Robust stability of switched positive linear systems

Nội dung text: Robust stability of switched positive linear systems

1. Dong Thap University Journal of Science, Vol. 10, No. 5, 2021, 15-24 ROBUST STABILITY OF SWITCHED POSITIVE LINEAR SYSTEMS Le Van Ngoc Department of Scientific Fundamentals, Posts and Telecommunications Institute of Technology Corresponding author: ngoclv@ptit.edu.vn Article history Received: 20/04/2021; Received in revised form: 24/05/2021; Accepted: 12/07/2021 Abstract The aim of the present paper is to give robust stability based on exponentially stable switched positive linear systems. Our theoretical analysis shows that if there exists a positive stable system of all positive subsystems, then a lower bound and upper bound for stability radius of the switched system under positive affine perturbations are established. In the particular case of two dimensional switched system, including two switching signals, we obtain a formula of stability radius. Several examples are provided to illustrate our approach. Keywords: Positive linear systems, robust stability, stability radii, switched linear systems. TÍNH ỔN ĐỊNH VỮNG CỦA HỆ CHUYỂN MẠCH TUYẾN TÍNH DƯƠNG Lê Văn Ngọc Khoa Cơ bản, Học viện Công nghệ Bưu chính Viễn thông Tác giả liên hệ: ngoclv@ptit.edu.vn Lịch sử bài báo Ngày nhận: 20/04/2021; Ngày nhận chỉnh sửa: 24/05/2021; Ngày duyệt đăng: 12/07/2021 Tóm tắt Mục tiêu của bài báo này nghiên cứu tính ổn định vững dựa trên ổn định mũ của hệ chuyển mạch tuyến tính dương. Phân tích lý thuyết của chúng tôi chỉ ra rằng nếu tồn tại một hệ tuyến tính dương ổn định là một chặn của tất cả các hệ dương con, thì cận dưới và cận trên cho bán kính ổn định này đối với các nhiễu cấu trúc affine dương của hệ thống được thiết lập. Trong trường hợp đặc biệt đối với hệ thống chuyển mạch hai chiều có hai tín hiệu chuyển mạch, chúng tôi thu được công thức bán kính ổn định. Một số ví dụ được cung cấp để minh họa cách tiếp cận của chúng tôi. Từ khóa: Hệ tuyến tính dương, ổn định vững, bán kính ổn định, hệ chuyển mạch tuyến tính DOI: Cite: Le Van Ngoc. (2021). Robust stability of switched positive linear systems. Dong Thap University Journal of Science, 10(5), 15-24. 15
2. Natural Sciences issue 1. Introduction (2) x( t ) Ak x ( t ), t 0, k N A switched system is described by a have a common quadratic Lyapunov function family of subsystems and a rule that controls (or QLF, for short) of the form V(x) xT Px or the switching between them. Switched (see Blanchini et al., 2015, Ding et al., 2011) a systems have gained attention from many common co-positive Lyapunov linear function scientists since they can be applied in a wide of the form V(x) vT x . variety of tasks, including mechanical engineering, the automotive industry, power The estimations of stability radii of systems, aircraft traffic, and many other switched linear systems and periodically fields. The books (Liberzon, 2003, Sun and switched linear systems were introduced in Ge, 2011) contain reports on various the paper (Nguyen Khoa Son et al., 2020, Do theoretical developments for switched Duc Thuan et al., 2019). The stability radii of systems as well as their applications in some the positive linear system proposed by Son- of these areas. In the mathematical setting, Hinrichsen (see Nguyen Khoa Son et al., such a system, in the case of a linear 1996) has a real stability radius equal to the continuous-time model, can be described by a complex stability radius, while the switched linear time-varying differential equation of positive linear system interested by many the form authors and given conditions stable (see Blanchini et al., 2015, Ding et al., 2011, x ( t ) A  (t) x ( t ), t 0,   (1) Gurvits et al., 2007, Mason et al., 2007, Le Where A : {A Kn×n ,k N},t 0, Van Ngoc et al., 2020, Sun., 2016), have  (t) k studied robust stability in (Le Van Ngoc et al., N : {1, 2, , N} a given family of N matrices 2020). However, with an inevitable limitation, with elements in K, K or and  is a the formula of stability radius has not been set of switching signals  :[0, ) N, which fully studied. In this paper, we seek to provide are piecewise constant right-side continuous a case with a stability radius. functions with points of discontinuity ti , 2. Preliminaries i 1, 2, satisfying  : inf (tt ) 0 . n×m n×m ii 1 Let denote , be a set of all n×m i Among qualitative properties of switched matrices with elements in , respectively. systems, stability and stabilization play a We adopt the notation A0 for the case that a pivotal role and have been most widely matrix A with entries is non-negative. A non- investigated. To mention a few, we refer the negative matrix with at least one positive entry reader to monographs (Liberzon, 2003, Sun is a positive matrix, denoted as (A>0). On the and Ge, 2011), survey papers (Lin and other hand, if all entries of matrix A are Antsaklis, 2009, Shorten et al., 2007), and the positive, then A is strictly positive ( A0). references therein. One of the basic problems Given two matrices A and B, of the same size in stability analysis of switched systems is to A B, A B and AB are synonymous of find conditions guaranteeing A B 0, A B 0 and A B 0, respectively. stability/stabilizability under arbitrary Throughout this article, unless otherwise switching. It has been well established, for n×m instance, that the zero solution of the switched stated, the norm of a matrix A K is linear system (1) is exponentially stable under understood as its operator norm induced by a n arbitrary switching signal   if all given pair of monotonic vector norms on K m subsystems and K that is ‖‖‖‖‖‖A max Axx : 1. 16
3. Dong Thap University Journal of Science, Vol. 10, No. 5, 2021, 15-24 For any matrix A nn , the spectral (i) (Perron-Frobenius) (A) is an abscissa of A is denoted by eigenvalue of A, and there exists a non-  A =R max e  :   ( A) , where negative eigenvector x 0 such that  (A): z : det(zI -A) 0 is the set of Axx = (A) . n  n×n all eigenvalues of A. A matrix A is (ii) Given , there exists a nonzero called Hurwitz if and only if  A. <0 A vector x 0 such that Axx if and only if matrix A nn is called a Metzler matrix if  (A) . all off-diagonal elements of A are non- (iii) (tI -A) 1 exists and is non-negative if negative. Consider a linear continuous-time n system in Kn the form: and only if t (A). The following result is immediate from x()At x t; x Kn ; t 0. (3) Theorem 2.1. Assume that the system (3) is Hurwitz nn Theorem 2.2. Let A be a stable that is  (0A) and is subjected to Metzler matrix. Then the following structured affine perturbations of the form: statements are equivalent: xt( ) (A+DΔE)x t ; x Kn ; t 0, (4) (i) (A) 0; n (ii) A0 p for some pp , 0; where ΔK lq× is unknown disturbance matrix, DK,EK nq× l × n are given matrices (iii)  is invertible and A 1 0. defining the structured perturbations. We have The following lemma is reused in that section. the well-known notion of structured stability radius (Hinrichsen and Pritchard., 1986) which Lemma 2.1 (Do Duc Thuan et al., 2019, is defined, for K= ; , as: Lemma 2.4). Let ,,   be given positive numbers, and l×q rK A;D;E : inf{ :ΔK, :(,) x y 2 :2 xy x  y  0,0,0 x y A + DΔE not Hurwitzstable}. (5) { }. Then, In particular, as the equation is shown in  (Nguyen Khoa Son and Hinrichsen, 1996), if  2  if  2  2 , 2 nn A is a Metzler Hurwitz stable and  2 nl qn min {xy } if  2   and   , (,)xy  DE ;, then formula (5) is   2 computed as follow: if  2   and   , 1 r(A,D,E) r ( A , D , E ) . (6) where  max{ ,  }. EA-1 D The following result is immediate from We repeat the following two theorems Lemma 2.2. about Metzler matrices. Lemma 2.2. Let , be given Theorem 2.1. (Nguyen Khoa Son et al., nonnegative numbers,  0 and nn 1996, Proposition 1). Suppose that A is :(,) {x y 2 :2 xy x  y  0,0,0 x y }, a Metzler matrix. Then set  max{ ,  }. Then, 17
4. Natural Sciences issue  linear system described by (1). If there exists  2  if  2  2; ,  0, 2 n vv ,0such that:  if  2  2 and   0,  A kv 0,  k N, (9)  then the switched positive linear system (1) is if  2  2 and   0, exponentially stable. Given Theorem 2.2 (ii), min {xy } (,)xy   the preceding result immediately implies. 2. if  0, 2 Corollary 2.1. If there exists a Hurwitz 2 2 if  0 and 2  0, stable Metzler matrix A0 such that: 2  Ak0 A ,  k N, (10) 2 if 0 and 2   2 0. 2 then the switched positive linear system (1) is Consider a continuous-time switched positive exponentially stable for any  . linear system in n described by (1). This Theorem 2.3. (Ding et al., 2011, Theorem ensures that if xx(0) belongs to the positive 0 4.2). Let A (aa1 ),A ( 2 ) 2 2 , ij, 1,2 orthant n , then for any switching signal 12ij ij be Metzler and Hurwitz matrices. The   the system (1) admits a unique solution following statements are equivalent. x t, x , , t 0. 0 (i) The switched positive linear system (1) Definition 2.1 (Ding et al., 2011, is exponentially stable. Definition 2.4). The switched linear system (1) (ii) The switched positive linear is said to be positive if x0 0 implies that subsystems (2) have a common linear copositive Lyapunov function (CLCLF). x t, x0 , 0 for all t 0. Definition 2.2 (Blanchini et al., 2015, (iii) There exists a diagonal matrix P > 0 T Definition 3.1). The switched positive linear such that Ak P+PAk 0 for k = 1, 2. system (1) is said to be exponentially stable if a1 a 2 a 2 a 1 there exist real constants M 0 and  0 (iv) 11 21 0, 11 21 0. a1 a 2 a 2 a 1 such that all the solutions of (1) satisfy 12 22 12 22 t x (,,), t x 00  Me x (7) We prove the following lemma. for every xt n ,0 and for all switching 22 0 Lemma 2.6. Let A be Metzler signal  . Hurwitz stable. Then, aii 0,i 1,2. Lemma 2.3 (Ding et al., 2011, Lemma 2.3). The switched positive linear system (1) is Proof. Consider is the determinant det(A-λI) positive if and only if A,k kN are Metzler matrices. aa  11 21 Lemma 2.4 (Blanchini et al., 2015, aa12 22  Definition 3.1). Consider a switched positive 2 linear system described by (1). If there exists  (a11 a 22 ) ( a 11 a 22 a 12 a 21 ). n vv ,0 such that: Assume det(A-λI) 0 , we have T v A k 0,  k N, (8) aa (a a )2 4( a a a a ) then the positive switched linear system (1) is  (A) 11 22 11 22 11 22 12 21 exponentially stable. 1, 2 22 Lemma 2.5 (Nguyen Khoa Son et al., Since A is Metzler Hurwitz stable, then  0 and 2020, Lemma 2). Consider a switched positive 1, 2 18
5. Dong Thap University Journal of Science, Vol. 10, No. 5, 2021, 15-24 22the system. To deal with this question, let us (a11 a 22 ) 4( a 11 a 22 a 12 a 21 ) ( a 11 a 22 ) 4 a 12 a 21 0. We obtain measure the size of perturbations : ( , , , ) l1 ×q 1 l 2 ×q 2 lNN ×q  aa 0 1 2N + + + by 1 2 11 22 N  a a a a 0 : Δ. (13) 1 2 11 22 12 21 S  k k=1 a a ( a a )2 4( a a a a ) 0 11 22 11 22 11 22 12 21 Definition 3.1. If the switched positive linear system (1) is exponentially stable and is a a ( a a )2 4( a a a a ) 0 (Obviousl y ). 11 22 11 22 11 22 12 21 subjected to affine perturbations of the form aa11 22 0 (12). Then its structured stability radius is Equivalently, . a11 a 22 a 12 a 21 defined as: r ( , , ) : inf{ : Δ=(Δ ,Δ , ,Δ ),Δ lkk ×q ,k N, Since A is Metzler Hurvitz stable, then S 1 2 N k +   such that (12) is not exponentially st a ble} 14 a12 0, a 21 0, a 11 a 22 a 12 a 21 0. where the norm . of perturbations is aa11 22 0 S We have . defined by (13). If D =E =I, k N (the case of aa11 22 0 kk unstructured perturbations) then we put This implies aa11 0, 22 0. The proof is rr( ) ( ,I,I). completed. Now, we will use Corollary 2.1 to get a 3. Stability radius of switched positive more explicit formula for computing a lower linear systems bound of the real stability radius r (,,) In this section, consider a continuous-time of the switched positive linear systems (1) with switched positive linear system in n a Metzler Hurwitz stable matrix. described by (1). We assume that the switched Theorem 3.1. Assume that the switched positive linear system (1) is exponentially positive linear system (1) is exponentially stable A , k N stable and the matrices k of positive and is subjected to affine perturbations of the subsystems (2) are subjected to affine positive form (11). Moreover, if there exists a Hurwitz perturbations of the form: nn stable Metzler matrix A0 such that: A k A k :=A k +D k Δ k E k , k N, (11) A A ,  k N. k0 (15) n×lk where : {Dkk , D }, Then the stability radius (14) satisfies the : {E , Eqk ×n }, kN are given matrices k k + following inequality: defining the structure of the perturbations and 1 rr( , , ) min (Ak ,D k ,E k ) , (16) lkk ×q -1 kN k+ , k N are unknown disturbances. max Ei A 0 D j i, j N Then the perturbed system is described by 1 where, for each kN , r A ,D ,E x ( t ) A x ( t ), t 0,   , (12) k k k -1  (t) EADkk k lqkk  (t) A k +D kΔE,Δ k k k ,k N, t 0. is the stability radius of the positive subsystems (2). An important question arising in the Proof. To prove the upper bound, assume robustness analysis of stability for the nominal to the contrary that system (1) subjected to parameter perturbations rr( , , ) min (Ak ,D k ,E k ). kN is how large perturbations Δk ,k N are It follows that there exists kN such that allowable in the perturbed perturbation (12), 0 without destroying the exponential stability of rr( , ,) (A,D,E). k0 k 0 k 0 19
6. Natural Sciences issue By definition (5) of structured stability stable under structured perturbation (12). Then radius, we can choose a positive perturbation the stability radius is Δ=Δ such that rr(,,) ‖‖ (A,D,E) r( , , ) min{ r12 , r }. (18) k0 k0k 0 k 0 k 0 then the subsystem x( t ) A x ( t ), t 0 is not k0 Proof. Using Theorem 2.3, item (iv), the exponentially stable. This implies, however, perturbed system (11) is not exponentially that the perturbed switched linear system (11) stable if and only if is not exponentially stable under the switching a1 d 1 1 e 1 a 2 d 2 2 e 2 11 11 11 11 21 21 21 21 signal  (t ) k0 , t 0 contradicting the 1 1 1 1 2 2 2 2 0 a d e a d e definition of stability radius (14). 12 12 12 12 22 22 22 22 a2 d 2 2 e 2 a 1 d 1 1 e 1 To prove the lower bound, based on the or 11 11 11 11 21 21 21 21 0. a2 d 2 2 e 2 a 1 d 1 1 e 1 article by Le Van Ngoc et al. (2020), Theorem 12 12 12 12 22 22 22 22 3.1, we also have the results in which is S Equivalently Δ replaced by max . The theorem is proved. 1 1 1 1 2 2 2 2 (a11 d 11 11 e 11 )( a 22 d 22 22 e 22 ) Remark 3.1. In case the two dimensions 1 1 1 1 2 2 2 2 (a12 d 12 12 e 12 )( a 21 d 21 21 e 21 ) 0; A ()ak 22 ,k 1, 2 are Metzler Hurvitz k ij (a2 d 2 2 e 2 )( a 1 d 1 1 e 1 ) stable matrices and D(), d k 2 lk 11 11 11 11 22 22 22 22 k ij 2 2 2 2 1 1 1 1 k q 2 (a d e )( a d e ) 0 E() e k are given matrices defining 12 12 12 12 21 21 21 21 k ij the structure of positive perturbations. We or equivalently, measure the size of the positive structured 1 2 1 2 2 2 2 1 1 1 l ×q l ×q a11 a 22 a 11 d 22 e 22 22 a 22 d 11 e 11 11 perturbations :(,), 1 1 2 2 1 2 + + 112212 12 1222 d11 e 11 d 22 e 22 11  22 () a 12 a 21 a 12 d 21 e 21  21 k lkk ×q k+ ();ij k, i , j 1,2 identified by 2 11 1 1 1 2 2 1 2 (a21 d 12 e 12 12 d 12 e 12 d 21 e 21  12  21) 0; a2 a 1 a 2 d 1 e 1 1 a 1 d 2 e 2 2 ‖‖‖‖‖‖ S : 12 . (17) 11 22 11 22 22 22 22 11 11 11 2 2 1 1 2 12 1 1 2 2 2 d11 e 11 d 22 e22 11  22 ()a 12 a 21 a 21 d 12 e 12  12 Put r1 min ‖‖S such that‖‖ 12 0, ‖‖ 0 , 2 1 1 1 1 1 2 2 1 2 1 1 2 2 1 1 2 2 ()a12 d 21 e 21 21 d 21 e 21 d 12 e 12  212  1 0. ()d11 e 11 d 22 e 22 d 12 e 12 d 21 e 21‖‖ 1.‖‖ 2 + nl k qk ×n lkk ×q 2 11 2 1 1 Since DE;kk , + Δk+ and +(a21 d 12 e 12 a 22 d 11 e 11)‖ 1 ‖ + 1 2 2 1 2 2 1 2 1 2 Ak , k=1,2 are Metzler Hurwitz matrices the +(a12 d 21 e 21 a 11 d 22 e 22 )‖‖ 2-( a 11 a 22 a 12 a 21) 0and k k aii 0 and aij 0,  i j 1,2. r2 min ‖‖S such that ‖‖ 1 0, ‖‖ 2 0, 2 2 1 1 1 1 2 2 Then ()d11 e 11 d 22 e 22 d 21 e 21 d 12 e 12‖‖ 1.‖‖ 2 + a1 a 2 a 1 d 2 e 2 2 a 2 d 1 e 1 1 +(a2 d 1 e 1 a 2 d 1 e 1 )‖‖ + 11 22 11 22 22 22 22 11 11 11 12 21 21 11 22 22 1 112212 12 1222 1 2 2 1 2 2 2 1 1 2 d11 e 11 d 22 e 22 11  22 a 12 a 21 a 12 d 21 e 21  21 (a21 d 12 e 12 a 22 d 11 e 11 )‖ 2‖ - ( a 11 a 22 a 12 a 21) 0. 2 11 1 1 1 2 2 1 2 a21 d 12 e 12 12 d 12 e 12 d 21 e 21  12  21 0; Then, we obtain the formula of the stable 2 1 2 1 1 1 1 2 2 2 radius of the system (1) by the following theorem. a11 a 22 a 11 d 22 e 22 22 a 22 d 11 e 11 11 2 2 1 1 2 121 1 2 2 2 Theorem 3.2. Assume that the switched d11 e 11 d 22 e 2211  22 a 12 a 21 a 21 d 12 e 12  12 2 1 1 1 1 1 2 2 1 2 positive linear system (1) is exponentially a12 d 21 e 21 21 d 21 e 21 d 12 e 12  21  12 0. 20
7. Dong Thap University Journal of Science, Vol. 10, No. 5, 2021, 15-24 By using matrix norm, we have: Case 1: Find r1 min ‖‖S such that 1 2 2 1 1 1 2 2 ‖‖ 12 0, ‖‖ 0 and a11 a 22 a 22 d 11 e 11‖‖‖‖ 1 a 11 d 22 e 22 2 - 1 1 2 2 1 1 2 2 1 1 2 2 1 2 1 2 2 ()d11 e 11 d 22 e 22 d 12 e 12 d 21 e 21‖‖ 1.‖‖ 2 + -d11 e 11 d 22 e 22‖‖ 1.‖‖‖‖ 2 a 12 a 21 a 12 d 21 e 21 2 - 2 11 2 1 1 2 11 1 1 2 2 +(a21 d 12 e 12 a 22 d 11 e 11)‖ 1 ‖ + -a21 d 12 e 12‖‖ 1- d 12 e 12 d 21 e 21‖‖ 1.‖‖ 2 0; 1 2 2 1 2 2 1 2 1 2 2 1 2 1 1 1 2 2 +(a12 d 21 e 21 a 11 d 22 e 22 )‖‖ 2-( a 11 a 22 a 12 a 21) 0. It a11 a 22 a 11 d 22 e 22‖‖ 1+ a 22 d 11 e 11‖‖ 2 - 2 2 1 1 1 2 1 2 2 is divided into two claims: -d11 e 11 d 22e22‖‖ 1.‖‖ 2- a 12 a 21 a 21 d 12 e 12‖‖ 2 - 1 1 2 2 1 1 2 2 2 1 1 1 1 2 2 d11 e 11 d 22 e 22 d 12 e 12 d 21 e 21 0 -a12 d 21 e 21‖‖ 1- d 21 e 21 d 12 e 12‖‖ 1.‖‖ 2 0. 1 1 2 2 1 1 2 2 Equivalently or d11 e 11 d 22 e 22 d 12 e 12 d 21 e 21 0 1 1 2 2 1 1 2 2 d11 e 11 d 22 e 22 d 12 e 12 d 21 e 21 ‖‖ 1.‖‖ 2 + 1 1 2 2 1 1 2 2 Claim 1: d12 e 12 d 21 e 21 d 11 e 11 d 22 e 22 0, 2 11 2 1 1 we get a21 d 12 e 12 a 22 d 11 e 11 ‖‖ 1 + 2 11 2 1 1 (a21 d 12 e 12 a 22 d 11 e 11)‖‖ 1 + 1 2 2 1 2 2- 1 2 1 2 a12 d 21 e 21 a 11 d 22 e 22 ‖‖ 2 a 11 a 22 a 12 a 21 0; 1 2 2 1 2 2 1 2 1 2 (a12 d 21 e 21 a 11 d 22 e 22 )‖‖ 2-( a 11 a 22 a 12 a 21) 0. 2 2 1 1 1 1 2 2 d e d e d e d e ‖‖ .‖‖+ Because (a2 d 1 e1 a 2 d 1 e 1 ) or 11 11 22 22 21 21 12 12 1 2 21 12 12 22 11 11 2 1 1 2 1 1 ()a1 d 2 e 2 a 1 d 2 e 2 are not simultaneously + a12 d 21 e 21 a 11 d 22 e 22 ‖ 1 ‖ + 12 21 21 11 22 22 zero. a1 d 2 e 2 a 1 d 2 e 2‖‖ - a 2 a 1 a 1 a 2 0. 21 12 12 22 11 11 2 11 22 12 21 2 1 1 2 1 1 - If a21 d 12 e 12 a 22 d 11 e 11 0 then According to the assumption that the switched a1 a 2 a 1 a 2 positive linear system (1) is exponentially r 11 22 12 21 . 22 1 1 2 2 1 2 2 stable with A,A12 being Metzler a12 d 21 e 21 a 11 d 22 e 22 Hurwitz stable matrices, then aa11 0, 0, 1 2 2 1 2 2 11 22 - If a12 d 21 e 21 a 11 d 22 e 22 0 then 22 aa 0, 0, 1 2 1 2 11 22 a11 a 22 a 12 a 21 r1 . a2 d 1 e1 a 2 d 1 e 1 a1 a 2 a 2 a 1 21 12 12 22 11 11 11 21 0, 11 21 0, 1 2 2 1 - If a2 d 1 e1 a 2 d 1 e 1 0 and a12 a 22 a 12 a 22 21 12 12 22 11 11 1 2 2 1 2 2 k a d e a d e 0 then akij 0, 1,2, ij 1,2 and D,D,E,E1 2 1 2 12 21 21 11 22 22 are given non-negative definite matrices. a1 a 2 a 1 a 2 a 1 a 2 a 1 a 2 r min11 22 12 21 , 11 22 12 21 . We have 1 122 122 211 211  ade12 21 21 ade 11 2222 ade 21 12 12 ade 22 1 1 11  1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 d12 e 12 d 21 e 21 d 11 e 11 d 22 e 22 0, Claim 2: d12 e 12 d 21 e 21 d 11 e 11 d 22 e 22 0 , we get 2 11 2 1 1 a2 d 1 e1 a 2 d 1 e 1 a21 d 12 e 12 a 22 d1 1 e 11 0, 21 12 12 22 11 11 ‖‖‖‖‖‖ 1 2 1 d1 e 1 d 2 e 2 d 1 e 1 d 2 e 2 a1 d 2 e 2 a 1 d 2 e 2 0, a 1 a 2 a 1 a 2 0, 11 11 22 22 12 12 21 2 1 12 21 21 11 22 22 12 21 11 22 1 2 2 1 2 2 a12 d 21 e 21 a 11 d 22 e 22 1 1 2 2 2 2 1 1 ‖‖ 2 - d21 e 21 d 12 e 12 d 11 e 11 d 22 e 22 0, 1 1 2 2 1 1 2 2 d11 e 11 d 22 e 22 d 12 e 12 d 21 e 21 a2 d 1 e 1 a 2de 1 1 0, 12 21 21 11 22 22 a1 a 2 a 1 a 2 - 11 22 12 21 0. 1 2 2 1 2 2 1 2 2 1 1 1 2 2 1 1 2 2 a21 d 12 e 12 a 22 d 11 e 11 0, a 12 a 21 a 11 a 22 0. d11 e 11 d 22 e 22 d 12 e 12 d 21 e 21 21
8. Natural Sciences issue 1 2 2 1 2 2 Using Lemma 2.2, we have r1 min{ x y } a d e a d e 21 12 12 22 11 11 ‖‖ - such that 2xy x  y  0, x 0, y 0, 1 1 2 2 2 2 1 1 2 d21 e 21 d 12 e 12 d 11 e 11 d 22 e 22 where xy ‖‖‖‖ ;; a2 a 1 a 1 a 2 12 - 11 22 12 21 0. 1 1 2 2 2 2 1 1 a2 d 1 e1 a 2 d 1 e 1 d21 e 21 d 12 e 12 d 11 e 11 d 22 e 22 21 12 12 22 11 11 0; 2 1 1 2 2 1 1 2 2 Using Lemma 2.2, we have r min{ x y } d12 e 12 d 22 e 22 d 12 e 12 d 21 e 21 2 such that 2xy x  y  0, x 0, y 0,  a1 d 2 e 2 a 1 d 2 e 2 12 21 21 11 22 22 0; 1 1 2 2 1 1 2 2 where, xy ‖‖‖‖ 12;; 2 d12 e 12 d 22 e 22 d 12 e 12 d 21 e 21 1 2 1 2 2 1 1 2 1 1  a a a a a12 d 21 e 21 a 11 d 22 e 22 11 22 12 21 0 . 0; 1 1 2 2 1 1 2 2 2 d1 e 1 d 2 e 2 d 2 e 2 d 1 e 1 2 d12 e 12 d 22 e 22 d 12 e 12 d 21 e 21 21 21 12 12 11 11 22 22  a1 d 2 e 2 a 1 d 2 e 2 Case 2: Find r2 min ‖‖S such that 21 12 12 22 11 11 0, 2 1 1 2 2 2 2 1 1 0, 0 d21 e 21 d 12 e 12 d 11 e 11 d 22 e 22 ‖‖ 12 ‖‖ and  a2 a 1 a 1 a 2 d1 e 1 d 2 e 2 d 2 e 2 d 1 e 1 ‖‖ .‖‖ + 11 22 12 21 0 . 21 21 12 12 11 11 22 22 1 2 1 1 2 2 2 2 1 1 2 d21 e 21 d 12 e 12 d 11 e 11 d 22 e 22 a2 d 1 e 1 a 2 d 1 e 1 ‖‖ 12 21 21 11 22 22 1 Combining the above two cases, we come a1 d 2 e 2 a 1 d 2 e 2‖‖- a 2 a 1 a 1 a 2 0. to a formula r( , , ) min{ r , r }. 21 12 12 22 11 11 2 11 22 12 21 12 It is divided into two parts. Example 3.1. Consider the switched 1 1 2 2 2 2 1 1 positive linear system (1) with N={1, 2}, Claim 1: d21 e 21 d 12 e 12 d 11 e 11 d 22 e 22 0,we get subjected to affine perturbations of the form a2 d 1 e 1 a 2 d 1 e 1 ‖‖ 12 21 21 11 22 22 1 (11), where 1 2 2 1 2 2 2 1 1 2 a21 d 12 e 12 a 22 d 11 e 11 ‖‖ 2- a 11 a 22 a 12 a 21 0. 2.02 1.01 0.11 1.21 2 1 1 2 1 1 1 2 2 1 2 2 A 1.13 1.03 0.12 , D 0.13 , Because a d e a d e or a d e a d e 11 12 21 21 11 22 22 21 12 12 22 11 11 are not simultaneously zero. 0.01 0.12 2.1 0.02 2 1 1 2 1 1 E 0.01 1.03 1.01 ; - If a12 d 21 e 21 a 11 d 22 e 22 0 then 1 2 1 1 2 a11 a 22 a 12 a 21 1.04 1.13 0.02 1.32 r2 . 1 2 2 1 2 2 A 0.01 2.01 1.01 , D 1.21 , a21 d 12 e 12 a 22 d1 1 e 11 22 1 2 2 1 2 2 1.01 0.12 2.05 0.01 - If a12 d 21 e 21 a 11 d 22 e 22 0 then a2 a 1 a 1 a 2 E 1.02 1.05 0.03 . r 11 22 12 21 . 2 2 2 1 1 2 1 1 a12 d 21 e 21 a 11 d2 2 e 22 It is easy to verify that the Metzler matrices 2 1 1 2 1 1 A ,k=1, 2 are Hurwitz stable and - If a12 d 21 e 21 a 11 d 22 e 22 0 and k a1 d 2 e 2 a 1 d 2 e 2 0 then 1.04 1.13 0.11 12 21 21 11 22 22 2 1 1 2 2 1 1 2 A0 1.13 1.03 1.01 . a11 a 22 a 12 a 21 a 11 a 22 a 12 a 21 r2 min , . 1.01 0.12 2.05 ade122 ade 122 ade 211 ade 211 21 12 12 22 11 11 12 21 21 11 22 22  From Theorem 3.1, we obtain Claim 2: 1 1 2 2 2 2 1 1 , we get d21 e 21 d 12 e 12 d 11 e 11 d 22 e 22 0 0.1788 r (,, ) 0.2350. 2 1 1 2 1 1 a12 d 21 e 21 a 11 d 22 e 22 ‖‖‖‖‖‖ 1 2 1 Example 3.2. Consider the switched d1 e 1 d 2 e 2 d 2 e 2 d 1 e 1 21 21 12 12 11 11 22 22 positive linear system (1) with 22
9. Dong Thap University Journal of Science, Vol. 10, No. 5, 2021, 15-24 subjected to affine perturbations of the form we obtained a formula of the stability radius by (11), where estimating the positive real stability radius. 0.5 0 0.2 Some examples are provided for illustrating A1 , D 1 , E 1 0.1 0.2 ; the result. Our future work is on the formulas 0.2 0.3 0.3 of stability radius for multi-dimensional 0.1 0.2 0.2 switched positive linear systems with multiple A , D , E 0.3 0.1 . 2 2 2 switching signals. 0 0.4 0.1 k References Since a 0,  i j 1,2;k 1,2 and ij (A ) 0.3, (A ) 0.5; Blanchini, B., Colaneri, P., and Valcher, M.E. 121 1 (2015). Switched positive linear systems. (A ) 0.1, (A ) 0.4. 1 2 2 2 Foundations and Trends in Systems and The switched positive linear system (1) is Control, 2, 101-273. exponentially stable, because Ding, X., Shu, L., and Wang, Z. (2011). On 12 aa11 12 0.5 0 stability for switched linear positive 0.2 0, systems. Mathematical and Computer aa12 0 0.4 21 22 Modelling, 47, 1044-1055. aa21 0.1 0 11 12 0.04 0. Gurvits, L., Shorten, R., and Mason, O. (2007). aa21 0.2 0.4 21 22 On the stability of switched positive linear We have. systems. IEEE Transactions on Automatic 1 1 2 2 1 1 2 2 Control, 52, 1099-1103. Case 1: d12 e 12 d 21 e 21 d 11 e 11 d 22 e 22 0, 2 11 2 1 1 1 2 2 1 2 2 a21 d 12 e 12 a 22 d 11 e1 1 0.08 0, a12 d 21 e 21 a 11 d 22 e 22 0 Horn, R.A., and Johnson,C.R. (1985). Matrix Analysis. Cambridge University Press: and a1 a 2 a 1 a 2 0.2 then 11 22 12 21 Cambridge. a1 a 2 a 1 a 2 0.2 r 11 22 12 21 2.5. Hinrichsen, D., and Pritchard, A.J. (1986). 1 2 1 2 2 1 1 0.08 a21 d 12 e 12 a 22 d 11 e 11 Stability radii of linear systems. Systems 1 1 2 2 2 2 1 1 Case 2: d21 e 21 d 12 e 12 d 11 e 11 d 22 e 22 0, & Control Lett, 8, 105-113. 2 1 1 2 1 1 1 2 2 1 2 2 a12 d 21 e 21 a 11 d 22 e 22 0, a21 d 12 e 12 a 22 d 11 e 11 0.018 0 Liberzon, D. (2003). Switching in systems and 2 1 1 2 control. Birkhauser: Boston. and a11 a 22 a 12 a 21 0.03 then a2 a 1 a 1 a 2 0.03 Lin, H., and Antsaklis, P.J. (2009). Stability r 11 22 12 21 1.6667. 2 2 1 1 2 1 1 and stabilizability of switched linear a12 d 21 e 21 a 11 d 22 e 22 0.018 systems: A survey of recent results. IEEE Using Theorem 3.2, we obtain Trans. Automat. Control, 54, 308-332. r( , , ) min{ r , r } 1.6667. 12 Mason, O., and Shorten, R. (2007). On linear 4. Conclusion copositive Lyapunov functions and the In this paper, based on conditions of stability of switched positive linear exponential stability of the switched positive systems. IEEE Trans. Automat.Control, linear system and the concept of a stability 52, 1346-1349. radius related to the structured affine of a Le Van Ngoc and Le Nhat Nam. (2020). On matrix of the subsystem, we propose a new the stability radius of switched positive approach to studying the robustness of a linear linear systems. Journal of Military system. In the case of a two-dimensional Science and Technology, Special Issue, switched system with two switching signals, 69A(11), 90-98. 23
10. Natural Sciences issue Shorten, R.,Wirth, F., Mason, O., Wulff, K., Do Duc Thuan and Le Van Ngoc. (2019). and King, C. (2007). Stability criteria for Robust stability and robust stabilizability switched and hybrid systems. SIAM for periodically switched linear systems. Review, 47, 545-592. Applied Mathematics and Computation, Nguyen Khoa Son and Hinrichsen, D. (1996). 15, 112-130. Robust stability of positive continuous- Sun, Z., and Ge, S.S. (2011). Stability theory time systems. Numerical functional of switched dynamical systems. Springer analysis and optimization, 17, 649-659. Verlag: London. Nguyen Khoa Son and Le Van Ngoc. (2020). Sun, Y. (2016). Stability analysis of positive On robust stability of switched linear switched systems via joint linear systems. IET Control Theory & copositive Lyapunov functions. Nonlinear Applications, 14, 19-29. Anal. Hybrid Syst, 19, 146-152. 24