Journal of Control Science and Engineering

Volume 2010 (2010), Article ID 482561, 8 pages

http://dx.doi.org/10.1155/2010/482561

## Static Output Feedback Control Design for Descriptor Systems

^{1}(Institut de Recherche en Communications et Cybernétique de Nantes) IRCCyN, UMR CNRS, 6597 at 1 rue de la Noë, B.P. 93101, 44432 Nantes Cedex 3, France^{2}(Ecole des Mines de Nantes) EMN, 4 rue Alfred Kastler La Chantrerie, 44307 Nantes, France

Received 9 December 2009; Accepted 19 May 2010

Academic Editor: L. Z. Yu

Copyright © 2010 Mohamed Yagoubi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 performance of the closed-loop system. These algorithms are then used to propose a (descriptor) observer-based 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, and 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 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 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 control problem is given. Section 5 tackles the observer-based control problem for descriptor systems. Some concluding remarks are given in Section 6.

*Notation 1. *The notation (resp.,) stands for definite positive (resp., semidefinite positive). The notation stands for and for terms that are induced by symmetry. (resp.) 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 where is the descriptor variable, is the disturbance, is the control input, is the controlled output, and is the measured output. The matrix may be singular and we denote its rank by. It is known that systems having direct transmission paths from and to and can be transformed by augmenting the descriptor variable as pointed out for example in [5].

If 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

*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 which are of full-column rank such that and and matrices of full-column rank such that .

Lemma 2 (see [18]). * System (1) is admissible if and only if there exist a positive definite matrix and a matrix satisfying the LMI:
*

Lemma 3 (see [18]). * For a given positive number , the system (1) is stable and
**
holds if and only if there exist a positive definite matrix and a matrix satisfying the LMI:
*

Lemma 4 (see [18]). * Let be symmetric such that and be nonsingular. Then, is nonsingular and its inverse is expressed as
**
where is symmetric and is nonsingular such that
**
Table 2 summarizes the results obtained for this example.*

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

In this section, an alternative technique for determining observer-based 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

The closed-loop system is given by where The suboptimal observer-based control problem is to find an observer gain and a feedback gain such that the closed-loop (42) is admissible and satisfies the norm constraint: Let us introduce the following augmented model: where and the structured SOF gain: The observer-based control problem given above is equivalent to finding a structured SOF gain of the form (47) such that the closed-loop is admissible and satisfies the norm constraint: Thus, using the proposed Algorithms 13, 14, and 6 in the previous section one can solve the aforementioned problem (i.e., control problem for descriptor systems).

*Example 17. *Consider the observer-based control problem of (1) with for the descriptor system given in Example 15. The problem consists in finding a structured SOF with the following fixed structure:
for an augmented model constructed following (45), (46), and (47). After 2 iterations of Algorithm 13, 3 iterations of Algorithm and finally 9 iterations of Algorithm , we obtain the following admissible structured SOF gain:

#### 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 performance of the closed-loop system are proposed. These algorithms are used in order to propose a (descriptor) observer-based controller design method. Although iterative the proposed approaches are appealing and find suitable solutions for some benchmarking numerical examples.

#### References

- L. Dai,
*Singular Control Systems*, Lectures Notes in Control and Information Sciences, Springer, Berlin, Germany, 1989. - M. Ikeda, T. W. Lee, and E. Uezato, “A strict LMI condition for ${\text{H}}_{2}$ control of descriptor systems,” in
*Proceedings of the IEEE Conference on Decision and Control*, vol. 1, pp. 601–604, Sydney, Australia, 2000. View at Scopus - J. Y. Ishihara and M. H. Terra, “On the Lyapunov theorem for singular systems,”
*IEEE Transactions on Automatic Control*, vol. 47, no. 11, pp. 1926–1930, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - F. L. Lewis, “A survey of linear singular systems,”
*Circuits, Systems, and Signal Processing*, vol. 5, no. 1, pp. 3–36, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - I. Masubuchi, Y. Kamitane, A. Ohara, and N. Suda, “${\text{H}}_{\infty}$ control for descriptor systems: a matrix inequalities approach,”
*Automatica*, vol. 33, no. 4, pp. 669–673, 1997. View at Google Scholar · View at Scopus - E. Uezato and M. Ikeda, “Strict LMI conditions for stability, robust stabilization, and ${\text{H}}_{\infty}$ control of descriptor systems,” in
*Proceedings of the 38th IEEE Conference on Decision and Control (CDC '99)*, pp. 4092–4097, Phoenix, Ariz, USA, December 1999. View at Scopus - S. Xu and J. Lam,
*Robust Control and Filtering of Singular Systems*, Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 2006. - A. Trofino-Neto and V. Kucera, “Stabilization via static output feedback,”
*IEEE Transactions on Automatic Control*, vol. 38, no. 5, pp. 764–765, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J.-T. Yu, “A convergent algorithm for computing stabilizing static output feedback gains,”
*IEEE Transactions on Automatic Control*, vol. 49, no. 12, pp. 2271–2275, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - C.-H. Kuo and C.-H. Fang, “An LMI approach to admissibilization of uncertain descriptor systems via static output feedback,” in
*Proceedings of the American Control Conference*, pp. 5104–5109, June 2003. View at Scopus - D. L. Chu and D. W. C. Ho, “Necessary and sufficient conditions for the output feedback regularization of descriptor systems,”
*IEEE Transactions on Automatic Control*, vol. 44, no. 2, pp. 405–412, 1999. View at Google Scholar · View at Scopus - V. Lovass-Nagy, D. L. Powers, and R. J. Schilling, “On regularizing descriptor systems by output feedback,”
*IEEE Transactions on Automatic Control*, vol. 39, no. 7, pp. 1507–1509, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - A. Fujimori, “Optimization of static output feedback using substitutive LMI formulation,”
*IEEE Transactions on Automatic Control*, vol. 49, no. 6, pp. 995–999, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. Yagoubi and P. Chevrel, “Structured ${\text{H}}_{2}$ controller synthesis via a dilated LMI based algorithm,” in
*Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference (CDC-ECC '05)*, pp. 5746–5751, December 2005. View at Publisher · View at Google Scholar · View at Scopus - Y. He and Q.-G. Wang, “An improved ILMI method for static output feedback control with application to multivariable PID control,”
*IEEE Transactions on Automatic Control*, vol. 51, no. 10, pp. 1678–1683, 2006. View at Google Scholar - M. Yagoubi and P. Chevrel, “An ILMI approach to structure constrained control design,” in
*Proceedings of the European Control Conference*, Porto, Portugal, 2001. - L. El Ghaoui, F. Oustry, and M. Aït Rami, “A cone complementarity linearization algorithm for static output-feedback and related problems,”
*IEEE Transactions on Automatic Control*, vol. 42, no. 8, pp. 1171–1176, 1997. View at Google Scholar - E. Uezato and M. Ikeda, “Strict LMI conditions for stability, robust stabilization, and ${\text{H}}_{\infty}$ control of descriptor systems,” in
*Proceedings of the 38th IEEE Conference on Decision and Control (CDC '99)*, pp. 4092–4097, Phoenix, Ariz, USA, December 1999. - D. Alazard and P. Apkarian, “Exact observer-based structures for arbitrary compensators,”
*International Journal of Robust and Nonlinear Control*, vol. 9, no. 2, pp. 101–118, 1999. View at Google Scholar · View at Scopus - M. Yagoubi and P. Chevrel, “Direct versus Indirect observer-based ${\text{H}}_{\infty}$ control,” in
*Proceedings of the IFAC World Congress*, Barcelona, Spain, 2002.