Mathematical Problems in Engineering

Volume 2008 (2008), Article ID 356124, 11 pages

http://dx.doi.org/10.1155/2008/356124

## LMI Conditions for Robust Stability of 2D Linear Discrete-Time Systems

^{1}Laboratoire d'Electronique, Signaux, Systemes et Informatique (LESSI), Département de Physique, Faculté des Sciences Dhar El Mehraz, B.P. 1796, 30000 Fes-Atlas, Morocco^{2}Research Unit, Constrained and Robust Regulation, Department of Physics, Faculty of Sciences Semlalia, B.P. 2390, 40000 Marrakech, Morocco^{3}Departamento de Ingenieria de Sistemas y Automática, Universidad de Valladolid, 47005 Valladolid, Spain

Received 11 May 2007; Revised 1 October 2007; Accepted 8 February 2008

Academic Editor: Pramod Khargonekar

Copyright © 2008 A. Hmamed et al. 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

Robust stability conditions are derived for uncertain 2D linear discrete-time systems, described by Fornasini-Marchesini second models with polytopic uncertainty. Robust stability is guaranteed by the existence of a parameter-dependent Lyapunov function obtained from the feasibility of a set of linear matrix inequalities, formulated at the vertices of the uncertainty polytope. Several examples are presented to illustrate the results.

#### 1. Introduction

Robust stability is an important issue for many applications, as it is the first requirement for any design. Methodologies to check stability are usually derived from Lyapunov functions. In particular, quadratic stability (QS) has played a central role, as it gives conditions simple to check. In fact, during the last decades, the QS concept has been widely used for robust analysis and control design for uncertain systems [1–4]. Using QS, the stability of a polytope of matrices can be checked by testing the feasibility of a set of linear matrix inequalities (LMIs). An important characteristic is that this set of LMIs involves only the vertices of the uncertain domain, which simplifies the robust stability tests [3].

Unfortunately, it is known that tests derived from QS might lead to very conservative results in several cases. Recently, other techniques have appeared, based on parameter-dependent Lyapunov functions [5–8] or piecewise Lyapunov functions [9, 10], providing less conservative results. In particular, [5] gives sufficient LMI conditions for robust stability of uncertain discrete-time systems. The generalization of these conditions to robust D-stability has been published in [11]. The key idea used in these papers, which will be used here, is to introduce new variables and increase the size of the LMIs, to obtain sufficient conditions for the existence of a parameter-dependent Lyapunov function.

This paper concentrates on two-dimensional (2D) systems because these 2D systems play important roles in image data processing and transformation, water stream heating, thermal processes, biomedical imaging, gas absorption, and many other areas [12], so the study of these systems has received much attention in past decades. Thus, a great number of stability and control results, related to 2D systems, have been reported in the literature , for example, using 2-D Lyapunov equations, some stability results have already been obtained in [13–16].

In particular, the problem of stability for uncertain 2D discrete-time systems is studied here. The class of systems under consideration is described by Fornasini-Marchesini models with polytopic uncertainty, which is frequent in image processing problems. However, it must be pointed out that the results are quite general, in the sense that with adequate modifications, parallel results can be obtained for other 2D systems (e.g., Roesser models, as presented in [17]). New sufficient conditions for robust stability are obtained from the feasibility of a set of LMIs formulated at the vertices of uncertainty polytope, extending the results proposed by the authors in [18]. Several examples will be presented and discussed to illustrate the results.

*Notation*

For real symmetric matrices and , the notation (resp., ) means that
the matrix is positive
semidefinite (resp., positive definite). is the identity
matrix with appropriate dimensions. means the
Euclidean norm. If no explicitly stated, matrices are assumed to have
compatible dimensions. For symmetric matrices, off-diagonal blocks are
abbreviated with as follows:

#### 2. Problem Formulation and Preliminary Results

Consider the following 2D linear discrete-time system described by a Fornasini-Marchesini second model [16]where is the state vector, is the dynamic matrix, and the boundary conditions are given byWe first introduce the notion of asymptotic stability of 2D discrete-time systems.

*Definition 2.1 (See [12]). *The 2D linear discrete-time system (2.1) is said to be asymptotically stable if
under , where and .

The following lemma gives a sufficient condition for the asymptotic stability of 2D linear discrete-time systems described by (2.1) in terms of an LMI.

Lemma 2.2 (See [13]). * The 2D linear discrete-time system (2.1) is asymptotically stable if there exist matrices ,
,
, and
such that the
following LMI holds:
**where *

*Remark 2.3. * As proven in [13], (2.4)
includes the previous results in [14, 15].

*Remark 2.4. *The LMI (2.4) is useful for stability analysis but not for synthesis of
controllers because it involves three variables , , and that render the
linearization of the problem a difficult task. Thus, for the synthesis problem,
the following alternative result will be used.

Lemma 2.5. *The 2D
linear discrete-time system (2.1) is asymptotically stable if there exist
matrices
,
,
, and
such that the
following LMI holds
**where or equivalently*

*Proof. *It is possible to obtain (2.5) substituting , , and in (2.4) and
using a Schur complement argument.

Suppose now that is not exactly known, but belongs to a given polytopic uncertain domain . In this way, any matrix inside the domain can be written as a convex combination of the vertices of the uncertainty polytope: We begin our discussion by defining Robust Stability of system (2.1) under the structured model (2.7).

*Definition 2.6. *System (2.1) is robustly stable in the
uncertainty domain (2.7) if there exist
, , and
such that
for all such that , where

To the authors knowledge, there is no general and systematic way to formally determine as a function of the uncertain parameter . Such a matrix is called a parameter-dependent Lyapunov matrix.

An effective way of addressing such problem is to look for a single Lyapunov matrix which solves inequality (2.8). Unfortunately, this approach is known to provide quite conservative results, but it constitutes one of the first results in the quadratic approach. The test for this kind of stability, also known as a quadratic stability (QS) test, is summarized in the following lemma.

Lemma 2.7. *The
uncertain system (2.1) is robustly stable in the uncertainty domain (2.7) if there
exist matrices
,
, , and such that
*

Theorem 2.8. *The following conditions are
equivalent.*

(i)*There exist matrices , , , and such that*(ii)*There exist matrices , , , , , and such that*

*Proof. *The proof is rather
straightforward. First, if (2.10) holds for some , by setting , , and applying the Schur complement, (2.12) is satisfied.
On the other hand, if (2.12) holds for some , multiplying (2.12) from the left by and from the
right by , where , then (2.10) follows.

*Remark 2.9. *Condition (ii) appears as a direct expansion of condition (i) via its “Schur
complement” formulation. An LMI is obtained in which the Lyapunov matrix is not involved
in any product with the dynamic matrix , thanks to the introduction of the additional
matrices and . This feature enables one to write new robust
stability conditions which, although sufficient, are not too conservative, as
shown later in some numerical examples, thanks to the presence of the extra
degrees of freedom provided by the introduction of matrices and .

The next three lemmas state sufficient conditions for
the existence of a parameter-dependent Lyapunov function given bysuch that (2.8) holds.

Lemma 2.10 (See [19]). * The
uncertain system (2.1) is robustly stable in the uncertainty domain (2.7) if there
exist matrices
, ,
, and such that
**then, for any , a parameter-dependent matrix given by (2.13)
satisfies (2.8).*

*Proof. *It is clear that given by (2.13)
is a positive-definite parameter Lyapunov matrix. Using (2.13), the definition of given in (2.7),
and the direct expansion, one gets
Such an expansion is also
adopted in [9]. Imposing conditions (3.1)–(3.4), one gets
Inequality (3.6) implies that (2.8)
holds, so the theorem is proven.

*Remark 3.2. *Theorem 3.1 provides a sufficient condition for the 2D system (2.1) with
polytopic uncertainty (2.7) to be asymptotically stable in terms of LMI
conditions. Note that if system (2.1) reduces to a 1D system with polytopic
uncertainty, Theorem 3.1 coincides with the asymmetric stability for 1D systems
considered in [19]. Therefore, Theorem 3.1 can be viewed as an extension of
existing results on the asymptotic stability for 1D systems to the 2D case.

*Remark 3.3. *Theorem 3.1 contains both Lemmas 2.7 and 2.10 as particular cases. In the
numerical examples, it will be shown that the condition proposed in Theorem 3.1
is the least conservative in comparison with the other four methods.

#### 4. Examples

*Example 4.1. *The first
example illustrates the main results for 2D uncertain system with three
vertices. The system is parameterized by given by the
triple with the
following vertices:where , , and .

The robust stability of this system has been
investigated by means of different lemmas cited above. The maximum value of such that the
system is stable, is
only using the QS
approach (Lemma 2.7), but increases to using Lemma
2.10, using Lemma
2.11, using Lemma
2.12, and using Theorem
3.1. Thus, there is a significant improvement using the proposed approach.

*Example 4.2. *The objective
is to check the stability of system (2.1) when the state matrix is within a
polytope (2.8) with the following vertices:where , , and .

This system has been selected so that it is possible
to prove the robust stability using the proposed Theorem 3.1, but the
previously proposed conditions in Lemmas 2.7, 2.10, 2.11, and 2.12 cannot prove
the robust stability, as the corresponding LMIs are not feasible. The vertices
of the parameter-dependent Lyapunov matrices that fulfill the conditions of
Theorem 3.1 are the following:

##### 4.1. Numerical Evaluation

Finally, a numerical evaluation procedure is considered to check the improvements obtained with the proposed method when testing the stability of plants randomly generated. Recall that the kind of systems we are dealing with is characterized by its order (), and the number of vertices in its polytopic description (). Thus, systems were randomly generated for each pair of values in the ranges , and , giving a total of stable polytopes. Each of these polytopes was evaluated using the different methods to check if the conditions successfully confirmed the robust stability. The results are given in Table 1, which exhibits the number of success of the different methods, which gives a measure of their performance and conservativeness. It can be seen that the tests given by Theorem 3.1 prove to be significantly less conservative than the original QS test (Lemma 2.7) and the modified versions derived in Lemmas 2.10, 2.11 and 2.12. Of course the computational cost increases, being the most computationally demanding the test corresponding to Theorem 3.1, but always within acceptable computational times.

#### 5. Conclusions

This paper has proposed conditions to determine the robust stability of 2D linear discrete-time systems described by Fornasini-Marchesini second models, under polytopic uncertainty. The conditions are formulated in terms of sets of LMIs described only in terms of the vertices of the uncertainty domain. Several examples have been presented that illustrate the results, showing the feasibility of the proposed approaches.

#### Acknowledgment

This work has been funded by AECI research projects A/5422/06, A/7882/07, and projects DPI2004-07444-C04-02 and DPI2007-66718-C04-02.

#### References

- B. R. Barmish, “Necessary and sufficient conditions for quadratic stabilizability of an uncertain system,”
*Journal of Optimization Theory and Applications*, vol. 46, no. 4, pp. 399–408, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Bernussou, P. L. D. Peres, and J. C. Geromel, “A linear programming oriented procedure for quadratic stabilization of uncertain systems,”
*Systems & Control Letters*, vol. 13, no. 1, pp. 65–72, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan,
*Linear Matrix Inequalities in System and Control Theory*, vol. 15 of*SIAM Studies in Applied Mathematics*, SIAM, Philadelphia, Pa, USA, 1994. View at Zentralblatt MATH · View at MathSciNet - H. R. Karimi, “Robust stabilization with ${H}_{\infty}$ performance for a class of linear parameter-dependent systems,”
*Mathematical Problems in Engineering*, vol. 2006, Article ID 59867, 15 pages, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - M. C. de Oliveira, J. C. Geromel, and L. H. Su, “LMI characterization of structural and robust stability: the discrete-time case,”
*Linear Algebra and Its Applications*, vol. 296, no. 1–3, pp. 27–38, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Feron, P. Apkarian, and P. Gahinet, “Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions,”
*IEEE Transactions on Automatic Control*, vol. 41, no. 7, pp. 1041–1046, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. J. S. Leite and P. L. D. Peres, “An improved LMI condition for robust D-stability of uncertain polytopic systems,”
*IEEE Transactions on Automatic Control*, vol. 48, no. 3, pp. 500–504, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - D. C. W. Ramos and P. L. D. Peres, “A less conservative LMI condition for the robust stability of discrete-time uncertain systems,”
*Systems & Control Letters*, vol. 43, no. 5, pp. 371–378, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Rantzer and M. Johansson, “Piecewise linear quadratic optimal control,”
*IEEE Transactions on Automatic Control*, vol. 45, no. 4, pp. 629–637, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Xie, S. Shishkin, and M. Fu, “Piecewise Lyapunov functions for robust stability of linear time-varying systems,”
*Systems & Control Letters*, vol. 31, no. 3, pp. 165–171, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Peaucelle, D. Arzelier, O. Bachelier, and J. Bernussou, “A new robust D-stability condition for real convex polytopic uncertainty,”
*Systems & Control Letters*, vol. 40, no. 1, pp. 21–30, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Kaczorek,
*Two-Dimensional Linear Systems*, vol. 68 of*Lecture Notes in Control and Information Sciences*, Springer, Berlin, Germany, 1985. View at Zentralblatt MATH · View at MathSciNet - H. D. Tuan, P. Apkarian, T. Q. Nguyen, and T. Narikiyo, “Robust mixed ${\mathscr{H}}_{2}/{\mathscr{H}}_{\infty}$ filtering of 2-D systems,”
*IEEE Transactions on Signal Processing*, vol. 50, no. 7, pp. 1759–1771, 2002. View at Publisher · View at Google Scholar - T. Hinamoto, “2-D Lyapunov equation and filter design based on the Fornasini-Marchesini second model,”
*IEEE Transactions on Circuits and Systems I*, vol. 40, no. 2, pp. 102–109, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. Ooba, “On stability analysis of 2-D systems based on 2-D Lyapunov matrix inequalities,”
*IEEE Transactions on Circuits and Systems I*, vol. 47, no. 8, pp. 1263–1265, 2000. View at Publisher · View at Google Scholar · View at MathSciNet - E. Fornasini and G. Marchesini, “Stability analysis of 2-D systems,”
*IEEE Transactions on Circuits and Systems*, vol. 27, no. 12, pp. 1210–1217, 1980. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Hmamed, M. Alfidi, A. Benzaouia, and F. Tadeo, “Robust stabilization under linear fractional parametric uncertainties of 2-D systems with Roesser models,”
*International Journal on Sciences and Techniques of Automatic Control*, 2008. View at Google Scholar - A. Hmamed, M. Alfidi, A. Benzaouia, and F. Tadeo, “LMI conditions for robust stability of 2-D uncertain discrete-time linear systems,” in
*Proceedings of the Conference on Systems and Control (CSC '07)*, Marrakech, Morocco, May 2007. - S.-W. Kau, Y.-S. Liu, L. Hong, C.-H. Lee, C.-H. Fang, and L. Lee, “A new LMI condition for robust stability of discrete-time uncertain systems,”
*Systems & Control Letters*, vol. 54, no. 12, pp. 1195–1203, 2005. View at Publisher · View at Google Scholar · View at MathSciNet