Stability radii of positive linear systems under fractional perturbations

$: Stability radii of positive linear systems under fractional perturbations$
INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust Nonlinear Control 2009; 19:1267–1277Published online 20 October 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/rnc.1383

Stability radii of positive linear systems underfractional perturbations

Bui The Anh1,∗,† and Nguyen Khoa Son2

1Department of Mathematics, University of Pedagogy, 280 An Duong Vuong Str., HoChiMinh City, Vietnam2Institute of Mathematics, 18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam

SUMMARY

In this paper we study stability radii of positive linear discrete-time systems under fractional perturbations.It is shown that real and complex stability radii coincide and can be computed by a simple formula.From the obtained results, we apply to derive estimates and computable formulae for the stability radiiof positive linear delay systems. Finally, a simple example is given to illustrate the obtained results.Copyright q 2008 John Wiley & Sons, Ltd.

Received 18 June 2008; Revised 14 August 2008; Accepted 25 August 2008

KEY WORDS: positive linear system; stability radius; fractional perturbation; nonnegative matrix

1. INTRODUCTION

A dynamic system is called positive if any trajectory of the system starting in the positive orthantRn+ =[0,∞)n remains in Rn+. In the time-invariant linear discrete-time case, such systems aredescribed by equations of the form

x(t+1)= Ax(t), t ∈N (1)

where A is a matrix with nonnegative entries. Positive systems play an important role in themodeling of dynamic phenomena whose variables are nonnegative. This model class is used inmany areas such as economics, populations dynamics, ecology. The mathematical theory of positivesystems is based on the theory of nonnegative matrices founded by Perron and Frobenius. Asreferences we mention [1, 2].

In this paper we study the asymptotic stability of uncertain positive systems of the form (1)where the system matrix A is subjected to fractional perturbations. We recall that the system (1)

∗Correspondence to: Bui The Anh, Department of Mathematics, University of Pedagogy, 280 An Duong Vuong Str.,HoChiMinh City, Vietnam.

†E-mail: bt [email protected]

Copyright q 2008 John Wiley & Sons, Ltd.

1268 B. T. ANH AND N. K. SON

is asymptotically stable if and only if A is Schur stable, i.e. the spectrum of A is contained inthe open unit disk. The stability of positive linear systems is well studied and a number of usefulstability criteria are available, see [1, 3].

Over the past decade, problems of model and parameter uncertainty have attracted a lot ofattention in control theory and this has a renewed interest in robustness issues for positive systems.In the study of these problems, the notion of stability radius was proved to be a very effectivetool, see [4–12]. In its simplest form the problem is to determine the maximal �>0 for which thefamily of systems

x(t+1)=(A+D�E)x(t), ‖�‖<�

is asymptotically stable. � is unknown disturbance matrix, D and E are given matrices definingthe structure of the perturbations. Depending upon whether complex or real disturbances areconsidered, this maximal � is called complex or real stability radius, respectively. It is importantto note that these two stability radii are in general different. The complex stability radius is moreeasily analyzed and computed than the real one. A detailed analysis of the complex stabilityradius of discrete systems can be found in [8, 12], an efficient algorithm is described in [8] andthe difference between the two stability radii is discussed in [8, 9]. The computation of the realstability radius, being a much more difficult problem, has been solved quite recently, see [13].Fortunately, the situation is much simpler for the class of positive systems. It has been shown in[11, 12] that if the system (1) is positive and D,E are nonnegative matrices, then the complex andthe real stability radii coincide and can be computed directly by a simple formula. Some similarresults for positive linear continuous-time systems can be found in [8, 11, 12].

In this present paper we go beyond the restricted class of single perturbations A→ A+D�E .We study the robust stability of the system (1) under fractional perturbations of the form

A→ A+N∑i=1

Di (I −�i Mi )−1�i Ei (2)

where the matrices Di ,Ei ,Mi , i ∈N :={1,2, . . . ,N } are given matrices defining the structure ofthe perturbations and �i , i ∈N are unknown matrices.

The organization of this paper is as follows. In the following section we recall some results onnonnegative matrices and present preliminary results for later use. In Section 3 we study robuststability of positive linear discrete systems under fractional perturbations: first, we establish theestimates for the complex stability radius; second, it is shown that the three stability radii: complex,real and positive radii coincide and can be computed by a simple formula. In Section 4, we applyobtained results to study the robust stability of positive delay systems. Finally, a simple exampleis given to illustrate the obtained results.

2. PRELIMINARIES

We first introduce some notations. Let n, l,q be positive integers, a matrix P=[pi j ]∈Rl×q is saidto be nonnegative (P�0) if all its entries pi j are nonnegative. It is said to be positive (P>0) if allits entries pi j are positive. For P,Q∈Rl×q , P>Q means that P−Q>0. The set of all nonnegative

l×q-matrices is denoted by Rl×q+ . A similar notation will be used for vectors. Let K=C or R, then

Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2009; 19:1267–1277DOI: 10.1002/rnc

STABILITY RADII OF POSITIVE LINEAR SYSTEMS 1269

for any x ∈Kn and P ∈Kl×q , we define |x |∈Rn+ and |P|∈Rl×q+ by |x |=(|xi |), |P|=[|pi j |]. For

any matrix A∈Kn×n , let �(A) denote the spectrum of A, �(A)=C\�(A) the resolvent set of A.The spectral radius and the spectral abscissa of A is defined by r(A)=max{|�| :�∈�(A)} and�(A)=max{�� :�∈�(A)}, respectively.

A norm ‖·‖ on Kn is said to be monotonic if it satisfies

|x |�|y|⇒‖x‖�‖y‖ ∀x, y∈Kn

It can be shown that a vector norm ‖·‖ on Kn is monotonic if and only if ‖x‖=‖|x |‖ for allx ∈Kn , see [11]. It follows that every p-norm on Kn , 1�p�∞, is monotonic. Recall that everyp-norms on Kn are defined by

‖x‖p =

⎧⎪⎪⎨⎪⎪⎩

(n∑

i=1|xi |p

)1/p

, 1�p<∞

max1�i�n

|xi |, p=∞

All norms on Kn we use in this paper are assumed to be monotonic. Throughout this paper ‖P‖denotes the operator norm of P ∈Kl×q induced by a given pair of norms on Kq and Kl .

‖P‖= max‖y‖=1

‖Py‖

We recall some useful results, see [1].Lemma 2.1Suppose that P ∈R

l×q+ and ‖·‖ denotes the operators norm with respect to monotonic norms on

Rl ,Rq . Then there exists e∈Rq+ of norm ‖e‖=1 such that

‖Pe‖=‖P‖Theorem 2.2 (Perron–Frobenius)Suppose that A∈Rn×n+ . Then,

(i) r(A) is an eigenvalue of A and there is a nonnegative eigenvector x�0, x =0 such thatAx=r(A)x .

(ii) If �∈�(A) and |�|=r(A) then, the algebraic multiplicity of � is not greater than thealgebraic multiplicity of the eigenvalue r(A).

(iii) Given �>0, there exists a nonzero vector x�0 such that Ax��x if and only if r(A)��.(iv) (t I −A)−1 exists and is nonnegative if and only if t>r(A).

3. STABILITY RADII OF POSITIVE LINEAR DISCRETE-TIME SYSTEMS

We consider a dynamic system described by the following linear difference equation:

x(t+1)= Ax(t), t ∈N (3)



where A∈Cn×n . We assume that the system (3) is (Schur) stable, i.e. the spectrum �(A) of Alies in the open unit disk D={s∈C : |s|<1}. We assume that the matrix A is subjected to theperturbations of the following type:

A→ A� := A+N∑i=1

Di (I −�i Mi )−1�i Ei (4)

where the matrices Di ∈Cn×li ,Ei ∈Cqi×n,Mi ∈Cqi×li , i ∈N are given matrices defining the struc-ture of the perturbations and �i ∈Cli×qi , i ∈N are unknown matrices. Now setting

D=[D1 . . . DN ], �=

⎡⎢⎢⎢⎣

�1 0

. . .

0 �N

⎤⎥⎥⎥⎦ , M=

⎡⎢⎢⎢⎣M1 0

. . .

0 MN

⎤⎥⎥⎥⎦ , E=

⎡⎢⎢⎢⎣E1

...

EN

⎤⎥⎥⎥⎦

we see that perturbations (4) can be written in the following form:

A→ A� = A+D(I −�M)−1�E

Associated with perturbations (4) we define the transfer functions Gi j :�(A)→Cqi×l j as follows:

Gi j (s)={Ei (s I −A)−1Dj , i = j

Ei (s I −A)−1Di +Mi , i= j

for all i, j ∈N and we set G(·)=[Gi j (·)]i, j∈N ∈Cq×l where q=∑Ni=1 qi , l=

∑Ni=1 li .

Assume that system (3) is stable. Then, by maximal principle we have

sup|s|�1

‖Gi j (s)‖= sup|s|=1

‖Gi j (s)‖ for all i, j ∈N (5)

Definition 3.1Let system (3) be stable. The complex, real and positive stability radii of system (3) underperturbations of the form (4) can be defined as

rC(A;D,E,M) = inf

{N∑i=1

‖�i‖:�i ∈Cli×qi , i ∈N either r(A�)�1 or (∃i ∈N :1∈�(�i Mi ))

}

rR(A;D,E,M) = inf

{N∑i=1

‖�i‖:�i ∈Rli×qi , i ∈N either r(A�)�1 or (∃i ∈N :1∈�(�i Mi ))

}

r+(A;D,E,M) = inf

{N∑i=1

‖�i‖:�i ∈Rli×qi+ , i ∈N either r(A�)�1 or (∃i ∈N :1∈�(�i Mi ))

}

respectively, where we set inf ∅=+∞.

Before studying the robust stability of system (3) under perturbations of the form (4), we needthe following lemmas.



Lemma 3.2Let �i ∈Cli×qi for all i ∈N . Then, one has

inf

{N∑i=1

‖�i‖:∃i ∈N ,1∈�(�i Mi )

}= 1

maxi∈N ‖Mi‖We set inf ∅=+∞.

ProofThe proof is not difficult thus, we omit it. �

Lemma 3.3 (Hinrichsen and Pritchard [8])Let �i ∈Cli×qi such that 1∈�(�i Mi ) for all i ∈N and s∈�(A). Then,

s∈�(A�) ⇔ 1∈�(G(s)�)

Proposition 3.4Let system (3) be stable,�i ∈Cli×qi such that 1∈�(�i Mi ) for all i ∈N and s∈C, |s|�1. If s∈�(A�)

then,

N∑i=1

‖�i‖� 1

maxi, j∈N sup|s|=1 ‖Gi j (s)‖ProofLet s∈C, |s|�1 such that s∈�(A�). Owing to Lemma 3.3, 1∈�(G(s)�). Therefore, there existsa vector x=(xi )i∈N , xi ∈Cqi , i ∈N satisfying x=G(s)�x , or equivalently,

N∑j=1

Gi j� j (s)x j = xi for all i ∈N

Assume that there exists an index k∈N such that ‖xk‖=max{‖xi‖: i ∈N }. Then, from the followingequations:

N∑j=1

Gkj (s)� j x j = xk

and (5) implies that

‖xk‖�N∑j=1

‖Gkj (s)‖‖� j‖‖x j‖�N∑j=1

‖� j‖ maxi, j∈N

sup|s|=1

‖Gi j (s)‖‖xk‖

This follows that

N∑i=1

‖�i‖� 1

maxi, j∈N sup|s|=1 ‖Gi j (s)‖The proof is complete. �



Thanks to Lemma 3.2 and Proposition 3.4 we deduce that

rC(A;D,E,M)� 1

max{maxi∈N ‖Mi‖,maxi, j∈N sup|s|=1 ‖Gi j (s)‖}

However, since lim|s|→∞ ‖Ei (s I −A)−1Di‖=0, one has sup|s|=1 ‖Gii (s)‖=sup|s|�1 ‖Gii (s)‖=sup|s|�1 ‖Ei (s I −A)−1Di +Mi‖�‖Mi‖. Thus, we obtain the following theorem about the lowerbound of the complex stability radius.

Theorem 3.5Let system (3) be stable. Then,

rC(A;D,E,M)� 1

maxi, j∈N sup|s|=1 ‖Gi j (s)‖Theorem 3.6Let system (3) be stable. Then,

rC(A;D,E,M)� 1

maxi∈N sup|s|=1 ‖Gii (s)‖ProofGiven s0∈C, |s0|=1, suppose that maxi∈N ‖Gii (s0)‖=‖Gkk(s0)‖ for some k∈N . By the definitionof ‖Gkk(s0)‖ there exists a vector u0∈Clk such that ‖u0‖=1 and ‖Gkk(s0)u0‖=‖Gkk(s0)‖.Then, by the Hahn–Banach Theorem, there exists y∗

0 ∈(Cqk )∗,‖y∗0‖=1 satisfying y∗

0 (Gkk(s0)u0)=‖Gkk(s0)u0‖. Define �k :=‖Gkk(s0)u0‖−1u0y∗

0 ∈Clk×qk . It is clear that ‖�k‖=‖Gkk(s0)u0‖−1 and�kGkk(s0)u0=u0. Setting a nonzero vector x0=Gkk(s0)u0=(Ek(s0 I −A)−1Dk+Mk)u0 and adisturbance �=(�1, . . . ,�N ) defined by �i =0, i =k and �k =�k , then we get x0=(Gkk(s0)�k)x0,which implies 1∈�(Gkk(s0)�k). Applying Lemma 5.7.2 in [8, p. 549], we obtain that s0∈�(A+Dk(I −�kMk)

−1�k Ek)=�(A�). This completes the proof. �

In general, these stability radii can be different (see e.g. [11]). As far as their computationconcerns, while Theorems 3.5 and 3.6 reduce the estimates of the complex stability radius toa global optimization problem over the unit disk of complex plane. The problem for the realstability radius is much more difficult and a very complicated solution is known in recent time, see[13]. This is therefore natural to investigate for which kind of systems these three radii coincide.Motivated by results of the robust stability problems of the positive linear dynamical systems (see,e.g. [10, 11]). The following theorem yields a simple formula for the three stability radii of positivesystem.

Theorem 3.7Let system (3) be stable and positive, i.e. A∈Rn×n+ . Assume that all matrices Di ,Ei ,Mi arenonnegative for i ∈N . If Di =Dj (or Ei =E j ) for all i, j ∈N , then,

rC(A;D,E,M)=rR(A;D,E,M)=r+(A;D,E,M)= 1

maxi∈N ‖Gii (1)‖



ProofLet s∈C, |s|=1 be given arbitrarily. Then, |s|=1>r(A) and thus we have

|(s I −A)−1|=∣∣∣∣ ∞∑n=0

An

sn+1

∣∣∣∣� ∞∑n=0

An =(I −A)−1

Therefore, by invoking Theorem 3.5 we get that

rC(A;D,E,M)� 1

maxi∈N ‖Gii (1)‖By the definition of the stability radii, it remains to show that

r+(A;D,E,M)� 1

maxi∈N ‖Gii (1)‖Indeed, assume that maxi∈N ‖Gii (1)‖=‖Gkk(1)‖ for some index k∈N . Owing to Theorem 2.2,(I −A)−1�0 which implies ‖Gkk(1)‖ is a nonnegative matrix. Thus, thanks to Lemma 2.1,there exists a vector u0∈R

lk+ such that ‖u0‖=1 and ‖Gkk(s0)u0‖=‖Gkk(s0)‖. Then, by theKrein–Rutman Theorem, see [14], there exists y∗

0 ∈(Rqk )∗+,‖y∗0‖=1 satisfying y∗

0 (Gkk(1)u0)=‖Gkk(1)u0‖. Define �k :=‖Gkk(1)u0‖−1u0y∗

0 ∈Rlk×qk+ . It is clear that ‖�k‖=‖Gkk(1)u0‖−1 and

�kGkk(1)u0=u0. Setting a nonzero vector x0=Gkk(1)u0=(Ek(I −A)−1Dk+Mk)u0 and a distur-bance �=(�1, . . . ,�N ) defined by �i =0, i =k and �k =�k , then we get x0=(Gkk(1)�k)x0, whichimplies 1∈�(Gkk(s0)�k). Applying Lemma 5.7.2 in [8, p. 549], we obtain that 1∈�(A+Dk(I −�kMk)

−1�k Ek)=�(A�). Therefore, by definition of the positive stability radius, we obtain that

r+(A;D,E,M)� 1

maxi∈N ‖Gii (1)‖This completes the proof. �

4. APPLICATION TO ROBUST STABILITY OF POSITIVE LINEAR DELAY SYSTEMS

Now we consider the difference system with delays of the form

x(t+K )= A1x(t+K −1)+·· ·+AK x(t), t,K ∈N (6)

where Ai ∈Kn×n, i ∈K ={1, . . . ,K }.Set

P(�)= A1+·· ·+�−K+1AK , � =0

We denote the resolvent set of P(·) by�(P(·))={�∈C :det[�I −P(�)] =0}

and the spectrum set of P(·) by �(P(·))=C\�(P(·)). The spectral radius of P(·) can be defined as

r(P(·))=max{|�| :�∈�(P(·))}



Now we will introduce to an equivalent state–space system easily determined by the state vectory(t)=(x(t+K −1), . . . , x(t))T

y(t+1)= AC y(t), t ∈N (7)

where

AC =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

A1 A2 . . . AK−1 AK

In 0 . . . 0 0

0 In . . . 0 0

... ... . . . ... ...

0 0 . . . In 0

⎤⎥⎥⎥⎥⎥⎥⎥⎦

By a simple calculation, we can check that

�(P(·))=�(AC )\{0} (8)

and

(�I −AC )−1(y,0, . . . ,0)T=

⎡⎢⎢⎢⎢⎢⎢⎣

(�I −P(�))−1y

�−1(�I −P(�))−1y

...

�−K+1(�I −P(�))−1y

⎤⎥⎥⎥⎥⎥⎥⎦

(9)

Owing to (8), it is easy to see that system (6) is stable if and only if system (7) is stable, i.e.�(AC )⊂D. Furthermore, this is equivalent to �(P(·))⊂D, or, r(P(·))=r(AC )<1.

Assume that system (6) is stable. Now we suppose that all matrices Ai , i ∈K are subjected toperturbations of the form

Ai → Ai +Di (I −�i Mi )−1�i Ei , i ∈K (10)

where the matrices Di ∈Cn×li ,Ei ∈Cqi×n,Mi ∈Cqi×li , i ∈K are given matrices defining the struc-ture of the perturbations and �i ∈Cli×qi , i ∈K are unknown matrices. For the perturbations deter-mined by above way the perturbed system can be written as follows:

x(t+K ) = (A1+D1(I −�1M1)−1�1E1)x(t+K −1)

+·· ·+ (AK +DK (I −�K MK )−1�K EK )x(t) (11)

Then, the perturbed polynomial associated with system (11) can be defined by

P�(�) = (A1+D1(I −�1M1)−1�1E1)+�−1(A2+D2(I −�2M2)

−1�2E2)

+·· ·+ �−K+1(AK +DK (I −�K MK )−1�K EK ) (12)

The size of each perturbation �=(�i )i∈K is measured by �(�)=∑i∈K ‖�i‖.



Definition 4.1Let system (6) be stable. The complex, real and positive stability radius of system (6) underperturbations of the form (10) can be defined by

rC = inf{�(�) :�i ∈Cli×qi , i ∈K either r(P�(·))�1 or (∃i ∈K :1∈�(�i Mi ))}rR = inf{�(�) :�i ∈Rli×qi , i ∈K either r(P�(·))�1 or (∃i ∈K :1∈�(�i Mi ))}r+ = inf{�(�) :�i ∈R

li×qi+ , i ∈K either r(P�(·))�1 or (∃i ∈K :1∈�(�i Mi ))}respectively, where we set inf ∅=+∞.

For i, j ∈K ,s∈�(P(·)) let

Hi j (s)={si−1Ei (s I −P(s))−1Dj , i = j

si−1Ei (s I −P(s))−1Di +Mi , i= j

be associated transfer functions of system (6).Now setting

Di =

⎡⎢⎢⎢⎢⎢⎣

Di

0

...

0

⎤⎥⎥⎥⎥⎥⎦ , Ei =[0 . . . Ei . . . 0] for all i ∈K

then, the equivalent state–space system associated with the perturbed system (11) can be definedas follows:

y(t+1) =(AC +

K∑i=1

Di (I −�i Mi )−1�iEi

)y(t)

= AC (�)y(t), t ∈N

Therefore, the stability radii of system (6) can be rewritten as follows:

rC = inf{�(�) :�i ∈Cli×qi , i ∈K either r(AC (�))�1 or (∃i ∈K :1∈�(�i Mi ))}rR = inf{�(�) :�i ∈Rli×qi , i ∈K either r(AC (�))�1 or (∃i ∈K :1∈�(�i Mi ))}r+ = inf{�(�) :�i ∈R

li×qi+ , i ∈K either r(AC (�))�1 or (∃i ∈K :1∈�(�i Mi ))}On the other hand, thanks to (9), one has

Ei (s I −AC )−1D j =si−1Ei (s I −P(s))−1Dj for all i, j ∈K , s∈�(P(·))Therefore, by invoking Theorems 3.5–3.7, we obtain the following results.



Theorem 4.2Let system (6) be stable. Then, we have

1

maxi, j∈K sup|s|=1 ‖Hi j (s)‖�rC� 1

maxi∈K sup|s|=1 ‖Hii (s)‖Theorem 4.3Let system (6) be stable and positive, i.e. Ai ∈Rn×n+ , i ∈K . Assume that Di ,Ei ,Mi are nonnegativematrices for all i ∈K . If Di =Dj (or Ei =E j ) for all i, j ∈K then,

rC =rR =r+ = 1

maxi∈K ‖Hii (1)‖Remark 4.4It is important to note that we can similarly consider the case of delay systems where operatorsAi , i ∈K are subjected to multi-perturbations of the form

Ai → Ai +N∑j=1

Di j (I −�i j Mi j )−1�i j Ei j , i ∈K

Now we consider the following example to illustrate the obtained results.

ExampleConsider the following delay system:

y(t+2)= A1y(t+1)+A2y(t), t ∈N (13)

where

A1=[ 1

413

14 0

], A2=

[0 1

6

14

13

]

Since A1, A2∈R2×2+ , system (13) is positive. Moreover, one can check that system (13) is stable.Assume that the matrices A1, A2 are subjected to perturbations of the form

A1 → A1+D1(I −�1M1)−1�1E1

A2 → A2+D2(I −�2M2)−1�2E2

where

D1=D2 =[1 0

1 0

], E1=

[1 1

0 0

], E1=

[1 0

1 0

]

M1 =[ 1

313

12

12

], M2=

[ 13

12

14 0

]



Then,

H11(1) = E1(I −A1−A2)−1D1+M1=

[10 1

3

12

12

]

H22(1) = E2(I −A1−A2)−1D2+M2=

[5 11

2

14 0

]

If R2 is provided with the norm defined by ‖(x, y)‖=|x |+|y|, then by Theorem 4.3, we have

rC =rR =r+ = 221

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Stability radii of positive linear systems under fractional perturbations

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