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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 396759, 11 pages

http://dx.doi.org/10.1155/2013/396759
Research Article

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

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

Received 1 November 2012; Accepted 21 March 2013

Academic Editor: Gani Stamov

Copyright © 2013 Octavian Pastravanu and Mihaela-Hanako Matcovschi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 -norm, , abbreviated as (resp., ). Our framework incorporates, as the particular case with , the known condition of quadratic stability satisfied by a diagonal positive-definite matrix, i.e. (resp., ) 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 (resp., HS and ), (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 -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 is a finite set of real 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 [113]. 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 , i.e. where the inequalities have a componentwise meaning. Significant results on Schur and Hurwitz stability of interval matrices have been reported in [10, 1424].

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 Hölder -norm, , and generalized the aforementioned diagonal stability concept to “Schur (resp., Hurwitz) diagonal stability relative to a Hölder -norm”—abbreviated as (resp., ). For , the framework proposed by [28] coincides with the classic approach presented by [27].

The (resp., ), , 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 [2729, 3135]. 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 and , the entries of matrix are considered fixed (not time varying); they are uncertain in the sense that their values are incompletely known but surely satisfy the condition . In other words, a single matrix is used for modeling a certain evolution of the process, whereas for modeling two different evolutions (taking place separately) two distinct matrices , 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 (resp., ), , unlike the case of an arbitrary matrix polytope (e.g., [30]) where (i) (resp., ) is more conservative than SS (resp., HS) and (ii) results on and (resp., and ) may be different for , , . 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 , respectively , 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 -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 (resp., ). 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 , be vectors.(i) is the Hölder vector -norm defined by for and by for .(ii) “ ”, “ ” mean componentwise inequalities, i.e.   , , .

Let , be square matrices.(iii) is the matrix norm induced by the vector -norm through .(iv) is the matrix measure [36, page 41], based on the matrix norm .(v) If , , , then the following regions of the complex plane , , are called the generalized Gershgorin's disks of defined with for columns.(vi) If , , , then the following regions of the complex plane , , are called the generalized Gershgorin’s disks of defined with for rows.(vii) is the spectrum of , and , , are the eigenvalues of .(viii) If , then matrix is said to be Schur stable (abbreviated as SS).(ix) If , then matrix is said to be Hurwitz stable (abbreviated as HS).(x) If is nonnegative (all entries are nonnegative), its spectral radius is a positive eigenvalue, denoted by , such that , .(xi) If is essentially nonnegative (all off-diagonal entries are nonnegative), then it has a real eigenvalue, denoted by , such that , —for example, Lemma 1 in [28].(xii)If is symmetrical, then all its eigenvalues are real and there exists an eigenvalue denoted by , such that , .(xiii) ", “ " mean that is a positive-definite, negative-definite matrix.(xiv) If the oriented graph of is strongly connected, then is called irreducible; otherwise is called reducible.(xv) For , the matrix norms and matrix measures have the following expressions: (xvi)   denotes the matrix built with the absolute values of the entries of .(xvii) ( superscript from Schur) denotes the nonnegative matrix defined by .(xviii) ( superscript from Hurwitz) denotes the essentially nonnegative matrix defined by , where and .(xix)  “ ”, “ ” mean componentwise inequalities, i.e. , , .

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 -norms .

2.1. Preliminaries

Definition 1 (S). Let us consider: ; an arbitrary matrix ; a matrix polytope of form (1); a nonsingular matrix .

(a) The inequality is called the Stein-type inequality relative to the -norm associated with matrix ; matrix is said to be a solution to this inequality.

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

Definition 1 (H). Let us consider: ; an arbitrary matrix ; a matrix polytope of form (1); a nonsingular matrix .

(a) The inequality is called the Lyapunov-type inequality relative to the -norm associated with matrix ; matrix is said to be a solution to this inequality.

(b) Matrix is said to be a solution to the Lyapunov-type inequality relative to the -norm associated with the polytope 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 (resp., ) with is equivalent to the standard Stein inequality respectively the standard Lyapunov inequality

Indeed, for with we may write that is equivalent to which means the fulfillment of .

Similarly, for with we may write that is equivalent to which means the fulfillment of .

(ii) The existence of solving the standard Stein inequality (resp., Lyapunov inequality is equivalent to Schur (resp., Hurwitz) stability of matrix .

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

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

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

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

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

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

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

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

(ii) Let . If is a proper polytope (i.e. , : 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 that satisfies inequality (resp., ) for all matrices .

(iii) Let . The (resp., ) is a property of stronger than SS (resp., HS). Indeed, each matrix is SS (resp., HS) once it is (resp., ) in accordance with Remark 2 in [28].

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

Definition 3 (S). Let be a matrix polytope of form (1). If there exists a subscript , such that the vertex fulfills one of the following two sets of componentwise inequalities: then is called a matrix polytope with an S-dominant vertex, and is called the S-dominant vertex of . (The notation is introduced in Section 1.3.)

Definition 3 (H). If there exists a subscript , such that the vertex fulfills the componentwise inequalities: then is called a matrix polytope with an H-dominant vertex, and is called the H-dominant vertex of . (The notation is introduced in Section 1.3.)

Remark 3. (i) If 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 or nonpositive—inequalities (resp., essentially nonnegative—inequalities ).

(ii) In the remainder of the text, we mainly address the case of the S-dominant vertex defined by inequalities . The case based on inequalities 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 may have two S-dominant vertices denoted as in the particular case when , satisfies inequalities , satisfies inequalities . We still can refer to as having “an S-dominant vertex,” since the stability properties of 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: ; a matrix polytope with an S-dominant vertex .

The following statements are equivalent.(i) is SS.(ii) is SS.(iii)There exists a , , such that is .(iv) is for all , .(v)There exists a diagonal matrix such that the union for all 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 such that the union for all 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: ; a matrix polytope with an H-dominant vertex .

The following statements are equivalent.(i) is HS.(ii) is HS.(iii)There exists a , , such that is .(iv) is for all , .(v)There exists a diagonal matrix such that the union for all 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 such that the union for all 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 can be used not only for testing the properties (resp., ) of but also for finding concrete diagonal matrices that satisfy the inequality in Definition 2(S) (resp., inequality in Definition 2(H)).

Theorem 2 (S). Let us consider: ; a matrix polytope with an S-dominant vertex ; a diagonal positive-definite matrix .

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

Proof. See the Appendix.

Theorem 2 (H). Let us consider: ; a matrix polytope with an H-dominant vertex ; a diagonal positive-definite matrix   .

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

Proof. See the Appendix.

Remark 4. Let . Whenever is SS (resp., HS) diagonal matrices that satisfy (resp., ) 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 in the dynamics of a polytopic system of form (resp., ).

2.4. Stability Margins

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

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

(a) If is SS, then is called the SS margin of .

(b) If is SS, then is called the SS margin of .

(c) Let . If is , then is called the margin of .

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

(a) If is HS, then is called the HS margin of .

(b) If is HS, then is called the HS margin of .

(c) Let . If is , then is called the margin of .

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

Proof. See the Appendix.

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

Proof. See the Appendix.

Remark 5. (i) For each stability property of the polytope discussed in Section 2.2, the corresponding margin (also called “degree” in the control-engineering literature) quantifies the distance between a matrix 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 relative to SS and SDSp (resp., HS and ) is defined by the S-dominant (resp., H-dominant) vertex.

(ii) For , Theorem 3(S) (resp., Theorem 3(H)) ensure the existence of a diagonal matrix such that the union for all 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 of the generalized Gershgorin’s disks written for rows , where the existence of the diagonal matrix is guaranteed by Theorem 3(S) (resp., Theorem 3(H)) with . 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 (resp., ) plays an important role in the characterization of the dynamic properties exhibited by the polytopic system (resp., ). Further details on this role are available in Remark 6 of the next section.

(iv) For an arbitrary polytope (without a dominant vertex), equality (resp., ) does not hold true, in general. If, for a given , , is (resp., ) 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 , , , is diagonally stable relative to both - and -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 of interval matrices of form (2) with or nonnegative, because these two types of interval matrices represent particular cases of matrix polytopes with an S-dominant vertex defined by inequalities or .

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

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:  ; a discrete-time polytopic system of form where polytope has an S-dominant vertex ; a positive-definite diagonal matrix    ; a constant   satisfying .

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

Proof. See the Appendix.

Theorem 4 (H). Let us consider:    ; a continuous-time polytopic system of form where polytope has an H-dominant vertex ; a positive-definite diagonal matrix   ; a constant   satisfying .

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

Proof. See the Appendix.

Remark 6. (i) Let . 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 (resp., ) in the right hand side of inequality (resp., ). This concrete value does not appear explicitly in the Stein-type inequality (resp., Lyapunov-type inequality ); the existence of a diagonal matrix that solves inequality (resp., ) represents a necessary and a sufficient condition for the (resp., ) of a matrix polytope . For a polytopic system (resp., ) a complete description of the dynamics implies the knowledge of pairs formed by and that satisfy Theorem 4(S) (resp., Theorem 4(H)).

(ii) The constant (resp., ) 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 , , 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 satisfying inequality (resp., ). For , we use Lemma 3 and Remark 3 in [28] that yield the following discussion on the irreducibility/reducibility of .

Case 1 (matrix is irreducible). (The definition of an irreducible matrix is introduced in Section 1.3.) Denote by and its right and left Perron eigenvectors, respectively. Given , , we construct the diagonal matrix , where (i) if ; (ii) , if ; (iii) , if . Matrix fulfills the following equality:(i) if is nonnegative, i.e. is defined by ; (ii) if is nonpositive, i.e. is defined by ; (iii) if is essentially nonnegative, i.e. is defined by .

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

Thus, for any , , we can construct a diagonal matrix such that inequality (resp., ) 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 and, concomitantly, allow the evaluation of the dynamical properties of the polytopic system (resp., ).

(iv) For an arbitrary polytope (without a dominant vertex), the fastest decreasing rate may depend on the -norm that defines the Lyapunov function and the invariant sets (if exist). If, for a given , is (resp., ) then the fastest decreasing rate corresponding to the -norm of the polytopic system can be expressed in terms of the stability margins as (resp., ). The fastest decreasing rate corresponding to the -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   ; (ii) the dynamical properties of the discrete-time polytopic system of form defined by the considered polytope. Example 2 explores (i) the stability of a matrix polytope with an H-dominant vertex of form (1) and ; (ii) the dynamical properties of the continuous-time polytopic system of form defined by the considered polytope.

Example 1. It is adapted from [27]. Let be the set of diagonal matrices whose elements are subunitary. is the convex hull generated by the set of vertices , also called the class of signature matrices.

Given a nonnegative matrix , the set is a matrix polytope of form (1) with the vertices , where , . The set defined by (23) is a matrix polytope with an S-dominant vertex because matrix is a vertex that satisfies condition .

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

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

Finally, we notice that the nonpositive matrix is also a vertex of the polytope and it satisfies the dominance condition . This means that the polytope 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 brings no supplementary information (since the matrix is nonnegative, there exists , and for all , , the equality holds true).

The above approach applies mutatis-mutandis to the investigation of the matrix polytope . In this case, Theorem 2(S) generalizes for the result that can be obtained when by using Proposition 2.5.9 in [27] for matrix and polytope .

Example 2. Let us consider the interval matrix [14]: that is a matrix polytope with vertices: has an H-dominant vertex which satisfies inequalities . Note that is essentially positive and has the Perron-Frobenius eigenvalue . Theorem 1(H) shows that the Hurwitz stability of ensures the Hurwitz diagonal stability of the polytope relative to any -norm, .

The stability margins of the polytope are given by relation in Theorem 2, i.e. , for any , .

For the qualitative analysis of the continuous-time polytopic system defined by and (24), we can apply Theorem 4(H). Remark 6(ii) shows that for any , , the fastest decreasing rate for the diagonal Lyapunov functions and for the contractive invariant sets is exactly , since is irreducible. We apply Case 1 of the procedure presented in Remark 6(ii) and relying on the right and left Perron eigenvectors of ( and ), we construct the diagonal matrices corresponding to the fastest decreasing rate. For , these diagonal matrices are , , and , and they satisfy Theorem 4(H) with .

Note that all the above results remain valid if instead of defined by (24), we consider the matrix polytope which has the same dominant vertex (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 -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 -norm; (iv) existence of diagonal-type Lyapunov functions and contractive invariant sets defined by a -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 -norm are equivalent for the considered class of matrix polytopes (fact which is not true for general matrix polytopes).

Appendix

Proof of Theorem 1(S). (i)  (ii).

It results from the following: since , 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 . Lemma 3 in [28] ensures the existence of a diagonal matrix 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 . It results from the equivalence

(vi)  (iii) with . It is similar to (v)  (iii) with .

Proof of Theorem 1(H). (i)  (ii).

It results from the following: Take so that . 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 . Lemma 3 in [28] ensures the existence of a diagonal matrix 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 . It results from the equivalence

(vi)  (iii) With . It is similar to (v)  (iii) with .

Proof of Theorem 2(S). From the proof (i)  (iv) of Theorem 1(S), we can write for all . Thus, if satisfies inequality (i.e. the Stein-type inequality relative to the -norm associated with the matrix ), then satisfies inequality (i.e. the Stein-type inequality relative to the -norm associated with the polytope ). 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 . Thus, if satisfies inequality (i.e. the Lyapunov-type inequality relative to the -norm associated with the matrix ), then satisfies inequality (i.e. the Lyapunov-type inequality relative to the -norm associated with the polytope ). 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 and . Lemma 3 in [28] ensures the existence of a diagonal matrix such that . At the same time, from the proof (i)  (iv) of Theorem 1(S) we have for all . 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 and . Lemma 3 in [28] ensures the existence of a diagonal matrix such that . At the same time, from the proof (i) (iv) of Theorem 1(H), we have for all . As , we may write   and, subsequently, .

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

Proof of Theorem 4(H). Inequality is equivalent to the statement for all . This results from the inequality , for all that was obtained in the proof for (i)  (iv) of Theorem 1(H). Then, we apply Theorem 2 in [28] to all matrices in , and we get the equivalence .

Acknowledgment

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

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