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

Authors: A. Hmamed; M. Alfidi; A. Benzaouia; F. Tadeo
Journal: Mathematical Problems in Engineering
 (2008)

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## 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.


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## Body

## 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 matricesX 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:(1.1)[X11X12X12TX22]=[X11X12*X22].


## 2. Problem Formulation and Preliminary Results

Consider the
following 2D linear discrete-time system described by a Fornasini-Marchesini
second model [16](2.1)x(i+1,j+1)=A[x(i+1,j)x(i,j+1)],where x(i,j)∈Rn is the state
vector, A∈R2n×n is the dynamic
matrix, and the boundary conditions are given by(2.2)x(i,0)=x0(i),x(0,j)=x0(j),i,j=0,1,2,….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
(2.3)limk→∞∥X(k)∥=0under supj∥X(0)∥<∞, where X(k)={x(i,j):i+j=k} and ∥X(k)∥=supx∈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=P12T≥0, and
Π>0
such that the
following LMI holds:
(2.4)[−ΠAT(P11+2P12+P22)*−(P11+2P12+P22)]<0,where Π=[P11P12P12TP22].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=RT≥0, and
Π>0
such that the
following LMI holds
(2.5)ATPA−Π<0,where Π=[P−Q−2RRRTQ], or equivalently(2.6)[ATAT]Π[AA]−Π<0.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 thatA 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 Ai of the
uncertainty polytope:(2.7)Ω={A(α):A(α)=∑i=1NαiAi,∑i=1Nαi=1,αi>0}.
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
(2.8)AT(α)P(α)A(α)−Π(α)<0for all α such that A(α)∈Ω, where (2.9)Π(α)=[P(α)−Q(α)−2R(α)R(α)RT(α)Q(α)]>0.To the authors
knowledge, there is no general and systematic way to formally determineΠ(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 matrixΠ(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=RT≥0, and	Π>0 such that
(2.10)AiTPAi−Π<0foralli=1,…,N.Theorem 2.8.
The following conditions are
equivalent.
(i)
There exist matricesP>0, Q>0, R=RT≥0, and Π>0 such that(2.11)ATPA−Π<0.(ii)
There exist matricesP>0, Q>0, R=RT≥0, Π>0, F∈R2n×n, and G∈Rn×n such that(2.12)[−Π+FA+ATFT−F+ATGT*P−G−GT]<0.Proof.
The proof is rather
straightforward. First, if (2.10) holds for some Π>0, by setting F=0, G=GT=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 ΓT and from the
right by Γ, where ΓT=[IAT], 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 matrixΠ 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 functionΠ(α)=ΠT(α)>0 given by(2.13)Π(α)=∑i=1NαiΠi=∑i=1Nαi[Pi−Qi−2RiRiRiTQi],whereαi≥0,i=1,…,N,∑i=1Nαi=1,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
Pi>0, Qi>0,
Ri=RiT≥0, and Πi=[Pi−Qi−2RiRiRiTQi]>0,i=1,…,N such that
(2.14)AiTPiAi−Πi<−I,(2.15)AiTPiAj+AjTPiAi+AiTPjAi−2Πi−Πj<1(N−1)2I,j=1,…,N,j≠i,(2.16)AjTPiAk+AkTPiAj+AiTPjAk+AkTPjAi+AiTPkAj+AjTPkAi−2(Πi+Πj+Πk)<6(N−1)2I,i=1,…,N−2,j=i+1,…,N−1,k=j+1,…,N,
then (2.8) holds with Π(α)=ΠT(α)>0 given by
(2.13).Lemma 2.11 (See [19]).
The
uncertain system (2.1) is robustly stable in the uncertainty domain (2.7) if there
exist matrices
Pi>0, Qi>0, Ri=RiT≥0, Πi>0, F, and G, for i=1,…,N such that
(2.17)[−Πi+FAi+AiTFT−F+AiTGT*Pi−Gi−GiT]<0,then (2.8) is verified with Π(α)>0 given by
(2.13).Lemma 2.12 (See [19]).
The uncertain system (2.1) is robustly
stable in the uncertainty domain (2.7) if there exist matrices Pi>0, Qi>0, Ri=RiT≥0, Πi>0, Fi, and Gi for i=1,…,N such that
(2.18)Mi≜[−Πi+FiAi+AiTFiT−Fi+AiTGiT*Pi−Gi−GiT]<−I,Mij≜[Γ11Γ12∗Γ22]<1(N−1)2I,j=1,…,N,j≠i,
where(2.19)Γ11=−(2Πi+Πj)+FiAj+FjAi+FiAi+AjTFiT+AjTFiT+AiTFiT,Γ12=−(2Fi+Fj)+AiTGjT+AjTGiT+AiTGiT,Γ22=2Pi+Pj−(2Gi+Gj)−(2Gi+Gj)T,Mijk≜[ϒ11ϒ12∗ϒ22]<6(N−1)2I,i=1,…,N−2,j=i+1,…,N−1,k=j+1,…,N,
where(2.20)ϒ11=−2(Πi+Πj+Πk)+(Fi+Fk)Aj+(Fi+Fj)Ak+(Fj+Fk)Ai+AjT(Fi+Fk)T+AkT(Fi+Fj)T+AiT(Fj+Fk)T,ϒ12=−2(Fi+Fj+Fk)+AjT(Gi+Gk)T+AkT(Gi+Gj)T+AiT(Gj+Gk)T,ϒ22=2(Pi+Pj+Pk)−2(Gi+Gj+Gk)−2(Gi+Gj+Gk)T.Remark 2.13.
It is easy to see that the LMI conditions of Lemmas2.10, 2.11, and 2.12 are less
conservative than those provided in Lemma 2.7. For example, condition (2.14)
contains (2.8) as a particular case when P1=P2=⋯=PN=P. This fact can be verified as follows. First, note
that in the quadratic stability case, (2.14) reduces to (2.8). Second, computing
the quadratic form with P>0 and j≠k, it is possible to see that (Aj−Ak)TP(Aj−Ak)≥0. Thus, one gets AjTPAk+AkTPAj≤AjTPAj+AkTPAk, implying, in this particular case, that the LMIs
(2.15) and (2.15) are always feasible.


## 3. Main Results

In the
following, we propose an LMI-based condition for finding a parameter-dependent
Lyapunov matrixΠ(α)>0 satisfying (2.8)
for every A(α)∈Ω, which is even less conservative than the results
stated in the previous lemmas.Theorem 3.1.
The uncertain system (2.1) is
robustly stable in the uncertainty domain (2.7) if there exist matrices Pi>0, Qi>0, Ri=RiT≥0, Yiii(1≤i≤N), Yiij=YjiiT, Yiji(1≤i,j≤N,i≠j), Yijl=YljiT, Yjil=YlijT(1≤i<j<k≤N), and Πi=[Pi−Qi−2RiRiRiTQi]>0,(1≤i≤N) such that
(3.1)AiTPiAi−Πi<Yiii,1≤i≤N,(3.2)AiTPiAj+AjTPiAi+AiTPjAi−2Πi−Πj≤Yiij+Yiji+YiijT,1≤i,j≤N,i≠j,(3.3)AjTPiAk+AkTPiAj+AiTPjAk+AkTPjAi+AiTPkAj+AjTPkAi−2(Πi+Πj+Πk)≤Yijk+Yikj+Yjik+YijkT+YikjT+YjikT,1≤i<j<k≤N,(3.4)[Y1i1Y1i2⋯Y1iNY2i1Y2i2⋯Y2iN⋮⋮⋱⋮YNi1YNi2⋯YNiN]≤0,1≤i≤N;then, for any A(α)∈Ω, a parameter-dependent matrix Π(α)>0 given by (2.13)
satisfies (2.8).Proof.
It is clear thatΠ(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 gets(3.5)A(α)P(α)A(α)−Π(α)=A(α)P(α)A(α)−(∑i=1Nαi)2P(α)=∑i=1Nαi3(AiTPiAi−Πi)+∑i=1N∑j=1,j≠iNαi2αj(AiTPiAj+AjTPiAi+AiTPjAi−2Πi−Πi)+∑i=1N−2∑j=i+1N−1∑k=j+1Nαiαjαk(AiTPjAk+AiTPkAj+AjTPiAk+AkTPjAi+AjTPkAi+AkTPiAj−2(Πi+Πj+Πk)).
Such an expansion is also
adopted in [9]. Imposing conditions (3.1)–(3.4), one gets(3.6)A(α)P(α)A(α)−Π(α)<∑i=1Nαi3Yiii+∑i=1N∑j=1,j≠iNαi2αj(Yiij+Yiji+YiijT)+∑i=1N−2∑j=i+1N−1∑k=j+1Nαiαjαk×(Yijk+Yikj+Yjik+YijkT+YikjT+YjikT)=α1[α1Iα2I⋮αNI]T[Y111Y112⋯Y11NY211Y212⋯Y21N⋮⋮⋱⋮YN11YN12⋯YN1N][α1Iα2I⋮αNI]+α2[α1Iα2I⋮αNI]T[Y121Y122⋯Y12NY221Y222⋯Y22N⋮⋮⋱⋮YN21YN22⋯YN2N][α1Iα2I⋮αNI]+⋯+αN[α1Iα2I⋮αNI]T[Y1N1Y1N2⋯Y1NNY2N1Y2N2⋯Y2NN⋮⋮⋱⋮YNN1YNN2⋯YNNN][α1Iα2I⋮αNI]=[α1Iα2I⋮αNI]T(∑i=1Nαi[Y1i1Y1i2⋯Y1iNY2i1Y2N2⋯Y2iN⋮⋮⋱⋮YNi1YNi2⋯YNiN])[α1Iα2I⋮αNI]≤0.
Inequality (3.6) implies that (2.8)
holds, so the theorem is proven.Remark 3.2.
Theorem3.1 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.
Theorem3.1 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 byr given by the
triple (rA1,rA2,rA3) with the
following vertices:(4.1)A11=[−0.0173−0.3258−0.22900.0935−0.10200.0159−0.00760.31380.1993],A21=[−0.3338−0.0043−0.37150.21280.58340.81340.0676−0.2660−0.1896],A12=[0.12530.31460.1192−0.0279−0.2348−0.1120−0.3263−0.10550.0595],A22=[0.0691−0.12050.10590.1916−0.01500.6133−0.03970.0357−0.0941],A13=[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],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 ofr 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:(4.2)A11=[−0.0304−0.5718−0.40190.1641−0.17900.0279−0.01330.55080.3498],A21=[−0.5859−0.0075−0.65210.37351.02401.42770.1187−0.4669−0.3328],A12=[0.21990.55220.2092−0.0490−0.4121−0.1966−0.5727−0.18520.1044],A22=[0.1213−0.21150.18590.3363−0.02631.0765−0.06970.0627−0.1652],A13=[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],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 Theorem3.1, 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:(4.3)P1=[0.24780000.24780000.2478],P2=[0.24780000.24780000.2478],P3=[0.24780000.24780000.2478],Q1=[0.26710.16290.40520.16290.34890.51720.40520.51721.0891],Q2=[0.22240.02420.27780.02420.04650.00270.27780.00270.6236],Q3=[0.0744−0.10240.0442−0.10240.72620.22070.04420.22070.1619],R1=[0.02010.00950.02630.00950.00640.01230.02630.01230.0397],R2=10−7×[0.37240.12350.12910.12350.6149−0.20360.1291−0.20360.5666],R3=10−6×[0.0141−0.00150.0098−0.00150.1565−0.05380.0098−0.05380.0608].


### 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 for2≤n≤4, 2≤N≤3 (from a maximum
of 1000 systems for each {n,N}).
nNLemma2.7Lemma2.10Lemma2.11Lemma2.12Theorem3.122582860928940945328969181783183432213709827828844325481656674683429344560862574732234429439456


## 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 for2≤n≤4, 2≤N≤3 (from a maximum
of 1000 systems for each {n,N}).
nNLemma2.7Lemma2.10Lemma2.11Lemma2.12Theorem3.122582860928940945328969181783183432213709827828844325481656674683429344560862574732234429439456


## 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.



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