Static Output Feedback Control Design for Descriptor Systems

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 even H_∞ 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.

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 SOF H_∞ 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-based H_∞ 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 Section 2. 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 notation 𝐴>0 (resp.,𝐴≥0) stands for 𝐴 definite positive (resp., semidefinite positive). The notation 𝐻𝑒{𝐴} stands for 𝐴^𝑇+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 by𝐸𝑥̇=𝐴𝑥+𝐵₁𝑤+𝐵₂𝑢,𝑧=𝐶₁𝑥,𝑦=𝐶₂𝑥, (1) where 𝑥∈ℝ^𝑛 is the descriptor variable, 𝑤∈ℝ^𝑞 is the disturbance, 𝑢∈ℝ^𝑚 is the control input, 𝑧∈ℝ^𝑙 is the controlled output, and 𝑦∈ℝ^𝑝 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].

If det(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 ifdeg(det(𝑠𝐸−𝐴))=rank(𝐸). (2)

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∈ℝ^{𝑛×(𝑛−𝑟)} which are of full-column rank such that 𝐸𝑉=0 and 𝐸^𝑇𝑈=0 and matrices 𝐸_𝐿,𝐸_𝑅∈ℝ^𝑛×𝑟 of full-column rank such that  𝐸=𝐸_𝐿𝐸^𝑇_𝑅.

Lemma 2 (see [18]). System (1) is admissible if and only if there exist a positive definite matrix 𝑃∈ℝ^𝑛×𝑛 and a matrix 𝑆∈ℝ^{(𝑛−𝑟)×(𝑛−𝑟)} satisfying the LMI: He{𝐴(𝑃𝐸^𝑇+𝑉𝑆𝑈^𝑇)}<0, (3)

Lemma 3 (see [18]). For a given positive number 𝛾, the system (1) is stable and ‖𝐶₁(𝑠𝐸−𝐴)⁻¹𝐵₁‖_∞<𝛾 (4) holds if and only if there exist a positive definite matrix 𝑃∈ℝ^𝑛×𝑛 and a matrix 𝑆∈ℝ^{(𝑛−𝑟)×(𝑛−𝑟)} satisfying the LMI: He{𝐴(𝑃𝐸^𝑇+𝑉𝑆𝑈^𝑇)}+𝐵₁𝐵^𝑇₁(𝑃𝐸^𝑇+𝑉𝑆𝑈^𝑇)^𝑇𝐶^𝑇₁•−𝛾²𝐼<0. (5)

Lemma 4 (see [18]). Let 𝑃∈ℝ^𝑛×𝑛 be symmetric such that 𝐸^𝑇_𝑅𝑃𝐸_𝑅>0 and 𝑆 be nonsingular. Then, 𝑃𝐸^𝑇+𝑉𝑆𝑈^𝑇 is nonsingular and its inverse is expressed as (𝑃𝐸^𝑇+𝑉𝑆𝑈^𝑇)⁻¹=𝑄𝐸+𝑈𝑅𝑉^𝑇, (6) where 𝑄∈ℝ^𝑛×𝑛 is symmetric and 𝑅∈ℝ^{(𝑛−𝑟)×(𝑛−𝑟)} is nonsingular such that (𝐸^𝑇_𝑅𝑃𝐸_𝑅)⁻¹=𝐸^𝑇_𝐿𝑄𝐸_𝐿,𝑅=(𝑈^𝑇𝑈)⁻¹𝑆⁻¹(𝑉^𝑇𝑉)⁻¹. (7)(8) Table 2 summarizes the results obtained for this example.

Table 2 

A structured SOF H_∞ control problem (𝛾=1).

5. H_∞ Observer-Based Control for Descriptor Systems

In this section, an alternative technique for determining observer-based H_∞ 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 by𝐸𝑥̇=𝐴𝑥+𝐵₂𝑢+𝐿(𝑦−𝐶₂𝑥),𝑢=𝐾𝑥, (41)

The closed-loop system is given by 𝐸_𝑐𝑙𝑥̇_𝑐𝑙=𝐴_𝑐𝑙𝑥_𝑐𝑙+𝐵_𝑐𝑙𝑤,𝑧=𝐶_𝑐𝑙𝑥_𝑐𝑙, (42) where 𝐸_𝑐𝑙=𝐸00𝐸,𝐴_𝑐𝑙=𝐴+𝐵₂𝐾−𝐵₂𝐾0𝐴−𝐿𝐶₂,𝐵_𝑐𝑙=𝐵₁𝐵₁,𝐶_𝑐𝑙=𝐶₁0,𝑥_𝑐𝑙=𝑥𝑥.𝑥− (43) 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:‖𝐶_𝑐𝑙(𝑠𝐸_𝑐𝑙−𝐴_𝑐𝑙)⁻¹𝐵_𝑐𝑙‖_∞<𝛾for𝛾>0. (44) Let us introduce the following augmented model: (Σ_𝑎)𝐸_𝑎𝑥̇_𝑎=𝐴_𝑎𝑥_𝑎+𝐵_𝑎1𝑤+𝐵_𝑎1𝑢_𝑎,𝑧=𝐶_𝑎1𝑥_𝑎,𝑦_𝑎=𝐶_𝑎2𝑥_𝑎, (45) where 𝐸_𝑎=𝐸_𝑐𝑙,𝐴_𝑎=𝐴00𝐴,𝐵_𝑎1=𝐵_𝑐𝑙,𝐵_𝑎2=𝐵₂00𝐼,𝐶_𝑎1=𝐶₁0,𝐶_𝑎2=𝐼−𝐼0𝐶₂ (46) and the structured SOF gain: 𝐹_𝑠𝑠∶=𝐾00𝐿. (47) 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 𝐹_𝑙(Σ_𝑎,𝐹_𝑠𝑠) is admissible and satisfies the H_∞ norm constraint: ‖𝐹_𝑙(Σ_𝑎,𝐹_𝑠𝑠)‖_∞<𝛾. (48) 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-based H_∞ 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: Λ_𝐹=1110000100010001, (49) 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: 𝐹₁=5.95272.361116.74260000−1.290400012.0856000−11.9409. (50)

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 and H_∞ 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.