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

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 𝑋 and 𝑌, the notation 𝑋≥𝑌 (resp., 𝑋>𝑌) means that the matrix 𝑋−𝑌 is positive semidefinite (resp., positive definite). 𝐼 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:𝑋₁₁𝑋₁₂𝑋^𝑇₁₂𝑋₂₂=𝑋₁₁𝑋₁₂∗𝑋₂₂. (1.1)

2. Problem Formulation and Preliminary Results

Consider the following 2D linear discrete-time system described by a Fornasini-Marchesini second model [16]𝑥(𝑖+1,𝑗+1)=𝐴𝑥(𝑖+1,𝑗)𝑥(𝑖,𝑗+1), (2.1)where 𝑥(𝑖,𝑗)∈𝑅^𝑛 is the state vector, 𝐴∈𝑅^2𝑛×𝑛 is the dynamic matrix, and the boundary conditions are given by𝑥(𝑖,0)=𝑥₀(𝑖),𝑥(0,𝑗)=𝑥₀(𝑗),𝑖,𝑗=0,1,2,…. (2.2)We 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 lim_𝑘→∞‖𝑋(𝑘)‖=0 (2.3)under sup_𝑗‖𝑋(0)‖<∞, where 𝑋(𝑘)={𝑥(𝑖,𝑗)∶𝑖+𝑗=𝑘} and ‖𝑋(𝑘)‖=sup_{𝑥∈𝑋(𝑘)}‖𝑥‖.

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 𝑃₁₁>0, 𝑃₂₂>0, 𝑃₁₂=𝑃^𝑇₁₂≥0, and Π>0 such that the following LMI holds: −Π𝐴^𝑇(𝑃₁₁+2𝑃₁₂+𝑃₂₂)∗−(𝑃₁₁+2𝑃₁₂+𝑃₂₂)<0, (2.4)where Π=[^𝑃11𝑃12_{𝑃𝑇12𝑃22}].

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 𝑃₁₁, 𝑃₁₂, and 𝑃₂₂ 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 𝑃>0, 𝑄>0, 𝑅=𝑅^𝑇≥0, and Π>0 such that the following LMI holds 𝐴^𝑇𝑃𝐴−Π<0, (2.5)where Π=[_𝑅^{𝑃−𝑄−2𝑅𝑅}_𝑇𝑄], or equivalently[𝐴^𝑇𝐴^𝑇𝐴𝐴]Π−Π<0. (2.6)

Proof. It is possible to obtain (2.5) substituting 𝑃₁₁=𝑃−𝑄−2𝑅, 𝑃₁₂=𝑅, and 𝑃₂₂=𝑄 in (2.4) and using a Schur complement argument.

Suppose now that 𝐴 is not exactly known, but belongs to a given polytopic uncertain domain Ω. In this way, any matrix inside the domain Ω can be written as a convex combination of the vertices 𝐴_𝑖 of the uncertainty polytope:Ω={𝐴(𝛼)∶𝐴(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝐴_𝑖,^𝑁_𝑖=1𝛼_𝑖=1,𝛼_𝑖>0}. (2.7) 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 𝑃(𝛼)>0, 𝑄(𝛼)>0, and 𝑅(𝛼)≥0 such that 𝐴^𝑇(𝛼)𝑃(𝛼)𝐴(𝛼)−Π(𝛼)<0 (2.8)for all 𝛼 such that 𝐴(𝛼)∈Ω, where 𝑅Π(𝛼)=𝑃(𝛼)−𝑄(𝛼)−2𝑅(𝛼)𝑅(𝛼)_𝑇(𝛼)𝑄(𝛼)>0. (2.9)

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

An effective way of addressing such problem is to look for a single Lyapunov matrix Π(𝛼)=Π 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 𝑃>0, 𝑄>0, 𝑅=𝑅^𝑇≥0, and Π>0 such that 𝐴^𝑇_𝑖𝑃𝐴_𝑖−Π<0 (2.10)forall𝑖=1,…,𝑁.

Theorem 2.8. The following conditions are equivalent.
(i)There exist matrices 𝑃>0, 𝑄>0, 𝑅=𝑅^𝑇≥0, and Π>0 such that𝐴^𝑇𝑃𝐴−Π<0. (2.11)(ii)There exist matrices 𝑃>0, 𝑄>0, 𝑅=𝑅^𝑇≥0, Π>0, 𝐹∈𝑅^2𝑛×𝑛, and 𝐺∈𝑅^𝑛×𝑛 such that−Π+𝐹𝐴+𝐴^𝑇𝐹^𝑇−𝐹+𝐴^𝑇𝐺^𝑇∗𝑃−𝐺−𝐺^𝑇<0. (2.12)

Proof. The proof is rather straightforward. First, if (2.10) holds for some Π>0, by setting 𝐹=0, 𝐺=𝐺^𝑇=𝑃, and applying the Schur complement, (2.12) is satisfied. On the other hand, if (2.12) holds for some (Π,𝐹,𝐺), multiplying (2.12) from the left by Γ^𝑇 and from the right by Γ, where Γ^𝑇=[𝐼𝐴^𝑇], 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 Π is not involved in any product with the dynamic matrix 𝐴, thanks to the introduction of the additional matrices 𝐹 and 𝐺. 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 𝐹 and 𝐺.
The next three lemmas state sufficient conditions for the existence of a parameter-dependent Lyapunov function Π(𝛼)=Π^𝑇(𝛼)>0 given byΠ(𝛼)=^𝑁_𝑖=1𝛼_𝑖Π_𝑖=^𝑁_𝑖=1𝛼_𝑖𝑃_𝑖−𝑄_𝑖−2𝑅_𝑖𝑅_𝑖𝑅^𝑇_𝑖𝑄_𝑖,where𝛼_𝑖≥0,𝑖=1,…,𝑁,^𝑁_𝑖=1𝛼_𝑖=1, (2.13)such 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 𝑃_𝑖>0, 𝑄_𝑖>0, 𝑅_𝑖=𝑅^𝑇_𝑖≥0, and Π_𝑖=[^{𝑃𝑖−𝑄𝑖−2𝑅𝑖𝑅𝑖}_{𝑅𝑇𝑖𝑄𝑖}]>0,𝑖=1,…,𝑁 such that 𝐴^𝑇_𝑖𝑃_𝑖𝐴_𝑖−Π_𝑖𝐴<−𝐼,_𝑇𝑖𝑃_𝑖𝐴_𝑗+𝐴^𝑇_𝑗𝑃_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑗𝐴_𝑖−2Π_𝑖−Π_𝑗<1(𝑁−1)²𝐴𝐼,𝑗=1,…,𝑁,𝑗≠𝑖,_𝑇𝑗𝑃_𝑖𝐴_𝑘+𝐴^𝑇_𝑘𝑃_𝑖𝐴_𝑗+𝐴^𝑇_𝑖𝑃_𝑗𝐴_𝑘+𝐴^𝑇_𝑘𝑃_𝑗𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑘𝐴_𝑗+𝐴^𝑇_𝑗𝑃_𝑘𝐴_𝑖−2(Π_𝑖+Π_𝑗+Π_𝑘)<6(𝑁−1)²𝐼,𝑖=1,…,𝑁−2,𝑗=𝑖+1,…,𝑁−1,𝑘=𝑗+1,…,𝑁, (2.14)(2.15)(2.16)then, for any 𝐴(𝛼)∈Ω, a parameter-dependent matrix Π(𝛼)>0 given by (2.13) satisfies (2.8).

Proof. It is clear that Π(𝛼) given by (2.13) is a positive-definite parameter Lyapunov matrix. Using (2.13), the definition of 𝐴(𝛼)∈Ω given in (2.7), and the direct expansion, one gets𝐴(𝛼)𝑃(𝛼)𝐴(𝛼)−Π(𝛼)=𝐴(𝛼)𝑃(𝛼)𝐴(𝛼)−(^𝑁_𝑖=1𝛼_𝑖)²=𝑃(𝛼)_𝑁𝑖=1𝛼³_𝑖(𝐴^𝑇_𝑖𝑃_𝑖𝐴_𝑖−Π_𝑖)+^𝑁𝑁_{𝑖=1𝑗=1,𝑗≠𝑖}𝛼²_𝑖𝛼_𝑗(𝐴^𝑇_𝑖𝑃_𝑖𝐴_𝑗+𝐴^𝑇_𝑗𝑃_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑗𝐴_𝑖−2Π_𝑖−Π_𝑖)+^𝑁−2_𝑖=1^𝑁−1𝑁_{𝑗=𝑖+1𝑘=𝑗+1}𝛼_𝑖𝛼_𝑗𝛼_𝑘(𝐴^𝑇_𝑖𝑃_𝑗𝐴_𝑘+𝐴^𝑇_𝑖𝑃_𝑘𝐴_𝑗+𝐴^𝑇_𝑗𝑃_𝑖𝐴_𝑘+𝐴^𝑇_𝑘𝑃_𝑗𝐴_𝑖+𝐴^𝑇_𝑗𝑃_𝑘𝐴_𝑖+𝐴^𝑇_𝑘𝑃_𝑖𝐴_𝑗−2(Π_𝑖+Π_𝑗+Π_𝑘)). (3.5) Such an expansion is also adopted in [9]. Imposing conditions (3.1)–(3.4), one gets<𝐴(𝛼)𝑃(𝛼)𝐴(𝛼)−Π(𝛼)_𝑁𝑖∑₌₁𝛼³_𝑖𝑌_𝑖𝑖𝑖+∑^𝑁𝑁_𝑖=1∑_{𝑗=1,𝑗≠𝑖}𝛼²_𝑖𝛼_𝑗(𝑌_𝑖𝑖𝑗+𝑌_𝑖𝑗𝑖+𝑌^𝑇_𝑖𝑖𝑗)+∑^𝑁−2∑_𝑖=1^𝑁−1𝑁_𝑗=𝑖+1∑_𝑘=𝑗+1𝛼_𝑖𝛼_𝑗𝛼_𝑘×(𝑌_𝑖𝑗𝑘+𝑌_𝑖𝑘𝑗+𝑌_𝑗𝑖𝑘+𝑌^𝑇_𝑖𝑗𝑘+𝑌^𝑇_𝑖𝑘𝑗+𝑌^𝑇_𝑗𝑖𝑘)=𝛼₁𝛼₁𝐼𝛼₂𝐼⋮𝛼_𝑁𝐼^𝑇𝑌₁₁₁𝑌₁₁₂⋯𝑌_11𝑁𝑌₂₁₁𝑌₂₁₂⋯𝑌_21𝑁𝑌⋮⋮⋱⋮_𝑁11𝑌_𝑁12⋯𝑌_𝑁1𝑁𝛼₁𝐼𝛼₂𝐼⋮𝛼_𝑁𝐼+𝛼₂𝛼₁𝐼𝛼₂𝐼⋮𝛼_𝑁𝐼^𝑇𝑌₁₂₁𝑌₁₂₂⋯𝑌_12𝑁𝑌₂₂₁𝑌₂₂₂⋯𝑌_22𝑁𝑌⋮⋮⋱⋮_𝑁21𝑌_𝑁22⋯𝑌_𝑁2𝑁𝛼₁𝐼𝛼₂𝐼⋮𝛼_𝑁𝐼+⋯+𝛼_𝑁𝛼₁𝐼𝛼₂𝐼⋮𝛼_𝑁𝐼^𝑇𝑌_1𝑁1𝑌_1𝑁2⋯𝑌_1𝑁𝑁𝑌_2𝑁1𝑌_2𝑁2⋯𝑌_2𝑁𝑁𝑌⋮⋮⋱⋮_𝑁𝑁1𝑌_𝑁𝑁2⋯𝑌_𝑁𝑁𝑁𝛼₁𝐼𝛼₂𝐼⋮𝛼_𝑁𝐼=𝛼₁𝐼𝛼₂𝐼⋮𝛼_𝑁𝐼^𝑇(∑^𝑁_𝑖=1𝛼_𝑖𝑌_1𝑖1𝑌_1𝑖2⋯𝑌_1𝑖𝑁𝑌_2𝑖1𝑌_2𝑁2⋯𝑌_2𝑖𝑁𝑌⋮⋮⋱⋮_𝑁𝑖1𝑌_𝑁𝑖2⋯𝑌_𝑁𝑖𝑁)𝛼₁𝐼𝛼₂𝐼⋮𝛼_𝑁𝐼≤0. (3.6) 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 𝑟 given by the triple (𝑟𝐴₁,𝑟𝐴₂,𝑟𝐴₃) with the following vertices:𝐴₁₁=−0.0173−0.3258−0.22900.0935−0.10200.0159−0.00760.31380.1993,𝐴₂₁=,𝐴−0.3338−0.0043−0.37150.21280.58340.81340.0676−0.2660−0.1896₁₂=0.12530.31460.1192−0.0279−0.2348−0.1120−0.3263−0.10550.0595,𝐴₂₂=,𝐴0.0691−0.12050.10590.1916−0.01500.6133−0.03970.0357−0.0941₁₃=0.47240.72530.60480.14140.01150.1843−0.3777−0.5544−0.4840,𝐴₂₃=,−0.41870.5585−0.2581−0.1361−0.1769−0.25640.27780.44860.5656 (4.1)where 𝐴₁=[𝐴₁₁𝐴₂₁], 𝐴₂=[𝐴₁₂𝐴₂₂], and 𝐴₃=[𝐴₁₃𝐴₂₃].
The robust stability of this system has been investigated by means of different lemmas cited above. The maximum value of 𝑟 such that the system (𝑟𝐴₁,𝑟𝐴₂,𝑟𝐴₃) is stable, is only 𝑟₁=0.0691 using the QS approach (Lemma 2.7), but increases to 𝑟₂=0.5745 using Lemma 2.10, 𝑟₃=0.7521 using Lemma 2.11, 𝑟₄=0.7551 using Lemma 2.12, and 𝑟₅=0.7552 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:𝐴₁₁=−0.0304−0.5718−0.40190.1641−0.17900.0279−0.01330.55080.3498,𝐴₂₁=,𝐴−0.5859−0.0075−0.65210.37351.02401.42770.1187−0.4669−0.3328₁₂=0.21990.55220.2092−0.0490−0.4121−0.1966−0.5727−0.18520.1044,𝐴₂₂=,𝐴0.1213−0.21150.18590.3363−0.02631.0765−0.06970.0627−0.1652₁₃=0.82921.27301.06150.24820.02020.3235−0.6629−0.9731−0.8495,𝐴₂₃=,−0.73490.9803−0.4530−0.2389−0.3105−0.45000.48760.78740.9927 (4.2)where 𝐴₁=[𝐴₁₁𝐴₂₁], 𝐴₂=[𝐴₁₂𝐴₂₂], and 𝐴₃=[𝐴₁₃𝐴₂₃].
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:𝑃₁=0.24780000.24780000.2478,𝑃₂=,𝑃0.24780000.24780000.2478₃=0.24780000.24780000.2478,𝑄₁=,𝑄0.26710.16290.40520.16290.34890.51720.40520.51721.0891₂=0.22240.02420.27780.02420.04650.00270.27780.00270.6236,𝑄₃=,𝑅0.0744−0.10240.0442−0.10240.72620.22070.04420.22070.1619₁=0.02010.00950.02630.00950.00640.01230.02630.01230.0397,𝑅₂=10⁻⁷×,𝑅0.37240.12350.12910.12350.6149−0.20360.1291−0.20360.5666₃=10⁻⁶×.0.0141−0.00150.0098−0.00150.1565−0.05380.0098−0.05380.0608 (4.3)

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 (𝑛), and the number of vertices in its polytopic description (𝑁). Thus, 1000 systems were randomly generated for each pair of values {𝑛,𝑁} in the ranges 2≤𝑛≤4, and 2≤𝑁≤3, giving a total of 6000 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.

Table 1 

Number of robustly stable systems correctly confirmed by the different methods for 2≤𝑛≤4, 2≤𝑁≤3 (from a maximum of 1000 systems for each {𝑛,𝑁}).

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.