectively;

• a turbocharger consisting of turbine, shaft and compressor;

• an intercooler (also referred to as charge air cooler);

• an engine; and

• an ori�ce, representing bypasses, wastegate and the e�ect of exhaustgas aftertreatement.

Thus, an air-system can be modelled by few distinct components if theyare re-used with di�erent parameters in one model . In the case of theair-system of a turbocharged engine, plenum chambers (the pipes) andthe shafts are storage models. Plenum chambers store mass and energywhich are represented by the pressure p and temperature T of the �uidinside and are designed to have mass �ows mi and enthalpy �ows Hi

of various coupling blocks as inputs. Similarly, shafts have powers Pjas inputs and yield their current speed as output. Figure A.1 shows allstorage models used to model the air-systems shown in Figures 6.1 and7.1. All other components of the air-system are coupling blocks and are

p

TV

(a) Environ-ment

Vim&

iH&

p

T

(b) Plenumchamber

V

p

T

outm&

outH&im&

iH&

(c) Intercooler connected toa plenum chamber

TP

CP

n

(d) Shaft

Fig. A.1: Storage component models.

shown in Fig.A.2.In order to illustrate the connection method, an ori�ce with constant

cross section is placed between two plenum chambers. The resulting con-nection signals are shown in Figure A.3. It shows the model structure of

168

A.2. CONNECTING INDIVIDUAL COMPONENT MODELS

ip

iT

im&

iH&

(a) Ori�cewith constantcross-section

ip

iT

im&

iH&

A

(b) Ori�cewith variablecross-section

En

fm&

inp

inT

im&

iH&

(c) Engine

ip

iT

im&

iH&

CPn

(d) Compressor

TP

ip

iT

im&

iH&

n

(e) Turbine

TP

ip

iT

im&

iH&

n

vntu

(f) VNT

Fig. A.2: Coupling component models.

1V

1p

1T

1H&

1m&

2V

2p

2T

2H&

2m&

Fig. A.3: Example: Modelling the equalisation between two tanks.

a pressure equalisation procedure between two tanks. Each plenum cham-ber supplies the ori�ce with the information of its �uid's current pressure pand temperature T . From this information, the ori�ce calculates the massand enthalpy �ow from one chamber to the other. The �ow information isgiven to each chamber according to the �ow direction: If �uid �ows awayfrom a chamber, its �ow signals will have a negative sign (e. g., m < 0 andH < 0) and if it �ows into a chamber the �ow signals will be positive.

An exception to the connection rules are intercoolers and environments(see Figure A.1(a)). Intercoolers only lower enthalpy �ow and, therefore,have �ow-variables as inputs and outputs. They may be regarded as aninput-extension of plenum chambers (see Figure A.1(c)). Environmentsare essentially in�nitely large chambers and, therefore, supply a constantpressure and temperature.

169


A.3 Component Models

A.3.1 Storage Models

Environment

An environment (Fig.A.1(a)) has the constant outputs p = pamb andT = Tamb, representing the ambient pressure pamb and temperature Tamb,which are supplied as parameters. Physically, there are always �ows intoand out of the environment. However, this information is needless from amodelling perspective, since it does not in�uence the environment.

Plenum chamber with optional intercooler

A derivation of the following equations is based on the energy balance ofopen systems [67] and the gas law Eq. (A.1a) and can be found e. g., in[92]. Here, a constant volume V is assumed.

Plenum chambers (Fig.A.1(b)) can have an arbitrary number of cou-pling models connected to them. Each coupling model i sends its �owinformations mi and Hi to the storage model. The expression

∑i signi-

�es the sum over all respective �ows. The behaviour of a �uid inside aplenum chamber with given parameters V,R, cv and the initial conditionsp0 and T0 is described by two di�erential equations:

m =∑i

mi, m(0) =p0V

RT0(A.2)

p =κ− 1V

∑i

Hi, p(0) = p0. (A.3)

The outputs are given by the pressure p from Eq. (A.3) and the tempera-ture can be obtained from Eq. (A.1a)

T =pV

mR, (A.4)

with m from Eq. (A.2).If an intercooler is connected to a chamber (see Figure A.1(c)), it

is assumed that it only changes the enthalpy �ow or temperature. Thus,the intercooler is assumed to have no pressure drop and does not in�uence

170

A.3. COMPONENT MODELS

mass �ow

pin = pout (A.5)∑i

mi = mout. (A.6)

The coolant has the temperature Tcool and lowers the �uid's temperatureT relative (by a factor η) to the di�erence between the two temperatures.

If �uid �ows into the chamber through the intercooler one �nds withEq. (A.1a)

Hout =∑i

Hi − η(∑

i

Hi − Q)

with Q = cpmTcool for∑i

mi > 0.

(A.7)

However, if �uid �ows out of a chamber through the intercooler, Eq. (A.7)does not hold. Then, according to Eq. (A.5) the pressure reported fromthe chamber is correct, but its reported temperature T has to be alteredfrom its actual temperature Tin

T = Tin − η(Tin − Tcool) for∑i

mi < 0 (A.8)

and the input signal Hi is given to the chamber model unaltered.

Shaft

A shaft (Fig.A.1(d)) has two inputs, namely the power of the turbine PT

and the power of compressor PC. With parameters inertia J , damping d,and its initial speed ω0, the dynamics of a rotating shaft is described by

Jω =1ω

(PT + PC)− ω d, ω(0) = ω0 with ω 6= 0, ∀t. (A.9)

The model output is its current speed ω in radians per second. Since theturbine and compressor models rely on a speed input with unit rpm, theoutput of the shaft can be chosen as n = 30

π ω.

A.3.2 Coupling Models

All coupling models receive the information about pressures and temper-atures from the plenum chambers they are connected to. They compute

171


mass and enthalpy �ows from and to those chambers. Except for the ori-�ce, the coupling models work only in one direction. The variables pin

and Tin signify the pressure and temperature of the chamber upstreamand pout and Tout signify the pressure and temperature of the chamberdownstream.

Ori�ce

A model derivation of the �ow through an ori�ce can be found e. g., in [8].If an ori�ce is placed between two chambers V1 and V2 (cf. Figure A.3)�uid �ow is always directed to the lower pressure level pout:

pin = max (p1, p2)pout = min (p1, p2) .

The expressions Tin and Tout refer to the temperatures in the chamberswith the pressures of the same subscript. With the pressure ratio Π =pinpout≥ 1, the mass �ow through the ori�ce is

m =ACqpin

√2

TinR·

√

κκ−1

(Π−2/κ −Π

κ+1−κ

), for Π ≤ Πcrit(

2κ+1

) 1κ−1

√κκ+1 , for Π > Πcrit

with Πcrit =(

2κ+ 1

) −κκ−1

,

(A.10)

where A is the open cross-section area and the parameter Cq (0 ≤ Cq ≤ 1)is the �ow coe�cient which accounts for losses in �uid velocities due to dif-ferent geometries. The open cross-section A can be variable, representinga continuously variable bypass.

The mass �ow signals given to the chambers are

mout = −mmin = m

(A.11)

Hout = cpmoutTout

Hin = cpminTout,(A.12)

where the subscript �in� and �out� determine which chamber the signal isgiven to. Equation (A.11) says that mass, which �ows from one chamber,

172


must �ow into the next. It is further assumed that an ori�ce is isentropicand, thus, enthalpy �ow Eq. (A.12) between the chambers is constant.

Diesel engine

The engine component model (Fig.A.2(c)) essentially covers two e�ects:It works as a pump and a heater. How much mass is �pumped� throughthe engine depends upon the air density ρin = pin

RTinin the intake manifold

V2 and on how much volume can be pushed into the engine at speed nE.With given engine displacement VE, the amount of mass which �ows intothe engine, on average, during one cycle is

mE = 1/2 ρinVE(a0 + a1nE). (A.13)

The factor 1/2 in Eq. (A.13) accounts for the fact that air is aspirated onlyevery second cycle. The expression (a0 + a1nE) with parameters a0 anda1 accounts for the air-e�ciency, which varies in dependence upon theengine speed nE. The linear approximation as presented here is ratherrough but has shown to yield su�cient accuracy.

The air mass and enthalpy �ow signal given to the intake manifold are

min = −mEnE (A.14)

Hin = mincpTin. (A.15)

The outgoing mass �ow carries the aspirated air and the injected fuel massmf . Combustion does not change mass and, thus, the exhaust mass �owis

mout = −min + mf . (A.16)

Combustion heats the outgoing exhaust, however, only a maximum amountof fuel can combust given a certain air mass. With the stoichiometric ratiorstoic and the mass ratio of oxygen rO2 in the air the maximum fuel mass�ow which can partake in combustion is

mf,maxburn = nEmErO2

rstoic(A.17)

mf,burn = min(mf , mf,maxburn), (A.18)

where mf,burn is the actual fuel contributing to combustion. The heatgenerated by combustion is assumed to be proportional (parameter heatvalue c) to this fuel mass �ow. Thus, the outgoing enthalpy �ow is

Hout = −Hin + c mf,burn with c > 0.

173


Turbine

A turbine (Fig.A.2(e)) cannot be modelled with satisfactory accuracyusing only physical or thermodynamic laws, unless a three dimensionalcomputational �uid dynamics approach is chosen. This is especially truefor the two-stage turbocharged air-system where the HPT has a large oper-ation range and saturation e�ects must be taken into account. Therefore,this approach uses a data driven model of a turbine and proposes a newmethod for extrapolating manufacturer measurement maps.

Turbocharger manufacturers supply measurements of mass �ow ande�ciency at some operation points of both turbine and compressor. How-ever, this data only represents a small part of the possible operation rangeand, thus, has to be extrapolated. In [70] several methods for extrapola-tion are given and compared. Here, a new method of extrapolation issuggested, which yields the necessary accuracy for the two-stage turbo-charged diesel engine. It surpasses the models presented in [70] in termsof accuracy, especially concerning the calculation of turbine e�ciency.

A turbine's pressure ratio is given as Π = pinpout

, where �in� and �out�refer to the upstream and downstream values, respectively. The isentropicenthalpy �ow His is corrected by the e�ciency η to obtain the actualoutgoing enthalpy �ow Hout

His = −HinΠ1−κκ (A.19)

Hout = −Hin − η(Hin + His

). (A.20)

The incoming enthalpy �ow Hin and the mass �ows are de�ned as

Hin = cpTinmin < 0 (A.21)

min = −mout < 0. (A.22)

The power PT a turbine supplies to the shaft depends on how much it canlower enthalpy during expansion. One gets

PT = −Hin − Hout ≥ 0. (A.23)

With equations (A.19)-(A.23), the ingoing and outgoing signals from theturbine component model are de�ned if the mass �ow mout and e�ciencyη can be determined.

174


Extrapolation of mass �ow. Manufacturer data for turbines are givenin �reduced� mass �ow and �corrected� speed, denoted by the subscript�red� and �corr� and are given by

mred = mout

√Tin

pin(A.24)

ncorr = n

√Tref

Tin, (A.25)

where Tref is a manufacturer supplied parameter and represents the tem-perature at which the measurements were recorded. Reduced mass �owmred is given over pressure ratio Π and is assumed to behave similarlyas mass �ow through an ori�ce described by Eq. (A.10). The followingfunction approximates mred:

mred =a(ncorr) ·

√

γγ−1

(Π−2/γ − Π

γ+1−γ

), for Π ≤ Πcrit;(

2γ+1

) 1γ−1

√γγ+1 , for Π > Πcrit.

with Πcrit =(

2γ + 1

) −γγ−1

(A.26)

and with Π = Π−b(ncorr). The functions a(ncorr) and b(ncorr) are de�nedas

a(ncorr) = a1 − a2n2corr

b(ncorr) = b1n2corr.

(A.27)

The four parameters γ, a1, a2 and b1 can be determined by �tting thefunction Eq. (A.26) to measurements. If a variable nozzle turbine (VNT)is used, the four parameters can be assumed to be a function of the noz-zle position and can be approximated by splines. Figure A.4 shows anexample of an extrapolated turbine mass �ow at di�erent speeds ncorr.

Extrapolation of e�ciency. E�ciency is plotted versus the blade-speed ratio Θ = u/c, where u is the circumferential speed of the rotorand c is the velocity of the �uid �owing out of the turbine

Θ =u

c=

πdt nred/60√2cpTref

(1−Π

1−κκ

) . (A.28)

175


1.5 2 2.5 3

0.1

0.11

0.12

0.13

0.14

0.15

0.16

Π

m[k

g√

Ksbar

]

ncorr

Fig. A.4: Reduced mass �ow mred over pressure ratio π at di�erentspeeds ncorr. The circles show the available data froma manufacturer supplied map and the lines show the be-haviour of the model.

The parameter dt signi�es the diameter of the turbine's rotor.In most publications, e�ciency η over blade-speed ratio Θ is approx-

imated by a parabola. However, this approach yields unsatisfactory re-sults outside of the provided measurement data since e�ciency η must begreater than zero at rotor speed u = 0⇔ u/c = Θ = 0 for c ≥ 0. Existingturbine models make a fundamental mistake by allowing the e�ciency atΘ = 0 to be zero (e. g., [37]) or even negative (e. g., [70]). If this werethe case, a turbine would never start rotating when the engine is started.Thus, a novel approximation is presented here. E�ciency is approximatedby

η =a(ncorr)

b1 · (Θ− c(ncorr))2 + 1

+ d1 (A.29)

with

a(ncorr) =a1

a2 (ncorr − a3)2 + 1c(ncorr) = c1 + c2ncorr,

(A.30)

where the parameters a1, a2, b1, c1, c2 and d1 are determined by �tting thefunction (A.29) to measurements. If a VNT is used, the six parameters

176


can be assumed to be a function of the nozzle position and can be ap-proximated by splines. Figure A.5 shows an example of an extrapolatedturbine e�ciency at di�erent speeds ncorr.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.2

0.3

0.4

0.5

0.6

0.7

Θ

η

ncorr

Fig. A.5: E�ciency η over blade-speed ratio Θ at di�erent speedsncorr. The circles show the available data from a manu-facturer supplied map and the lines show the behaviour ofthe model.

Compressor

A compressor's pressure ratio is given as Π = poutpin

. The isentropic enthalpy

�ow His is corrected by the e�ciency η to obtain the actual outgoingenthalpy �ow Hout

His = −HinΠκ−1κ (A.31)

Hout = −Hin −1η

(Hin + His

). (A.32)

The incoming enthalpy �ow Hin and the mass �ows are de�ned as

Hin = cpTinmin < 0 (A.33)

min = −mout < 0. (A.34)

The power PC ≤ 0 a compressor supplies to the shaft depends on howmuch it can lower enthalpy during expansion. One gets

PC = Hout + Hin ≤ 0. (A.35)

177


With Eqns. (A.31)-(A.35) the ingoing and outgoing signals from the tur-bine component model are de�ned if the mass �ow mout and e�ciency ηcan be determined.

Extrapolation of mass �ow. Compressor mass �ow is given �cor-

rected�, that is mcorr = m√

TinTref

prefpin

. The corrected mass �ow is plotted

over the pressure ratio Π and is approximated by

mcorr = a1 tan( −Πb(ncorr)

π

2

)+ c(ncorr), (A.36)

with the functions

b(ncorr) = b1 tanh(b2ncorr + b3) + b4

c(ncorr) = c1n2corr + 1,

(A.37)

where the parameters a1, b1, b2, b3, b4 and c1 are determined by �tting thefunction Eq. (A.36) to measurements. Figure A.6 shows an example of anextrapolated compressor mass �ow.

0 0.05 0.1 0.15 0.21

1.5

2

2.5

3

m [kg/s]

Π

ncorr

Fig. A.6: Corrected mass �ow mcorr over pressure ratio Π at dif-ferent speeds ncorr. The circles show the available datafrom a manufacturer supplied map and the lines show thebehaviour of the model.

Extrapolation of e�ciency. A compressor's e�ciency is plotted overa normalised �ow rate Φ with

Φ =mcorr

prefRTref

π4 d

2cu

with u = π/60dcncorr.

178


The diameter dc of the compressor rotor is given as a parameter.Similar to the turbine, most publications suggest a parabola. Again,

this approach yields unsatisfactory results. Here, it is proposed to modele�ciency by a lower branch of a hyperbola

η = −

√a2b2 + a2(Φ− c)2

b+ d, (A.38)

which is rotated by angle α using the substitution

Φ 7→ cos(α)Φ− sin(α)η (A.39)

η 7→ sin(α)Φ + cos(α)η. (A.40)

The parameter a is assumed constant whereas the other variables b, c, dand α are functions of ncorr. They can be approximated by splines. FigureA.7 shows an example of an extrapolated compressor e�ciency.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.60.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

Φscaled

η

ncorr

Fig. A.7: E�ciency η over normalised �ow rate Φ at di�erent speedsncorr. The circles show the available data from a manu-facturer supplied map and the lines show the behaviour ofthe model.

Figures A.8 and A.9 show the resulting one-stage and two-stage air-system model, respectively, composed using the component models. Thearrows indicate the direction of signal �ows. Signal �ow direction is notnecessarily identical to �uid �ow direction. Bold arrows indicate vectorsignals.

Figures A.8 and A.9 present the model structure of a one-stage andtwo-stage turbocharged air-system. Together with the component modelsintroduced in Section A.3 and matching initial conditions, Fig. A.8 andA.9 present simulation models.

179


2V

3V

4V 5V

0V

00,Tp

55,Tp44,Tp

inExh,m&

inExh,H&

1V

vntu

TP

44,Tp33

,Tp

inT,m&

inT,H&

outT,m&

outT,H&

nCP

22,Tp

outC,m&

outC,H&

11,Tp

inC,m&

inC,H&

Enfm&

22,Tp

outE,H&

outE,m&

inE,H&

inE,m&

n

Fig. A.8: Implementation and signal direction of a one-stage turbo-charged air-system. Bold arrows indicate vector signals.

A.4 Parameter Estimation and Model Veri�cation

It is desired for the resulting model of an air-system to represent the be-haviour of a speci�c real air-system. Parameter estimation of a continuous-time dynamic model deals with �nding the values for some (or all) of themodel parameters such that the simulation model (with appropriate ini-tial conditions) represents the behaviour of the real plant. In this work,the criterion used to judge how well the model and real plant �t, is basedon the least squares error between the two. Since this work is not con-cerned with the details of the various optimisation techniques, the readeris referred to [17, 66, 77, 97] and the references therein.

In order to estimate the parameters of the model, at least one mea-surement of the real system is required. Since such a measurement can berecorded at a test bed, additional sensors can be implemented that shouldprovide data of

• the pressures p0, . . . , p6, the temperature T3 and optionally the tem-peratures T0, . . . , T6 in all pipes;

• the turbocharger speed(s) n or nLP, nHP;

180

A.4. PARAMETER ESTIMATION AND MODEL VERIFICATION

• the mass�ow through the �rst compressor mC,out or mLPC,out;

• and the stimuli of the plant, namely the engine speed nE, fuel mass-�ow mf as well as the chosen VNT position uvnt or the bypass crosssections AHPC, AHPT, ALPT.

The following explains the procedure of parameter identi�cation of an air-system. This procedure was found to work well on several engines. On astandard PC, the following algorithm takes a maximum of two hours tocomplete the parameter estimation for any air-system.

Algorithm A.1 (Parameter estimation of an air-system model[xvi, xvii]). .

Given: A test bed measurement of the real plant and the model structureshown in Fig. A.8 or Fig. A.9.

Step 1: Identify the engine component model using steady-state signals.At steady-state conditions, the recorded mass�ow mC,out or mLPC,out

is equal to the �uid �ow −mE,in into the engine. The temperatureT3 at steady-state results from engine combustion. Thus, the steady-state measurements su�ce to identify all parameters of the enginecomponent model.

Step 2: Identify the turbine and compressor models. The turbochargermanufacturer provides the mass�ow and e�ciency maps of the com-pressor(s) and turbine(s). With these maps, the turbine and com-pressor component models can be identi�ed. The inertia of a tur-bocharger is also given in the speci�cations.

Step 3: Identify the remaining parameters in the fresh air path. Themodel structure can by split into two parts: The fresh air path andthe exhaust path. This can be done by regarding the measured tur-bocharger speed(s) as stimuli to the fresh air path. Therewith, thefresh air path is independent of the exhaust path. It is proposed toestimate the remaining parameters of the fresh air path by using nu-merical parameter estimation methods. It was found that a GeneticAlgorithm [15] works well.

Step 4: Identify the remaining parameters in the exhaust path. Similarlyto Step 3, with the turbocharger speed(s) as stimuli and the fullyidenti�ed fresh air path, the exhaust path component models can nowbe identi�ed using numerical parameter estimation methods.

181


Step 5: Identify turbocharger shaft(s) using the full model. When com-bining the identi�ed fresh air path with the identi�ed exhaust path,the turbocharger shafts account for the coupling. The parametersof the shaft component model(s) can be estimated using numericalestimation.

Result: A model of either the one-stage or two-stage turbocharged air-system with a good �t to measurements.

Figure A.10 shows the comparison of simulation and measurement ofa one-stage turbocharged engine. Only a small selection of the availablesignals are shown. However, the portrayed accuracy of the simulationis representative for all signals. The stimuli shown in the �rst subplotof Fig. A.10 (i. e., engine speed nE, fuel mass �ow mf and VNT positionuvnt) was chosen such that the whole operation range of the engine wascovered. The second subplot of Fig.A.10 shows the comparison of simu-lated (dashed) and measured (solid) boost pressure p2. Similarly, subplotsthree and four portray the mass�ow through the compressor mC,out (whichis equal to the mass�ow into the system) and the turbocharger speed n,respectively. Overall, the resulting model accuracy is good in both dy-namical as well as steady-state behaviour. It is well suited for controllerdevelopment as it accurately represents the behaviour of the plant at thetest bed and is modelled using di�erential equations.

Figures A.11 to A.13 show the comparison of a two-stage air-systemmodel and the test bed measurements. Again, dashed lines indicate sim-ulated and solid lines indicate measured signals. The available test bedmeasurement from the two-stage turbocharged diesel engine lacks someimportant data. First, the engine is not stimulated in its whole operatingrange. Second, important pressure signals, namely p5 and p6 were notavailable. Third, the bypass of the HPC was not closing correctly andthe bypass cross sections were measured poorly. Unfortunately, bettermeasurement data is not available for this work. Despite the bad qual-ity of the available measurement data, the resulting model is chosen forcontroller development.

The computational e�ort for simulating the models is low: On a 3GhzPentium, the one-stage simulation model needs below one second compu-tation time for about 500 seconds of simulation time. The two-stage modelneeds about three seconds for the same simulation time interval. There-fore, one concludes that it is possible to simulate both models in real-timeon a controller that is about 100 times slower than a 3Ghz Pentium.

182

A.5. MODEL SIMPLIFICATION

A.5 Model Simpli�cation

This section is concerned with reducing the order of the air-system models.Model reduction is an important step, because a model of low order greatlysimpli�es nonlinear IMC design and reduces the order of the resulting IMCcontroller.

The order of a coupled air-system model results from the number ofchamber component models (two states per chamber, see equations (A.2)and (A.3)) and the number of shafts (one state per shaft, see Eq. (A.9)).This yields a ninth order model for the one-stage turbocharged air-system(cf. Fig.A.8) and a twelfth order model for the two-stage turbochargedair-system (cf. Fig.A.9).

Considering the dynamics of the storage component models with re-alistic parameterisation, it becomes clear that the chamber models havesigni�cantly faster dynamics than the shaft models: Tests using the sim-ulation models have shown that the volume of any chamber model hasvirtually no e�ect on the dynamics of the simulation result in the consid-ered low-frequency domain. This holds for chamber volumes lower thanabout �fty liters, where �fty liters is an unrealistically high number forthe volume of a chamber in any air-system. In conclusion, the air-systemlow-frequency dynamics solely stem from the dynamics of the shaft com-ponent models. Since this work is only interested in the low-frequencybehaviour of the air-system, it is proposed to reduce the model order byreplacing either one or both states of the chamber component models withalgebraic relationships.

A.5.1 Simpli�ed Chamber Model

Steady-state temperature dynamics in the plenum chamber

During driving, �uid is constantly �owing through the pipes in the samedirection. This simpli�cation assumes that the outgoing �uid's temper-ature is equal to the incoming �uid's temperature. Thus, the simpli�edchamber component model is described by

T =∑

in Hi,in

cp ·∑

in mi,in(A.41)

p =κ− 1V

∑i

Hi, p(0) = p0, (A.42)

183


where the subscript �in� indicates that only incoming �uid �ow is con-sidered. Thus, the terms

∑in Hi,in and

∑in mi,in mean the sum over

all positive enthalpy �ows Hi (with Hi > 0) and positive mass �ows mi

(with mi > 0), respectively. By replacing the original chamber componentmodel, described by Eqns. (A.2)-(A.4) with Eqns. (A.41) and (A.42),each plenum chamber will consist of only one state, signi�cantly reducingthe number of states in the implementation of an air-system model.

Steady-state model of a plenum chamber

This simpli�cation aims at replacing the pressure dynamics (A.42) by analgebraic relationship. Together with the simpli�ed temperature calcula-tion (A.41) this step results in a completely algebraic model of a plenumchamber [xvi].

It is assumed that temperature and pressure in a chamber are directlyrelated to each other by the polytropic relationship

p =(T

c

) kk−1

(A.43)

where c and k are parameters which vary for each chamber and have tobe determined separately for each chamber.

The assumption in Eq. (A.43) only holds in the fresh air path but notin the exhaust path. This is due to the e�ect of fuel combustion on thepressure in a chamber in the exhaust path, for which no good algebraicapproximation was found. In the fresh air path, this simpli�cation yieldsa good approximation of the behaviour of a chamber.

A.5.2 Model Simpli�cation Procedure

The following develops the simpli�cation of a two-stage turbocharged air-system and applies to Fig.A.9. With only minor changes, the steps intro-duced below are also applicable for the one-stage turbocharged air-systemgiven in Fig.A.8.

Simpli�cation of the fresh air path

The fresh air path does not model the turbocharger dynamics and ratherassumes the turbocharger speeds as given exogenous signals. It followsthat the chambers are the only dynamic components in the fresh air path

184

A.5. MODEL SIMPLIFICATION

and that their dynamics can be neglected, one concludes that the com-plete fresh air path from environment up to and including the enginecould essentially be represented algebraically. However, such an approachyields algebraic loops which have to be solved numerically. Therefore, thefollowing develops a simpli�cation of the fresh air path which keeps theboost pressure p2 as a state and does not yield an algebraic loop:

1. The boost pressure p2 and boost temperature T2 are calculated bythe equations (A.42) and (A.43), which means that p2 remains astate. This gives the signals p2 and T2.

2. From Eq. (A.1b), the density in the chamber V2 is available by ρ2 =p2/(RT2) and the engine model can compute the �uid and enthalpy�ows mE,out or HE,out, respectively.

3. The �uid �ow through the engine mE,out, without the injected fuelmass mf is assumed to �ow through the LPC:

mLPC,out ≈ mE,out − mf

With the output mLPC,out of the compressor model from the equa-tion above, the pressure ratio Π of the compressor can be computedfrom Eq. (A.36). With the known ambient pressure p0, the pressurep1 results from p1 = Πp0. Now, all input signals of the LPC areknown and its output signal HLPC,out can be computed.

4. The temperature T1 follows using Eq. (A.41).

5. All input signals for the HPC are known and its remaining outputsignals mHPC,out and HHPC,out follow from the component model.

6. Finally, all inputs signals for the HPC bypass are known and its out-puts mOHPC,out and HOHPC,out follow from the component model.

This results in a fresh air path, consisting of a single state (namely p2)instead of four states. Moreover, no algebraic loop is present.

Simpli�cation of the exhaust path

In the exhaust path, a similar reduction of the states is not possible,since Eq. (A.43) does not hold. Thus, in order to obtain a reduced modelwithout any algebraic loops, more states will have to be retained.

185


1. The temperature T3 in the exhaust manifold follows from Eq. (A.41)using the known output signals mE,out, HE,out from the engine. Theexhaust back pressure p3 is kept as a state and computed accordingto Eq. (A.42). Therewith, the signals p3 and T3 are given.

2. In order to obtain an input-a�ne model of a two-stage turbochargedengine, the chamber V4 is not simpli�ed. Thus, both p4 and T4

remain states.

3. Experience shows that the temperature T5 varies only slightly. Thus,T5 is assumed to be constant. Finally, with the assumptions that thepressure in the exhaust pipe p5 is higher than the ambient pressurep5 > p6 and that the �ow out of the exhaust pipe is approximatelyequal to the mass�ow out of the engine

−mExh,in ≈ mE,out,

the pressure p5 can now be computed from Eq. (A.10).

As a result, the simpli�ed exhaust path consists of three states (p3, p4, T4)instead of the original six.

The coupling of the simpli�ed fresh air path with the simpli�ed exhaustpath is done using the shaft component models which are not changed.In summary, the simpli�ed model of a two-stage turbocharged air-systemconsists of six states, namely p2, p3, p4, T4, ωLP, and ωHP. Thus, six statesof the original model have been removed.

A.6 Control Design Models of One- and Two-StageTurbocharged Diesel Engines

A.6.1 Simpli�ed Model of a One-Stage Turbocharged DieselEngine

A similar simpli�cation as the one developed for the two-stage turbo-charged air-system can be employed to simplify the one-stage turbochargedair-system (cf. Fig.A.8). An extensive review of the simpli�cation can befound in [76] and will not be treated here in detail.

This simpli�ed model is used as IMC model Σ and is the basis fordeveloping a feedforward controller. It is described by using the statevector x = [ω, p2]T and the input u = uvnt, where ω is the speed of theturbocharger and p2 is the boost pressure. The input u is a function of the

186

A.6. CONTROL DESIGN MODELS OF ONE- AND TWO-STAGETURBOCHARGED DIESEL ENGINES

nozzle position of the variable nozzle turbine (VNT). The engine speed nE

and the injected fuel mass mf are considered to be measured disturbancesdm = [nE, mf ]T . The model Σ is given by

Σ : x1 =k1k2

x1(k3(dm) + k6(dm)k4x2)ϕ1(x2)− k1k5ϕ3(x2)

x1ϕ2(x1, x2)

− k1k2

x1(k3(dm) + k6(dm)k4x2)ϕ1(x2)u

x2 =ϕ4(x2)k7

ϕ2(x1, x2)− k9

k7k6x2

(A.44a)

with the initial condition x(0) = x0 and the output equation

y =x2 = p2 (A.44b)

with

ϕ1(x2) =k9

k4+

k17(dm)k14(dm) + k4x2

, T3

ϕ2(x1, x2) = k8k10x

21 − ϕ3(x2)k11x1

, mC,out

ϕ3(x2) = k16

((x2

pamb

)k12− 1

),HC,out + HC,in

mC,out

ϕ4(x2) = k15ϕ3(x2) + k13 , T2,

where the coe�cients ki are system parameters (e. g., diameters, inertiaof the turbocharger, etc.), pamb is the ambient pressure and the ϕj(·) aresome nonlinear functions of the states.

Figure A.14 shows the simpli�ed model versus the original model andmeasurement data. Although the sixth order model was reduced to secondorder, the accuracy is still good. In conclusion, the reduced-order model(6.1) can be used for controller design.

A.6.2 Simpli�ed Model of a Two-Stage Turbocharged DieselEngine

The six state variables of the reduced model of a two-stage turbochargedengine are explained in Tab.A.1 and Tab.A.2 introduces the parameters.

187


Tab. A.1: State variables of the simpli�ed plant model of the two-stage turbocharged engine (cf. Fig. A.9).

x1: Boost pressure p2

x2: Exhaust back pressure p3

x3: Pressure between turbines p4

x4: Fluid mass m4 contained in pipe V4

x5: Speed ωLP of LP shaftx6: Speed ωHP of HP shaft

Tab. A.2: Nomenclature for the two-stage turbocharged air-system(cf. Fig. A.9).

dm: Measured disturbances, engine speed and fuel mass �ow(dm = [nE, mf ]T )

κi: Chamber parameter polytropic exponent of chamber iVi: Chamber parameter volume of chamber i

JLP, JHP: Shaft parameter inertia of low-pressure orhigh-pressure shaft, respectively

dLP, dHP: Shaft parameter damping of low-pressure orhigh-pressure shaft, respectively

The reduced order, input-a�ne model Σ, with which the IMC con-troller is developed, is given by

Σ : x = f(x,dm) +G(x,dm)u, x(0) = x0, x ∈ R6 u ∈ R3 (A.45a)

y = h(x) =[x1, x2, x3

]T, y ∈ R3, (A.45b)

with model output y. The vector �eld f is de�ned as

f(x,dm) =

κ2−1V2

(HHPC,out + HE,in

)κ3−1V3

(HE,out + HHPT,in

)κ4−1V4

(HHPT,out + HLPT,in

)mHPT,out + mLPT,in

1JLPωLP

(PLPT − PLPC − dLPω

2LP

)1

JHPωHP

(PHPT − PHPC − dHPω

2HP

)

(A.45c)

188

A.6. CONTROL DESIGN MODELS OF ONE- AND TWO-STAGETURBOCHARGED DIESEL ENGINES

and the matrix G has the following structure:

G(x,dm)=

g11(x,dm) 0 0

0 g22(x,dm) 00 g32(x,dm) g33(x,dm)0 g42(x,dm) g43(x,dm)0 0 00 0 0

· 1/m2 (A.45d)

with

g11 =κ2 − 1V2

H∗OHPC,out

g22 =κ3 − 1V3

H∗OHPT,in

g32 =κ4 − 1V4

H∗OHPT,out

g33 =κ4 − 1V4

H∗OLPT,in

g42 = m∗OHPT,out

g43 = m∗OLPT,in,

(A.45e)

where the asterisk `∗' indicates that the respective �ow is to be calcu-lated assuming a unitary (i. e., 1m2) open cross section of the respectivebypass. For example, H∗OHPC,out presents the output enthalpy �ow ofthe high-pressure compressor bypass with a cross section of 1m2. FromHOHPC,out = H∗OHPC,out ·1/m2 ·u1 one �nds that the unit correction 1/m2

is necessary since the input u1 has units of m2.All enthalpy �ows H(·) and mass �ows m(·) in Eqns. (7.4c)-(A.45e) are

given from the coupling component models as introduced in Section A.3.Thus, they are functions dependent upon the states of their neighbouringstorage component models and upon the measured disturbances dm =[nE, mf ]T .

This leads to an interesting result: The model structure shown inFig.A.9 and the model equations (7.4) are both independent of the speci�csof the coupling or storage component models. Thus, di�erent models ofturbines, compressors, ori�ces, chambers do not change the model struc-ture as long as no additional states are introduced.

The quality of the reduced-order model is shown in Fig.A.15 by com-paring simulation results of the full and reduced-order model. The reduced-order model is such a good approximation that e. g., the boost pressure

189


never deviates more than 0.02 bar from the full order model. In conclusion,the model (7.4) is feasible for controller development.

A.7 Summary

This chapter has introduced a compositional model library for turbo-charged air-systems which is able to represent their low-frequency be-haviour well. With the library and the presented algorithms of parameteridenti�cation and model reduction, it is possible to obtain coupled modelsfor virtually any turbocharging solution and use the resulting model forcontroller design.

The library components are divided into storage and coupling mod-els where only storage models have a dynamic behaviour. Thus, storagemodels are represented by di�erential equations and coupling models arerepresented by algebraic equations. The connection philosophy allows tobuild air-systems by alternatingly connecting storage models to couplingmodels. Therefore, an algorithm has been introduced which simpli�esparameter estimation of a model, given an appropriate test bed measure-ment and the turbocharger maps. The test bed measurement must containall pressures, the temperature in the exhaust manifold, the turbochargerspeeds and the stimulation signals of engine speed, fuel mass �ow andthe VNT position or bypass cross section signals. The main idea of theparameter identi�cation is to split the air-system into the fresh air pathand the exhaust path by using the measured turbocharger speeds as stim-uli. Therewith, the fresh air path and the exhaust path can be identi�edseparately, reducing the number of parameters to be identi�ed at once.

The resulting model quality of both a one-stage and a two-stage tur-bocharged air-system is very good. However, the models are of ninth andtwelfth order for the one- and two-stage model, respectively. It is pro-posed to reduce the model order by using simpli�ed component modelsfor the plenum chambers since their dynamics can be neglected. An algo-rithm for model reduction is given and the resulting models show almostno deviation from the models they originated from. Finally, it is proposedto use the simpli�ed models for a design of an internal model controller.

190

A.7. SUMMARY

LPC

LPT

LPT

A

2V

3V

4V

5V

6V

0V

00,Tp

LP

nLPC

P

LP

nLPT

P

55,T

p4

4,Tp

55,Tp

inLPT,

m&in

LPT,

H&out

LPT,

m&out

LPT,

H&

out

OLPT,

m&

out

OLPT,

H&

66,T

p5

5,T

p

inExh,

m&in

Exh,

H&

11,T

p

out

LPC,

m&out

LPC,

H&

1V

44,Tp

inOLPT,

m&

inOLPT,

H&

HPT

HPT

A

HP

nHPT

P

44,T

p3

3,T

p

44,Tp

inHPT,

m&in

HPT,

H&out

HPT,

m&out

HPT,

H&

out

OHPT,

m&out

OHPT,

H&

33,Tp

inOHPT,

m&in

OHPT,

H&

HPC

HP

nHPC

P

22,T

p

inHPC,

m&in

HPC,

H&

11,T

p

out

HPC,

m&out

HPC,

H&

HPC

A

11,T

p

inOHPC,

m&in

OHPC,

H&

22,Tp

out

OHPC,

m&

out

OHPC,

H&

Enfm&

22,Tp

out

E,

H&

out

E,

m&

inE,

H&

inE,

m&

Fig. A.9: Implementation and signal direction of a two-stage turbo-charged air-system. Bold arrows indicate vector signals.

191


0

1

2

3

4

5

1.2

1.4

1.6

1.8

2

2.2

Pre

ssur

ep

2[b

ar]

0 50 100 150 200 250 300 350

1

1.5

2

time [s]

Turb

ocha

rger

spee

dn

[100

00ra

d/s]

40

60

80

100

120

Mas

sflow

mC

,out[g

/s]

nE [1000 rpm]mf [g/s]uvnt

Fig. A.10: Model validation of a one-stage turbocharged diesel en-gine. Dashed lines indicate simulation results and solidlines refer to measurement.

192

A.7. SUMMARY

0 2 4 6 8

050

100150

200250

300350

400450

5000 2 4 6 8 10 12 14

time [s]

Engine

speedn

E[1000

rpm]

Fuelm

assflowm

f[g/s]

AH

PC

[cm2]

AH

PT

[cm2]

ALPT

[cm2]

Fig. A.11: Model validation of a two-stage turbocharged air-system(1 of 3). Engine speed nE and fuel mass �ow mf areshown.

193


0 50

100

150

200MassflowmHPC,out [g/s]

1

1.5 2

2.5

Pressurep2 [bar]

050

100150

200250

300350

400450

5001

1.5 2

2.5 3

time [s]

Pressurep3 [bar]

Fig. A.12: Model validation of a two-stage turbocharged air-system(2 of 3). The mass�ow through the LPC mLPC,out, theboost pressure p2 and the exhaust back pressure p3 areshown. Dashed lines indicate simulation results and solidlines refer to measurement.

194

A.7. SUMMARY

0

0.5 1

1.5 2

x 104

Turbocharger speednHP [rad/s]

050

100150

200250

300350

400450

5000

0.5 1

1.5 2

x 104

time [s]

Turbocharger speednLP [rad/s]

Fig. A.13: Model validation of a two-stage turbocharged air-system(3 of 3). The turbocharger speeds nHP and nLP areshown. Dashed lines indicate simulation results and solidlines refer to measurement.

195


0.5

1

1.5

2

2.5Full ModelSimplified ModelMeasurement

0 2 4 6 8 10 12 140

0.5

1

1.5

2x 10 5

Full ModelSimplified ModelMeasurement

time (s)

Boost

Pressurep2[bar]

Turbochargerspeedω

[rpm]

Fig. A.14: Accuracy of the simpli�ed model of a one-stage turbo-charged air-system.

196

A.7. SUMMARY

1

1.2

1.4

1.6

1.8

2

2.2

Boo

stpr

essu

rep

2[b

ar]

160 170 180 190 200 210 220 230 240 250 260

1

1.2

1.4

1.6

1.8

2x 10

4

time [s]

HPC

spee

dω

HP

[rad

/s]

Reduced Model

Full Model

Reduced Model

Full Model

Fig. A.15: Accuracy of the simpli�ed model of a two-stage turbo-charged air-system.

197


198

B. OPEN PROBLEMS

B.1 Two-Degrees-of-Freedom IMC

B.1.1 Linear Case

A two-degrees-of-freedom control structure may improve results if bothdisturbance rejection and command tracking is desired. Therefore, a twodegrees-of-freedom IMC control structure is reviewed and its design isbrie�y discussed. Note that this work mainly focuses on a one-degree-of-freedom design. The reader is referred to [9, 72] for more detail ontwo-degrees-of-freedom IMC design.

A two-degrees-of-freedom IMC structure is shown in Fig. B.1. The

w u

Σ~

Σdy

y~

Q

FBQ

−

−

Fig. B.1: Two-degrees-of-freedom IMC structure.

control error is de�ned as e = w − y. From Fig. B.1 one gets

e(s) =I − Σ(s)QFB(s)

I +QFB(s)(

Σ(s)− Σ(s))d(s)

−

I − Σ(s)Q(s)

1 +QFB(s)(

Σ(s)− Σ(s))w(s).

(B.1)

Considering an exact model (Σ(s) = Σ(s)), one �nds from Eq. (B.1) the

B. OPEN PROBLEMS

relationship

e(s) =(I − Σ(s)QFB(s)

)d(s)−

(I − Σ(s)Q(s)

)w(s). (B.2)

From the two equations above, one concludes that Q(s) is to be designedfor reference tracking and the feedback controller QFB(s) should be de-signed for disturbance rejection. Moreover, both can be designed indepen-dently of each other in the presence of small modelling errors.

Design. It is proposed to design Q(s) as discussed in Section 2.2 withthe exception that the designer does not need to consider a gain restrictionon measurement noise. Further, QFB(s) must have the property QFB(0) =1 for the closed-IMC loop to guarantee zero steady-state o�set. It isproposed to design QFB(s) as a low-pass �lter

QFB(s) =1

(s/λFB + 1)k, (B.3)

where both k and λFB are to be chosen by the designer. The following ruleof thumb applies: A slow feedback partQFB(s) (i. e., high k and small λFB)results in greater closed-loop robustness but slower command tracking inthe presence of modelling errors and disturbances. A fast feedback part(i. e., small k and high λFB) results in aggressive disturbance rejectionand good command tracking, but also increases noise ampli�cation andthe risk of instability.

Remark B.1. Assume a two-degrees-of-freedom IMC, where Q(s) andQFB(s) are chosen to be identical (i. e., Q(s) = QFB(s)). Then theresulting two-degrees-of-freedom IMC closed-loop behaviour is identical

to the one-degree-of-freedom IMC closed-loop behaviour (as shown inFig. 2.1) with IMC controller Q(s).

B.1.2 Nonlinear Case

For the linear IMC, a two-degrees-of-freedom structure has been reviewedin Section B.1.1. For nonlinear IMC, it is proposed to exploit the separateimplementation of IMC Filter F and right inverse Σr (cf. Fig. 3.6).

Figure B.2 shows the proposed two-degrees-of-freedom IMC structurefor nonlinear systems. The structure in Fig. B.2 is similar to the two-degrees-of-freedom structure for linear IMC in Fig. B.1. The two di�er in

200

B.2. AD HOC NON-MINIMUM PHASE IMC DESIGN FOR FLAT SYSTEMSUSING A MINIMUM PHASE MODEL

Σy

y~

d

w Fd~y

(r)d~y

urΣ~

FBFL

FB~y

(r)FB~y

FFy~

(r)FFy~

−−

−

Fig. B.2: Two-degrees-of-freedom structure for a nonlinear IMC.

the location of the model inverse Σr. Due to the nonlinearity of the rightinverse Σr, a summation of control inputs as presented in Fig. B.1 is nota feasible step since the principle of superposition does not hold in thiscase.

With the implementation in Fig. B.2, it is only necessary to imple-ment the right inverse once. The blocks F and FFB both represent state-variable-�lters (cf. Fig. 3.7). However, their parameterisation may di�erfrom each other. If both �lters are chosen to be identical (F = FFB) thenthe two-degrees-of-freedom structure in Fig. B.2 behaves exactly as theone-degree-of-freedom IMC structure from Fig. 3.8.

Assuming an exact model Σ = Σ and assuming the presence of outputdisturbances (i. e., y = y + d) one �nds

y = Fw + (I − FFB)d. (B.4)

Hence, the �lter F can be designed for feedforward control (as in one-degree-of-freedom) and FFB is to be designed for disturbance rejection.Both �lters need to be of order r and need to be implemented as SVF.Note that the two �lters can be designed independently of each other.

Since this work focuses on the one-degree-of-freedom structure, thetwo-degrees-of-freedom design is not discussed further.

B.2 Ad Hoc Non-Minimum Phase IMC Design forFlat Systems using a Minimum Phase Model

This section presents a ad hoc method by which NMP systems can becontrolled using IMC. It is regarded as a �hands-on� (ad-hoc) solutiondue to the lack of a generally applicable algorithm.

201

B. OPEN PROBLEMS

Main idea. It is proposed to deal with NMP nonlinear systems in thesame manner as it was introduced for linear systems in Section 2.2.2.Hence, it is proposed to do the following:

• Invert only a part ΣMP of the model Σ which leads to a stable inverse.Hence, the part ΣMP is MP.

• The steady-state gain of the part ΣMP to be inverted must beequal to the steady-state gain of the NMP model Σ, i. e., g(Σss) =g(ΣMP,ss).

• In the IMC structure, the internal model Σ should not be replacedby a direct connection of the �lter output yd.

It is proposed to obtain the IMC controller Q by

Q = ΣrMP F , (B.5)

where the IMC �lter F is designed as proposed in Section 3.5.1. Figure

w u

Q

Σ~

Σ

dy

y~

w~ F rMP~Σ−

−

Fig. B.3: IMC structure for NMP nonlinear systems where the rightinverse Σr

MP has been derived from a minimum-phase ap-

proximation of the non-minimum phase model Σ.

B.3 shows the proposed IMC structure for the control of NMP nonlinearsystems.

Interpretation. The altered model ΣMP needs to be chosen by thedesigner. The result is an inverse Σr

MP that is stable, but

Σ ΣrMP 6= I

holds. That is, the MP inverse ΣrMP is not a perfect inverse of the (full)

NMP model Σ. As in the linear case, this approach does not changethe properties of nominal stability (Property 3.1) and steady-state o�set

202

B.2. AD HOC NON-MINIMUM PHASE IMC DESIGN FOR FLAT SYSTEMSUSING A MINIMUM PHASE MODEL

(Property 3.3). Naturally, the IMC design in Eq. (B.5) does in�uence theproperty of robust stability (Property 3.4), which is to be expected, sincethis is also true for linear systems. As in the linear case, the closed loophas an NMP I/O behaviour (cf. Eq. (2.29)). In this case, however, theNMP I/O behaviour may be nonlinear.

Unfortunately, the proposed procedure is di�cult to follow, since nogeneral algorithm can be given for the removal of the NMP behaviour:In the nonlinear case, a system representation as a transfer function doesnot exist. Thus, removal of the NMP behaviour of a nonlinear model isnon-trivial.

Application to �at NMP models. A �atness-based IMC (see Theo-rem 4.1) for an NMP system yields an unstable IMC controller becausethe solution, obtained from the di�erential equation (4.10b) (called Fy→z),is unstable. Thus, the following method is proposed:

If the solution of Fy→z is unstable, one should change Fy→z such that

1. the altered di�erential equation Fy→z,MP yields a stable solution,

2. the steady-state gain of Fy→z is retained (i. e., Fy→z,ss =Fy→z,MP,ss), and

3. the order of Fy→z is retained.

The resulting inverse ΣrMP is stable and has the same steady-state gain as

Σ. Thus, the properties of nominal stability and zero steady-state o�setof the IMC structure still hold.

Although it seems unnatural to directly alter the right inverse of amodel, it is admissible since it is equivalent to inverting a di�erent (mini-mum phase) model. Unfortunately, the method of how the operator Fy→z

is changed to obtain a stable Fy→z,MP is completely left to the engineer.Deriving a generally applicable algorithm to do this task is an unsolvedproblem.

The following presents an example of this method to �at systems. Inthe example, a linear system is discussed. Nevertheless, the steps proposedfor nonlinear systems are followed to demonstrate their feasibility and togive the reader a means for comparison to the linear case. A nonlinearexample can be obtained by altering the following model (B.6) accordinglyand then proceed by taking the same steps.

203

B. OPEN PROBLEMS

Example B.1 (Flatness-based IMC of an NMP system):Consider the linear plant model

x =

[0 1−5 −2

]x+

[01

]u (B.6a)

y =[5 −5

]x (B.6b)

with poles at p1/2 = −1± 2j and a RHP zero at z = +1. Thus, the modelis stable and NMP.

With the �at output z = x1, one �nds the control law

ψu : u = zd + 2zd + 5zd. (B.7)

Further, one �nds the transformation from yd to zd as

Fy→z : yd = 5zd − 5zd

⇔ ˙yd = 5zd − 5zd.(B.8)

As expected, Fy→z is unstable due to a pole at +1 which is the location ofthe model zero. Thus, Eq. (B.8) is changed to

Fy→z,MP : yd = 5zd + 5zd

⇔ ˙yd = 5zd + 5zd,(B.9)

which yields a stable pole at −1 and is equivalent to inverting a di�erentmodel, which had a zero at −1 to begin with.

The IMC �lter is chosen as F (s) = 1s/λ+1

since the model's relative de-

gree is r = 1. Its implementation as SVF is shown in Fig. B.4(a). Theimplementation of Fy→z,MP is shown in Fig. B.4(b). With the output of

−w~ ∫λ

)(d~ ry

dy~

(a) IMC Filter F

dy~

dy&~

51

51

∫−

−

dz

dz&dz&&

MP,zyF →

(b) Fy→z,MP

Fig. B.4: IMC �lter F and Fy→z,MP for the example.

Fy→z,MP, namely zd, zd, zd, the input u can be computed with Eq. (B.7).

204

Bibliography

0 2 4 6 8 100

0.5

1

time (s)

Inpu

tu

0

0.5

1

Out

put A

mpl

itude

y

yd

Fig. B.5: Closed-loop response of the IMC controller of ExampleB.1.

Finally, Fig. B.5 shows the response of the model output y and the �lteroutput yd. As expected, the two di�er from each other in such that themodel output y still exhibits an inverse response which is characteristic foran NMP behaviour resulting from one right-half plane zero. However, theyboth reach the same steady-state value. As expected, the input u is a stablesignal which indicates an internally stable feedforward controller Q. �

205

206

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Conferences.

[i] J. Hanschke, R. Nitsche, and D. Schwarzmann. Nonlinear internalmodel control of diesel air systems. In E-COSM � Rencontres Sci-enti�ques de l'IFP, pages 121�131. Institut Francais du Petrole,October 2006.

[ii] D. Schwarzmann, J. Lunze, and R. Nitsche. A �atness-based ap-proach to internal model control of linear systems. In Proceedingsof the American Control Conference, 2006.

[iii] D. Schwarzmann, R. Nitsche, and J. Lunze. Diesel boost-pressurecontrol using �atness-based internal model control. SAE Spe-cial Publication Papers, SP-2003(2006-01-0855), April 2006. Pre-sented at SAE-World Conference 2006.

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[iv] D. Schwarzmann, R. Nitsche, and J. Lunze. Modelling of theairsystem of a two-stage turbocharged diesel engine. In MATH-MOD, 2006.

[v] D. Schwarzmann, R. Nitsche, J. Lunze, and A. Schanz. Pressurecontrol of a two-stage turbocharged diesel engine using a novelnonlinear imc approach. In Proceedings of the Conference onControl Applications, 2006.

[vi] D. Schwarzmann, R. Nitsche, J. Lunze, and M. Schmidt. Non-linear multivariable robust internal model control of a two-stageturbocharged diesel engine. In Fifth IFAC Symposium on Ad-vances in Automotive Control, 2007.

[vii] D. Schwarzmann, Nitsche R., and J. Lunze. Robuste Ladedruck-regelung eines Pkw-Dieselmotors mittels �achheitsbasierter IMC-Regelung. In VDI AUTOREG, 2006.

[viii] R. Nitsche and D. Schwarzmann. Flachheitsbasierte IMC-Regelung einer elektronisch kommutierten Synchronmaschine. InMechatronik, May 2007.

Journals.

[ix] D. Schwarzmann, R. Nitsche, and J. Hanschke. Nonlinear inter-nal model control of diesel air systems. Oil & Gas Science andTechnology-Revue de l'Institut Francais du Petrole (OGST), 63,2007.

Internal Reports and Patents.

[x] D. Schwarzmann. Modellbasierte Funktionsentwicklung amBeispiel eines Pkw-Luftsystems. Internal report, Robert BoschGmbH, Juni 2004.

[xi] D. Schwarzmann. (Nonlinear) Internal-Model-Control: Unter-suchung bestehender Konzepte und Einführung einer neuen Idee.Internal report, Robert Bosch GmbH, März 2005.

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Bibliography

[xii] D. Schwarzmann. Verfahren zum Betreiben einer Brennkraftmas-chine, Computerprogramm-Produkt, Computerprogramm undSteuer- und/oder Regeleinrichtung. Patent 0710028.5-1263,Robert Bosch GmbH, 2007.

[xiii] D. Schwarzmann. Extrapolation of trubine mass �ow. Appliedfor Patent, Robert Bosch GmbH, 2006.

Student Theses and Internships

[xiv] S. Hesse. Modellierung und Identi�kation von Lusftsystementurboaufgeladener Dieselmotoren. Diploma Thesis, UniversitätStuttgart, July 2004.

[xv] R. Huck. Analyse eines turboaufgeladenen Diesel-Luftsystemsund methodischer Entwurf einer robusten Ladedruckregelung.Diploma Thesis, Universität Stuttgart, May 2005.

[xvi] A. Schanz. Entwurf einer Vorsteuerung für das zweistu�g aufge-ladene Luftsystem. Diploma Thesis, Universität Stuttgart, Octo-ber 2005.

[xvii] M. Schmidt. Erstellung und Durchführung eines modellbasiertenGesamtkonzeptes zur Identi�kation, Funktionsentwicklung undImplementierung für die zweistu�ge Au�adung. Diploma Thesis,Universität Stuttgart, Januar 2006.

[xviii] B. Ahrens. Steer-by-Wire für Landmaschinen bis 60km/h.Diploma Thesis, Universität Karlsruhe, March 2007.

[xix] D. Daumiller. Positionsregelung eines hydraulischen arbeit-sarms mittels nichtlinearem IMC. Diploma Thesis, UniversitätStuttgart, June 2007.

[xx] K. Boualem. Extrapolation of compressor-data. Internschip,Robert Bosch GmbH, August 2004.

[xxi] T. Schlage. Modellierung eines Turboladers. Internship, RobertBosch GmbH, August 2005.

[xxii] R. Huck. Automatisierte Applikation der Ladedruckregelungeines Diesel-Luftsystems. Internship, Robert Bosch GmbH, Au-gust 2004.

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[xxiii] F. Kroll. Werkzeuge zur Modellierung von Turbinen und Kom-pressoren auf Basis von Herstellerdaten. Internship, Robert BoschGmbH, Januar 2006.

[xxiv] J. Hanschke. Flachheitsbasierte IMC-Regelung von Diesel Luft-systemen mit Stellgröÿenbeschränkungen. Internship, RobertBosch GmbH, July 2006.

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Dieter Schwarzmann [email protected] +49 172 6241420

Home Address:

Eckenerstr. 63 74081 Heilbronn

Germany Personal Nationality: German Education 7/2003 – Present Ruhr-University, Bochum, Germany

Ph.D. Student • Dissertation: “Nonlinear Internal Model Control with

Automotive Applications”

8/2001 – 1/2003 Rose-Hulman Institute of Technology, Terre Haute, IN, USA• Master of Science in Mechanical Engineering, • Title of Masters' Thesis: "Optimal Pulse-Jet Control of an

Atmospheric Rocket"

10/1997 – 5/2003 Universität Stuttgart, Germany • Dipl.-Ing. Engineering Cybernetics

Work Experience 7/2003 – Present R&D Control Engineer

Robert Bosch GmbH, Stuttgart, Germany • Modeling of turbocharged engines using Simulink and C++• Control design Since 06.2006: Project leader “Control of mobile hydraulic robots”

2/2003 – 6/2003 Intern Robert Bosch GmbH, Stuttgart, Germany • Design and implementation of a positional controller for an

electric throttle

Skills Languages German: Native Speaker

English: Fluent French: Intermediate

Technical C/C++, Visual C++, Matlab/Simulink, Mathematica, AMESim, Maple, Ansys, EES, Turbo Pascal, Oberon, SQL, PHP

Documents

Nonlinear Internal Model Control With Automotive Applications