Robust stability of positive linear systems under time varying perturbations

This article was downloaded by: [University of Newcastle (Australia)] On: 14 September 2014, At: 03:46 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Numerical Functional Analysis and Optimization Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/lnfa20 Robust Stability of Positive Linear Systems Under TimeVarying Perturbations Pham Huu Anh Ngoc a a Department of Mathematics , Vietnam National University—HCMC, International University , Saigon , Vietnam Accepted author version posted online: 01 Jul 2013.Published online: 01 Apr 2014 To cite this article: Pham Huu Anh Ngoc (2014) Robust Stability of Positive Linear Systems Under Time-Varying Perturbations, Numerical Functional Analysis and Optimization, 35:6, 739-751, DOI: 10.1080/01630563.2013.816319 To link to this article: http://dx.doi.org/10.1080/01630563.2013.816319 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content This article may be used for research, teaching, and private study purposes Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions Numerical Functional Analysis and Optimization, 35:739–751, 2014 Copyright © Taylor & Francis Group, LLC ISSN: 0163-0563 print/1532-2467 online DOI: 10.1080/01630563.2013.816319 Downloaded by [University of Newcastle (Australia)] at 03:46 14 September 2014 ROBUST STABILITY OF POSITIVE LINEAR SYSTEMS UNDER TIME-VARYING PERTURBATIONS Pham Huu Anh Ngoc Department of Mathematics, Vietnam National University—HCMC, International University, Saigon, Vietnam By a novel approach, we get explicit robust stability bounds for positive linear differential systems subject to time-varying multi-perturbations and time-varying affine perturbations Our approach is based on the celebrated Perron-Frobenius theorem and ideas of the comparison principle An example is given to illustrate the obtained results Keywords Exponential stability; Robust stability; Time-varying perturbation Mathematics Subject Classification 34D20; 93D09 INTRODUCTION Roughly speaking, a dynamical system is called positive if for any nonnegative initial condition, the corresponding solution of the system is also nonnegative Positive dynamical systems play an important role in modelling of dynamical phenomena whose variables are restricted to be nonnegative They are often encountered in applications, for example, networks of reservoirs, industrial processes involving chemical reactors, heat exchangers, distillation columns, storage systems, hierarchical systems, compartmental systems used for modelling transport and accumulation phenomena of substances, see, for example, [4, 7, 12] Motivated by many applications in control engineering, problems of robust stability of dynamical systems have attracted much attention from researchers during the past 30 years, see, for example, [2, 3, 5, 9, 14, 24] Received 28 April 2012; Revised and Accepted 11 June 2013 Address correspondence to Pham Huu Anh Ngoc, Department of Mathematics, Vietnam National University—HCMC International University, Thu Duc District, Saigon 84-08, Vietnam; E-mail: phangoc@hcmiu.edu.vn 739 740 P H A Ngoc and references therein In particular, problems of robust stability of the linear time-invariant differential system x(t ˙ ) = Ax(t ), t ≥ 0, (1) under time-invariant perturbations of the form A A + D E, A A+ (structured perturbation), (2) Downloaded by [University of Newcastle (Australia)] at 03:46 14 September 2014 N Di i Ei , (multi-perturbation), (3) i=1 and N A+ A k Ak , (affine perturbation), (4) k=1 have been studied intensively, see, for example, [8, 9, 22, 23] Recently, problems of stability and robust stability of positive systems have attracted a lot of attention from researchers, see, for example, [2, 6, 7, 11, 13–19] In this article, we give explicit robust stability bounds for positive linear differential systems of the form (1) subject to one of the following time-varying perturbations N A+ A Dk (t ) k (t )Ek (t ) (time-varying multi-perturbation) (5) k=1 N A A+ k (t )Ak (t ) (time-varying affine perturbation) (6) k=1 Although there are many works devoted to the study of robust stability of the differential system (1), however, to the best of our knowledge, the problems of robust stability of the positive linear differential system (1) under the time-varying multi-perturbations (5) and the time-varying affine perturbations (6), have not yet been studied in the literature and the main purpose of this article is to fill this gap In contrast to the traditional approach to stability analysis of timevarying differential systems (Lyapunov’s method and its variants), see, for example, [3, 5, 20, 21], we present in this article a novel approach to the problems of robust stability of positive systems of the form (1) under the time-varying perturbations of the form (5) and (6) Our approach is based on the celebrated Perron-Frobenius theorem and ideas of the comparison principle To the best of our knowledge, the obtained results of this article are new 741 Robust Stability of Positive Linear Systems Downloaded by [University of Newcastle (Australia)] at 03:46 14 September 2014 PRELIMINARIES Let be the set of all natural numbers For given m ∈ , let us denote m := 1, 2, , m and m := 0, 1, 2, , m For integers l , q ≥ 1, l denotes the l -dimensional vector space over and l ×q stands for the set of all l × q-matrices with entries in Inequalities between real matrices or vectors will be understood componentwise, that is, for two real matrices A = (aij ) and B = (bij ) in ł×q , we write A ≥ B iff aij ≥ bij for i = 1, , l , j = 1, , q In particular, if aij > bij for i = 1, , l , j = 1, , q, l ×q then we write A B instead of A ≥ B We denote by + the set of all nonnegative matrices A ≥ Similar notations are adopted for vectors For x ∈ n and P ∈ l ×q we define |x| = (|xi |) and |P | = (|pij |) Then one has |PQ | ≤ |P ||Q |, ∀P ∈ l ×q , ∀Q ∈ q×r (7) A norm · on n is said to be monotonic if x ≤ y whenever x, y ∈ n , |x| ≤ |y| Every p-norm on n ( x p = (|x1 |p + |x2 |p + · · · + |xn |p ) p , ≤ p < ∞ and x ∞ = maxi=1,2, ,n |xi |), is monotonic Throughout this article, if otherwise not stated, the norm of vectors on n is monotonic and the norm of a matrix P ∈ l ×q is understood as its operator norm associated with a given pair of monotonic vector norms on l and q , that is P = max Py : y = Note that P ∈ l ×q , Q ∈ l ×q + , |P | ≤ Q ⇒ P ≤ |P | ≤ Q , (8) see, for example, [9] In particular, if n is endowed with · or · ∞ then A = |A| for any A = (aij ) ∈ n×n More precisely, one has A = n n |A| = max1≤j ≤n i=1 |aij | and A ∞ = |A| ∞ = max1≤i≤n j =1 |aij | n×n the spectral abscissa of M is denoted by For any matrix M ∈ (M ) = max : ∈ (M ) , where (M ) := z ∈ : det(zIn − M ) = is the spectrum of M A matrix M ∈ n×n is said to be Hurwitz stable if (M ) < A matrix M ∈ n×n is called a Metzler matrix if all off-diagonal elements of M are nonnegative We now summarize in the following theorem some properties of Metzler matrices Theorem 2.1 ([9]) Suppose M ∈ n×n is a Metzler matrix Then (i) (Perron-Frobenius) (M ) is an eigenvalue of M and there exists a nonnegative eigenvector x = such that Mx = (M )x (ii) Given ∈ , there exists a nonzero vector x ≥ such that Mx ≥ x if and only if (M ) ≥ (iii) (tIn − M )−1 exists and is nonnegative if and only if t > (M ) 742 P H A Ngoc (iv) Given B ∈ n×n + , C ∈ n×n |C | ≤ B Then ⇒ (M + C ) ≤ (M + B) The following is immediate from Theorem 2.1 and is used in what follows Downloaded by [University of Newcastle (Australia)] at 03:46 14 September 2014 Theorem 2.2 Let M ∈ statements are equivalent (i) (ii) (iii) (iv) (v) n×n be a Metzler matrix Then the following (M ) < 0; Mp for some p ∈ n , p 0; −1 M is invertible and M ≤ 0; For given b ∈ n , b 0, there exists x ∈ +n , such that Mx + b = n For any x ∈ + \ , the row vector x T M has at least one negative entry Let J be an interval of Denote C (J , n ) the set of all continuous functions on J with values in n In particular, if n is endowed with the norm · then C ([ , ], n ) is a Banach space with the maximum norm = max ∈[ , ] ( ) ROBUST STABILITY OF POSITIVE LINEAR DIFFERENTIAL SYSTEMS Consider a linear time-invariant differential system of the form (1), where A ∈ n×n is a given matrix For given x0 ∈ n , (1) has a unique solution satisfying the initial condition x(0) = x0 This solution is denoted by x(·; x0 ) Then (1) is said to be (uniformly) exponentially stable if there are positive numbers , M such that ∀t ∈ +, ∀x0 ∈ n : x(t ; x0 ) ≤ Me − t x0 Note that (1) is exponentially stable if, and only if, det(zIn − A) = 0, ∀z ∈ + , or equivalently, (A) < 0, see, for example, [10, Theorem 3.3.20] Definition 3.1 The system (1) is said to be positive if x(t ; x0 ) ≥ 0, ∀t ∈ n , x0 ≥ + for any x0 ∈ It is well known that (1) is positive if, and only if, A ∈ n×n is a Metzler matrix, see, for example, [4] We now deal with robust stability of the positive linear differential system (1) under the time-varying multiperturbations (5) and the time-varying affine perturbations (6) 743 Robust Stability of Positive Linear Systems 3.1 Time-Varying Multi-Perturbations Suppose (1) is exponentially stable Consider perturbed systems of the form N x(t ˙ )= A+ Dk (t ) k (t )Ek (t ) x(t ), t≥ , (9) Downloaded by [University of Newcastle (Australia)] at 03:46 14 September 2014 k=1 where N is a given positive integer, Dk (·) ∈ C ( + , n×lk ), Ek (·) ∈ C ( + , qk ×n ), k ∈ N are given and k (·) ∈ C ( + , lk ×qk ) (k ∈ N ) are unknown perturbations For fixed ≥ and given x0 ∈ n , (9) has a unique solution satisfying the initial value condition x( ) = x0 (10) This solution is denoted by x(·; , x0 ) Recall that x(·; , x0 ) is continuously differentiable on [ , ∞) and satisfies (9) for any t ∈ [ , ∞) Then (9) is said to be (globally) exponentially stable if there exist M , > such that ∀x0 ∈ n , ∀t ≥ ≥ 0: x(t ; , x0 ) ≤ Me − (t − ) x0 , see, for example, [10] The main problem here is to find a positive number such that an arbitrary perturbed system of the form (9) remains exponentially stable whenever the size of perturbations is less than We are now in the position to state the first result of this article Theorem 3.2 Let (1) be positive and exponentially stable Suppose there exist q ×n l ×q n×l Dk ∈ + k , Ek ∈ +k and k ∈ +k k for k ∈ N such that |Dk (t )| ≤ Dk , |Ek (t )| ≤ Ek and | k (t )| ≤ k for any t ∈ + and any k ∈ N Then (9) remains exponentially stable if N k k=1 < maxi,j ∈N Ei (−A)−1 Dj (11) Proof We divide the proof into two steps Step I: We claim that (A + k=1 Dk k Ek ) < Since (1) is positive, it follows that A is a Metzler matrix Thus, (A + N k are nonnegative k=1 Dk k Ek ) is also a Metzler matrix because Dk , Ek , N for any k ∈ N We show that := (A + k=1 Dk k Ek ) < Assume on the N 744 P H A Ngoc contrary that ≥ By the Perron-Frobenius theorem (Theorem 2.1 (i)), there exists x ∈ +n , x = such that N A+ Dk x= k Ek 0x k=1 Let Q (t ) = tIn − A, t ∈ Since (1) is exponentially stable, Thus, Q ( ) is invertible and this implies (A) < Downloaded by [University of Newcastle (Australia)] at 03:46 14 September 2014 N −1 Q ( 0) Dk k Ek x =x (12) k=1 Let i0 be an index such that Ei0 x = maxk∈N Ek x It follows from (12) that Ei0 x > Multiply both sides of (12) from the left by Ei0 , to get N Ei0 Q ( )−1 Dk k Ek x = Ei0 x k=1 It follows that N Ei0 Q ( )−1 Dk Ek x ≥ Ei0 x k k=1 Thus, N max Ei Q ( )−1 Dj k i,j ∈N Ei0 x ≥ Ei0 x , k=1 or equivalently, N max Ei Q ( )−1 Dj i,j ∈N i ≥1 (13) k=1 On the other hand, the resolvent identity gives Q (0)−1 − Q ( )−1 = −1 −1 Q (0) Q ( ) (14) Since A is a Metzler matrix with (A) < and ≥ 0, Theorem 2.1 (iii) yields Q (0)−1 ≥ and Q ( )−1 ≥ It follows from (14) that Q (0)−1 ≥ Q ( )−1 ≥ Hence, Ei Q (0)−1 Dj ≥ Ei Q ( )−1 Dj ≥ 0, for any i, j ∈ N By (8), we have Ei Q (0)−1 Dj ≥ Ei Q ( )−1 Dj , for any i, j ∈ N Then (13) implies N k k=1 ≥ maxi,j ∈N However, this conflicts with (11) Ei Q (0)−1 Dj 745 Robust Stability of Positive Linear Systems Step II: Let x0 ∈ n be given and let x(t ) := x(t ; , x0 ), t ∈ [ , ∞) be a solution of (9)–(10) We show that there exist K , > such that for any ≥ and any x0 ∈ n with x0 ≤ 1, x(t ; , x0 ) ≤ Ke − By step I, (A + N k=1 k Ek ) Dk (t − ) , ∀t ≥ (15) 0, (16) < and then Downloaded by [University of Newcastle (Australia)] at 03:46 14 September 2014 N A+ Dk k Ek p k=1 for some p := ( , (16) implies that 2, T n) , , i > 0, ∀i ∈ n, by Theorem 2.2 By continuity, N A+ Dk k Ek p − ( 1, , T n) , (17) k=1 for some sufficiently small > Choose K > such that |x| Kp for any x ∈ n with x ≤ Define u(t ) := Ke − (t − ) p, t ∈ [ , ∞) Set x(t ) := u( ) We claim that |x(t )| ≤ u(t ) x(t ; , x0 ), t ≥ Then, we have |x( )| for any t > Assume on the contrary that there exists t0 > such that |x(t0 )| u(t0 ) Set t1 := inf t ∈ ( , ∞) : |x(t )| By continuity, t1 > u(t ) and there is i0 ∈ n such that |x(t )| ≤ u(t ), ∀t ∈ [ , t1 ); |xi0 (t1 )| = ui0 (t1 ), |xi0 (t )| > ui0 (t ), ∀t ∈ (t1 , t1 + ), (18) for some > Let A := (aij) ∈ n×n , Dk k Ek = (bij(k) ) ∈ n×n and let |Dk (t )|| k (t )||Ek (t )| = (k) (bij (t )) ∈ n×n , ∀t ≥ 0, for k ∈ N Since A is a Metzler matrix and Dk k Ek ≥ for k ∈ N , we have for any i ∈ n, n d |xi (t )| = sgn(xi (t ))x˙i (t ) ≤ aii |xi (t )| + aij |xj (t )| + dt j =1,j =i N n bij(k) (t )|xj (t )|, k=1 j =1 for almost any t ∈ [ , ∞) Thus, n d |xi (t )| ≤ aii |xi (t )| + aij |xj (t )| + dt j =1,j =i N n bij(k) |xj (t )|, k=1 j =1 746 P H A Ngoc for almost any t ∈ [ , ∞) Thus, we have for any t ∈ [ , ∞), D + |xi (t )| := lim sup h→0+ n N ≤ aii |xi (t )| + t d |xi (s)|ds ds n aij |xj (t )| + j =1,j =i Downloaded by [University of Newcastle (Australia)] at 03:46 14 September 2014 t +h |xi (t + h)| − |xi (t )| = lim sup h h h→0+ bij(k) |xj (t )|, k=1 j =1 where D + denotes the Dini upper-right derivative In particular, it follows from (17) and (18) that n (18) D + |xi0 (t1 )| ≤ ai0 i0 Ke − (t1 − ) i0 N ai0 j Ke − + (t1 − ) j j =1,j =i0 n = a i0 j j (17) < − Ke − (t1 − ) i0 (t1 − ) j n + j =1 bi(k) Ke − 0j k=1 j =1 N Ke − (t1 − ) n + bi(k) 0j j k=1 j =1 = D + ui0 (t1 ) However, this conflicts with (18) Therefore, |x(t ; , x0 )| ≤ u(t ) = Ke − (t − ) p, ∀t ∈ [ , ∞), for any ≥ and any x0 ∈ n with x0 ≤ By the monotonicity of vector norms, there exists K1 > such that x(t ; , x0 ) ≤ K1 e − for any ≥ and any x0 ∈ n x(t ; , x) ≤ K1 e − (t − ) , ∀t ∈ [ , ∞), with x0 ≤ By linearity of (9), (t − ) x , ∀t ≥ ≥ 0; ∀x ∈ n Hence, (9) is exponentially stable This completes the proof Remark 3.3 The problem of robust stability of the positive linear differential system (1) under the time-invariant structured perturbations (2) has been studied in [9, 23] More precisely, it has been shown in [23] that if (1) is exponentially stable and positive and D, E are given nonnegative constant matrices then a perturbed system of the form x(t ˙ ) = (A + D E )x(t ), remains exponentially stable whenever < EA −1 D t ≥ 0, 747 Robust Stability of Positive Linear Systems Furthermore, robust stability of the positive system (1) under the timeinvariant multi-perturbations (3) has been analyzed in [9] by techniques of -analysis However, to the best of our knowledge, the problem of robust stability of the positive system (1) under the time-varying multiperturbations (5) has not yet studied and a result like Theorem 3.2 cannot be found in the literature Downloaded by [University of Newcastle (Australia)] at 03:46 14 September 2014 3.2 Time-Varying Affine Perturbations Suppose (1) is exponentially stable and the system matrix A is now subject to time-varying affine perturbations of the form N x(t ˙ )= A+ k (t )Ak (t ) t≥ x(t ), ≥0 (19) k=1 Here N ∈ and Ak (·) ∈ C ( + , n×n ) (k ∈ N ) are given and C ( + , ) (k ∈ N ) are unknown perturbations Furthermore, we assume that (H1 ) ∀k ∈ N , ∃ Ak ∈ (H2 ) ∀k ∈ N , ∃ k ∈ n×n + + : |Ak (t )| ≤ Ak , : | k (t )| ≤ k, k (·) ∈ ∀t ≥ 0; ∀t ≥ The problem is now to find a positive number such that a perturbed system of the form (19) remains exponentially stable whenever the size of := ( , , , N ) ∈ N is less than Theorem 3.4 Let (1) be positive and exponentially stable Suppose (H1 ) and (H2 ) hold Then a perturbed system of the form (19) remains exponentially stable if < where := maxk∈N ((−A)−1 N k=1 Ak ) , (20) k Proof We first show that := (A + k=1 k Ak ) < Since (1) is N positive, it follows that A is a Metzler matrix Thus, A + k=1 k Ak , is also a Metzler matrix because Ak ≥ and k are nonnegative for any k ∈ N Assume on the contrary that ≥ By the Perron-Frobenius theorem (Theorem 2.1 (i)), there exists x ∈ +n , x = such that N N A+ k Ak k=1 x= 0x 748 P H A Ngoc Let Q (t ) = tIn − A, t ∈ Since (1) is exponentially stable, Thus, Q ( ) is invertible and it follows that (A) < N −1 Q ( 0) x =x k Ak k=1 Downloaded by [University of Newcastle (Australia)] at 03:46 14 September 2014 Taking (7) into account, we get N x ≤ |Q ( )−1 | (21) x k Ak k=1 As shown in the proof of Theorem 3.2, (A) < and Q (0)−1 ≥ Q ( )−1 ≥ Then (21) implies ≥ imply that N x ≤ |Q (0)−1 | k Ak x k=1 N = Q (0)−1 k Ak x k=1 N ≤ Q (0) −1 Ak x k=1 Since Q (0)−1 N k=1 Ak is a nonnegative matrix, it follows that N Q (0) −1 −1 ≥ Ak >0 k=1 or equivalently ≥ ((−A)−1 N k=1 Ak ) , by Theorem 2.1 (ii) However, this conflicts with (20) Let x0 ∈ n be given and let x(t ) := x(t ; , x0 ), t ∈ [ , ∞) be a solution of (19) satisfying the initial condition (10) We show that there exists > such that for any ≥ and any x0 ∈ n with x0 ≤ 1, x(t ; , x0 ) ≤ Ke − (t − ) , ∀t ≥ , (22) 749 Robust Stability of Positive Linear Systems where K depends on Since (A + N k=1 k Ak ) p 0, < 0, it follows that N A+ k Ak (23) k=1 for some p := ( , , , n )T , continuity, (23) implies that > 0, ∀i ∈ n, by Theorem 2.2(ii) By i Downloaded by [University of Newcastle (Australia)] at 03:46 14 September 2014 N A+ k Ak − ( 1, p T n) , , (24) k=1 > The remainder of the proof is similar to for some sufficiently small that of Theorem 3.2 Remark 3.5 The problem of robust stability of the positive linear differential system (1) under the time-invariant affine perturbations (4) has been studied in [9] Theorem 3.4 gives an extension of [9, Theorem 18] to the class of time-varying perturbations of the form (6) ILLUSTRATIVE EXAMPLE Consider a linear differential equation in x(t ˙ ) = Ax(t ), t∈ defined by +, (25) where A := −1 1 −2 Clearly, (25) is positive and exponentially stable Consider a perturbed system given by x(t ˙ ) = (A + D1 (t ) (t )E1 (t ) + D2 (t ) (t )E2 (t ))x(t ), (26) where D1 (t ) := − sin t , E1 (t ) := −e −t t ≥ 0; D2 (t ) := − t 22t+1 , and (t ) := (a(t ), b(t )) ∈ C ( are unknown perturbations cos t +1 t ≥ 0; E2 (t ) := +, 1×2 ), , 1+t t ≥ 0; , − t 21+1 (t ) := (c(t ), d(t )) ∈ C ( t ≥ 0, +, 1×2 ), 750 P H A Ngoc Note that for any t ∈ +, we have |D1 (t )| ≤ D1 := ; |E1 (t )| ≤ E1 := ; 1 |D2 (t )| ≤ D2 := ; |E2 (t )| ≤ E2 := 0 Downloaded by [University of Newcastle (Australia)] at 03:46 14 September 2014 and E1 A −1 D1 = 1 −2 −1 −1 −1 −3 ; = −5 E1 A −1 D2 = 1 −2 −1 −1 −1 −1 ; = −2 E2 A −1 D1 = 0 −2 −1 −1 −1 −3 = ; −2 E2 A −1 D2 = 0 −2 −1 −1 −1 −1 = −1 Let be endowed with 2-norm By Theorem 3.2, (26) is exponentially stable if a(·), b(·), c(·), d(·) are bounded and satisfy sup |a(t )| + sup |b(t )| t∈ + t∈ + + sup |c(t )| t∈ + + sup |d(t )| t∈ +