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International Journal of Control

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Improved synthesis conditions for mixed H2/H∞ gain-scheduling control subject to uncertainscheduling parameters

Ali Khudhair Al-Jiboory & Guoming G. Zhu

To cite this article: Ali Khudhair Al-Jiboory & Guoming G. Zhu (2017) Improved synthesisconditions for mixed H2/H∞ gain-scheduling control subject to uncertain scheduling parameters,International Journal of Control, 90:3, 580-598, DOI: 10.1080/00207179.2016.1186843

To link to this article: https://doi.org/10.1080/00207179.2016.1186843

Accepted author version posted online: 12May 2016.Published online: 17 Jun 2016.

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INTERNATIONAL JOURNAL OF CONTROL, VOL. , NO. , –http://dx.doi.org/./..

Improved synthesis conditions for mixedH2/H∞ gain-scheduling control subjectto uncertain scheduling parameters

Ali Khudhair Al-Jiboory and Guoming G. Zhu

Department of Mechanical Engineering, Michigan State University, East Lansing, MI, USA

ARTICLE HISTORYReceived June Accepted May

KEYWORDSLinear parameter-varyingsystems; gain-scheduling;linear matrix inequality;robust control

ABSTRACTThe vast majority of the existing work in gain-scheduling (GS) control literature assumes perfectknowledge of scheduling parameters. Generally, this assumption is not realistic since for practicalcontrol applications measurement noises are unavoidable. In this paper, novel synthesis conditionsare derived to synthesise robust GS controllers with mixed H2/H∞ performance subject to uncer-tain scheduling parameters. The conditions are formulated in terms of parameterised bilinear matrixinequalities (PBMIs) that depend on varying parameters insidemulti-simplex domain. The conditionsprovide practicalGS controllers independent of the derivatives of scheduling parameters. That is, thedesigned controllers are feasible for implementation. Since bilinear matrix inequality problems areintractable, an iterative PBMI algorithm is developed to solve the developed synthesis conditions.By the virtue of this algorithm, conservativeness reduction is achieved with few iterations. Examplesare presented to illustrate the effectiveness of the developed conditions. Compared to other designmethods from literature, the developed conditions achieve better performance.

1. Introduction

One of the main objectives in control theory is the devel-opment of synthesis conditions and control strategiesthat not only guarantee closed-loop stability but alsoachieve multi-objective performance. Since H2 perfor-mance is well known to handle stochastic aspects of ran-dom disturbances and H∞ performance is important toclosed-loop system robustness, mixed H2/H∞ perfor-mance is widely used in robust control to achieve bothobjectives simultaneously. For linear parameter-varying(LPV) systems, the mixed performance becomes moreinvolved since the system matrices depend on time-varying scheduling parameters. The interest in LPV con-trol theory grew rapidly over the past two decades (seeHoffmann & Werner, 2015; Leith & Leithead, 2000; andreferences therein) due to the fact that the dynamicsof many physical systems can be efficiently modelledas functions of varying parameters. In addition, a wideclass of nonlinear systems can be modelled as quasi-LPV (qLPV) systems such that linear control theories canbe used for controller synthesis instead of using sophis-ticated nonlinear methodologies. Although schedulingparameters are unknown a priori during controller syn-thesis stage, they are implicitly assumed to be measurablein real time for feedback control to achieve better perfor-mance. Practical examples that prove the effectiveness ofgain-scheduling (GS) control include automotive engines

CONTACT Guoming G. Zhu zhug@egr.msu.edu

(White, Ren, Zhu, & Choi, 2013), electro-hydraulic actu-ators (Németh et al., 2015), wind turbine (Shirazi, Grigo-riadis, & Viassolo, 2012), and biomedical applications(Luspay & Grigoriadis, 2015).

Recent literature also deals with the analysis and syn-thesis of robust controllers in the LPV/LFT (linear frac-tional transformation) framework. For instance, switch-ing control scheme was used in Deaecto, Geromel, andDaafouz (2011) and Geromel and Deaecto (2009) to syn-thesise switching state-feedback controllers, where theswitching rules are synthesised simultaneously throughconvex optimisation formulation with linear matrixinequality (LMI) constraints. In Yuan, Wu, and Chang(2015) and Yuan and Wu (2015), robust H2 and H∞switched feed-forward control of uncertain LFT systemsis synthesised based onpiecewise Lyapunov functionwiththe min-switching control technique. Other approachesthat deal with polytopic LPV systems can be foundin Oliveira and Peres (2009) and Montagner, Oliveira,Peres, and Bliman (2009). However, the vast majority ofthe existing LPV framework is assuming that the exactknowledge of scheduling parameters are available in realtime for feedback control. Generally, this assumptionis not practical in control applications due to sensorcalibration errors and measurement noises. Therefore,uncertainties in scheduling parameters are inevitable andperfect knowledge of scheduling parameters are very dif-ficult to obtain.Although there are several techniques that

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INTERNATIONAL JOURNAL OF CONTROL 581

handlemeasurement noise in scheduling parameters, thiscontrol problem is still under-investigated. In Kose andJabbari (1999), only some of the scheduling parametersare assumed to be available for feedback control with-out considering uncertainties in scheduling parameters.Synthesis conditions with guaranteed performance in thepresence of uncertainties in scheduling parameters areproposed by Daafouz, Bernussou, and Geromel (2008);however, these conditions are impractical since uncer-tainties aremodelled to be proportional to the schedulingparameters, which is not common to any measurementsystems. Furthermore, the designed controllers are notonly conservative but also very sensitive to measurementnoises. Then, several papers that address the same controlproblem have appeared in Sato (2010a, 2010b), Sato, Ebi-hara, and Peaucelle (2010), and Sato and Peaucelle (2013).In Sato (2010a, 2010b), quadratic stability (constant Lya-punov matrix) approach is used to derive the synthesisconditions. As pointed out in Wu, Yang, Packard, andBecker (1996), this approach is extremely conservativeand cannot deal with many systems that are not quadrat-ically stabilisable. To alleviate this problem, parameter-dependent Lyapunov function approach is used to syn-thesise scheduling controllers in Sato et al. (2010) andSato andPeaucelle (2013) as a remedy for quadratic stabil-ity approach. However, the approach in Sato et al. (2010)and Sato and Peaucelle (2013) has an implementationdrawback since the synthesised controller requires notonly the real-timemeasurement of the scheduling param-eters but also their time derivatives. Hence, the synthe-sised controller is not practically implementable (Apkar-ian & Adams, 1997). From the practical point of view,the derivatives of the scheduling parameters should notbe used for feedback control in real time due to the factthat derivatives are very sensitive to measurement noises.Based on the review of the existing work, the motivationof this paper is to overcome these shortcomings by devel-oping new approach to handle this control problem.

The main contribution of this paper is the characteri-sation of novel synthesis conditions formulated in termsof parameterised bilinear matrix inequalities (PBMIs)to synthesise robust GS controllers with guaranteedmixed H2/H∞ performance subject to noisy schedul-ing parameters. The conditions can handle the moregeneral case when the varying-parameters affects boththe state matrix and the control matrix. The developedconditions utilise slack variable approach (Ebihara &Hagiwara, 2005; Oliveira, Geromel, & Bernussou, 2002)with a scalar line search to introduce additional degreesof freedom for performance improvement. An impor-tant feature of the developed conditions is the utilisa-tion of different (parameter-dependent) Lyapunovmatri-ces to handle the H2 and H∞ performance bounds

independently. Thus, reducing conservativeness associ-ated with imposing single Lyapunovmatrix for the mixedperformance in previous works (Khargonekar & Rotea,1991; Meisami-Azadt, Mohammadpour, & Grigoriadis,2009; Scherer, Gahinet, & Chilali, 1997). We believe thatsuch an interesting feature is extremely useful to dealwith multi-objective control synthesis problems withoutintroducing additional conservativeness. Moreover, sincePBMI problems are non-convex in general, an iterativealgorithm is developed to solve the synthesis conditions.By the virtue of this algorithm, considerable performanceimprovement is achieved with few iterations.

The rest of the paper is organised as follows: pre-liminary concepts and definitions are given in the nextsection. Section 3 presents a modelling approach to con-vert the measured scheduling parameters and their ratesof changes into a (convex) multi-simplex domain. Prob-lem formulation of the robust GS control design withuncertain scheduling parameters is given in Section 4.Section 5 characterises PBMI conditions for synthesisingGS dynamic output-feedback controllers with mixedH2/H∞ performance. PBMI algorithm is developed atthe end of Section 5 to solve the developed synthesis con-ditions. Section 6 includes two illustrative examples todemonstrate the effectiveness of the developed approach.Finally, conclusions are drawn in the last section.

Notations used in this paper are fairly standard. Thepositive definiteness of a matrix A is denoted by A> 0. Rdenotes the set of real numbers. The symbol � is used torepresent the transpose of the off-diagonal matrix block.trace(A) denotes the trace of matrix A, which representsthe sum of diagonal elements of thematrixA. In is used torepresent an identity matrix of size n × n. Zero matricesof size n × p are referred as 0n × p. These subscripts willbe omitted when the size of the correspondingmatrix canbe inferred from the context. The transpose of matrix Ais referred as A′; and A + (•)′ = A + A′. Other notationswill be explained in due course.

2. Preliminaries

2.1 Linear parameter-varying systems

Consider the following LPV system:

G(θ (t )) :=⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

x(t ) = A(θ (t ))x(t ) + Bu(θ (t ))u(t ) + B2(θ (t ))w2(t )+ B∞(θ (t ))w∞(t ),

z∞(t ) = C∞(θ (t ))x(t ) + D∞u(θ (t ))u(t )+ D∞∞(θ (t ))w∞(t ),

z2(t ) = C2(θ (t ))x(t ) + D2u(θ (t ))u(t ),y(t ) = Cyx(t ) + Dy2(θ (t ))w2(t ) + Dy∞(θ (t ))w∞(t ),

(1)

582 A. K. AL-JIBOORY AND G. G. ZHU

where x(t ) ∈ Rn is the state, u(t ) ∈ R

nu is the controlinput, w2(t ) ∈ R

nw2 and w∞(t ) ∈ Rnw∞ are the dis-

turbance inputs, y(t ) ∈ Rny is the measured output,

z∞(t ) ∈ Rnz∞ and z2(t ) ∈ R

nz2 are the controlled (per-formance) outputs related to the H2 and H∞ channels,respectively. The systemmatrices have the following com-patible dimensions: A(θ (t )) ∈ R

n×n, Bu(θ (t )) ∈ Rn×nu ,

B2(θ (t )) ∈ Rn×w2 , B∞(θ (t )) ∈ R

n×nw∞ , C2(θ (t )) ∈R

nz2×n, D2u(θ (t )) ∈ Rnz2×nu , C∞(θ (t )) ∈ R

nz∞×n,D∞u(θ (t )) ∈ R

nz∞×nu , D∞∞(θ (t )) ∈ Rnz∞×w∞ , Cy ∈

Rny×n, Dy2(θ (t )) ∈ R

ny×nw2 , and Dy∞(θ (t )) ∈ Rny×nw∞ .

Note that θ(t) represents a real vector containing thetime-varying scheduling parameters, where

θ (t ) = [θ1(t ), θ2(t ), . . . , θq(t )

]′. (2)

The system matrices in (1) are assumed to be affineparameter-dependent, i.e. each of these matrices can berepresented by the following parameterisation:

A(θ (t )) = A0 +q∑

i=1

θi(t )Ai. (3)

The scheduling parameters in (2) are assumed to beinexactly measured (corrupted with noise) denoted byθ (t ) when they are used for feedback control, such that

θ (t ) =[θ1(t ), θ2(t ), . . . , θq(t )

]′,

δ(t ) = [δ1(t ), δ2(t ), . . . , δq(t )

]′,

θ (t ) = θ (t ) + δ(t ),

or in scalar form

θi(t ) = θi(t ) + δi(t ), i = 1, 2, . . . , q, (4)

where δi(t) represents uncertainty in the ith schedul-ing variable and θ i(t) is the exact value. These schedul-ing parameters and its uncertainties are assumed to beindependent and time-varying with the following knownbounds:

−θi ≤ θi(t ) ≤ θi, −δi ≤ δi(t ) ≤ δi, i = 1, 2, . . . , q.(5)

Furthermore, these parameters are assumed to havebounded rates of variation given by

− bθi ≤ θi(t ) ≤ bθi,

−bδi ≤ δi(t ) ≤ bδi, i = 1, 2, . . . , q, (6)

and without loss of generality, these bounds are assumedto be symmetric.

G(θ(t))

Closed-loop system

K(θ(t))

z2(t)

z∞(t)

w2(t)w∞(t)

y(t) u(t)

Figure . Closed-loop system with output-feedback GS control.

Remark 2.1: Note that (5) and (6) are not restrictive,since (5) can always be achieved by a simple changeof variables; while (6) represents realistic hypothesisbecause the rates of variations of the parameters are natu-rally limited in practical engineering applications. How-ever, the rate of change of the uncertainties in (6) is noteasy to obtain. Therefore, in case that the spectrum ofthe uncertainty (or measurement noise) can be knownthrough experiments before controller design, the ratebounds can be estimated based on the bandwidth ofthe noise spectrum. If this is not the case, setting thesebounds sufficiently, a large number is recommended.

The goal is to synthesise strictly proper full-orderdynamic output-feedback robust GS controller of the fol-lowing form:

K(θ (t )) :={xc(t ) = Ac(θ (t ))xc(t ) + Bc(θ (t ))y(t ),u(t ) = Cc(θ (t ))xc(t ),

(7)that robustly stabilises the closed-loop system and min-imises the H2 performance while satisfying predefinedconstraint on H∞ performance (see Figure 1). Further-more, the synthesised controller should be robust tomeasurement uncertainties of the scheduling parameters.Augmenting the open-loop system G(θ (t )) in (1) withthe controller K(θ (t )) in (7), the following closed-loopsystem matrices are obtained:

ξ (t ) = A(θ (t ), θ (t ))ξ (t ) + B2(θ (t ), θ (t ))w2(t )+ B∞(θ (t ), θ (t ))w∞(t ),

z∞(t ) = C∞(θ (t ), θ (t ))ξ (t ) + D∞(θ (t ), θ (t ))w(t ),z2(t ) = C2(θ (t ), θ (t ))ξ (t ), (8)

where ξ (t) = [x(t)′ xc(t)′]′, and

A(θ (t ), θ (t )) =[

A(θ (t )) Bu(θ (t ))Cc(θ (t ))Bc(θ (t ))Cy(θ (t )) Ac(θ (t ))

],

(9)

INTERNATIONAL JOURNAL OF CONTROL 583

B2(θ (t ), θ (t )) =[

B2(θ (t ))Bc(θ (t ))Dy2(θ (t ))

], (10)

B∞(θ (t ), θ (t )) =[

B∞(θ (t ))Bc(θ (t ))Dy∞(θ (t ))

], (11)

C2(θ (t ), θ (t )) = [C2(θ (t )) D2u(θ (t ))Cc(θ (t ))

],

(12)

C∞(θ (t ), θ (t )) = [C∞(θ (t )) D∞u(θ (t ))Cc(θ (t ))

],

(13)

D∞(θ (t ), θ (t )) = D∞∞(θ (t )). (14)

The controller matrices in (7) are assumed to have affineparameterisation with respect to the measured schedul-ing parameters. In other words, these matrices (Ac(θ ),Bc(θ ), andCc(θ )) are parameterised as

Ac(θ ) = Ac0 +q∑

i=1

θiAci

Bc(θ ) = Bc0 +q∑

i=1

θiBci

Cc(θ ) = Cc0 +q∑

i=1

θiCci .

(15)

Therefore, the goal is to obtain the controller coefficientsAci , Bci , andCci (i= 0, 1, …, q) for real-time implementa-tion using only the measured (noisy) scheduling param-eters.

Following are definitions that form the foundation ofthe developed approach.

Definition 2.1 (Unit-simplex) (Oliveira & Peres, 2007):A unit-simplex is defined as follows:

�r ={α(t ) ∈ R

r :r∑

i=1

αi(t ) = 1, αi(t ) ≥ 0, i = 1, 2, . . . , r

},

where αi(t) varies in the unit-simplex �r that has r ver-tices.Definition 2.2 (Multi-simplex) (Oliveira, Bliman, &Peres, 2008): Amulti-simplex � is the Cartesian productof a finite number of q simplexes that is defined as

�N1 × �N2 × · · · × �Nq =q∏

i=1

�Ni � �.

The dimension of the multi-simplex � is defined as theindexN= (N1,N2, …,Nq) and for simplicity of notation,

RN denotes for the space R

N1+N2+···+Nq . Thus, any vari-able α(t) in the multi-simplex domain � can be decom-posed as (α1(t), α2(t), …, αq(t)), and each αi(t), belong-ing to a unit-simplex �Ni , can be further decomposed as(αi1(t ), αi2(t ), . . . , αiNi (t )) for i = 1, 2, …, q.Lemma 2.1: For a given parameter-dependent symmetricmatrix�(α)andmatrices�1(α)and�2(α)with compati-ble dimensions, if one of the two following conditions holds:

[�(α)

[�1(α) H(α)�2(α)] −H(α)

]< 0, (16)[

�(α)

[H(α)�1(α) �2(α)] −H(α)

]< 0 (17)

for some parameter-dependent symmetric positive-definitematrix H(α), then the condition

�(α) +[

0

�2(α)′ �1(α) 0

]< 0, (18)

holds.Proof: Applying Schur compliment to (16) gives

�(α) +[

�1(α)′

�2(α)′ H(α)′

]× H(α)−1 [�1(α) H(α)�2(α)

]< 0,

which can be written as

�(α) +[

0

�2(α)′ �1(α) 0

]

< −[

�1(α)′H(α)−1�1(α) 00 �2(α)′H(α)�2(α)

].

(19)

Since the right-hand side (RHS) is negative semi-definite,(18) holds. The proof for (17) can be done in a similarway. �

3. Modelling approach

The developed synthesis approach can be illustratedby Figure 2. First, appropriate transformation is usedto convert the scheduling parameters and uncertaintiesfrom their original parameter domain into multi-simplexdomain. Then, the rates of variations of the schedulingparameters and their uncertainties aremodelled in a con-vex set. The details of these two steps are presented inthis section.Most of the notations used in this section canbe found in Oliveira et al. (2008) and Lacerda, Tognetti,Oliveira, and Peres (2014). Other steps of Figure 2 will becovered in the subsequent sections.

584 A. K. AL-JIBOORY AND G. G. ZHU

Affine to multi-simplex transformation

Rate of variation modeling

PBMI synthesis conditions derivation

PBMI algorithm and relaxation

Inverse Transformation (Multi-simplex to affine)

Controller implementation

Figure . The developed synthesis approach.

3.1 Affine tomulti-simplex transformation

The goal of this subsection is to develop a suitable changeof variables to transform all the time-varying parame-ters (scheduling ones and uncertainties) from their origi-nal parameter space into a convexmulti-simplex domain.This transformation is dictated by the relaxation tech-nique (matrix coefficient check using Pólya’s theorem)used to solve the parameter-dependent LMIs (Oliveira &Peres, 2007; Oliveira, Bliman, & Peres, 2008). Supposethat the actual scheduling parameters θ i(t) are affectedby a time-varying measurement noise δi(t) as given by(4), and suppose further that θ(t), δ(t), θ (t ), and δ(t ) arebounded as defined in (5) and (6). Since δi(t) is associ-ated with the corresponding θ i(t), two unit-simplexes arerequired to model each measured scheduling parameter.Each unit-simplex has two vertices due to the fact thateach parameter has upper and lower bounds as defined in(5). Thus, each of those (time-varying) parameters (θ i(t)and δi(t)) can be modelled independently in their ownunit-simplexes with two vertices. Following the approachdepicted in Lacerda et al. (2014), the actual schedulingparameters and their uncertainties can be modelled asfollows:

(1) Actual scheduling parameters (θi(t ) ⇒ αi(t )),

αi1(t ) = θi(t ) + θi

2θi⇒ θi(t ) = 2θiαi1(t ) − θi,

(20)then,

αi2(t ) = 1 − αi1(t ) = 1 − θi(t ) + θi

2θi= θi − θi(t )

2θi,

where,

αi(t ) = (αi1(t ), αi2(t )) ∈ �2, ∀i = 1, 2, . . . , q,α(t ) = (α1(t ), α2(t ), . . . , αq(t )).

(2) Uncertainties (δi(t ) ⇒ αi(t )),

αi1(t ) = δi(t ) + δi

2δi⇒ δi(t ) = 2δiαi1(t ) − δi,

(21)then,

αi2(t ) = 1 − αi1(t ) = 1 − δi(t ) + δi

2δi= δi − δi(t )

2δi,

where

αi(t ) = (αi1(t ), αi2(t )) ∈ �2, ∀i = 1, 2, . . . , q,α(t ) = (α1(t ), α2(t ), . . . , αq(t )).

Thus, using this change of variables, the original affineparameter-dependent system (1) as well as the GS con-troller (7) can be converted from θ(t) and θ (t ) into newmulti-simplex (convex) variables α(t ) and α(t ). There-fore, the multi-simplex variables α(t ) can be defined as

α(t ) = (αi(t ), αi(t )), i = 1, 2, . . . , q,α(t ) ∈ �, where � = �2 × �2 × · · · × �2︸ ︷︷ ︸

2q unit-simplexes

.

(22)

Consider the case that q = 1 (one scheduling param-eter), α1(t ) = (α11(t ), α12(t )), and α1(t ) = (α11(t ),α12(t )), then the homogeneous terms in the multi-simplex variables can be written in terms of the newconvex variables as α(t ) = (α11(t ), α12(t ), α11(t ),α12(t )).

Let Z(θ (t )) be a parameter-dependent matrix thatrepresents any of the controller matrices defined in (15).This matrix can be expressed affinely in terms of themea-sured scheduling parameter for one scheduling parame-ter as

Z(θ (t )) = Z0 + θ1(t )Z1 = Z0 + (θ1(t ) + δ1(t ))Z1.

(23)Substituting for θ1(t) and δ1(t) using (20) and (21)yields,

Z(θ (t )) = Z0 + (2θ1α11 − θ1 + 2δ1α11 − δ1)Z1

= Z(α(t )). (24)

Therefore, (24) illustrates the equivalence between θ (t )and α(t ). However, (24) is not homogeneous at thisstage, applying homogenisation procedure proposed inOliveira, Bliman, & Peres (2008) to obtain

Z(α(t )) = α11α11Z1,1 + α11α12Z1,2

+ α21α11Z2,1 + α21α21Z2,2, (25)

INTERNATIONAL JOURNAL OF CONTROL 585

where the coefficients Z1,1,Z1,2,Z2,1, and Z2,2 can begenerated for any number of scheduling parametersq � 1 as

Z j1, j2,..., jq,k1,k2,...,kq

= Z0 +q∑

i=1

{(−1) ji+1θi + (−1)ki+1δi}Zi, (26)

with ji = 1, 2, ki = 1, 2, and i= 1, 2, …, q. Thus, all the syn-thesis variables (that will be given shortly in Theorem5.1)should be treated by this procedure to have the parame-terisation of (25), as a consequence, the controller matri-ces can be written in terms of the multi-simplex parame-ters α(t ) as Ac(α(t )), Bc(α(t )), andCc(α(t )).

Remark 3.1: Note that the open-loop system matri-ces in (1) are independent of measurement uncertain-ties δi(t). They only depend on the exact schedulingparameters θ i(t). However, the same procedure describedabove should be used to transform them from the origi-nal parameter space θ(t) intomulti-simplex space α(t ) byimposing δi = 0 in (26). In this case, for notational sim-plicity, we will denote the multi-simplex variables as α(t)instead of α(t ) to distinguish it from other synthesis vari-ables. Therefore, the open-loop system matrices will bewritten in terms of the multi-simplex variables as

A(α(t )),Bu(α(t )),B2(α(t )),B∞(α(t )),C∞(α(t )),C2(α(t )),D∞u(α(t )),D∞∞(α(t )),D2u(α(t )),Dy2(α(t )), and Dy∞(α(t )).

3.2 Rate of variationmodel

The objective of this subsection is to construct a convexparameter space η(t) to model the derivatives of the vary-ing parameters in a convex domain. The rates of varia-tions of each parameter and uncertainty are assumed tobe bounded as defined in (6) for all t� 0. Since each vary-ing parameter belongs to a unit-simplex, it is clear that thefollowing relation should be satisfied:

αi1(t ) + αi2(t ) = 0 i = 1, 2, . . . , q. (27)

Since αi(t) � �2, the time derivatives of the parametersαi can assume values that are modelled by a convex poly-tope �i (Chesi, Garulli, Tesi, & Vicino, 2007; Geromel &

Colaneri, 2006):

�i ={

φ ∈ R2 : φ =

2∑k=1

ηikH (k)i ,

2∑k=1

Hi(k, j) = 0, ηi ∈ �2

}, j = 1, 2,

i = 1, 2, . . . , 2q. (28)

Given the bounds bθi and bδi in (6), H (k)i represents the

kth column of matrix Hi. Since simplexes with two ver-tices have been considered for each varying parameter,the matrices Hi will have size of 2 × 2. Consequently,

α(t ) ∈ � = �1 × �2 × · · · × �2q =2q∏i=1

�i. (29)

The relationship between the bounds of the rates of varia-tions of scheduling parameters and the rates of variationsof themulti-simplex variables can be easily obtained from(6) and (20) as follows:

−bθi

2θi≤ αi1(t ) ≤ bθi

2θi,

with αi2(t ) = −αi1(t ) due to (27). Therefore, the trans-formation of the rates of variations from θ (t ) and δ(t )into αi(t ) is exact as well.

Thus, the derivative of the parametric Lyapunovmatrix that depends on time-varying parameters inmulti-simplex can be computed through this procedureas

P(α) = ∂P(α)

∂αα =

q∑i=1

2∑j=1

∂P(α)

∂αi jαi j

=q∑

i=1

2∑j=1

∂P(α)

∂αi j

2∑k=1

ηikHi( j, k)

= ∂P(α)

∂αi j(ηi1Hi( j, 1) + ηi2Hi( j, 2))

:= Π(α, η), ηi ∈ �2. (30)

After all scheduling parameters and their uncertainties(with their rates of variations) aremodelled to varywithinconvex sets (�, �), we are ready to define the mixedH2/H∞ control problem for LPV systems in the next sec-tion.

4. Problem formulation

Problem 4.1: Suppose that the scheduling parametersθ(t) are provided as θ (t ) with uncertainties δ(t) and the

586 A. K. AL-JIBOORY AND G. G. ZHU

systemmatrices are transformed into multi-simplex vari-ablesα(t) and α(t ) (as described in Section 3). For a givenpositive scalar γ , find a dynamic output-feedback con-troller in the form of (7) for any pair (α, ˙α) ∈ � × � thatstabilises the closed-loop system (8) and satisfies

sup(α, ˙α)∈�×�

supw∞∈L2,w∞=0

‖z∞(t )‖2‖w∞(t )‖2

< γ . (31)

Additionally, the controller minimises the upper boundon theH2 norm ν from w2(t) to the performance outputz2(t) (see Figure 1) such that

sup(α, ˙α)∈�×�

E{∫ T

0z2(t )′z2(t )dt

}< ν2, (32)

for the disturbance input w2(t) given by

w2(t ) = w0δ(t ),

where δ(t) is theDirac’s delta function andw0 is a randomvariable satisfying

E{w0w

′0} = Inw

and E{ · } denotes the mathematical expectation.

The next two lemmas will be used in the derivation ofthe PBMI synthesis conditions.

Lemma 4.1 (Sato, 2008): If there exists a continuouslydifferentiable parameter-dependent symmetric positive-definite matrix P(α) ∈ R

n×n for any pair (α(t ), ˙α(t )) ∈� × � such that the following parameterised linearmatrixinequality (PLMI) is satisfied:

⎡⎣A(α, α)P(α) + (•)′ − P(α)

C∞(α, α)P(α) −Inz

B∞(α, α)′ D∞(α, α)′ −γ 2Inw

⎤⎦ < 0,

(33)

then the closed-loop system (8) is asymptotically stable and(31) is satisfied, whereA(α, α), B∞(α, α), C∞(α, α) andD∞(α, α) are the closed-loop system matrices defined in(8) with θ and θ replaced by α and α using the change ofvariables defined in (20) and (21).

Lemma 4.2 (De Souza & Trofino, 2006): For a givenpositive scalar ν, if there exist a continuously differen-tiable positive-definite matrix P(α) = P(α)′ ∈ R

n×n andparameter-dependent matrix W (α) = W (α)′ ∈ R

nz2×nz2

such that the following PLMIs are satisfied:

[A(α, α)P(α) + P(α)A(α, α)′ − P(α)

B2(α, α)′ −I

]< 0,

(34)[P(α)

C2(α, α)P(α) W (α)

]> 0, (35)

trace(W (α)) < ν2, (36)

then the closed-loop system defined in (8) is asymptoticallystable for any pairs (α, ˙α) ∈ � × � and (32) is satisfied,whereA(α, α),B2(α, α), and C2(α, α) are the closed-loopsystem matrices defined in (8) with θ and θ replaced byα and α using the change of variables defined in (20) and(21).

5. Parameter-dependent synthesis conditions

The next theorem provides PBMI conditions to synthe-sise robust GSH2/H∞ controller.

Theorem 5.1: For the LPV system defined in (1), givena scalar γ > 0 and a sufficiently small positive scalarε > 0, there exists a GS dynamic output-feedback con-troller K(θ (t )) in the form of (7) that minimises ν

such that the closed-loop system (8) is exponentiallystable and satisfies (31) and (32), if there exist con-tinuously differentiable parameter-dependent matricesP11(α) = P11(α)′ ∈ R

n×n, P22(α) = P22(α)′ ∈ Rn×n,

P21(α) ∈ Rn×n, P11∞(α) = P11∞(α)′ ∈ R

n×n,P22∞(α) = P22∞(α)′ ∈ R

n×n, P21∞(α) ∈ Rn×n, and

parameter-dependent matrices W (α) = W (α)′ ∈R

nz2×nz2 , R(α) ∈ Rn×n, S(α) ∈ R

n×n, T (α) ∈ Rn×n,

K1(α) ∈ Rn×n, K2(α) ∈ R

n×ny , K3(α) ∈ Rnu×ny ,

M(α) = M(α)′ ∈ Rn×n, and N(α) = N(α)′ ∈ R

n×n

satisfying the following PBMIs from (a) to (c):(a)

[�∞(α, α)

[�1(α) M(α)�2(α)] −M(α)

]< 05n+nz∞+nw∞

(37)or

[�∞(α, α)

[M(α)�1(α) �2(α)] −M(α)

]< 05n+nz∞+nw∞ ,

(38)with

�1(α) = [�A(α, α)R(α) + �Bu(α, α)K3(α) 0n×n] ,

�2(α) = [S(α)′ 0n×n 0n×nz∞ 0n×nw∞

],

�A(α, α) := A(α) − A(α), �Bu(α, α) := Bu(α) − Bu(α)

(39)

INTERNATIONAL JOURNAL OF CONTROL 587

and

�∞(α, α) :=⎡⎢⎢⎢⎢⎢⎢⎣

A(α)R(α) + Bu(α)K3(α) + (•)′ − P11∞(α)

K1(α) + A(α)′ − P21∞(α) S(α)A(α) + K2(α)Cy(α) + (•)′ − P22∞(α)

P11∞(α) − R(α) + ε(R(α)′A(α)′ + K3(α)′Bu(α)′) P21∞(α)′ − In + εK1(α)′

P21∞(α) − T (α) + εA(α)′ P22∞(α) − S(α) + ε(A(α)′S(α)′ +Cy(α)′K2(α)′)C∞(α)R(α) + D∞u(α)K3(α) C∞(α)

B∞(α)′ B∞(α)′S(α)′ + Dy∞(α)′K2(α)′

−ε(R(α) + R(α)′)

−ε(T (α) + In) −ε(S(α) + S(α)′)

ε(C∞(α)R(α) + D∞u(α)K3(α)) εC∞(α) −Inz∞

0nw∞×n 0nw∞×n D∞∞(α)′ −γ 2Inw∞

⎤⎥⎥⎥⎥⎥⎥⎦

(40)

[P11∞(α)

P21∞(α) P22∞(α)

]> 0, (41)

(b) [�2(α, α)

[�1(α) N(α)�3(α)] −N(α)

]< 05n+nw2

(42)

or [�2(α, α)

[N(α)�1(α) �3(α)] −N(α)

]< 05n+nw2

(43)

with

�3(α) = [S(α)′ 0n×n 0n×nw2

](44)

and

�2(α, α) :=

⎡⎢⎢⎢⎢⎣

A(α)R(α) + Bu(α)K3(α) + (•)′ − P11(α)

A(α)′ + K1(α) − P21(α) S(α)A(α) + K2(α)Cy(α) + (•)′ − P22(α)

P11(α) − R(α) + ε(R(α)′A(α)′ + K3(α)′Bu(α)′) P21(α)′ − In + εK1(α)′

P21(α) − T (α) + εA(α)′ P22(α) − S(α) + ε(A(α)′S(α)′ +Cy(α)′K2(α)′)B2(α)′ B2(α)′S(α)′ + Dy2(α)′K2(α)′

−ε(R(α) + R(α)′)

−ε(T (α) + In) −ε(S(α) + S(α)′)

0nw2×n 0nw2×n −Inw2

⎤⎥⎥⎥⎥⎦

(45)(c)

trace (W (α)) < ν2 (46)[P11(α)

P21(α) P22(α)

]> 0, (47)

588 A. K. AL-JIBOORY AND G. G. ZHU

⎡⎣ R(α) + R(α)′ − P11(α)

In + T (α) − P21(α) S(α) + S(α)′ − P22(α)

C2(α)R(α) + D2u(α)K3(α) C2(α) W (α)

⎤⎦ > 02n+nz2

(48)for any pair (α, ˙α) ∈ � × �, then the robust GSH2/H∞dynamic output feedback controller matrices in (7) aregiven as

Cc(α) = K3(α)F(α)−1,

Bc(α) = X (α)−1K2(α),

Ac(α) = X (α)−1[K1(α) − S(α)A(α)R(α)

− S(α)Bu(α)K3(α)F(α)

− X (α)K2(α)CyR(α)]F(α)−1, (49)

where F(α) ∈ Rn×n and X (α) ∈ R

n×n can be obtainedusing any full-rank matrix factorisation satisfyingX (α)F(α) = T (α) − S(α)R(α).Proof: See Appendix 1. �Remark 5.1: Once a feasible solution exists for Theo-rem 5.1, the controller coefficients in (15) (which areused for real-time implementation) can be retrieved fromthe coefficients of the controller matrices in (49) (inthe multi-simplex domain) using inverse transformation(multi-simplex to affine transformation). More specifi-cally, the coefficients Zj for i = 0, 1, …, q in (26) canbe obtained (from the multi-simplex coefficients in (25))using the following inverse transformation (for any num-ber of scheduling parameters q):

Z0 = 122q

2∑j1=1

2∑j2=1

· · ·2∑

jq=1

2∑ki=1

2∑k2=1

· · ·

2∑kq=1

Z j1, j2,..., jq,k1,k2,...,kq,

Zi = 122qθi

2∑j1=1

2∑j2=1

· · ·2∑

jq=1

2∑ki=1

2∑k2=1

· · ·

2∑kq=1

(−1) ji+iZ j1, j2,..., jq,k1,k2,...,kq . (50)

Remark 5.2: Based on (37), (38), (42), and (43), fourdifferent formulations of Theorem 5.1 can be obtained.At this stage, it is not clear which formulation achievesthe best performance. Therefore, for a given design prob-lem, these four formulations should be evaluated inde-pendently to obtain the best possible performance.

Remark 5.3: The conditions of Theorem 5.1 can be cus-tomised to deal with H2 or H∞ performance individu-ally. For example,H2 controller can be designed by using

inequalities (46), (47), and (48), along with (42) or (43).On the other hand,H∞ controller can be synthesised byusing inequalities (41) and (40), along with (37) or (38).

Remark 5.4: Theorem 5.1 handles the general case whenthe varying parameters affect both state matrix A(θ) andcontrol input matrix Bu(θ). On the other hand, the meth-ods described in Sato and Peaucelle (2013) and Daafouzet al. (2008) can also be extended to deal with the samecase by combining them with the approach presented inXie (2008) that utilises a three-step design procedure.

Remark 5.5: Exact knowledge of scheduling parameterscan be viewed as a special case of the developed condi-tions. To illustrate, setting δ(t)= 0 in (4) and deleting thelast row and last column of inequalities (37), (38), (42),and (43), perfect knowledge of scheduling parameters canbe recovered as a special case of Theorem 5.1.

5.1 PBMI algorithm

As shown above, due to themultiplications between deci-sion variables (M(α)S(α),N(α)S(α),M(α)R(α), and/orM(α)R(α) from one hand, and ε and other variables atthe other hand), the synthesis conditions of Theorem 5.1are formulated as PBMIs in terms of time-varying param-eters inside multi-simplex domain. This type of synthe-sis problem can be viewed as a special type of non-convex optimisation problem. In other words, PBMIsare equivalent to infinite-dimensional BMI constraintswhich is numerically non-tractable. To handle this prob-lem, numerical algorithm (Algorithm 1) is developed tosolve this type of optimisation problem. This algorithmassumes the possibility of solving PLMIs (for fixed ε)which is indeed possible with the advent of powerfultheoretical and computational tools (Oliveira & Peres,2007; Oliveira, de Oliveira, & Peres, 2008; Scherer, 2005).Thus, a specialised parser ROLMIP (Agulhari, Oliveira, &Peres, 2012), recently developed as a tool to perform suchPLMI manipulation, and LMI relaxations, works jointlywith the LMI parser YALMIP (Löfberg, 2004) and theLMI solver SeDuMi (Sturm, 1999), have been used inthis work to implement the conditions of Theorem 5.1 inAlgorithm1 to obtain the suboptimal solution of the non-convex optimisation problem. Although this algorithmdoes not guarantee the convergence to the global optima,significant conservativeness reduction can be expectedsince ν is monotonically non-increasing as demonstratedin the next section

INTERNATIONAL JOURNAL OF CONTROL 589

Algorithm 1: Parameter-Dependent Bilinear MatrixInequality AlgorithmInitialisation:

• SetM0(α) = In, N0(α) = In, ε0 = 0.1, and γ = 9.5.• GivenM0(α), N0(α), ε0 and γ , minimise ν underthe PLMI conditions to obtain R(α), S(α), T (α),K1(α), K2(α), and K3(α).

• Set R0(α) = R(α), S0(α) = S(α), T0(α) = T (α),K01 (α) = K1(α), K0

2 (α) = K2(α) andK03 (α) = K3(α).

• Set imax and Tolarance.• Set i = 1.

while i < imax OR |νi − νi−1| > Tolarance do

• Given Ri−1(α), Si−1(α), Ti−1(α), Ki−11 (α),

Ki−12 (α), Ki−1

3 (α), minimise ν under the PLMIconditions to obtainM(α), N(α), and ε.

• SetMi−1(α) = M(α), Ni−1(α) = N(α), εi−1 = ε.• GivenMi−1(α), Ni−1(α) and εi−1, minimise ν

under the PLMI conditions to obtain R(α), S(α),T (α), K1(α), K2(α), and K3(α).

• Set Ri(α) = R(α), Si(α) = S(α), Ti(α) = T (α),Ki1(α) = K1(α), Ki

2(α) = K2(α), andKi3(α) = K3(α).

• Set i = i + 1.

end

6. Numerical examples

The objective of the numerical examples presented in thissection is to verify the effectiveness of the developed con-ditions. Two illustrative examples have been borrowedfrom literature to facilitate comparisons with other syn-thesis methods.

Example 6.1: Consider the LPV system defined by thestate-space representation Xie (2012):

⎡⎢⎢⎣

A(θ ) Bu(θ ) B2(θ ) B∞(θ )

C∞(θ ) D∞u(θ ) D∞∞(θ )

C2(θ ) D2u(θ )

Cy(θ ) Dy2(θ ) Dy∞(θ )

⎤⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎣

0 11 + θ (t ) 2 0 1 1−1 1 0 1 0 00 2 −6 + θ (t ) 0 1 11 0 0 1 00 1 1 11 0 0 2 0

⎤⎥⎥⎥⎥⎥⎥⎦ ,

ζ = 0.1

ζ = 0.3

ζ = 0.2

1 2 3 4 5 61

1.5

2

2.5

3

3.5

4

Number of Iterations

ν

Figure . Algorithm convergence for different bounds on mea-surement noise.

where x(t), u(t), w(t), and y(t) are defined in (1). θ(t) isthe time-varying parameter that varies between −1 and1 with measurement noise bound �δ(t)� � ζ . The goalis to synthesise dynamic output-feedback GS controllerthat minimises the H2 bound ν from w2 to z2 subject toH∞ bound constraint γ from w� to z� (see Figure 1).Algorithm 1 is used to solve the PBMI conditions of The-orem 5.1 to synthesise mixedH2/H∞ controllers for thisexample. Figure 3 illustrates the algorithm convergencefor different bounds onmeasurement noiseswith the con-straint γ = 9.5. It is clear from Figure 3 that a few itera-tions are sufficient to reduce conservativeness.

Since the developed conditions encompass exactknowledge of scheduling parameters as a special case (seeRemark 5.5), this assumption is considered here to makea fair comparison with the existing results from literaturethat handle mixed H2/H∞ performance without mea-surement noise. Table 1 provides a comparison of theachieved performance using different methods from lit-erature with and without measurement noise. It can beseen that the developed conditions are about 10 timesless conservative than the best existing result (Xie, 2012)with exact scheduling parameter, and when measure-ment noise is present with a noise bound of ζ = 0.2, thedesign is still about 7 times less conservation than that ofXie (2012). Furthermore, Table 2 shows the closed-loopH2 norms at the vertices of the multi-simplex domainfor a controller synthesised with the H∞ constraint γ

= 9.5 and one can see that they are very close to theguaranteed bound of 1.197 (for ζ = 0) and 1.890 (forζ = 0.2), indicating the design is fairly tight and notconservative.

To study the conservativeness ofH∞ control, anothercontroller is synthesised with a tighter constraint on the

590 A. K. AL-JIBOORY AND G. G. ZHU

Table . Comparison of the mixed performance with and withoutmeasurement noise (�δ(t)� < ζ , γ = .).

ν (ζ = ) ν (ζ = .)Theorem . .

Conditions of Xie () . −Conditions of Apkarian, Tuan, and Bernussou () . −Conditions of Scherer et al. () . −

Table . Closed-loop H2 norm at the vertices of themulti-simplex domain withoutmeasurement noise (ζ =, the upper bound is ν = .) and with measurementnoise (ζ = ., the upper bound is ν = .).

Vertex# Vertex# Vertex# Vertex#

H2 (ζ = ) . . . .H2 (ζ = .) . . . .

Table . Closed-loop H2 and H∞ norms at thevertices of the multi-simplex domain with theconstraint γ = . (the achieved upper bound ontheH2 norm is ν = .).

Vertex# Vertex# Vertex# Vertex#

H2 . . . .H∞ . . . .

H∞ norm, γ = 1.5. The achieved bound on the H2norm for this controller is ν = 1.882. Table 3 providesthe closed-loop norms at the vertices of themulti-simplexdomain associated with this controller. It can be observedthat the vertexH2 andH∞ norms are tight to the designbounds, respectively, indicating the design is not conser-vative for bothH2 andH∞ norm bounds. Also, note thatfrom both tables (Table 2 and Table 3) the constraint onthe H∞ norm is satisfied with a much less conservativebound on theH2 norm.

Example 6.2: Consider the LPV system of a missilemodel that has been studied widely in literature (Apkar-ian & Tuan, 2000; Daafouz et al., 2008; Sato, 2010b). Thisrepresentative example is used to demonstrate the effec-tiveness of the developed synthesis conditions. To facil-itate comparisons with other design methods that canhandle uncertainties in scheduling parameters,H∞ con-troller is studied here as special case of Theorem 5.1 (seeRemark 5.3). The state-space model that represents thedynamic of the pitch-axis motion for a missile model is

Developed conditions

[Sato(2010b)]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

ζ

γ

Figure . Comparison of H∞ performance vs. uncertaintybetween the developed conditions and Sato (b).

given by

⎡⎢⎢⎣

A(θ ) Bu(θ ) B2(θ ) B∞(θ )

C∞(θ ) D∞u(θ ) D∞∞(θ )

C2(θ ) D2u(θ )

Cy(θ ) Dy2(θ ) Dy∞(θ )

⎤⎥⎥⎦

=

⎡⎢⎢⎢⎢⎣

−0.89 − 0.89θ (t ) 1 −0.119 0.01 0.01−142.6 − 178.25θ (t ) 0 −130.8 0 0

0 1 1 00 1 1

−1.52 0 0.01 0.01

⎤⎥⎥⎥⎥⎦ ,

with the bounds �θ(t)�� 1 and |θ (t )|≤ 1. The uncertaintyδ(t) in themeasured scheduling parameter is supposed tobe bounded by ζ , i.e. �δ(t)� � ζ with |δ(t )|≤ 10 × ζ .

The procedure developed in Section 3 is used to con-vert the system matrices and optimisation variables intomulti-simplex domain. Then, the PBMI (37) with (41)and (40) are solved using Algorithm 1 to obtain GS H∞controller (PBMI (38) with (41) and (40) could also beused). Figure 4 illustrates considerable robustness of thesynthesised controller over the method by Sato (2010b).

Another comparison between our synthesisedH∞ controllers and those based on the method of

INTERNATIONAL JOURNAL OF CONTROL 591

Table . Comparison of γ with Daafouz et al. ().

ζ

. . . . . . .

Developed conditions . . . . . . . . .Daafouz et al. () . . . . . . . − −Note:−means no feasible solution.

A

0 5 10 15 20 25 30 35−1.5

−1

−0.5

0

0.5

1

1.5

Measured scheduling parameterActual scheduling parameter

0 5 10 15 20 25 30 35−1

0

1

2

3

4

5

Time (sec.)

Disturbance inputPerformance output

B

-0.150 5

0.1

Figure . Simulation: (A) measured and actual scheduling param-eters; (B) disturbance attenuation.

Daafouz et al. (2008) is given in Table 4. For perfectmeasurement (ζ = 0), the achievedH∞ performance forboth controllers is fairly close as shown in Table 4, whereour controllers achieve a little worse performance boundthan those based on the method of Daafouz et al. (2008).As ζ increases, the controllers provided by Daafouz et al.(2008) show considerable sensitivity to the uncertaintyand the H∞ performance degrades exponentially. It isclear from Table 4 that when ζ > 0.48, the conditionsof Daafouz et al. (2008) failed to provide any stabilising

controller while our synthesis conditions provide con-trollers for a much wider range of ζ .

Furthermore, closed-loop simulations are conductedwith actual scheduling parameter defined by θ(t) =cos (0.25t) and their bounds �θ(t)� � 1 and | θ (t ) |≤ 1.Measurement noisewith bounds given by �δ(t)�� 0.2 and| δ(t ) |≤ 2 is added to imitate noisy scheduling parame-ter. The detailed resulting controller coefficients are pre-sented in Appendix 2. Figure 5 (A) shows the measured(noisy) scheduling parameter and the actual schedulingparameter, respectively. L2 disturbance signal defined byw(t) = 15exp ( − 0.3t)sin (0.3t) is generated as distur-bance input to the closed-loop system. Figure 5(B) illus-trates the corresponding system response to the distur-bance input with noisy measurement scheduling param-eter. These simulations show that good robustness againstmeasurement noise in scheduling parameter is achieved,so does good disturbance attenuation.

These comparisons’ results and simulations provideevidence that the proposed control design methodachieves considerable robustness against uncertainties inthe scheduling parameters.

7. Conclusion

New synthesis conditions are derived for synthesisingrobust GS dynamic output-feedback controllers withmixed H2/H∞ performance. These conditions are for-mulated in terms of PBMIs with varying parametersinside a multi-simplex domain. The scheduling param-eters are assumed to be noisy, which is a realistic assump-tion. The synthesised controller not only guaranteesrobust stability and mixed H2/H∞ performance butalso is robust against scheduling parameter uncertain-ties. A multi-simplex modelling approach is used tomodel the scheduling parameters and their uncertain-ties. Parameter-dependent Lyapunov function approachis utilised to guarantee robust stability. Two indepen-dent Lyapunov matrices are utilised for H2 and H∞performance indexes, respectively, which significantlyreduces conservativeness associated with using the sin-gle Lyapunov matrix. We believed that this feature canbe extended in future research to handle other objec-tive criteria with reduced conservativeness. Since PBMIproblems are non-convex, a numerical algorithm is alsodeveloped to solve the synthesis conditions iteratively.

592 A. K. AL-JIBOORY AND G. G. ZHU

With the aid of this algorithm, considerable performanceimprovement can be attained with a few iterations. Oursynthesis conditions are compared with other designmethods from literature through two illustrative exam-ples. Comparison results demonstrate that the developedconditions significantly reduce design conservativeness.

Acknowledgement

Ali Khudhair Al-Jiboory would like to thank the Higher Com-mittee for Education Development (HCED) and UniversityofDiyala in Iraq for their financial support during his graduatestudy at MSU.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. To be precise, since we are dealing with LPV systems,the H2 and/or H∞ performances are not well defined inthe addressed problem yet. However, we use the mixedH2/H∞ performance here with slightly abused terminol-ogy so that the reader can easily grasp our problem setting.We will postpone the strict definition of the control prob-lem until Section 4 since necessary definitions and trans-formations need to be introduced in the next two sections.

2. Sometimes the dependency on t will be omitted for nota-tional simplicity.

3. Equation (50) is derived from Equation (26) using sim-ple algebraic manipulations; however, the proof is omittedhere for brevity.

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Appendices

Appendix 1. Proof of Theorem 5.1

To simplify notations in the proof, closed-loop sys-tem matrices A(α, α), B∞(α, α), B2(α, α), C∞(α, α),C2(α, α), and D∞(α, α) in Lemmas 4.1 and 4.2 will bedenoted by A, B∞, B2, C∞, C2, and D∞, respectively.Using slack variable approach (Pipeleers, Demeulenaere,Swevers, &Vandenberghe, 2009), additional optimisationvariables U (α) can be introduced to inequality (33) viaFinsler’s lemma (de Oliveira & Skelton, 2001) to decou-ple the dynamic matrix A from Lyapunov matrix P(α).This leads to the following improved condition of (33):

�(α) +U (α)V (α) +V (α)′U (α)′ < 0 (A1)

with

�(α) =

⎡⎢⎢⎣

−P(α) P(α) 0 0P(α) 0 0 00 0 −Inz D(α)

0 0 D(α)′ −γ 2Inw

⎤⎥⎥⎦ (A2)

and

U (α) =

⎡⎢⎢⎣

G(α)′ IεG(α)′ 0

0 00 0

⎤⎥⎥⎦ ,

V (α) =[A(α)′ −I C(α)′ 00 0 0 B(α)

]

such thatV (α)⊥′�(α)V (α)⊥ < 0, with

V (α)⊥ =

⎡⎢⎢⎣

I 0A(α)′ C(α)′

0 I0 0

⎤⎥⎥⎦ .

SubstitutingU (α), V(α), and (A2) into (A1), inequal-ity (33) can be written in terms of the closed-loop matri-ces as

594 A. K. AL-JIBOORY AND G. G. ZHU

⎡⎢⎢⎣

AG(α) + (•)′ − P∞(α)

P∞(α) − G(α) + ε(G(α)′A′) −ε(G(α) + G(α)′)

C∞G(α) εC∞G(α) −Inz∞

B′∞ 0nw∞×n D′

∞ −γ 2Inw∞

⎤⎥⎥⎦ < 02n+nz∞+nw∞ . (A3)

Define

P∞(α) :=[P11∞(α)

P21∞(α) P22∞(α)

]> 0.

Block(2, 2) of (A3) implies G(α) + G(α)′ > 0. Therefore, the matrix G(α) is invertible and can be partitioned as

G(α) =[R(α) G1(α)

F(α) G2(α)

],

V (α) := G(α)−1 =[S(α)′ V1(α)

X (α)′ V2(α)

].

Define the following non-singular congruence transformation matrices:

Qg(α) =[R(α) IF(α) 0

], Qv (α) =

[I S(α)′

0 X (α)′

], (A4)

such that

G(α)Qv (α) = Qg(α), V (α)Qg(α) = Qv (α). (A5)

In order to guarantee that congruence transformations in (A4) have full rank, block matrices F(α) and X (α) shouldbe non-singular. If this is not the case, we can always perturb F(α) and X (α) with sufficiently small matrices in termsof norms such that F(α) + �F(α) and X (α) + �X (α) are invertible.

Define also

Qv (α)′P∞(α)Qv (α) := P∞(α) =[P11∞(α)

P21∞(α) P22∞(α)

]> 0. (A6)

Multiplying (A3) by T1(α) from right and by T1(α)′ from left with

T1(α) =

⎡⎢⎢⎣Qv (α) 0 0 0

0 Qv (α) 0 00 0 Inz∞ 00 0 0 Inw∞

⎤⎥⎥⎦ ,

and using (A5) and (A6), we obtain⎡⎢⎢⎣

Q′v (α)AQg(α) + (•)′ − P∞(α)

P∞(α) − Q′v (α)Qg(α) + ε(Q′

v (α)AQg(α))′ −ε(Q′v (α)Qg(α) + (•)′) 0

C∞Qg(α) εC∞Qg(α) −Inz∞

B′∞Qv (α) 0 D∞ −γ 2Inw∞

⎤⎥⎥⎦ < 0, (A7)

Then, substituting closed-loop matrices (9),(11),(13), and (14), and considering (A4) with the following relations:

Qv (α)′Qg(α) =[R(α) IT (α) S(α)

], T (α) := S(α)R(α) + X (α)F(α), (A8)

INTERNATIONAL JOURNAL OF CONTROL 595

Qv (α)′AQg(α) =[A(α)R(α) + Bu(α)K3(α) A(α)

K1(α) S(α)A(α) + K2(α)Cy

]:= �(α),

Qv (α)′B∞ =[

B∞(α)

S(α)B∞(α) + K2(α)Dy∞(α)

],

C∞Qg(α) = [C∞(α)R(α) + Dzu(α)K3(α) C∞(α)

],

D∞ = D∞∞(α),

inequality (A7) can be written as

⎡⎢⎢⎢⎢⎢⎢⎣

�(α) + (•)′ −[P11∞(α) P21∞(α)′

P21∞(α) P22(α)

]

P∞(α) −[R(α) InT (α) S(α)

]+ ε�(α)′ −ε

([R(α) InT (α) S(α)

]+ (•)′

) [

C∞(α)R(α) + Dzu(α)K3(α) Cz(α)]

�1(α) −Inz∞ [B∞(α)′ B∞(α)′S(α)′ + Dyw(α)′K2(α)′

]0nw∞×2n D∞∞(α)′ −γ 2Inw∞

⎤⎥⎥⎥⎥⎥⎥⎦ < 04n+nz∞+nw∞ ,

(A9)

with

�1(α) := ε[C∞(α)R(α) + D∞u(α)K3(α) C∞(α)

],

where K1(α), K2(α), and K3(α) are intermediate con-troller variables defined as

[K1(α) K2(α)

K3(α) 0

]:=[

X (α) S(α)Bu(α)

0 I

] [Ac(α) Bc(α)

Cc(α) 0

] [F(α) 0

CyR(α) I

]

+[S(α)

0

]A(α)

[R(α) 0

]. (A10)

This definition of controller intermediate variablesallows us to construct controller matrices based onlyon the measured scheduling parameters. Thus, we allowopen-loop matrices in (A10) to be dependent on themulti-simplex variables α not α. However, A(α) andBu(α) can be written as

A(α) = A(α) + A(α) − A(α) = A(α) + �A(α, α)

Bu(α) = Bu(α) + Bu(α) − Bu(α) = Bu(α) + �Bu(α, α),

(A11)

where �A(α, α) := A(α) − A(α), and �Bu(α, α) :=Bu(α) − Bu(α). Substituting (A11) into (A10) to obtain

[K1(α) K2(α)

K3(α) 0

]=[

X (α) S(α)(Bu(α) + �Bu(α, α))

0 I

] [Ac(α) Bc(α)

Cc(α) 0

]

×[

F(α) 0CyR(α) I

]+[S(α)A(α)R(α) 0

0 0

]

+[S(α)�A(α, α)R(α) 0

0 0

], (A12)

Therefore, (A12) can be written as

[K1(α) K2(α)

K3(α) 0

]=

[X (α) S(α)Bu(α)

0 I

] [Ac(α) Bc(α)

Cc(α) 0

] [F(α) 0

CyR(α) I

]

+[S(α)A(α)R(α) 0

0 0

]+ � (A13)

� :=[S(α)�A(α, α)R(α) + S(α)�Bu(α, α)Cc(α)F(α) 0

0 0

].

SubstitutingCc(α) = K3(α)F(α)−1 into � yields

� =[S(α)�A(α, α)R(α) + S(α)�Bu(α, α)K3(α) 0

0 0

].

(A14)

596 A. K. AL-JIBOORY AND G. G. ZHU

Substituting (A14) into (A13), then into (A9), leads to

⎡⎢⎢⎢⎢⎢⎢⎣

�(α) + (•)′ −[P11∞(α) P21∞(α)′

P21∞(α) P22∞(α)

]

P∞(α) −[R(α) InT (α) S(α)

]+ ε�(α)′ −ε

([R(α) InT (α) S(α)

]+ (•)′

) [

C∞(α)R(α) + D∞u(α)K3(α) C∞(α)]

�1(α) −Inz∞ [B∞(α)′ B∞(α)′S(α)′ + Dy∞(α)′K2(α)′

]0nw∞×2n D∞∞(α)′ −γ 2Inw∞

⎤⎥⎥⎥⎥⎥⎥⎦

+[

02n×2n

�2(α)′�1(α) 02n+nz∞+nw∞×2n

]< 04n+nz∞+nw∞ ,

(A15)

which is in the form of (18) of Lemma 2.1 with

�2(α)′�1(α) =⎡⎢⎢⎣S(α)�A(α, α)R(α) + S(α)�Bu(α, α)K3(α) 0n×n

0n×n 0n×n0nz∞ ×n 0nz∞ ×n0nw∞ ×n 0nw∞ ×n

⎤⎥⎥⎦

(A16)

which can be factorised into

�1(α) = [�A(α, α)R(α) + �Bu(α, α)K3(α) 0n×n] ,

�2(α) = [S(α)′ 0n×n 0n×nz∞ 0n×nw∞

],

(A17)which directly leads to (37) or (38).

Consider now (34) of theH2 performance. In a similarway, inequality (34) can be written in the form of (A1)with

�(α) :=⎡⎣−P(α) P(α) 0

P(α) 0 00 0 I

⎤⎦ ,

U (α) :=⎡⎣ G(α)′ 0

εG(α)′ 00 I

⎤⎦ , V (α) :=

[A′ −I 0B′2 0 −I

].

(A18)

Thus, after substituting (A18) into (A1), inequality (34)can be written as

⎡⎣ AG(α) + (•)′ − P(α)

P(α) − G(α) + ε(G(α)′A′) −ε(G(α) + G(α)T )

B′2 0nw2×n −Inw2

⎤⎦

< 02n+nw2.

(A19)

Then, following similar steps to the proof of (37) and(38), the proof of(42) and (43) can be done in a the same

way. Therefore, it will be omitted here for brevity.Consider now (35) that can be written as[

G(α) + (•)′ − P(α)

C2G(α) W (α)

]> 02n+nz2 . (A20)

Multiplying (A20) by T2(α) from the right and by T2(α)′

from the left with

T2(α) =[Qv (α) 0

0 Inz2

],

leads to[Qv (α)′Qg(α) + (•)′ − Qv (α)′P(α)Qv (α)

C2Qg(α) W (α)

]> 02n+nz2 . (A21)

Considering (A5) and using

C2Qg(α) = [C2(α)R(α) + D2u(α)K3(α) C2(α)

],

(A22)yield

⎡⎣[R(α) IT (α) S(α)

]+ (•)′ −

[P11(α) P21(α)′

P21(α) P22(α)

][

C2(α)R(α) + D2u(α)K3(α) C2(α)]

W (α)

⎤⎦ > 02n+nz2 ,

(A23)

which leads to (48). On the other hand, solving (A10)for the intermediate variables K1(α), K2(α), and K3(α),yields the following relations:

K1(α) = X (α)Ac(α)F(α) + S(α)A(α)R(α)

+ S(α)Bu(α)Cc(α)F(α) + K2(α)CyR(α),

K2(α) = X (α)Bc(α),

K3(α) = Cc(α)F(α).

Controller matrices can be solved in the following order:Cc(α), Bc(α), and Ac(α), which leads to (49).

INTERNATIONAL JOURNAL OF CONTROL 597

Appendix 2. Controller coefficients

This appendix presents the optimal solution of Theo-rem 5.1 for the controller used in the time-domain simu-lation of Example 6.2. Synthesis variables, controller coef-ficients at the vertices of multi-simplex domain, and con-troller coefficients at the original parameter space aregiven here.

(1) Optimal solution of Theorem 5.1 (using Algo-rithm 1) that provides synthesis variables at thevertices of the multi-simplex domain:

R11 =[

0.0633 −0.7227−0.0776 16.292

],

R12 =[

0.0605 −0.6691−0.0378 15.625

],

R21 =[

0.0629 −0.7147−0.0716 16.192

],

R22 =[

0.0600 −0.661−0.0318 15.525

]

S11 =[

95.285 −0.4786−1.8761 0.0424

],

S12 =[

101.13 −0.5076−1.8567 0.0402

],

S21 =[

96.162 −0.4829−1.8732 0.0421

],

S22 =[

102.01 −0.5119−1.8537 0.0399

]

T11 =[

1.0377 8.9478−0.0047 −0.2907

],

T12 =[

1.1564 10.021−0.0070 −0.3249

],

T21 =[

1.0555 9.1087−0.0050 −0.2958

],

T22 =[

1.1742 10.182−0.0074 −0.3300

]

X11 =[−5.0306 85.611

0.1174 −2.3377

],

X12 =[−4.9777 85.615

0.1068 −2.1957

],

X21 =[−5.0249 85.654

0.1158 −2.3161

],

X22 =[−4.9667 85.559

0.1052 −2.1749

]

K111 =[ −7.1765 −2624.6

−0.79143 40.015

],

K112 =[−9.2562 −2556.9

−0.7661 39.474

],

K121 =[−7.4884 −2614.4

−0.7876 39.934

],

K122 =[−9.5681 −2546.7

−0.7623 39.393

]

K211 =[

5242−26.172

],

K212 =[

4806.6−18.356

],

K221 =[

5176.7−24.999

],

K222 =[

4741.3−17.184

]

K311 = [0.0589 30.865

],

K312 = [−0.0528 32.274],

K321 = [0.0422 31.077

],

K322 = [−0.0696 32.486]

(2) Controller coefficients at the vertices of the multi-simplex domain (obtained from Theorem 5.1(49)):

Ac11 =[−521.69 2472

−24.784 10.574

],

Ac12 =[−432.03 1567.3

−19.976 −34.917

],

Ac21 =[−507.28 2318.2

−24.004 2.7346

],

Ac22 =[−419.86 1454.6

−19.333 −40.464

]

Bc11 =[−5849.5

−282.49

],

Bc12 =[−5018.9

−235.66

],

Bc21 =[−5716.3

−274.91

],

Bc22 =[ −4906

−229.38

]

Cc11 = [0.0589 30.865

],

Cc12 = [−0.0528 32.274],

Cc21 = [0.0422 31.077

],

Cc22 = [−0.0696 32.486]

598 A. K. AL-JIBOORY AND G. G. ZHU

(3) Controller coefficients in the original parame-ter space (obtained using inverse transformation(50)). These coefficients are used in (15) to imple-ment the controller in real time:

Ac0 =[−216.16 398.96

−4.8781 −276.32

],

Ac1 =[0.64916 135.700.00393 1.2816

]

Bc0 =[ −16863

−432.06

],

Bc1 =[71.2401.1735

]

Cc0 = [0.00679 22.01

],

Cc1 = [0.00137 0.03769

]