An improved approach to controller design of positive systems using controller gain decomposition

Author’s Accepted Manuscript An improved approach to controller design of positive systems using controller gain decomposition Junfeng Zhang, Xudong Zhao, Ridong Zhang www.elsevier.com/locate/jfranklin

PII: DOI: Reference:

S0016-0032(16)30445-8 http://dx.doi.org/10.1016/j.jfranklin.2016.11.026 FI2815

To appear in: Journal of the Franklin Institute Received date: 30 June 2016 Revised date: 27 September 2016 Accepted date: 27 November 2016 Cite this article as: Junfeng Zhang, Xudong Zhao and Ridong Zhang, An improved approach to controller design of positive systems using controller gain d e c o m p o s i t i o n , Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2016.11.026 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

An improved approach to controller design of positive systems using controller gain decomposition Junfeng Zhanga,b,∗ , Xudong Zhaoc , Ridong Zhangb a b c

Institute of Information and Control, Automation, Hangzhou Dianzi University, 310018, China

Key Lab for IOT and Information Fusion Technology of Zhejiang, Hangzhou Dianzi University, 310018, China

Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, China

Abstract This paper aims to propose an improved controller design approach for positive systems in both continuous-time and discrete-time contexts. First, by decomposing the controller gain matrix into components, compact conditions that is simple and easy to be computed for guaranteeing the positivity and stability of positive systems is formulated. Based on the obtained conditions, new stabilization results and subsequent controller design are formulated with improvements in both the feasibility of conditions and the generality of the controller. All conditions are solvable in terms of linear programming. Compared with existing approaches in the literature, the new design approach constructs a controller with simpler forms and less restrictions. Meanwhile, the efficiency and advantage of the present approach are verified by application to the typically studied problems in the literature. Some discussions on the proposed approach are provided to show its potential applications on positive systems. Finally, three comparison examples are given to verify the merits of the theoretical findings. Keywords: Positive systems, controller design, controller gain decomposition, linear programming 1. Introduction Positive systems were first introduced in 1979 [1] and have drawn a lot of attention in the control community. Positive systems are a class of special systems with many interesting features since then. For instance, the state of a positive system is always confined in the positive orthant if the initial conditions are nonnegative; a positive system with bounded time-delay is stable if its corresponding system without time-delay is stable; a Lyapunov function of positive systems can be chosen as the linear form, etc. These features have attracted the attention of researchers and also pave the way for new approaches to appear for positive systems. In recent years, the topics on positive systems mainly focus on controllability [2, 3], reachability [4, 5], realization [6], and so on [7–10]. Like general systems, stabilization is still a fundamental issue for positive systems. However, there are relatively fewer results on the stabilization of positive systems [11]. In 2007, Ait Rami et al. presented a linear programming approach to controller design of positive systems [12, 13]. Subsequently, the linear programming approach was applied to output-feedback stabilization [14] of positive systems. These developments verify that linear programming is an effective approach to control synthesis of positive systems. The problem of 1 -induced state-feedback controller design for positive systems was investigated by using a linear copositive Lyapunov function in [15]. In [16], a static output-feedback controller design was presented, where an iterative linear matrix inequality algorithm was provided ∗

E-mail address: [email protected] (J. Zhang).

Preprint submitted to Journal of The Franklin Institute

November 28, 2016

to compute the feedback gain matrix. In [17], the output-feedback controller in [14] was improved by virtue of an iterative convex optimization algorithm. More results on stabilization of positive systems can be referred to in [18–22] and references cited therein. First of all, the existing results mentioned above are interesting, significant and they have inspired many more results on positive systems. However, there are still ways of new methods for improved control performance since there exist some restrictions in the literature. To satisfy the positivity of the resulting closed-loop systems x(t) ˙ = Ax(t) + Bu(t) in [12], a condition aij dj + bi zj ≥ 0 ∀i = j is used, where A = [aij ], B T = [bT1 , . . . , bTn ]. However, a compact form of this condition may be better such that it may be extended to hybrid positive systems. Some iterative algorithms are introduce in [15–17] to compute the controller gain matrices, however, these algorithms will in fact increase the computation burden. In [14], the rank of the controller gain matrix is confined to be 1. Furthermore, as far as the stabilization of positive systems is concerned, there is still room left for improvements in the aforementioned works, which motivate us to carry out the paper. This paper will further provide a new controller design approach to remove some restrictions in existing literature. The new approach is based on linear programming and provides a construction method of a linear copositive Lyapunov function for the systems under consideration. The developed approach is efficient in solving the control synthesis problems of positive systems and some applications of the obtained results are also given to show the efficiency of the proposed approach. The rest of the paper is organized as follows: Section 2 provides the problem statements. Section 3 gives the main results. Section 4 provides extensions and applications of the obtained results. In Section 5, some discussions on the presented approach are addressed. Two illustrative examples are given in Section 6. Section 7 concludes the paper. Notations Let , n , n×n be the sets of real numbers, n-dimensional vectors and n × n matrices, respectively. Denote by N, N+ the sets of nonnegative and positive integers. For a vector x = (x1 , . . . , xn )T , x  0 ( 0) means that xi ≥ 0 (xi > 0) ∀i = 1, . . . , n. Similarly, x  0 (≺ 0) means that xi ≤ 0 (xi < 0) ∀i = 1, . . . , n. For a matrix A = [aij ] ∈ n×n , A  0 ( 0) means that aij ≥ 0 (aij > 0) ∀i, j = 1, . . . , n. Similarly, A  0 (≺ 0) means that aij ≤ 0 (aij < 0) ∀i, j = 1, . . . , n. A matrix A is called Metzler if all of its non-diagonal elements are nonnegative. I is the identical matrix with proper dimensions. n+  {x|x ∈ n , x  0}. Let 1n = (1, . . . , 1)T and    n n (i) T T 1n = (0, . . . , 0, 1, 0, . . . , 0) . For a vector x = (x1 , . . . , xn ) , its 1-norm is defined by x 1 = i=1 |xi |. For the state       i−1 n−i ∞ x(t) of a continuous-time system, the 1 -norm of x(t) is defined by x(t) 1 = t=0 ||x(t)||1 dt. For the state x(k)  of a discrete-time system, the 1 -norm of x(k) is defined by x(k) 1 = ∞ k=0 ||x(k)||1 . Throughout the paper, the dimensions of vectors and matrices are assumed to be compatible if not stated. 2. Problem formulation Consider the following system: δx(t) = Ax(t) + Bu(t), y(t) = Cx(t),

(1)

where x(t) ∈ n , u(t) ∈ r and y(t) ∈ m are system state, control input, and output, respectively, δ denotes the derivative operator in the continuous-time context (δx(t) = 2

d dt x(t), t

≥ 0) and the shift forward operator in the

discrete-time context (δx(t) = x(t + 1), t ∈ N). Assume that A ∈ n×n is a Metzler matrix, B  0 with B ∈ n×r , and C  0 with C ∈ m×n in the continuous-time system (1), and A  0 with A ∈ n×n , B  0 with B ∈ n×r , and C  0 with C ∈ m×n in the discrete-time system (1). The following preliminaries are first introduced for later use. Definition 1 ([2, 3]) System (1) is positive if its state and output are nonnegative for all time t if the initial condition x(t0 ) and the control input u(t) are nonnegative. Lemma 1 ([2, 3]) System (1) in the continuous-time context is positive if and only if A is a Metzler matrix, B  0 and C  0; System (1) in the discrete-time context is positive if and only if A  0 and B  0. Noting the assumptions for system (1), it follows that system (1) is positive from Lemma 1. Lemma 2 ([2, 3]) Let A be a Metzler matrix, then the following statements are equivalent: (i) The matrix A is Hurwitz; (ii) There is a vector v  0 such that Av ≺ 0. Let A  0, then the following statements are equivalent: (a) The matrix A is Schur; (b) There is a vector v  0 in n such that (A − I)v ≺ 0. Remark 1 By Lemma 2, it follows that, system (1) in the continuous-time context is positive if the matrix A is Metzler and the term (ii) hods; system (2) in the discrete-time context is positive if the matrix A  0 and the term (b) holds. Lemma 3 A matrix M is Metzler if and only if there exists a constant ς such that M + ςI  0. 3. Main results In this section, we will address the main results on the stabilization of positive systems. We first consider the stabilization of the continuous-time system (1) and then discuss the stabilization of the discrete-time system (1). 3.1. Continuous-time case For convenience, we rewrite the continuous-time system (1) as follows: x(t) ˙ = Ax(t) + Bu(t), y(t) = Cx(t).

(2)

Theorem 1 If there exist a constant ς > 0 and vectors v  0 with v ∈ n , zi ∈ m , z ∈ m such that AT v + C T z ≺ 0, A1Tr B T v + B

r  i=1

(i)

1r ziT C + ςI  0,

(3a) (3b) (3c)

zi  z, hold, then under the output-feedback control law (i)

Σr 1r ziT y(t) u(t) = Ky(t) = i=1 1Tr B T v

3

(4)

the resulting closed-loop system (2) is positive and exponentially stable. Proof Since 1r  0 with 1r ∈ r , B  0 with B ∈ n×r , and v  0 with v ∈ n , then we have 1Tr B T v > 0. This together with (3b) gives that A + B

(i)

Σri=1 1r ziT T 1T rB v

C+

ς T I 1T r B v

 0. Using (4), it follows that A + BKC +

ς T I 1T rB v

 0.

From Lemma 3, A + BKC is a Metzler matrix. Then, the closed-loop system (2) is positive based on Lemma 1, that is, x(t)  0 ∀t ≥ 0. Next, choose a linear copositive Lyapunov function candidate V (x(t)) = x(t)T v. Then, the derivative of V (x(t)) along the trajectories of system (2) is V˙ (x(t)) = x(t)T (AT v + C T K T B T v). (i)

(i)

(i)

(5)

(i)

From (3c), we get 1r ziT  1r z T . Then, Σri=1 1r ziT  Σri=1 1r z T = 1r z T . Therefore, K T B T v  z. Noting the fact that C  0 and x(t)  0, one can obtain from (5) that V˙ (x(t)) ≤ x(t)T (AT v + C T z).

(6)

From (3a), we have V˙ (x(t)) < 0. This completes the proof.



Remark 2 In Theorem 1, a matrix decomposition technique is used to transform present condition into linear programming, that is, ⎛ k11 k12 · · · ⎜ ⎜ ⎜ k21 k22 · · · K =⎜ ⎜ .. .. .. ⎜ . . . ⎝ kr1 kr2 · · · ⎛ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎝

k11 k12 · · ·

(1)

0 .. .

0 .. .

··· .. .

0

0

···

k1n k2n .. .

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

krn ⎞

⎛

⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ .. ⎟ + ⎜ ⎜ . ⎟ ⎠ ⎜ ⎜ ⎝ 0

k1n

0

0

···

k21 k22 · · · 0 .. .

0 .. .

··· .. .

0

0

···

0

⎞

⎟ ⎟ k2n ⎟ ⎟ ⎟ 0 ⎟ + ··· ⎟ .. ⎟ . ⎟ ⎠ 0

(1)

⎛ ⎜ ⎜ ⎜ +⎜ ⎜ ⎜ ⎝

0

0

···

0 .. .

0 .. .

··· .. .

kr1 kr2 · · ·

0

⎞

⎟ ⎟ 0 ⎟ ⎟ .. ⎟ . ⎟ ⎠ krn

(7)

(1)

= 1r × (k11 k12 · · · k1n ) + 1r × (k21 k22 · · · k2n ) + · · · + 1r × (kr1 kr2 · · · krn ). Based on (7), the conditions (3a) and (3b) are presented. Remark 3 In [14], an output-feedback controller for positive systems was designed. To transform their conditions into linear programming, a fixed parameter v was introduced that will lead to the fact that the rank of the designed controller gain matrix should be 1. The literature [17] removed this restriction, where the controller gain matrix is obtained by using an iterative convex optimization algorithm. In Theorem 1, a controller without the rank constraint is constructed and the algorithm of constructing the controller is easily implemented. Remark 4 In [16], an output-feedback controller was designed on the basis of linear matrix inequality. To guarantee the positivity of the systems, an iterative algorithm based on linear matrix inequality was introduced. In Theorem 1, the positivity is easily satisfied via condition (3b) rather than imposing additional algorithms. The following corollary gives a design method for the state-feedback controller of positive systems. Corollary 1 If there exist a constant ς > 0 and vectors v  0 with v ∈ n , zi ∈ n , z ∈ n such that AT v + z ≺ 0, 4

(8a)

A1Tr B T v + B

r  i=1

(i)

1r ziT + ςI  0,

(8b) (8c)

zi  z, hold, then under the state-feedback control law (i)

u(t) = Kx(t) =

Σri=1 1r ziT x(t) 1Tr B T v

(9)

the resulting closed-loop system (2) is positive and exponentially stable. The proof of Corollary 1 can be obtained by following Theorem 1 and is omitted. Remark 5 In [12], an effective approach to the stabilization of positive systems was proposed for the first time, and a state-feedback controller was designed by using linear programming algorithm. The approach in [12] has been commonly used to solve the corresponding issues of positive systems. As stated in the introduction, a condition aij dj + bi zj ≥ 0, i = j is given to guarantee the positivity of the considered systems, where aij is the ith row j column of the system matrix A. The condition will be complex. In contrast, Corollary 1 provides a compact form of the conditions that guarantees the positivity and stability of the systems. Meanwhile, the conditions in Corollary 1 are also simpler and can be obtained easily. Furthermore, a linear copositive Lyapunov function for positive systems can be directly constructed through the established conditions. As we all know, how to construct a linear copositive Lyapunov function is always important for control system design. The proposed results certainly provides an effective way to investigate this issue. In particular, the Lyapunov function approach is effective in control synthesis of hybrid positive systems such as switched positive systems, stochastic positive systems, and so on. The method developed in Theorem 1 can provide a framework to further investigate on some control problems related to positive systems and also applications. In Theorem 1 and Corollary 1, the open-loop system (2) is assumed to be positive. In the following criterion, we will provide a controller design for general system (2), that is, the system is non-positive. Corollary 2 If there exist a constant ς > 0 and vectors v  0 with v ∈ n , zi ∈ n , z ∈ n , z ∈ n such that AT v + z ≺ 0, A1Tr B T v + B

r  i=1

(i)

1r ziT + ςI  0,

1Tr B T v > 0, (i)T

(10a) (10b) (10c)

1r

B T v < 0, i = 1, 2, . . . , j < r,

(10d)

(i)T

B T v > 0, i = j + 1, j + 2, . . . , r,

(10e)

1r

z ≺ zi  z, i = 1, 2, . . . , r,

(10f)

hold, then under the state-feedback control law (9) the resulting closed-loop system (2) is positive and exponentially stable.

5

Proof Using (10a) and (10c)-(10f) gives (i)T

Σri=1 zi 1r B T v 1Tr B T v (i)T (i)T j Σi=1 zi 1r B T v Σri=j+1 zi 1r B T v T + =A v+ 1Tr B T v 1Tr B T v (i) (i)T T j r zΣi=1 1r B v zΣi=j+1 1r B T v ≺ AT v + + 1Tr B T v 1Tr B T v (i)T (i)  Σri=j+1 1r B T v  Σri=j+1 1r B T v T +z =A v+z 1− 1Tr B T v 1Tr B T v (i) T r Σi=j+1 1r B v = AT v + z + (z − z) 1Tr B T v

AT v + K T B T v = AT v +

(11)

≺ AT v + z ≺ 0. The rest of the proof can be given by using a similar derivation in Theorem 1 and is omitted.



Corollary 2 is an extension of Corollary 1 for general systems. It gives a feasible approach to the positive stabilization of general systems. In addition, Corollary 2 verifies that the approach in Theorem 1 possesses potential applications in control synthesis of general systems. 3.2. Discrete-time case For convenience, we rewrite the discrete-time system (1) as x(k + 1) = Ax(k) + Bu(k), y(k) = Cx(k), k ∈ N.

(12)

Theorem 2 If there exist vectors v  0 with v ∈ n , zi ∈ m , z ∈ m such that AT v + C T z − v ≺ 0, A1Tr B T v + B

r  i=1

(i)

1r ziT C  0,

(13a) (13b) (13c)

zi  z, hold, then under the output-feedback control law (i)

u(k) = Ky(k) =

Σri=1 1r ziT y(k) 1Tr B T v

(14)

the resulting closed-loop system (12) is positive and exponentially stable. Proof By (13b) and (14), we have that A + BKC  0. Then, the resulting closed-loop system (12) is positive by Lemma 1, that is, x(k)  0 ∀k ∈ N. Choose a linear copositive Lyapunov function V (x(k)) = x(k)T v, then ΔV = V (x(k + 1)) − V (x(k)) = x(k)T (AT v + C T K T B T v − v).

(15)

By (13c), we get that K T B T v  z. Using C  0 and x(k)  0 gives ΔV ≤ x(k)T (AT v + z − v). 6

(16)



By (13a), we have ΔV < 0. This completes the proof. Corollary 3 If there exist vectors v  0 with v ∈ n , zi ∈ n , z ∈ n such that AT v + z − v ≺ 0, A1Tr B T v + B

r  i=1

(17a)

(i)

1r ziT  0,

(17b) (17c)

zi  z, hold, then under the state-feedback control law (i)

Σr 1r ziT x(k) u(k) = Kx(k) = i=1 1Tr B T v

(18)

the resulting closed-loop system (12) is positive and exponentially stable. Remark 6 In [13], the discrete-time version of the controller in [12] was addressed. Similar to the statements in Remark 5, the conditions in [13] might be complex when the system dimension is large. Compared with the conditions in [13], the conditions in Theorem 2 and Corollary 2 can be computed easily. 4. Extension and application To clearly show the effectiveness of the addressed approach in Subsection 3.2, we apply the approach to positive systems with disturbance input. Consider the systems x(t) ˙ = Ax(t) + Bu(t) + Bw w(t),

(19)

y(t) = Cx(t) + Du(t) + Dw w(t), and x(k + 1) = Ax(k) + Bu(k) + Bw w(k),

(20)

y(k) = Cx(k) + Du(k) + Dw w(k),

where w(t) ∈ q+ (or w(k) ∈ q+ ) is the disturbance input, Bw ∈ n×q , Dw ∈ m×q , and the other parameters are defined as those in system (1). Throughout this section, we assume that A is a Metzler matrix, B  0, Bw  0, C  0, D  0, and Dw  0 for system (19), and A  0, B  0, Bw  0, C  0, D  0, and Dw  0 for system (20). We introduce some preliminaries on systems (19) and (20). Definition 2 ([2, 3]) System (19) or (20) is positive if the state and output are nonnegative for all times if the initial condition x(t0 ) (or x(k0 )), control input u(t) (or u(k)), and disturbance input w(t) (or w(k)) are nonnegative. Lemma 4 ([2, 3]) System (19) is positive if and only if A is a Metzler matrix, B  0, Bw  0, C  0, and D  0, Dw  0; System (20) is positive if and only if A  0, B  0, Bw  0, C  0, D  0, and Dw  0. Definition 3 System (19) (or System (20)) is said to be asymptotically stable with L1 -gain performance (1 -gain performance) if (i) system (19) (or System (20)) is asymptotically stable with w(t) = 0 (or w(k) = 0), and (ii) there exists a constant γ such that ∞ 0

||y(t)||1 dt ≤ γ

∞ 0

||w(t)||1 dt (or

7

∞

k=0 ||y(k)||1

≤γ

∞

k=0 ||w(k)||1 )

(21)

holds for x(t0 ) = 0 (or x(k0 ) = 0). Theorem 3 If there exist a constant ς > 0 and vectors v  0 with v ∈ n , zi ∈ n , z ≺ 0 with z ∈ n such that AT v + z + C T 1m ≺ 0,

(22a)

T v + D T 1 − γ1 ≺ 0, Bw m w m

(22b)

A1Tr B T v + B

r 

(i)

i=1

C1Tr B T v + D

1r ziT + ςI  0,

r  i=1

(i)

1r ziT  0,

(22c) (22d) (22e)

zi  z, hold, then under the state-feedback control law (i)

Σr 1r ziT x(t) u(t) = Kx(t) = i=1 1Tr B T v

(23)

the resulting closed-loop system (19) is positive and asymptotically stable with L1 -gain performance. Proof First, we prove that the closed-loop system (19) is positive. From (22c) and (22d), we have (i)

A+B

Σri=1 1r ziT ς + T T I  0, 1Tr B T v 1r B v (i)

Σr 1r ziT  0. C + D i=1 1Tr B T v Noticing (23), (24) is rewritten as A + BK +

ς 1Tr B T v

I  0,

C + DK  0.

(24a)

(24b)

(25a) (25b)

It is clear that A + BK is a Metzler matrix by Lemma 3. Then, the closed-loop system (19) is positive by Lemma 1, that is, x(t)  0 and y(t)  0 ∀t ≥ 0. Choose a linear copositive Lyapunov function as V (x(t)) = x(t)T v. Then, T v. V˙ (x(t)) = x(t)T (AT v + K T B T v) + w(t)T Bw

(26)

From (23) and (22e), it follows that (i)

K=

(i)

Σri=1 1r z T 1r z T Σri=1 1r ziT  = . 1Tr B T v 1Tr B T v 1Tr B T v

(27)

Thus, we have K T B T v  z. So, V˙ (x(t)) < 0 by (22a) when w(t) = 0. This implies that the closed-loop system (19) with w(t) = 0 is asymptotically stable. Let J

= ||y(t)||1 − γ||w(t)||1 + V˙ (x(t)) − V˙ (x(t)) = y(t)T 1m − γw(t)T 1q + V˙ (x(t)) − V˙ (x(t))   = x(t)T AT v + K T B T v + C T 1m + K T D T 1m  T  T 1 − γ1 ˙ v + Dw + w(t)T Bw m m − V (x(t)).

8

(28)

Using (27) gives K T B T v + K T D T 1m 

z1Tr (B T v + D T 1m ) . 1Tr B T v

(29)

By 1Tr (B T v + D T 1m ) ≥ 1Tr B T v and z ≺ 0, we have K T B T v + K T D T 1m  z.

(30)

   T  T 1 − γ1 ˙ v + Dw J ≤ x(t)T AT v + z + C T 1m + w(t)T Bw m m − V (x(t)).

(31)

Then, (28) is transformed into

By (22a) and (22b), we get J ≤ −V˙ (x(t)).

(32)

Integrating both sides of (32) from 0 to ∞ yields ∞ 0

||y(t)||1 dt − γ

∞ 0

||w(t)||1 dt ≤ V (x(0)) − V (x(∞)) ≤ V (0).

(33)

Noticing the fact that V (0) = 0 when x(0) = 0, then ∞ 0

||y(t)||1 dt ≤ γ

∞ 0

||w(t)||1 dt.

(34) 

This completes the proof. Theorem 4 If there exist vectors v  0 with v ∈ n , zi ∈ n , z ≺ 0 with z ∈ n such that AT v + z − v + C T 1m ≺ 0,

(35a)

T v + D T 1 − γ1 ≺ 0, Bw m w m

(35b)

r 

A1Tr B T v + B

i=1

C1Tr B T v + D

r  i=1

(i)

1r ziT  0, (i)

1r ziT  0,

(35c) (35d) (35e)

zi  z, hold, then under the state-feedback control law (i)

u(k) = Kx(k) =

Σri=1 1r ziT x(k) 1Tr B T v

(36)

the resulting closed-loop system (20) is positive and asymptotically stable with 1 -gain performance. Proof First, we prove that the closed-loop system (20) is positive. Using (35c) and (35d) gives (i)

Σr 1r ziT  0, A + B i=1 1Tr B T v (i)

C+D

Σri=1 1r ziT  0. 1Tr B T v

(37a)

(37b)

It follows from (36) and (37) that A + BK  0,

(38a)

C + DK  0.

(38b)

9

Then, the closed-loop system (20) is positive by Lemma 1, that is, x(k)  0 and y(k)  0 ∀k ≥ 0. Choose a linear copositive Lyapunov function as V (x(k)) = x(k)T v. Then, T v. ΔV = V (x(k + 1)) − V (x(k)) = x(k)T (AT v + K T B T v − v) + w(k)T Bw

(39)

Based on the proof of Corollary 3, we know that K T B T v  z. So, ΔV < 0 is true based on (35a) when w(t) = 0. This implies that the closed-loop system (20) with w(t) = 0 is asymptotically stable. Let J

= ||y(k)||1 − γ||w(k)||1 + ΔV − ΔV = y(k)T 1m − γw(k)T 1q + ΔV − ΔV   = x(k)T AT v + K T B T v + C T 1m + K T D T 1m − v  T  T 1 − γ1 + w(t)T Bw v + Dw m m − ΔV.

(40)

Combining (30) and (40) gives    T  T 1 − γ1 v + Dw J ≤ x(k)T AT v + z + C T 1m − v + w(k)T Bw m m − ΔV.

(41)

By (35a) and (35b), we have (42)

J ≤ −ΔV. Summing both sides of (42) from 0 to ∞ yields ∞  k=0

||y(k)||1 − γ

∞  k=0

||w(k)||1 ≤ V (x(0)) − V (x(∞)) ≤ V (x(0)).

(43)

Utilizing the fact V (0) = 0 when x(0) = 0, one has ∞  k=0

||y(k)||1 ≤ γ

∞  k=0

||w(k)||1 .

(44) 

This completes the proof.

Remark 7 In [15], an 1 -induced controller for system (20) was presented and an iterative algorithm was given to compute the controller gain matrix K. In Step 1 of the iterative algorithm, a gain matrix K1 ensuring the positivity and stability of the closed-loop system is computed by solving the following inequalities: ⎡ ⎤ −P AP + BQ ⎣ ⎦<0 ∗ −P aij pj + chj pj +

l  z=1 l  z=1

(45a)

biz qzj ≥ 0,

(45b)

dhz qzj ≥ 0,

(45c)

where A = [aij ], B = [biz ], C = [chj ], D = [dhz ] are system matrices and P = diag[p1 , . . . , pn ], Q = [qij ] are variables to be determined (see (24)-(26) in [15]). In Theorem 4, the condition (35) is solvable by directly using the standard Linprog toolbox in Matlab. In addition, the optimal value of 1 -gain γ can be easily solved by using the optimization problem:

min γ s.t. (35).

v,zi ,z,γ

10

(46)

5. Discussions Aiming to overcome the restrictions in the literature, we have proposed the new approaches in the paper and three advantages are witnessed as follows: (i) a general design of controllers with looser restrictions for positive systems is proposed, (ii) the stabilization conditions can be easily solved via linear programming, (iii) the approach can be easily extended to other control issues of positive systems. For the issues (i) and (ii), we have provided some detailed statements in Section 3. In the following, we will give a discussion on issue (iii). In Section 4, we apply the presented approaches to two class of positive systems with disturbance inputs. Theorems 3 and 4 verify that the approaches are effective for constructing L1 /1 -induced controllers of positive systems. In fact, the presented approaches can also be applied to some other types of positive systems. For instance, the approaches in the paper can be used for the systems in [23, 24]. Next, we will give a simple corollary for the controller design of continuous-time positive systems with time-delay. Consider the following system: x(t) ˙ = A0 x(t) + A1 x(t − τ ) + Bu(t), t ≥ 0 x(t) = ϕ(t), t ∈ [−τ, 0]

(47)

where x(t) ∈ n and u(t) ∈ r are system state and control input, and τ represents the constant time-delay. A0 ∈ n×n , A1 ∈ n×n , and B ∈ n×r . ϕ(t) is a vector-valued initial condition function. Assume that A0 is a Metzler matrix, A1  0, and B  0. System (47) is positive if and only if A0 is a Metzler matrix, A1  0, and B  0. The objective is to design a state-feedback control law u(t) = K0 x(t) + K1 x(t − τ ) such that the resulting closed-loop system (47) is positive and asymptotically stable. Theorem 5 If there exist a constant λ > 0 and vectors v  0 with v ∈ n , μ  0 with μ ∈ n , ρ  0 with ρ ∈ n , zi ∈ n , z ∈ n , η (i) ∈ n , η ∈ n such that AT0 v + z + μ + τ ρ + λv ≺ 0,

(48a)

AT1 v + η − e−λτ μ ≺ 0,

(48b)

A0 1Tr B T v + B

r 

(i)

i=1

A1 1Tr B T v + B

1r ziT + ςI  0,

r  i=1

(48c)

(i)

1r ziT  0,

(48d)

zi  z,

(48e)

η (i)  η,

(48f)

hold, then under the state-feedback control law u(t) = K0 x(t) + K1 x(t − τ ) =

(i)

(i)

Σri=1 1r η (i)T Σri=1 1r ziT x(t) + x(t − τ ) 1Tr B T v 1Tr B T v

(49)

the resulting closed-loop system (47) is positive and exponentially stable. Proof From (48c), (48d), and (49), it follows that A0 + BK0 is a Metzler matrix and A1 + BK1  0. Therefore, the resulting closed-loop system is positive. A linear copositive Lyapunov function can be chosen as follows: V (x(t)) = x(t)T v +

t

t−τ

eλ(−t+s) x(s)T μds + 11

0 t −τ

t+θ

eλ(−t+s) x(s)T ρdsdθ.

(50)

Then

  V˙ (x(t)) = −λV (x(t)) + x(t)T AT0 v + K0T B T v + μ + τ ρ + λv  t  + x(t − τ )T AT1 v + K1T B T v − e−λτ μ − t−τ e−λτ x(s)T ρds.

(51)

From (48e), (48f), and (49), we can get K0T B T v  z and K1T B T v  η. These together with (48a) and (48b) imply that V˙ (x(t)) ≤ −λV (x(t)).



Remark 8 The stabilization of continuous-time positive systems with time-delay was previously investigated in [23]. To guarantee the positivity of the resulting closed-loop systems, a condition similar to aij dj + bi zj ≥ 0, i = j in [12] is used. However, the computation burden will be heavy with the increasement of the dimension of the systems. Theorem 5 provides a compact form (see (48c) and (48d)) for the positivity conditions. All conditions in (48) can be directly computed via liner programming and the computation burden is not heavy regardless of the system dimension. Moreover, a linear copositive Lyapunov function is constructed in Theorem 5. The construction of the Lyapunov function is helpful for applying the approach in Theorem 5 to other issues of positive systems. In addition, Theorem 5 can be developed to discrete-time positive systems with time-delay. Here, we omit the detailed deduction. Remark 9 From the results in Sections 3, 4, and 5, the advantages of the presented approach lie in the following facts: (i) it proposes a simple controller form; (ii) the conditions in the approach can be easily computed; and (iii) it is easily extended to other issues of positive systems. So, the proposed controller framework holds unified property. When designing a controller (state-feedback, output-feedback, L1 -induced, finite-time state- and output-feedback, (i) Σr 1r ziT . Furthermore, when designing etc.), one can choose a framework of the controller gain matrix as: K = i=1 1Tr B T v (i) Σr 1n z T an observer for positive systems, one can also choose a framework of the observer gain matrix as: L = i=1 T i or 1n v (i)T r Σ zi 1m , where L, C, v, zi possess compatible dimensions. L = i=1T 1m Cv Recently, hybrid positive systems arise in the control community of positive systems. Switched positive systems are a class of typical and important hybrid positive systems. In [25–27], some practical models based on switched positive systems were constructed. The stability of switched positive systems was investigated in [28–30]. In our previous works [31, 32], the stabilization of switched positive linear and nonlinear systems was discussed, respectively. The stability and stabilization of switched positive systems with time-delay was investigated in [33] and [34], respectively. It should be pointed out that some previously mentioned constraints and drawbacks exist in the algorithms in [31, 32, 34]. In this paper, the presented approaches in Theorems 1, 2, and 5 can be applied to the systems in [31, 32, 34] and improve existing results. In short, the presented approach possesses many potential applications in the area of control synthesis of positive systems. Remark 10 All stabilization conditions in the paper are sufficient but not necessary. For some systems, they are stabilizable but the conditions do not bear feasible solutions. It is a pity to say that we fail to take one of such systems. How to find such systems and establish necessary and sufficient conditions by using the presented controller framework is an interesting topic in the future. 6. Illustrative examples Three examples are provided to show the effectiveness of the proposed strategy. The first one illustrates that the controller gain matrix has a more general form than that in [14], the second one reveals that the approach in the 12

paper can be more easily implemented than that in [12], and the third one provides a comparison between Theorem 4 and the design in [15].

⎛

⎜ ⎜ Example 1 Consider system (2) with A = ⎜ ⎝

−0.5 0 0.4

⎞

⎛

0 0 1

⎞

⎛

0.5 0

0

⎞

⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ 0 ⎟, B = ⎜ 1 0 0 ⎟, C = ⎜ 0 1 0 ⎟. ⎠ ⎝ ⎠ ⎝ ⎠ 0 0 0 0 1 0 0 0 0.2 ⎞ ⎛ k11 k12 k13 ⎟ ⎜ ⎟ ⎜ Choose a controller gain matrix with rank 1 as follows: K = ⎜ 1 k11 1 k12 1 k13 ⎟ , where k11 , k12 , k13 and ⎠ ⎝ 2 k11 2 k12 3 k13 ⎞ ⎛ −0.5 + 0.52 k11 2 k12 0.4 + 0.22 k13 ⎟ ⎜ ⎟ ⎜ 1 , 2 are constants. Then A + BKC = ⎜ ⎟ . Clearly, the closed-loop 0.5k11 k12 0.2k13 ⎠ ⎝ 0.52 k11 2 k12 0.23 k13 system is unstable and the system cannot be stabilized using a control law whose gain matrix rank is 1. This 0

0

implies that the design in [14] cannot be applied to the system. Now, we apply Theorem ⎛ ⎞ the proposed ⎛ ⎞ ⎛ ⎞ ⎛ above ⎞ 0.0000 0.0000 0.0000 0.0000 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 1 to the system and obtain v = ⎜ 96.0676 ⎟ , z = ⎜ 0.0000 ⎟ , z1 = ⎜ −91.1345 ⎟ , z2 = ⎜ 0.0000 ⎟ , ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 0.0000 −94.4864 96.0676 0.0000 ⎛ ⎞ ⎛ ⎞ −78.2419 0.0000 −0.4743 0.0000 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ z3 = ⎜ 0.0000 ⎟ , ς = 206.7501. Then, K = ⎜ 0.0000 0.0000 −0.4918 ⎟ and the closed-loop system is ⎝ ⎠ ⎝ ⎠ −81.3213 −0.4072 0.0000 −0.4233 ⎛ ⎞ −0.7036 0.0000 0.3153 ⎜ ⎟ ⎜ ⎟ A + BKC = ⎜ 0.0000 −0.4743 0.0000 ⎟ . ⎝ ⎠ 0.0000 0.0000 −0.0984 Remark 11 It is necessary to note two points. First, the solutions obtained by Theorem 1 are feasible. From the form of A + BKC, we know that the designed controller gain matrix K ensures the positivity and stability of the −6 closed-loop system. Second, the elements 0.0000 actually positive numbers ⎛ in v and z are ⎞ ⎛ ⎞ at the level of 10 . −0.15 0.8 0.4 0.3 ⎠, B = ⎝ ⎠ . By Theorem 3.1 in [12], Example 2 Consider system (2) with A = ⎝ 0.4 −0.4 0.1 0.2 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ −101.5332 −45.5435 97.6458 ⎠ , z2 = ⎝ ⎠ . Then, the controller gain matrix is ⎠ , z1 = ⎝ we get d = ⎝ −66.1549 −64.3958 92.0156 ⎛ ⎞ ⎛ ⎞ −1.0398 −0.4950 −0.7692 0.3921 ⎠. Thus, we can obtain the system matrix as A + BK = ⎝ ⎠, and K=⎝ −0.6775 −0.6998 0.1605 −0.5895 its eigenvalues are −0.9458 and −0.4128. ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

Now, we apply Corollary 1 to the system and get v = ⎝ ⎛ z2 = ⎝

−133.9479 −15.6336

⎞

101.7447 114.2294

⎠, z = ⎝

−112.8542

⎛

⎠ , ς = 171.0137. Then, the controller gain matrix is K = ⎝ ⎛

the system matrix as A + BK = ⎝

−1.0752

⎞ 0.7098

82.6576

⎠ , z1 = ⎝

−1.3607 −0.1143 −1.2698 −0.1482

⎞

−143.5393 −12.0571

⎠,

⎠. Furthermore,

⎠, and its eigenvalues are −1.0862 and −0.4301. 0.0100 −0.4411 Remark 12 From the eigenvalues of the closed-loop system, we can observe that the closed-loop system possess a

13

faster convergence speed. Based on this point and Remark 5, the approach in this paper both effective and easily to be computed. It should be pointed out that the solution of Theorem 3.1 in [12] is unique by virtue of Matlab, whereas the solution of Corollary 1 is not. The reason lies in the fact that we can get some closed-loop systems with much faster convergence speed since one can introduce a term λv into the first condition of (9), that is, AT v + z + λv ≺ 0, where λ > 0. Example 3 In [15], the structured population dynamics of a certain pest was described by a Leslie model. To provide a comparison with the design in [15], we ⎛ 0.2 ⎜ ⎜ A = ⎜ 0.8 ⎝ 0

also consider the model ⎞ ⎛ 0.3 2 0.5 ⎟ ⎜ ⎟ ⎜ 0 0 ⎟, B = ⎜ 0 ⎠ ⎝ 0.7 0 0

C= ⎞



in [15]. Consider system (20) with ⎛ ⎞ ⎞ 0.1 ⎜ ⎟ ⎟ ⎜ ⎟ ⎟ ⎟ , Bw = ⎜ 0.05 ⎟ , ⎝ ⎠ ⎠ 0.1



1 1 1 , D = 0.5, Dw = 0. ⎛ ⎞ ⎛ ⎞ ⎛ ⎛ ⎞ −1.0727 0.0000 5.3636 −1.0727 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ By Theorem 4, we get v = ⎜ 5.4545 ⎟ , z = ⎜ −1.6091 ⎟ , z1 = ⎜ −1.6091 ⎟ , z2 = ⎜ 0.0000 ⎟ . Then, ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ −5.3636 −94.4864 6.3636 −5.3636 ⎛ ⎞ 0.0000 0.0000 1.0000 ⎜ ⎟   ⎜ ⎟ K = −0.4000 −0.6000 −2.0000 and the system matrix is A + BK = ⎜ 0.8000 ⎟. 0 0 ⎝ ⎠ 0 0.7000 0 Remark 13 In Example 3, we get the same controller gain matrix and the closed-loop system as those in [15]. The controller gain matrix is optimal based on the strict proof in [15] (see Theorem 4 therein). Based on this point and Remark 7, we can get that the design in Theorem 4 is effective and easily to be computed. 7. Conclusions This paper has addressed new approaches to control synthesis of positive systems. Sufficient conditions for the output-feedback stabilization of positive systems in both the continuous-time and discrete-time contexts are respectively established by using a linear copositive Lyapunov function associated with linear programming technique. Then, the obtained approaches are applied to L1 /1 -introduced controller design of positive systems. Some discussions on the present approaches are provided to state the potential applications to the corresponding issues of positive systems. Several comparison examples illustrate the effectiveness of the results. Acknowledgements This work was supported in part by the National Nature Science Foundation of China (61503107, 61203123), the Zhejiang Provincial Natural Science Foundation of China (LY16F030005), the Liaoning Excellent Talents in University (LR2014035), and the National Natural Science Major Foundation of Research Instrumentation of China under Grants 61427808.

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