# Static Output Feedback Control Design for Descriptor Systems

Authors: Mohamed Yagoubi
Journal: Journal of Control Science and Engineering
 (2010)

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## Abstract

The static output feedback (SOF) synthesis problem for descriptor systems is considered in this paper. LMI-based algorithms are proposed to find potentially structured SOF gains ensuring admissibility and evenH∞ performance of the closed-loop system. These algorithms are then used to propose a (descriptor) observer-based H∞ controller design method. An alternative technique for determining such separated estimation/control structure, after the design step, is also proposed. Several numerical examples, throughout the paper, demonstrate the effectiveness of the proposed algorithms.


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## Body

## 1. Introduction

The descriptor framework encompasses a wide class of systems [1]. The so-called descriptor representation describes the dynamic part, static part, and improper part of the system. Even in the nonsingular case, descriptor systems are very useful to manipulate physical models without losing a physical parameterization or to describe controller/filter implementation. Strict LMI characterizations for admissibility, H∞ and H2 norms of descriptor systems are established in [2–7].The static output feedback (SOF) control problem plays a central role in control theory and applications (see, e.g., [8, 9] and references therein). Because of its generality, several LMI-based SOF design methods have been proposed for state-space systems over last decade whereas there are few papers discussing the SOF control problem for descriptor systems [10]. Some existing results on the subject deal with only the problem of the closed-loop system regularity or impulse immunity [11, 12].In this paper, a novel coordinate-descent iterative numerical procedure that can synthesize a SOF, ensuring the admissibility of the closed-loop system, is derived in the continuous time setting. The proposed algorithm does not require any special decomposition of the considered descriptor system. Contrary to some existing iterative LMI algorithms (see, e.g., [13] or [14]) no additional variables are needed for the proposed method. The idea underlying this method can be seen as an extension to the descriptor case of some LMI-based algorithms previously proposed in [15] or [16] in the state-space case.Furthermore, it is well known in control theory that an important problem in control arises when a specific structure on the overall control scheme is considered, especially when dealing with complex or distributed systems. Thus, we extend the proposed algorithms to deal with SOFH∞ controllers with a fixed structure for descriptor systems. The algorithm developed here is used to compute local optimal solutions. Decomposition in consecutive subproblems of the initial problem, following the same lines as in [14] or [15], is adopted.In addition, the observer-basedH∞ control problem for descriptor systems is considered in this paper as a direct application of the proposed approach.This paper is organized as follows. some preliminary results on descriptor systems are recalled in Section2. In Section 3, some LMI-based algorithms for ensuring the admissibility of closed-loop descriptor systems, via potentially structured SOF gains are presented. Then, in Section 4, the extension of these algorithms to the H∞ control problem is given. Section 5 tackles the observer-based H∞ control problem for descriptor systems. Some concluding remarks are given in Section 6.Notation 1.
The notationA>0 (resp.,A≥0) stands for A definite positive (resp., semidefinite positive). The notation He{A} stands for AT+A and • for terms that are induced by symmetry. I (resp.,Γ) indicates the identity matrix (resp., the matrix with all entries set to 1) with proper dimensions. The symbol ⊗ denotes the direct product of matrices.


## 2. Preliminaries

Let us consider the descriptor system given by(1)Eẋ=Ax+B1w+B2u,z=C1x,y=C2x,
where x∈ℝn is the descriptor variable, w∈ℝq is the disturbance, u∈ℝm is the control input, z∈ℝl is the controlled output, and y∈ℝp is the measured output. The matrix E may be singular and we denote its rank byrank(E)=r≤n. It is known that systems having direct transmission paths from w and u to z and y can be transformed by augmenting the descriptor variable as pointed out for example in [5].Ifdet(sE-A)≠0 for some complex numbers, then system (1) is said to be regular [1]. A regular system of the form (1) is said to be impulse-free if(2)deg(det(sE-A))=rank(E).Definition 1.
System (1) is admissible if it is regular and has neither impulsive modes nor unstable finite modes.As defined above, the admissibility of descriptor systems can be formulated as an LMI feasibility problem as stated in [17]. Let us introduce matrices V,U∈ℝn×(n-r) which are of full-column rank such that EV=0 and ETU=0 and matrices EL,ER∈ℝn×r of full-column rank such that  E=ELERT.Lemma 2 (see [18]).
System (1) is admissible if and only if there exist a positive definite matrix P∈ℝn×n and a matrix S∈ℝ(n-r)×(n-r) satisfying the LMI:
(3)He{A(PET+VSUT)}<0,Lemma 3 (see [18]).
For a given positive numberγ, the system (1) is stable and
(4)∥C1(sE-A)-1B1∥∞<γ
holds if and only if there exist a positive definite matrix P∈ℝn×n and a matrix S∈ℝ(n-r)×(n-r) satisfying the LMI:
(5)[He{A(PET+VSUT)}+B1B1T(PET+VSUT)TC1T•-γ2I]<0.Lemma 4 (see [18]).
LetP∈ℝn×n be symmetric such that ERTPER>0 and S be nonsingular. Then, PET+VSUT is nonsingular and its inverse is expressed as
(6)(PET+VSUT)-1=QE+URVT,
where Q∈ℝn×n is symmetric and R∈ℝ(n-r)×(n-r) is nonsingular such that
(7)(ERTPER)-1=ELTQEL,(8)R=(UTU)-1S-1(VTV)-1.


## 3. SOF Admissibility Control

### 3.1. LMI-Based Algorithm for the SOF Admissibility Control Problem

In this section, the static output feedback (SOF) admissibility control for descriptor systems given by (1) is investigated. The following assumption is made on system (1).Assumption.
(i)(E,A,B2) is finite dynamics stabilizable and impulse controllable [1]. (ii)(E,A,C2) is finite dynamics detectable and impulse observable [1].The SOF admissibility control problem is to find an SOF controller:(9)u=Fy,
where F∈ℝm×p such that the closed-loop system with w=0 given by
(10)Eẋ=(A+B2FC2)x
is admissible.According to Lemma2, (10) is admissible if and only if there exist a positive definite matrix P and a matrix S such that the following condition:(11)He{(A+B2FC2)(PET+VSUT)}<0
holds.Recall that the matrixP is related with the finite dynamics behavior and the matrix S is related with regularity and the impulsive behavior. The main point of an iterative LMI algorithm concerning the problem stated above is to find initial variables P and S. In this note, Algorithm 6 will consider the problem of finding initializing variables, namely, Ps and Ss. Then, Algorithm 7 will consider the problem of deriving a feasible solution F of the SOF admissibility control problem stated above starting with the variables Ps and Ss derived by Algorithm 6.By setting:K1:=FC2P and K2:=FC2VS, (11) becomes(12)He{A(PET+VSUT)+B2(K1ET+K2UT)}<0.
Note that (12) is an LMI, however, this inequality does not take into account all the information since it ignores matrix  C2. Thus, pre- and postmultiplying (11) by (PET+VSUT)-T and (PET+VSUT)-1 leads to the following dual LMI: (13)He{(QE+URVT)T(A+B2FC2)}<0,whereQandRsatisfy (7) and (8).Following the same lines as above, we can set:L1:=QB2F and L2:=RTUTB2F and transform (13) to (14)He{(QE+URVT)TA+(ETL1+VL2)C2}<0.However, equalities (7) and (8) are not linear. According to the idea of the cone complementary linearization method [17], (7) can be relaxed to an LMI as follows. Minimizing a linearized version of trace(ELTQEPER) under the following LMI:
(15)[ERTPERI•ELTQEL]≥0.
On the other hand, (8) can be relaxed by minimizing the following norm:(16)∥Si+1-Si∥F:=trace((Si+1-Si)T(Si+1-Si))and fixingRi+1=(UTU)-1Si-1(VTV)-1, where the superscript i means the number of iteration. This can be done by minimizing trace(Y) under the following constraint:
(17)[Y(Si+1-Si)T•I]>0.Based on this, an iterative algorithm is proposed hereafter to find an initial pair(Ps,Ss).Algorithm 6.
We have the following steps.Step 1.
Seti=1, P0=I, S0=I, and R1=(UTU)-1S0-1(VTV)-1. Choose the computation parameter μ>0.Step 2.
Solve the following optimization problem forPi,Qi,Si and K1,K2,L1,L2. Minimize trace(ELTQiEPi-1ER+ELTQi-1EPiER)+μtrace(Y) subject to the following LMI constraints:
(18)He{A(PiET+VSiUT)+B2(K1ET+K2UT)}<0,He{(QiE+URiVT)TA+(ETL1+VL2)C2}<0,[ERTPiERI•ELTQiEL]>0,Pi>0,Qi>0,[Yi(Si-Si-1)T•I]>0.Step 3.
If trace(ELTQiEPiER)-r<ε1and trace(Yi)<ε2,where ε1,ε2some prescribed tolerances, initial variables Psand Ssare found, stop.Step 4.
If the difference of two iterations is small that is trace(ELTQiEPiER)-trace(ELTQi-1EPi-1ER)<ε3 andtrace(Yi)-trace(Yi-1)<ε4 where ε3,ε4 are some prescribed tolerances, initial variables Ps and Ss may not be found, stop.Step 5.
Seti=i+1, go back to Step 2.Recall that the closed-loop system (10) is admissible and all the finite eigenvalues of the pair (E,A+B2FC2) are in the strict left side of the line α/2 in the complex plane if and only if the following condition:
(19)He{(A+B2FC2)(PET+VSUT)}-αEPET<0
holds for a definite positive matrix P∈ℝn×n and a matrix S∈ℝ(n-r)×(n-r). This condition will be used in Algorithm 7 to find a SOF gain ensuring admissibility of (10).Algorithm 7.
We have the following steps.Step 1.
Seti=1, P1=Ps and S1=Ss, as obtained from Algorithm 6.Step 2.
Solve the following optimization problem forF with given  Pi,Si.
Minimizeαi subject to the following LMI constraints:
(20)Pi>0,He{(A+B2FC2)(PiET+VSiUT)}-αiEPiET<0.Step 3.
Ifαi≤0, F ensures admissibility of the closed-loop (10), stop.Step 4.
Seti=i+1. Solve the optimization problem (20) for Pi,Si with given F.Step 5.
Ifαi≤0, F ensures admissibility of the closed-loop (10), stop.Step 6.
Solve the following optimization problem forPi,Si with given F and αi.
Minimizetrace(Pi) subject to the LMI constraints (20).Step 7.
If∥Pi-Pi-1∥/∥Pi∥<ε, a prescribed tolerance, a solution may not be found, stop, else set i=i+1; go back to Step 2.Remarks 8.
(i) To shed light on the role of the computation parameterμ, let us say roughly that by increasing μ we can reduce more rapidly the difference between (ERTPER)-1 and ELTQEL. On the other hand, by decreasing μ we can reduce more rapidly the difference between R and (UTU)-1S-1(VTV)-1.
(ii) In Algorithm6, the selection of parameters ε1 and ε2 is not crucial since the variables Ps and Ss are not necessarily feasible solutions of (11) but only some suitable initializing solutions. The feasible solutions will be derived by Algorithm 7.
(iii) The discussion on the iterative procedure and convergence, particularly, of Algorithm7 follow those in [13].
(iv) As iterative LMI methods, when the proposed procedure fails to find a solution it does not mean that no solution can be found.Example 9.
Consider the SOF admissibility control problem of system (1) with w=0 and the following data:
(21)E=[1000001000101000000000000],A=[1-3-3160-2700-5105800600-65480],B2=[120251211-3],C2=[100113-1-735]T.
The open-loop system is not admissible (unstable finite dynamics and impulsive behavior). Note that the Algorithm given in [10] could not find a solution for this system. Algorithm 6 yields initial variables Ps and Ss after 2 iterations. Then, an SOF gain F is found as
(22)F=[-6.72312.94360.0571-0.1467]
using Algorithm 7 after 3 iterations. In this case, α=-5.5599 and the closed-loop pair (E,A+B2FC2) has 3 finite eigenvalues: {-8694.5;-6.2;-2.8}.Example 10.
Consider the SOF admissibility control problem of (1) with w=0 and the following data:
(23)E=[100000010000101000000100000000000000],A=[1-3-32169-27700-51-3-158306-1904-20-2-10-654-280],B2=[61210951-19121-35498],C2=[102-1340148-75].
The open-loop system is clearly not admissible. Algorithm 6 yields initial variables Ps and Ss after 2 iterations. Using Algorithm 7 after 5 iterations, an SOF gain F is found as
(24)F=[4.06830.77162.76220.5292-7.2871-1.3928].
In this case, α=-0.3943 and the closed-loop pair (E,A+B2FC2) has 4 finite eigenvalues:
(25){-0.7328±7.7547i;-1.0350±3.3930i}.


### 3.2. An LMI-Based Algorithm for the Structured SOF Admissibility Control Problem

In this section, the structured SOF admissibility control problem is investigated. The structure constraint on the SOF controller is defined by(26)F=ΛF⊗F,ΛF∈{0,1}m×p,
where ΛF∈{0,1}m×p.It is easy to see that(27)ΩΛF:={F∈ℝm×p/F=ΛF⊗F,ΛF∈{0,1}m×p}
is a convex set.The main idea underlying Algorithm11 comes from an obvious fact: any SOF gain Fs solving the SOF admissibility control problem may be decomposed as(28)Fs=ΛF⊗Fs+Λ̃F⊗Fs=F1+F2,
where Λ̃F=Γm×p-ΛF. A structured SOF (associated to the desired structure represented by ΛF) may then be obtained when annulling F2.Algorithm 11.
We have the following steps.Step 1.
SetF=Fs (Fs is the SOF gain obtained by Algorithm 7) and find matrices P1>0 and S1 such that
(29)He{(A+B2FsC2)(P1ET+VS1UT)}<0
holds.Step 2.
Solve the following optimization problem forF1 and F2 with given Pi and Si.
(30)minF1,F2β,He{(A+B2(F1+F2)C2)(PiET+VSiUT)}<0,trace(F2TF2)<β,F1=ΛF⊗F1,F2=Λ̃F⊗F2.Step 3.
Ifβ<ε5, a prescribed tolerance, a feasible solution is found and (E,A+B2F1C2) is admissible, stop.Step 4.
Solve the following optimization problem forPi,Si with given F1,F2:
(31)mintrace(Pi),Pi>0,He{(A+B2(F1+F2)C2)(PiET+VSiUT)}<0.Step 5.
If∥Pi-Pi-1∥/∥Pi∥<ε, a prescribed tolerance, a solution may not be found, stop. Else set i=i+1, go back to Step 2.Example 12.
Consider the structured SOF admissibility control problem of (1) with w=0 and the following data:
(32)E=[1000010000000000],A=[12-10-42020-13-90-400],B2=[21100015],C2=[100-12001]T,ΛF=[1001].
Table 1 summarizes the results obtained when applying successively the proposed three algorithms. The notation l+k in Table 1 means that l is the iteration number of Algorithm 6 and k is the iteration number of Algorithm 7.Table 1
A structured SOF admissibility control problem.
AlgorithmIterations numberThe obtained SOFAlgorithms6 and 71+3Fs=[7.5519-45.2884-1.50829.0436]Algorithm117F1=[0.750900-0.8472]


## 3.1. LMI-Based Algorithm for the SOF Admissibility Control Problem

In this section, the static output feedback (SOF) admissibility control for descriptor systems given by (1) is investigated. The following assumption is made on system (1).Assumption.
(i)(E,A,B2) is finite dynamics stabilizable and impulse controllable [1]. (ii)(E,A,C2) is finite dynamics detectable and impulse observable [1].The SOF admissibility control problem is to find an SOF controller:(9)u=Fy,
where F∈ℝm×p such that the closed-loop system with w=0 given by
(10)Eẋ=(A+B2FC2)x
is admissible.According to Lemma2, (10) is admissible if and only if there exist a positive definite matrix P and a matrix S such that the following condition:(11)He{(A+B2FC2)(PET+VSUT)}<0
holds.Recall that the matrixP is related with the finite dynamics behavior and the matrix S is related with regularity and the impulsive behavior. The main point of an iterative LMI algorithm concerning the problem stated above is to find initial variables P and S. In this note, Algorithm 6 will consider the problem of finding initializing variables, namely, Ps and Ss. Then, Algorithm 7 will consider the problem of deriving a feasible solution F of the SOF admissibility control problem stated above starting with the variables Ps and Ss derived by Algorithm 6.By setting:K1:=FC2P and K2:=FC2VS, (11) becomes(12)He{A(PET+VSUT)+B2(K1ET+K2UT)}<0.
Note that (12) is an LMI, however, this inequality does not take into account all the information since it ignores matrix  C2. Thus, pre- and postmultiplying (11) by (PET+VSUT)-T and (PET+VSUT)-1 leads to the following dual LMI: (13)He{(QE+URVT)T(A+B2FC2)}<0,whereQandRsatisfy (7) and (8).Following the same lines as above, we can set:L1:=QB2F and L2:=RTUTB2F and transform (13) to (14)He{(QE+URVT)TA+(ETL1+VL2)C2}<0.However, equalities (7) and (8) are not linear. According to the idea of the cone complementary linearization method [17], (7) can be relaxed to an LMI as follows. Minimizing a linearized version of trace(ELTQEPER) under the following LMI:
(15)[ERTPERI•ELTQEL]≥0.
On the other hand, (8) can be relaxed by minimizing the following norm:(16)∥Si+1-Si∥F:=trace((Si+1-Si)T(Si+1-Si))and fixingRi+1=(UTU)-1Si-1(VTV)-1, where the superscript i means the number of iteration. This can be done by minimizing trace(Y) under the following constraint:
(17)[Y(Si+1-Si)T•I]>0.Based on this, an iterative algorithm is proposed hereafter to find an initial pair(Ps,Ss).Algorithm 6.
We have the following steps.Step 1.
Seti=1, P0=I, S0=I, and R1=(UTU)-1S0-1(VTV)-1. Choose the computation parameter μ>0.Step 2.
Solve the following optimization problem forPi,Qi,Si and K1,K2,L1,L2. Minimize trace(ELTQiEPi-1ER+ELTQi-1EPiER)+μtrace(Y) subject to the following LMI constraints:
(18)He{A(PiET+VSiUT)+B2(K1ET+K2UT)}<0,He{(QiE+URiVT)TA+(ETL1+VL2)C2}<0,[ERTPiERI•ELTQiEL]>0,Pi>0,Qi>0,[Yi(Si-Si-1)T•I]>0.Step 3.
If trace(ELTQiEPiER)-r<ε1and trace(Yi)<ε2,where ε1,ε2some prescribed tolerances, initial variables Psand Ssare found, stop.Step 4.
If the difference of two iterations is small that is trace(ELTQiEPiER)-trace(ELTQi-1EPi-1ER)<ε3 andtrace(Yi)-trace(Yi-1)<ε4 where ε3,ε4 are some prescribed tolerances, initial variables Ps and Ss may not be found, stop.Step 5.
Seti=i+1, go back to Step 2.Recall that the closed-loop system (10) is admissible and all the finite eigenvalues of the pair (E,A+B2FC2) are in the strict left side of the line α/2 in the complex plane if and only if the following condition:
(19)He{(A+B2FC2)(PET+VSUT)}-αEPET<0
holds for a definite positive matrix P∈ℝn×n and a matrix S∈ℝ(n-r)×(n-r). This condition will be used in Algorithm 7 to find a SOF gain ensuring admissibility of (10).Algorithm 7.
We have the following steps.Step 1.
Seti=1, P1=Ps and S1=Ss, as obtained from Algorithm 6.Step 2.
Solve the following optimization problem forF with given  Pi,Si.
Minimizeαi subject to the following LMI constraints:
(20)Pi>0,He{(A+B2FC2)(PiET+VSiUT)}-αiEPiET<0.Step 3.
Ifαi≤0, F ensures admissibility of the closed-loop (10), stop.Step 4.
Seti=i+1. Solve the optimization problem (20) for Pi,Si with given F.Step 5.
Ifαi≤0, F ensures admissibility of the closed-loop (10), stop.Step 6.
Solve the following optimization problem forPi,Si with given F and αi.
Minimizetrace(Pi) subject to the LMI constraints (20).Step 7.
If∥Pi-Pi-1∥/∥Pi∥<ε, a prescribed tolerance, a solution may not be found, stop, else set i=i+1; go back to Step 2.Remarks 8.
(i) To shed light on the role of the computation parameterμ, let us say roughly that by increasing μ we can reduce more rapidly the difference between (ERTPER)-1 and ELTQEL. On the other hand, by decreasing μ we can reduce more rapidly the difference between R and (UTU)-1S-1(VTV)-1.
(ii) In Algorithm6, the selection of parameters ε1 and ε2 is not crucial since the variables Ps and Ss are not necessarily feasible solutions of (11) but only some suitable initializing solutions. The feasible solutions will be derived by Algorithm 7.
(iii) The discussion on the iterative procedure and convergence, particularly, of Algorithm7 follow those in [13].
(iv) As iterative LMI methods, when the proposed procedure fails to find a solution it does not mean that no solution can be found.Example 9.
Consider the SOF admissibility control problem of system (1) with w=0 and the following data:
(21)E=[1000001000101000000000000],A=[1-3-3160-2700-5105800600-65480],B2=[120251211-3],C2=[100113-1-735]T.
The open-loop system is not admissible (unstable finite dynamics and impulsive behavior). Note that the Algorithm given in [10] could not find a solution for this system. Algorithm 6 yields initial variables Ps and Ss after 2 iterations. Then, an SOF gain F is found as
(22)F=[-6.72312.94360.0571-0.1467]
using Algorithm 7 after 3 iterations. In this case, α=-5.5599 and the closed-loop pair (E,A+B2FC2) has 3 finite eigenvalues: {-8694.5;-6.2;-2.8}.Example 10.
Consider the SOF admissibility control problem of (1) with w=0 and the following data:
(23)E=[100000010000101000000100000000000000],A=[1-3-32169-27700-51-3-158306-1904-20-2-10-654-280],B2=[61210951-19121-35498],C2=[102-1340148-75].
The open-loop system is clearly not admissible. Algorithm 6 yields initial variables Ps and Ss after 2 iterations. Using Algorithm 7 after 5 iterations, an SOF gain F is found as
(24)F=[4.06830.77162.76220.5292-7.2871-1.3928].
In this case, α=-0.3943 and the closed-loop pair (E,A+B2FC2) has 4 finite eigenvalues:
(25){-0.7328±7.7547i;-1.0350±3.3930i}.


## 3.2. An LMI-Based Algorithm for the Structured SOF Admissibility Control Problem

In this section, the structured SOF admissibility control problem is investigated. The structure constraint on the SOF controller is defined by(26)F=ΛF⊗F,ΛF∈{0,1}m×p,
where ΛF∈{0,1}m×p.It is easy to see that(27)ΩΛF:={F∈ℝm×p/F=ΛF⊗F,ΛF∈{0,1}m×p}
is a convex set.The main idea underlying Algorithm11 comes from an obvious fact: any SOF gain Fs solving the SOF admissibility control problem may be decomposed as(28)Fs=ΛF⊗Fs+Λ̃F⊗Fs=F1+F2,
where Λ̃F=Γm×p-ΛF. A structured SOF (associated to the desired structure represented by ΛF) may then be obtained when annulling F2.Algorithm 11.
We have the following steps.Step 1.
SetF=Fs (Fs is the SOF gain obtained by Algorithm 7) and find matrices P1>0 and S1 such that
(29)He{(A+B2FsC2)(P1ET+VS1UT)}<0
holds.Step 2.
Solve the following optimization problem forF1 and F2 with given Pi and Si.
(30)minF1,F2β,He{(A+B2(F1+F2)C2)(PiET+VSiUT)}<0,trace(F2TF2)<β,F1=ΛF⊗F1,F2=Λ̃F⊗F2.Step 3.
Ifβ<ε5, a prescribed tolerance, a feasible solution is found and (E,A+B2F1C2) is admissible, stop.Step 4.
Solve the following optimization problem forPi,Si with given F1,F2:
(31)mintrace(Pi),Pi>0,He{(A+B2(F1+F2)C2)(PiET+VSiUT)}<0.Step 5.
If∥Pi-Pi-1∥/∥Pi∥<ε, a prescribed tolerance, a solution may not be found, stop. Else set i=i+1, go back to Step 2.Example 12.
Consider the structured SOF admissibility control problem of (1) with w=0 and the following data:
(32)E=[1000010000000000],A=[12-10-42020-13-90-400],B2=[21100015],C2=[100-12001]T,ΛF=[1001].
Table 1 summarizes the results obtained when applying successively the proposed three algorithms. The notation l+k in Table 1 means that l is the iteration number of Algorithm 6 and k is the iteration number of Algorithm 7.Table 1
A structured SOF admissibility control problem.
AlgorithmIterations numberThe obtained SOFAlgorithms6 and 71+3Fs=[7.5519-45.2884-1.50829.0436]Algorithm117F1=[0.750900-0.8472]


## 4. SOFH∞ control

The objective of the SOFH∞ control problem is to find an SOF controller F∈ℝm×p such that the closed-loop system given by (33)Ecẋc=Acxc+Bcw,z=Ccxc
with Ac=A+B2FC2, Bc=B1, Cc=C1is admissible and satisfies the H∞ norm constraint: (34)∥Cc(sEc-Ac)-1Bc∥∞<γforγ>0.
According to Lemma 3, this is possible if and only if there exist a positive definite matrix P and a matrix S such that(35)[He{Ac(PET+VSUT)}+BcBcT(PET+VSUT)TCcT•-γ2I]<0.
As done for the SOF admissibility control problem, we propose here two iterative LMI algorithms, namely, Algorithms 13 and 14 that are similar to Algorithms 6 and 7. The main idea is to find first initial variables  Ps and Ss and based on them derive a feasible solution Fs.We sum up briefly the construction of these algorithms in the following.Algorithm 13.
It is constructed by using the LMI constraints:(36)[𝒜+B1B1T(PiET+VSiUT)TC1T•-γ2I]<0,[ℬ+C1TC1(QiE+URiVT)TB1•-γ2I]<0,[ERTPiERI•ELTQiEL]>0,Pi>0,Qi>0,[Yi(Si-Si-1)T•I]>0,
where 𝒜=He{A(PiET+VSiUT)+B2(K1ET+K2UT)}  and ℬ=He{(QiE+URiVT)TA+(ETL1+VL2)C2}, instead of constraints (18) in Algorithm 6.Algorithm 14.
It is constructed by using the LMI constraints:(37)Pi>0,[Θ(PiET+VSiUT)TC1T•-γ2I]<0
with Θ:=He{(A+B2FC2)(PiET+VSiUT)+B2(K1ET+K2UT)}+B1B1T-αiEPiETinstead of constraints (20) in Algorithm 7.Example 15.
Consider the SOFH∞ control problem of (1) with γ=1 and the following data:
(38)E=[300020000],A=[-2-110-10110],B1=[01-1],C1=[00-1]T,B2=[01-1],C2=[011]T.
Algorithm 13 yields initial variables Ps and Ss after one iteration. Then, an SOF gain F satisfying γ=1 is found as F=2.4162 using Algorithm 14 after 3 iterations. In this case, α=-0.2763 and the closed-loop pair (E,A+B2FC2) has two finite eigenvalues: {-0.2521±0.4342i}.
Following the same lines as in Algorithm11, we can propose an algorithm associated to the structured SOF H∞ control problem. Let us denote it Algorithm 6. (For brevity reasons this algorithm is not given here, however it can be easily constructed using Algorithm 11.)Example 16.
Consider the structured SOFH∞ control problem of (1) with γ=1 and the following data:
(39)E=[100010000],A=[10-100-8-11000],B1=[101],C1=[011],(40)B2=[211001],C2=[10210-1],ΛF=[1001].
Table 2 summarizes the results obtained for this example.Table 2
A structured SOFH∞ control problem (γ=1).
AlgorithmIterations numberThe obtained SOFAlgorithms13 and 141+7F=[-3.1507-8.20521.26163.2928]62F=[0.4677001.6957]


## 5.H∞ Observer-Based Control for Descriptor Systems

In this section, an alternative technique for determining observer-basedH∞ controllers for descriptor systems is discussed. Motivated by the practical appeal of such forms, many papers have dealt with this topic in the state-space case (see, e.g., [19] or [20] and references therein). The following results can be seen as an extension to the descriptor case of those derived in the state-space case.Consider the descriptor system (1) under Assumption 5 and the observer-based feedback controller described by(41)Ex̂̇=Ax̂+B2u+L(y-C2x̂),u=Kx̂,The closed-loop system is given by(42)Eclẋcl=Aclxcl+Bclw,z=Cclxcl,
where
(43)Ecl=[E00E],Acl=[A+B2K-B2K0A-LC2],Bcl=[B1B1],Ccl=[C10],xcl=[xx-x̂].
The suboptimal observer-based H∞ control problem is to find an observer gain L and a feedback gain K such that the closed-loop (42) is admissible and satisfies the H∞ norm constraint:(44)∥Ccl(sEcl-Acl)-1Bcl∥∞<γforγ>0.
Let us introduce the following augmented model: (45)(Σa){Eaẋa=Aaxa+Ba1w+Ba1ua,z=Ca1xa,ya=Ca2xa,
where
(46)Ea=Ecl,Aa=[A00A],Ba1=Bcl,Ba2=[B200I],Ca1=[C10],Ca2=[I-I0C2]
and the structured SOF gain: (47)Fss:=[K00L].
The observer-based H∞ control problem given above is equivalent to finding a structured SOF H∞ gain of the form (47) such that the closed-loop Fl(Σa,Fss) is admissible and satisfies the H∞ norm constraint:
(48)∥Fl(Σa,Fss)∥∞<γ.
Thus, using the proposed Algorithms 13, 14, and 6 in the previous section one can solve the aforementioned problem (i.e., H∞ control problem for descriptor systems).Example 17.
Consider the observer-basedH∞ control problem of (1) with γ=1 for the descriptor system given in Example 15. The problem consists in finding a structured H∞ SOF with the following fixed structure:
(49)ΛF=[1110000100010001],
for an augmented model constructed following (45), (46), and (47). After 2 iterations of Algorithm 13, 3 iterations of Algorithm 5 and finally 9 iterations of Algorithm 6, we obtain the following admissible structured SOF gain:
(50)F1=[5.95272.361116.74260000-1.290400012.0856000-11.9409].


## 6. Conclusion

In this note, the static output feedback synthesis problem for descriptor systems is considered. Some LMI–based algorithms to find potentially structured SOF gains ensuring admissibility andH∞ performance of the closed-loop system are proposed. These algorithms are used in order to propose a (descriptor) observer-based H∞ controller design method. Although iterative the proposed approaches are appealing and find suitable solutions for some benchmarking numerical examples.



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*Source: 482561-2010-08-30.xml*