Robust stability of positive linear systems under time varying perturbations

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Numerical Functional Analysis and Optimization

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Robust Stability of Positive Linear Systems Under Time-Varying 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

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ISSN: 0163-0563 print/1532-2467 online

DOI: 10.1080/01630563.2013.816319

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.

1 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

and references therein In particular, problems of robust stability of the linear time-invariant differential system

under time-invariant perturbations of the form

A  A +

N

i=1

D i
i E i, (multi-perturbation), (3)

and

A  A +

N

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

A  A +

N

k=1

D k (t )
k (t )E k (t ) (time-varying multi-perturbation) (5)

A  A +

N

k=1

k (t )A k (t ) (time-varying affine perturbation) (6)

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 time-varying 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

2 PRELIMINARIES Let
be the set of all natural numbers For given m ∈
, let us denote m := 1, 2,    , m and m0:= 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 = (a ij ) and B = (b ij) inł×q, we write A ≥ B iff a ij ≥ b ij for i =

1,    , l , j = 1,    , q In particular, if a ij > b ij for i = 1,    , l, j = 1,    , q, then we write A  B instead of A ≥ B We denote by  l ×q

+ the set of all

nonnegative matrices A≥ 0 Similar notations are adopted for vectors

For x ∈ n and P ∈ l ×q we define |x| = (|x i |) and |P | = (|p ij|) 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 (xp = (|x1|p + |x2|p + · · · + |x n|p)p1, 1≤

p < ∞ and x∞= maxi =1,2,,n |x i|), 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 onl andq, that isP  =

maxPy : y = 1 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  · 1 or  · ∞ thenA = |A| for any A = (a ij)∈ n ×n More precisely, one has A1 =

|A|1= max1≤j≤nn

i=1|a ij | and A∞= |A|∞= max1≤i≤nn

j=1|a ij|

For any matrix M ∈ n ×n the spectral abscissa of M is denoted by

(M ) = max  :  ∈ (M ), where (M ) := z ∈  : det(zI n − M ) = 0 is the spectrum of M A matrix M ∈ n ×n is said to be Hurwitz stable if

(M ) < 0.

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

(ii) Given ∈ , there exists a nonzero vector x ≥ 0 such that Mx ≥ x if and only if (M ) ≥

(iii) (tI − M )−1 exists and is nonnegative if and only if t > (M ).

(iv) Given B ∈ n ×n

+ , C ∈ n ×n Then

|C| ≤ B ⇒ (M + C) ≤ (M + B)

The following is immediate from Theorem 2.1 and is used in what follows

Theorem 2.2 Let M ∈ n ×n be a Metzler matrix Then the following statements are equivalent

(i) (M ) < 0;

(ii) Mp 0 for some p ∈  n , p  0;

(iii) M is invertible and M−1≤ 0;

(iv) For given b ∈ n , b  0, there exists x ∈  n

+, such that Mx + b = 0 (v) For any x ∈ n

+\ 0, 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, ifn is endowed with the

 = max ()

3 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(zI n

+, 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 ∈

+ for any x0 ∈ n , x0 ≥ 0

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)

3.1 Time-Varying Multi-Perturbations Suppose (1) is exponentially stable Consider perturbed systems of the form

˙x(t) =



A+

N

k=1

D k (t )
k (t )E k (t )



where N is a given positive integer, D k(·) ∈ C(+,n ×l k ), E k(·) ∈ C(+,

q k ×n ), k ∈ N are given and
k(·) ∈ C(+,l k ×q k ) (k ∈ N ) are unknown

perturbations

For fixed ≥ 0 and given x0 ∈ n, (9) has a unique solution satisfying the initial value condition

This solution is denoted by x(·; , x0) Recall that x(·; , x0) is continuously differentiable on [, ∞) and satisfies (9) for any t ∈ [, ∞) Then (9) is

∀x0 ∈ n, ∀ t ≥  ≥ 0: x(t; , x0) ≤ Me x0, see, for example, [10]

perturbed system of the form (9) remains exponentially stable whenever the size of

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

Dk∈ n ×l k

+ , Ek∈ q k ×n

+ and
k ∈ l k ×q k

+ for k ∈ N such that |D k (t )| ≤ D k,

|E k (t )| ≤ E k and|
k (t )| ≤
k for any t ∈ +and any k ∈ N Then (9) remains exponentially stable if

N

k=1

maxi,j ∈N E i(−A)−1Dj (11) Proof We divide the proof into two steps

Step I: We claim that (A+N

k=1Dk
kEk) < 0

Since (1) is positive, it follows that A is a Metzler matrix Thus, (A+

N

k=1D k
k E k ) is also a Metzler matrix because D k , E k,
k are nonnegative

for any k ∈ N We show that 0:= (A +N

k=1D k
k E k) < 0 Assume on the

contrary that 0 ≥ 0 By the Perron-Frobenius theorem (Theorem 2.1 (i)),

there exists x ∈ n

+, x



A+

N

k=1

Dk
k Ek



x = 0x

Let Q (t ) = tI n − A, t ∈  Since (1) is exponentially stable, (A) < 0 Thus, Q (0) is invertible and this implies

Q (0)−1

N

k=1

Let i0 be an index such that E i0x = maxk ∈N E kx It follows from (12) that E i0x  > 0 Multiply both sides of (12) from the left by E i0, to get

N

k=1

E i0Q (0)−1D k
k E k x = E i0x

It follows that

N

k=1

E i0Q (0)−1Dk
k E kx  ≥ E i0x

Thus,

max

i,j ∈N E i Q (0)−1D j


N

k=1


k



E i0x  ≥ E i0x,

or equivalently,

max

i,j ∈N E i Q (0)−1D j

N

k=1

On the other hand, the resolvent identity gives

Q (0)−1− Q (0)−1= 0Q (0)−1Q (0)−1 (14)

Since A is a Metzler matrix with (A) < 0 and 0 ≥ 0, Theorem 2.1 (iii)

yields Q (0)−1≥ 0 and Q (0)−1≥ 0 It follows from (14) that Q (0)−1≥

Q (0)−1≥ 0 Hence, E i Q (0)−1D j ≥ E i Q (0)−1D j ≥ 0, for any i, j ∈ N By (8),

we haveE i Q (0)−1D j  ≥ E i Q (0)−1D j , for any i, j ∈ N Then (13) implies

N

k=1

maxi,j ∈N E i Q (0)−1D j However, this conflicts with (11)

Step II: Let x0 ∈ n be given and let x(t ) := x(t; , x0), t ∈ [, ∞) be

≥ 0 and any x0 ∈ n with x0 ≤ 1,

By step I, (A+N

k=1Dk
kEk) < 0 and then



A+

N

k=1

D k
k E k



for some p:= ( 1, 2,    , n)T, i > 0,∀i ∈ n, by Theorem 2.2 By continuity,

(16) implies that



A+

N

k=1

Dk
kEk



|x| Kp for any x ∈ n with x ≤ 1 Define u(t) := Ke p, t ∈ [, ∞) Set x(t) := x(t ; , x0), t ≥  Then, we have |x()| u() We claim that |x(t)| ≤ u(t) for any t > .

Assume on the contrary that there exists t0 >  such that|x(t0)|  u(t0) Set

t1 := inft ∈ (, ∞) : |x(t)|  u(t)

By continuity, t1 >  and there is i0 ∈ n such that

|x(t)| ≤ u(t), ∀t ∈ [, t1);

|x i0(t1)| = u i0(t1), |x i0(t )| > u i0(t ), ∀t ∈ (t1, t1+ ), (18) for some  > 0

Let A :=(a ij)∈n ×n , D k
kEk =(b (k)

ij )∈n ×n and let|D k (t )||
k (t )||E k (t )|= (b ij (k) (t ))∈ n ×n, ∀t ≥ 0, for k ∈ N Since A is a Metzler matrix and

Dk
kEk ≥ 0 for k ∈ N , we have for any i ∈ n,

d

dt |x i (t )| = sgn(x i (t )) ˙x i (t ) ≤ a ii |x i (t )| +

n

j aij |x j (t )| +

N

k=1

n

j=1

b ij (k) (t )|x j (t )|,

for almost any t∈ [, ∞) Thus,

d

dt |x i (t )| ≤ a ii |x i (t )| +

n

j

a ij |x j (t )| +

N

k=1

n

j=1

b ij (k) |x j (t )|,

for almost any t ∈ [, ∞) Thus, we have for any t ∈ [, ∞),

D+|x i (t )| := lim sup

h→0 +

|x i (t + h)| − |x i (t )|

h→0 +

1

h

 t +h

t

d

ds |x i (s)|ds

≤ a ii |x i (t )| +

n

j aij |x j (t )| +

N

k=1

n

j=1

b ij (k) |x j (t )|,

where D+ denotes the Dini upper-right derivative In particular, it follows from (17) and (18) that

D+|x i0(t1)|(18)≤ a i0i0Ke 1 −) i0+

n

a i0j Ke 1 −) j+

N

k=1

n

j=1

b i (k)0j Ke 1 −) j

= Ke 1 −)


n

j=1

ai0j j +

N

k=1

n

j=1

b i (k)0j j



(17)

< 1 −) i0= D+ui0(t1)

However, this conflicts with (18) Therefore,

|x(t; , x0)| ≤ u(t) = Ke p, ∀t ∈ [, ∞),

for any ≥ 0 and any x0 ∈ n withx0 ≤ 1 By the monotonicity of vector

norms, there exists K1 > 0 such that

x(t; , x0) ≤ K1e , ∀t ∈ [, ∞),

for any ≥ 0 and any x0 ∈ n with x0 ≤ 1 By linearity of (9),

x(t; , x) ≤ K1e 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), t ≥ 0,

remains exponentially stable whenever


 < EA1−1

D

Furthermore, robust stability of the positive system (1) under the time-invariant 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 multi-perturbations (5) has not yet studied and a result like Theorem 3.2 cannot

be found in the literature

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

˙x(t) =



A+

N

k=1

k (t )A k (t )



Here N ∈
and A k(·) ∈ C(+,n ×n ) (k ∈ N ) are given and  k(·) ∈

C (+,) (k ∈ N ) are unknown perturbations.

Furthermore, we assume that

(H1) ∀k ∈ N , ∃ A k ∈ n ×n

+ : |A k (t )| ≤ A k, ∀t ≥ 0;

(H2) ∀k ∈ N , ∃ k∈ +: |k (t )| ≤  k, ∀t ≥ 0

system of the form (19) remains exponentially stable whenever the size of := (1, 2,    , N)∈ N

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

((−A)−1N

where := maxk ∈N k

Proof We first show that 0 := (A +N

k=1kAk) < 0 Since (1) is

positive, it follows that A is a Metzler matrix Thus, A+N

k=1kAk, is also

a Metzler matrix because A k ≥ 0 and k are nonnegative for any k ∈ N

Assume on the contrary that 0≥ 0 By the Perron-Frobenius theorem

(Theorem 2.1 (i)), there exists x ∈ n

+, x



A+

N

k=1

k A k



x = 0x

3 ROBUST STABILITY OF POSITIVE LINEAR DIFFERENTIAL SYSTEMS

Consider a linear time- invariant differential system of the form (1),

where A∈... 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]... data-page="10">

Furthermore, robust stability of the positive system (1) under the time- invariant multi -perturbations (3) has been analyzed in [9] by techniques

of -analysis However, to the best of our knowledge,