Internal Report No 20060117OB v. 4

LMI Approach to Stability of some Class of nD-Polynomial matrices:

Application to the Stability of nD-Systems

Submited to Je ne sais pas encore

       Olivier Bachelier^{a}, Olivier.Bachelier@univ-poitiers.fr
            Driss Mehdi^{a}, Driss.Mehdi@univ-poitiers.fr

aLaii-Esip, Bâtiment de Mécanique, 40 avenue du recteur Pineau, 86022 Poitiers Cedex, France

Version 4

Abstract

This paper addresses the hard problem of the D-stability of some class of nD-polynomial matrices. An LMI approach is used to derive sufficient conditions for matrix D-stability when D is a region complying with some specific description. The proposed conditions aim at encompassing several strong results related to the stability of multidimensional models. Thus, they open a new range for possible further investigations in the realm of multidimensional systems.

Index Terms

D-stability, nD-polynomial matrices, LMI, Roesser nD-models.

1 Introduction

There has been a growing interest in the study of multidimensional systems (or nD-systems) in the past decades [1–4]. It is not only due to purely theoretical problems induced by nD-systems but also to their practical applications in various industrial domains such as acoustics, signal or image processing, multipass manufacturing processes, traffic control problems, etc [1, 5–9]. Among nD-systems, the most frequently encountered are the 2D-systems [10, 11] that encompass, for instance, the repetitive processes [12] known for their utility in mining and steel processing, metal rolling [13]. Nevertheless, nD-systems with n > 2 also deserve to be studied.

A possible approach to test the stability of such systems is to adapt Lyapunov’s approach [14] and the associated celebrated matrix equality or, in the discrete-time case, Stein’s result [15], to nD-models such as in [16–18]. It is also possible to consider inequalities rather than equalities thanks to the now well-known LMI machinery (LMI : Linear Matrix Inequality [19]), as highlighted in [20–22].

Because those nD-systems are clearly of interest, it matters to take benefit of all various tools related to nD-models. Particularly, since nD-models can be handled through nD-polynomial matrices, it is important to derive theoretical results relevant to those matrices. Literature offers interesting results on the stability of multivariate polynomial matrices [23,24]. Besides, LMIs can be a good tool to tackle problems induced by those polynomial matrices [25, 26]. This paper lies in this framework. It focuses on a special class of multivariate first degree polynomial matrices complying with

A(s) = ⊕_{i=1}^{k}(s_{i}I_{n_{i}}) − A  (1)

(see notations below), where A is a known matrix and s_{i} are the various Laplace variables. The property to be tested is the D-stability. Here, a multivariate polynomial matrix is said to be D-stable when none of its zeros belongs to some multi region D to be defined in the body of the paper. A sufficient LMI condition for the matrix A(s) to be D-stable is proposed. It is shown that several known results relevant to the stability of nD-models are special cases of this condition.

The paper is organized as follows. After this introduction, Section 2 presents the LMI condition for the D-stability of matrix A(s). Section 3 highlights the relevance of this result for many applications requiring a stability analysis of a (continuous-time, discrete-time, hybrid, ...) nD-model before the paper is concluded.

Notations

M′ denotes the transpose conjugate of matrix M. Hence, λ′ is the conjugate of complex number λ. I_{n} is the identity matrix of order n and 0 is a null matrix of appropriate dimensions. Matrix inequalities are considered in the sense of Löwner i.e. > 0 (resp. < 0) means positive (resp. negative) definite. Also let the following matrices be defined:

⊕_{i=1}^{k} M_{i} = diag_{i=1,...k} {M_{i}}  (2)

For a given subset S of the complex plane, the set S^{C} is the complementary set of S such that S ∪ S^{C} = l C & S ∩ S^{C} = ∅. At last, symbol ⊗ denotes the matrix Kronecker product [27].

2 D-stability of some polynomial matrices

In this section, the description of region D which is considered in this work is introduced. Then, the definition of the D-stability of a multivariate-polynomial matrix proposed in [25] is recalled. At last, a sufficient LMI condition for the polynomial matrix A(s) complying with (1) to be D-stable is presented.

Let k Hermitian matrices R_{i} be defined by

R_{i} = R′_{i} = [ \begin{matrix}R_{i_{11}} & R′_{i_{10}} \\ R_{i_{10}} & R_{i_{00}} \end{matrix} ] ∈ l C^{2d×2d} & R_{i_{11}} ∈ l C^{d×d} ∀i ∈ \{1, ..., k\}.  (3)

Also let the multi-region D of order d be defined by

D = (D_{1} × D_{2} × … D_{k}) ⊂ l C^{k},  (4)

where each region D_{i} is described through the matrix R_{i} by

D_{i} = \{λ ∈ l C |f_{D_{i}}(λ) = R_{i_{00}} + R_{i_{10}}λ + R′_{i_{10}}λ′ + R_{i_{11}}λ′λ = [ \begin{matrix}λI_{d} \\ I_{d} \end{matrix} ]′ R_{i} [ \begin{matrix}λI_{d} \\ I_{d} \end{matrix} ] ≥ 0\} ∀i ∈ \{1, ..., k\}.

(5)

Note that, when d = 1 and R_{i_{11}} ≥ 0, D_{i} is expressed as the complementary set of an open EMI-region (EMI for Ellipsoidal Matrix Inequality region: see [28,29]), i.e., in practice, a closed half plane or the closed exterior of a disc.

Remark 1 There is no loss of generality in considering the same order d for each subregion D_{i}. If it is not the case, it suffices to arbitrarily increase the dimension of the matrices R_{i} with suitable zero entries. The only interest in keeping the same order d for all subregions is to avoid heavy notations in the sequel.

Let the vector of complex variables s be written

s = [ s_{1} ; ⋮ ; s_{k} ] ∈ l C^{k}.  (6)

A multivariate polynomial matrix A(s) is also defined by

A(s) = ⊕_{i=1}^{k}(s_{i}I_{n_{i}}) − A ∈ l C^{n×n},  (7)

where of course, ∑_{i=1}^{k}(n_{i}) = n.

Definition 2 The nD-polynomial matrix A(s) is said to be D-stable if and only if

(p(s) = det(A(s)) = 0 ⇒ s ∉ D) ⇔ (p(s) ≠ 0 ∀s ∈ D) .  (8)

This definition means that A(s) is D-stable if and only if its determinant has no zero inside D. It is the same definition as the one used in [25]. Based upon this basic definition, some LMI condition for matrix D-stability is now proposed in the following theorem which must be considered as the main result of the paper.

Theorem 3 Let the polynomial matrix A(s) and the multi-region D be defined by (7) and (4) with (3), (5) respectively. Matrix A(s) is D-stable if there exist k Hermitian positive definite matrices P_{i} ∈ l C^{n_{i}×n_{i}}, i = 1, ..., n such that

M = [ \begin{matrix}A ⊗ I_{d} \\ I_{dn} \end{matrix} ]′ Q [ \begin{matrix}A ⊗ I_{d} \\ I_{dn} \end{matrix} ] < 0,  (9)

with

Q = [ \begin{matrix}⊕_{i=1}^{k}(P_{i} ⊗ R_{i_{11}}) & ⊕_{i=1}^{k}(P_{i} ⊗ R_{i_{10}}) \\ ⊕_{i=1}^{k}(P_{i} ⊗ R′_{i_{10}}) & ⊕_{i=1}^{k}(P_{i} ⊗ R_{i_{00}}) \end{matrix} ] .  (10)

Proof. From [30, Corollary 2.3.5] (particular case of elimination lemma), noting that the columns of

[ \begin{matrix}A ⊗ I_{d} \\ I_{dn} \end{matrix} ] ,  (11)

span the kernel of

[ I_{dn}  −A ⊗ I_{d} ] ,  (12)

there exists a matrix G = G′ such that

N = Q + [ \begin{matrix}I_{dn} \\ −A′ ⊗ I_{d} \end{matrix} ] G [ I_{dn} −A ⊗ I_{d} ] < 0.  (13)

Let a vector s (as introduced in (6)) comply with p(s) = 0, where p(.) is defined in (8). It will be proven that s cannot lie in D. Indeed, if p(s) = 0, there exists a vector v ∈ l C^{n} such that

A(s)v = 0.  (14)

Let the matrix q be defined by

q = [ \begin{matrix}⊕_{i=1}^{k}(s_{i}I_{n_{i}})v \\ v \end{matrix} ] ⊗ I_{d}.  (15)

Note that, if v is written

v = [ v_{1} ; ⋮ ; v_{k} ] , with v_{i} ∈ l C^{n_{i}} ∀i ∈ {1, …, k},  (16)

then q also writes

q = [ v_{1} ⊗ (s_{1}I_{d}) ; ⋮ ; v_{k} ⊗ (s_{k}I_{d}) ; v_{1} ⊗ I_{d} ; ⋮ ; v_{k} ⊗ I_{d} ] .  (17)

Now, from (13), it can be deduced that

q′Nq < 0  (18)

which can be rewritten

∑_{i=1}^{k} ((v′_{i}P_{i}v_{i} ⊗ f_{D_{i}}(s_{i})) + ((v′A′(s)) ⊗ I_{d})G((A(s)v) ⊗ I_{d})) < 0.  (19)

From (14), it can be seen that the second term in the above inequality is zero and if s belongs to D, then the first one is positive or zero (since s ∈ D ⇔ s_{i} ∈ D_{i}∀i ∈ {1, …, k}), which contradicts (19). Thus, s cannot belong to D. □

Remark 4 Conditions (9) or (13) consist in LMI systems that are very tractable since they can be solved with classical LMI software.

Theorem 3 might seem quite theoretic. It involves a particular structure of multivariate polynomial matrix as well as a special description of region. Besides, the practical interest of such a proposition might not seem obvious. However, the next section aims at highlighting the connection to the the realm of multidimensional systems. More precisely, it is shown that several results form the literature related to the stability analysis of multidimensional systems fall out as special cases of Theorem 3.

3 Application to the analysis of multidimensional models

This section is devoted to some possible applications of the above result to the analysis of nD-models corresponding to various cases encountered in the literature. For each case, the relationship with Theorem 3 is emphasized.

3.1 Stability of continuous-time Roesser models

In this subsection, the so-called state-space Roesser-form of continuous-time nD-models, adapted from the original discrete model [31], is recalled. It is very useful to describe the behaviour of systems with multidimensional continuous-time dynamics. Let the following autonomous continuous linear model be given by

[ \frac{∂}{∂t_{1}} x_{1}(t_{1}, …, t_{k}) ; ⋮ ; \frac{∂}{∂t_{k}} x_{k}(t_{1}, …, t_{k}) ] = A [ x_{1}(t_{1}, …, t_{k}) ; ⋮ ; x_{k}(t_{1}, …, t_{k}) ] .  (20)

where the state subvectors x_{i} ∈ IR^{n_{i}}, i = 1, ..., k, are such that x = [x′_{1} … x′_{k}]′ ∈ IR^{n}. A stability condition for the above Roesser model is provided in the next lemma.

Lemma 5 [32] The system (20) is asymptotically stable if and only if

a(s) = det( ⊕_{i=1}^{k}(s_{i}I_{n_{i}}) − A) ≠ 0 ∀s = [ s_{1} ; ⋮ ; s_{k} ] |{Re(s_{i}) ≥ 0, i = 1, ..., k}.  (21)

This lemma simply means that (20) is stable if and only if the characteristic polynomial a(s) has no zero in the multi right half plane ( l C^{+^{k}}).

It is now shown in few lines how the mathematical result stated in Section 2 can be applied to find a sufficient LMI condition for the one proposed in Lemma 5.

Theorem 6 The model (20) is asymptotically stable if there exist symmetric positive definite matrices P_{i}, i = 1, …, k, such that

M_{c} = [ \begin{matrix}A \\ I_{n} \end{matrix} ]′ Q_{c} [ \begin{matrix}A \\ I_{n} \end{matrix} ] < 0,  (22)

with

Q_{c} = [ \begin{matrix}0 & ⊕_{i=1}^{k} P_{i} \\ ⊕_{i=1}^{k} P_{i} & 0 \end{matrix} ] .  (23)

Proof: it suffices to apply Theorem 3 with

R_{i} = [ \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ] ∀i ∈ \{1, …, k\},  (24)

and with D denoting l C^{+k}. Besides, p(s) introduced in (8) is nothing else than a(s) given by (21). □

Such a result is known since it was proposed in a slightly different formulation [33, 34]. If k = 1 (conventional one dimensional linear systems), the well-known Lyapunov inequality [14] is recovered and thus the condition is also necessary.

3.2 Stability of discrete-time Roesser models

Let now the autonomous discrete linear shift-invariant model be defined by

[ x_{1}(j_{1} + 1, …, j_{k}) ; ⋮ ; x_{k}(j_{1}, …, j_{k} + 1) ] = A [ x_{1}(j_{1}, …, j_{k}) ; ⋮ ; x_{k}(j_{1}, …, j_{k}) ] .  (25)

with x having the same dimensions as in (20). The above model can be referred to as the discrete Roesser model [31]. The discrete counterpart of Lemma 5 is now recalled.

Lemma 7 [35] The system (25) is asymptotically stable if and only if

b(s) = det (I_{n} − ( ⊕_{i=1}^{k} s_{i}^{−1}I_{n_{i}}) A) ≠ 0 ∀s = [ s_{1} ; ⋮ ; s_{k} ] |{|s_{i}| ≥ 1, i = 1, ..., k}.  (26)

This lemma means that (25) is stable if and only if b(z), the multivariate polynomial in s_{i}^{−1}, has no root in the multi-exterior of unit disc(D^{C^{k}}). This work can also be connected to the seminal work of [36].

Now, let the discrete-counterpart of Theorem 6 be given by the next theorem

Theorem 8 The model (25) is asymptotically stable if there exist symmetric positive definite matrices P_{i}, i = 1, …, k, such that

M_{d} = [ \begin{matrix}A \\ I_{n} \end{matrix} ]′ Q_{d} [ \begin{matrix}A \\ I_{n} \end{matrix} ] < 0,  (27)

with

Q_{d} = [ \begin{matrix}⊕_{i=1}^{k} P_{i} & 0 \\ 0 & − ⊕_{i=1}^{k} P_{i} \end{matrix} ] .  (28)

Proof: it suffices to apply Theorem 3 with

R_{i} = [ \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ] ∀i ∈ \{1, …, k\},  (29)

and with D denoting D^{C^{k}}. Besides, since each s_{i} is outside the unit disc, its reciprocal exists and defining the nonsingular matrix S by

S = ⊕_{i=1}^{k} s_{i}I_{n_{i}},  (30)

enables to deduce that

p(s) = det(A(s)) ≠ 0 ⇔ det(S^{−1}) det(A(s)) = det(S^{−1}A(s)) = b(s) ≠ 0,  (31)

with b(s) given by (21). □

Such a result was established by another way in [35]. For k = 1 (conventional 1D-systems), Stein’s inequality [15], most known as “Lyapunov’s discrete inequality” is recovered and thus the condition becomes also necessary. It is noticeable that some so-called linear repetitive processes can often be handled as special class of discrete 2D-models, once they are completely discretized [37]. Such models are expected to be “stable along the pass” [37]. A necessary and sufficient condition for a discrete linear repetitive model to be stable along the pass is given by Lemma 7 with a special structure for A and then Theorem 8 can be used to derive a tractable sufficient condition. The reader is also invited to pay attention to [38] to have further details on this possible application. Another class of systems which can be studied through nD-models is some class of discrete fractional systems. More precisely, the stability of some k-commensurate discrete fractional order polynomials can be analyzed (with additional conservatism) through Lemma 7 and thus, with sufficiency, through Theorem 8. This idea was originally proposed in [39] and also becomes an application of Theorem 3.

3.3 Stability of hybrid Roesser models

This kind of models involving both discrete and continuous dynamics was introduced in [5] for the 2D-case. The contribution [40] highlights the relevance of such 2D hybrid models to provide a mathematical description of the so-called differential repetitive processes. But the hybrid Roesser model, in its generality, is particularly well introduced in [34]. The next autonomous model is borrowed from this paper.

[ \frac{∂}{∂t_{1}} x^{1}(t_{1}, …, t_{p}, j_{p+1}, …, j_{k}) ; ⋮ ; \frac{∂}{∂t_{p}} x^{p}(t_{1}, …, t_{p}, j_{p+1}, …, j_{k}) ; x^{p+1}(t_{1}, …, t_{p}, j_{p+1} + 1, …, j_{k}) ; ⋮ ; x^{k}(t_{1}, …, t_{p}, j_{p+1}, …, j_{k} + 1) ] = [ \begin{matrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix} ] [ x^{1}(t_{1}, …, t_{p}, j_{p+1}, …, j_{k}) ; ⋮ ; x^{p}(t_{1}, …, t_{p}, j_{p+1}, …, j_{k}) ; x^{p+1}(t_{1}, …, t_{p}, j_{p+1}, …, j_{k}) ; ⋮ ; x^{k}(t_{1}, …, t_{p}, j_{p+1}, …, j_{k}) ]  (32)

where x^{i}(t_{1}, …, t_{p}, j_{p+1}, …, j_{k}) ∈ IR^{n_{i}} and ∑_{i=1}^{k} n_{i} = n. Let the matrices 𝕁, 𝕀 and ℤ be defined by

𝕁 = ⊕_{i=1}^{p} s_{i}I_{n_{i}}, 𝕀 = I_{∑_{i=p+1}^{k} n_{i}}, ℤ = ⊕_{i=p+1}^{k} s_{i}^{−1}I_{n_{i}},  (33)

respectively. Also let the function c(s) be defined by

c(s) = det [ \begin{matrix}𝕁 − A_{11} & −A_{12} \\ −ℤA_{21} & 𝕀 − ℤA_{22} \end{matrix} ] .  (34)

Inspired from Lemmas 5 and 7, the stability condition is given by the following lemma.

Lemma 9 [34] The system (20) is asymptotically stable if and only if

c(s) ≠ 0 ∀s = [ s_{1} ; ⋮ ; s_{k} ] |{Re(s_{i}) ≥ 0, i = 1, ..., p & |s_{i}| ≥ 1, i = p + 1, ..., k},  (35)

where c(s) is given by (34).

In [34], an interesting sufficient LMI condition for model (32) to be stable is proposed. An simple alternative formulation that, once again, falls out as a special case of Theorem 3 is now presented.

Theorem 10 The model (32) is asymptotically stable if there exist symmetric positive definite matrices P_{i}, i = 1, …, k, such that

M_{h} = [ \begin{matrix}A \\ I_{n} \end{matrix} ]′ Q_{h} [ \begin{matrix}A \\ I_{n} \end{matrix} ] < 0,  (36)

with

A = [ \begin{matrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix} ]  (37)

and

Q_{h} = [ \begin{matrix}0 & 0 & ⊕_{i=1}^{p} P_{i} & 0 \\ 0 & ⊕_{i=p+1}^{k} P_{i} & 0 & 0 \\ ⊕_{i=1}^{p} P_{i} & 0 & 0 & 0 \\ 0 & 0 & 0 & − ⊕_{i=p+1}^{k} P_{i} \end{matrix} ] .  (38)

Proof: it suffices to apply Theorem 3 with

R_{i} = [ \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ] ∀i ∈ \{p + 1, …, k\} & R_{i} = [ \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ] ∀i ∈ \{p + 1, …, k\},  (39)

and with D denoting (D^{C})^{p} × ( l C^{+})^{(k−p)}. Besides, since each s_{i} is outside the unit disc, its reciprocal exists and defining the nonsingular matrix 𝕊 by

𝕊 = [ \begin{matrix}I_{∑_{i=1}^{p} n_{i}} & 0 \\ 0 & ⊕_{i=1}^{k} s_{i}I_{n_{i}} \end{matrix} ] ,  (40)

enables to deduce that

p(s) = det(A(s)) ≠ 0 ⇔ det(𝕊^{−1}) det(A(s)) = det(𝕊^{−1}A(s)) = c(s) ≠ 0,  (41)

with c(s) given by (34). □

The authors insist on the 2D-case with p = 1 which enables to deal with the differential repetitive processes [40].

3.4 Stability of some continuous fractional polynomials

This subsection adresses the relationship between some result on the stability of continuous fractional polynomials [41] and Theorem 3. Here again, a result of [41] is briefly recalled and then it is shown that this result is encompassed by Theorem 3.

Firstly, let the fractional degree polynomial (the notion of polynomial is actually extended, see [42])

r(σ) = σ^{α_{1}} + r_{2}σ^{α_{2}} + … + r_{m}σ^{α_{m}}  (42)

be considered, where r_{h} are real numbers and α_{h} are positive rational numbers, h = 1, 2, …, m. Such a fractional degree polynomial is called k-commensurate if there exist natural numbers q_{h}^{i} and coprime natural numbers l_{i}, i = 1, …, k, h = 1, …, m, such that α_{h} = ∑_{i=1}^{k} \frac{q_{h}^{i}}{l_{i}} ∀h ∈ \{1, ..., m\}. Therefore, a k-commensurate fractional polynomial r(σ) as defined in (42) can be written as a multivariate polynomial

r(σ) = ξ(s) = ξ(s_{1}, …, s_{k}) = ∑_{h=1}^{m} (ξ_{h} (∏_{i=1}^{n} si^{\frac{q^{i}_{h}}{l_{i}}} )) .  (43)

where s_{i} = σ^{\frac{1}{l_{i}}} , ∀i ∈ \{1, …, k\}.

Secondly, let the complex sets V^{l} be introduced as follows:

V^{l} ˆ=\{λ =| λ | e^{jϕ} ∈ C : ϕ ∈ [−\frac{π}{2l}, \frac{π}{2l}]\}.  (44)

Clearly for l = 1, the set V^{1} reduces to the right half complex plane (together with the imaginary axis). For higher values of l, V^{l} becomes a right-open symmetric sector starting from the origin and including the boundary. Besides, involving natural parameters l_{i} introduced above, let the multi-region D be defined by

D = V^{l_{1}} × … × V^{l_{k}}.  (45)

In [41], a relation is established between the Hurwitz-stability of r(σ) and the root-location of kDpolynomial ξ(s).

Theorem 11 Let a k-commensurate fractional degree polynomial r(σ) comply with (42), its associated kD-polynomial ξ(s) be given by (43) and a k-region D be defined by (45) with (44). The polynomial r(σ) is Hurwitz if

ξ(s) ≠ 0 ∀s ∈ D.  (46)

Note here that some degree of conservatism is introduced. Theorem 11 is interesting but does not provide any tractable test. However, among the class of k-commensurate fractional degree polynomials r(σ) = ξ(s), some can be associated, owing to a procedure proposed in [43] to a companion matrix A so as to check

ξ(s) = p(s) = det(A(s))  (47)

where A(s) is given by (7) and thus involves the companion matrix A.

Remark 12 For other polynomials that do not enable the procedure to be performed (those polynomials are such that ξ(s) is not clearly monic), another procedure allows ones to derive a singular companion decomposition slightly different from (47) [41] but this case is not addressed here.

Owing to this matrix A, it is possible to derive a test which consists in a sufficient LMI condition.

Theorem 13 [41] Let a k-commensurate fractional degree polynomial r(σ) comply with (42), its associated kD-polynomial ξ(s) be given by (43) and a k-region D be defined by (45) with (44). The polynomial r(σ) is Hurwitz if there exists k symmetric positive definite matrices P_{i}, i = 1, …, n such that

([ \begin{matrix}A \\ I_{n} \end{matrix} ] ⊗ I_{2})^{T} Q ([ \begin{matrix}A \\ I_{n} \end{matrix} ] ⊗ I_{2}) < 0,  (48)

with

Q = [ \begin{matrix}0 & ⊕_{i=1}^{n} (P_{i} ⊗ R_{i_{10}}) \\ ⊕_{i=1}^{n} (P_{i} ⊗ R′_{i_{10}}) & 0 \end{matrix} ] .  (49)

where

\{ \begin{matrix}R_{i_{10}} = I_{2} & if & l_{i} = 1 \\ R_{i_{10}} = [ \begin{matrix}tan (\frac{π}{2l_{i}} ) & 1 \\ −1 & tan (\frac{π}{2l_{i}} ) \end{matrix} ] & if & l_{i} > 1 \end{matrix}  (50)

It is completely straightforward to see that Theorem 13 is a special case of Theorem 3 by noting that each sector V^{l_{i}}, complies with the description of D_{i} in (5) with the choice (50), with

R_{i_{00}} = 0 & R_{i_{11}} = 0 ∀i ∈ {1, …, k},  (51)

and by checking that (48) is nothing but (9).

4 Conclusion

In this paper, a sufficient LMI condition for the D-stability of any first degree polynomial matrix complying with (7) when D = D_{1}^{C} × … × D_{k}^{C} is a composition of subregions D_{i} that belong to specified class of regions was proposed. This condition is such that many conditions of the literature for the stability of continuous and discrete multidimensional systems, repetitive processes and even some fractional-order models fall out as special cases of this unified result. Paricularly, continuous and discrete multidimensional versions of Lyapunov and Stein inequalities are presented under one single formulation.

Not only a unique and generic condition is proposed for many stability analysis problems but it shall bring many insights for the robust stability analysis, or even for nominal and robust synthesis. As another possible investigation, a larger class of polynomial matrices, slightly different from the one considered in the paper, and corresponding to singular systems, could be studied. It would then correspond to a further generalization of the present work and would allow ones, for example, to handle a larger class of k-commensurate fractional degree polynomials, encompassing some other results in [41].

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