# Further Results on Diagonally Invariant Exponential Stability of Switching Linear Systems

Authors: Mihaela-Hanako Matcovschi; Octavian Pastravanu
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

The concept ofdiagonally invariant exponential stability (DIES) was originally introduced for single-model linear systems and subsequently expanded in the study of linear systems with interval-type uncertainties and linear systems with arbitrary switching. The results presented in this article refer to new approaches to DIES characterization for arbitrary switching systems, which exploit mathematical tools completely different from earlier work. The previous papers are based on the properties of matrix norms and measures applied to the constituent matrices defining the switching system, while the present paper uses the eigenvalues and eigenvectors of the column and row representatives built for a set of matrices derived from the constituent matrices of the switching system. The applicability of previous and new results, respectively, is illustrated by case studies (in both continuous- and discrete-time) that lead to relevant comparisons between the two classes of analysis methods.


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## Body

## 1. Introduction

### 1.1. Research Framework

The current paper considers the dynamics of switching linear systems described in discrete-time by1-Sxt+1=Aυtxt,xt0=x0,Aυt∈A,t,t0∈Z+,t≥t0,and in continuous-time by1-Hx˙t=Aυtxt,xt0=x0,Aυt∈A,t,t0∈R+,t≥t0,where2A=A1,A2,…,AN⊂Rn×nis a set of matrices characterizing the N linear modes exhibited by the switching system, and υ:Z+→{1,…,N} (discrete-time) and, respectively, υ:R+→{1,2,…,N} (continuous-time) are, respectively, an arbitrarily switching signal. At any time instant t∈Z+ (discrete-time) or t∈R+ (continuous-time), theactive mode of the switching system is defined by the subscript θ=υ(t) associated with the matrix Aθ∈A that, respectively, uniquely generates the dynamics3-Sxt+1=Aθxt,t∈Z+,θ∈1,…,Nand3-Hx˙t=Aθxt,t∈R+,θ∈1,…,N.A system trajectory that starts from the initial condition x(t0)=x0∈Rn depends on the switching signal υ(·), reason for which this dependence is outlined by the notation x(t)=x(t;t0,x0,υ).The continuous-time operation of the switching system1-H is assumed to preserve the values of the state-space variables at any commutation time τ∈R+; i.e., the equality limh↓0⁡x(τ-h)=x(τ) is satisfied.The notations “S” and “H” used above, as extension in equations numbering, come from the stability analysis intended by our work, which is addressed in the sense of Schur (abbreviated “S”) for discrete-time dynamics and in the sense of Hurwitz (abbreviated “H”) for continuous-time dynamics. The meaning of this notations will also be preserved when used as superscripts for some matrices.Throughout the text the notation “X//Y” is used in place of “X” and, respectively, “Y” in order to organize a parallel presentation of the statements referring to analogous properties exhibited by discrete- and continuous-time systems.The study of stability and stabilizability ofswitching andswitched systems constituted the research interests of numerous control engineering groups during the past fifteen years. The monograph [1] presents the background existing in the early 2000, created by the pioneering works in this field. Later on, the picture is enlarged by the survey paper [2] and the monographs [3, 4], as well as by the numerous works cited therein. The recent period, meaning results reported after 2010, can be globally characterized as focusing on specialized classes of dynamics (such as dynamics with dwell time, e.g., [5–7]; delayed dynamics, e.g., [7–9]; positive dynamics, e.g., [10–12]—to mention just a few of the representative publications of the period we are referring to).The nomenclature used by most of the cited works considers that the commutations of aswitching system are determined by external signals, whereas the commutations of aswitched system are controlled by internal signals. In accordance with this terminology, system 3-S//3-H described above exhibits an arbitrary switching, where the commutations between modes are driven by the signal υ:Z+→{1,…,N} (discrete-time), υ:R+→{1,2,…,N} (continuous-time).The concept ofdiagonally invariant exponential stability (DIES) to be analyzed by the current article was introduced by paper [13] for linear time-invariant systems whose discrete- and continuous-time dynamics was described by a unique matrix. Paper [13] shows that DIES characterizes a particular type of exponential stability, reason for which the test instruments rely on algebraic properties stronger than the eigenvalue location, namely, inequalities involving matrix norms and matrix measures. The DIES concept was extended to interval systems by [14] and to switching systems of form 1-S//1-H by [15] for discrete-time and [16] for continuous-time.


### 1.2. Research Objectives and Exposition Plan

The current research is founded on the DIES concept and its characterization, by equivalence, presented in our previous works [15] for discrete-time and [16] for continuous-time. The developments proposed in this article refer to novel approaches to DIES characterization that exploit mathematical instruments completely different from [15, 16].Briefly speaking, the old results, corresponding to [15, 16], rely on the properties of matrix norms and matrix measures applied to the matrices Aθ∈Rn×n,θ=1,…,N, defining A  2, whereas the new results (herein developed) explore the spectra of the column and row representatives built for the matrix set A  2. It is worth mentioning that the new results have been derived only for the DIES analysis with respect to the Hölderp-norms, with p∈{1,2,∞}, although the DIES concept is defined for 1≤p≤∞. This fact should not be seen as a limitation of the new results, since most of the concrete problems are formulated relatively to the above-mentioned norms.The exposition plan for the remainder of the text takes into consideration a short visit of the already existing results (also called old results) that are summarized by Section2, by using a presentation style that unifies the discrete- and continuous-time cases (separately addressed by [15, 16], respectively). Section 3 develops the novel results in full accordance with the background created by the previous section and by using the same unified presentation style. Section 4 illustrates the applicability of the old and new results in parallel for concrete numerical examples and carefully compares similarities and differences between the two types of results. Some concluding remarks on the role and importance of our new results are formulated in Section 5.The notations used throughout the whole text are presented by the appendix accompanying this article.


## 1.1. Research Framework

The current paper considers the dynamics of switching linear systems described in discrete-time by1-Sxt+1=Aυtxt,xt0=x0,Aυt∈A,t,t0∈Z+,t≥t0,and in continuous-time by1-Hx˙t=Aυtxt,xt0=x0,Aυt∈A,t,t0∈R+,t≥t0,where2A=A1,A2,…,AN⊂Rn×nis a set of matrices characterizing the N linear modes exhibited by the switching system, and υ:Z+→{1,…,N} (discrete-time) and, respectively, υ:R+→{1,2,…,N} (continuous-time) are, respectively, an arbitrarily switching signal. At any time instant t∈Z+ (discrete-time) or t∈R+ (continuous-time), theactive mode of the switching system is defined by the subscript θ=υ(t) associated with the matrix Aθ∈A that, respectively, uniquely generates the dynamics3-Sxt+1=Aθxt,t∈Z+,θ∈1,…,Nand3-Hx˙t=Aθxt,t∈R+,θ∈1,…,N.A system trajectory that starts from the initial condition x(t0)=x0∈Rn depends on the switching signal υ(·), reason for which this dependence is outlined by the notation x(t)=x(t;t0,x0,υ).The continuous-time operation of the switching system1-H is assumed to preserve the values of the state-space variables at any commutation time τ∈R+; i.e., the equality limh↓0⁡x(τ-h)=x(τ) is satisfied.The notations “S” and “H” used above, as extension in equations numbering, come from the stability analysis intended by our work, which is addressed in the sense of Schur (abbreviated “S”) for discrete-time dynamics and in the sense of Hurwitz (abbreviated “H”) for continuous-time dynamics. The meaning of this notations will also be preserved when used as superscripts for some matrices.Throughout the text the notation “X//Y” is used in place of “X” and, respectively, “Y” in order to organize a parallel presentation of the statements referring to analogous properties exhibited by discrete- and continuous-time systems.The study of stability and stabilizability ofswitching andswitched systems constituted the research interests of numerous control engineering groups during the past fifteen years. The monograph [1] presents the background existing in the early 2000, created by the pioneering works in this field. Later on, the picture is enlarged by the survey paper [2] and the monographs [3, 4], as well as by the numerous works cited therein. The recent period, meaning results reported after 2010, can be globally characterized as focusing on specialized classes of dynamics (such as dynamics with dwell time, e.g., [5–7]; delayed dynamics, e.g., [7–9]; positive dynamics, e.g., [10–12]—to mention just a few of the representative publications of the period we are referring to).The nomenclature used by most of the cited works considers that the commutations of aswitching system are determined by external signals, whereas the commutations of aswitched system are controlled by internal signals. In accordance with this terminology, system 3-S//3-H described above exhibits an arbitrary switching, where the commutations between modes are driven by the signal υ:Z+→{1,…,N} (discrete-time), υ:R+→{1,2,…,N} (continuous-time).The concept ofdiagonally invariant exponential stability (DIES) to be analyzed by the current article was introduced by paper [13] for linear time-invariant systems whose discrete- and continuous-time dynamics was described by a unique matrix. Paper [13] shows that DIES characterizes a particular type of exponential stability, reason for which the test instruments rely on algebraic properties stronger than the eigenvalue location, namely, inequalities involving matrix norms and matrix measures. The DIES concept was extended to interval systems by [14] and to switching systems of form 1-S//1-H by [15] for discrete-time and [16] for continuous-time.


## 1.2. Research Objectives and Exposition Plan

The current research is founded on the DIES concept and its characterization, by equivalence, presented in our previous works [15] for discrete-time and [16] for continuous-time. The developments proposed in this article refer to novel approaches to DIES characterization that exploit mathematical instruments completely different from [15, 16].Briefly speaking, the old results, corresponding to [15, 16], rely on the properties of matrix norms and matrix measures applied to the matrices Aθ∈Rn×n,θ=1,…,N, defining A  2, whereas the new results (herein developed) explore the spectra of the column and row representatives built for the matrix set A  2. It is worth mentioning that the new results have been derived only for the DIES analysis with respect to the Hölderp-norms, with p∈{1,2,∞}, although the DIES concept is defined for 1≤p≤∞. This fact should not be seen as a limitation of the new results, since most of the concrete problems are formulated relatively to the above-mentioned norms.The exposition plan for the remainder of the text takes into consideration a short visit of the already existing results (also called old results) that are summarized by Section2, by using a presentation style that unifies the discrete- and continuous-time cases (separately addressed by [15, 16], respectively). Section 3 develops the novel results in full accordance with the background created by the previous section and by using the same unified presentation style. Section 4 illustrates the applicability of the old and new results in parallel for concrete numerical examples and carefully compares similarities and differences between the two types of results. Some concluding remarks on the role and importance of our new results are formulated in Section 5.The notations used throughout the whole text are presented by the appendix accompanying this article.


## 2. Background: Results Available on DIES

### 2.1. Definition and Characterization of DIES

The DIES concept was introduced by paper [13] for linear time-invariant systems whose discrete- and continuous-time dynamics are described by a unique matrix. Papers [15, 16] extended the DIES concept to switching linear systems in accordance with the following definition.Definition 1.
Let1≤p≤∞, D=diag⁡{d1,…,dn}, di>0, i=1,…,n, and 0<r<1//r<0. The switching linear system 1-S//1-H is calleddiagonally invariant exponentially stable relative to the p-norm and parametersD, r (abbreviated as DIESpD,r) under arbitrary switching if4-S∀ε>0,∀t0∈Z+,∀x0=xt0∈Rn,∀υ:Z+⟶1,2,…,N:x0pD≤ε⟹xt;t0,x0,υpD≤εrt-t0,∀t∈Z+,t≥t0.and, respectively,4-H∀ε>0,∀t0∈R+,∀x0=xt0∈Rn,∀υ:R+⟶1,2,…,N:x0pD≤ε⟹xt;t0,x0,υpD≤εert-t0,∀t∈R+,t≥t0.In 4-S and 4-H, the vector norm xpD=D-1xp is used in accordance with the details given in the appendix.A qualitative analysis developed in terms of invariant sets reveals thatDIESpD,r property expresses the invariance of theexponentially decreasing time-dependent sets (or, equivalently, contractive sets)5-SXpD,rε;t,t0=x∈Rn∣xpD≤εrt-t0,t,t0∈Z+,t≥t0,ε>0,and, respectively,5-HXpD,rε;t,t0=x∈Rn∣xpD≤εert-t0,t,t0∈R+,t≥t0,ε>0,with respect to the trajectories of the switching linear system 1-S//1-H for any switching sequence υ.For further details on exploring the connection between set invariance and the standard definition of (local and global) exponential stability, the reader is referred to the discussions in paper [13]. For instance, the DIESpD,r concept implies the satisfaction of the classical δ-ε definition of the exponential stability of the equilibrium {0} of system 1-S//1-H, by using the vector norm pD and the precise value δ(ε)=ε for satisfying the inequality δ(ε)≥ε. In other words, the equilibrium {0} may be exponentially stable, but the switching system 1-S//1-H does not have invariant sets of form 5-S//5-H.In relation5-S//5-H the diagonal entries of matrix  D scale the state variables, and the constant 0<r<1//r<0 represents the contraction rate of the considered time-dependent sets. For the usualp-norms defined by p∈{1,2,∞}, the contractive sets 5-S//5-H have well-known geometric shapes: hyper-rhombus, ellipses, and rectangles, respectively.TheDIESpD,r property of switching linear systems of form 1-S//1-H is characterized by the following theorem that employs the matrix norms and measures defined in the appendix. This theorem merges, in a unified form, the results proven in [15] for the discrete-time case and in [16] for the continuous-time case.Theorem 2.
Let1≤p≤∞, D≻0 diagonal, and 0<r<1//r<0. The following statements are equivalent:
(i) The switching system1-S//1-H is DIESpD,r under arbitrary switching.
(ii) The function6V:Rn⟶R+,Vx=xpDis a strong Lyapunov function for the switching system 1-S//1-H, which exhibits the decreasing rater along each nontrivial trajectory of the considered system; i.e.,7-S∀t∈Z+:Vxt+1≤rVxt,and, respectively,7-H∀t∈R+:Dt+Vxt=limh↓0⁡Vxt+h-Vxth≤rVxt.(iii) The matrices Aθ,θ=1,…,N, defining the component subsystems of the switching system 1-S//1-H satisfy the inequalities8-SAθpD≤r,θ=1,…,N,and, respectively,8-HμpDAθ≤r,θ=1,…,N.Proof.
See the proofs of Theorem 2 in [15] for the discrete-time case and Theorem 1 in [16] for the continuous-time case.Inequalities8-S//8-H may be utilized in testing the DIESpD,r of system 1-S//1-H. It is important to notice the necessary and sufficient role of these inequalities in the DIESpD,r characterization, because other works, such as [2, 17], mention their use just as sufficient conditions for uniform asymptotic stability.For a single-model system, the DIES characterization via8-S//8-H operates in the particular form ||A||pD≤r // μpD(A)≤r presented in our previous work [13]. This matrix-norm//matrix-measure inequality implies the Schur//Hurwitz property of matrix A, a fact which offers a supplementary motivation for the discussion on “DIES stronger compared to exponential stability” developed above, after Definition 1. An interesting consequence refers to the dynamics of the arbitrary switching system 1-S//1-H, analyzed from the perspective of the individual dynamics exhibited by theN constituent subsystems. Thus, it is well known that system 1-S//1-H is not necessarily stable if all its constituent subsystems are exponentially stable. On the other hand, if there exist a matrix D≻0 diagonal andN constants 0<r1,…,rN<1//r1,…,rN<0, such that theN constituent subsystems are DIESpD,r1,…,DIESpD,rN, respectively, then system 1-S//1-H is DIESpD,r, for r=max⁡{r1,…,rN}.


### 2.2. Particular Forms of Inequalities8-S//8-H in Theorem 2 for p∈{1,2,∞}

The easy to handle, concrete forms of the matrix norms//matrix measures corresponding top∈{1,2,∞} (see Appendix) allow a convenient reformulation of inequalities 8-S//8-H in Theorem 2, as shown by Theorem 3 stated below, which joins results proven in [15] for the discrete-time case and in [16] for the continuous-time case.Consider the set of matricesA=A1,A2,…,AN  2 that generates the dynamics of switching system 1-S//1-H. For the discrete-time case, define the set of matrices AS=A1S,A2S,…,ANS, where9-SAθSij=Aθij,i,j=1,…,n,θ=1,…,N,and for the continuous-time case, the set of matrices AH=A1H,A2H,…,ANH, where9-HAθHij=AθijAθij,i,j=1,…,n,θ=1,…,N.Theorem 3.
LetD=diag⁡{d1,…,dn}, di>0, i=1,…,n, and r∈R.
(i) Forp=1, inequalities 8-S//8-H are equivalent to10-SvTAθS≤rvT,θ=1,…,N,and10-HvTAθH≤rvT,θ=1,…,N,respectively, where v=[1/d1⋯1/dn]T∈Rn is a positive vector formed with the inverses of the diagonal entries of matrix D.
(ii) Forp=2, inequalities 8-S//8-H are equivalent to11-SAθTQAθ-r2Q⪯0,θ=1,…,N,and11-HAθTQ+QAθ-2rQ⪯0,θ=1,…,Nrespectively, where Q=(D-1)2 is a positive definite diagonal matrix,  formed with the square values of the inverses of the diagonal entries of matrix D.
(iii) Forp=∞, inequalities 8-S//8-H are equivalent to12-SAθSv≤rv,θ=1,…,N,and12-HAθHv≤rv,θ=1,…,N,respectively, where v=[d1⋯dn]T∈Rn is a positive vector formed with the diagonal entries of matrix D.Proof.
See the proofs of Corollary 1 in [15] for the discrete-time case and Corollary 1 in [16] for the continuous-time case.Papers [15, 16] recommend the use of the above results in the sense that, for a chosen (or given) value of 0<r<1//r<0, Theorem 3 permits the search for a diagonal matrix D≻0. Thus, inequalities 10-S//10-H and 12-S//12-H can be numerically approached as LP problems, and 11-S//11-H as LMI problems.It is worth noticing thatDIESpD,r testing in the manner of the concomitant search for the parameters D, r requires a different point of view on the numerical exploitation of Theorem 3. For instance, one can devise algorithms that involve the iterative use of the LP or LMI problems (providing matrix D), correlated to an optimization scheme looking for the value r, e.g., a bisection method [18]. Another numerical approach may regard inequalities 10-S//10-H to 12-S//12-H as BMI problems, in accordance with the key principles discussed by [19].


## 2.1. Definition and Characterization of DIES

The DIES concept was introduced by paper [13] for linear time-invariant systems whose discrete- and continuous-time dynamics are described by a unique matrix. Papers [15, 16] extended the DIES concept to switching linear systems in accordance with the following definition.Definition 1.
Let1≤p≤∞, D=diag⁡{d1,…,dn}, di>0, i=1,…,n, and 0<r<1//r<0. The switching linear system 1-S//1-H is calleddiagonally invariant exponentially stable relative to the p-norm and parametersD, r (abbreviated as DIESpD,r) under arbitrary switching if4-S∀ε>0,∀t0∈Z+,∀x0=xt0∈Rn,∀υ:Z+⟶1,2,…,N:x0pD≤ε⟹xt;t0,x0,υpD≤εrt-t0,∀t∈Z+,t≥t0.and, respectively,4-H∀ε>0,∀t0∈R+,∀x0=xt0∈Rn,∀υ:R+⟶1,2,…,N:x0pD≤ε⟹xt;t0,x0,υpD≤εert-t0,∀t∈R+,t≥t0.In 4-S and 4-H, the vector norm xpD=D-1xp is used in accordance with the details given in the appendix.A qualitative analysis developed in terms of invariant sets reveals thatDIESpD,r property expresses the invariance of theexponentially decreasing time-dependent sets (or, equivalently, contractive sets)5-SXpD,rε;t,t0=x∈Rn∣xpD≤εrt-t0,t,t0∈Z+,t≥t0,ε>0,and, respectively,5-HXpD,rε;t,t0=x∈Rn∣xpD≤εert-t0,t,t0∈R+,t≥t0,ε>0,with respect to the trajectories of the switching linear system 1-S//1-H for any switching sequence υ.For further details on exploring the connection between set invariance and the standard definition of (local and global) exponential stability, the reader is referred to the discussions in paper [13]. For instance, the DIESpD,r concept implies the satisfaction of the classical δ-ε definition of the exponential stability of the equilibrium {0} of system 1-S//1-H, by using the vector norm pD and the precise value δ(ε)=ε for satisfying the inequality δ(ε)≥ε. In other words, the equilibrium {0} may be exponentially stable, but the switching system 1-S//1-H does not have invariant sets of form 5-S//5-H.In relation5-S//5-H the diagonal entries of matrix  D scale the state variables, and the constant 0<r<1//r<0 represents the contraction rate of the considered time-dependent sets. For the usualp-norms defined by p∈{1,2,∞}, the contractive sets 5-S//5-H have well-known geometric shapes: hyper-rhombus, ellipses, and rectangles, respectively.TheDIESpD,r property of switching linear systems of form 1-S//1-H is characterized by the following theorem that employs the matrix norms and measures defined in the appendix. This theorem merges, in a unified form, the results proven in [15] for the discrete-time case and in [16] for the continuous-time case.Theorem 2.
Let1≤p≤∞, D≻0 diagonal, and 0<r<1//r<0. The following statements are equivalent:
(i) The switching system1-S//1-H is DIESpD,r under arbitrary switching.
(ii) The function6V:Rn⟶R+,Vx=xpDis a strong Lyapunov function for the switching system 1-S//1-H, which exhibits the decreasing rater along each nontrivial trajectory of the considered system; i.e.,7-S∀t∈Z+:Vxt+1≤rVxt,and, respectively,7-H∀t∈R+:Dt+Vxt=limh↓0⁡Vxt+h-Vxth≤rVxt.(iii) The matrices Aθ,θ=1,…,N, defining the component subsystems of the switching system 1-S//1-H satisfy the inequalities8-SAθpD≤r,θ=1,…,N,and, respectively,8-HμpDAθ≤r,θ=1,…,N.Proof.
See the proofs of Theorem 2 in [15] for the discrete-time case and Theorem 1 in [16] for the continuous-time case.Inequalities8-S//8-H may be utilized in testing the DIESpD,r of system 1-S//1-H. It is important to notice the necessary and sufficient role of these inequalities in the DIESpD,r characterization, because other works, such as [2, 17], mention their use just as sufficient conditions for uniform asymptotic stability.For a single-model system, the DIES characterization via8-S//8-H operates in the particular form ||A||pD≤r // μpD(A)≤r presented in our previous work [13]. This matrix-norm//matrix-measure inequality implies the Schur//Hurwitz property of matrix A, a fact which offers a supplementary motivation for the discussion on “DIES stronger compared to exponential stability” developed above, after Definition 1. An interesting consequence refers to the dynamics of the arbitrary switching system 1-S//1-H, analyzed from the perspective of the individual dynamics exhibited by theN constituent subsystems. Thus, it is well known that system 1-S//1-H is not necessarily stable if all its constituent subsystems are exponentially stable. On the other hand, if there exist a matrix D≻0 diagonal andN constants 0<r1,…,rN<1//r1,…,rN<0, such that theN constituent subsystems are DIESpD,r1,…,DIESpD,rN, respectively, then system 1-S//1-H is DIESpD,r, for r=max⁡{r1,…,rN}.


## 2.2. Particular Forms of Inequalities8-S//8-H in Theorem 2 for p∈{1,2,∞}

The easy to handle, concrete forms of the matrix norms//matrix measures corresponding top∈{1,2,∞} (see Appendix) allow a convenient reformulation of inequalities 8-S//8-H in Theorem 2, as shown by Theorem 3 stated below, which joins results proven in [15] for the discrete-time case and in [16] for the continuous-time case.Consider the set of matricesA=A1,A2,…,AN  2 that generates the dynamics of switching system 1-S//1-H. For the discrete-time case, define the set of matrices AS=A1S,A2S,…,ANS, where9-SAθSij=Aθij,i,j=1,…,n,θ=1,…,N,and for the continuous-time case, the set of matrices AH=A1H,A2H,…,ANH, where9-HAθHij=AθijAθij,i,j=1,…,n,θ=1,…,N.Theorem 3.
LetD=diag⁡{d1,…,dn}, di>0, i=1,…,n, and r∈R.
(i) Forp=1, inequalities 8-S//8-H are equivalent to10-SvTAθS≤rvT,θ=1,…,N,and10-HvTAθH≤rvT,θ=1,…,N,respectively, where v=[1/d1⋯1/dn]T∈Rn is a positive vector formed with the inverses of the diagonal entries of matrix D.
(ii) Forp=2, inequalities 8-S//8-H are equivalent to11-SAθTQAθ-r2Q⪯0,θ=1,…,N,and11-HAθTQ+QAθ-2rQ⪯0,θ=1,…,Nrespectively, where Q=(D-1)2 is a positive definite diagonal matrix,  formed with the square values of the inverses of the diagonal entries of matrix D.
(iii) Forp=∞, inequalities 8-S//8-H are equivalent to12-SAθSv≤rv,θ=1,…,N,and12-HAθHv≤rv,θ=1,…,N,respectively, where v=[d1⋯dn]T∈Rn is a positive vector formed with the diagonal entries of matrix D.Proof.
See the proofs of Corollary 1 in [15] for the discrete-time case and Corollary 1 in [16] for the continuous-time case.Papers [15, 16] recommend the use of the above results in the sense that, for a chosen (or given) value of 0<r<1//r<0, Theorem 3 permits the search for a diagonal matrix D≻0. Thus, inequalities 10-S//10-H and 12-S//12-H can be numerically approached as LP problems, and 11-S//11-H as LMI problems.It is worth noticing thatDIESpD,r testing in the manner of the concomitant search for the parameters D, r requires a different point of view on the numerical exploitation of Theorem 3. For instance, one can devise algorithms that involve the iterative use of the LP or LMI problems (providing matrix D), correlated to an optimization scheme looking for the value r, e.g., a bisection method [18]. Another numerical approach may regard inequalities 10-S//10-H to 12-S//12-H as BMI problems, in accordance with the key principles discussed by [19].


## 3. New Results: Column and Row Representatives-Based Approach to DIES

This section shows that theDIESpD,r characterization, for p∈{1,2,∞}, expressed by inequalities 8-S//8-H in Theorem 2 can also be addressed in terms ofeigenvaluesof some special matrices. These matrices are constructed from the set A=A1,A2,…,AN that generates the dynamics of switching system 1-S//1-H, relying on the concepts ofcolumn and row representatives of a matrix set.The eigenvalue employment in theDIESpD,r testing for switching linear systems represents an important (and, in principle, expected) generalization of the results reported in [13] for DIESpD,r, p∈{1,2,∞}, of linear time-invariant systems (single mode systems).


### 3.1. Column and Row Representatives for a Set of Matrices as Analytical Tools

LetM=M1,…,MN⊂Rn×n be a set of matrices. Given a function σ:{1,…,n}→{1,…,N}, denote by σ=(σ(1),…,σ(n)) the correspondingn-tuple and represent by C the set of all then-tuples with elements from {1,…,N}.For anyσ∈C, we construct the matrix M_σ∈Rn×n, whose first column is the first column of Mσ(1) (denoted as [Mσ(1)](:,1)), second column is the second column of Mσ(2) (denoted as [Mσ(2)](:,2)), and so on; that is, M_σ=[Mσ1]:,1⋯[Mσ(n)](:,n)∈Rn×n. A matrix M_σ∈Rn×n, σ∈C, constructed as above is called acolumn representativeof the matrix set M. We say that σ∈C labels the column representatives.For anyσ∈C, we construct the matrix M-σ∈Rn×n, whose first row is the first row of Mσ(1) (denoted as [Mσ(1)](1,:)), second row is the second row of Mσ(2) (denoted as [Mσ(2)](2,:)), and so on; that is, M-σ=[Mσ(1)](1,:)⋯[Mσ(n)](n,:)∈Rn×n. A matrix M-σ∈Rn×n, σ∈C, constructed as above is called arow representativeof the matrix set M. We say that σ∈C labels the row representatives.If the matricesMθ∈Rn×n,θ=1,…,N, are (essentially) nonnegative, then all column representatives M_σ∈Rn×n, σ∈C, and all row representatives M-σ∈Rn×n, σ∈C, are (essentially) nonnegative matrices. Subsequently, relying on Perron-Frobenius theory, one can define13λ_∗=maxσ∈C⁡λmaxM_σ,C_∗=σ∗∈C∣λmaxM_σ∗=λ_∗,where λ_∗ represents the dominant eigenvalue of the column representatives and C_∗ the set of labels associated with λ_∗. Similarly, one can define14λ-∗=maxσ∈C⁡λmaxM-σ,C-∗=σ∗∈C∣λmaxM-σ∗=λ-∗,where λ-∗ represents the dominant eigenvalue of the row representatives and C-∗ the set of labels associated with λ-∗.In the next two subsections, regarding the switching system1-S//1-H and 2, we are going to exploit the following:(a) For the discrete-time case, the representatives of the matrix setAS=A1S,A2S,…,ANS are defined by 9-S, the dominant eigenvalue of the column representatives is defined by15-Sλ_S∗=maxσ∈C⁡λmaxA_σS,C_S∗=σ∗∈C∣λmaxA_σ∗S=λ_S∗,and the dominant eigenvalue of the row representatives is defined by16-Sλ-S∗=maxσ∈C⁡λmaxA-σS,C-S∗=σ∗∈C∣λmaxA-σ∗S=λ-S∗.(b) For the continuous-time case, the representatives of the matrix set AH=A1H,A2H,…,ANH are defined by 9-H, the dominant eigenvalue of the column representatives is defined by15-Hλ_H∗=maxσ∈C⁡λmaxA_σH,C_H∗=σ∗∈C∣λmaxA_σ∗H=λ_H∗,and the dominant eigenvalue of the row representatives is defined by16-Hλ-H∗=maxσ∈C⁡λmaxA-σH,C-H∗=σ∗∈C∣λmaxA-σ∗H=λ-H∗.


### 3.2. Necessary and Sufficient Conditions forDIESpD,r with p∈{1,∞}

Theorem 4.
(a) There exist D≻0 diagonal and 0<r<1 such that system 1-S is DIES1D,r under arbitrary switching,if and only if0<(λ_S)∗<1.
(b) There exist D≻0 diagonal and r<0 such that system 1-H is DIES1D,r under arbitrary switching,if and only if(λ_H)∗<0.Proof.
We give the proof only for the continuous-time case (b); the proof of discrete-time case (a) may be addressed along the same main lines.
Theorem2 shows that system 1-H is DIES1D,r under arbitrary switching, if and only if inequalities 8-H are satisfied. On the other hand, in accordance with Theorem 3(i), inequalities 8-H are equivalent to inequalities 10-H, and therefore the proof relies on the use of the latter.
Only If. We first show that if inequalities 10-H have solutions v>0, r∈R, thenr satisfies the inequality r≥(λ_H)∗. Indeed, assume that r<(λ_H)∗ and v>0; r solves the inequality (inequalities) vTA_σ∗H≤rvT, σ∗∈C_H∗. By taking r=(λ_H)∗-ε, ε>0 and referring to one of the matrices A_σ∗H, we get the inequality ((λ_H)∗I-A_σ∗H)Tv≥εv with solutions v>0. If we use the notation Mσ∗=((λ_H)∗I-A_σ∗H)T, then Mσ∗ is an M-matrix, and, for all the nonzero components of the vector v>0, we have (Mσ∗v)i≥εvi>0. This contradicts Theorem 6.4.6 (A5) from ([20], p. 149). Thus, the conditions r<0 and r≥(λ_H)∗ imply the inequality (λ_H)∗<0.
If. Define the matrices Mθ=rI-AθH, θ=1,…,N, with (λ_H)∗≤r<0. In accordance with Exercise 1.3.7 ([20], p. 9), the following two algebraic systems are dual:17SM1T⋯MNTv≥0v>0,and18S~M1…MNx1⋮xN≪0xθ≫0,θ=1,…,N.We write the first inequality in (S~) as M1x1+⋯+MNxN=-x0, x0≫0, which is equivalent to the equality (IX0+M1X1+⋯+MNXN)e=0, where Xθ=diag⁡xθ has all diagonal entries positive, θ=0,1,…,N, and e=[1⋯1]T∈Rn. All the column representatives of M={I,M1,…,MN} are M-matrices and their determinants are nonnegative (Theorem 6.4.6 (A1) in ([20], p. 149). Since det(I)>0, M has the column-W0-property and det(IX0+M1X1+⋯+MNXN)≠0 for any diagonal matrix Xθ with positive diagonal entries, θ=0,1,…,N, as per Theorem 7, [21]. Hence, the equality (IX0+M1X1+⋯+MNXN)e=0 is not true and (S~) is not consistent. Subsequently, (S) is consistent and inequalities 10-H have the solution v>0 – as being equivalent to the algebraic system (S). Implicitly, the semipositive vector v>0 satisfies the inequality ((λ_H)∗I-A_σ∗H)Tv≥0.
Case (1). If there exists σ∗∈C_∗ such that the matrix A_σ∗H isirreducible, then ((λ_H)∗I-A_σ∗H)T is a singular, irreducible M-matrix. Thus, Theorem 6.4.16 ([20], p. 156) applied to ((λ_H)∗I-A_σ∗H)Tv≥0 shows that the only semipositive vector v>0, v1=1, satisfying 10-H is the left eigenvector wL(A_σ∗H)≫0 (that is strictly positive). Thus, there exist D=diag⁡{wL(A_σ∗H)}≻0 diagonal and r=(λ_H)∗<0 ensuring μ1D(Aθ)=μ1D(AθH)≤r,θ=1,…,N, a fact which shows that inequalities 8-H are satisfied.
Case (  2). If for all σ∗∈C_∗ the matrices A_σ∗H arereducible, then consider the matrix E∈Int(R+n×n) and c>0. Define the set of essentially positive matrices AθH(c)=AθH+cE, θ=1,…,N, and their column representatives A_σH(c)=A_σH+cE, σ∈C. For any σ∈C, λmax(A_σH(c)) is a continuous and increasing function with respect to c>0, so that for any small ε>0 we can define the interval Jε=(0,cε]={c>0∣maxσ∈C⁡λmax(A_σH(c))≤(λ_H)∗+ε<0}. For any c∈Jε, introduce (similarly to 15-H) (λ_H(c))∗=maxσ∈C⁡λmax(A_σH(c)), C_cH∗=σc∗∈C∣λmax(A_σc∗H(c))=(λ_H(c))∗ and notice that there exists σc∗∈C_cH∗ such that matrix A_σc∗H(c) is irreducible (since all A_σH(c), σ∈C are essentially positive and, inherently, irreducible). Hence, for any c∈Jε, we apply case (1) detailed above to the inequality (v(c))TA_σc∗H(c)≤(λ_H(c))∗(v(c))T, and we show that v(c)=wL(A_σc∗H(c))≫0, r(c)=(λ_H(c))∗<0 solve the inequalities vTA_θH(c)≤rvT, θ=1,…,N. For these v(c), r(c), we can write (v(c))TAθH<(v(c))TAθH(c)≤r(c)(v(c))T, θ=1,…,N, meaning that v(c), r(c) solve inequalities 10-H. Thus, there exist D=diag⁡{wL(A_σc∗H(c))}≻0 diagonal and r=(λ_H(c))∗<0 ensuring μ1D(Aθ)=μ1D(AθH)≤r, θ=1,…,N, a fact which shows that inequalities 8-H are satisfied.
The analysis of both irreducible and reducible cases forA_σ∗H completes the proof.Theorem 5.
(a) There exist D≻0 diagonal and 0<r<1 such that system 1-S is DIES∞D,r under arbitrary switching,if and only if0<(λ-S)∗<1.
(b) There exist D≻0 diagonal and r<0 such that system 1-H is DIES∞D,r under arbitrary switching,if and only if(λ-H)∗<0.Proof.
It is similar to the proof of Theorem4, taking into account the fact that, in accordance with Theorem 3(iii), inequalities 8-S//8-H are equivalent to inequalities 12-S//12-H.Remark 6.
The results in Theorems4 and 5 focus on thequalitative aspect of the DIES analysis based on the dominant eigenvalues of the representatives, in the sense of the traditional threshold values 1 for discrete-time and 0 for continuous-time dynamics.
However the proof of Theorem4 together with Lemma 1 in [22] also allows addressing quantitativeaspects, in the sense that there exist pairs (v,r), with v≫0, satisfying inequalities 10-S, with r≥(λ_S)∗ as close to (λ_S)∗ as we want, and, respectively, inequalities 10-H, with r≥(λ_H)∗ as close to (λ_H)∗ as we want. Equivalently, this means that (λ_S)∗ and (λ_H)∗, respectively, represent the best (fastest) contraction rate for all the invariant sets defined by inequality 8-S and 8-H, respectively, with p=1. Similar reasoning supports the conclusion that (λ-S)∗ and (λ-H)∗, respectively, represent the best (fastest) contraction rate for all the invariant sets defined by inequality 8-S and 8-H, respectively, with p=∞.


### 3.3. Sufficient Conditions forDIES2D,r

Theorem 7.
(a) If 0 < ( λ _ S ) ∗ ( λ - S ) ∗ < 1, then there exist D≻0 diagonal and 0<r<1, such that the discrete-time system 1-S is DIES2D,runder arbitrary switching.
(b) If ( λ _ H ) ∗ + ( λ - H ) ∗ < 0, then there exist D≻0 diagonal and r<0, such that the continuous-time system 1-H is DIES2D,r under arbitrary switching.Proof.
We give the proof only for the continuous-time case (b); the proof of discrete-time case (a) may be addressed along the same main lines, by using matrix-norm properties instead of matrix-measure properties.
Ifε>0 is arbitrarily small, then there exist r_ with (λ_H)∗<r_ and r- with (λ-H)∗<r-, such that r_+r-=2r=(λ_H)∗+(λ-H)∗+ε<0. Since (λH_)∗<r_, from the proof of Theorem 4 (“if” part) we can find D_≻0 diagonal, such that μ1(D_-1AθD_)=μ1(D_-1AθHD_)≤r_,θ=1,…,N. By a similar reasoning (relative to Theorem 5), we can get D-≻0 diagonal, such that μ∞(D--1AθD-)=μ∞(D--1AθHD-)≤r-,θ=1,…,N.
Now, if we use the notationsD_=(U_)2, D-=(U-)2, we can write192r=r_+r-≥μ1D_-1AθD_+μ∞D--1AθD-=μ1D_-1AθD_+μ1D-AθTD--1≥μ1D_-1AθD_+D-AθTD--1≥λmaxD_-1AθD_+D-AθTD--1=λmaxU_-12AθU_2+U-2AθTU--12=λmaxU--1U_U_-12AθU_2+U-2AθTU--12U--1U_-1=λmaxU--1U_-1AθU_U-+U_U-AθTU--1U_-1=λmaxU_U--1AθU_U-+U_U-AθTU_U--1=2μ2U_U--1AθU_U-,θ=1,…,N.Thus, we have proven the existence of D=U_U-≻0 diagonal and r=(r_+r-)/2<0 ensuring μ2D(Aθ)≤r, θ=1,…,N, a fact which shows that inequalities 8-H are satisfied.Remark 8.
Unlike Theorems4 and 5, Theorem 7 expresses only a sufficient condition for DIES. It is formulated inqualitative terms by referring to the dominant eigenvalues of the representatives, as well as to the traditional threshold values 1 for discrete-time and 0 for continuous-time.
However the proofs of Theorems4 and 7 also allow a refined formulation inquantitative terms, in the sense that, for p=2, there exist pairs (D,r), with D≻0, satisfying inequalities 8-S, with r≥(λ_S)∗(λ-S)∗ as close to (λ_S)∗(λ-S)∗ as we want, and, respectively, inequalities 8-H, with r≥1/2(λ_H)∗+(λ-H)∗ as close to 1/2(λ_H)∗+(λ-H)∗ as we want. Obviously, this formulation in quantitative terms is still limited just to sufficiency (similarly to Theorem 7). Subsequently, the corresponding value, i.e., (λ_S)∗(λ-S)∗ and 1/2(λ_H)∗+(λ-H)∗, respectively, does not necessarily represent the best (fastest) contraction rate for all the invariant sets defined by inequality 8-S and 8-H, respectively, with p=2.


## 3.1. Column and Row Representatives for a Set of Matrices as Analytical Tools

LetM=M1,…,MN⊂Rn×n be a set of matrices. Given a function σ:{1,…,n}→{1,…,N}, denote by σ=(σ(1),…,σ(n)) the correspondingn-tuple and represent by C the set of all then-tuples with elements from {1,…,N}.For anyσ∈C, we construct the matrix M_σ∈Rn×n, whose first column is the first column of Mσ(1) (denoted as [Mσ(1)](:,1)), second column is the second column of Mσ(2) (denoted as [Mσ(2)](:,2)), and so on; that is, M_σ=[Mσ1]:,1⋯[Mσ(n)](:,n)∈Rn×n. A matrix M_σ∈Rn×n, σ∈C, constructed as above is called acolumn representativeof the matrix set M. We say that σ∈C labels the column representatives.For anyσ∈C, we construct the matrix M-σ∈Rn×n, whose first row is the first row of Mσ(1) (denoted as [Mσ(1)](1,:)), second row is the second row of Mσ(2) (denoted as [Mσ(2)](2,:)), and so on; that is, M-σ=[Mσ(1)](1,:)⋯[Mσ(n)](n,:)∈Rn×n. A matrix M-σ∈Rn×n, σ∈C, constructed as above is called arow representativeof the matrix set M. We say that σ∈C labels the row representatives.If the matricesMθ∈Rn×n,θ=1,…,N, are (essentially) nonnegative, then all column representatives M_σ∈Rn×n, σ∈C, and all row representatives M-σ∈Rn×n, σ∈C, are (essentially) nonnegative matrices. Subsequently, relying on Perron-Frobenius theory, one can define13λ_∗=maxσ∈C⁡λmaxM_σ,C_∗=σ∗∈C∣λmaxM_σ∗=λ_∗,where λ_∗ represents the dominant eigenvalue of the column representatives and C_∗ the set of labels associated with λ_∗. Similarly, one can define14λ-∗=maxσ∈C⁡λmaxM-σ,C-∗=σ∗∈C∣λmaxM-σ∗=λ-∗,where λ-∗ represents the dominant eigenvalue of the row representatives and C-∗ the set of labels associated with λ-∗.In the next two subsections, regarding the switching system1-S//1-H and 2, we are going to exploit the following:(a) For the discrete-time case, the representatives of the matrix setAS=A1S,A2S,…,ANS are defined by 9-S, the dominant eigenvalue of the column representatives is defined by15-Sλ_S∗=maxσ∈C⁡λmaxA_σS,C_S∗=σ∗∈C∣λmaxA_σ∗S=λ_S∗,and the dominant eigenvalue of the row representatives is defined by16-Sλ-S∗=maxσ∈C⁡λmaxA-σS,C-S∗=σ∗∈C∣λmaxA-σ∗S=λ-S∗.(b) For the continuous-time case, the representatives of the matrix set AH=A1H,A2H,…,ANH are defined by 9-H, the dominant eigenvalue of the column representatives is defined by15-Hλ_H∗=maxσ∈C⁡λmaxA_σH,C_H∗=σ∗∈C∣λmaxA_σ∗H=λ_H∗,and the dominant eigenvalue of the row representatives is defined by16-Hλ-H∗=maxσ∈C⁡λmaxA-σH,C-H∗=σ∗∈C∣λmaxA-σ∗H=λ-H∗.


## 3.2. Necessary and Sufficient Conditions forDIESpD,r with p∈{1,∞}

Theorem 4.
(a) There exist D≻0 diagonal and 0<r<1 such that system 1-S is DIES1D,r under arbitrary switching,if and only if0<(λ_S)∗<1.
(b) There exist D≻0 diagonal and r<0 such that system 1-H is DIES1D,r under arbitrary switching,if and only if(λ_H)∗<0.Proof.
We give the proof only for the continuous-time case (b); the proof of discrete-time case (a) may be addressed along the same main lines.
Theorem2 shows that system 1-H is DIES1D,r under arbitrary switching, if and only if inequalities 8-H are satisfied. On the other hand, in accordance with Theorem 3(i), inequalities 8-H are equivalent to inequalities 10-H, and therefore the proof relies on the use of the latter.
Only If. We first show that if inequalities 10-H have solutions v>0, r∈R, thenr satisfies the inequality r≥(λ_H)∗. Indeed, assume that r<(λ_H)∗ and v>0; r solves the inequality (inequalities) vTA_σ∗H≤rvT, σ∗∈C_H∗. By taking r=(λ_H)∗-ε, ε>0 and referring to one of the matrices A_σ∗H, we get the inequality ((λ_H)∗I-A_σ∗H)Tv≥εv with solutions v>0. If we use the notation Mσ∗=((λ_H)∗I-A_σ∗H)T, then Mσ∗ is an M-matrix, and, for all the nonzero components of the vector v>0, we have (Mσ∗v)i≥εvi>0. This contradicts Theorem 6.4.6 (A5) from ([20], p. 149). Thus, the conditions r<0 and r≥(λ_H)∗ imply the inequality (λ_H)∗<0.
If. Define the matrices Mθ=rI-AθH, θ=1,…,N, with (λ_H)∗≤r<0. In accordance with Exercise 1.3.7 ([20], p. 9), the following two algebraic systems are dual:17SM1T⋯MNTv≥0v>0,and18S~M1…MNx1⋮xN≪0xθ≫0,θ=1,…,N.We write the first inequality in (S~) as M1x1+⋯+MNxN=-x0, x0≫0, which is equivalent to the equality (IX0+M1X1+⋯+MNXN)e=0, where Xθ=diag⁡xθ has all diagonal entries positive, θ=0,1,…,N, and e=[1⋯1]T∈Rn. All the column representatives of M={I,M1,…,MN} are M-matrices and their determinants are nonnegative (Theorem 6.4.6 (A1) in ([20], p. 149). Since det(I)>0, M has the column-W0-property and det(IX0+M1X1+⋯+MNXN)≠0 for any diagonal matrix Xθ with positive diagonal entries, θ=0,1,…,N, as per Theorem 7, [21]. Hence, the equality (IX0+M1X1+⋯+MNXN)e=0 is not true and (S~) is not consistent. Subsequently, (S) is consistent and inequalities 10-H have the solution v>0 – as being equivalent to the algebraic system (S). Implicitly, the semipositive vector v>0 satisfies the inequality ((λ_H)∗I-A_σ∗H)Tv≥0.
Case (1). If there exists σ∗∈C_∗ such that the matrix A_σ∗H isirreducible, then ((λ_H)∗I-A_σ∗H)T is a singular, irreducible M-matrix. Thus, Theorem 6.4.16 ([20], p. 156) applied to ((λ_H)∗I-A_σ∗H)Tv≥0 shows that the only semipositive vector v>0, v1=1, satisfying 10-H is the left eigenvector wL(A_σ∗H)≫0 (that is strictly positive). Thus, there exist D=diag⁡{wL(A_σ∗H)}≻0 diagonal and r=(λ_H)∗<0 ensuring μ1D(Aθ)=μ1D(AθH)≤r,θ=1,…,N, a fact which shows that inequalities 8-H are satisfied.
Case (  2). If for all σ∗∈C_∗ the matrices A_σ∗H arereducible, then consider the matrix E∈Int(R+n×n) and c>0. Define the set of essentially positive matrices AθH(c)=AθH+cE, θ=1,…,N, and their column representatives A_σH(c)=A_σH+cE, σ∈C. For any σ∈C, λmax(A_σH(c)) is a continuous and increasing function with respect to c>0, so that for any small ε>0 we can define the interval Jε=(0,cε]={c>0∣maxσ∈C⁡λmax(A_σH(c))≤(λ_H)∗+ε<0}. For any c∈Jε, introduce (similarly to 15-H) (λ_H(c))∗=maxσ∈C⁡λmax(A_σH(c)), C_cH∗=σc∗∈C∣λmax(A_σc∗H(c))=(λ_H(c))∗ and notice that there exists σc∗∈C_cH∗ such that matrix A_σc∗H(c) is irreducible (since all A_σH(c), σ∈C are essentially positive and, inherently, irreducible). Hence, for any c∈Jε, we apply case (1) detailed above to the inequality (v(c))TA_σc∗H(c)≤(λ_H(c))∗(v(c))T, and we show that v(c)=wL(A_σc∗H(c))≫0, r(c)=(λ_H(c))∗<0 solve the inequalities vTA_θH(c)≤rvT, θ=1,…,N. For these v(c), r(c), we can write (v(c))TAθH<(v(c))TAθH(c)≤r(c)(v(c))T, θ=1,…,N, meaning that v(c), r(c) solve inequalities 10-H. Thus, there exist D=diag⁡{wL(A_σc∗H(c))}≻0 diagonal and r=(λ_H(c))∗<0 ensuring μ1D(Aθ)=μ1D(AθH)≤r, θ=1,…,N, a fact which shows that inequalities 8-H are satisfied.
The analysis of both irreducible and reducible cases forA_σ∗H completes the proof.Theorem 5.
(a) There exist D≻0 diagonal and 0<r<1 such that system 1-S is DIES∞D,r under arbitrary switching,if and only if0<(λ-S)∗<1.
(b) There exist D≻0 diagonal and r<0 such that system 1-H is DIES∞D,r under arbitrary switching,if and only if(λ-H)∗<0.Proof.
It is similar to the proof of Theorem4, taking into account the fact that, in accordance with Theorem 3(iii), inequalities 8-S//8-H are equivalent to inequalities 12-S//12-H.Remark 6.
The results in Theorems4 and 5 focus on thequalitative aspect of the DIES analysis based on the dominant eigenvalues of the representatives, in the sense of the traditional threshold values 1 for discrete-time and 0 for continuous-time dynamics.
However the proof of Theorem4 together with Lemma 1 in [22] also allows addressing quantitativeaspects, in the sense that there exist pairs (v,r), with v≫0, satisfying inequalities 10-S, with r≥(λ_S)∗ as close to (λ_S)∗ as we want, and, respectively, inequalities 10-H, with r≥(λ_H)∗ as close to (λ_H)∗ as we want. Equivalently, this means that (λ_S)∗ and (λ_H)∗, respectively, represent the best (fastest) contraction rate for all the invariant sets defined by inequality 8-S and 8-H, respectively, with p=1. Similar reasoning supports the conclusion that (λ-S)∗ and (λ-H)∗, respectively, represent the best (fastest) contraction rate for all the invariant sets defined by inequality 8-S and 8-H, respectively, with p=∞.


## 3.3. Sufficient Conditions forDIES2D,r

Theorem 7.
(a) If 0 < ( λ _ S ) ∗ ( λ - S ) ∗ < 1, then there exist D≻0 diagonal and 0<r<1, such that the discrete-time system 1-S is DIES2D,runder arbitrary switching.
(b) If ( λ _ H ) ∗ + ( λ - H ) ∗ < 0, then there exist D≻0 diagonal and r<0, such that the continuous-time system 1-H is DIES2D,r under arbitrary switching.Proof.
We give the proof only for the continuous-time case (b); the proof of discrete-time case (a) may be addressed along the same main lines, by using matrix-norm properties instead of matrix-measure properties.
Ifε>0 is arbitrarily small, then there exist r_ with (λ_H)∗<r_ and r- with (λ-H)∗<r-, such that r_+r-=2r=(λ_H)∗+(λ-H)∗+ε<0. Since (λH_)∗<r_, from the proof of Theorem 4 (“if” part) we can find D_≻0 diagonal, such that μ1(D_-1AθD_)=μ1(D_-1AθHD_)≤r_,θ=1,…,N. By a similar reasoning (relative to Theorem 5), we can get D-≻0 diagonal, such that μ∞(D--1AθD-)=μ∞(D--1AθHD-)≤r-,θ=1,…,N.
Now, if we use the notationsD_=(U_)2, D-=(U-)2, we can write192r=r_+r-≥μ1D_-1AθD_+μ∞D--1AθD-=μ1D_-1AθD_+μ1D-AθTD--1≥μ1D_-1AθD_+D-AθTD--1≥λmaxD_-1AθD_+D-AθTD--1=λmaxU_-12AθU_2+U-2AθTU--12=λmaxU--1U_U_-12AθU_2+U-2AθTU--12U--1U_-1=λmaxU--1U_-1AθU_U-+U_U-AθTU--1U_-1=λmaxU_U--1AθU_U-+U_U-AθTU_U--1=2μ2U_U--1AθU_U-,θ=1,…,N.Thus, we have proven the existence of D=U_U-≻0 diagonal and r=(r_+r-)/2<0 ensuring μ2D(Aθ)≤r, θ=1,…,N, a fact which shows that inequalities 8-H are satisfied.Remark 8.
Unlike Theorems4 and 5, Theorem 7 expresses only a sufficient condition for DIES. It is formulated inqualitative terms by referring to the dominant eigenvalues of the representatives, as well as to the traditional threshold values 1 for discrete-time and 0 for continuous-time.
However the proofs of Theorems4 and 7 also allow a refined formulation inquantitative terms, in the sense that, for p=2, there exist pairs (D,r), with D≻0, satisfying inequalities 8-S, with r≥(λ_S)∗(λ-S)∗ as close to (λ_S)∗(λ-S)∗ as we want, and, respectively, inequalities 8-H, with r≥1/2(λ_H)∗+(λ-H)∗ as close to 1/2(λ_H)∗+(λ-H)∗ as we want. Obviously, this formulation in quantitative terms is still limited just to sufficiency (similarly to Theorem 7). Subsequently, the corresponding value, i.e., (λ_S)∗(λ-S)∗ and 1/2(λ_H)∗+(λ-H)∗, respectively, does not necessarily represent the best (fastest) contraction rate for all the invariant sets defined by inequality 8-S and 8-H, respectively, with p=2.


## 4. Illustrative Examples and Numerical Considerations

The theoretical results on DIES are applied to two numerical examples referring to both continuous-time (example 1) and discrete-time (example 2) switching dynamics. Examples1 and 2 have a similar architecture, in the sense that the constituent matrices are parameterized by two parameters a, b. In both examples, the construction of the DIES domains corresponding to p∈{1,2,∞}, is addressed in a dual manner, by using the older results (summarized by Section 2), in parallel with the new ones (developed by Section 3). Relevant comparisons are permitted for the applicability of Theorem 3(i) versus Theorem 4, of Theorem 3(ii) versus Theorem 7, and of Theorem 3(iii) versus Theorem 5, respectively. Moreover, these comparative discussions on the DIES domains can include information about subdomains corresponding to various values selected for the contraction rate r<0 (continuous-time) and 0<r<1 (discrete-time).Both considered examples are inspired by the literature of arbitrary switching linear systems, as detailed below. Unlike the organization of the exposition for theoretical results, in the previous sections, the current section starts with the continuous-time case—example 1—continued by the discrete-time case—example 2. This is because example 1 presents a higher numerical complexity and requires more meticulous explanations. The computations are performed in MATLAB R2016a, running on a laptop equipped with an Intel(R) Core™ i7-4710HQ CPU @2.50 GHz x64-based processor and 16 GB RAM.Example 1.
Consider a continuous-time arbitrary switching linear system of form1-H defined by the matrices20A=A1,A2,A3⊂R3×3,A1=-1262a1-102b+13-10,A2=-12-40-6-10-9-b-a-13,A3=-9-2-8-6-10-b-a0-11,where a,b∈[1,10] are parameters, inspired by the numerical example presented in [23].We are interested in finding the domain of the parameters(a,b)∈[1,10]×[1,10] that guarantees DIES for p∈{1,2,∞}. Given a certain p∈{1,2,∞}, the corresponding DIES domain is represented as a union of subdomains corresponding to different decreasing ratesr, namely, -0.25≤r<0, -0.50≤r<-0.25, etc. The subdomain role is understood as a refinement of the global results with r<0, which can take into consideration quantitative information about the contraction of the invariant sets defined by 5-H. For instance, if a point (a∗,b∗) belongs to the DIES1D,r subdomain corresponding to -0.25≤r<0, then, for the parameter values a∗,b∗, system 1-H cannot have invariant sets of form 5-H with p=1 and a decreasing rate faster than -0.25 (regardless of the considered positive definite diagonal matrix D≻0).The DIES analysis is meant to permit comparisons between the employment of the already known results reproduced in Section2 and the employment of the new results proposed by Section 3, for all the points belonging to the grid constructed for (a,b)∈[1,10]×[1,10], with a step of 0.05. When applying the results from Section 2 for p∈{1,2,∞}, in each point of the grid we look for theminimum value of r such that there exists a positive vector v∈R4, v≫0, fulfilling 8-H or the particular forms derived in Theorem 3, namely, 10-H, 11-H, and 12-H, respectively. This problem can be solved numerically by using thefmincon solver from the Optimization Toolbox for MATLAB or thega solver from the Global Optimization Toolbox for MATLAB, that performs the minimization of a function subject to nonlinear constraints. For the particular case when p=2, the constraints 11-H are in the form of BMIs (bilinear matrix inequalities) and may be solved, for example, using the PENBMI solver commercially available from TOMLAB Optimization [24]. A second strategy that can be employed uses a bisection strategy for r∈maxλmax(A1H),λmax(A2H),λmax(A3H),10 that yields (for every feasibler) a linear programming problem for p∈{1,∞} that can be approached using thelinprog solver from the Optimization Toolbox for MATLAB or an LMI (linear matrix inequality) problem for p=2 that can be solved using CVX, a MATLAB package for specifying and solving convex programs [25, 26]. In the current paper we have adopted the first strategy depicted previously. The numerical implementation of the results from Section 3 requires standard linear algebra facilities for the representative construction and eigenvalue computation using MATLAB built-in functions.Let us begin the DIES analysis forp=1. The DIES1D,r parametric domain in Figure 1(a) is obtained applying Theorems 2 and 3(i). The DIES1D,r parametric domain in Figure 1(b) was obtained applying Theorem 4(b) and Remark 6. In each point of the grid, we compute the dominant eigenvalue (λ_H)∗ for all column representatives of the matrix family21AH=A1H,A2H,A3H⊂R3×3,A1H=-1262a1-102b+13-10,A2H=-12406-109ba-13,A3H=-9286-10ba0-11.The plot presented in Figure 1(b) offers the same graphic information as Figure 1(a)—a fact expected from the theoretical point of view, in accordance with the proof of Theorem 4 and the comments included in Remark 6. Nevertheless, it is extremely important to notice that the computation effort corresponding to the old approach (summarized in Section 2) is significantly greater than the effort corresponding to the new one (developed in Section 3). For the considered example (studied on the machine with the characteristics presented above), the requested computation time is 8,826 seconds for the first (old) procedure and 10.85 seconds for the second (new) one. This major difference is caused by the great number of points in the grid, but even at the level of a single point the faster operation of the new procedure is directly visible. This fact may be regarded as a general advantage of the new approach, not strictly related to the considered example. However, generally speaking, the cardinality of the column-representative set associated with AH must be reasonable, because the existence of a high number of representatives may severely increase the computation time for (λ_H)∗.Figure 1
Comparative plots ofDIES1D,r parametric domain for example 1 provided by (a) the MATLAB implementation of the old results summarized in Section 2 (Theorems 2 and 3(i)); (b) the MATLAB implementation of the new results developed in Section 3 (Theorem 4(b) and Remark 6).
(a) (b)Here we may also point out an interesting connection with the numerical example from paper [23] that corresponds to the arbitrary switching positive system defined by the essentially nonnegative matrices 21, with the particular values a∈{3,5} and b=4. The cited example studies the stability by means of copositive linear Lyapunov functions, an investigation which is equivalent to DIES1 analysis of the switching system defined by the matrices 20. Our results are in agreement with the findings in [23]. For a=3 and b=4, the switching system defined by matrices 20 is DIES1, meaning the existence of a common copositive linear Lyapunov function for the switching positive system defined by matrices 21. For a=5 and b=4 the switching system defined by matrices 20 is not DIES1, meaning that no common copositive linear Lyapunov function can be found for the switching positive system defined by matrices 21.Forp=∞ the DIES analysis is addressedmutatis mutandis along the same lines as the previously discussed case of p=1. The old procedure relies on the use of Theorems 2 and 3(iii) in each point of the grid and provides the DIES∞D,r parametric domain in Figure 2(a). The new procedure applies Theorem 5(b) and Remark 6 in each point of the grid and computes the dominant eigenvalue (λ-H)∗ for all row representatives of the matrix family AH=A1H,A2H,A3H; the resulting DIES∞D,r parametric domain is depicted in Figure 2(b). The substantial difference between the computation times requested by the two methods (commented above for p=1) is also evident here, 9,026 seconds being necessary for the first (old) procedure and 10.89 seconds for the second (new) one.Figure 2
Comparative plots ofDIES∞D,r parametric domain for example 1 provided by (a) the MATLAB implementation of the old results summarized in Section 2 (Theorems 2 and 3(iii)); (b) the MATLAB implementation of the new results developed in Section 3 (Theorem 5(b) and Remark 6).
(a) (b)The DIES analysis forp=2 reveals noticeable variations in the structure of the DIES2D,r parametric domains provided by the old results summarized in Section 2 (i.e., Theorems 2 and 3(ii)) and by the new results developed in Section 3 (i.e., Theorem 7(b) and Remark 8). Theorems 2 and 3(ii) are applied in each point of the grid and provide the domain presented in Figure 3(a). Theorem 7(b) and Remark 8 are applied in each point of the grid for computing (λ_H)∗ associated with all column representatives and (λ-H)∗ associated with all row representatives, as well as the value (λ_H)∗+(λ-H)∗. The resulting domain is presented in Figure 3(b) and one can simply notice that the DIES2D,r region in this figure is included into (but not identical to) the DIES2D,r region in Figure 3(a). This is because Theorem 7(b) expresses just a sufficient condition, unlike the necessary and sufficient condition formulated by Theorems 2 and 3(ii), a fact already mentioned in Remark 8. On the other hand, the advantage of a much shorter computation time requested by the new procedure, compared to the old one, still exists (as in the previously studied cases of p∈{1,∞}). The concrete duration for this example is 20,652 seconds for the first approach and 21.75 seconds for the second one. Under such circumstances, the use of the new procedure may be preferable for some types of DIES2D,r problems, compared to the old procedure, despite the inconvenience of the new one that ensures only sufficiency.Figure 3
Comparative plots ofDIES2D,r parametric domain for example 1 provided by (a) the MATLAB implementation of the old results summarized in Section 2 (Theorems 2 and 3(ii)); (b) the MATLAB implementation of the new results developed in Section 3 (Theorem 7(b) and Remark 8).
(a) (b)This example, via Figure3, shows that relatively large differences can occur for DIES2D,r when using the sufficient condition in Theorem 7, unlike the necessary and sufficient condition in Theorem 3(ii). Generally speaking, these differences cannot be anticipated from the simple examination of the switching system model. For instance our example 2 illustrates a visibly smaller difference between the applications of the two theorems.Example 2.
Consider a discrete-time arbitrary switching linear system of form1-S defined by the matrices22A=A1,A2,A1=a0.80.30.3,A2=b0.20.60.2,where a,b∈[0,1] are parameters, inspired by the numerical example presented in [27]. Note that matrices A1, A222 are nonnegative, and therefore AθS=Aθ, θ=1,2; i.e., AS=A. The DIES analysis is organizedmutatis mutandis as in example 1, by taking a grid constructed for (a,b)∈[0,1]×[0,1], with a step of 0.02.Figures4 and 5 present the DIES1D,r and the DIES∞D,r parametric domain, respectively, obtained by applying Theorems 4(a) and 4(a), whose numerical implementation required 0.115 seconds and 0.105 seconds, respectively. As mentioned in the comment on example 1, the application of Theorem 2 together with Theorem 3(i) or 2(iii) would provide the same domains but at the expense of much longer computation times.Figure 4
Plot ofDIES1D,r parametric domain for example 2 provided by the MATLAB implementation of Theorem 4(a) and Remark 6.Figure 5
Plot ofDIES∞D,r parametric domain for example 2 provided bythe MATLAB implementation of Theorem 5(a) and Remark 6.The application of Theorem2 together with Theorem 3(ii) yields the DIES2D,r parametric domain depicted in Figure 6(a) and requires a computation time of 765.9 seconds. On the other hand, the application of Theorem 7 requires only 0.23 seconds and yields the DIES2D,r parametric domain depicted in Figure 6(b). A direct visual inspection of Figures 3 and 6 shows that the differences between the DIES2D,r parametric domains provided by the two theorems (Theorem 7 versus Theorem 3(ii)) is significantly smaller in example 2 than in example 1. This comparison consolidates our point of view on the practical usefulness of Theorem 7, offering a DIES2D,r sufficient condition, whose computational advantages cannot be ignored.Figure 6
Comparative plots ofDIES2D,r parametric domain for example 2 provided by (a) the MATLAB implementation of the old results summarized in Section 2 (Theorems 2 and 3(ii)); (b) the MATLAB implementation of the new results developed in Section 3 (Theorem 7(a) and Remark 8).
(a) (b)


## 5. Conclusions

The current paper presents a novel approach to the characterization ofDIESpD,rwith p∈{1,2,∞}, for arbitrary switching linear systems modelled by 1-S//1-H and 2, in both discrete- and continuous-time, based on the usage of row and column representatives built for the set of nonnegative matrices AS (discrete-time case) and AH (continuous-time case), respectively. This approach explores a theoretical potential that has not been exploited by previous works, whose main results are stated in Theorems 2 and 3. For p∈{1,∞} Theorems 4 and 5 provide necessary and sufficient conditions for DIESpD,r, whereas for p=2 the condition presented in Theorem 7 is only sufficient for DIES2D,r. The applicability of the results is illustrated for concrete numerical examples which also compare similarities and differences between the current approach and the older results depicted in Theorems 2 and 3. At the same time, these comparisons highlight a number of advantages referring to the computational effort required by the numerical implementation based on the new results paralleled to previous ones. The use of representatives theory in DIESpD,r analysis for p∈{1,2,∞} is expected to be also applicable to linear dynamics with polytopic uncertainties. Our further research will focus on this class of systems, aimed at the possible development of results relying on the same principles as proposed by the current paper for switching linear systems.



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