From LPV Control Towards a General Synthesis FrameworkCarsten Scherer From LPV Control Towards a General Synthesis Framework Supported by the German Research Foundation (DFG) within

CarstenScherer

From LPV ControlTowards a General

Synthesis Framework

Supported bythe German Research Foundation (DFG)

within the Cluster of ExcellenceData-Integrated Simulation Science (EXC 2075)

1/54

Commemorating Andy Packard

Prestigious Berkeley Citation Award (May 2019)

for contributions to UC Berkeley that go beyond the call of duty and whoseachievements exceed the standards of excellence in their fields.

1/54

Commemorating Andy Packard

Andy had been valiantly battling an extremely aggressive cancerfor the last several years. As everything that he did, he put in agood fight and he did not let the cancer interrupt his teaching,research and family life ... until only two weeks ago.

Roberto Horowitz on October 1, 2019

1/54

Some Key Achievments of Andy Packard

ProcNdlnga ol the 30th Conlerence on Declllon and Control Brighton, England• Decomber 11191 T1-7 • 10:00

John DoyJe•t

Review of LFTs, LMis, and Jt

Andy Packard t Kemin Zhou §

Abstract The purpose of this paper is to present a tutorial overview of Linear Fra.ctional Transformations (LFT) and the role of the Structured Singular Value, µ, and Linear Matrix Inequalities (LMI) in solving LFT problems. 1 Introduction LFTs and LMis play a very important role in postmodern control theory by providing a framework that unifies many concepts and generalizes transfer functions and their state-space realizations to include uncertainty. The focus of this paper is on reviewing known results on robust stability and performa.nce and establishing a common and unified framework for the companion papers in this session, which consider generalizations and extensions of balanced realizations and model reduction [WDBG], stabilization [LuZD], optimal control [PZPB), mixed real/complex µ [YoND], model validation [New!), and LMI computation [Beck]. Section 2 introduces the notation for LFTs and briefly discusses some of their properties. Section 3 describes µ and it 's connections with LFTs. Section 4 focuses on two standard notions of robust stability a.nd performance, µ stability and performance and Q stability and performance, a.nd their relationship is discussed. Comparisons with the new and exciting L1 theory of robust performance with structured uncertainty are also considered. 2 Linear Fractional Transformations (LFTs)

2.1 Definitions of LFTs Suppose M is a complex matrix partitioned as M = [ Mn M12 ] E c(p,+p,)x(q,+q,) M21 M22 (2.1)

The resulting closed-loop transfer functions from w to z are, respectively, F1(M,Ll1) and Fu(M,Llu)-2.2 Redheffer Star-Products Suppose that Q and M are complex matrices, partitioned as

Q = [ �:: �:: ] M = [ ::: ::: ] with the matrix product Q22M11 weil defined, a.nd in fact, square. lf I - Q22M11 is invertible, define the star product of Q and M, with respect to this partition to be S(Q,M):= [ F1(Q, M11) Q12(I-M11Q22)-1 Mu] M21 (I -Q22M11)-1

Q21 Fu (M,Q22) Note that this definition is dependent on the partitioning of the matrices Q and M above. In fact it may be weil defined for one partition and not weil defined for another. However, we will not explicitly show this dependence, as it is always clear from the context. In a block diagram, this appears as

Recall the definitions of linear fractional transformations. Note that for a matrix [( of the appropriate dimension, if all of the necessary matrices are invertible (implied by the loop equations) then Fl (S (Q,M) , K) = F1 (Q, F1 (M,K)) We can also use the S notation for LFTs, as in 2.3 Examples of LFTs and Jet D, c cq,xp, and o2 C c•2x1>2, then we define the linear State Space, Transfer Functions and LFTs fractional transformations (LFTs) as the maps: Given the state space realization of a discrete time system

with F1(M, Ll1) :=Mn+ M12Ll1(I M22Ll1)-1 M21 (2.2) Fu(M, Llu) := M22 + M21Llu(I - M11Llu)-1 M12 (2.3) Clearly the existence of the inverses is necessary for the LFT's to be weil defined. We can also define the LFTs more generally, say with respect to a real rational matrix ß E anxm(s), with the other related matrices also being defined as real rational. The LFT forrnulae arise naturally when describing feedback systems as shown in the following figures.

�

�

(2.4)

then its transfer matrix is G(z) = D + C(zl - A)-1 B = Fu( [ � � ] , ;I) =: [ � 1 � ]

This last notation is deliberately somewhat ambiguous, and can be viewed as both a transfer matrix a.nd its realization. The ambiguity is benign and convenient and can always be resolved from the context. We also use this notation for arbitrary LFTs when the arguments are clear from context, for example, Frequency Transformation The bilinear tranformation between the z-domain and s-domain

•Edited by John Doyle from material by Andy Packard and Kemin Zhou. \Vith is

help from Peter Young, C:arolyn Beck, Jorge Tierno, and \,\'ay Lu. and support from NSF, ONR, NASA, and AFOSR.

1Electrical Engineering, M/S 116-81. Caltech, Pasadena, CA 91125 1Mechanica.l Engineering, UC Berkeley, Berkeley, CA 94720 §Electrical and Computer Engineering, LSU, Baton Ronge, LA 70803

CH3076-7/91/0000-1227$01.00 © 1991 IEEE 1227

z+l s= z-1' where 1�" =

�1 = 1 - ./21 z-11 (I + z-' I)-1 ./2! = :Fu(N, z-1 J) s

[ r h1]-./21 -1

Automatica, Vol. 29, No. 1, pp. 71-109, 1993 Printed in Great Britain.

0005-1098/93 $6.00 + 0.00 © 1992 Pergamon Press Ltd

The Complex Structured Singular Value*

A. PACKARDt and J. DOYLE:j:

A tutorial introduction to the complex structured singular value (µ) is presented, with an emphasis on computable bounds and robust st�bil�ty and performance tests for transfer functions and their state space realtzatwns.

Key Words-Computational methods; control system analysis; disturbance rejection; frequency domain; matrix algebra; multivariable control systems; performance bounds; robust control; sensitivity analysis; state space methods.

Ahstract-A tutorial introduction to the complex structured singular value (µ) is presented, with an emphasis. on themathematical aspects of µ. The µ-based methods d1scussed here have been useful for analysing the performance and robustness properties of linear feedback systems. Several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared, and a simple synthesis _probl�m is studied. Uncertain systems are represented usmg Lmear Fractional Transformations (LFTs) which naturally unify the frequency-domain and state space methods.

1. INTRODUCTIONTms PAPER GIVES a fairly complete introduction to the Structured Singular Value (µ) for complex perturbations. This paper is intended to be of tutorial value on the mathematical aspects of µ,

and it is assumed that the reader is familiar with the control engineering motivation. The µ-based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems. The more elementary methods are now available in commercial software products and the manual (Balas et al., 1991) for one such product �ouldserve as a tutorial introduction to the engmeering motivation. The interested reader might also consult the tutorial in Stein and Doyle (1991) or other application-oriented papers, such as Skogestad et al. (1988). We present very few new results in this paper, although many of the results have appeared only in reports and conference proceedings. The paper is reasonably self-contained, skipping only those proofs which are readily available in the literature.

Section 3 begins with the definition of µ and some of its elementary properties, including

• Received 13 February 1992; received in final form 23 July1992. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Guest Editor H. Kimura.

t Mechanical Engineering, University of Caiifornia, Berkeley, CA 94720, U.S.A.

:j: Electrical Engineering, 116-81, Caltech, Pasadena, CA 91125, U.S.A.

71

simple bounds that form the basis for computational schemes. This section also introduces the relationship between the upper bound for µ and Linear Matrix Inequalities (LMis) which results in a simple characterization of the convexity properties of the upper bound. The connections between µ and Linear Fractional Transformations are introduced in Section 4. These connections, especially the Main Loop Theorem, form the basis for most of the applications of µ to linear systems. In Section 5, using the definition of µ, and the Main Loop theorem, robust stability and robust performance theorems are derived for linear systems with structured linear fractional uncertainty.

Section 6 covers a maximum-modulus theorem for linear fractional transformations. Section 7 presents a generalization of the standard power algorithms for computing the spectral radius or maximum singular value of a matrix to the computation of µ. This power algorithm provides an attractive method for computing lower bounds for µ. Sections 8 and 9 consider issues associated with the upper bound, focusing particularly on conditions under which the upper bound is equal to µ. For certain simple block structures, this equality is guaranteed.

The remainder of the paper discusses applications of µ to problems motivated by control systems. Section 10 considers how various µ problems can be viewed in transfer function and state-space formulations. This leads to a variety of tests for robust performance, each with an interesting and useful interpretation. In Section 11, many (computable) necessary and sufficient conditions for quadratic stability of uncertain systems are given, for a wide variety of uncertainty structures. The proof techniques used in each different case are identical, giving a unifying treatment of many known and new results.

2/54

Outline

The Standard Robustness Framework

Dissipativity-Based Stability and Performance Analysis

Scheduled Controller Synthesis and its Flexibility

General IQC theorem: Dynamic Multipliers

Ramifications

Conclusions and Outlook

2/54

Classical Control Loop

Classical multi-input multi-output feedback loop:

PlantK+�

d e

Saturation at plant input:

PlantsatK+�

d e

Are stability and performance preserved?

3/54


View complicating block as uncertainty �:

Plant�K+�

d e

Give names to input and output of � and disconnect:

PlantK+z

�

d e w

4/54

Standard Configuration: Analysis

System compactly written as z

e

!=M

w

d

!:

Mzw

ed

Original system obtained by reconnecting � as w = �(z):

M

�

w z

ed

Classical configuration of absolute stability and robust control!

5/54


Saturation at plant input and delay at plant output:

Plantsat

delay

K+�

d e

Now rewrite with two uncertainties as

Plant�1

�2

K+z1

z2w2

�

d e w1

6/54

Standard Configuration: Analysis

After disconnecting uncertainties get

M

zw

edw =

w1

w2

!; z =

z1z2

!

Original system obtained with w1

w2

!=

�1(z1)

�2(z2)

!:=

�1 0

0 �2

! z1z2

!:

M

�1 0

0 �2

!w z

ed

7/54

Motivation

M

�

w z

edThis configuration is extremely flexible:

• M comprises information about specific control configuration

• � represents complicating elements or uncertainties

• Is MIMO loop: Can capture structured systems/uncertainties

Provides unified framework for developing theory/algorithms:

• Just one configuration for multitude of interconnections

• M typically is linear time-invariant system

• � captured by input-output properties (abstraction)

• Highly modular

8/54

Outline





Ramifications


8/54

Large Variety of Techniques

• Input-Output Approaches- Small-gain, passivity, conic separation (Zames)- Topological separation (Safonov)- Stability multipliers (Desoer, Vidyasagar)- Integral quadratic constraints (Megretski, Rantzer)

• Dissipativity Approaches- Absolute stability (Popov, Yakubovich, Brockett, J.L. Willems)- Theory of dissipative dynamical systems (J.C. Willems)- Abundance of LMI results in literature

Jan Willems developed dissipativity theory with the explicit goal ofarriving at a more fundamental understanding of the stability

properties of complex feedback interconnections.

9/54

Example I

Time-varying uncertain system:

_x =

0@ �1 2�1(t)

� 12+�1(t)

�0:1 + 3�2(t)

1Ax with j�1(t)j � 1; j�2(t)j � 1:

Can be written as nominal system

_x =

�1 0

�:5 �0:1

!x+

0 2 0

�:5 �2 1:5

! w1

w2

!

z1z2

!=

0BB@�:5 �4

0 1

0 2

1CCAx+

0BB@�:5 �2 1:5

0 0 0

0 1 0

1CCA w1

w2

!

with the time-varying feedback gains

w1(t) = �1(t)z1(t) and w2(t) = �2(t)z2(t):

9/54

Example I

Time-varying uncertain system:

_x =

0@ �1 2�1(t)

� 12+�1(t)

�0:1 + 3�2(t)

1Ax with j�1(t)j � 1; j�2(t)j � 1:

Compactly expressed as a nominal linear system

_x = Ax+Bw; z = Cx+Dw

in feedback with the uncertainty

w = �(z):

The uncertainty � is a system which takes the input signal z(:) into theoutput signal w(:) according to the law

w(t) =

0BB@�1(t) 0 0

0 �1(t) 0

0 0 �2(t)

1CCA z(t):

10/54

Example II

Nonlinear system

_x = Ax+B sat�(Cx)

with saturation function

sat�(z) =(

�z for jzj � 1

� sign(z) for jzj > 1

Graphs of saturation functions:

1

1� = 1 � = 1:5

1

1:5

10/54

Example II

Compactly described as feedback interconnection_x = Ax+Bw

z = Cx

)and w = �(z)

with � taking the input z(:) into the output w(:) as

w(t) = sat�(z(t)):

Question of absolute stability theory:

Is loop stable for all

w(t) = '(z(t))

with a static nonlinearity ' which satisfiesthe sector condition

'(z)(�z � '(z)) � 0 for all z 2 R:

w

z w = �z

w = 0z

11/54

Setup

w = �(z)

_x = Ax+Bw

z = Cx+Dw M

�

zw

Classical feedback interconnection with linear system in forward pathand uncertainty in feedback path.

• Disturbance: Non-zero initial condition x(0).

• Stability of interconnection:

Signals w and z have finite energy for all trajectories.

• Uncertainty � is very general. Diagonal combination of systems:Linear, nonlinear, dynamic, infinite-dimensional, ...

12/54

IQC-Theorem

w = �(z)

_x = Ax+Bw

z = Cx+Dw M

�

zw

Suppose all input-output trajectories w = �(z) of the uncertainty satisfyIntegral Quadratic Constraint (IQC) with multiplier P = P T :

Z T

0

z(t)

w(t)

!TP

z(t)

w(t)

!dt � 0 for all T > 0:

IQC Theorem. Stability guaranteed if dissipation LMIs are feasible:

X � 0;

A B

I 0

!T 0 X

X 0

! A B

I 0

!+

C D

0 I

!TP

C D

0 I

!� 0:

Willems (72)

13/54

Dissipativity Proof

Consider any trajectory of interconnection. Get for t � 0: _x(t)

x(t)

!=

A B

I 0

! x(t)

w(t)

!;

z(t)

w(t)

!=

C D

0 I

! x(t)

w(t)

!:

LMI implies for some small " > 0: A B

I 0

!T 0 X

X 0

! A B

I 0

!+

C D

0 I

!T[P + "I]

C D

0 I

!� 0

and hence get for t � 0: _x(t)

x(t)

!T 0 X

X 0

! _x(t)

x(t)

!+

z(t)

w(t)

!T[P+"I]

z(t)

w(t)

!� 0:

13/54

Dissipativity Proof

Consider any trajectory of interconnection. Get for t � 0: _x(t)

x(t)

!=

A B

I 0

! x(t)

w(t)

!;

z(t)

w(t)

!=

C D

0 I

! x(t)

w(t)

!:

LMI implies for some small " > 0: A B

I 0

!T 0 X

X 0

! A B

I 0

!+

C D

0 I

!T[P + "I]

C D

0 I

!� 0

and hence get for t � 0: _x(t)

x(t)

!T 0 X

X 0

! _x(t)

x(t)

!| {z }

d

dtx(t)TXx(t)

+

z(t)

w(t)

!T[P+"I]

z(t)

w(t)

!� 0:

Integration over [0; T ] gives for T > 0:

x(T )TXx(T )� x(0)TXx(0) +Z T

0

z(t)

w(t)

!T[P+"I]

z(t)

w(t)

!dt � 0:

13/54

Dissipativity Proof

Now exploit IQC for uncertainty:Z T

0

z(t)

w(t)

!TP

z(t)

w(t)


Conclude

x(T )TXx(T )� x(0)TXx(0) + "Z T

0

z(t)

w(t)

!T z(t)

w(t)

!dt � 0:

Since X � 0 and letting T !1 we getZ1

0kz(t)k2 + kw(t)k2 dt �

1

"x(0)TXx(0):

Hence w and z have finite energy.

Various further safety properties for x(:), w(:), z(:) can be extractedfrom these dissipativity arguments!

14/54

Example: The System

Time-varying uncertain system saturated system:

_x =

0@ �1 2�1(t)

� 12+�1(t)

�0:1 + 3�2(t)

1Ax+

sat�(x1)

0

!:

Rewrite as linear system

_x =

�1 0

�:5 �0:1

!| {z }

A

x+

0 2 0 1

�:5 �2 1:5 0

!| {z }

B

0BB@w1

w2

w3

1CCA

0BB@z1z2z3

1CCA =

0BBBBB@�:5 �4

0 1

0 2

1 0

1CCCCCA

| {z }C

x+

0BBBBB@�:5 �2 1:5 0

0 0 0 0

0 1 0 0

0 0 0 0

1CCCCCA

| {z }D

0BB@w1

w2

w3

1CCA

in feedback with

w1(t) = �1(t)z1(t); w2(t) = �2(t)z2(t) and w3(t) = sat�(z3(t)):

14/54

Example: Multipliers for Parameter

Now observe that

w1(t) = �1(t)z1(t) with j�1(t)j � 1 for all t � 0

impliesZ T

0

z1(t)

w1(t)

!T I 0

0�I

!| {z }

P

z1(t)

w1(t)

!dt =

Z T

0kz1(t)k

2 � kw1(t)k2 dt =

=Z T

0kz1(t)k

2�1� j�1(t)j

2�dt � 0 for all T > 0:

Same arguments works with arbitrary Q � 0 and S + ST = 0:Z T

0

z1(t)

w1(t)

!T Q S

ST �Q

!| {z }

P

z1(t)

w1(t)


14/54

Example: Multipliers for Nonlinearity

For z 2 R the sector bound

'(z)(�z � '(z)) � 0

can be expressed as z

'(z)

!T 0@ 0 1

1 � 2�

1A z

'(z)

!� 0:

w

z w = �z

w = 0z

Then w3(t) = '(z3(t)) implies with arbitrary scalar s > 0:Z T

0

z3(t)

w3(t)

!T 0@ 0 s

s � 2�s

1A z3(t)

w3(t)


Constraints on different i/o channels can be easily combined!

14/54

Example: Combination of Multipliers

If j�1(t)j � 1 and j�2(t)j � 1 the trajectories with

w1(t) = �1(t)z1(t); w2(t) = �2(t)z2(t) and w3(t) = sat�(z3(t))

satisfy the IQC

Z T

0

0BBBBBBBBBB@

z1(t)

z2(t)

z3(t)

w1(t)

w2(t)

w3(t)

1CCCCCCCCCCA

T0BBBBBBBBBBB@

Q 0 0 S 0 0

0 q 0 0 0 0

0 0 0 0 0 s

ST 0 0 �Q 0 0

0 0 0 0 �q 0

0 0 s 0 0 � 2�s

1CCCCCCCCCCCA

| {z }P2P

0BBBBBBBBBB@

z1(t)

z2(t)

z3(t)

w1(t)

w2(t)

w3(t)

1CCCCCCCCCCAdt � 0 for all T > 0

in case that

Q � 0; S + ST = 0 and q > 0 and s > 0:

Found a whole family P of multipliers with LMI description!

15/54

Example: Robust Stability Test

Stability guaranteed if there exist X and P 2 P such that

X � 0;

A B

I 0

!T 0 X

X 0

! A B

I 0

!+

C D

0 I

!TP

C D

0 I

!� 0:

Is standard LMI problem. Very easy to implement e.g. using Yalmip.

• Illustrated how to build multiplier classes for individual uncertainties.

• Modularity: Have seen how to routinely combine multipliers.

• IQC theorem generates computational stability test.Encompasses a very wide variety of results in literature!

Can trade computational complexity for conservatism through P!

16/54

Outline





Ramifications


16/54

Straightforward Extension: Performance

w = �(z)0B@ _xze

1CA =

0B@ A B1 B2

C1 D11 D12

C2 D21 D22

1CA0B@ xwd

1CA M

�

w z

ed

Performance specification (energy gain bound, passivity, ...):Z T

0

e(t)

d(t)

!TPp

e(t)

d(t)


Left-upper block of Pp = P Tp is positive semidefinite.

Theorem. Stability & performance assured if exist X � 0, P 2 P with �

�

!T 0 X

X 0

! A B1 B2

I 0 0

!+

�

�

!TP

C1 D11 D12

0 I 0

!+

+

�

�

!TPp

C2 D21 D22

0 0 I

!� 0:

17/54

Example: Compositional Performance Certification

�1

�2

�3

�4

�5 �6

�7

�8d1

e2d2

e1

Collect given systems as

w = �z where � = diag(�1; : : : ;�N):

Describe coupling as static interconnection z

e

!=

D11 D12

D21 D22

! w

d

!:

18/54

Example: Compositional Performance Certification

Let wk = �k(zk) satisfy individual IQCZ T

0

zk(t)

wk(t)

!T Qk SkSTk Rk

! zk(t)

wk(t)


Multiplier for full � by conic combination with �1; : : : ; �N � 0:

P (�) :=

diag(�1Q1; : : : ; �NQN) diag(�1S1; : : : ; �NSN)

diag(�1ST1 ; : : : ; �NS

TN) diag(�1R1; : : : ; �NRN)

!:

Stability & performance guaranteed if there exist �1; : : : ; �N � 0 with D11 D12

I 0

!TP (�)

D11 D12

I 0

!+

D21 D22

0 I

!TPp

D21 D22

0 I

!� 0:

Immediate extension to dynamic or uncertain couplings!

19/54

General Framework

• Exhibits one mechanism behind huge variety of stability tests:- to handle structured uncertainties in robust control- allowing general operator uncertainties- for networked interconnected systems

• Extends seamlessly to performance

• Basis for synthesis. Many problems open!

• Recent developments:Novel IQCs, switched couplings, ODE-PDE systems, data-integration, ...

Unfortunately, the mechanism is not often made clearly visible.

Scherer, Weiland (LMIs in Control) (99-15), Scherer (SIAM Book) (00)

20/54

Outline





Ramifications


20/54

Variable Speed Wind Turbine System

Nyuyen-Tien, Scherer, Scherpen, Muller, IEEE Trans. Ind. Electron. (16)

20/54

Variable Speed Wind Turbine System

• Rotor-current controller Krc

Fast and robust tracking of rotor-side currents ir

• Electrical torque controller Kg

Tracking optimal electrical torques and power factorReference values determined by look-up table

21/54

Linear Parameter Varying Model

State, disturbance input, control input and output:

xr =

0BBBBB@irdirqsd

sq

1CCCCCA ; vs =

vsdvsq

!; vr =

vrdvrq

!; yr =

irdirq

!

Model Grc depends on mechanical angular speed !(t) 2 [0:7!s; 1:3!s]:

_xr = A(!s + 0:3�(t))xr +Bsvs +Brvr; j�(t)j � 1

yr = Cxr

A(!) =

0BBBBB@��a+1Tr

+ aTs

�!s � !

aLmTs

� aLm!

! � !s ��a+1Tr

+ aTs

�aLm! a

LmTsLmTs

0 � 1Ts

!s0 Lm

Ts�!s � 1

Ts

1CCCCCA

22/54


Classical uncertain control loop:

Plant

�Im

+�

d e

zw

Closed loop for LPV control with LTI part K:

Plant

�Im�Ik

K+�

d e

zwzcwc

23/54

Generalized Plant: Synthesis for Stability

Consider only stability:

w(t) = �(t)z(t) with j�(t)j � 1 for all t � 0:

G

K

�(t)Im

�(t)Ik

yu

zw

wczc

M

�(t)Im+k z

zc

! w

wc

!

Robust synthesis for this interconnection is convex!Packard (94), Apkarian, Gahinet (95)

24/54

System DescriptionsDescription of G:

_x = Ax+B1w +B2u

z = C1x+D11w +D12u

y = C2x+D21w

Gw z

yu

Description of K:_xc = Ac

cxc +Bc1y +Bc

2wc

u = Cc1xc +Dc

11y +Dc12wc

zc = Cc2xc +Dc

21y +Dc22wc

K

yu

wczc

Description of interconnection:_xe = Axe + Bwe

ze = Cxe +Dwe

xe =

x

xc

!; we =

w

wc

!; ze =

z

zc

!G

K

yu

w z

wczc

25/54

Mathematical Problem

Recall that

we(t) = �(t)ze(t) for all j�(t)j � 1:

Any Q � 0 leads to the valid IQCZ T

0

ze(t)

we(t)

!T Q 0

0 �Q

! ze(t)

we(t)


• Can guarantee stability of controlled system with IQC theorem.• Leads to following synthesis problem.

LPV Synthesis: There exist K and X � 0, Q � 0 with A B

I 0

!T 0 X

X 0

! A B

I 0

!+

C D

0 I

!T Q 0

0 �Q

! C D

0 I

!� 0:

Very non-convex!

26/54

Mathematical Problem

LPV Synthesis: Do there exist K and X � 0, Q � 0 with A B

I 0

!T 0 X

X 0

! A B

I 0

!+

C D

0 I

!T Q 0

0 �Q

! C D

0 I

!� 0?

Inequality rewritten to0BBBBB@A B

C D

I 0

0 I

1CCCCCA

T0BBBBB@0 0 X 0

0 Q 0 0

X 0 0 0

0 0 0 �Q

1CCCCCA

| {z }P

0BBBBB@A B

C D

I 0

0 I

1CCCCCA � 0:

With controller matrices Z this has the format F +GTZH

I

!TP

F +GTZH

I

!� 0:

27/54

Intermezzo: Elimination Lemma

Consider the inequality in the variable Z: F +GTZH

I

!TP

F +GTZH

I

!� 0:

Is equivalent to I

�F T �HTZTG

!TP�1

I

�F T �HTZTG

!� 0:

Lemma. A solution Z exists iff

HT?

F

I

!�P

F

I

!H? � 0 and GT

?

I

�F T

!TP�1

I

�F T

!G? � 0:

Here H?, G? are basis matrices of ker(H), ker(G).

28/54

Zoom-In

Recall descriptions of G and K:0BB@

_x

z

y

1CCA =

0BB@A B1 B2

C1 D11 D12

C2 D21 D22

1CCA0BB@x

w

u

1CCA ;

0BB@

_xc

u

zc

1CCA =

0BB@Ac Bc

1 Bc2

Cc1 D

c11 D

c12

Cc2 D

c21 D

c22

1CCA

| {z }Z

0BB@xc

y

wc

1CCA

Matrices of controlled system:

A B

C D

!=

0BBBBB@A 0 B1 0

0 0 0 0

C1 0 D1 0

0 0 0 0

1CCCCCA

| {z }F

+

0BBBBB@

0 B2 0

Inxc 0 0

0 D12 0

0 0 Inzc

1CCCCCA

| {z }GT

Z

0BB@

0 Inxc 0 0

C2 0 D21 0

0 0 0 Inwc

1CCA

| {z }H

Natural partitions of X and Q and their inverses:

X =

X �

� �

!; X�1 =

Y �

� �

!and Q =

Q �

� �

!; Q�1 =

R �

� �

!:

29/54

LMIs for Synthesis

Results in convex constraints

�C2 D21

�T?

0BBBBB@A B1

C1 D1

I 0

0 I

1CCCCCA

T0BBBBB@0 0 X 0

0 Q 0 0

X 0 0 0

0 0 0 �Q

1CCCCCA

0BBBBB@A B1

C1 D1

I 0

0 I

1CCCCCA�C2 D21

�?� 0

�BT2 D

T12

�T?

0BBBBB@

I 0

0 I

�AT �CT1

�BT1 �D

T1

1CCCCCA

T0BBBBB@0 0 Y 0

0 R 0 0

Y 0 0 0

0 0 0 �R

1CCCCCA

0BBBBB@

I 0

0 I

�AT �CT1

�BT1 �D

T1

1CCCCCA�BT2 D

T12

�?� 0

and convex couplings X I

I Y

!� 0 and

Q I

I R

!� 0:

Feasibility of LMIs is necessary and sufficient for LPV Synthesis!

30/54

Controller Construction

Step 1: Multiplier Dilation X I

I Y

!� 0 and

Q I

I R

!� 0

permit to construct X � 0 and Q � 0 with

X =

X �

� �

!; X�1 =

Y �

� �

!and Q =

Q �

� �

!; Q�1 =

R �

� �

!:

Step 2: Determine LTI part of Controller

Once X and Q are given, can find controller matrices Z with

F +GTZH

I

!TP

F +GTZH

I

!=

0BBBBB@A B

C D

I 0

0 I

1CCCCCA

T0BBBBB@0 0 X 0

0 Q 0 0

X 0 0 0

0 0 0 �Q

1CCCCCA

0BBBBB@A B

C D

I 0

0 I

1CCCCCA � 0:

31/54

General Gain-Scheduling Problem

Problem: Determine K and schedulingfunction �c(:) to guarantee stability andperformance for all � 2∆.

Known: Class of multipliers P 2 P such that

� 2∆ satisfies IQC for all P 2 P:

Question: Problem solvable by LMIs?

G

K

�

�c(�)

yu

zw

wczc

d e

• D-scalings and L2-gain Packard (94), Apkarian, Gahinet (95)

• D=G scalings and L2-gain Helmersson (95), Scorletti, El-Ghaoui (98)

• Full-block scalings Scherer (00), Wu, Dong (04)

• Distributed Control D’Andrea, Dullerud (03), Langbort, et al. (04)

Very nice survey: Hoffmann, Werner (15)

32/54

Outline





Ramifications


32/54

General Gain-Scheduling Problem

Problem: Determine K and schedulingfunction �c(:) to guarantee stability andperformance for all � 2∆.

Known: Class of multipliers P 2 P such that

� 2∆ satisfies IQC for all P 2 P:

Question: Problem solvable by LMIs?

G

K

�

�c(�)

yu

zw

wczc

d e

Paradigm permits scheduling onparameters, linear dynamics, nonlinearities, delays, PDEs ...

that enter � according to a graph to generate networked controllers.

Ideal setting towards General Synthesis Framework.

33/54

Example: Hybrid Powertrain Control

powertrain

combustionengine

electricmotors

controllerref

Mechatronic model of uncontrolled system:

_x = A(Tmotors)x+Bu; y = Cx with x =

0BBBBBBB@

!engine!wheelTmotors

TengineE

1CCCCCCCA

Joint work with Ton van der Weiden (TU Delft) and Renault

33/54

References and Tracked Outputs

0 2 4 6 8 10 12 14 16 18 200

50

100

150

200

250References: To/3 (blue) omice (red) E (green)

0 2 4 6 8 10 12 14 16 18 20−50

0

50

100

150

200

250Response: To/3 (blue) omice (red) E (green)

33/54

Errors and Control Inputs

0 5 10 15−10

−5

0

5

10Relative Error %: To (b) omice (r) E (g)

0 5 10 15−60

−40

−20

0

20

40

60

80Control: Te1 (b) Te2 (r) Tice (g)

34/54

Anti-Windup Design

G+

sat�

K

Kawp

�awp(�; sat�)

�

�c(�)

y

u

zw

d e

Encapsulating saturation by parametric uncertainty as

w(t) = sat�(z(t)) = �s(t)z(t) with �s(t) 2 [0; �]

allows to use LPV synthesis for convex design of anti-windup controller.

34/54

Errors and Control Inputs with Anti-Windup

0 5 10 15−10

−5

0

5


0 5 10 15−100

−50

0

50


34/54

Clipping Does not Work

0 1 2 3 4 5 6 7 8 9 10−10

−5

0

5


0 1 2 3 4 5 6 7 8 9 10−100

−50

0

50

100

150


35/54

Outline





Ramifications


35/54

Example

Time-invariant uncertain system saturated system:

_x =

�1 2�

� 12+�

�0:1

!x+

sat�(x1)

0

!; � 2 [0; r]:

Reduce conservatism with Zames-Falb multipliers for sat�.

Let h(i!) := g � f(i!) where f : R! R satisfies

f(x) � 0 andZ1

�1

f(x) dx < g:

Then z 2 L2 and w = sat�(z) imply validity of frequency domain IQCZ1

�1

z(i!)

w(i!)

!�0@ 0 h(i!)�

h(i!) � 1�[h(i!)� + h(i!)]

1A z(i!)

w(i!)

!d! � 0:

Not so well-known that it’s due to convexity! Zames, Falb (68)

36/54

Parametrization of Multipliers

Ratinonal Zames-Falb multipliers can be parameterized as

h = �S with S in LMI set.

Here is a fixed stable “basis” transfer matrix (filter).

Observe that0@ 0 h�

h � 1�[h� + h]

1A =

0@ � 1

�

0

1A� 0 ST

S 0

!| {z }

P2P

0@ � 1

�

0

1A

| {z }

:

Many general dynamic multipliers admit the structure

�P

with a fixed stable filter and a real symmetric structured P

contained in a set of matrices P described by LMIs.

37/54

IQC with Terminal Cost

w = �(z)

_x = Ax+Bw

z = Cx+DwM

�

zw

Filtered trajectories of uncertainty:

_x = Ax +B

z

�(z)

!

y = Cx +D

z

�(z)

! �

zw

y

Suppose � satisfies a finite-horizon IQC with terminal cost Z:Z T

0y(t)TPy(t) dt+ x(T )

TZx(T ) � 0 for all T > 0

38/54

IQC with Terminal Cost

w = �(z)

_x = Ax+Bw

z = Cx+DwM

�

zw

Filtered trajectories of system:

y =

M

I

!w

M

zw

y

Introduce state-space realization

M

I

!=

266664A B

C

0

!B

D

I

!

0 A B

C D

C

0

!D

D

I

!

377775 =:

24Af Bf

Cf Df

35

39/54

A Recent Encompassing IQC Theorem

Theorem. The feedback interconnection is stable if

• � satisfies a finite-horizon IQC with terminal cost Z:Z T

0y(t)TPy(t) dt+ x(T )

TZx(T ) � 0 for all T > 0

holds along all filtered trajectories y =

z

�(z)

!.

• There exists a solution X of the dissipation LMI Af Bf

I 0

!T 0 X

X 0

! Af Bf

I 0

!+�Cf Df

�TP�Cf Df

�� 0

that is coupled to the terminal cost as

X �

Z 0

0 0

!� 0:

Veenman, Scherer (13), Seiler (15), Scherer, Veenman (18)

40/54

Dynamic IQCs and Dissipativity

M

�

w z

ed

Theorem fully genuinely extends the celebrated IQC result by Megretskiand Rantzer (97) for rational multipliers �P.

• No technical assumptions required (well-posedness, homotopy)

• Straightforward extension to performance

• Straightforward extension to time-varying/LPV systems- Use time-varying/LPV versions of dissipation inequalities- Can combine all classically known multipliers (modularity)

• Dissipativity arguments permit local stability/performance analysis- Guarantee robust ellipsoidal bounds on output- Exploit locality to reduce conservatism

41/54

Example

Uncertain system saturated system:

_x(t) =

�1 2�

� 12+�

�0:1

!x(t) +

sat�(x1(t)) + d

0

!; e = x1

Rewrite as linear system

_x =

�1 0

�:5 �0:1

!| {z }

A

x+

0 2 0 1

�:5 �2 1:5 0

!| {z }

B

0BB@w1

w2

d

1CCA

0BB@z1z2

e

1CCA =

0BBBBB@�:5 �4

0 1

0 2

1 0

1CCCCCA

| {z }C

x+

0BBBBB@�:5 �2 1:5 0

0 0 0 0

0 1 0 0

0 0 0 0

1CCCCCA

| {z }D

0BB@w1

w2

d

1CCA

in feedback with w1(t) = �z1(t) and w2(t) = sat�(z2(t)).

42/54

Example: Results

• Handle � 2 [0; � ] with dynamic D-scalings• Handle � sat�(:) with Zames-Falb multipliers• Plot guaranteed L2-gain bounds of d 7! e for � 2 [0; 1]:

0 0.2 0.4 0.6 0.8 1

0

10

20

30

40

50

Gains for static versus dynamic multipliers

43/54

Lessons: Combination of Disspativity and IQCs

• is encompassing classical and modern approaches:- absolute stability theory- �-theory- dissipativity theory

• is highly flexible and modular:- easy to combine uncertainties of diverse nature- permits compositional safety verification of complex systems

• has close links to standard Lyapunov approach:- via dissipativity theory- often more insightful and easier to generalize

• It is not difficult to apply!

Many interesting open questions for analysis and synthesis!

44/54

Example: Less Conservative Anti-Windup Design

G+

sat�

K

Kawp

�awp(�; sat�)

�

�c(�)

y

u

zw

d e

Convex synthesis based on Zames-Falb multipliers?

45/54

Outline





Ramifications


45/54

Extended IQC-Theorem: Recent Result

w(t) 2 '(z(t)) z

e

!=M

w

d

!M

'

w z

ed

' : R ⇒ R is subdifferential of convex f : R! R with 0 2 '(0).

Example: f(x) = jxj leads to '(x) =

8>><>>:

1 for all x > 0

[� 1; 1] for all x = 0

�1 for all x < 0

Can use Zames-Falb multipliers in IQC Theorem.

Scherer, Holicki (18)

46/54

Example

Relay systems:

• Switching control

• Unilateral constraints

• Complementarity systems

z

wIdeal Relay' = @j:j

0 2 4 6 8 10 12 14 16 18 20Parameter

0

2

4

6

Ene

rgy

gain

bou

nd

Brogliato(04)ZF ( = 2)ZF ( = 3)

Family of systems depending on � 2 [0; 20]

Heemels, Camlibel, Schumacher (00), Brogliato (04)

47/54

Outline





Ramifications


47/54

Controller Synthesis for 2D Systems

For t � 0 and k = 0; 1; 2; : : : consider0BB@@x(t; k)

�w(t; k)

e(t; k)

1CCA =

0BB@A B B2

C D D12

C2 D21 D22

1CCA0BB@x(t; k)

w(t; k)

d(t; k)

1CCA

with states x, w and disturbance input d, error e.

Derivative @ acts on first variable. Left-shift � acts on second variable.

Features

• Mixed one-sided continuous and discrete time axes• Finite-dimensional, time- and shift-invariant

• Signal norm kxk22 :=1Xk=0

Z1

0kx(t; k)k2 dt

48/54

Applications

• Control of repetitive processes as in steel rolling:

E.g Rogers, Galkowski, Owens (07)

• More examples:- Control of disturbance propagation in vehicle platoons- Control of irrigation channels

E.g Sebek, Hurak (11), Li, Cantoni, Weyer (05)

49/54

H1-Synthesis for 2D Systems

Full solution of two-time axes H1-design problem:

2D plant P

2D controller K

d e

yu

Optimal feedback control:

• K stabilizes P

• Minimize H1-normof d! e

• Have finite-dimensional convex optimization hierarchy for design• Can guarantee convergence to optimal achievable attenuation level

Scherer (16)

Related: Computation of H1-norm and (non-convex) static output-feedbackChesi, Middleton (15,17)

50/54

Outline





Ramifications


50/54

Extended IQC-Theorem: Hybrid Controller Synthesis

System with jumps at times 0 = t0 < t1 < t2 < : : : and piecewiseconstant parametric uncertainty �(:):

w(t) = �(t)z(t)

_x(t) = Ax(t) +B1w(t) +B2u(t)

z(t) = C1x(t) +D11w(t) +D12u(t)

y(t) = C2x(t) +D21w(t)

x(tk) = AJx(t�

k ) +BJuJ(k)

yJ(k) = CJx(t�

k )

Can solve hybrid LPV synthesis problem:

• Joint design of flow and jump controller component• IQC Analysis based on dynamic multipliers with resetting filters:Z

1

0(�)TP

z

w

!dt =

1Xk=0

Z tk+1

tk

(�)TP

z

w

!dt

• Involves solution of finite-horizon differential LMIs with SOSHolicki, Scherer (19)

51/54

Example: Formation Control

Output-feedback of relative positions/Switched communication topologies

52/54

IQC-Theorem in Discrete-Time

xt+1 = Axt +Bwt

zt = Cxt

rf

zw

Optimization algorithms for strongly convex f : Rn ! R:

• Gradient descent is a first order linear system

• Nesterov proposed accelerated gradient descent:- Much better practical performance- Proved fast convergence by estimation sequence

• Better convergence rate shown with causal Zames-Falb multiplierLessard, Recht, Packard (16)

• General Zames-Falb multipliers and H2-performanceMichalowsky, Scherer, Ebenbauer (provisionally accepted)

53/54


• Surveyed classical and recent insights

- Clarified flexibility of LFTs and IQCs in dissipativity

- Reviewed key mechanisms for controller design

- Showcased crucial benefits of paradigm

• Interesting issues

- Scalability: Exploit interconnection structure

- Solvers: Dedicated and stable algorithms

- Synthesis: For general dynamic multipliers

• Publications related to this talk:https://www.imng.uni-stuttgart.de/mst/publications/

https://www.imng.uni-stuttgart.de/mst/publications/

https://www.imng.uni-stuttgart.de/mst/publications/

54/54

Thanks for your fantastic contributions to robust control!You will be dearly missed.

Documents

From LPV Control Towards a General Synthesis FrameworkCarsten Scherer From LPV Control Towards a General Synthesis Framework Supported by the German Research Foundation (DFG) within