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 stable positive linear systemsI Zheng Chen ∗ , Yan Gao School of Management, University of Shanghai for Science and Technology, 516 Jungong Road, Shanghai 200093, China

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Article history: Received 24 January 2009 Accepted 16 March 2009 Keywords: Positive linear systems Switched systems Common linear copositive Lyapunov functions

abstract 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 give another proof of this theorem by means of properties of geometry. Based on the process of the proof, we extended the results to a finite number of second order positive linear systems. Then we extend this result to third order systems. Finally, for higher order systems, 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 variables are 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 all subsequent times, is known as a positive system. Positive systems are of fundamental importance to many applications in areas such as Ecology, Biology, Economics and in the study of communication systems [1–4]. Specially, the theory of positive linear time-invariant (LTI) systems has assumed a position of great importance in systems theory and has been applied in the study of a wide variety of dynamic systems [1,5]. In recent years, new studies in hybrid systems have highlighted the importance of switched positive linear systems. In the last few years, considerable effort has been expended on gaining an understanding of the properties of general positive switched LTI systems [6–11]. However, most of the results currently available 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 very easy to check. In this paper, we give another proof of this theorem and then a method of finding common linear copositive Lyapunov functions is presented explicitly. The advantage of this new proof is that we can easily extend this result to a finite number 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 the main results of the paper. In Section 3, we present another proof of the theorem and extend the result. Some examples are given finally. In Section 4 we prove some properties of higher order switched systems. Finally, our conclusions are presented in 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).

1751-570X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2009.03.004

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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 is the space of n × n matrices 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) (ii) (iii) (iv)

for x ∈ Rn , x  0 (x  0) means that xi > 0 (xi ≥ 0) for 1 ≤ i ≤ n; for A ∈ Rn×n , A  0 (A  0) means that aij > 0 (aij ≥ 0) for 1 ≤ i, j ≤ n; for x, y ∈ Rn , x  y (x  y) means that x − y  0 (x − y  0); 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 > 0 means that the matrix P is positive definite. We shall denote the maximal real part of any eigenvalue of A by µ(A). If µ(A) < 0, A is said to be Hurwitz or Hurwitz-stable. Obviously, if A is Hurwitz, all the eigenvalues of 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 ) = x0

is 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 is obtained by switching between a number of LTI systems ΣAi , 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. A possible approach to establishing the stability of positive switched linear systems is to look for a common linear copositive Lyapunov 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 T x 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. (k)

Theorem 3.1.1. Let A1 , A2 be Metzler, Hurwitz matrices in R2×2 . Then, writing aij 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 (1) a

(2)

(2)

a22

21

a(2) 11 (2) a



a12

,

(1)



a12 (1)

a22

21

,

are positive. Proof. Firstly, based on Definition 2.4.1, ΣA1 , ΣA2 have a common linear copositive Lyapunov function if and only if there v

 

exists a single vector v  0 such that −ATi v  0, i = 1, 2. We denote v = v1 , then we obtain the following inequities 2 (1)

(1)

(1)

(1)

(2)

(2)

(2)

(2)

a11 v1 + a21 v2 < 0

(1)

a12 v1 + a22 v2 < 0

(2)

a11 v1 + a21 v2 < 0

(3)

a12 v1 + a22 v2 < 0

(4)

v1 > 0 v2 > 0.

(5) (6)

Obviously ΣA1 , ΣA2 have a common linear copositive Lyapunov function which is equivalent to the existence of feasible solutions of (1)–(6). Next we consider the inequities (1)–(6). Note that A1 , A2 are Metzler, Hurwitz. The following facts can be obtained easily. (i) det A1 > 0, det A2 > 0; (k) (ii) aii < 0, i, k = 1, 2; (k)

(iii) aij ≥ 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 v1 ov2 . We can only consider the first quadrant because of (5) and (6). From (1) and (2) we can obtain v2 < − (1)

(1)

(1)

(1) a11 (1) a21

v1 , v2 > −

(1) a12 (1) a22

v1 . Firstly, we know −

(1) a11 (1) a21

(1)

graphs of a11 v1 + a21 v2 = 0 and a12 v1 + a22 v2 = 0 lie in the first quadrant. Moreover, − (1)

(1)

(1)

(1)

and − (1)

a11 (1) a21

(1) a12 (1) a22

>−

are both positive, so the

(1) a12 (1) a22

because of det A1 > 0.

That is to say, the slope of a11 v1 + a21 v2 = 0 is greater than the slope of a12 v1 + a22 v2 = 0. From the discussion above, we can get

(   ) (   ) (1) (1) a11 a12 v1 v1 M = v < − (1) v1 , v1 > 0, v2 > 0 ∩ v > − (1) v1 , v1 > 0, v2 > 0 6= φ. v2 2 v2 2 a21

a22

(1)

(1)

(1)

(1)

In conclusion, the area composed of set M is below the line a11 v1 + a21 v2 = 0 and above the line a12 v1 + a22 v2 = 0, which lies in the first quadrant. We deal with (3) and (4) similarly. We get

(   ) (   ) (2) (2) a11 a12 v1 v1 N = v < − (2) v1 , v1 > 0, v2 > 0 ∩ v > − (2) v1 , v1 > 0, v2 > 0 6= φ. v2 2 v2 2 a a 21

22

(2)

(2)

(2)

(2)

Similarly, the area composed of set N is below the line a11 v1 + a21 v2 = 0 and above the line a12 v1 + a22 v2 = 0, which lies in the first quadrant. Consider M ∩ N 6= φ , we have (1)

M ∩ N 6= φ ⇔

−

a12

(1)

a22

,−

(1)

a11

(1)

(2)

! ∩ −

a21

a12

(2)

a22

,−

(2)

a11

!

(2)

a21

6= φ.

Thus, (1)

−

(1)

a12

a11

a22

a21

,− (1)

(1)

(2)

! ∩ −

(2)

a12

a11

a22

a21

,− (2)

(2)

! 6= φ ⇔ −

(1)

a11

(1)

a21

>−

(2)

(1)

a12

a12

a22

a22

,− (2)

(1)

<−

(2)

a11

(2)

a21

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

(1)

a11

−

(1)

a21

>−

(2)

a12

(2)

a22

,−

(1)

a12

<−

(1)

a22

a(1) ⇔ 11 a(1)

(2)

a11

(2)

a21

(2)



a12 (2)

a22

21

> 0 and

a(2) 11 (2) a 21

(1)



a12 (1)

a22

> 0.

Next, if A1 or A2 have the sign pattern



 + −

− 0

 or



−

0

0

−

.

The proof will be a slight different from above.

 (i) A1 =

(1) a11

(1) a12

0

(1) a22

(1) a In this case, 11 (1) a

, A2 =

21

a12

, +∞ ∩ − (1)

a22

(1)

a12

−

(1)

a22 (ii) A1 =

<−

(1)



(1)

a11

a12

(1) a21

(1) a22

21

−

(2) a11

(2) a12

(2) a21

(2) a22

(2)

a11

(2)

a21

(2)

a11

a22

a21

,− (2)

a(2) ⇔ 11 a(2) 21



(1)

a22

(1)

>−

a(1) ⇔ 11 a(1)

(1)

(1)

 (iii) A1 =

∩ −

(1)

a11

a12

0

(1) a22

(1) a In this case, 11 0

(2) a12

(1)

a22

<−

(2)

a11

(2)

a21



(2)

a21

a21

(2)

(1)

a12

22

a11

a22

(1)

6= φ ⇔ −

(1) a12 > 0. (1) a

(2)



a11

,− (1)

a11

(2)

a12

a12

−



, A2 = (2) 0 a22 (1) a12 > 0 is always true. The inequities (1)–(6) are equivalent to (1) a22 ! !

(2) a In this case, 11 (2) a (1)



(2) a12 > 0 is always true. The inequities (1)–(6) are equivalent to (2) a22 ! !

(1)

−



(2)

a12

(2)

a22

a12

, +∞ 6= φ ⇔ − (2)

21





(1)

a11

(1)

a21

>−

(2)

a12

(2)

a22

(2) a12 > 0. (2) a 22

(2 ) a11

(2 ) a12

, A2 = 0 (2) (2) a12 > 0 and a11 (2) 0 a



(2 ) a22 (1) a12

> 0 are always true. This completes the proof. (1) a



22

22

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





−

+ −

(i) Both A1 and A2 have the sign pattern + (1)

Consider the four positive numbers − (2)

−

a12 (2) a22

<−

(2) a11 (2) . a21

a12 (1) a22

,−



−

0

or + (1)

a11 (1) a21

,−



− (2)

a12 (2) a22

,−

( 2) a11 (2) . a21

Without loss of generality, we assume −

(1) a12 (1) a22

<−

(1) a11 (1) a21

<

Then, we can get a common linear copositive Lyapunov function V (x) = x1 − (1) a11

(1) a12

0

(1) a22

 (ii) A1 =

(1)

a11

(1)

(2)

+

2a21



a12

! x2 .

(2)

2a22

, A2 =

(2) a11

(2) a12

(2) a21

(2) a22





Consider the three positive numbers −

(1) a12 (1) a22

,−

(2) a12 (2) a22

,−

(2) a11 (2) . a21

Without loss of generality, we assume −

Then, we can get a common linear copositive Lyapunov function V (x) = x1 −

(2)

a12

(2)

2a22

(2)

+

a11

(2)

2a21

! x2 .

(1) a12 (1) a22

<−

(2) a12 (2) a22

<−

(2) a11 (2) . a21

Z. Chen, Y. Gao / Nonlinear Analysis: Hybrid Systems 3 (2009) 467–474 (1) a11

a12

(1)

(1)

 (1)

 (iii) A1 =

a21

, A2 =

a22



(2) a11

a12

0

(2) a22

471

 (2)

Consider the three positive numbers −

(1) a12 (1) a22

,−

(1) a11 (1) a21

,−

(2) a12 (2) . a22

Without loss of generality, we assume −

Then, we can get a common linear copositive Lyapunov function (1)

V (x) = x1 −

 (iv) A1 =

(2)

a11

(1)

+

(1) a11

(1) a12

0

(1) a22



(2) a11

(2) a12

0

(2) a22



(1)

a12

(1)

a22

<−

(2)

a12

(2)

a22

<−

(2) a12 (2) . a22



Consider the two positive numbers −

−

(1) a11 (1) a21

x2 .

(2)

2a22

, A2 =

<−

!

a12

2a21

(1) a12 (1) a22

(1) a12 (1) a22

,−

(2) a12 (2) . a22

Without loss of generality, we assume

.

Then, we can get a common linear copositive Lyapunov function (2)

a12

V (x) = x1 −

(2)

a22

! − 1 x2 .

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

Corollary 3.1.1. Let A1 , . . . , Ak be Metzler, Hurwitz matrices in R2×2 . Then, writing aij 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 Lyapunov function is that k \

(n)

−

n =1

where b(n) =

a12

(n)

a22

 

! ,b

a

(n)

(n)

− (11 , n)

(n) a21 > 0

+∞,

(n) a21 = 0

a21



6= φ,

, n = 1, . . . , k.

Corollary 3.1.1 provides the simple test for the existence of a common linear copositive Lyapunov function for a finite number of second order positive LTI systems. 3.2. Third order positive linear systems Now we give the result as follows. (k)

Theorem 3.2.1. Let A1 , A2 be Metzler, Hurwitz matrices in R3×3 . Then, writing aij for the (i, j) entry of Ak , if the positive LTI systems ΣA1 , ΣA2 have a common linear copositive Lyapunov function, both of the determinants

a(1) 11 (1) a 21

(2)

a(2) 11 (2) a



a12 (2)

a22

,

(1)



a12 (1)

a22

21

,

are positive. Proof. To begin with, we suppose a common linear copositive Lyapunov function V (x) = v T x. Ak = (k)



A11 =

 (k)

( k) a11

a12

( k) a21

(k) a22

(k)



, A12 =

 ( k)

a13

( k) a23

(k)



and A21 =

 ( k) T

a31

( k) a32

 0 v



( k) A11

( k) A12

(2) A21

(2 ) a33

 , where

v

 

, k = 1, 2. Denote v = v , where v 0 = v1 . 2 3

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



( k) A11

( k) A12

(2) A21

(2)

  0 v v3

a33

=−

( k) ( k) A11 v 0 + A12 v3





∗ ∗

 0, where ∗ stands for some uncertain elements.  (k)  a (k) 0 (k) (k) So we have −A11 v − A12 v3  0. We know that A12 = 13  0 and v3 > 0, then we can get −A(11k) v 0  0. It has found (k) −

∗

a23

a common linear copositive Lyapunov function V (x) = v 0 x for the positive LTI systems ΣA(1) , ΣA(2) . From Theorem 3.1.1, T

11

11

472

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

we get both of the determinants

a(1) 11 (1) a 21

(2)

a(2) 11 (2) a



a12 (2)

a22

,

21

(1)



a12 (1)

a22

,

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 systems ΣA5 , i = 1, . . . , k are upper triangular form and lower triangular form. Since the system matrices are upper triangular form or lower triangular form, we can write ai11



0  Ai =  .  .

ai12

···

ai1n

ai22

··· .. . ···

ai2n  

.

.. .

0

0

ai11





..  . 



0

···

0

ai  21 or Ai =  .

ai22

0  

ain1

ain2

··· .. . ···

 ..

ainn

.. .

, ..  . 

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

ainn

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 the systems ΣAi : x˙ = Ai x, i = 1, 2, . . . , k share a common linear copositive Lyapunov function V (x) = v T x. 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. ai11



ai  −ATi v = −  12  ... ai1n

0

···

0

ai22

··· .. . ···

0

.. .

ai2n

.. .

ainn

    ai11 v1 v1  v2    ai12 v1 + ai22 v2       ..  = −    0. ..  .   . vn ai1n v1 + ai2n v2 + · · · + ainn vn

It is equivalent to ai11 v1 < 0 ai12 v1 + ai22 v2 < 0

.. . ai1n v1 + ai2n v2 + · · · + ainn vn < 0. We can get v in the following method.

n

ai

Firstly, let v1 = 1, and ai11 v1 < 0. Next let v2 > max1≤i≤k − 12 ai

o

such that ai12 v1 + ai22 v2 < 0. Fix v2 = v2∗ > 22   ai +ai v ∗ ai max1≤i≤k − 12 , then let v3 > max1≤i≤k − 13 i 23 2 such that ai13 v1 + ai23 v2 + ai33 v3 < 0. This progress repeated ai22 a33  i i ∗  a1n +a2n v2 +···+ai(n−1)n vn∗−1 such that ai1n v1 + ai2n v2 + · · · + ainn vn < 0. Finally we fix the until let vn > max1≤i≤k − i ann  i i ∗  a +a v +···+ai v∗ vn = vn∗ > max1≤i≤k − 1n 2n 2 ai (n−1)n n−1 . Thus, we have found the v and got a common linear copositive Lyapunov

n

o

nn

function V (x) = (1, v2∗ , . . . , vn∗ )T x. 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 the systems ΣAi : x˙ = Ai x, i = 1, 2, . . . , k share a common linear copositive Lyapunov function V (x) = v T x. 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. ai11

ai21

···

0

ai22

0

0

··· .. . ···

   −ATi v = −  

.. .

.. .

ain1

    i a11 v1 + ai21 v2 + · · · + ain1 vn v1   v2  ai22 v2 + · · · + ain2 vn ain2     .  = −     0. . ..   .  ..   . . i i v n a v a nn n nn

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

473

It is equivalent to ai11 v1 + ai21 v2 + · · · + ain1 vn < 0 ai22 v2 + · · · + ain2 vn < 0

.. .

ainn vn < 0. Let vn = 1, and ainn vn < 0. We can get v in the same way as the proof of Theorem 4.1.1. Thus, we have found the v and got a common linear copositive Lyapunov function V (x) = (v1∗ , . . . vn∗−1 , 1)T x. 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 copositive Lyapunov 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 Ai11 0 

Ai12 Ai22

 .

.. .

··· ··· .. .

Ai1n ai2n  

0

0

···

Ainn



Ai =  . .



..  , . 

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 any n for both necessity and sufficiency. Sufficiency: denote by

 Ai =

Xi 0

Yi Zi



,

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

Assume dim(Xi ) = p and dim(Zi ) = q. Let v1T x and v2T x be the common linear copositive Lyapunov function of {Xi } and {Yi }, respectively.



v

We claim that for large enough µ > 0, µv1 2

−

ATi



v1 µv2



XT = − iT Yi



0 ZiT

v1 µv2



T



x is a common linear copositive Lyapunov function of Ai . Calculate

 =−

XiT v1



YiT v1 + µZiT v2

−XiT v1  0 because v1T x is the common linear copositive Lyapunov function of {Xi }.  i b1

. YiT v1  0 because Ai is Metzler and v1  0. Set YiT v1 =  ..   0and ZiT v2 ≺ 0 because v2T x is the common linear biq

 i c1

. copositive Lyapunov function of {Zi }. Then, denote by ZiT v2 =  ..  ≺ 0. We want −(YiT v1 + µZiT v2 )  0. Choosing i

µ > max1≤i≤k

cq  i   T bj v max1≤1≤q − i x is a common linear copositive , then it is obvious that −(YiT v1 + µZiT v2 )  0. So µv1 2 c



j

Lyapunov function of Ai . Necessity: Assume Ai , i = 1, 2, . . . , k share a common linear copositive Lyapunov function, v T x. According to the v structure of Ai , we split v as v1  0. Then 2

−ATi



v1 µv2



 =−

XiT YiT

0 ZiT

    v1 X Tv =− T i 1T  0. v2 Yi v1 + Zi v2

We have −XiT v1  0, v1  0. {Xi } share a common linear copositive Lyapunov function v1T x. Since −YiT v1 − ZiT v2  0 and −YiT v1  0, −ZiT v2  0. It is easy to see that {Zi } share a common linear copositive Lyapunov function v2T x. This completes the proof. 

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5. Conclusions In this paper, we give a new proof of the theorem in [12]. The geometrical explanation on the relationship of the line of the slope 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 methods of the proof. Then we give a necessary condition of the existence of common linear copositive Lyapunov functions of third order 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 some block upper triangular forms, a necessary and sufficient condition of the existence of common linear copositive Lyapunov functions is presented. References [1] L. Farina, S. Rinaldi, Positive linear systems, in: Wiley Interscience Series, 2000. [2] L. Benvenuti, L. Farina, Positive and compartmental systems, IEEE Trans. Automat. Control 47 (2) (2002) 370–373. [3] W. Haddad, V. Chellaboina, E. August, Stability and dissipativity theory for non-negative dynamical system: A thermodynamic framework for biological and physiological systems, in: Proceedings of IEEE Conference on Decision and Control, 2001. [4] L. Metzler, Stability of multiple markets: the Hicks condition, Econometrica 13 (4) (1945) 277–292. [5] A. Berman, R. Plemmons, Non-negative matrices in the mathematical science, SIAM Classics in Appl. Math. (1994). [6] R. Shorten, K. Narendra, Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for a finite number of stable second order linear time-invariant systems, Internat J. Adapt. Control Signal Process. 16 (2002) 709–728. [7] C. King, R. Shorten, A singularity test for the existence of common quadratic Lyapunov functions for pairs of stable LTI systems, in: Proc. Amer. Control Conf. 2004, pp. 3881–3884. [8] L. Gurvits, A. Samorodnitsky, A note on common quadratic Lyapunov functions for linear inclusions: Exact results and open problems, in: Proc. of IEEE Conf. Decision and Control, 2005, pp. 2350–2355. [9] D. Cheng, L. Guo, J. Hunag, On quadratic Lyapunov functions, IEEE Trans. Automat. Control. 48 (2003) 885–890. [10] O. Mason, R. Shorten, On linear copositive Lyapunov functions and the stability of switched positive linear systems, IEEE Trans. Automat. Control 52 (7) (2007) 1346–1349. [11] O. Mason, R. Shorten, On the simultaneous diagonal stability of a pair of positive linear systems, Linear Algebra Appl. 413 (2006) 13–23. [12] O. Mason, Switched systems, convex cones and common Lyapunov functions, Ph.D. Thesis, 2004.