Max-type copositive Lyapunov functions for switching positive linear systems

Automatica 50 (2014) 3323–3327

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Max-type copositive Lyapunov functions for switching positive linear systems✩ Octavian C. Pastravanu, Mihaela-Hanako Matcovschi Department of Automatic Control and Applied Informatics, ‘‘Gheorghe Asachi’’ Technical University of Iasi, Blvd. Prof. Mangeron 27, 700050 Iasi, Romania

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Article history: Received 10 February 2014 Received in revised form 9 June 2014 Accepted 6 September 2014 Available online 29 October 2014

The paper aims to expand the stability analysis framework based on copositive Lyapunov functions (CLFs) for arbitrarily switching positive systems with continuous- or discrete-time dynamics. The first part focuses on max-type CLFs. It provides an algebraic characterization (in terms of weak quasi-linear inequalities), as well as an existence criterion relying on a concrete construction procedure (in terms of Perron–Frobenius eigenstructure). The second part explores the connections between max-type and two other classes of CLFs, namely linear and quadratic-diagonal. New qualitative and quantitative features are revealed for the CLFs belonging to the two mentioned classes. © 2014 Elsevier Ltd. All rights reserved.

Keywords: Switching positive linear systems Copositive Lyapunov function Perron–Frobenius eigenstructure

1. Introduction 1.1. Switching positive linear systems and copositive Lyapunov function candidates Consider a family of N positive linear time-invariant systems

ΣAθ : x′ (t ) = Aθ x(t ), x(t0 ) = x0 ∈ Rn+ , t0 , t ∈ T, θ = 1, . . . , N ,

(1)

with continuous-time dynamics (T = R+ ) or discrete-time dynamics (T = Z+ ), and the operator (◦)′ acting accordingly. In the continuous-time case, matrices Aθ ∈ Rn×n are essentially nonnegative and Hurwitz stable; in the discrete-time case, Aθ ∈ Rn×n are nonnegative and Schur stable. By using the family of matrices

A = A1 , . . . , A



 N

,

(2)

we define the dynamics of the arbitrarily switching positive system:

ΣA : x′ (t ) = Aν(t ) x(t ), x(t0 ) = x0 ∈ Rn+ , t ≥ t0 , (3) where ν : T → {1, . . . , N } is the switching signal (piecewise continuous for T = R+ ). n

Since the nonnegative orthant R+ is invariant with respect to the dynamics of system (3), one can search for copositive Lyapunov

✩ The authors acknowledge the support of UEFISCDI Romania under Grant PN-

II-ID-PCE-2011-3-1038. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Jie Chen under the direction of Editor André L. Tits. E-mail addresses: [email protected] (O.C. Pastravanu), [email protected] (M.-H. Matcovschi). http://dx.doi.org/10.1016/j.automatica.2014.10.043 0005-1098/© 2014 Elsevier Ltd. All rights reserved.

functions (abbreviated CLFs), defined as V : Rn+ → R+ , V (x) > 0 for x ̸= 0. By using the concept of ‘‘decreasing rate’’ of a Lyapunov function, e.g. Blanchini and Miani (2008), we say that V is a ‘‘CLF with the decreasing rate r’’ (abbreviated CLF(r )), if along each nontrivial trajectory of system (3) the following condition is satisfied:

• for the continuous-time case, with r < 0: D+ t V (x(t )) = lim

1 h↓0 h

(V (x(t + h)) − V (x(t )))

≤ r V (x(t )),

t ≥ t0 ,

(4)

• for the discrete-time case, with 0 < r < 1: V (x(t + 1)) ≤ r V (x(t )),

t ≥ t0 .

(5)

Let v = [v1 . . . vn ]⊤ ≫ 0 be a positive vector, vi > 0, i = 1, . . . , n. As CLF candidates for both continuous- and discrete-time cases, we consider functions with the following forms:

• linear: V : Rn+ → R+ ,

V (x) =

n 

vi xi ,

(6)

i=1

• max-type: V : Rn+ → R+ ,

V (x) = max {xi /vi } , i=1,...,n

(7)

• quadratic-diagonal: Vqd : Rn+ → R+ ,

Vqd (x) =

n  i=1

vi x2i .

(8)

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O.C. Pastravanu, M.-H. Matcovschi / Automatica 50 (2014) 3323–3327

1.2. Research context and objectives The research reported on CLFs for system (3) focuses on the linear form (6). For existence, necessary and sufficient conditions are presented in Fornasini and Valcher (2010), Knorn, Mason, and Shorten (2009), Mason, Bokharaie, and Shorten (2009), Mason and Shorten (2007) and Moldovan and Gowda (2010) — for continuous-time systems; (Ding, Shu, & Liu, 2011; Fornasini & Valcher, 2012) — for discrete-time systems. For construction, the use of the Perron–Frobenius eigenstructure is proposed by our paper (Pastravanu, Matcovschi, & Voicu, 2011) — in the continuous-time case, and by paper (Doan, Kalauch, & Siegmund, 2013) — in the discrete-time case. For the discrete-time case, paper (Liu, 2009) explores the existence of switched linear CLFs that are less conservative than a single linear form. The current paper enlarges the CLF analysis framework for system (3), by studying the max-type CLFs of form (7), as well as the connections to linear and quadratic-diagonal CLFs of form (6) and (8), respectively. Both continuous- and discrete-time cases are considered. The exposition of our main developments is organized in two parts (Sections 3 and 4), preceded by some preliminaries on (essentially) nonnegative matrices (Section 2). Section 3 presents the key results on max-type CLFs, providing:

• an algebraic characterization in terms of weak quasi-linear inequalities; • an existence criterion based on a concrete construction procedure in terms of Perron–Frobenius eigenstructure. Section 4 explores the connections between max-type, linear and quadratic-diagonal CLFs, showing that:

• the existence conditions are different for max-type and linear CLFs, meaning they may have complementary use in stability testing; • the max-type or linear CLFs with the fastest decreasing rate are unique, under a mild hypothesis on the row- or columnrepresentatives of (2); • the existence of either max-type or linear CLFs (not necessarily both) ensures the existence of quadratic-diagonal CLFs for a large class of switching positive systems. To improve readability of the text, the proofs of the main results are given in Appendix.

and X ≫ Y , if X − Y ≥ 0, X − Y > 0, and X − Y ≫ 0, respectively. A square matrix X = [xij ] ∈ Rn×n is called essentially nonnegative (positive) if xij ≥ 0, (xij > 0), i, j = 1, . . . , n, i ̸= j. Let A ∈ Rn×n be (essentially) nonnegative and let λi (A), i = 1, . . . , n, denote its eigenvalues. If A is nonnegative, then it has a real eigenvalue λmax (A) such that |λi (A)| ≤ λmax (A), i = 1, . . . , n. If A is essentially nonnegative, then it has a real eigenvalue λmax (A), such that Re{λi (A)} ≤ λmax (A), i = 1, . . . , n. If A is (essentially) nonnegative, then it has a nonnegative right eigenvector wR (A) > 0, satisfying ∥wR (A)∥1 = 1, and a nonnegative left eigenvector wL (A) > 0, satisfying ∥wL (A)∥1 = 1, that correspond to the eigenvalue λmax (A). If A is (essentially) nonnegative and irreducible (i.e. its associated graph is strongly connected (Berman & Plemmons, 1994, Chapter 2)), then λmax (A) is a simple eigenvalue, called the Perron–Frobenius eigenvalue, and the associated eigenvectors are positive wR (A) ≫ 0, wL (A) ≫ 0, being called the Perron–Frobenius right eigenvector and left eigenvector, respectively. If A is nonnegative, then B = α I − A with α ≥ λmax (A) is called an M-matrix (Berman & Plemmons, 1994). Consider a set of essentially nonnegative matrices A of form (2). For a function s : {1, . . . , n} → {1, . . . , N } we denote by σ = (s(1), . . . , s(n)) the corresponding n-tuple and by S the set of all the n-tuples with elements from {1, . . . , N }. For any σ ∈ S, we construct the matrix

 s(1)  [A ](1,:) Aσ =  · · ·  ∈ Rn×n , [As(n) ](n,:)

(9)

called a row-representative of the matrix set A (2), where the first row is the first row of As(1) , the second row is the second row of As(2) , # and so on. The set of all row-representatives is denoted by A . For #

any σ ∈ S, Aσ ∈ A is an (essentially) nonnegative matrix, implying that λmax (Aσ ) is well defined. Thus, we can define the dominant ∗

(10)

A similar construction can be developed for column representatives. For any σ ∈ S, we build the matrix



Our developments are going to exploit some properties of a set of (essentially) nonnegative matrices A (2) that defines the dynamics of the arbitrarily switching system ΣA (3). The notations and nomenclature introduced by the current section are used throughout the paper. For a vector x ∈ Rn , we denote its entries by xi or [x](i) , i = 1, . . . , n, and its Hölder p-norm by ∥x∥p , p ∈ {1, 2, ∞}. For a square matrix A ∈ Rn×n , we denote its entries by aij or [A](i,j) , its columns by [A](:,j) , its rows by [A](i,:) , i, j = 1, . . . , n, and its induced norm by ∥A∥p . The notation µp (A) = limh↓0 1h (∥I + hA∥p − 1) is used for the matrix measure based on the induced matrix norm ∥ ∥p (Bernstein, 2009). Let X = [xij ] ∈ Rn×m , Y = [yij ] ∈ Rn×m , be real matrices (in particular, real vectors). X is called nonnegative (notation X ≥ 0) if xij ≥ 0, i = 1, . . . , n, j = 1, . . . , m; semi-positive (notation X > 0) if X ≥ 0 and X ̸= 0; positive (notation X ≫ 0) if xij > 0, i = 1, . . . , n, j = 1, . . . , m. We write X ≥ Y , X > Y ,

#

∗

λ = max λmax (Aσ ), σ ∈S  ∗  ∗ ∗ ∗ # A = Aσ ∈ A | λmax (Aσ ) = λ .

Aσ = [As(1) ](:,1) · · · [As(n) ](:,n) ∈ Rn×n , 2. Preliminaries — notations and properties for (essentially) nonnegative matrices

∗

eigenvalue of the row representatives λ and the subset A ⊆ A ∗ corresponding to λ , by:



(11)

called a column-representative of the matrix set A (2), where the first column is the first column of As(1) , the second column is the second column of As(2) , and so on. The set of all columnrepresentatives is denoted by A# . We define the dominant eigenvalue of the column representatives λ∗ and the subset A∗ ⊆ A# corresponding to λ∗ , by:

λ∗ = max λmax (Aσ ), σ∈S   ∗ A = A∗σ ∈ A# | λmax (A∗σ ) = λ∗ .

(12)

3. Key results on max-type CLFs 3.1. Continuous-time case Algebraic characterization of max-type CLFs Theorem 1. Let v = [v1 , . . . , vn ]⊤ ≫ 0 be a positive vector and r < 0 a constant. The function V of form (7) is a max-type CLF(r) for

O.C. Pastravanu, M.-H. Matcovschi / Automatica 50 (2014) 3323–3327

system (3) iff the pair (v, r ) satisfies the quasi-linear inequalities θ

A v ≤ r v,

θ = 1, . . . , N .

(13)

Relying on Theorem 1, we show that no max-type CLF can decrease with a faster rate than the dominant eigenvalue of the row ∗ representatives λ defined by (10). Corollary 1. Let r < 0. If V of form (7) is a max-type CLF(r) for ∗ system (3), then r ≥ λ . Existence (by construction) of max-type CLFs. Starting from the family of constituent matrices A (2), we define the family of slightly perturbed constituent matrices

A(c ) = A1 (c ), . . . , AN (c ) ,





Aθ (c ) = Aθ + cE ,

θ = 1, . . . , N ,

4. Connections to linear and quadratic-diagonal CLFs 4.1. Max-type vs. linear CLFs Complementarity in stability testing. The conditions for the existence of max-type CLFs (Corollary 2) and linear CLFs (e.g. Knorn et al., 2009) do not overlap. In general, the dominant eigenvalue of the ∗ row representatives λ defined by (10) differs from the dominant eigenvalue of the column representatives λ∗ defined by (12), fact meaning that the search for max-type and linear CLFs may complement each other in stability analysis. Example 1 (Mason et al., 2009). Consider a continuous-time switching system of form (3) defined by

(14)

where c > 0 and E is the n × n matrix with all entries 1. Use the notations A(c )σ , σ ∈ S, for the row-representatives of A(c ) (14), #

and A(c ) for the whole set. Similarly to (10), let us define

  ∗ λ(c ) = max λmax A(c )σ , σ∈S     ∗ ∗ # ∗ ∗ A(c ) = A(c )σ ∈ A(c ) | λmax A(c )σ = λ(c ) .

(15)

∗

Theorem 2. Let λ < 0.

∗

∗

(i) Assume that there exists a row representative Aσ ∈ A that is irreducible and consider its eigenvector ∗

∗ wR (Aσ ).

The function V

of form (7) with v = wR (Aσ ) is the unique (up to a multiplicative ∗

positive constant) max-type CLF (λ ) of (3). ∗ (ii) Let ε > 0 be arbitrarily small so as λ + ε < 0. There exists c (ε) > 0 such that the following statements (a), (b) hold true, for any c , 0 < c ≤ c (ε). ∗ ∗ ∗ (a) λ < λ(c ) ≤ λ + ε ; ∗ ∗ ∗ (b) Let A(c )σ ∈ A(c ) , and consider its eigenvector wR (A(c )σ ). ∗

The function V of form (7) with v = wR (A(c )σ ) is a max-type ∗

CLF (λ(c ) ) for system (3). ∗

Remark 1. If the set A (10) includes two or more irreducible matrices, then any of them can be used for the CLF construction ∗ ∗ ∗ procedure presented by Theorem 2(i). Indeed, if Aσˆ , Aσ˜ ∈ A ∗ Aσˆ

∗ are irreducible and ̸= Aσ˜ , then the proof of Theorem 2(i) via ∗ ∗ Lemma 1(ii) shows that wR (Aσˆ ) ≡ wR (Aσ˜ ). Similar comments can ∗

be made for Theorem 2(ii) when the set A(c ) (15) includes two or more representatives. Theorem 2 omits the case when all repre∗ sentatives in A are reducible, as requiring a laborious analysis of the communication classes.  The result of Corollary 1 can be refined in the sense that the existence of max-type CLFs is fully characterized by the negativeness ∗ of the dominant eigenvalue λ . ∗

Corollary 2. System (3) has max-type CLFs iff λ < 0. 3.2. Discrete-time case The results of the previous subsection can be restated, with appropriate changes, for discrete-time systems. All constituent matrices from A (2) are nonnegative, implying that the row representatives are nonnegative. Inequalities (13) preserve their form, with 0 < r < 1, and are equivalently derived by using matrix norms instead of matrix measures (see the proof of Theorem 1). ∗ Theorem 2 and Corollary 2 consider λ < 1.

3325

A1 =





−1

2 , −3

1

A2 =

 −6 2



6 . −6

Since λ∗ = 0, there exists no linear CLF and no decision can be ∗ made on the stability of system (3). Instead, λ = −0.2679 < 0 ensures the existence of max-type CLFs.  CLFs with the fastest decreasing rate. Uniqueness. Theorem 2(i) guides the construction of a max-type CLF with the fastest decreas∗ ing rate r = λ and proves its uniqueness, under a relatively mild ∗ hypothesis on A (10). For the research reported on linear CLFs, ∗ the case r = λ remained obscure; both papers (Doan et al., 2013; Pastravanu et al., 2011) show that linear CLFs can be constructed with r > λ∗ arbitrarily close, and the latter analyzes the case r = λ∗ under the restrictive assumption that A∗ (12) contains a single column-representative. The adaptation mutatis mutandis of Theorem 2(i) and its proof allows a deeper exploration of the case r = λ∗ for linear CLFs. ∗

Example 1 (ctd.). The λ -row-representative A(1,1) is irreducible with wR (A(1,1) ) = [0.7321 0.2679]⊤ . Theorem 2(i) yields

V [x1 x2 ]⊤ = max







x1 0.7321

x2 , 0.2679



is the unique max-type CLF with the fastest decreasing rate, ∗ namely λ = −0.2679.  CLFs of dual system. Let v ≫ 0 be a positive vector and r < 0 (continuous-time) or 0 < r < 1 (discrete-time). By the adaptation mutatis mutandis of Theorem 1 and its proof, one can show that the function V of form (6) expressed as

V (x) = ∥V x∥1 ,

V = diag{v1 , . . . , vn },

(16)

is a linear CLF(r ) for system (3) iff the quasi-linear inequalities

v ⊤ Aθ ≤ r v ⊤ ,

θ = 1, . . . , N ,

(17)

hold true. Thus, the function V of form (6) is a linear CLF(r ) for system (3), iff the function V of form (7) is a max-type CLF(r ) for the dual of system (3), defined as

ΣA⊤ : y′ (t ) = (Aν(t ) )⊤ y(t ),   A⊤ = (A1 )⊤ , . . . , (AN )⊤ .

y(t0 ) = y0 ∈ Rn+ ,

t ≥ t0 ,

(18)

4.2. Quadratic-diagonal CLFs For a class of switching positive systems, the existence of either max-type CLFs or linear CLFs (not necessarily both) ensures the existence of quadratic-diagonal CLFs.

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O.C. Pastravanu, M.-H. Matcovschi / Automatica 50 (2014) 3323–3327

∗

∗

Theorem 3. If λ + λ∗ < 0 (continuous-time) or λ λ∗ < 1 (discrete-time), then system (3), as well as its dual (18) have quadratic-diagonal CLFs of form (8). Remark 2. For proving the existence of a quadratic-diagonal CLF, Remark 25.20 in Moldovan and Gowda (2010) needs a much ∗ stronger hypothesis, namely λ < 0, λ∗ < 0, because the decreasing rate of the CLF is not exploited. For constructing a quadraticdiagonal CLF(r ) of form (8), the coefficients can be calculated as vi = v i /v i , i = 1, . . . , n (see the proof of Theorem 3), with v i , v i , i = 1, . . . , n, obtained from Lemma 1 (see Appendix) applied to inequalities (17) and (13), respectively. 

inequalities (13). Hence, vˆ ≫ 0 is a solution to the inequality ∗ ∗ ∗ (λ I − Aσ )v ≥ 0 discussed above and we get vˆ ≡ wR (Aσ ). (iii) (a) For any σ ∈ S, λmax (A(c )σ ) is a continuous and increasing function with respect to c > 0. Subsequently,

  ∗ ∗ λ(c ) = max λmax A(c )σ > λ , σ∈S

and we can find c (ε) > 0 such that ∗

∗

λ(c ) ≤ λ + ε,

∀c ≤ c (ε). ∗

(b) For any 0 < c ≤ c (ε), we can apply(ii), since A(c )σ ∈  ∗

A(c ) is irreducible. Thus, the pair

∗

Example 1 (ctd.). There exist quadratic-diagonal CLFs,since λ + λ∗ = −0.2679 < 0. Lemma 1(ii) ensures that the pairs wR (A(1,1) ),

∗

(Aθ + cE )v ≤ λ(c ) v,

  λ , wL (A(1,2) ), λ∗ are solutions to inequalities (13) and (17). Hence,

θ = 1, . . . , N ,

and, implicitly, inequalities (13).





0.5 0.7321

.5 x21 + 0.02679 x22 = 0.6830 x21 + 1.8664 x22

is a quadratic-diagonal CLF.

V (x) = ∥V −1 x∥∞ = ∥x∥V∞ , where V = diag{v1 , . . . , vn } is a diagonal matrix. The vector norm ∥ • ∥V∞ induces a matrix norm that defines the matrix measure

Lemma 1. Consider the sets of essentially nonnegative matrices A (2) and A(c ) (14), as well as the quasi-linear inequalities (13). ∗

(i) If r ≥ λ , then inequalities (13) have solutions  (v, r ) withv > 0. ∗

∗

∗

∗

(ii) If there exists Aσ ∈ A irreducible, then wR (Aσ ), λ

is the

∗

unique pair (v, λ ) with v ≫ 0 that solves inequalities (13). (iii) Let ε > 0 be arbitrarily small. There exists c (ε) > 0 such that the following statements (a), (b) hold true, for any c , 0 < c ≤ c (ε). ∗ ∗ ∗ (a) λ < λ(c ) ≤ λ + ε ;   ∗

∗

∗

∗

(b) If A(c )σ ∈ A(c ) , then the pair wR (A(c )σ ), λ(c )

solves

µV∞ (Aθ ) = µ∞ (V −1 Aθ V ) and, after some simple calculation, inequalities (13) can be equivalently written as

µV∞ (Aθ ) ≤ r , ∗∗

Sufficiency: for any solution x of this constituent system and any t ∈ [τ ∗ , τ ∗∗ ), as per Bernstein (2009, Fact 11.15.7.iii) we get

  ∥x(t + h)∥V∞ − ∥x(t )∥V∞   θ = lim 1h ∥ehA x(t )∥V∞ − ∥x(t )∥V∞ h↓0    hAθ V 1 ≤ lim h ∥e ∥∞ − 1 ∥x(t )∥V∞ 1 h↓0 h

∗

Proof. (i) Let r > λ . The matrices (rI − Aθ )⊤ , θ = 1, . . . , N, are positive stable Z-matrices (equivalently, nonsingular Mmatrices) and there exists v ≫ 0 such that

(rI − Aθ )v ≫ 0,

θ = 1, . . . , N ,

h ↓0

in accordance with Theorem 25.16 in Moldovan and Gowda (2010). Subsequently the pair (v, r ) satisfies quasi-linear inequalities (13). Thus, for all ε > 0, there exists v(ε) ≫ 0 such that ∗

(εI + λ I − Aθ )v(ε) ≫ 0,

θ = 1, . . . , N .

If we assume ∥v(ε)∥1 = 1, the norm continuity for ε ↘ 0 ensures v(0) > 0, meaning that ∗

∃v > 0 : (λ I − Aθ )v ≥ 0,

θ = 1, . . . , N .

∗

(ii) Let (v, λ ) with v > 0 be a pair that solves inequalities (13). Implicitly, vector v > 0 satisfies ∗

(λ I − Aσ )v ≥ 0, ∗

∗

∗

∗

6.4.16 in Berman and Plemmons (1994) applied to λ I − Aσ shows that

(λ I −

∗

≥ 0 ⇒ (λ I −

∗ Aσ )v

=0

Necessity: the inequality V V D+ t ∥x(t )∥∞ ≤ r ∥x(t )∥∞

implies that the system trajectory x(t ; t0 , x0 ) satisfies the inequality

∥x(t ; t0 , x0 )∥V∞ ≤ er (t −t0 ) ∥x0 ∥V∞ , for all t0 , t ∈ [τ ∗ , τ ∗∗ ) and x0 ∈ Rn , as per Michel, Wang, and Hu (2001, Theorem 4.2.11). Hence,

∗ ∗ wR (Aσ ), λ

. Assume (by contradiction) there

∗

θ

∥x(t0 + τ ; t0 , x0 )∥V∞ ∥eτ A x0 ∥V∞ = sup ≤ er τ , = sup V ∥x0 ∥V∞ x0 ̸=0 x0 ̸=0 ∥x(t0 )∥∞

that yields

    θ µV∞ (A) = lim 1h ∥ehA ∥V∞ − 1 ≤ lim 1h erh − 1 = r .  h↓0

whose only solution satisfying v > 0, ∥v∥1 = 1, is the eigen∗ vector wR (Aσ) ≫ 0. Westill have to prove the uniqueness for the pair

= µV∞ (Aθ )∥x(t )∥V∞ ≤ r V (x(t )).

θ ∥eτ A ∥V∞

where λ I − Aσ is a singular, irreducible M-matrix. Theorem

∗ Aσ )v

θ = 1, . . . , N .

Let τ , τ , be successive switching moments, and assume that the switching signal is ν(t ) = θ for all t ∈ [τ ∗ , τ ∗∗ ); consider the corresponding constituent system ΣAθ for t ∈ [τ ∗ , τ ∗∗ ). ∗

D+ t V (x(t )) = lim

inequalities (13).

∗



Proof of Theorem 1. Express the function V defined by (7) as



Appendix

∗

∗

satisfies

∗

Vqd [x1 x2 ]⊤ =

∗

wR (A(c )σ ), λ(c )

∗

exists vˆ ≫ 0, vˆ ̸= wR (Aσ ) such that the pair (ˆv , λ )solves

h↓0

Proof of Corollary 1. Take a positive α ≥ −r so that α I + Aσ is #

nonnegative, for any Aσ ∈ A . Theorem 1 ensures the validity of inequalities (13), which imply that inequalities Aσ v ≤ r v are true #

∗

∗

∗

for any Aσ ∈ A . Thus, for any Aσ ∈ A , we have (α I + Aσ )v ≤

O.C. Pastravanu, M.-H. Matcovschi / Automatica 50 (2014) 3323–3327

∗ (α+r ) v and λmax (α I +Aσ )

∗

∗

= α+λ , fact that yields α+r ≥ α+λ , as per Berman and Plemmons (1994, Theorem 2.1.11).  Proof of Theorem 2. It results from Theorem 1 (Sufficiency) and Lemma 1(ii), (iii).  Proof of Corollary 2. Sufficiency results from Theorem 2 and necessity from Corollary 1. 

Proof of Theorem 3 (Continuous-Time Case). Lemma 1 applied to ∗ inequalities (13) and (17) shows that for any r > λ , r > λ∗ , there exist the positive vectors v = [v 1 , . . . , v n ]⊤ ≫ 0, v = [v 1 , . . . , v n ]⊤ ≫ 0 such that the inequalities below hold true Aθ v ≤ r v ⇔ µ∞ (V

−1 θ

A V ) ≤ r , V = diag{v 1 , . . . , v n }, v ⊤ Aθ ≤ r v ⊤ ⇔ µ1 (V Aθ V −1 ) ≤ r , V = diag{v 1 , . . . , v n }. Some calculations based on the matrix measure properties yield the inequality

µ2 [(UU −1 )−1 Aθ (UU −1 )] ≤ (r + r )/2, ∗

with (U )2 = V , (U )2 = V , θ = 1, . . . , N. Thus, if λ + λ∗ < 0, ∗ we can take r > λ , r > λ∗ , such that r = r + r < 0, and a proof similar to Theorem 1 shows that

Vqd (x) = ∥(UU −1 )−1 x∥22 is a quadratic-diagonal CLF(r ) of form (8), with the coefficients vi = v i /v i , i = 1, . . . , n. For the dual system, the coefficients are vi−1 . The discrete-time case exploits the matrix norm (instead of measure) properties.



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References Berman, A., & Plemmons, R. J. (1994). Nonnegative matrices in the mathematical sciences. SIAM. Bernstein, D. S. (2009). Matrix mathematics: theory, facts, and formulas. Princeton University Press. Blanchini, F., & Miani, S. (2008). Set-theoretic methods in control. Boston: Birkhäuser. Ding, Xiuyong, Shu, Lan, & Liu, Xiu (2011). On linear copositive Lyapunov functions for switched positive systems. Journal of the Franklin Institute, 348(8), 2099–2107. Doan, T. S., Kalauch, A., & Siegmund, S. (2013). A constructive approach to linear Lyapunov functions for positive switched systems using Collatz–Wielandt sets. IEEE Transactions on Automatic Control, 58(3), 748–751. Fornasini, E., & Valcher, M. E. (2010). Linear copositive Lyapunov functions for continuous-time positive switched systems. IEEE Transactions on Automatic Control, 55(8), 1933–1937. Fornasini, E., & Valcher, M. E. (2012). Stability and stabilizability criteria for discretetime positive switched systems. IEEE Transactions on Automatic Control, 57(5), 1208–1221. Knorn, F., Mason, O., & Shorten, R. (2009). On linear co-positive Lyapunov functions for sets of linear positive systems. Automatica, 45(8), 1943–1947. Liu, Xingwen (2009). Stability analysis of switched positive systems: a switched linear copositive lyapunov function method. IEEE Transactions on Circuits and Systems II: Express Briefs, 56(5), 414–418. Mason, O., Bokharaie, V. S., & Shorten, R. (2009). Stability and D-stability for switched positive systems. In R. Bru, & S. Romero-Viv (Eds.), Positive systems (pp. 101–109). Springer. Mason, O., & Shorten, R. (2007). On linear copositive Lyapunov functions and the stability of switched positive linear systems. IEEE Transactions on Automatic Control, 52(7), 1346–1349. Michel, A., Wang, K., & Hu, B. (2001). Qualitative theory of dynamical systems. CRC Press. Moldovan, M. M., & Gowda, M. S. (2010). On common linear/quadratic Lyapunov functions for switched linear systems. In P. Pardalos, Th. M. Rassias, & A. A. Khan (Eds.), Nonlinear analysis and variational problems (pp. 415–429). Springer. Pastravanu, O., Matcovschi, M.H., & Voicu, M. Qualitative analysis results for arbitrarily switching positive systems. In Prepr. 18th IFAC world congress, 2011 (pp. 1326–1331).