Volume 2013, Issue 1 396759

Research Article

Open Access

Stability of Matrix Polytopes with a Dominant Vertex and Implications for System Dynamics

Octavian Pastravanu

Department of Automatic Control and Applied Informatics, Technical University “Gheorghe Asachi” of Iasi, Boulevard Mangeron 27, 700050 Iasi, Romania tuiasi.ro

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Mihaela-Hanako Matcovschi,

Corresponding Author

Mihaela-Hanako Matcovschi

[email protected]

Department of Automatic Control and Applied Informatics, Technical University “Gheorghe Asachi” of Iasi, Boulevard Mangeron 27, 700050 Iasi, Romania tuiasi.ro

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Octavian Pastravanu,

Octavian Pastravanu

Department of Automatic Control and Applied Informatics, Technical University “Gheorghe Asachi” of Iasi, Boulevard Mangeron 27, 700050 Iasi, Romania tuiasi.ro

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Mihaela-Hanako Matcovschi,

Corresponding Author

Mihaela-Hanako Matcovschi

[email protected]

Department of Automatic Control and Applied Informatics, Technical University “Gheorghe Asachi” of Iasi, Boulevard Mangeron 27, 700050 Iasi, Romania tuiasi.ro

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First published: 05 June 2013

https://doi.org/10.1155/2013/396759

Citations: 6

Academic Editor: Gani Stamov

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Abstract

The paper considers the class of matrix polytopes with a dominant vertex and the class of uncertain dynamical systems defined in discrete time and continuous time, respectively, by such polytopes. We analyze the standard concept of stability in the sense of Schur—abbreviated as SS (resp., Hurwitz—abbreviated as HS), and we develop a general framework for the investigation of the diagonal stability relative to an arbitrary Hölder p-norm, 1 ≤ p ≤ ∞, abbreviated as SDS_p (resp., HDS_p). Our framework incorporates, as the particular case with p = 2, the known condition of quadratic stability satisfied by a diagonal positive-definite matrix, i.e. SDS₂ (resp., HDS₂) means that the standard inequality of Stein (resp., Lyapunov) associated with all matrices of the polytope has a common diagonal solution. For the considered class of matrix polytopes, we prove the equivalence between SS and SDS_p (resp., HS and HDS_p), 1 ≤ p ≤ ∞ (fact which is not true for matrix polytopes with arbitrary structures). We show that the dominant vertex provides all the information needed for testing these stability properties and for computing the corresponding robustness indices. From the dynamical point of view, if an uncertain system is defined by a polytope with a dominant vertex, then the standard asymptotic stability ensures supplementary properties for the state-space trajectories, which refer to special types of Lyapunov functions and contractive invariant sets (characterized through vector p-norms weighted by diagonal positive-definite matrices). The applicability of the main results is illustrated by two numerical examples that cover both discrete- and continuous-time cases for the class of uncertain dynamics studied in our paper.

1. Introduction

1.1. Research Context and Objective

Consider the matrix polytope

()

where {A₁, A₂, …, A_K} is a finite set of real n × n matrices. The Schur (resp., Hurwitz) stability—abbreviated as SS (resp., HS)—has been investigated for matrix polytope (1) starting with the 80s, by papers 4such as [1–13]. Research was strongly motivated by the dynamics analysis of linear systems with model uncertainties (which inherently occur due to incomplete or approximate information on process parameters). Description (1) is also referred to as a “polytopic matrix”, and from the modeling-power point of view, it incorporates the class of “interval matrices,” defined by hyperrectangles in ℝ^n×n, i.e.

()

where the inequalities have a componentwise meaning. Significant results on Schur and Hurwitz stability of interval matrices have been reported in [10, 14–24].

Also starting with the 80s the linear algebra literature developed studies on a stronger type of matrix stability, called “diagonal stability”; pioneering works such as [25, 26] should be mentioned. In accordance with the monograph [27], a square matrix is Schur (resp., Hurwitz) diagonally stable if the Stein (resp., Lyapunov), inequality associated with that matrix has diagonal positive-definite solutions. As a natural expansion, our work [28] introduced the Stein (resp., Lyapunov), inequalities relative to a Hölderp-norm, 1 ≤ p ≤ ∞, and generalized the aforementioned diagonal stability concept to “Schur (resp., Hurwitz) diagonal stability relative to a Hölder p-norm”—abbreviated as SDS_p (resp., HDS_p). For p = 2, the framework proposed by [28] coincides with the classic approach presented by [27].

The SDS_p (resp., HDS_p), 1 ≤ p ≤ ∞, has been recently explored by our papers [29, 30] for interval matrices and for arbitrary polytopic matrices, respectively. It is worth saying that the monograph [27] addressed the standard case of diagonal stability (i.e. SDS₂ and HDS₂ in our nomenclature) for interval matrices.

During the last decade, diagonal stability ensured a visible research potential for systems and control engineering, mainly related to the simpler form of the Lyapunov function candidates, as outlined by works such as [27–29, 31–35]. These works use the same terminology “diagonal stability” in the sense of a system property that is induced by the original matrix property discussed in the previous paragraphs.

Our current paper focuses on the stability of a class of matrix polytopes of form (1) called “with a dominant vertex”—the concept is to be rigorously introduced by Definition 3 in the next section. We also study the dynamics of the polytopic systems associated with this class of matrix polytopes, described by the following equations:

(i)
in the discrete-time case,
(ii)
in the continuous-time case,

In both models (3-S) and (3-H), the entries of matrix A are considered fixed (not time varying); they are uncertain in the sense that their values are incompletely known but surely satisfy the condition A ∈ 𝒜. In other words, a single matrix A is used for modeling a certain evolution of the process, whereas for modeling two different evolutions (taking place separately) two distinct matrices A₁ ≠ A₂, A₁, A₂ ∈ 𝒜 may be needed.

1.2. Paper Structure

For a matrix polytope (1) with a dominant vertex, we prove that SS (resp., HS) is equivalent to SDS_p (resp., HDS_p), 1 ≤ p ≤ ∞, unlike the case of an arbitrary matrix polytope (e.g., [30]) where (i) SDS_p (resp., HDS_p) is more conservative than SS (resp., HS) and (ii) results on and (resp., and ) may be different for p₁ ≠ p₂, 1 ≤ p₁, p₂ ≤ ∞. These aspects are discussed by Section 2 of our work. Section 3 analyzes the implications of Section 2 for the dynamics of a polytopic system (3-S), respectively (3-H), defined by a matrix polytope with a dominant vertex. We show that the asymptotic stability of such a system is equivalent to the existence of Lyapunov functions and contractive invariant sets expressed in terms of any Hölder p-norm, by using an appropriate weighting matrix of diagonal form (whose positive entries depend on the chosen norm). The utility of our main results is illustrated in Section 4 by numerical examples, covering Schur (resp., Hurwitz) stability for matrix polytopes with a dominant vertex, as well as the implications for the dynamics of discrete-time (resp., continuous-time), polytopic systems.

Throughout the text, in equation numbering we use the extension S (resp., H), for referring to Schur (resp., Hurwitz) stability and/or to discrete-time (resp., continuous-time), dynamics—as in the above equation (3-S) (resp., (3-H)). The extensions (S) and (H) play the same role for the labels of definitions and theorems.

To ensure the fluent presentation of our results, their proofs are given in the Appendix.

1.3. Notations and Nomenclature

Let x = [x₁ ⋯ x_n] ^T,

be vectors.

(i)
∥x∥_p is the Hölder vector p-norm defined by for 1 ≤ p < ∞ and by ∥x∥_∞ = max_1≤i≤n|x_i| for p = ∞.
(ii)
“x ≤ y”, “x < y” mean componentwise inequalities, i.e. x_i ≤ y_i, x_i < y_i, i = 1, …, n.

Let M = [m_ij], Q = [q_ij] ∈ ℝ^n×n be square matrices.

(iii)
∥M∥_p is the matrix norm induced by the vector p-norm through .
(iv)
μ_p(M) = lim _h↓0h⁻¹(∥I+hM∥_p − 1) is the matrix measure [36, page 41], based on the matrix norm ∥ ∥_p.
(v)
If D = diag {d₁, …, d_n}, d_i > 0, i = 1, …, n, then the following regions of the complex plane , j = 1, …, n, are called the generalized Gershgorin′s disks of M defined with D for columns.
(vi)
If D = diag {d₁, …, d_n}, d_i > 0, i = 1, …, n, then the following regions of the complex plane , i = 1, …, n, are called the generalized Gershgorin’s disks of M defined with D for rows.
(vii)
σ(M) = {z ∈ ℂ∣det (zI − M) = 0} is the spectrum of M, and λ_i(M) ∈ σ(M), i = 1, …, n, are the eigenvalues of M.
(viii)
If σ(M) ⊂ ℂ_S = {z ∈ ℂ∣ | z | < 1}, then matrix M is said to be Schur stable (abbreviated as SS).
(ix)
If σ(M) ⊂ ℂ_H = {z ∈ ℂ∣Rez < 0}, then matrix M is said to be Hurwitz stable (abbreviated as HS).
(x)
If M is nonnegative (all entries are nonnegative), its spectral radius is a positive eigenvalue, denoted by λ_max(M), such that |λ_i(M)| ≤ λ_max(M), i = 1, …, n.
(xi)
If M is essentially nonnegative (all off-diagonal entries are nonnegative), then it has a real eigenvalue, denoted by λ_max(M), such that Re{λ_i(M)} ≤ λ_max(M), i = 1, …, n—for example, Lemma 1 in [28].
(xii)
If M is symmetrical, then all its eigenvalues are real and there exists an eigenvalue denoted by λ_max(M), such that λ_i(M) ≤ λ_max(M), i = 1, …, n.
(xiii)
“M≻0", “M≺0" mean that M is a positive-definite, negative-definite matrix.
(xiv)
If the oriented graph of M is strongly connected, then M is called irreducible; otherwise M is called reducible.
(xv)
For p ∈ {1,2, ∞}, the matrix norms ∥M∥_p and matrix measures μ_p(M) have the following expressions:
(xvi)
|M| denotes the matrix built with the absolute values of the entries of M.
(xvii)
M^S ∈ ℝ^n×n (S superscript from Schur) denotes the nonnegative matrix defined by M^S = |M|.
(xviii)
M^H ∈ ℝ^n×n (H superscript from Hurwitz) denotes the essentially nonnegative matrix defined by M^H = M^d+|M^o|, where M^d = diag {m₁₁, …, m_nn} and M^o = M − M^d.
(xix)
“M ≤ Q”, “M < Q” mean componentwise inequalities, i.e. m_ij ≤ q_ij, m_ij < q_ij, i, j = 1, …, n.

Throughout the text we shall write “X (resp., Y)” wherever “X” and “Y” are referred to in parallel.

2. Results on Matrix Polytopes

The current section explores the stability of matrix polytopes with a dominant vertex. For this class of polytopes, the standard Schur (Hurwitz) stability is proved to be equivalent to stronger stability properties, namely, diagonal stability relative to arbitrary Hölder p-norms 1 ≤ p ≤ ∞.

2.1. Preliminaries

Definition 1 (S). Let us consider: 1 ≤ p ≤ ∞; an arbitrary matrix A ∈ ℝ^n×n; a matrix polytope 𝒜 of form (1); a nonsingular matrix Q ∈ ℝ^n×n.

(a) The inequality

is called the Stein-type inequality relative to the p-norm associated with matrix A; matrix Q is said to be a solution to this inequality.

(b) Matrix Q is said to be a solution to the Stein-type inequality relative to the p-norm associated with the polytope 𝒜 if the following condition is fulfilled:

Definition 1 (H). Let us consider: 1 ≤ p ≤ ∞; an arbitrary matrix A ∈ ℝ^n×n; a matrix polytope 𝒜 of form (1); a nonsingular matrix Q ∈ ℝ^n×n.

(a) The inequality

is called the Lyapunov-type inequality relative to the p-norm associated with matrix A; matrix Q is said to be a solution to this inequality.

(b) Matrix Q is said to be a solution to the Lyapunov-type inequality relative to the p-norm associated with the polytope 𝒜 if the following condition is fulfilled:

Remark 1. (i) The terminology introduced by Definition 1(S)(a) (resp., Definition 1(H)(a)) is motivated by the fact that inequality (5-S) (resp., (5-H)) with p = 2 is equivalent to the standard Stein inequality

respectively the standard Lyapunov inequality

Indeed, for (5-S) with p = 2 we may write that is equivalent to

which means the fulfillment of (7-S).

Similarly, for (5-H) with p = 2 we may write that μ₂(Q⁻¹AQ) < 0 is equivalent to

which means the fulfillment of (7-H).

(ii) The existence of P≻0 solving the standard Stein inequality (7-S) (resp., Lyapunov inequality (7-H) is equivalent to Schur (resp., Hurwitz) stability of matrix A.

(iii) Conditions (6-S) (resp., (6-H)) with p = 2 in Definition 1(S)(b) (resp., Definition 1(H)(b)) represent the definition of Schur (resp., Hurwitz) quadratic stability of the matrix polytope 𝒜, for example, [37, page 213].

Definition 2 (S). Let 𝒜 be a matrix polytope of form (1).

(a) 𝒜 is called Schur stable (abbreviated as SS) if

(b) Let 1 ≤ p ≤ ∞. 𝒜 is called Schur diagonally stable relative to the p-norm (abbreviated as SDS_p) if there exists a diagonal positive-definite matrix D≻0 that satisfies the Stein-type inequality relative to the p-norm associated with the polytope 𝒜, i.e.

Definition 2 (H). Let 𝒜 be a matrix polytope of form (1).

(a) 𝒜 is called Hurwitz stable (abbreviated as HS) if

(b) Let 1 ≤ p ≤ ∞. 𝒜 is called Hurwitz diagonally stable relative to the p-norm (abbreviated as HDS_p) if there exists a diagonal positive-definite matrix D≻0 that satisfies the Lyapunov-type inequality relative to the p-norm associated with the polytope 𝒜, i.e.

Remark 2. (i) If 𝒜 is a trivial polytope defined by a single matrix A (i.e. A₁ = A, K = 1, in (1)), then Definition 2(S)(b) (resp., Definition 2(H)(b)) coincides with Definition 1 in [28].

(ii) Let 1 ≤ p ≤ ∞. If 𝒜 is a proper polytope (i.e. ∃k₁ ≠ k₂, k₁, k₂ ∈ {1, …, K}: in (1)), then Definition 2(S)(b) (resp., Definition 2(H)(b)) proposes a meaningful extension of Definition 1 in [28]. Indeed, the simple use of Definition 1 in [28] does not necessarily imply the existence of a unique diagonal matrix D≻0 that satisfies inequality (10-S) (resp., (10-H)) for all matrices A ∈ 𝒜.

(iii) Let 1 ≤ p ≤ ∞. The SDS_p (resp., HDS_p) is a property of 𝒜 stronger than SS (resp., HS). Indeed, each matrix A ∈ 𝒜 is SS (resp., HS) once it is SDS_p (resp., HDS_p) in accordance with Remark 2 in [28].

(iv) Let p = 2. Definition 2(S)(b) (resp., Definition 2(H)(b)) expresses a particular case of Schur (resp., Hurwitz) quadratic stability of the matrix polytope 𝒜—see Remark 1(iii). Subsequently, the quadratic stability is a property of 𝒜 stronger than SS (resp., HS) but, at the same time, weaker than SDS₂ (resp., HDS₂).

Definition 3 (S). Let 𝒜 be a matrix polytope of form (1). If there exists a subscript k^* ∈ {1, …, K}, such that the vertex fulfills one of the following two sets of componentwise inequalities:

then 𝒜 is called a matrix polytope with an S-dominant vertex, and

is called the S-dominant vertex of 𝒜. (The notation M^S is introduced in Section 1.3.)

Definition 3 (H). If there exists a subscript k^* ∈ {1, …, K}, such that the vertex fulfills the componentwise inequalities:

then 𝒜 is called a matrix polytope with an H-dominant vertex, and

is called the H-dominant vertex of 𝒜. (The notation M^H is introduced in Section 1.3.)

Remark 3. (i) If 𝒜 is a matrix polytope with an S-dominant (resp., H-dominant) vertex , then Definition 3(S) (resp., Definition 3(H)) shows that matrix is nonnegative—inequalities (11-S-1) or nonpositive—inequalities (11-S-2) (resp., essentially nonnegative—inequalities (11-H)).

(ii) In the remainder of the text, we mainly address the case of the S-dominant vertex defined by inequalities (11-S-1). The case based on inequalities (11-S-2) does not require a separate approach, since all the results we are going to use for nonnegative remain valid for nonnegative.

(iii) A matrix polytope 𝒜 may have two S-dominant vertices denoted as in the particular case when , satisfies inequalities (11-S-1), satisfies inequalities (11-S-2). We still can refer to 𝒜 as having “an S-dominant vertex,” since the stability properties of 𝒜 induced by and by are identical, as resulting from the further development of our paper.

2.2. Stability Analysis

Theorem 1 (S). Let us consider: 1 ≤ p ≤ ∞; a matrix polytope 𝒜 with an S-dominant vertex .

The following statements are equivalent.

(i)
is SS.
(ii)
𝒜 is SS.
(iii)
There exists a p, 1 ≤ p ≤ ∞, such that 𝒜 is SDS_p.
(iv)
𝒜 is SDS_p for all p, 1 ≤ p ≤ ∞.
(v)
There exists a diagonal matrix D≻0 such that the union for all A ∈ 𝒜 of the generalized Gershgorin’s disks written for columns is located inside the unit circle of the complex plane, i.e. .
(vi)
There exists a diagonal matrix D≻0 such that the union for all A ∈ 𝒜 of the generalized Gershgorin’s disks written for rows is located inside the unit, circle of the complex plane, i.e. .

Proof. See the Appendix.

Theorem 1 (H). Let us consider: 1 ≤ p ≤ ∞; a matrix polytope 𝒜 with an H-dominant vertex .

The following statements are equivalent.

(i)
is HS.
(ii)
𝒜 is HS.
(iii)
There exists a p, 1 ≤ p ≤ ∞, such that 𝒜 is HDS_p.
(iv)
𝒜 is HDS_p for all p, 1 ≤ p ≤ ∞.
(v)
There exists a diagonal matrix D≻0 such that the union for all A ∈ 𝒜 of the generalized Gershgorin’s disks written for columns is located in the left half plane of the complex plane, i.e. .
(vi)
There exists a diagonal matrix D≻0 such that the union for all A ∈ 𝒜 of the generalized Gershgorin’s disks written for rows is located in the left half plane of the complex plane, i.e. .

Proof. See the Appendix.

2.3. Diagonal Solutions to Stein-Type and Lyapunov-Type Inequalities

The S-dominant (resp., H-dominant) vertex of a matrix polytope 𝒜 can be used not only for testing the properties SDS_p (resp., HDS_p) of 𝒜 but also for finding concrete diagonal matrices D≻0 that satisfy the inequality (10-S) in Definition 2(S) (resp., inequality (10-H) in Definition 2(H)).

Theorem 2 (S). Let us consider: 1 ≤ p ≤ ∞; a matrix polytope 𝒜 with an S-dominant vertex ; a diagonal positive-definite matrix D≻0.

Matrix D is a solution to (10-S) (i.e. D satisfies the Stein-type inequality relative to the p-norm associated with the polytope 𝒜) if and only if

(i.e. D satisfies the Stein-type inequality relative to the p-norm associated with the S-dominant vertex

Proof. See the Appendix.

Theorem 2 (H). Let us consider: 1 ≤ p ≤ ∞; a matrix polytope 𝒜 with an H-dominant vertex ; a diagonal positive-definite matrix D≻0.

Matrix D is a solution to (10-H) (i.e. D satisfies the Lyapunov-type inequality relative to the p-norm associated with the polytope 𝒜) if and only if

(i.e. D satisfies the Lyapunov-type inequality relative to the p-norm associated with the H-dominant vertex

Proof. See the Appendix.

Remark 4. Let 1 ≤ p ≤ ∞. Whenever is SS (resp., HS) diagonal matrices D≻0 that satisfy (12-S) (resp., (12-H)<?cmd?>) can be built along the lines of Lemma 3 and Remark 3 in [28]. Further comments are available in the next section that discloses the role of D≻0 in the dynamics of a polytopic system of form (3-S) (resp., (3-H)).

2.4. Stability Margins

The S-dominant (resp., H-dominant) vertex of a matrix polytope 𝒜 also allows one to develop a robustness analysis for SS and SDS_p of 𝒜 defined by (1) and (11-S-1) or (11-S-2) (resp., HS and HDS_p of 𝒜 defined by (1) and (11-H)).

Definition 4 (S). Let 𝒜 be a matrix polytope with an S-dominant vertex .

(a) If is SS, then

is called the SS margin of

(b) If 𝒜 is SS, then

is called the SS margin of 𝒜.

is called the SDS_p margin of 𝒜.

Definition 4 (H). Let 𝒜 be a matrix polytope with an H-dominant vertex .

(a) If is HS, then

is called the HS margin of

(b) If 𝒜 is HS, then

is called the HS margin of 𝒜.

is called the HDS_p margin of 𝒜.

Theorem 3 (S). Let 𝒜 be a matrix polytope with an S-dominant vertex . For any p, 1 ≤ p ≤ ∞, the following equalities hold:

Proof. See the Appendix.

Theorem 3 (H). Let 𝒜 be a matrix polytope with an H-dominant vertex . For any p, 1 ≤ p ≤ ∞, the following equalities hold:

Proof. See the Appendix.

Remark 5. (i) For each stability property of the polytope 𝒜 discussed in Section 2.2, the corresponding margin (also called “degree” in the control-engineering literature) quantifies the distance between a matrix A ∈ 𝒜 representing the “worst case” relative to that property and the “limit situation” where that property is generically lost for an arbitrary matrix. Theorem 3(S) (resp., Theorem 3(H)) shows that the “worst case” of 𝒜 relative to SS and SDS_p (resp., HS and HDS_p) is defined by the S-dominant (resp., H-dominant) vertex.

(ii) For p = 1, Theorem 3(S) (resp., Theorem 3(H)) ensure the existence of a diagonal matrix D≻0 such that the union for all A ∈ 𝒜 of the generalized Gershgorin’s disks written for columns is located in the region of the complex plane defined by (resp., ). The same location also corresponds to the union for all A ∈ 𝒜 of the generalized Gershgorin’s disks written for rows , where the existence of the diagonal matrix D≻0 is guaranteed by Theorem 3(S) (resp., Theorem 3(H)) with p = ∞. Obviously, the region (resp., ) refines the condition formulated by Theorem 1(S)(v)-(vi) (resp., Theorem 1(H)(v)-(vi)) for the location of the generalized Gershgorin’s disks.

(iii) The equality (16-S) (resp., (16-H)) plays an important role in the characterization of the dynamic properties exhibited by the polytopic system (3-S) (resp., (3-H)). Further details on this role are available in Remark 6 of the next section.

(iv) For an arbitrary polytope 𝒜 (without a dominant vertex), equality (16-S) (resp., (16-H)) does not hold true, in general. If, for a given p, 1 ≤ p ≤ ∞, 𝒜 is SDS_p (resp., HDS_p) then (resp., ), fact which was anticipated by Remark 2(iii) in general terms, without using this specific language of “stability margins.” Moreover, if for given p₁ ≠ p₂, 1 ≤ p₁, p₂ ≤ ∞, 𝒜 is diagonally stable relative to both p₁- and p₂-norm, then we may have (resp., ), as already suggested by our recent paper [30].

2.5. Particular Case of Interval Matrices with a Dominant Vertex

Theorems 1(S), 2(S), and 3(S) generalize the results reported in [27, Lemma 3.4.18], [29] for SS and SDS_p of interval matrices of form (2) with A₀ or −A₀ nonnegative, because these two types of interval matrices represent particular cases of matrix polytopes with an S-dominant vertex defined by inequalities (11-S-1) or (11-S-2).

Similarly, Theorems 1(H), 2(H), and 3(H) generalize results reported in [29] for HS and HDS_p of interval matrices of form (2) with A₀ essentially nonnegative, since such interval matrices represent a particular case of matrix polytopes with an H-dominant vertex defined by inequalities (11-H).

3. Results on Polytopic Systems

The current section shows that a polytopic system defined by a matrix polytope with a dominant vertex may exhibit dynamical properties stronger than the standard concept of asymptotic stability; these dynamical properties are correlated, by equivalence, to the algebraic properties of the dominant vertex.

Theorem 4 (S). Let us consider: 1 ≤ p ≤ ∞; a discrete-time polytopic system of form (3-S) where polytope 𝒜 has an S-dominant vertex ; a positive-definite diagonal matrix D≻0; a constant r satisfying 0 < r < 1.

The following statements are equivalent:

(i)
(ii)
For the polytopic system (3-S), the functions
are strong diagonal Lyapunov functions, with the decreasing rate r, i.e.
(iii)
The contractive sets
are invariant with respect to the state-space trajectories (solutions) of the polytopic system (3-S), i.e.

Proof. See the Appendix.

Theorem 4 (H). Let us consider: 1 ≤ p ≤ ∞; a continuous-time polytopic system of form (3-H) where polytope 𝒜 has an H-dominant vertex ; a positive-definite diagonal matrix D≻0; a constant r satisfying r < 0.

The following statements are equivalent:

(i)
(ii)
For the polytopic system (3-H), the functions
are strong diagonal Lyapunov functions, with the decreasing rate r, i.e.
(iii)
The contractive sets
are invariant with respect to the state-space trajectories (solutions) of the polytopic system (3-H), i.e.

Proof. See the Appendix.

Remark 6. (i) Let 1 ≤ p ≤ ∞. The exploration of the dynamical properties of a polytopic system via Theorem 4(S) (resp., Theorem 4(H)) outlines the importance of the concrete value 0 < r < 1 (resp., r < 0) in the right hand side of inequality (17-S) (resp., (17-H)). This concrete value r does not appear explicitly in the Stein-type inequality (12-S) (resp., Lyapunov-type inequality (12-H)); the existence of a diagonal matrix D≻0 that solves inequality (12-S) (resp., (12-H)) represents a necessary and a sufficient condition for the SDS_p (resp., HDS_p) of a matrix polytope 𝒜. For a polytopic system (3-S) (resp., (3-H)) a complete description of the dynamics implies the knowledge of pairs formed by r and D that satisfy Theorem 4(S) (resp., Theorem 4(H)).

(ii) The constant 0 < r < 1 (resp., r < 0) in Theorem 4(S) (resp., Theorem 4(H)) represents a decreasing rate for the diagonal Lyapunov functions and for the contractive invariant sets. We are going to prove that for any p, 1 ≤ p ≤ ∞, the value of the fastest decreasing rate is given by the , regardless of the discrete-time or continuous-time nature of the dynamics. Indeed, for , there exists no diagonal matrix D≻0 satisfying inequality (17-S) (resp., (17-H)). For , we use Lemma 3 and Remark 3 in [28] that yield the following discussion on the irreducibility/reducibility of .

Case 1 (matrix Ak* is irreducible). (The definition of an irreducible matrix is introduced in Section 1.3.) Denote by v = [v₁ ⋯ v_n] ^T > 0 and w = [w₁ ⋯ w_n] ^T > 0 its right and left Perron eigenvectors, respectively. Given p, 1 ≤ p ≤ ∞, we construct the diagonal matrix , where (i) (1/p) + (1/q) = 1 if 1 < p < ∞; (ii) 1/p = 1, 1/q = 0 if p = 1; (iii) 1/p = 0, 1/q = 1 if p = ∞. Matrix D_p≻0 fulfills the following equality:

(i)
if is nonnegative, i.e. is defined by (11-S-1);
(ii)
if is nonpositive, i.e. is defined by (11-S-2);
(iii)
if is essentially nonnegative, i.e. is defined by (11-H).

Case 2 (matrix Ak* is reducible). For any , there exists ε > 0 such that , where J is the n by n matrix with all its entries 1. We apply the procedure presented by Case 1 to the irreducible matrix , and the resulting diagonal matrix D_p ≻ 0 fulfills , , respectively. On the other hand, we have the norm inequality , the measure inequality , respectively. Subsequently, we obtain , , respectively.

Thus, for any p, 1 ≤ p ≤ ∞, we can construct a diagonal matrix D_p≻0 such that inequality (17-S) (resp., (17-H)) is fulfilled with —for irreducible and with but as close to as we want—for reducible.

(iii) The fastest decreasing rate of the diagonal Lyapunov functions and of the contractive invariant sets can be expressed in terms of the stability margins (discussed in Section 2.4), as for the discrete-time case and for the continuous-time case. This point of view shows that the stability margins provide an algebraic characterization for the polytope 𝒜 and, concomitantly, allow the evaluation of the dynamical properties of the polytopic system (3-S) (resp., (3-H)).

(iv) For an arbitrary polytope 𝒜 (without a dominant vertex), the fastest decreasing rate may depend on the p-norm that defines the Lyapunov function and the invariant sets (if exist). If, for a given p, 1 ≤ p ≤ ∞, 𝒜 is SDS_p (resp., HDS_p) then the fastest decreasing rate corresponding to the p-norm of the polytopic system can be expressed in terms of the stability margins as (resp., ). The fastest decreasing rate corresponding to the p-norm can be fairly estimated by a computational procedure based on a bisection method presented in [30].

4. Illustrative Examples

This section presents two examples that illustrate the usefulness of the theoretical results developed by our work. Example 1 explores (i) the stability of a matrix polytope with an S-dominant vertex of form (1) and (11-S-1); (ii) the dynamical properties of the discrete-time polytopic system of form (3-S) defined by the considered polytope. Example 2 explores (i) the stability of a matrix polytope with an H-dominant vertex of form (1) and (11-H); (ii) the dynamical properties of the continuous-time polytopic system of form (3-H) defined by the considered polytope.

Example 1. It is adapted from [27]. Let

()

be the set of diagonal matrices whose elements are subunitary. 𝒮^[−1,1] is the convex hull generated by the set of 2ⁿ vertices 𝒮_sgn = {S ∈ ℝ^n×n∣ | S | = I_n}, also called the class of signature matrices.

Given a nonnegative matrix A^* ∈ ℝ^n×n, the set

()

is a matrix polytope of form (1) with the vertices A_i = A^*S_i, where S_i ∈ 𝒮_sgn, i = 1, …, 2ⁿ. The set 𝒜 = ℛ(A^*) defined by (23) is a matrix polytope with an S-dominant vertex because matrix A^* is a vertex that satisfies condition (11-S-1).

Theorem 1(S) shows that the Schur stability of A^* is a necessary and a sufficient condition for the Schur diagonal stability of the polytope 𝒜 = ℛ(A^*) relative to any p-norm, 1 ≤ p ≤ ∞. According to Theorem 2(S), a diagonal positive-definite matrix D_p≻0 satisfies the Stein-type inequality relative to the p-norm associated with 𝒜 = ℛ(A^*) if and only if D_p≻0 satisfies the Stein-type inequality relative to the p-norm associated with A^*. This property of 𝒜 = ℛ(A^*) is guaranteed for any p-norm by Theorem 2(S), whereas Proposition 2.5.8 in [27] can guarantee only the particular case corresponding to p = 2.

If A^* is Schur stable, then the eigenvalue λ_max(A^*) allows one to investigate the following properties: (i) for the polytope 𝒜 = ℛ(A^*), the SDS_p margins are given by relation (16-S) in Theorem 3(S) i.e. , for any p, 1 ≤ p ≤ ∞; (ii) for the discrete-time polytopic system of form (3-S) defined by 𝒜 = ℛ(A^*), regardless of the p-norm considered in Theorem 4(S) for the Lyapunov function (18-S), and for the contractive invariant sets (20-S), the fastest decreasing rate is λ_max(A^*) if A^* is irreducible and arbitrarily close to λ_max(A^*) if A^* is reducible—as per Remark 6(ii).

Finally, we notice that the nonpositive matrix A^** = −A^* is also a vertex of the polytope 𝒜 = ℛ(A^*) and it satisfies the dominance condition (11-S-2). This means that the polytope 𝒜 = ℛ(A^*) fits in the particular context commented by Remark 3(iii). It is obvious that the analysis presented by the current example is complete, in the sense that the vertex A^** ≤ 0 brings no supplementary information (since the matrix −A^** is nonnegative, there exists λ_max(−A^**) = λ_max(A^*), and for all p, 1 ≤ p ≤ ∞, the equality holds true).

The above approach applies mutatis-mutandis to the investigation of the matrix polytope ℒ(A^*) = {A = SA^*∣S ∈ 𝒮^[−1,1]}. In this case, Theorem 2(S) generalizes for 1 ≤ p ≤ ∞ the result that can be obtained when p = 2 by using Proposition 2.5.9 in [27] for matrix A^* and polytope ℒ(A^*).

Example 2. Let us consider the interval matrix [14]:

()

that is a matrix polytope with K = 2⁴ vertices:

()

𝒜 has an H-dominant vertex

()

which satisfies inequalities (11-H). Note that A^* is essentially positive and has the Perron-Frobenius eigenvalue λ_max(A^*) = −2.6456. Theorem 1(H) shows that the Hurwitz stability of A^* ensures the Hurwitz diagonal stability of the polytope 𝒜 relative to any p-norm, 1 ≤ p ≤ ∞.

The stability margins of the polytope 𝒜 are given by relation (16-H) in Theorem 2, i.e. , for any p, 1 ≤ p ≤ ∞.

For the qualitative analysis of the continuous-time polytopic system defined by (3-H) and (24), we can apply Theorem 4(H). Remark 6(ii) shows that for any p, 1 ≤ p ≤ ∞, the fastest decreasing rate for the diagonal Lyapunov functions and for the contractive invariant sets is exactly λ_max(A^*) = −2.6456, since A^* is irreducible. We apply Case 1 of the procedure presented in Remark 6(ii) and relying on the right and left Perron eigenvectors of A^* (v = [1 0.5562] ^T and w = [0.7822 1] ^T), we construct the diagonal matrices D_p≻0 corresponding to the fastest decreasing rate. For p ∈ {1,2, ∞}, these diagonal matrices are D₁ = diag {1.2785,1}, D₂ = diag {1.1307,0.7458}, and D_∞ = diag {1,0.5562}, and they satisfy Theorem 4(H) with r = λ_max(A^*) = −2.6456.

Note that all the above results remain valid if instead of 𝒜 defined by (24), we consider the matrix polytope

()

which has the same dominant vertex A^* (26).

5. Conclusions

The paper provides analysis instruments for the stability of matrix polytopes with a dominant vertex, as well as for the dynamics of discrete- and continuous-time uncertain systems defined by such polytopes. These analysis instruments are formulated as necessary and sufficient conditions exclusively based on the characteristics of the dominant vertex. Thus, the dominant vertex represents the only test matrix used for studying the following properties of a matrix polytope and its associated dynamical system: (i) Schur (resp., Hurwitz) stability (including the computation of the corresponding margin); (ii) Schur (resp., Hurwitz) diagonal stability relative to a p-norm (including the computation of the corresponding margin); (iii) existence of diagonal positive-definite matrices solving the Stein-type (resp., Lyapunov-type) inequalities relative to a p-norm; (iv) existence of diagonal-type Lyapunov functions and contractive invariant sets defined by a p-norm and decreasing with a given rate. A global result of our work is the proof that stability and diagonal stability relative to an arbitrary p-norm are equivalent for the considered class of matrix polytopes (fact which is not true for general matrix polytopes).

Acknowledgment

The authors acknowledge the support of UEFISCDI Romania, Grant no. PN-II-ID-PCE-2011-3-1038.

Appendix

Proof of Theorem 1(S). (i) ⇒(ii).

It results from the following:

()

since A^S,

are nonnegative and we can apply Theorem 8.1.18 and Corollary 8.1.19 in [38].

(ii) ⇒(i) It is obvious, because .

(i) ⇒(iv) Let 1 ≤ p ≤ ∞. Lemma 3 in [28] ensures the existence of a diagonal matrix D≻0 such that . On the other hand, we have the implication

()

as per Lemma 4 in [28]. We conclude that for all

(iv) ⇒(iii) It is obvious.

(iii) ⇒(ii) It follows from Remark 2(iii).

(v) ⇔(iii) with p = 1. It results from the equivalence

()

(vi) ⇔(iii) with p = ∞. It is similar to (v) ⇔(iii) with p = 1.

Proof of Theorem 1(H). (i) ⇒(ii).

It results from the following:

()

Take s > 0 so that sI + A^H ≥ 0. Hence,

. By applying Theorem 8.1.18 and Corollary 8.1.19 in [38], we get

, which implies that

(ii) ⇒(i) It is obvious, because .

(i) ⇒(iv) Let 1 ≤ p ≤ ∞. Lemma 3 in [28] ensures the existence of a diagonal matrix D≻0 such that . On the other hand, we have the implication

()

as per Lemma 4 in [28]. We conclude that for all A ∈ 𝒜 : μ_p(D⁻¹AD) < 0.

(iv) ⇒(iii) It is obvious.

(iii) ⇒(ii) It follows from Remark 2(iii).

(v) ⇔(iii) with p = 1. It results from the equivalence

()

(vi) ⇔(iii) With p = ∞. It is similar to (v) ⇔(iii) with p = 1.

Proof of Theorem 2(S). From the proof (i) ⇒(iv) of Theorem 1(S), we can write for all A ∈ 𝒜. Thus, if D≻0 satisfies inequality (12-S) (i.e. the Stein-type inequality relative to the p-norm associated with the matrix ), then D≻0 satisfies inequality (10-S) (i.e. the Stein-type inequality relative to the p-norm associated with the polytope 𝒜). The converse part is obvious, since .

Proof of Theorem 2(H). From the proof (i) ⇒(iv) of Theorem 1(H), we can write for all A ∈ 𝒜. Thus, if D≻0 satisfies inequality (12-H) (i.e. the Lyapunov-type inequality relative to the p-norm associated with the matrix ), then D≻0 satisfies inequality (10-H) (i.e. the Lyapunov-type inequality relative to the p-norm associated with the polytope 𝒜). The converse part is obvious, since .

Proof of Theorem 3(S). From the proof (i) ⇒(ii) of Theorem 1(S), we have , and from , we get , such that we can conclude that . Let 1 ≤ p ≤ ∞ and ε > 0. Lemma 3 in [28] ensures the existence of a diagonal matrix D≻0 such that . At the same time, from the proof (i) ⇒(iv) of Theorem 1(S) we have for all A ∈ 𝒜. As , we may write and, subsequently, .

Proof of Theorem 3(H). From the proof (i) ⇒(ii) of Theorem 1(H), we have , and from , we get , such that we can conclude that . Let 1 ≤ p ≤ ∞ and ε > 0. Lemma 3 in [28] ensures the existence of a diagonal matrix D≻0 such that . At the same time, from the proof (i)⇒(iv) of Theorem 1(H), we have for all A ∈ 𝒜. As , we may write and, subsequently, .

Proof of Theorem 4(S). Inequality (17-S) is equivalent to the statement for all A ∈ 𝒜. This results from the inequality , for all A ∈ 𝒜 that was obtained in the proof for (i) ⇒(iv) of Theorem 1(S). Then we apply Theorem 2 in [28] to all matrices A in 𝒜, and we get the equivalence (17-S) ⇔(19-S) ⇔(21-S).

Proof of Theorem 4(H). Inequality (17-H) is equivalent to the statement μ_p(D⁻¹AD) ≤ r for all A ∈ 𝒜. This results from the inequality , for all A ∈ 𝒜 that was obtained in the proof for (i) ⇒(iv) of Theorem 1(H). Then, we apply Theorem 2 in [28] to all matrices A in 𝒜, and we get the equivalence (17-H) ⇔(19-H) ⇔(21-H).

References

1 Białas S., A necessary and sufficient condition for the stability of convex combinations of stable polynomials or matrices, Bulletin of the Polish Academy of Sciences. (1985) 33, no. 9-10, 473–480, MR939418, ZBL0607.93044.
Google Scholar
2 Barmish B. R., Fu M., and Saleh S., Stability of a polytope of matrices: counterexamples, IEEE Transactions on Automatic Control. (1988) 33, no. 6, 569–572, https://doi.org/10.1109/9.1254, MR940777, ZBL0644.93053.
10.1109/9.1254
Web of Science® Google Scholar
3 Fu M. and Barmish B. R., Maximal unidirectional perturbation bounds for stability of polynomials and matrices, Systems & Control Letters. (1988) 11, no. 3, 173–179, https://doi.org/10.1016/0167-6911(88)90056-4, MR960664, ZBL0666.93028.
10.1016/0167-6911(88)90056-4
Web of Science® Google Scholar
4 Soh C. B., Schur stability of convex combination of matrices, Linear Algebra and its Applications. (1990) 128, 159–168, https://doi.org/10.1016/0024-3795(90)90290-S, MR1049080, ZBL0703.15014.
10.1016/0024-3795(90)90290-S
Web of Science® Google Scholar
5 Wang Q. G., Necessary and sufficient conditions for stability of a matrix polytope with normal vertex matrices, Automatica. (1991) 27, no. 5, 887–888, https://doi.org/10.1016/0005-1098(91)90047-6, MR1126574.
10.1016/0005-1098(91)90047-6
Web of Science® Google Scholar
6 Cohen N. and Lewkowicz I., A necessary and sufficient criterion for the stability of a convex set of matrices, IEEE Transactions on Automatic Control. (1993) 38, no. 4, 611–615, https://doi.org/10.1109/9.250532, MR1220745, ZBL0777.93071.
10.1109/9.250532
Web of Science® Google Scholar
7 Monov V. V., On the spectrum of convex sets of matrices, IEEE Transactions on Automatic Control. (1999) 44, no. 5, 1009–1012, https://doi.org/10.1109/9.763218, MR1690546, ZBL0956.93049.
10.1109/9.763218
Web of Science® Google Scholar
8 Zahreddine Z., Matrix measure and application to stability of matrices and interval dynamical systems, International Journal of Mathematics and Mathematical Sciences. (2003) 2, 75–85, https://doi.org/10.1155/S0161171203202295, MR1961641, ZBL1016.15017.
10.1155/S0161171203202295
Google Scholar
9 Bliman P.-A., A convex approach to robust stability for linear systems with uncertain scalar parameters, SIAM Journal on Control and Optimization. (2004) 42, no. 6, 2016–2042, https://doi.org/10.1137/S0363012901398691, MR2080927, ZBL1069.93027.
10.1137/S0363012901398691
Web of Science® Google Scholar
10 Chesi G., Establishing stability and instability of matrix hypercubes, Systems & Control Letters. (2005) 54, no. 4, 381–388, https://doi.org/10.1016/j.sysconle.2004.08.016, MR2123711, ZBL1129.93467.
10.1016/j.sysconle.2004.08.016
Web of Science® Google Scholar
11 Grman L., Rosinová D., Veselý V., and Kozáková A., Robust stability conditions for polytopic systems, International Journal of Systems Science. (2005) 36, no. 15, 961–973, https://doi.org/10.1080/00207720500389592, MR2192005, ZBL1126.93409.
10.1080/00207720500389592
Web of Science® Google Scholar
12 Polyak B. T. and Gryazina E. N., Stability regions in the parameter space: D-decomposition revisited, Automatica. (2006) 42, no. 1, 13–26, https://doi.org/10.1016/j.automatica.2005.08.010, MR2183067, ZBL1121.93028.
10.1016/j.automatica.2005.08.010
Web of Science® Google Scholar
13 Gurvits L. and Olshevsky A., On the NP-hardness of checking matrix polytope stability and continuous-time switching stability, IEEE Transactions on Automatic Control. (2009) 54, no. 2, 337–341, https://doi.org/10.1109/TAC.2008.2007177, MR2491961.
10.1109/TAC.2008.2007177
Web of Science® Google Scholar
14 Kolev L. and Petrakieva S., Assessing the stability of linear time-invariant continuous interval dynamic systems, IEEE Transactions on Automatic Control. (2005) 50, no. 3, 393–397, https://doi.org/10.1109/TAC.2005.843857, MR2123102.
10.1109/TAC.2005.843857
Web of Science® Google Scholar
15 Liao L. X., Necessary and sufficient conditions for stability of a class of interval matrices, International Journal of Control. (1987) 45, no. 1, 211–214, https://doi.org/10.1080/00207178708933721, MR872981, ZBL0611.15017.
10.1080/00207178708933721
Web of Science® Google Scholar
16 Lin S. H., Juang Y. T., Fong I. K., Hsu C. F., and Kuo T. S., Dynamic interval systems analysis and design, International Journal of Control. (1988) 48, no. 5, 1807–1818, https://doi.org/10.1080/00207178808906288, MR973757, ZBL0659.93023.
10.1080/00207178808906288
Web of Science® Google Scholar
17 Chen J., Sufficient conditions on stability of interval matrices: connections and new results, IEEE Transactions on Automatic Control. (1992) 37, no. 4, 541–544, https://doi.org/10.1109/9.126595, MR1153123, ZBL0752.15015.
10.1109/9.126595
Web of Science® Google Scholar
18 Bauer P. H. and Premaratne K., M. Jamshidi, M. Mansour, and B. D. O. Anderson, Time-invariant versus time-variant stability of interval matrix systems, Fundamentals of Discrete Time Systems: A Tribute to Professor E.I. Jury, 1993, TSI Press, Albuquerque, NM, USA, 181–188.
Google Scholar
19 Sezer M. E. and Šiljak D. D., On stability of interval matrices, IEEE Transactions on Automatic Control. (1994) 39, no. 2, 368–371, https://doi.org/10.1109/9.272336, MR1265423, ZBL0800.93981.
10.1109/9.272336
Web of Science® Google Scholar
20 Wang K., Michel A. N., and Liu D. R., Necessary and sufficient conditions for the Hurwitz and Schur stability of interval matrices, IEEE Transactions on Automatic Control. (1994) 39, no. 6, 1251–1255, https://doi.org/10.1109/9.293189, MR1283701, ZBL0810.93051.
10.1109/9.293189
Web of Science® Google Scholar
21 Liu D. and Molchanov A., Criteria for robust absolute stability of time-varying nonlinear continuous-time systems, Automatica. (2002) 38, no. 4, 627–637, https://doi.org/10.1016/S0005-1098(01)00243-6, MR2131470, ZBL1013.93044.
10.1016/S0005-1098(01)00243-6
Web of Science® Google Scholar
22 Molchanov A. P. and Liu D., Robust absolute stability of time-varying nonlinear discrete-time systems, IEEE Transactions on Circuits and Systems. I. Fundamental Theory and Applications. (2002) 49, no. 8, 1129–1137, https://doi.org/10.1109/TCSI.2002.801275, MR1929294.
10.1109/TCSI.2002.801275
Google Scholar
23 Mao W.-J. and Chu J., Quadratic stability and stabilization of dynamic interval systems, IEEE Transactions on Automatic Control. (2003) 48, no. 6, 1007–1012, https://doi.org/10.1109/TAC.2003.812784, MR1986268.
10.1109/TAC.2003.812784
Web of Science® Google Scholar
24 Alamo T., Tempo R., Ramírez D. R., and Camacho E. F., A new vertex result for robustness problems with interval matrix uncertainty, Systems & Control Letters. (2008) 57, no. 6, 474–481, https://doi.org/10.1016/j.sysconle.2007.11.003, MR2413743, ZBL1154.93023.
10.1016/j.sysconle.2007.11.003
Web of Science® Google Scholar
25 Barker G. P., Berman A., and Plemmons R. J., Positive diagonal solutions to the Lyapunov equations, Linear and Multilinear Algebra. (1978) 5, no. 4, 249–256, MR0469939.
10.1080/03081087808817203
Google Scholar
26 Berman A. and Hershkowitz D., Matrix diagonal stability and its implications, Society for Industrial and Applied Mathematics. (1983) 4, no. 3, 377–382, https://doi.org/10.1137/0604038, MR711345, ZBL0547.15009.
10.1137/0604038
Google Scholar
27 Kaszkurewicz E. and Bhaya A., Matrix Diagonal Stability in Systems and Computation, 2000, Birkhäuser, Boston, Mass, USA, https://doi.org/10.1007/978-1-4612-1346-8, MR1733604.
10.1007/978-1-4612-1346-8
Google Scholar
28 Pastravanu O. and Voicu M., Generalized matrix diagonal stability and linear dynamical systems, Linear Algebra and its Applications. (2006) 419, no. 2-3, 299–310, https://doi.org/10.1016/j.laa.2006.04.021, MR2277971, ZBL1125.93058.
10.1016/j.laa.2006.04.021
Web of Science® Google Scholar
29 Pastravanu O. and Matcovschi M. H., Diagonal stability of interval matrices and applications, Linear Algebra and its Applications. (2010) 433, no. 8–10, 1646–1658, https://doi.org/10.1016/j.laa.2010.06.016, MR2718226, ZBL1204.15028.
10.1016/j.laa.2010.06.016
Web of Science® Google Scholar
30 Matcovschi M.-H., Pastravanu O., and Voicu M., On some properties of diagonally stable polytopic systems, Proceedings of the 15th International Conference on System Theory, Control and Computing (ICSTCC ′11), 2011, 353–358, 6085752.
Google Scholar
31 Arcak M. and Sontag E. D., Diagonal stability of a class of cyclic systems and its connection with the secant criterion, Automatica. (2006) 42, no. 9, 1531–1537, https://doi.org/10.1016/j.automatica.2006.04.009, MR2246842, ZBL1132.39002.
10.1016/j.automatica.2006.04.009
Web of Science® Google Scholar
32 Mason O. and Shorten R., On the simultaneous diagonal stability of a pair of positive linear systems, Linear Algebra and its Applications. (2006) 413, no. 1, 13–23, https://doi.org/10.1016/j.laa.2005.07.019, MR2202089, ZBL1094.93021.
10.1016/j.laa.2005.07.019
Web of Science® Google Scholar
33 Pastravanu O. and Matcovschi M. H., Matrix measures in the qualitative analysis of parametric uncertain systems, Mathematical Problems in Engineering. (2009) 2009, 17, 841303, https://doi.org/10.1155/2009/841303, MR2562902, ZBL1179.93154.
10.1155/2009/841303
Web of Science® Google Scholar
34 Wimmer H. K., Diagonal stability of matrices with cyclic structure and the secant condition, Systems & Control Letters. (2009) 58, no. 5, 309–313, https://doi.org/10.1016/j.sysconle.2008.11.009, MR2512483, ZBL1159.93022.
10.1016/j.sysconle.2008.11.009
Web of Science® Google Scholar
35 Arcak M., Diagonal stability on cactus graphs and application to network stability analysis, IEEE Transactions on Automatic Control. (2011) 56, no. 12, 2766–2777, https://doi.org/10.1109/TAC.2011.2125130, MR2906667.
10.1109/TAC.2011.2125130
Web of Science® Google Scholar
36 Coppel W. A., Stability and Asymptotic Behavior of Differential Equations, 1965, D. C. Heath, Boston, Mass, USA, MR0190463.
Google Scholar
37 Blanchini F. and Miani S., Set-Theoretic Methods in Control, 2008, Birkhäuser, Boston, Mass, USA, MR2359816.
10.1007/978-0-8176-4606-6
Google Scholar
38 Horn R. A. and Johnson C. R., Topics in Matrix Analysis, 1991, Cambridge University Press, Cambridge, UK, https://doi.org/10.1017/CBO9780511840371, MR1091716.
10.1017/CBO9780511840371
Google Scholar

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Stability of Matrix Polytopes with a Dominant Vertex and Implications for System Dynamics

Abstract

1. Introduction

1.1. Research Context and Objective

1.2. Paper Structure

1.3. Notations and Nomenclature

2. Results on Matrix Polytopes

2.1. Preliminaries

2.2. Stability Analysis

2.3. Diagonal Solutions to Stein-Type and Lyapunov-Type Inequalities

2.4. Stability Margins

2.5. Particular Case of Interval Matrices with a Dominant Vertex

3. Results on Polytopic Systems

4. Illustrative Examples

5. Conclusions

Acknowledgment

Appendix

References

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Stability of Matrix Polytopes with a Dominant Vertex and Implications for System Dynamics

Abstract

1. Introduction

1.1. Research Context and Objective

1.2. Paper Structure

1.3. Notations and Nomenclature

2. Results on Matrix Polytopes

2.1. Preliminaries

2.2. Stability Analysis

2.3. Diagonal Solutions to Stein-Type and Lyapunov-Type Inequalities

2.4. Stability Margins

2.5. Particular Case of Interval Matrices with a Dominant Vertex

3. Results on Polytopic Systems

4. Illustrative Examples

5. Conclusions

Acknowledgment

Appendix

References

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