- Email: [email protected]
Exponential stability for positive systems with bounded time-varying delays and static output feedback stabilization
Available online at www.sciencedirect.com
Journal of the Franklin Institute 350 (2013) 617–636 www.elsevier.com/locate/jfranklin
Exponential stability for positive systems with bounded time-varying delays and static output feedback stabilization Shuqian Zhua,n, Min Menga, Chenghui Zhangb a
School of Mathematics, Shandong University, Jinan 250100, PR China School of Control Science and Engineering, Shandong University, Jinan 250061, PR China
b
Received 29 February 2012; received in revised form 15 December 2012; accepted 27 December 2012 Available online 6 January 2013
Abstract This paper investigates the problem of exponential stability analysis and static output feedback stabilization for discrete-time and continuous-time positive systems with bounded time-varying delays. Based on the relationship between the solution to the system with time-varying delay and that to the corresponding system with constant delay under specific conditions, the equivalence between the a-exponential stability of such two types of systems is established. Then some necessary conditions and sufficient conditions are provided for a-exponential stability of positive systems with bounded time-varying delays. It is shown that, for such systems, the exponential stability with given decay rate is closely related to the bound of the delay. Then by using the singular value decomposition approach, sufficient conditions for the existence of static output feedback controllers are established in terms of linear programming (LP) problems. Some illustrative examples are given to show the correctness of the obtained theoretical results. & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction A common property of positive systems is that the states are nonnegative whenever the initial conditions are nonnegative. Note that many real world physical systems involve nonnegative variables, for example, population levels, absolute temperature, concentration n
Corresponding author. Tel.: þ86 13853145197. E-mail addresses: [email protected] (S. Zhu), [email protected] (M. Meng), [email protected] (C. Zhang). 0016-0032/$32.00 & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2012.12.022
618
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
Nomenclature R reals N set of all nonnegative integers Rn n-dimensional linear vector space over the reals Rn0,þ ðRnþ Þ nonnegative (positive) orthant of Rn In n-dimensional identity matrix Ak0ð$0,g0,!0Þ all elements of A are nonnegative (nonpositive, positive, negative) AkBðB$AÞ ABk0 A is Metzler matrix all off-diagonal elements of A are nonnegative: Aij Z0,iaj sðAÞ spectrum of matrix A mðAÞ spectral abscissa of matrix A, that is, mðAÞ ¼ maxfRe l : l 2 sðAÞg rðAÞ spectral radius of matrix A, that is, rðAÞ ¼ maxfjlj : l 2 sðAÞg JxJ maximum modulus norm of x with JxJ ¼ maxi ¼ 1,2,...,n jxi j,x 2 Rn T A transpose of matrix A
of substances, and positive system behavior seems to be intrinsic for many real-life dynamic systems. Hence, positive system models can be found in almost all fields, for instance, engineering, ecology, economics, biomedicine, social science and so on. Since the states of positive systems are confined within a ‘‘cone’’ located in the positive orthant rather than in linear spaces, many well-established results for general linear systems cannot be readily applied to positive systems. This feature makes the analysis and synthesis of positive systems challenging and interesting. The mathematical theory of positive systems is based on the theory of nonnegative matrices founded by Perron and Frobenius [1,2] and many results have been obtained [3–7]. On the other hand, for many practical control systems, time delays are frequently encountered and recently, there has been increasing interest in time-delay systems [8–14]. General linear systems, even nominal stable systems when are affected by delays, may inherit very complex behaviors such as oscillations, instability and bad performances. In contrast, for positive linear systems with constant delays or bounded time-varying delays, it has been shown that the asymptotic stability has nothing to do with the magnitudes of delays and only dependent of the system matrices [15–21]. Such systems are asymptotically stable if and only if the sum of the system matrices is a Schur matrix (for discrete-time case) or a Hurwitz matrix (for continuous-time case). All the aforementioned references are concerned with the asymptotic stability. However, in practice, it is desirable that the system can converge quickly (that is, has a certain decay rate). Hence, it is necessary to investigate the exponential stability of positive delayed systems. Recently, such problem has been discussed for positive systems with constant delays [22–24]. In [22,23], it has been shown that positive linear time-delay differential systems are exponentially stable if and only if the corresponding systems without delay are exponentially stable, that is, whether such systems are exponentially stable is also independent of delays. However, in [22,23], the decay rate is only required to exist and can be sufficiently small. It is not further considered. In [24], the a-exponential stability analysis problem has been discussed for both continuous-time and discrete-time positive systems
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
619
with constant delays and it is demonstrated that, unlike asymptotic stability of positive time-delay systems, the exponential stability with given decay rate of positive constantdelay systems is dependent of the system matrices as well as the magnitudes of delays. However, to the best of our knowledge, little attention has been paid to the exponential stability of positive systems with time-varying delays. It should also be pointed out that in most literature aforementioned for control of positive systems, the controller takes the form of state feedback. However, in practical applications, it is often impossible to obtain the full information on the state variables. Hence, it is necessary to investigate the output feedback stabilization problem. The positive stabilization problem with dynamic output feedback controllers has been investigated in [25] with sufficient conditions for the existence of the designed controller proposed, and an iteration linear matrix inequality (ILMI) algorithm provided to solve them. In [26,27], necessary and sufficient conditions for the existence of static and dynamic output feedback controllers have been established in terms of LMIs together with a matrix equality constraint, which can be solved via the cone complementarity linearization technique. In comparison with LMI approach, the linear programming (LP) approach is simpler and posses a numerical advantage since the existing LMI softwares cannot handle large size problems and are not numerically stable [6,28]. Recently, the LP approach has been used to handle the state feedback stabilization [6,20,29] and output feedback stabilization [28] for positive systems. In this paper, the exponential stability analysis problem with given decay rate is investigated for discrete-time and continuous-time positive systems with bounded timevarying delays, with necessary conditions and sufficient conditions provided. Then based on the obtained exponential stability results and by using the singular value decomposition approach, sufficient conditions for the existence of static output feedback controllers are established in terms of standard LP problems. 2. Exponential stability analysis 2.1. Discrete-time case Consider the following discrete-time delayed system: ( xðt þ 1Þ ¼ AxðtÞ þ At xðttðtÞÞ, xðsÞ ¼ fðsÞ, s ¼ t,ðt1Þ, . . . ,0,
ð1Þ
where t 2 N, x(t) 2 Rn is the state, A and At are known real constant matrices, the delay tðtÞ 2 N is the bounded time-varying delay which satisfies 0rtðtÞrt,
t2N
ð2Þ
with the constant t 2 N, and f : ft,ðt1Þ, . . . ,0g-Rn0,þ is a sequence specifying the initial state of the system. Definition 1 (Liu et al. [19]). System (1) is said to be positive if for any f : ft,ðt1Þ, . . . ,0g-Rn0,þ , the corresponding trajectory satisfies xðtÞk0 for all t 2 N. Lemma 1 (Liu et al. [19]). System (1) is positive if and only if Ak0 and At k0.
620
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
Lemma 2 (Liu et al. [19]). Assume that system (1) is positive. Let xa(t) and xb(t) be the trajectories of Eq. (1) under the initial conditions fa ðÞ and fb ðÞ, respectively. Then, fa ðsÞ$fb ðsÞ, s 2 ft,ðt1Þ, . . . ,0g implies that xa ðtÞ$xb ðtÞ for all t 2 N. In practice, it is often not enough demanding that the system is asymptotically stable. Then it is desirable that the system can converge quickly (that is, has a certain decay rate). To emphasize the decay rate and discuss whether and how the stability changes with the delay’s changing, similarly to [24], we give the definition of a-exponential stability for system (1) as follows. Definition 2. For given scalar 0oao1, system (1) is called a-exponentially stable if there exists a scalar G40 such that the solution of system (1) satisfies JxðtÞJrGJfJd at for all t 2 N, where JfJd :¼ sups2ft,ðt1Þ,...,0g JfðsÞJ. Since in many cases, the time-varying delay tðtÞ is unknown and the upper bound t is known, in this paper, we focus on the a-exponential stability analysis for all tðtÞ satisfying given bound conditions. Now, inspired by [19], we introduce the following system closely related to system (1) and (2) with constant delay: ( yðt þ 1Þ ¼ AyðtÞ þ At yðttÞ, ð3Þ yðsÞ ¼ cðsÞ, s ¼ t,ðt1Þ, . . . ,0, where all the system matrices are the same as in Eq. (1), and t is the upper bound of tðtÞ in Eq. (1), as shown in Eq. (2). For system (3), we have: Lemma 3 (Liu [29]). Assume that system (3) is positive, then the following statements are equivalent. (i) System (3) is asymptotically stable. (ii) There exists a vector pg0 such that ðA þ At In Þp!0. (iii) A þ At is a Schur matrix, i.e., rðA þ At Þo1. Lemma 4 (Zhu et al. [24]). If the positive system (3) is a-exponentially stable, then (i) rða1 A þ aðtþ1Þ At Þr1; (ii) there exists a nonzero vector pk0 such that ða1 A þ aðtþ1Þ At In ÞT p$0:
ð4Þ
Lemma 5 (Zhu et al. [24]). If one of the following conditions holds: (i) rða1 A þ aðtþ1Þ At Þo1; (ii) there exists a vector pg0 satisfying Eq. (4), then the positive system (3) is a-exponentially stable. The next lemma reveals the relationship between the solution to system (1) and that to system (3) under certain conditions.
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
621
Lemma 6 (Liu et al. [19]). Suppose that there exists a vector pg0 satisfying ðA þ At In Þp!0
ð5Þ
and that the initial conditions for system (1) and (3) satisfy: fðsÞ cðsÞ p, s 2 ft, ðt1Þ, . . . ,0g. Then xðtÞ$yðtÞ
ð6Þ
holds for all t 2 N, where x(t) and y(t) are solutions to Eqs. (1) and (3), respectively. The equivalence between the a-exponential stability of system (1) for all tðtÞ satisfying Eq. (2) and that of system (3) will be established in the theorem as follows. Theorem 1. The positive system (1) is a-exponentially stable for all tðtÞ satisfying Eq. (2) if and only if the positive system (3) is a-exponentially stable. Proof. Necessity. Suppose that the positive system (1) is a-exponentially stable for all tðtÞ satisfying Eq. (2). Particularly, let tðtÞ ¼ t, then the positive system (3) is necessarily a-exponentially stable. Sufficiency. Suppose that the positive system (3) is a-exponentially stable. Thus there ~ exists a scalar G40 such that the solution of system (3) satisfies ~ JyðtÞJrGJcJd at , t 2 N: ð7Þ Furthermore, since system (3) is a-exponentially stable, it is asymptotically stable. Then by ~ ~ Lemma 3, there exists a vector pg0 satisfying ðA þ At In Þp!0. In particular, choose the initial condition cðÞ mp~ with scalar m40, then the corresponding solution ymp~ ðtÞ must satisfy ~ pJa ~ t , t 2 N: Jymp~ ðtÞJrGmJ ð8Þ It follows from Lemma 6 that xmp~ ðtÞ$ymp~ ðtÞ,
t 2 N,
ð9Þ
where xmp~ ðtÞ is the corresponding solution of system (1) when the initial condition ~ fðÞ mp. For arbitrary initial condition fðÞ 2 Rn0,þ , let p~ ¼ ½p~ 1 p~ 2 p~ n T and m ¼ JfJd =mini ¼ 1,2,...,n p~ i , then ~ fðsÞ$mp,
s 2 ft,ðt1Þ, . . . ,0g:
ð10Þ
Thus it is obtained from Lemma 2 that for any initial condition fðÞ 2 Rn0,þ , the corresponding solution satisfies ~ pJa ~ t ¼ G~ Jxf ðtÞJ$Jxmp~ ðtÞJ$Jymp~ ðtÞJ$GmJ
JfJd mini ¼ 1,2,...,n p~ i
~ t ¼: GJfJd at JpJa
ð11Þ
~ pJ=min ~ ~ i 40: Therefore, the positive system (1) is a-exponentially with G ¼ GJ i ¼ 1,2,...,n p stable for all tðtÞ in Eq. (2). & Remark 1. From Theorem 1 it can be seen that the a-exponential stability of the positive system (1) for all tðtÞ : 0rtðtÞrt is determined by the a-exponential stability of the positive system (3). This result is still valid for a more general case: tm rtðtÞrt,tm Z0, which is called interval time-varying delay [30,31]. When the lower bound tm is considered, introduce system (3) with t replaced by tm , which is denoted by system (3)0 , the conclusion
622
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
corresponding to Theorem 1 is: the positive system (1) is a-exponentially stable for some tðtÞ : tðtÞZtm if and only if the positive system (3)0 is a-exponentially stable. Obviously, this result gives us a condition, which is that the positive system (3)0 is not a-exponentially stable, to conclude that if tðtÞZtm , then the positive system (1) is not a-exponentially stable. Based on Lemmas 4 and 5 and Theorem 1, we are now ready to provide the necessary conditions and sufficient conditions for the a-exponential stability of the positive system (1) for all tðtÞ satisfying Eq. (2). Theorem 2. If the positive system (1) is a-exponentially stable for all tðtÞ satisfying Eq. (2), then the conditions (i) and (ii) in Lemma4 hold. Theorem 3. If the condition (i) or (ii) in Lemma5 holds, then the positive system (1) is a-exponentially stable for all tðtÞ satisfying Eq. (2). Remark 2. It is known from [19] that the positive system (1) is asymptotically stable for all tðtÞ satisfying Eq. (2) if and only if A þ At is a Schur matrix, which shows that the magnitude of the delay does not affect the asymptotic stability of Eq. (1). However, as shown in Theorems 2 and 3, the a-exponential stability of the positive system (1) for all tðtÞ in Eq. (2) not only depends on the system matrices but also is closely related to the upper bound of the time-varying delay. Therefore, Theorems 2 and 3 reveal the difference between the asymptotical stability and the exponential stability with given decay rate for positive discrete-time systems with bounded time-varying delays. Remark 3. If n¼ 1, that is, system (1) is a scalar system, it follows from Theorems 2 and 3 that the positive system (1) is a-exponentially stable for all tðtÞ satisfying Eq. (2) if and only if a1 A þ aðtþ1Þ At 1r0:
ð12Þ
Lemma 7 (Horn and Johnson [32]). For two matrices A,B 2 Rnn , the following statements hold: (i) if AkBk0, then rðAÞZrðBÞ. (ii) if AgBk0, then rðAÞ4rðBÞ. Remark 4. It should be noticed that the obtained conditions in Theorems 2 and 3 are necessary conditions and sufficient conditions, respectively, rather than necessary and sufficient conditions. Let t0 2 N satisfy rða1 A þ aðt0 þ1Þ At Þ ¼ 1. Since by Lemma 7, rða1 A þ aðtþ1Þ At Þorða1 A þ aðt0 þ1Þ At Þ ¼ 1 for all t : 0rtot0 if Ak0,At k0, it follows from Theorem 3 that the positive system (1) is a-exponentially stable for all tðtÞ satisfying Eq. (2) if the upper bound t satisfying tot0 . Similarly, for all t4t0 , rða1 A þ aðtþ1Þ At Þ4rða1 A þ aðt0 þ1Þ At Þ ¼ 1 implies that the positive system (1) is not a-exponentially stable for some tðtÞ satisfying Eq. (2) if t4t0 according to Theorem 2. However, as for the critical case when t ¼ t0 , the a-exponential stability cannot be justified by the results in this paper except for the positive scalar system.
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
623
Remark 5. The results in Theorems 2 and 3 can be easily extended to the positive system with multiple time-varying delays of the form 8 m X > < xðt þ 1Þ ¼ AxðtÞ þ Ati xðtti ðtÞÞ, ð13Þ i¼1 > : xðsÞ ¼ fðsÞ, s ¼ t,ðt1Þ, . . . ,0, where t 2 N, x(t) 2 Rn , Ak0 and Ati k0, i ¼ 1,2, . . . ,m, are known real constant matrices, delays ti ðtÞ 2 N,i ¼ 1,2, . . . ,m are bounded time-varying delays which satisfy 0rti ðtÞrti ,
t2N
ð14Þ
with constants ti 2 N, t ¼ maxi ¼ 1,2,...,m ti and f : ft,ðt1Þ, . . . ,0g-Rn0,þ P is a sequence ðti þ1Þ specifying the initial state of the system. In this case, only by substituting m Ati i¼1a ðtþ1Þ for a At in Theorems 2 and 3, the corresponding results of the a-exponential stability of system (13) for all ti ðtÞ satisfying Eq. (14) can be obtained. Next, based on Theorem 3, we will provide the robust stability analysis results for two kinds of uncertain cases of system (1). Corollary 1. Consider the uncertain time-delay system (1) with unknown matrices A and At belonging to the sets L1 ¼ fA : Am $A$AM g,
L2 ¼ fAt : Atm $At $AtM g,
respectively, where Am ,AM ,Atm ,AtM are known bounds. If Am k0,Atm k0 and there exists a vector pg0 such that ða1 AM þ aðtþ1Þ AtM In ÞT p$0, then the uncertain system (1) is positive and a-exponentially stable for all tðtÞ satisfying (2) and A 2 L1 ,At 2 L2 . Proof. It is directly proved by using Theorem 3 and Lemma 7.
&
Corollary 2. Consider the uncertain time-delay system (1) with unknown matrices ½A At belonging to the convex set ( ) l l X X i i Y¼ bi ½A At j bi ¼ 1,bi Z0 , i¼1
i¼1
where ½A1 A1t , . . . ,½Al Alt are known matrices. If Ai k0,Ait k0, i ¼ 1,2, . . . ,l and there exists a vector pg0 such that ða1 Ai þ aðtþ1Þ Ait In ÞT p$0,
i ¼ 1,2, . . . ,l,
then the uncertain system (1) is positive and a-exponentially stable for all tðtÞ satisfying Eq. (2) and ½A At 2 Y.
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
624
2.2. Continuous-time case Consider the continuous-time delayed system: ( _ ¼ AxðtÞ þ At xðttðtÞÞ, xðtÞ xðsÞ ¼ fðsÞ,
s 2 ½t,0,
ð15Þ
where tZ0, x(t) 2 Rn is the state, A and At are known real constant matrices. The delay tðtÞZ0 is the bounded time-varying delay which is continuous with respect to t and satisfies 0rtðtÞrt,
tZ0
ð16Þ ½t,0-Rn0,þ
with the constant tZ0. f : the initial state of the system.
is a continuous vector valued function specifying
Definition 3 (Liu et al. [21]). System (15) is said to be positive if for any f : ½t,0-Rn0,þ , the state trajectory satisfies xðtÞk0 for all tZ0. Lemma 8 (Liu et al. [21]). System (15) is positive if and only if A is Metzler and At k0. Similar to [24], the a-exponential stability for continuous-time system (15) is proposed as follows. Definition 4. For given scalar a40, system (15) is called a-exponentially stable if there exists a scalar G40 such that the solution of system (15) satisfies JxðtÞJrGJfJc eat for all tZ0, where JfJc :¼ suptrsr0 JfðsÞJ. Inspired by [21], we introduce the following system closely related to system (15) with constant delay: ( _ ¼ AyðtÞ þ At yðttÞ, yðtÞ ð17Þ yðsÞ ¼ cðsÞ, s 2 ½t,0, where all the system matrices are the same as in Eq. (15), and t is the upper bound of tðtÞ in Eq. (15), as shown in Eq. (16). Next, similarly to the discrete-time case, based on the results given in [21,24], we will present in the following theorems some necessary conditions and sufficient conditions for the a-exponential stability of the continuous-time positive system (15). Theorem 4. The positive system (15) is a-exponentially stable for all tðtÞ satisfying Eq. (16) if and only if the positive system (17) is a-exponentially stable. Theorem 5. If the positive system (15) is a-exponentially stable for all tðtÞ satisfying Eq. (16), then (i) mðaIn þ A þ At eat Þr0; (ii) there exists a nonzero vector pk0 such that ðaIn þ A þ At eat ÞT p$0: Theorem 6. If one of the following conditions holds: (i) mðaIn þ A þ At eat Þo0;
ð18Þ
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
625
(i) there exists a vector pg0 satisfying Eq. (18), then the positive system (15) is a-exponentially stable for all tðtÞ satisfying Eq. (16). Remark 6. It has been shown in [21] that the positive system (15) is asymptotically stable for all tðtÞ satisfying Eq. (16) if and only if A þ At is a Hurwitz matrix, that is, the magnitude of tðtÞ has no any impact on its asymptotic stability. However, it can be seen from Theorems 5 and 6 that the a-exponential stability of the positive system (15) for all tðtÞ in Eq. (16) is closely related to the upper bound of tðtÞ. This result is similar to that of the discrete-time case given in Remark 2. Remark 7. If n¼ 1, then the positive system (15) is a-exponentially stable for all tðtÞ satisfying Eq. (16) if and only if a þ A þ At eat r0:
ð19Þ
P ati Remark 8. By substituting m for At eat in Theorems 5 and 6, we can obtain the i ¼ 1 Ati e corresponding results of a-exponential stability for system with multiple delays: 8 m X > < xðtÞ _ ¼ AxðtÞ þ Ati xðtti ðtÞÞ, ð20Þ i¼1 > : xðsÞ ¼ fðsÞ, s 2 ½t,0, where tZ0, xðtÞ 2 Rn , A is Metzler and Ati k0, i ¼ 1,2, . . . ,m, delays ti ðtÞZ0, i ¼ 1,2, . . . ,m are bounded time-varying delays which are continuous with respect to t and satisfy 0rti ðtÞrti ,
tZ0
ð21Þ
with constants ti Z0 and t ¼ maxi ¼ 1,2,...,m ti . 3. Static output feedback stabilization 3.1. Discrete-time case In this subsection, we consider the following discrete-time delayed system: 8 > < xðt þ 1Þ ¼ AxðtÞ þ At xðttðtÞÞ þ BuðtÞ, yðtÞ ¼ CxðtÞ, > : xðsÞ ¼ fðsÞ, s ¼ t,ðt1Þ, . . . ,0,
ð22Þ
where t 2 N, xðtÞ 2 Rn , uðtÞ 2 Rr , yðtÞ 2 Rq are, respectively, the state, the control input and the measurable output. A, At , B and C are known real constant matrices with appropriate dimensions and At k0. The delay tðtÞ and the initial function fðtÞ are the same as in Eqs. (1) and (2). The purpose of this subsection is to design a static output feedback controller uðtÞ ¼ KyðtÞ
ð23Þ
for system (22) such that for given a : 0oao1, the resultant closed-loop system ( xðt þ 1Þ ¼ ðA þ BKCÞxðtÞ þ At xðttðtÞÞ, xðsÞ ¼ fðsÞ,
s ¼ t,ðt1Þ, . . . ,0
ð24Þ
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
626
is positive and a-exponentially stable for all tðtÞ in Eq. (2), where K 2 Rrq is the feedback gain matrix. First, we consider the single-output case, i.e., q ¼ 1. The existence condition of the controller (23) is provided in the next theorem by using the singular value decomposition approach. Theorem 7. For system (22) with Ck0, make the singular value decomposition of C as CV T ¼ ½s 0 0 with s40 being the singular value of C and V being an orthogonal matrix. If there exist vectors l ¼ ½l1 l2 ln T 2 Rn and z 2 Rr such that the following LP problem is solvable: ½a1 A þ aðtþ1Þ At In V T l þ a1 sBz$0,
ð25Þ
V T lg0,
ð26Þ
aij l1 þ bi zcj Z0,
i,j ¼ 1,2, . . . ,n,
ð27Þ
where A ¼ ½aij ,
B ¼ ½bT1 bT2 bTn T ,
C ¼ ½c1 c2 cn ,
then there exists a static output feedback controller (23) such that the closed-loop system (24) is positive and a-exponentially stable for all tðtÞ in Eq. (2). In this case, the feedback gain matrix K can be designed as K ¼ l1 1 z:
ð28Þ
Proof. Let V ¼ ½vT1 vT2 vTn T and p ¼ V T l, then pg0 by Eq. (26). And it gets from l ¼ Vp and C ¼ ½s 0 0V that l1 ¼ v1 p and v1 ¼ C=s. Since s40,pg0, Ck0 and C contains at least a nonzero component, it can get easily that l1 40. Define K ¼ l1 1 z, then substituting z ¼ l1 K into Eq. (27) yields aij l1 þ bi l1 Kcj ¼ ðaij þ bi Kcj Þl1 Z0,
ð29Þ
which implies that aij þ bi Kcj Z0 since l1 40. Thus A þ BKCk0 and the closed-loop system (24) is positive. Now, substituting p ¼ V T l, z ¼ l1 K and CV T ¼ ½s 0 0 into Eq. (25) concludes that ½a1 A þ aðtþ1Þ At In p þ a1 sBKl1 ¼ ½a1 A þ aðtþ1Þ At In p þ a1 BK½s 0 0½l1 l2 ln T ¼ ½a1 A þ aðtþ1Þ At In p þ a1 BKCV T l ¼ ½a1 ðA þ BKCÞ þ aðtþ1Þ At In p$0:
ð30Þ
By Theorem 3, we have that the closed-loop system (24) is a-exponentially stable for all tðtÞ in Eq. (2). Thus the proof is complete. & Remark 9. In [28], the static output feedback stabilization problem has been solved for positive linear systems without delay with necessary and sufficient conditions obtained for single-output case. It can be seen that to get the standard LP formulation of the proposed approach in [28], additional vec operation and the Kronecker product are further used. However, in Theorem 7, via another approach: the singular value decomposition approach, the obtained conditions can be solved as a standard LP problem. The detailed design procedure is presented as follows:
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
627
Step 1: Make the singular decomposition of C as CV T ¼ ½s 0 0 to obtain V and s. Step 2: Rewrite conditions (25)–(27) as 3 2 1 a A þ aðtþ1Þ At In V T a1 sB 3 2 O 7 6 6 V T O 7 6 Eg 7 7 6 7 6 7 6 7 6 AE Bc 1 1 7 6 7 O ð31Þ 7b$6 6 7 6 6 AE 2 Bc2 7 7 6 7 6 4 ^ 5 6 ^ ^ 7 5 4 O AE n Bcn where the vector to be computed is b ¼ ½lT zT T with l ¼ ½l1 l2 ln T , O represents a matrix with compatible dimension whose elements are all 0, Ej is a matrix with 1 in the jth row, 1st column position and zeros elsewhere, g is a n-dimensional vector whose elements are all 1, and E40 is a given small scalar (which is introduced to guarantee the strict positivity constraint V T lg0). Then solve the above standard LP problem by using linprog Toolbox in Matlab. Step 3. If the above LP problem is solvable, then the static output feedback gain K can be designed as K ¼ l1 1 z. The corresponding results are stated as follows for the general case when the matrix C is not sign restricted. Theorem 8. For system (22), make the singular value decomposition of C and denote A,B,C as shown in Theorem7. Then there exists a static output feedback controller (23) such that the closed-loop system (24) is positive and a-exponentially stable for all tðtÞ in Eq. (2), if there exist vectors l ¼ ½l1 l2 ln T 2 Rn and z 2 Rr such that at least one of the following two LP problems is feasible: LP1 (25), (27) and V T lg0,
l1 40,
ð32Þ
LP2 (25), and V T lg0,
l1 o0,
ð33Þ
aij l1 þ bi zcj r0,
i,j ¼ 1,2, . . . ,n:
ð34Þ
Moreover, the gain matrix K can be designed as Eq. (28). For the case of multi-output system, we design the control which has the following structure as in [28] uðtÞ ¼ KnyðtÞ,
K 2 Rr ,
ð35Þ
1q
is a fixed parameter design. By using this kind of controls which involve where n 2 R one rank gains, we can maintain the resultant closed-loop system positive and a-exponentially stable. This is shown in the following result. Theorem 9. Assume that n 2 R1q be a fixed parameter design such that nCk0. Make the singular value decomposition of nC as nCV T ¼ ½s 0 0 with s40 being the singular value
628
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
of nC and V being an orthogonal matrix. Denote A, B and C as shown in Theorem 7. Then there exists a static output feedback controller (35) such that the closed-loop system is positive and a-exponentially stable for all tðtÞ in Eq. (2), if the following LP problem is feasible in the variables l ¼ ½l1 l2 ln T 2 Rn and z 2 Rr : (25), (26) and aij l1 þ bi zncj Z0,
i,j ¼ 1,2, . . . ,n:
ð36Þ
Moreover, the matrix K can be calculated as K ¼ l1 1 z. 3.2. Continuous-time case All the results presented in Section 3.1 can be easily extended to the continuous-time system 8 _ ¼ AxðtÞ þ At xðttðtÞÞ þ BuðtÞ, > < xðtÞ yðtÞ ¼ CxðtÞ, ð37Þ > : xðsÞ ¼ fðsÞ, s 2 ½t,0: Only by substituting aij l1 þ bi zcj ZðrÞ0, iaj,i,j ¼ 1,2, . . . ,n for aij l1 þ bi zcj ZðrÞ0, i,j ¼ 1,2, . . . ,n, and ½aIn þ A þ At eat V T l þ sBz$0 for ½a1 A þ aðtþ1Þ At In V T l þ a1 sBz$0, we can get the existence conditions of the static output feedback controller (23) for system (37). Remark 10. It should be noted that in this section, the positivity restrictions are not imposed on the open-loop system matrices A,B,C and the feedback matrix K. We only require that the closed-loop system is positive and a-exponentially stable. Therefore, the synthesis problem considered in this section can be interpreted as enforcing the system to be positive. In fact, take the discrete-time case for example, if the open-loop system is positive (i.e., Ak0,At k0, Bk0, Ck0) and rða1 A þ aðtþ1Þ At Þ41, then there exists no nonnegative feedback uðtÞ ¼ KyðtÞ such that the closed-loop system is positive and a-exponentially stable for all tðtÞ in Eq. (2) since rða1 ðA þ BKCÞþ aðtþ1Þ At Þ Z rða1 A þ aðtþ1Þ At Þ41. 4. Numerical examples Example 1. Consider the discrete-time delayed system (1) given in [19] with a 0:1 0:1 0:1 A¼ , At ¼ , 0:2 0:3 0:2 0:5 where t ¼ 1, a is a nonnegative parameter, by Lemma 1, which implies that system (1) is positive. Computation shows that if a ¼ 0.1635, the two eigenvalues of the matrix ð0:951 A þ 0:95ð1þ1Þ At Þ are 1 and 0.1526, and both of them lie inside the unit circle if ao0:1635 and at least one of them lies outside if a40:1635. By Theorems 2 and 3, one can conclude that system (1) is 0:95-exponentially stable for all delay tðtÞ satisfying 0rtðtÞr1 if 0rao0:1635, and not 0:95-exponentially stable for some tðtÞ satisfying 0rtðtÞr1 if a40:1635. And since when a ¼ 0.16, the eigenvalues of the matrix ð0:951 A þ 0:95ð2þ1Þ At Þ are 0.1561 and 1.0315, one gets that system (1) is not 0:95-exponentially stable for some tðtÞ satisfying 0rtðtÞr2 if a ¼ 0.16.
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
629
Figs. 1–3 give the simulation results of x1 ðtÞ,x2 ðtÞ and X ðtÞ ¼ 0:95t when a¼ 0.16 (Figs. 1 and 2) and a¼ 0.25 (Fig. 3), respectively. The initial functions randomly take values on the interval ½0,1, the delay tðtÞ randomly takes values in the set f0,1g in Fig. 1, tðtÞ ¼ 2 in Fig. 2 and tðtÞ ¼ 1 in Fig. 3. From the three figures, one can see that x(t) satisfies JxðtÞJr0:95t for all tZ0 when a ¼ 0:16o0:1635 and tðtÞr1, while in the case of a ¼ 0:16,tðtÞ ¼ 241 and the case of a ¼ 0:2540:1635,tðtÞ ¼ 1, when t is large enough, JxðtÞJ will be larger than 0:95t . 1
x1 x2 X
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
20
40
60
80
100
120
time t
Fig. 1. The simulation results of Example 1 when a ¼ 0:16,tðtÞr1. 1
x1 x2 X
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
20
40
60
80
100
120
time t
Fig. 2. The simulation results of Example 1 when a ¼ 0:16,tðtÞ ¼ 2.
630
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636 1
x1 x2 X
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
20
40
60
80
100
120
time t
Fig. 3. The simulation results of Example 1 when a ¼ 0:25,tðtÞ ¼ 1.
Example 2. Consider the continuous-time delayed system (15) given in [21] with 5 1:3 a 0:8 A¼ , At ¼ , 1:5 3 1 1:2 where t ¼ 3:9, a is a nonnegative parameter, by Lemma 8, which implies that system (15) is positive. Based on Theorems 5–6 and by computation, one can conclude that system (15) is 0.05-exponentially stable for all delay tðtÞ satisfying 0rtðtÞr3:9 if 0rao0:6695, not 0.05exponentially stable for some tðtÞ satisfying 0rtðtÞr3:9 if a40:6695, and not 0.05exponentially stable for some tðtÞ satisfying 0rtðtÞr4:9 if a ¼ 0.66. Figs. 4–6 give the simulation results of x1 ðtÞ,x2 ðtÞ and X ðtÞ ¼ 0:35 e0:05t when a ¼ 0.66 (Figs. 4 and 5) and a¼ 1.2 (Fig. 6), respectively. The initial function is fðtÞ ¼ ½0:3 0:2, the delay tðtÞ ¼ 4:9 in Fig. 5, and tðtÞ ¼ 3 þ 0:9 sinðtÞ in Figs. 4 and 6. The three figures show that x(t) satisfies JxðtÞJr0:35 e0:05t for all tZ0 when a ¼ 0:66o0:6695 and tðtÞr3:9, while in the case of a ¼ 0:66,tðtÞ ¼ 4:943:9 and the case of a ¼ 1:240:6695, tðtÞ ¼ 3 þ 0:9 sinðtÞ, when t is large enough, JxðtÞJ will be larger than 0:35 e0:05t . Example 3. Consider a practical example about a certain pest’s structured population dynamics described by the following Leslie matrix model: 8 2 3 2 3 f1 f2 f3 b1 b2 > > > > 6 7 6 < 0 07 xðt þ 1Þ ¼ 4 p1,2 5xðtÞ þ 4 0 0 5uðtÞ, 0 p2,3 0 > 0 0 > > > : yðtÞ ¼ ½0 c cxðtÞ, where xðtÞ ¼ ½x1 ðtÞ x2 ðtÞ x3 ðtÞT and x1 ðtÞ represents the number of juvenile pests at time t, x2 ðtÞ represents the number of immature pests at time t, x3 ðtÞ represents the number of adult pests at time t. Assume that the parameters for this model are given in [25] (without
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636 x1 x2 X
0.3 0.25 0.2 0.15 0.1 0.05 0
0
20
40
60
80
100
120
times t [s]
Fig. 4. The simulation results of Example 2 when a ¼ 0:66,tðtÞ ¼ 3 þ 0:9 sinðtÞ.
x1 x2 X
0.3 0.25 0.2 0.15 0.1 0.05 0
0
20
40
60
80
100
120
times t [s]
Fig. 5. The simulation results of Example 2 when a ¼ 0:66,tðtÞ ¼ 4:9.
uncertainties) as f1 ¼ 0:3011,
f2 ¼ 0:5915,
p2,3 ¼ 0:7894,
f3 ¼ 0:5235,
b1 ¼ 0:9000,
p1,2 ¼ 0:8868,
b2 ¼ 0:5000,
c ¼ 1:000:
Obviously, this model is a special case of system (22) when At ¼ 0.
631
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
632
x1 x2 X
0.3 0.25 0.2 0.15 0.1 0.05 0
0
20
40
60
80
100
120
times t [s]
Fig. 6. The simulation results of Example 2 when a ¼ 1:2,tðtÞ ¼ 3 þ 0:9 sinðtÞ.
Since the eigenvalues of A are 1:0904,0:394670:4247i, the open-loop system is not stable. Making the singular value decomposition of C yields 2 3 0 0:7071 0:7071 6 7 V ¼ 4 0:7071 0:5000 0:5000 5 and s ¼ 1:4142: 0:7071
0:5000
0:5000
Choose a ¼ 0:8, E ¼ 1:0 1010 , and solve the standard LP problem (31) by using linprog Toolbox in Matlab. A feasible solution is obtained as 2 3 163:8517 66:5035 6 7 : l ¼ 4 66:1464 5, z ¼ 36:9464 21:3024 Thus the static output feedback gain matrix is designed as K¼
0:4059 0:2255
:
Figs. 7 and 8 give the trajectory simulations of the open-loop system and the closed-loop system, respectively, with xð0Þ ¼ ½300 250 200T and X ðtÞ ¼ 350 0:8t . We can see that the open-loop system is not stable and the number of the pests increases, while the close-loop system is 0.8-exponentially stable and the number of the pests decays more quickly than X(t) with the designed output feedback law.
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636 6000
x1 x2 x3
5000 4000 3000 2000 1000 0
0
5
10
15
20
25
30
Fig. 7. The open-loop simulation results of Example 3.
350
x1 x2 x3 X
300 250 200 150 100 50 0
0
5
10
15
20
25
30
Fig. 8. The closed-loop simulation results of Example 3.
Example 4. Consider the single-output system (37) 2 2 3 0:002 0:15 1:90 1:55 6 6 7 A ¼ 4 0:50 0:3 0:10 5, At ¼ 4 0:01 0:01 0:20 0:50 2:55
with tðtÞ ¼ 3 þ 0:5 sinðtÞ and 3 0:01 0:03 7 0:01 0:04 5, 0:03 0:04
633
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
634
1000
x1 x2 x3
900 800 700 600 500 400 300 200 100 0
0
2
4
6
8
10 12 times t [s]
14
16
18
20
Fig. 9. The open-loop simulation results of Example 4.
2
1:05 6 B¼4 0 0:09
3 1:04 7 0:8 5,
C ¼ ½1 1 0:
0:50
Note that the matrix A þ At is not a Hurwitz matrix since its eigenvalues are 0.9033, 1:2422 and 2:6091, thus the open-loop system is not stable. Choose a ¼ 0:2, E ¼ 1:0 1010 and solve the standard LP problem got in Section 3.2 by using linprog Toolbox in Matlab. The static output feedback gain matrix is designed as 1:5512 K¼ : 0:1065 The state trajectories of the open-loop system and the closed-loop system are shown in Figs. 9 and 10, respectively, where x ¼ ½x1 x2 x3 T , the initial condition is xðtÞ ¼ ½0:9 0:2 0:1T ,t 2 ½3:5,0, and X ðtÞ ¼ e0:2t . From Figs. 9 and 10 we can see that the open-loop system is not stable while the closed-loop system is 0.2-exponentially stable with the designed static output feedback uðtÞ ¼ KyðtÞ. 5. Conclusions and future works Based on the relation between the solution to the system with time-varying delay and that to the corresponding system with constant delay under specific conditions, as well as the a-exponential stability conditions for constant-delay positive systems, some necessary conditions and sufficient conditions have been established for a-exponential stability of continuous-time and discrete-time positive systems with bounded time-varying delays. It is demonstrated that the a-exponential stability of positive systems with bounded timevarying delays is equivalent to that of the positive systems with constant delay which is just
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636 1
635
x1 x2 x3 X
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
2
4
6
8
10 12 times t [s]
14
16
18
20
Fig. 10. The closed-loop simulation results of Example 4.
the upper bound of the time-varying delays, and thus is dependent of delays. Then based on the obtained sufficient conditions and by using the singular value decomposition approach, the static output feedback stabilization problem is solved. The controller for single-output case, which guarantees that the resultant closed-loop system is not only positive, but also a-exponentially stable, can be efficiently designed by solving standard LP problems. The controller with one rank gain for the multi-output case has also been discussed. The a-exponential stability analysis problem for positive systems with time-varying delays involving the delay-derivatives, such as fast varying and slow varying time-delays, as well as the problem of how to design the general static output feedback controller for multi-output systems by using the LP based approach, can be further investigated. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grants 61004011 and 61034007.
References [1] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, PA, 1994. [2] L. Farina, S. Rinaldi, Positive Linear Systems: Theory and its Applications, Wiley, New York, 2000. [3] H. Gao, J. Lam, C. Wang, S. Xu, Control for stability and positivity: equivalent conditions and computation, IEEE Transactions on Circuits and Systems II: Express Briefs 52 (9) (2005) 540–544. [4] M.A. Rami, F. Tadeo, Positive observation problem for linear discrete positive systems, in: Proceedings of the 45th Conference on Decision and Control, San Diego, CA, 2006, pp. 4729–4733.
636
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
[5] O. Mason, R. Shorten, On linear copositive Lyapunov functions and the stability of switched positive linear systems, IEEE Transactions on Automatic Control 52 (7) (2007) 1346–1349. [6] M.A. Rami, F. Tadeo, Controller synthesis for positive linear systems with bounded controls, IEEE Transactions on Circuits and Systems II: Express Briefs 54 (2) (2007) 151–155. [7] X. Ding, L. Shu, X. Liu, On linear copositive Lyapunov functions for switched positive systems, Journal of the Franklin Institute 348 (8) (2011) 2099–2107. [8] J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. [9] J.-P. Richard, Time-delay systems: an overview of some recent advances and open problems, Automatica 39 (10) (2003) 1667–1694. [10] Q.-L. Han, Absolute stability of time-delay systems with sector-bounded nonlinearity, Automatica 41 (12) (2005) 2171–2176. [11] S. Zhu, C. Zhang, Z. Cheng, J. Feng, Delay-dependent robust stability criteria for two classes of uncertain singular time-delay systems, IEEE Transactions on Automatic Control 52 (5) (2007) 880–885. [12] X. Jiang, Q.-L. Han, New stability criteria for linear systems with interval time-varying delay, Automatica 44 (10) (2008) 2680–2685. [13] S. Ma, E.-K. Boukas, Y. Chinniah, Stability and stabilization of discrete-time singular Markov jump systems with time-varying delay, International Journal of Robust Nonlinear Control 20 (5) (2010) 531–543. [14] M.S. Mahmoud, Y. Xia, Improved exponential stability analysis for delayed recurrent neural networks, Journal of the Franklin Institute 348 (2) (2011) 201–211. [15] W.M. Haddad, V. Chellaboina, Stability theory for nonnegative and compartmental dynamical systems with time delays, System & Control Letters 51 (5) (2004) 355–361. [16] M.A. Rami, U. Helmke, F. Tadeo, Positive observation problem for linear time-delay positive systems, in: Proceedings of the 15th Mediterranean Conference on Control & Automation, Athens, Greece, 2007, pp. 1–6. [17] M. Buslowicz, Simple stability conditions for linear positive discrete-time systems with delays, Bulletin of the Polish Academy of Sciences: Technical Sciences 56 (4) (2008) 325–328. [18] T. Kaczorek, Stability of positive continuous-time linear systems with delays, Bulletin of the Polish Academy of Sciences: Technical Sciences 57 (4) (2009) 395–398. [19] X. Liu, W. Yu, L. Wang, Stability analysis of positive systems with bounded time-varying delays, IEEE Transactions on Circuits and Systems II: Express Briefs 56 (7) (2009) 600–604. [20] M.A. Rami, Stability analysis and synthesis for linear positive systems with time-varying delays, in: Positive Systems, Springer-Verlag, Berlin, 2009. [21] X. Liu, W. Yu, L. Wang, Stability analysis for continuous-time positive systems with time-varying delays, IEEE Transactions on Automatic Control 55 (4) (2010) 1024–1028. [22] P.H.A. Ngoc, A Perron–Frobenius theorem for a class of positive quasi-polynomial matrices, Applied Mathematics Letters 19 (8) (2006) 747–751. [23] A. Ilchmann, P.H.A. Ngoc, Stability and robust stability of positive Volterra systems, International Journal of Robust Nonlinear Control 22 (6) (2012) 604–629. [24] S. Zhu, Z. Li, C. Zhang, Exponential stability analysis for positive systems with delays, in: Proceedings of the 30th Chinese Control Conference Yantai, China, 2011, pp. 1234–1239. [25] Z. Shu, J. Lam, B. Du, L. Wu, Positive observers and dynamic output-feedback controllers for interval positive linear systems, IEEE Transactions on Circuits and Systems I: Regular Papers 55 (10) (2008) 3209–3222. [26] J. Feng, J. Lam, P. Li, Z. Shu, Delay rate constrained stabilization of positive systems using static output feedback, International Journal of Robust Nonlinear Control 21 (1) (2011) 44–54. [27] Z. Li, S. Zhu, Dynamic output feedback stabilization of controlled positive discrete-time systems with delays, Discrete Dynamics in Nature and Society (2010), http://dx.doi.org/10.1155/2010/640806. [28] M.A. Rami, Solvability of static output-feedback stabilization for LTI positive systems, System & Control Letters 60 (9) (2011) 704–708. [29] X. Liu, Constrained control of positive systems with delays, IEEE Transactions on Automatic Control 54 (7) (2009) 1596–1600. [30] D. Wang, W. Wang, P. Shi, Exponential H1 filtering for switched linear systems with interval time-varying delay, International Journal of Robust Nonlinear Control 19 (5) (2009) 532–551. [31] D. Wang, W. Wang, P. Shi, Delay-dependent exponential stability for switched delay systems, Optimal Control Applications and Methods 30 (4) (2009) 383–397. [32] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1958.
Journal of the Franklin Institute 350 (2013) 617–636 www.elsevier.com/locate/jfranklin
Exponential stability for positive systems with bounded time-varying delays and static output feedback stabilization Shuqian Zhua,n, Min Menga, Chenghui Zhangb a
School of Mathematics, Shandong University, Jinan 250100, PR China School of Control Science and Engineering, Shandong University, Jinan 250061, PR China
b
Received 29 February 2012; received in revised form 15 December 2012; accepted 27 December 2012 Available online 6 January 2013
Abstract This paper investigates the problem of exponential stability analysis and static output feedback stabilization for discrete-time and continuous-time positive systems with bounded time-varying delays. Based on the relationship between the solution to the system with time-varying delay and that to the corresponding system with constant delay under specific conditions, the equivalence between the a-exponential stability of such two types of systems is established. Then some necessary conditions and sufficient conditions are provided for a-exponential stability of positive systems with bounded time-varying delays. It is shown that, for such systems, the exponential stability with given decay rate is closely related to the bound of the delay. Then by using the singular value decomposition approach, sufficient conditions for the existence of static output feedback controllers are established in terms of linear programming (LP) problems. Some illustrative examples are given to show the correctness of the obtained theoretical results. & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction A common property of positive systems is that the states are nonnegative whenever the initial conditions are nonnegative. Note that many real world physical systems involve nonnegative variables, for example, population levels, absolute temperature, concentration n
Corresponding author. Tel.: þ86 13853145197. E-mail addresses: [email protected] (S. Zhu), [email protected] (M. Meng), [email protected] (C. Zhang). 0016-0032/$32.00 & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2012.12.022
618
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
Nomenclature R reals N set of all nonnegative integers Rn n-dimensional linear vector space over the reals Rn0,þ ðRnþ Þ nonnegative (positive) orthant of Rn In n-dimensional identity matrix Ak0ð$0,g0,!0Þ all elements of A are nonnegative (nonpositive, positive, negative) AkBðB$AÞ ABk0 A is Metzler matrix all off-diagonal elements of A are nonnegative: Aij Z0,iaj sðAÞ spectrum of matrix A mðAÞ spectral abscissa of matrix A, that is, mðAÞ ¼ maxfRe l : l 2 sðAÞg rðAÞ spectral radius of matrix A, that is, rðAÞ ¼ maxfjlj : l 2 sðAÞg JxJ maximum modulus norm of x with JxJ ¼ maxi ¼ 1,2,...,n jxi j,x 2 Rn T A transpose of matrix A
of substances, and positive system behavior seems to be intrinsic for many real-life dynamic systems. Hence, positive system models can be found in almost all fields, for instance, engineering, ecology, economics, biomedicine, social science and so on. Since the states of positive systems are confined within a ‘‘cone’’ located in the positive orthant rather than in linear spaces, many well-established results for general linear systems cannot be readily applied to positive systems. This feature makes the analysis and synthesis of positive systems challenging and interesting. The mathematical theory of positive systems is based on the theory of nonnegative matrices founded by Perron and Frobenius [1,2] and many results have been obtained [3–7]. On the other hand, for many practical control systems, time delays are frequently encountered and recently, there has been increasing interest in time-delay systems [8–14]. General linear systems, even nominal stable systems when are affected by delays, may inherit very complex behaviors such as oscillations, instability and bad performances. In contrast, for positive linear systems with constant delays or bounded time-varying delays, it has been shown that the asymptotic stability has nothing to do with the magnitudes of delays and only dependent of the system matrices [15–21]. Such systems are asymptotically stable if and only if the sum of the system matrices is a Schur matrix (for discrete-time case) or a Hurwitz matrix (for continuous-time case). All the aforementioned references are concerned with the asymptotic stability. However, in practice, it is desirable that the system can converge quickly (that is, has a certain decay rate). Hence, it is necessary to investigate the exponential stability of positive delayed systems. Recently, such problem has been discussed for positive systems with constant delays [22–24]. In [22,23], it has been shown that positive linear time-delay differential systems are exponentially stable if and only if the corresponding systems without delay are exponentially stable, that is, whether such systems are exponentially stable is also independent of delays. However, in [22,23], the decay rate is only required to exist and can be sufficiently small. It is not further considered. In [24], the a-exponential stability analysis problem has been discussed for both continuous-time and discrete-time positive systems
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
619
with constant delays and it is demonstrated that, unlike asymptotic stability of positive time-delay systems, the exponential stability with given decay rate of positive constantdelay systems is dependent of the system matrices as well as the magnitudes of delays. However, to the best of our knowledge, little attention has been paid to the exponential stability of positive systems with time-varying delays. It should also be pointed out that in most literature aforementioned for control of positive systems, the controller takes the form of state feedback. However, in practical applications, it is often impossible to obtain the full information on the state variables. Hence, it is necessary to investigate the output feedback stabilization problem. The positive stabilization problem with dynamic output feedback controllers has been investigated in [25] with sufficient conditions for the existence of the designed controller proposed, and an iteration linear matrix inequality (ILMI) algorithm provided to solve them. In [26,27], necessary and sufficient conditions for the existence of static and dynamic output feedback controllers have been established in terms of LMIs together with a matrix equality constraint, which can be solved via the cone complementarity linearization technique. In comparison with LMI approach, the linear programming (LP) approach is simpler and posses a numerical advantage since the existing LMI softwares cannot handle large size problems and are not numerically stable [6,28]. Recently, the LP approach has been used to handle the state feedback stabilization [6,20,29] and output feedback stabilization [28] for positive systems. In this paper, the exponential stability analysis problem with given decay rate is investigated for discrete-time and continuous-time positive systems with bounded timevarying delays, with necessary conditions and sufficient conditions provided. Then based on the obtained exponential stability results and by using the singular value decomposition approach, sufficient conditions for the existence of static output feedback controllers are established in terms of standard LP problems. 2. Exponential stability analysis 2.1. Discrete-time case Consider the following discrete-time delayed system: ( xðt þ 1Þ ¼ AxðtÞ þ At xðttðtÞÞ, xðsÞ ¼ fðsÞ, s ¼ t,ðt1Þ, . . . ,0,
ð1Þ
where t 2 N, x(t) 2 Rn is the state, A and At are known real constant matrices, the delay tðtÞ 2 N is the bounded time-varying delay which satisfies 0rtðtÞrt,
t2N
ð2Þ
with the constant t 2 N, and f : ft,ðt1Þ, . . . ,0g-Rn0,þ is a sequence specifying the initial state of the system. Definition 1 (Liu et al. [19]). System (1) is said to be positive if for any f : ft,ðt1Þ, . . . ,0g-Rn0,þ , the corresponding trajectory satisfies xðtÞk0 for all t 2 N. Lemma 1 (Liu et al. [19]). System (1) is positive if and only if Ak0 and At k0.
620
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
Lemma 2 (Liu et al. [19]). Assume that system (1) is positive. Let xa(t) and xb(t) be the trajectories of Eq. (1) under the initial conditions fa ðÞ and fb ðÞ, respectively. Then, fa ðsÞ$fb ðsÞ, s 2 ft,ðt1Þ, . . . ,0g implies that xa ðtÞ$xb ðtÞ for all t 2 N. In practice, it is often not enough demanding that the system is asymptotically stable. Then it is desirable that the system can converge quickly (that is, has a certain decay rate). To emphasize the decay rate and discuss whether and how the stability changes with the delay’s changing, similarly to [24], we give the definition of a-exponential stability for system (1) as follows. Definition 2. For given scalar 0oao1, system (1) is called a-exponentially stable if there exists a scalar G40 such that the solution of system (1) satisfies JxðtÞJrGJfJd at for all t 2 N, where JfJd :¼ sups2ft,ðt1Þ,...,0g JfðsÞJ. Since in many cases, the time-varying delay tðtÞ is unknown and the upper bound t is known, in this paper, we focus on the a-exponential stability analysis for all tðtÞ satisfying given bound conditions. Now, inspired by [19], we introduce the following system closely related to system (1) and (2) with constant delay: ( yðt þ 1Þ ¼ AyðtÞ þ At yðttÞ, ð3Þ yðsÞ ¼ cðsÞ, s ¼ t,ðt1Þ, . . . ,0, where all the system matrices are the same as in Eq. (1), and t is the upper bound of tðtÞ in Eq. (1), as shown in Eq. (2). For system (3), we have: Lemma 3 (Liu [29]). Assume that system (3) is positive, then the following statements are equivalent. (i) System (3) is asymptotically stable. (ii) There exists a vector pg0 such that ðA þ At In Þp!0. (iii) A þ At is a Schur matrix, i.e., rðA þ At Þo1. Lemma 4 (Zhu et al. [24]). If the positive system (3) is a-exponentially stable, then (i) rða1 A þ aðtþ1Þ At Þr1; (ii) there exists a nonzero vector pk0 such that ða1 A þ aðtþ1Þ At In ÞT p$0:
ð4Þ
Lemma 5 (Zhu et al. [24]). If one of the following conditions holds: (i) rða1 A þ aðtþ1Þ At Þo1; (ii) there exists a vector pg0 satisfying Eq. (4), then the positive system (3) is a-exponentially stable. The next lemma reveals the relationship between the solution to system (1) and that to system (3) under certain conditions.
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
621
Lemma 6 (Liu et al. [19]). Suppose that there exists a vector pg0 satisfying ðA þ At In Þp!0
ð5Þ
and that the initial conditions for system (1) and (3) satisfy: fðsÞ cðsÞ p, s 2 ft, ðt1Þ, . . . ,0g. Then xðtÞ$yðtÞ
ð6Þ
holds for all t 2 N, where x(t) and y(t) are solutions to Eqs. (1) and (3), respectively. The equivalence between the a-exponential stability of system (1) for all tðtÞ satisfying Eq. (2) and that of system (3) will be established in the theorem as follows. Theorem 1. The positive system (1) is a-exponentially stable for all tðtÞ satisfying Eq. (2) if and only if the positive system (3) is a-exponentially stable. Proof. Necessity. Suppose that the positive system (1) is a-exponentially stable for all tðtÞ satisfying Eq. (2). Particularly, let tðtÞ ¼ t, then the positive system (3) is necessarily a-exponentially stable. Sufficiency. Suppose that the positive system (3) is a-exponentially stable. Thus there ~ exists a scalar G40 such that the solution of system (3) satisfies ~ JyðtÞJrGJcJd at , t 2 N: ð7Þ Furthermore, since system (3) is a-exponentially stable, it is asymptotically stable. Then by ~ ~ Lemma 3, there exists a vector pg0 satisfying ðA þ At In Þp!0. In particular, choose the initial condition cðÞ mp~ with scalar m40, then the corresponding solution ymp~ ðtÞ must satisfy ~ pJa ~ t , t 2 N: Jymp~ ðtÞJrGmJ ð8Þ It follows from Lemma 6 that xmp~ ðtÞ$ymp~ ðtÞ,
t 2 N,
ð9Þ
where xmp~ ðtÞ is the corresponding solution of system (1) when the initial condition ~ fðÞ mp. For arbitrary initial condition fðÞ 2 Rn0,þ , let p~ ¼ ½p~ 1 p~ 2 p~ n T and m ¼ JfJd =mini ¼ 1,2,...,n p~ i , then ~ fðsÞ$mp,
s 2 ft,ðt1Þ, . . . ,0g:
ð10Þ
Thus it is obtained from Lemma 2 that for any initial condition fðÞ 2 Rn0,þ , the corresponding solution satisfies ~ pJa ~ t ¼ G~ Jxf ðtÞJ$Jxmp~ ðtÞJ$Jymp~ ðtÞJ$GmJ
JfJd mini ¼ 1,2,...,n p~ i
~ t ¼: GJfJd at JpJa
ð11Þ
~ pJ=min ~ ~ i 40: Therefore, the positive system (1) is a-exponentially with G ¼ GJ i ¼ 1,2,...,n p stable for all tðtÞ in Eq. (2). & Remark 1. From Theorem 1 it can be seen that the a-exponential stability of the positive system (1) for all tðtÞ : 0rtðtÞrt is determined by the a-exponential stability of the positive system (3). This result is still valid for a more general case: tm rtðtÞrt,tm Z0, which is called interval time-varying delay [30,31]. When the lower bound tm is considered, introduce system (3) with t replaced by tm , which is denoted by system (3)0 , the conclusion
622
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
corresponding to Theorem 1 is: the positive system (1) is a-exponentially stable for some tðtÞ : tðtÞZtm if and only if the positive system (3)0 is a-exponentially stable. Obviously, this result gives us a condition, which is that the positive system (3)0 is not a-exponentially stable, to conclude that if tðtÞZtm , then the positive system (1) is not a-exponentially stable. Based on Lemmas 4 and 5 and Theorem 1, we are now ready to provide the necessary conditions and sufficient conditions for the a-exponential stability of the positive system (1) for all tðtÞ satisfying Eq. (2). Theorem 2. If the positive system (1) is a-exponentially stable for all tðtÞ satisfying Eq. (2), then the conditions (i) and (ii) in Lemma4 hold. Theorem 3. If the condition (i) or (ii) in Lemma5 holds, then the positive system (1) is a-exponentially stable for all tðtÞ satisfying Eq. (2). Remark 2. It is known from [19] that the positive system (1) is asymptotically stable for all tðtÞ satisfying Eq. (2) if and only if A þ At is a Schur matrix, which shows that the magnitude of the delay does not affect the asymptotic stability of Eq. (1). However, as shown in Theorems 2 and 3, the a-exponential stability of the positive system (1) for all tðtÞ in Eq. (2) not only depends on the system matrices but also is closely related to the upper bound of the time-varying delay. Therefore, Theorems 2 and 3 reveal the difference between the asymptotical stability and the exponential stability with given decay rate for positive discrete-time systems with bounded time-varying delays. Remark 3. If n¼ 1, that is, system (1) is a scalar system, it follows from Theorems 2 and 3 that the positive system (1) is a-exponentially stable for all tðtÞ satisfying Eq. (2) if and only if a1 A þ aðtþ1Þ At 1r0:
ð12Þ
Lemma 7 (Horn and Johnson [32]). For two matrices A,B 2 Rnn , the following statements hold: (i) if AkBk0, then rðAÞZrðBÞ. (ii) if AgBk0, then rðAÞ4rðBÞ. Remark 4. It should be noticed that the obtained conditions in Theorems 2 and 3 are necessary conditions and sufficient conditions, respectively, rather than necessary and sufficient conditions. Let t0 2 N satisfy rða1 A þ aðt0 þ1Þ At Þ ¼ 1. Since by Lemma 7, rða1 A þ aðtþ1Þ At Þorða1 A þ aðt0 þ1Þ At Þ ¼ 1 for all t : 0rtot0 if Ak0,At k0, it follows from Theorem 3 that the positive system (1) is a-exponentially stable for all tðtÞ satisfying Eq. (2) if the upper bound t satisfying tot0 . Similarly, for all t4t0 , rða1 A þ aðtþ1Þ At Þ4rða1 A þ aðt0 þ1Þ At Þ ¼ 1 implies that the positive system (1) is not a-exponentially stable for some tðtÞ satisfying Eq. (2) if t4t0 according to Theorem 2. However, as for the critical case when t ¼ t0 , the a-exponential stability cannot be justified by the results in this paper except for the positive scalar system.
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
623
Remark 5. The results in Theorems 2 and 3 can be easily extended to the positive system with multiple time-varying delays of the form 8 m X > < xðt þ 1Þ ¼ AxðtÞ þ Ati xðtti ðtÞÞ, ð13Þ i¼1 > : xðsÞ ¼ fðsÞ, s ¼ t,ðt1Þ, . . . ,0, where t 2 N, x(t) 2 Rn , Ak0 and Ati k0, i ¼ 1,2, . . . ,m, are known real constant matrices, delays ti ðtÞ 2 N,i ¼ 1,2, . . . ,m are bounded time-varying delays which satisfy 0rti ðtÞrti ,
t2N
ð14Þ
with constants ti 2 N, t ¼ maxi ¼ 1,2,...,m ti and f : ft,ðt1Þ, . . . ,0g-Rn0,þ P is a sequence ðti þ1Þ specifying the initial state of the system. In this case, only by substituting m Ati i¼1a ðtþ1Þ for a At in Theorems 2 and 3, the corresponding results of the a-exponential stability of system (13) for all ti ðtÞ satisfying Eq. (14) can be obtained. Next, based on Theorem 3, we will provide the robust stability analysis results for two kinds of uncertain cases of system (1). Corollary 1. Consider the uncertain time-delay system (1) with unknown matrices A and At belonging to the sets L1 ¼ fA : Am $A$AM g,
L2 ¼ fAt : Atm $At $AtM g,
respectively, where Am ,AM ,Atm ,AtM are known bounds. If Am k0,Atm k0 and there exists a vector pg0 such that ða1 AM þ aðtþ1Þ AtM In ÞT p$0, then the uncertain system (1) is positive and a-exponentially stable for all tðtÞ satisfying (2) and A 2 L1 ,At 2 L2 . Proof. It is directly proved by using Theorem 3 and Lemma 7.
&
Corollary 2. Consider the uncertain time-delay system (1) with unknown matrices ½A At belonging to the convex set ( ) l l X X i i Y¼ bi ½A At j bi ¼ 1,bi Z0 , i¼1
i¼1
where ½A1 A1t , . . . ,½Al Alt are known matrices. If Ai k0,Ait k0, i ¼ 1,2, . . . ,l and there exists a vector pg0 such that ða1 Ai þ aðtþ1Þ Ait In ÞT p$0,
i ¼ 1,2, . . . ,l,
then the uncertain system (1) is positive and a-exponentially stable for all tðtÞ satisfying Eq. (2) and ½A At 2 Y.
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
624
2.2. Continuous-time case Consider the continuous-time delayed system: ( _ ¼ AxðtÞ þ At xðttðtÞÞ, xðtÞ xðsÞ ¼ fðsÞ,
s 2 ½t,0,
ð15Þ
where tZ0, x(t) 2 Rn is the state, A and At are known real constant matrices. The delay tðtÞZ0 is the bounded time-varying delay which is continuous with respect to t and satisfies 0rtðtÞrt,
tZ0
ð16Þ ½t,0-Rn0,þ
with the constant tZ0. f : the initial state of the system.
is a continuous vector valued function specifying
Definition 3 (Liu et al. [21]). System (15) is said to be positive if for any f : ½t,0-Rn0,þ , the state trajectory satisfies xðtÞk0 for all tZ0. Lemma 8 (Liu et al. [21]). System (15) is positive if and only if A is Metzler and At k0. Similar to [24], the a-exponential stability for continuous-time system (15) is proposed as follows. Definition 4. For given scalar a40, system (15) is called a-exponentially stable if there exists a scalar G40 such that the solution of system (15) satisfies JxðtÞJrGJfJc eat for all tZ0, where JfJc :¼ suptrsr0 JfðsÞJ. Inspired by [21], we introduce the following system closely related to system (15) with constant delay: ( _ ¼ AyðtÞ þ At yðttÞ, yðtÞ ð17Þ yðsÞ ¼ cðsÞ, s 2 ½t,0, where all the system matrices are the same as in Eq. (15), and t is the upper bound of tðtÞ in Eq. (15), as shown in Eq. (16). Next, similarly to the discrete-time case, based on the results given in [21,24], we will present in the following theorems some necessary conditions and sufficient conditions for the a-exponential stability of the continuous-time positive system (15). Theorem 4. The positive system (15) is a-exponentially stable for all tðtÞ satisfying Eq. (16) if and only if the positive system (17) is a-exponentially stable. Theorem 5. If the positive system (15) is a-exponentially stable for all tðtÞ satisfying Eq. (16), then (i) mðaIn þ A þ At eat Þr0; (ii) there exists a nonzero vector pk0 such that ðaIn þ A þ At eat ÞT p$0: Theorem 6. If one of the following conditions holds: (i) mðaIn þ A þ At eat Þo0;
ð18Þ
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
625
(i) there exists a vector pg0 satisfying Eq. (18), then the positive system (15) is a-exponentially stable for all tðtÞ satisfying Eq. (16). Remark 6. It has been shown in [21] that the positive system (15) is asymptotically stable for all tðtÞ satisfying Eq. (16) if and only if A þ At is a Hurwitz matrix, that is, the magnitude of tðtÞ has no any impact on its asymptotic stability. However, it can be seen from Theorems 5 and 6 that the a-exponential stability of the positive system (15) for all tðtÞ in Eq. (16) is closely related to the upper bound of tðtÞ. This result is similar to that of the discrete-time case given in Remark 2. Remark 7. If n¼ 1, then the positive system (15) is a-exponentially stable for all tðtÞ satisfying Eq. (16) if and only if a þ A þ At eat r0:
ð19Þ
P ati Remark 8. By substituting m for At eat in Theorems 5 and 6, we can obtain the i ¼ 1 Ati e corresponding results of a-exponential stability for system with multiple delays: 8 m X > < xðtÞ _ ¼ AxðtÞ þ Ati xðtti ðtÞÞ, ð20Þ i¼1 > : xðsÞ ¼ fðsÞ, s 2 ½t,0, where tZ0, xðtÞ 2 Rn , A is Metzler and Ati k0, i ¼ 1,2, . . . ,m, delays ti ðtÞZ0, i ¼ 1,2, . . . ,m are bounded time-varying delays which are continuous with respect to t and satisfy 0rti ðtÞrti ,
tZ0
ð21Þ
with constants ti Z0 and t ¼ maxi ¼ 1,2,...,m ti . 3. Static output feedback stabilization 3.1. Discrete-time case In this subsection, we consider the following discrete-time delayed system: 8 > < xðt þ 1Þ ¼ AxðtÞ þ At xðttðtÞÞ þ BuðtÞ, yðtÞ ¼ CxðtÞ, > : xðsÞ ¼ fðsÞ, s ¼ t,ðt1Þ, . . . ,0,
ð22Þ
where t 2 N, xðtÞ 2 Rn , uðtÞ 2 Rr , yðtÞ 2 Rq are, respectively, the state, the control input and the measurable output. A, At , B and C are known real constant matrices with appropriate dimensions and At k0. The delay tðtÞ and the initial function fðtÞ are the same as in Eqs. (1) and (2). The purpose of this subsection is to design a static output feedback controller uðtÞ ¼ KyðtÞ
ð23Þ
for system (22) such that for given a : 0oao1, the resultant closed-loop system ( xðt þ 1Þ ¼ ðA þ BKCÞxðtÞ þ At xðttðtÞÞ, xðsÞ ¼ fðsÞ,
s ¼ t,ðt1Þ, . . . ,0
ð24Þ
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
626
is positive and a-exponentially stable for all tðtÞ in Eq. (2), where K 2 Rrq is the feedback gain matrix. First, we consider the single-output case, i.e., q ¼ 1. The existence condition of the controller (23) is provided in the next theorem by using the singular value decomposition approach. Theorem 7. For system (22) with Ck0, make the singular value decomposition of C as CV T ¼ ½s 0 0 with s40 being the singular value of C and V being an orthogonal matrix. If there exist vectors l ¼ ½l1 l2 ln T 2 Rn and z 2 Rr such that the following LP problem is solvable: ½a1 A þ aðtþ1Þ At In V T l þ a1 sBz$0,
ð25Þ
V T lg0,
ð26Þ
aij l1 þ bi zcj Z0,
i,j ¼ 1,2, . . . ,n,
ð27Þ
where A ¼ ½aij ,
B ¼ ½bT1 bT2 bTn T ,
C ¼ ½c1 c2 cn ,
then there exists a static output feedback controller (23) such that the closed-loop system (24) is positive and a-exponentially stable for all tðtÞ in Eq. (2). In this case, the feedback gain matrix K can be designed as K ¼ l1 1 z:
ð28Þ
Proof. Let V ¼ ½vT1 vT2 vTn T and p ¼ V T l, then pg0 by Eq. (26). And it gets from l ¼ Vp and C ¼ ½s 0 0V that l1 ¼ v1 p and v1 ¼ C=s. Since s40,pg0, Ck0 and C contains at least a nonzero component, it can get easily that l1 40. Define K ¼ l1 1 z, then substituting z ¼ l1 K into Eq. (27) yields aij l1 þ bi l1 Kcj ¼ ðaij þ bi Kcj Þl1 Z0,
ð29Þ
which implies that aij þ bi Kcj Z0 since l1 40. Thus A þ BKCk0 and the closed-loop system (24) is positive. Now, substituting p ¼ V T l, z ¼ l1 K and CV T ¼ ½s 0 0 into Eq. (25) concludes that ½a1 A þ aðtþ1Þ At In p þ a1 sBKl1 ¼ ½a1 A þ aðtþ1Þ At In p þ a1 BK½s 0 0½l1 l2 ln T ¼ ½a1 A þ aðtþ1Þ At In p þ a1 BKCV T l ¼ ½a1 ðA þ BKCÞ þ aðtþ1Þ At In p$0:
ð30Þ
By Theorem 3, we have that the closed-loop system (24) is a-exponentially stable for all tðtÞ in Eq. (2). Thus the proof is complete. & Remark 9. In [28], the static output feedback stabilization problem has been solved for positive linear systems without delay with necessary and sufficient conditions obtained for single-output case. It can be seen that to get the standard LP formulation of the proposed approach in [28], additional vec operation and the Kronecker product are further used. However, in Theorem 7, via another approach: the singular value decomposition approach, the obtained conditions can be solved as a standard LP problem. The detailed design procedure is presented as follows:
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
627
Step 1: Make the singular decomposition of C as CV T ¼ ½s 0 0 to obtain V and s. Step 2: Rewrite conditions (25)–(27) as 3 2 1 a A þ aðtþ1Þ At In V T a1 sB 3 2 O 7 6 6 V T O 7 6 Eg 7 7 6 7 6 7 6 7 6 AE Bc 1 1 7 6 7 O ð31Þ 7b$6 6 7 6 6 AE 2 Bc2 7 7 6 7 6 4 ^ 5 6 ^ ^ 7 5 4 O AE n Bcn where the vector to be computed is b ¼ ½lT zT T with l ¼ ½l1 l2 ln T , O represents a matrix with compatible dimension whose elements are all 0, Ej is a matrix with 1 in the jth row, 1st column position and zeros elsewhere, g is a n-dimensional vector whose elements are all 1, and E40 is a given small scalar (which is introduced to guarantee the strict positivity constraint V T lg0). Then solve the above standard LP problem by using linprog Toolbox in Matlab. Step 3. If the above LP problem is solvable, then the static output feedback gain K can be designed as K ¼ l1 1 z. The corresponding results are stated as follows for the general case when the matrix C is not sign restricted. Theorem 8. For system (22), make the singular value decomposition of C and denote A,B,C as shown in Theorem7. Then there exists a static output feedback controller (23) such that the closed-loop system (24) is positive and a-exponentially stable for all tðtÞ in Eq. (2), if there exist vectors l ¼ ½l1 l2 ln T 2 Rn and z 2 Rr such that at least one of the following two LP problems is feasible: LP1 (25), (27) and V T lg0,
l1 40,
ð32Þ
LP2 (25), and V T lg0,
l1 o0,
ð33Þ
aij l1 þ bi zcj r0,
i,j ¼ 1,2, . . . ,n:
ð34Þ
Moreover, the gain matrix K can be designed as Eq. (28). For the case of multi-output system, we design the control which has the following structure as in [28] uðtÞ ¼ KnyðtÞ,
K 2 Rr ,
ð35Þ
1q
is a fixed parameter design. By using this kind of controls which involve where n 2 R one rank gains, we can maintain the resultant closed-loop system positive and a-exponentially stable. This is shown in the following result. Theorem 9. Assume that n 2 R1q be a fixed parameter design such that nCk0. Make the singular value decomposition of nC as nCV T ¼ ½s 0 0 with s40 being the singular value
628
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
of nC and V being an orthogonal matrix. Denote A, B and C as shown in Theorem 7. Then there exists a static output feedback controller (35) such that the closed-loop system is positive and a-exponentially stable for all tðtÞ in Eq. (2), if the following LP problem is feasible in the variables l ¼ ½l1 l2 ln T 2 Rn and z 2 Rr : (25), (26) and aij l1 þ bi zncj Z0,
i,j ¼ 1,2, . . . ,n:
ð36Þ
Moreover, the matrix K can be calculated as K ¼ l1 1 z. 3.2. Continuous-time case All the results presented in Section 3.1 can be easily extended to the continuous-time system 8 _ ¼ AxðtÞ þ At xðttðtÞÞ þ BuðtÞ, > < xðtÞ yðtÞ ¼ CxðtÞ, ð37Þ > : xðsÞ ¼ fðsÞ, s 2 ½t,0: Only by substituting aij l1 þ bi zcj ZðrÞ0, iaj,i,j ¼ 1,2, . . . ,n for aij l1 þ bi zcj ZðrÞ0, i,j ¼ 1,2, . . . ,n, and ½aIn þ A þ At eat V T l þ sBz$0 for ½a1 A þ aðtþ1Þ At In V T l þ a1 sBz$0, we can get the existence conditions of the static output feedback controller (23) for system (37). Remark 10. It should be noted that in this section, the positivity restrictions are not imposed on the open-loop system matrices A,B,C and the feedback matrix K. We only require that the closed-loop system is positive and a-exponentially stable. Therefore, the synthesis problem considered in this section can be interpreted as enforcing the system to be positive. In fact, take the discrete-time case for example, if the open-loop system is positive (i.e., Ak0,At k0, Bk0, Ck0) and rða1 A þ aðtþ1Þ At Þ41, then there exists no nonnegative feedback uðtÞ ¼ KyðtÞ such that the closed-loop system is positive and a-exponentially stable for all tðtÞ in Eq. (2) since rða1 ðA þ BKCÞþ aðtþ1Þ At Þ Z rða1 A þ aðtþ1Þ At Þ41. 4. Numerical examples Example 1. Consider the discrete-time delayed system (1) given in [19] with a 0:1 0:1 0:1 A¼ , At ¼ , 0:2 0:3 0:2 0:5 where t ¼ 1, a is a nonnegative parameter, by Lemma 1, which implies that system (1) is positive. Computation shows that if a ¼ 0.1635, the two eigenvalues of the matrix ð0:951 A þ 0:95ð1þ1Þ At Þ are 1 and 0.1526, and both of them lie inside the unit circle if ao0:1635 and at least one of them lies outside if a40:1635. By Theorems 2 and 3, one can conclude that system (1) is 0:95-exponentially stable for all delay tðtÞ satisfying 0rtðtÞr1 if 0rao0:1635, and not 0:95-exponentially stable for some tðtÞ satisfying 0rtðtÞr1 if a40:1635. And since when a ¼ 0.16, the eigenvalues of the matrix ð0:951 A þ 0:95ð2þ1Þ At Þ are 0.1561 and 1.0315, one gets that system (1) is not 0:95-exponentially stable for some tðtÞ satisfying 0rtðtÞr2 if a ¼ 0.16.
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
629
Figs. 1–3 give the simulation results of x1 ðtÞ,x2 ðtÞ and X ðtÞ ¼ 0:95t when a¼ 0.16 (Figs. 1 and 2) and a¼ 0.25 (Fig. 3), respectively. The initial functions randomly take values on the interval ½0,1, the delay tðtÞ randomly takes values in the set f0,1g in Fig. 1, tðtÞ ¼ 2 in Fig. 2 and tðtÞ ¼ 1 in Fig. 3. From the three figures, one can see that x(t) satisfies JxðtÞJr0:95t for all tZ0 when a ¼ 0:16o0:1635 and tðtÞr1, while in the case of a ¼ 0:16,tðtÞ ¼ 241 and the case of a ¼ 0:2540:1635,tðtÞ ¼ 1, when t is large enough, JxðtÞJ will be larger than 0:95t . 1
x1 x2 X
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
20
40
60
80
100
120
time t
Fig. 1. The simulation results of Example 1 when a ¼ 0:16,tðtÞr1. 1
x1 x2 X
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
20
40
60
80
100
120
time t
Fig. 2. The simulation results of Example 1 when a ¼ 0:16,tðtÞ ¼ 2.
630
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636 1
x1 x2 X
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
20
40
60
80
100
120
time t
Fig. 3. The simulation results of Example 1 when a ¼ 0:25,tðtÞ ¼ 1.
Example 2. Consider the continuous-time delayed system (15) given in [21] with 5 1:3 a 0:8 A¼ , At ¼ , 1:5 3 1 1:2 where t ¼ 3:9, a is a nonnegative parameter, by Lemma 8, which implies that system (15) is positive. Based on Theorems 5–6 and by computation, one can conclude that system (15) is 0.05-exponentially stable for all delay tðtÞ satisfying 0rtðtÞr3:9 if 0rao0:6695, not 0.05exponentially stable for some tðtÞ satisfying 0rtðtÞr3:9 if a40:6695, and not 0.05exponentially stable for some tðtÞ satisfying 0rtðtÞr4:9 if a ¼ 0.66. Figs. 4–6 give the simulation results of x1 ðtÞ,x2 ðtÞ and X ðtÞ ¼ 0:35 e0:05t when a ¼ 0.66 (Figs. 4 and 5) and a¼ 1.2 (Fig. 6), respectively. The initial function is fðtÞ ¼ ½0:3 0:2, the delay tðtÞ ¼ 4:9 in Fig. 5, and tðtÞ ¼ 3 þ 0:9 sinðtÞ in Figs. 4 and 6. The three figures show that x(t) satisfies JxðtÞJr0:35 e0:05t for all tZ0 when a ¼ 0:66o0:6695 and tðtÞr3:9, while in the case of a ¼ 0:66,tðtÞ ¼ 4:943:9 and the case of a ¼ 1:240:6695, tðtÞ ¼ 3 þ 0:9 sinðtÞ, when t is large enough, JxðtÞJ will be larger than 0:35 e0:05t . Example 3. Consider a practical example about a certain pest’s structured population dynamics described by the following Leslie matrix model: 8 2 3 2 3 f1 f2 f3 b1 b2 > > > > 6 7 6 < 0 07 xðt þ 1Þ ¼ 4 p1,2 5xðtÞ þ 4 0 0 5uðtÞ, 0 p2,3 0 > 0 0 > > > : yðtÞ ¼ ½0 c cxðtÞ, where xðtÞ ¼ ½x1 ðtÞ x2 ðtÞ x3 ðtÞT and x1 ðtÞ represents the number of juvenile pests at time t, x2 ðtÞ represents the number of immature pests at time t, x3 ðtÞ represents the number of adult pests at time t. Assume that the parameters for this model are given in [25] (without
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636 x1 x2 X
0.3 0.25 0.2 0.15 0.1 0.05 0
0
20
40
60
80
100
120
times t [s]
Fig. 4. The simulation results of Example 2 when a ¼ 0:66,tðtÞ ¼ 3 þ 0:9 sinðtÞ.
x1 x2 X
0.3 0.25 0.2 0.15 0.1 0.05 0
0
20
40
60
80
100
120
times t [s]
Fig. 5. The simulation results of Example 2 when a ¼ 0:66,tðtÞ ¼ 4:9.
uncertainties) as f1 ¼ 0:3011,
f2 ¼ 0:5915,
p2,3 ¼ 0:7894,
f3 ¼ 0:5235,
b1 ¼ 0:9000,
p1,2 ¼ 0:8868,
b2 ¼ 0:5000,
c ¼ 1:000:
Obviously, this model is a special case of system (22) when At ¼ 0.
631
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
632
x1 x2 X
0.3 0.25 0.2 0.15 0.1 0.05 0
0
20
40
60
80
100
120
times t [s]
Fig. 6. The simulation results of Example 2 when a ¼ 1:2,tðtÞ ¼ 3 þ 0:9 sinðtÞ.
Since the eigenvalues of A are 1:0904,0:394670:4247i, the open-loop system is not stable. Making the singular value decomposition of C yields 2 3 0 0:7071 0:7071 6 7 V ¼ 4 0:7071 0:5000 0:5000 5 and s ¼ 1:4142: 0:7071
0:5000
0:5000
Choose a ¼ 0:8, E ¼ 1:0 1010 , and solve the standard LP problem (31) by using linprog Toolbox in Matlab. A feasible solution is obtained as 2 3 163:8517 66:5035 6 7 : l ¼ 4 66:1464 5, z ¼ 36:9464 21:3024 Thus the static output feedback gain matrix is designed as K¼
0:4059 0:2255
:
Figs. 7 and 8 give the trajectory simulations of the open-loop system and the closed-loop system, respectively, with xð0Þ ¼ ½300 250 200T and X ðtÞ ¼ 350 0:8t . We can see that the open-loop system is not stable and the number of the pests increases, while the close-loop system is 0.8-exponentially stable and the number of the pests decays more quickly than X(t) with the designed output feedback law.
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636 6000
x1 x2 x3
5000 4000 3000 2000 1000 0
0
5
10
15
20
25
30
Fig. 7. The open-loop simulation results of Example 3.
350
x1 x2 x3 X
300 250 200 150 100 50 0
0
5
10
15
20
25
30
Fig. 8. The closed-loop simulation results of Example 3.
Example 4. Consider the single-output system (37) 2 2 3 0:002 0:15 1:90 1:55 6 6 7 A ¼ 4 0:50 0:3 0:10 5, At ¼ 4 0:01 0:01 0:20 0:50 2:55
with tðtÞ ¼ 3 þ 0:5 sinðtÞ and 3 0:01 0:03 7 0:01 0:04 5, 0:03 0:04
633
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
634
1000
x1 x2 x3
900 800 700 600 500 400 300 200 100 0
0
2
4
6
8
10 12 times t [s]
14
16
18
20
Fig. 9. The open-loop simulation results of Example 4.
2
1:05 6 B¼4 0 0:09
3 1:04 7 0:8 5,
C ¼ ½1 1 0:
0:50
Note that the matrix A þ At is not a Hurwitz matrix since its eigenvalues are 0.9033, 1:2422 and 2:6091, thus the open-loop system is not stable. Choose a ¼ 0:2, E ¼ 1:0 1010 and solve the standard LP problem got in Section 3.2 by using linprog Toolbox in Matlab. The static output feedback gain matrix is designed as 1:5512 K¼ : 0:1065 The state trajectories of the open-loop system and the closed-loop system are shown in Figs. 9 and 10, respectively, where x ¼ ½x1 x2 x3 T , the initial condition is xðtÞ ¼ ½0:9 0:2 0:1T ,t 2 ½3:5,0, and X ðtÞ ¼ e0:2t . From Figs. 9 and 10 we can see that the open-loop system is not stable while the closed-loop system is 0.2-exponentially stable with the designed static output feedback uðtÞ ¼ KyðtÞ. 5. Conclusions and future works Based on the relation between the solution to the system with time-varying delay and that to the corresponding system with constant delay under specific conditions, as well as the a-exponential stability conditions for constant-delay positive systems, some necessary conditions and sufficient conditions have been established for a-exponential stability of continuous-time and discrete-time positive systems with bounded time-varying delays. It is demonstrated that the a-exponential stability of positive systems with bounded timevarying delays is equivalent to that of the positive systems with constant delay which is just
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636 1
635
x1 x2 x3 X
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
2
4
6
8
10 12 times t [s]
14
16
18
20
Fig. 10. The closed-loop simulation results of Example 4.
the upper bound of the time-varying delays, and thus is dependent of delays. Then based on the obtained sufficient conditions and by using the singular value decomposition approach, the static output feedback stabilization problem is solved. The controller for single-output case, which guarantees that the resultant closed-loop system is not only positive, but also a-exponentially stable, can be efficiently designed by solving standard LP problems. The controller with one rank gain for the multi-output case has also been discussed. The a-exponential stability analysis problem for positive systems with time-varying delays involving the delay-derivatives, such as fast varying and slow varying time-delays, as well as the problem of how to design the general static output feedback controller for multi-output systems by using the LP based approach, can be further investigated. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grants 61004011 and 61034007.
References [1] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, PA, 1994. [2] L. Farina, S. Rinaldi, Positive Linear Systems: Theory and its Applications, Wiley, New York, 2000. [3] H. Gao, J. Lam, C. Wang, S. Xu, Control for stability and positivity: equivalent conditions and computation, IEEE Transactions on Circuits and Systems II: Express Briefs 52 (9) (2005) 540–544. [4] M.A. Rami, F. Tadeo, Positive observation problem for linear discrete positive systems, in: Proceedings of the 45th Conference on Decision and Control, San Diego, CA, 2006, pp. 4729–4733.
636
S. Zhu et al. / Journal of the Franklin Institute 350 (2013) 617–636
[5] O. Mason, R. Shorten, On linear copositive Lyapunov functions and the stability of switched positive linear systems, IEEE Transactions on Automatic Control 52 (7) (2007) 1346–1349. [6] M.A. Rami, F. Tadeo, Controller synthesis for positive linear systems with bounded controls, IEEE Transactions on Circuits and Systems II: Express Briefs 54 (2) (2007) 151–155. [7] X. Ding, L. Shu, X. Liu, On linear copositive Lyapunov functions for switched positive systems, Journal of the Franklin Institute 348 (8) (2011) 2099–2107. [8] J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. [9] J.-P. Richard, Time-delay systems: an overview of some recent advances and open problems, Automatica 39 (10) (2003) 1667–1694. [10] Q.-L. Han, Absolute stability of time-delay systems with sector-bounded nonlinearity, Automatica 41 (12) (2005) 2171–2176. [11] S. Zhu, C. Zhang, Z. Cheng, J. Feng, Delay-dependent robust stability criteria for two classes of uncertain singular time-delay systems, IEEE Transactions on Automatic Control 52 (5) (2007) 880–885. [12] X. Jiang, Q.-L. Han, New stability criteria for linear systems with interval time-varying delay, Automatica 44 (10) (2008) 2680–2685. [13] S. Ma, E.-K. Boukas, Y. Chinniah, Stability and stabilization of discrete-time singular Markov jump systems with time-varying delay, International Journal of Robust Nonlinear Control 20 (5) (2010) 531–543. [14] M.S. Mahmoud, Y. Xia, Improved exponential stability analysis for delayed recurrent neural networks, Journal of the Franklin Institute 348 (2) (2011) 201–211. [15] W.M. Haddad, V. Chellaboina, Stability theory for nonnegative and compartmental dynamical systems with time delays, System & Control Letters 51 (5) (2004) 355–361. [16] M.A. Rami, U. Helmke, F. Tadeo, Positive observation problem for linear time-delay positive systems, in: Proceedings of the 15th Mediterranean Conference on Control & Automation, Athens, Greece, 2007, pp. 1–6. [17] M. Buslowicz, Simple stability conditions for linear positive discrete-time systems with delays, Bulletin of the Polish Academy of Sciences: Technical Sciences 56 (4) (2008) 325–328. [18] T. Kaczorek, Stability of positive continuous-time linear systems with delays, Bulletin of the Polish Academy of Sciences: Technical Sciences 57 (4) (2009) 395–398. [19] X. Liu, W. Yu, L. Wang, Stability analysis of positive systems with bounded time-varying delays, IEEE Transactions on Circuits and Systems II: Express Briefs 56 (7) (2009) 600–604. [20] M.A. Rami, Stability analysis and synthesis for linear positive systems with time-varying delays, in: Positive Systems, Springer-Verlag, Berlin, 2009. [21] X. Liu, W. Yu, L. Wang, Stability analysis for continuous-time positive systems with time-varying delays, IEEE Transactions on Automatic Control 55 (4) (2010) 1024–1028. [22] P.H.A. Ngoc, A Perron–Frobenius theorem for a class of positive quasi-polynomial matrices, Applied Mathematics Letters 19 (8) (2006) 747–751. [23] A. Ilchmann, P.H.A. Ngoc, Stability and robust stability of positive Volterra systems, International Journal of Robust Nonlinear Control 22 (6) (2012) 604–629. [24] S. Zhu, Z. Li, C. Zhang, Exponential stability analysis for positive systems with delays, in: Proceedings of the 30th Chinese Control Conference Yantai, China, 2011, pp. 1234–1239. [25] Z. Shu, J. Lam, B. Du, L. Wu, Positive observers and dynamic output-feedback controllers for interval positive linear systems, IEEE Transactions on Circuits and Systems I: Regular Papers 55 (10) (2008) 3209–3222. [26] J. Feng, J. Lam, P. Li, Z. Shu, Delay rate constrained stabilization of positive systems using static output feedback, International Journal of Robust Nonlinear Control 21 (1) (2011) 44–54. [27] Z. Li, S. Zhu, Dynamic output feedback stabilization of controlled positive discrete-time systems with delays, Discrete Dynamics in Nature and Society (2010), http://dx.doi.org/10.1155/2010/640806. [28] M.A. Rami, Solvability of static output-feedback stabilization for LTI positive systems, System & Control Letters 60 (9) (2011) 704–708. [29] X. Liu, Constrained control of positive systems with delays, IEEE Transactions on Automatic Control 54 (7) (2009) 1596–1600. [30] D. Wang, W. Wang, P. Shi, Exponential H1 filtering for switched linear systems with interval time-varying delay, International Journal of Robust Nonlinear Control 19 (5) (2009) 532–551. [31] D. Wang, W. Wang, P. Shi, Delay-dependent exponential stability for switched delay systems, Optimal Control Applications and Methods 30 (4) (2009) 383–397. [32] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1958.