On common linear copositive Lyapunov functions for pairs of stable positive linear systems

Nonlinear Analysis: Hybrid Systems 3 (2009) 467–474

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Nonlinear Analysis: Hybrid Systems

journal homepage: www.elsevier.com/locate/nahs

On common linear copositive Lyapunov functions for pairs of stablepositive linear systemsI

Zheng Chen ∗, Yan GaoSchool of Management, University of Shanghai for Science and Technology, 516 Jungong Road, Shanghai 200093, China

a r t i c l e i n f o

Article history:Received 24 January 2009Accepted 16 March 2009

Keywords:Positive linear systemsSwitched systemsCommon linear copositive Lyapunovfunctions

a b s t r a c t

We study the common linear copositive Lyapunov functions of positive linear systems.Firstly, we present a theorem on pairs of second order positive linear systems, and giveanother proof of this theorem by means of properties of geometry. Based on the processof the proof, we extended the results to a finite number of second order positive linearsystems. Then we extend this result to third order systems. Finally, for higher ordersystems, we give some results on common linear copositive Lyapunov functions.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

There are numerous examples of practically important dynamical systems where negative values of their state variablesare physically meaningless and for which, effectively the state vector of the system can only take on non-negative values.Such a system, where any state trajectory starting from non-negative initial conditions must remain non-negative for allsubsequent times, is known as a positive system. Positive systems are of fundamental importance to many applications inareas such as Ecology, Biology, Economics and in the study of communication systems [1–4]. Specially, the theory of positivelinear time-invariant (LTI) systems has assumed a position of great importance in systems theory and has been applied inthe study of a wide variety of dynamic systems [1,5]. In recent years, new studies in hybrid systems have highlighted theimportance of switched positive linear systems. In the last few years, considerable effort has been expended on gaining anunderstanding of the properties of general positive switched LTI systems [6–11]. However, most of the results currentlyavailable in the literature give abstract conditions that do not have a meaningful dynamical interpretation.Recently, the author of [12] gives verifiable conditions that are necessary and sufficient for the existence of a common

linear copositive Lyapunov function for a pair of second order exponentially stable positive LTI systems, and the result is veryeasy to check. In this paper, we give another proof of this theorem and then a method of finding common linear copositiveLyapunov functions is presented explicitly. The advantage of this new proof is that we can easily extend this result to a finitenumber of systems. Next we extend the result to third order positive LTI systems. Finally, for higher order switched systems,we will give some interesting results.This paper is organized as follows. In Section 2 we present the mathematical background and notation necessary to state

themain results of the paper. In Section 3,we present another proof of the theoremand extend the result. Some examples aregiven finally. In Section 4we prove some properties of higher order switched systems. Finally, our conclusions are presentedin Section 5.

I This work was supported by the National Science Foundation of China (under grant: 10671126), Shanghai leading discipline project (under grant:S30501) and The Innovation Fund Project for Graduate Student of Shanghai (under grant: JWCXSL0901).∗ Corresponding author.E-mail address: [email protected] (Z. Chen).

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468 Z. Chen, Y. Gao / Nonlinear Analysis: Hybrid Systems 3 (2009) 467–474

2. Mathematical preliminaries

In this section, we present a number of notations and results that will be needed in the following sections.

2.1. Notations

Throughout this paper, R denotes the real numbers, Rn is for the vector space of all n-tuples of real numbers and Rn×n isthe space of n× nmatrices with real entries. If x ∈ Rn, xi denotes the ith component of x, and aij denotes the element in the(i, j) position of A. The following notation is adopted:(i) for x ∈ Rn, x � 0 (x � 0)means that xi > 0 (xi ≥ 0) for 1 ≤ i ≤ n;(ii) for A ∈ Rn×n, A � 0 (A � 0)means that aij > 0 (aij ≥ 0) for 1 ≤ i, j ≤ n;(iii) for x, y ∈ Rn, x � y (x � y)means that x− y � 0 (x− y � 0);(iv) for A, B ∈ Rn×n, A � B (A � B)means that A− B � 0 (A− B � 0).

For a matrix A ∈ Rn×n, we write AT for the transpose of A.If P ∈ Rn×n, the notation P > 0means that the matrix P is positive definite. We shall denote the maximal real part of any

eigenvalue of A byµ(A). Ifµ(A) < 0, A is said to be Hurwitz or Hurwitz-stable. Obviously, if A is Hurwitz, all the eigenvaluesof A are in the open left plane.

2.2. Positive LTI systems

We now introduce the mathematical definitions of continuous-time positive LTI systems.

Definition 2.2.1. The LTI system

ΣA : x = Ax, x(t0) = x0is positive if x0 � 0⇒ x(t) � 0 for all t ≥ t0.

Definition 2.2.2. If all of the off-diagonal elements of a matrix A are non-negative, A is known as Metzler matrices.It is well known that the LTI system ΣA is positive if and only if A is a Metzler matrix [12]. Furthermore, a matrix A is

Metzler if and only if its transpose is also Metzler.

2.3. Positive switched linear systems and stability

We define positive switched linear systems as follows.

Definition 2.3.1. A switched linear system is a system of the form

x = A(t)x,

where A(t) can switch between some given finite collection of matrix A1, . . . , Ak in Rn×n. Thus, a switched linear system isobtained by switching between a number of LTI systemsΣAi , i = 1, . . . , k, referred to as its constitute systems or modes.If the constitute systems are all positive LTI systems, the switched linear systems is known as positive switched linear

systems.

2.4. Common linear copositive Lyapunov functions

As we know, it is possible for an unstable trajectory to result from switching between stable positive LTI systems. Apossible approach to establishing the stability of positive switched linear systems is to look for a common linear copositiveLyapunov function.

Definition 2.4.1. Given Metzler, Hurwitz matrices A1, . . . , Ak in Rn×n. If there exists a single vector v � 0 such that−ATi v � 0, i = 1, . . . , k, V (x) = v

Tx is a common linear copositive Lyapunov function.

Next we give a lemma.

Lemma 2.4.1. Let A ∈ Rn×n be Metzler. Then A is Hurwitz if and only if there is some v � 0 in Rn such that −Av � 0.

3. Second order and third order positive linear systems

3.1. Second order positive linear systems

We give the theorem in [12] in this subsection, but we give a different proof.

Theorem 3.1.1. Let A1, A2 be Metzler, Hurwitz matrices in R2×2. Then, writing a(k)ij for the (i, j) entry of Ak, a necessary and

sufficient condition for the positive LTI systems ΣA1 ,ΣA2 to have a common linear copositive Lyapunov function is that both of

Z. Chen, Y. Gao / Nonlinear Analysis: Hybrid Systems 3 (2009) 467–474 469

the determinants∣∣∣∣∣a(1)11 a(2)12a(1)21 a(2)22

∣∣∣∣∣ ,∣∣∣∣∣a(2)11 a(1)12a(2)21 a(1)22

∣∣∣∣∣ ,are positive.

Proof. Firstly, based on Definition 2.4.1, ΣA1 ,ΣA2 have a common linear copositive Lyapunov function if and only if thereexists a single vector v � 0 such that−ATi v � 0, i = 1, 2. We denote v =

(v1v2

), then we obtain the following inequities

a(1)11 v1 + a(1)21 v2 < 0 (1)

a(1)12 v1 + a(1)22 v2 < 0 (2)

a(2)11 v1 + a(2)21 v2 < 0 (3)

a(2)12 v1 + a(2)22 v2 < 0 (4)

v1 > 0 (5)v2 > 0. (6)

Obviously ΣA1 ,ΣA2 have a common linear copositive Lyapunov function which is equivalent to the existence of feasiblesolutions of (1)–(6).Next we consider the inequities (1)–(6). Note that A1, A2 areMetzler, Hurwitz. The following facts can be obtained easily.

(i) det A1 > 0, det A2 > 0;(ii) a(k)ii < 0, i, k = 1, 2;(iii) a(k)ij ≥ 0, i, j, k = 1, 2; i 6= j.

So we mainly consider both A1 and A2 have the sign pattern(− +

+ −

)or

(− 0+ −

).

Set up the planar rectangular coordinate v1ov2. We can only consider the first quadrant because of (5) and (6).

From (1) and (2) we can obtain v2 < −a(1)11a(1)21v1, v2 > −

a(1)12a(1)22v1. Firstly, we know −

a(1)11a(1)21and − a

(1)12

a(1)22are both positive, so the

graphs of a(1)11 v1+a(1)21 v2 = 0 and a

(1)12 v1+a

(1)22 v2 = 0 lie in the first quadrant. Moreover,−

a(1)11a(1)21

> −a(1)12a(1)22because of det A1 > 0.

That is to say, the slope of a(1)11 v1 + a(1)21 v2 = 0 is greater than the slope of a

(1)12 v1 + a

(1)22 v2 = 0. From the discussion above,

we can get

M =

{(v1v2

)∣∣∣∣ v2 < −a(1)11a(1)21 v1, v1 > 0, v2 > 0}∩

{(v1v2

)∣∣∣∣ v2 > −a(1)12a(1)22 v1, v1 > 0, v2 > 0}6= φ.

In conclusion, the area composed of setM is below the line a(1)11 v1+ a(1)21 v2 = 0 and above the line a

(1)12 v1+ a

(1)22 v2 = 0, which

lies in the first quadrant.We deal with (3) and (4) similarly. We get

N =

{(v1v2

)∣∣∣∣ v2 < −a(2)11a(2)21 v1, v1 > 0, v2 > 0}∩

{(v1v2

)∣∣∣∣ v2 > −a(2)12a(2)22 v1, v1 > 0, v2 > 0}6= φ.

Similarly, the area composed of set N is below the line a(2)11 v1+ a(2)21 v2 = 0 and above the line a

(2)12 v1+ a

(2)22 v2 = 0, which lies

in the first quadrant.ConsiderM ∩ N 6= φ, we have

M ∩ N 6= φ ⇔

(−a(1)12a(1)22

,−a(1)11a(1)21

)∩

(−a(2)12a(2)22

,−a(2)11a(2)21

)6= φ.

Thus, (−a(1)12a(1)22

,−a(1)11a(1)21

)∩

(−a(2)12a(2)22

,−a(2)11a(2)21

)6= φ ⇔ −

a(1)11a(1)21

> −a(2)12a(2)22

,−a(1)12a(1)22

< −a(2)11a(2)21


−a(1)11a(1)21

> −a(2)12a(2)22

,−a(1)12a(1)22

< −a(2)11a(2)21⇔

∣∣∣∣∣a(1)11 a(2)12a(1)21 a(2)22

∣∣∣∣∣ > 0 and

∣∣∣∣∣a(2)11 a(1)12a(2)21 a(1)22

∣∣∣∣∣ > 0.Next, if A1 or A2 have the sign pattern(− +

0 −

)or

(− 00 −

).

The proof will be a slight different from above.

(i) A1 =(a(1)11 a(1)12

0 a(1)22

), A2 =

(a(2)11 a(2)12

a(2)21 a(2)22

)In this case,

∣∣∣∣a(1)11 a(2)12

a(1)21 a(2)22

∣∣∣∣ > 0 is always true. The inequities (1)–(6) are equivalent to(−a(1)12a(1)22

,+∞

)∩

(−a(2)12a(2)22

,−a(2)11a(2)21

)6= φ ⇔ −

a(1)12a(1)22

< −a(2)11a(2)21

−a(1)12a(1)22

< −a(2)11a(2)21⇔

∣∣∣∣∣a(2)11 a(1)12a(2)21 a(1)22

∣∣∣∣∣ > 0.(ii) A1 =

(a(1)11 a(1)12

a(1)21 a(1)22

), A2 =

(a(2)11 a(2)12

0 a(2)22

)In this case,

∣∣∣∣a(2)11 a(1)12

a(2)21 a(1)22

∣∣∣∣ > 0 is always true. The inequities (1)–(6) are equivalent to(−a(1)12a(1)22

,−a(1)11a(1)21

)∩

(−a(2)12a(2)22

,+∞

)6= φ ⇔ −

a(1)11a(1)21

> −a(2)12a(2)22

−a(1)11a(1)21

> −a(2)12a(2)22⇔

∣∣∣∣∣a(1)11 a(2)12a(1)21 a(2)22

∣∣∣∣∣ > 0.(iii) A1 =

(a(1)11 a(1)12

0 a(1)22

), A2 =

(a(2)11 a(2)12

0 a(2)22

)In this case,

∣∣∣∣a(1)11 a(2)12

0 a(2)22

∣∣∣∣ > 0 and ∣∣∣∣a(2)11 a(1)12

0 a(1)22

∣∣∣∣ > 0 are always true. This completes the proof. �

Comments: Based on the process of the proof above, a common linear copositive Lyapunov function will be givenexplicitly.

(i) Both A1 and A2 have the sign pattern(− +

+ −

)or(− 0+ −

)Consider the four positive numbers − a

(1)12

a(1)22,−

a(1)11a(1)21,−

a(2)12a(2)22,−

a(2)11a(2)21. Without loss of generality, we assume − a

(1)12

a(1)22< −

a(1)11a(1)21

<

−a(2)12a(2)22

< −a(2)11a(2)21.

Then, we can get a common linear copositive Lyapunov function

V (x) = x1 −

(a(1)112a(1)21

+a(2)122a(2)22

)x2.

(ii) A1 =(a(1)11 a(1)12

0 a(1)22

), A2 =

(a(2)11 a(2)12

a(2)21 a(2)22

)Consider the three positive numbers− a

(1)12

a(1)22,−

a(2)12a(2)22,−

a(2)11a(2)21. Without loss of generality, we assume− a

(1)12

a(1)22< −

a(2)12a(2)22

< −a(2)11a(2)21.


V (x) = x1 −

(a(2)122a(2)22

+a(2)112a(2)21

)x2.


(iii) A1 =(a(1)11 a(1)12

a(1)21 a(1)22

), A2 =

(a(2)11 a(2)12

0 a(2)22

)Consider the three positive numbers− a

(1)12

a(1)22,−

a(1)11a(1)21,−

a(2)12a(2)22. Without loss of generality, we assume− a

(1)12

a(1)22< −

a(1)11a(1)21

< −a(2)12a(2)22.


V (x) = x1 −

(a(1)112a(1)21

+a(2)122a(2)22

)x2.

(iv) A1 =(a(1)11 a(1)12

0 a(1)22

), A2 =

(a(2)11 a(2)12

0 a(2)22

)Consider the two positive numbers− a

(1)12

a(1)22,−

a(2)12a(2)22. Without loss of generality, we assume

−a(1)12a(1)22

< −a(2)12a(2)22

.


V (x) = x1 −

(a(2)12a(2)22− 1

)x2.

Furthermore, based on the process of the proof, the following result will be obtained easily.

Corollary 3.1.1. Let A1, . . . , Ak be Metzler, Hurwitz matrices in R2×2. Then, writing a(n)ij for the (i, j) entry of An, n = 1, . . . , k,

a necessary and sufficient condition for the positive LTI systems ΣA1 , . . . ,ΣAk to have a common linear copositive Lyapunovfunction is that

k⋂n=1

(−a(n)12a(n)22

, b(n))6= φ,

where b(n) =

− a(n)11

a(n)21, a(n)21 > 0

+∞, a(n)21 = 0

, n = 1, . . . , k.

Corollary 3.1.1 provides the simple test for the existence of a common linear copositive Lyapunov function for a finitenumber of second order positive LTI systems.

3.2. Third order positive linear systems

Now we give the result as follows.

Theorem 3.2.1. Let A1, A2 be Metzler, Hurwitz matrices in R3×3. Then, writing a(k)ij for the (i, j) entry of Ak, if the positive LTI

systemsΣA1 ,ΣA2 have a common linear copositive Lyapunov function, both of the determinants∣∣∣∣∣a(1)11 a(2)12a(1)21 a(2)22

∣∣∣∣∣ ,∣∣∣∣∣a(2)11 a(1)12a(2)21 a(1)22

∣∣∣∣∣ ,are positive.

Proof. To begin with, we suppose a common linear copositive Lyapunov function V (x) = vTx. Ak =(A(k)11 A(k)12

A(2)21 a(2)33

), where

A(k)11 =(a(k)11 a(k)12

a(k)21 a(k)22

), A(k)12 =

(a(k)13

a(k)23

)and A(k)21 =

(a(k)31

a(k)32

)T, k = 1, 2. Denote v =

(v′

v3

), where v′ =

(v1v2

).

From Definition 2.4.1, we have v � 0 and−ATkv � 0. Calculate−ATkv � 0 as follows.

−

(A(k)11 A(k)12

A(2)21 a(2)33

)(v′

v3

)= −

(A(k)11 v

′+ A(k)12 v3 ∗

∗ ∗

)� 0, where ∗ stands for some uncertain elements.

So we have−A(k)11 v′− A(k)12 v3 � 0. We know that A

(k)12 =

(a(k)13

a(k)23

)� 0 and v3 > 0, then we can get−A

(k)11 v′� 0. It has found

a common linear copositive Lyapunov function V (x) = v′Tx for the positive LTI systems ΣA(1)11,ΣA(2)11

. From Theorem 3.1.1,


we get both of the determinants∣∣∣∣∣a(1)11 a(2)12a(1)21 a(2)22

∣∣∣∣∣ ,∣∣∣∣∣a(2)11 a(1)12a(2)21 a(1)22

∣∣∣∣∣ ,are positive. This completes the proof. �

4. Higher dimensional systems

4.1. Upper triangular form and lower triangular form

Here we consider the system matrices A1, . . . , Ak of the constitute systemsΣA5 , i = 1, . . . , k are upper triangular formand lower triangular form.Since the system matrices are upper triangular form or lower triangular form, we can write

Ai =

ai11 ai12 · · · a

i1n

0 ai22 · · · ai2n

......

. . ....

0 0 · · · ainn

or Ai =

ai11 0 · · · 0

ai21 ai22 · · · 0...

.... . .

...

ain1 ain2 · · · ainn

, i = 1, 2, . . . , k.

Theorem 4.1.1. Let Ai, i = 1, 2, . . . , k be Metzler, Hurwitz matrices in Rn×n such that Ai are upper triangular matrices. Then thesystemsΣAi : x = Aix, i = 1, 2, . . . , k share a common linear copositive Lyapunov function V (x) = v

Tx.

Proof. Noticing that Ai is Hurwitz and Metzler, we get aipp < 0 and aipq ≥ 0, p 6= q, p, q = 1, 2, . . . , n.

Next, we will find v � 0 such that−ATi v � 0, i = 1, . . . , k.

−ATi v = −

ai11 0 · · · 0

ai12 ai22 · · · 0...

.... . .

...

ai1n ai2n · · · ainn

v1v2...vn

= −

ai11v1ai12v1 + a

i22v2

...

ai1nv1 + ai2nv2 + · · · + a

innvn

� 0.It is equivalent to

ai11v1 < 0ai12v1 + a

i22v2 < 0...

ai1nv1 + ai2nv2 + · · · + a

innvn < 0.

We can get v in the following method.Firstly, let v1 = 1, and ai11v1 < 0. Next let v2 > max1≤i≤k

{−ai12ai22

}such that ai12v1 + a

i22v2 < 0. Fix v2 = v∗2 >

max1≤i≤k{−ai12ai22

}, then let v3 > max1≤i≤k

{−ai13+a

i23v∗2

ai33

}such that ai13v1 + a

i23v2 + a

i33v3 < 0. This progress repeated

until let vn > max1≤i≤k

{−ai1n+a

i2nv∗2+···+a

i(n−1)nv

∗n−1

ainn

}such that ai1nv1 + a

i2nv2 + · · · + a

innvn < 0. Finally we fix the

vn = v∗n > max1≤i≤k

{−ai1n+a

i2nv∗2+···+a

i(n−1)nv

∗n−1

ainn

}. Thus, we have found the v and got a common linear copositive Lyapunov

function V (x) = (1, v∗2 , . . . , v∗n)Tx. This completes the proof. �

Theorem 4.1.2. Let Ai, i = 1, 2, . . . , k be Metzler, Hurwitz matrices in Rn×n such that Ai are lower triangular matrices. Then thesystemsΣAi : x = Aix, i = 1, 2, . . . , k share a common linear copositive Lyapunov function V (x) = v

Tx.

Proof. Noticing that Ai is Hurwitz and Metzler, we get aipp < 0 and aipq ≥ 0, p 6= q, p, q = 1, 2, . . . , n.

Next, we will find v � 0 such that−ATi v � 0, i = 1, . . . , k.

−ATi v = −

ai11 ai21 · · · a

in1

0 ai22 · · · ain2

......

. . ....

0 0 · · · ainn

v1v2...vn

= −ai11v1 + a

i21v2 + · · · + a

in1vn

ai22v2 + · · · + ain2vn

...

ainnvn

� 0.


It is equivalent to

ai11v1 + ai21v2 + · · · + a

in1vn < 0

ai22v2 + · · · + ain2vn < 0

...

ainnvn < 0.

Let vn = 1, and ainnvn < 0. We can get v in the same way as the proof of Theorem 4.1.1. Thus, we have found the v andgot a common linear copositive Lyapunov function V (x) = (v∗1 , . . . v

∗

n−1, 1)Tx. This completes the proof. �

4.2. System of same block upper triangular form

We show that when a set of matrices have some block upper triangular forms, finding a common linear copositiveLyapunov function will become more easily. The main result is as follows.

Theorem 4.2.1. Assume a finite set of block triangular matrices with same diagonal block structure as

Ai =

Ai11 Ai12 · · · A

i1n

0 Ai22 · · · ai2n

......

. . ....

0 0 · · · Ainn

, i = 1, 2, . . . , k

where the same lth diagonal blocks Aill have same dimensions for all i. Then, Ai share a common linear copositive Lyapunov function,if and only if for every l the diagonal blocks Aill, i = 1, 2, . . . , k share a common linear copositive Lyapunov function.

Proof. Without loss of generality, we have only to prove it for n = 2. Then, by mathematical induction we prove it for anyn for both necessity and sufficiency.Sufficiency: denote by

Ai =(Xi Yi0 Zi

), i = 1, 2, . . . , k.

Assume dim(Xi) = p and dim(Zi) = q. Let vT1x and vT2x be the common linear copositive Lyapunov function of {Xi} and {Yi},

respectively.

We claim that for large enough µ > 0,(v1µv2

)Tx is a common linear copositive Lyapunov function of Ai. Calculate

−ATi

(v1µv2

)= −

(XTi 0Y Ti ZTi

)(v1µv2

)= −

(XTi v1

Y Ti v1 + µZTi v2

)−XTi v1 � 0 because v

T1x is the common linear copositive Lyapunov function of {Xi}.

Y Ti v1 � 0 because Ai is Metzler and v1 � 0. Set YTi v1 =

bi1...

biq

� 0and ZTi v2 ≺ 0 because vT2x is the common linearcopositive Lyapunov function of {Zi}. Then, denote by ZTi v2 =

ci1...

ciq

≺ 0. We want −(Y Ti v1 + µZTi v2) � 0. Choosingµ > max1≤i≤k

{max1≤1≤q

{−bijcij

}}, then it is obvious that−(Y Ti v1+µZ

Ti v2) � 0. So

(v1µv2

)Tx is a common linear copositive

Lyapunov function of Ai.Necessity: Assume Ai, i = 1, 2, . . . , k share a common linear copositive Lyapunov function, vTx. According to the

structure of Ai, we split v as(v1v2

)� 0. Then

−ATi

(v1µv2

)= −

(XTi 0Y Ti ZTi

)(v1v2

)= −

(XTi v1

Y Ti v1 + ZTi v2

)� 0.

We have−XTi v1 � 0, v1 � 0. {Xi} share a common linear copositive Lyapunov function vT1x. Since−Y

Ti v1− Z

Ti v2 � 0 and

−Y Ti v1 � 0,−ZTi v2 � 0. It is easy to see that {Zi} share a common linear copositive Lyapunov function v

T2x. This completes

the proof. �


5. Conclusions

In this paper, we give a newproof of the theorem in [12]. The geometrical explanation on the relationship of the line of theslope shows the advantage of this new proof. So we can extend this result to a finite number of second order systems easily.Another advantage of the proof is that we can find a common linear copositive Lyapunov function based on the methodsof the proof. Then we give a necessary condition of the existence of common linear copositive Lyapunov functions of thirdorder systems. For higher order systems, when the system matrices are upper triangular form and lower triangular form,we show a way to find a common linear copositive Lyapunov function. In another case, when a set of matrices have someblock upper triangular forms, a necessary and sufficient condition of the existence of common linear copositive Lyapunovfunctions is presented.

References

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On common linear copositive Lyapunov functions for pairs of stable positive linear systems