Finite-time L1 control for positive Markovian jump ...

Original Manuscript

Finite-time L1 control for positive Markovian jump systems with constant time delay and partly known transition rates

Transactions of the Institute of Measurement and Control 1–8 Ó The Author(s) 2015 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0142331215585881 tim.sagepub.com

Wenhai Qi and Xianwen Gao

Abstract The paper is concerned with finite-time L1 control for positive Markovian jump systems with constant time delay and partly known transition rates. By constructing appropriate co-positive type Lyapunov function, sufficient conditions for finite-time boundedness and L1 finite-time boundedness of the underlying system are firstly given. Based on the results obtained, a state feedback controller is designed such that the closed-loop Markovian jump system is positive and L1 finite-time bounded. All the proposed conditions are given in linear programming. Finally, an example is given to demonstrate the validity of the main results.

Keywords Positive Markovian jump systems, partly known transition rates, finite-time boundedness, linear programming

Introduction Positive systems are dynamic systems whose common property is that the states and outputs are nonnegative whenever initial conditions and inputs are nonnegative (Farina and Rinaldi, 2000; Kaczorek, 2002). For this special class of systems, there are many applications in practice, such as communication networks (Shorten et al., 2006), industrial engineering (Caccetta et al., 2004), system control theory (Ait Rami and Tadeo, 2007; Chen et al., 2013; Ebihara et al., 2014), etc. As a special class of hybrid systems, Markovian jump systems have been a focus of study during the past few decades, since they can model many actual control processes and systems, such as (Song, et al., 2013), manufacturing systems (Shen and Buscher, 2012) and fault-detection systems (Ge and Han, 2014) and other aspects (Kao et al., 2014, 2015; Li et al., 2014). Very recently, some results on positive Markovian jump system have been obtained (Bolzern et al., 2014; Zhang et al., 2014b; Zhu et al., 2014). To mention a few, by employing appropriate co-positive type Lyapunov function, necessary and sufficient conditions for stochastic stability of positive Markovian jump system (Bolzern et al., 2014) were given in linear programming. A state feedback controller was designed to guarantee the closed-loop Markovian jump system (Zhang et al., 2014b) positive and stochastically stable. And a positive filter was constructed in Zhu et al. (2014). On the other hand, it is widely recognized that the reaction of real-world systems to exogenous signals is always not instantaneous, and is frequently affected by certain time delays, such as networked control systems, transportation

systems, etc. Time delay is frequently a source of instability, which gives rise to undesirable performance of dynamic system (Xu and Chen, 2003; Xu et al., 2006). Recently, studies on positive system with time delays have become a topic of major interest, including stability and stabilization analysis (Liu and Dang, 2011; Li et al., 2013). Furthermore, most existing results related to the stability analysis in the literature focus on the Lyapunov stability that is defined in an infinite-time interval. Compared with the Lyapunov stability, the finite-time stability requires that the states do not exceed a certain bound during fixed finite-time interval. It should be pointed out that a finite-time stable system may not be Lyapunov stable, and a Lyapunov stable system may not be finite-time stable since the transient behavior of the system response may exceed the prescribed bound. Very recently, some available results on finite-time control of positive system have been reported in Chen and Yang (2014), Xiang and Xiang (2013), and Zhang et al. (2014a, 2014c). It is well known that many previous results for positive systems are based on quadratic Lyapunov functions in linear matrix inequality (Ebihara et al., 2014; Fornasini and Valcher 2012; Li et al., 2010). However, some results for positive

College of Information Science and Engineering, Northeastern University, PR China Corresponding author: Wenhai Qi, College of Information Science and Engineering, Northeastern University, No. 3–11, Wenhua Road, Heping District, Shenyang 110819, PR China. Email: [email protected]

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systems with linear Lyapunov functions have been reported (Bolzern et al., 2014; Chen and Yang 2014; Xiang and Xiang 2013; Zhang et al., 2014a, 2014b; Zhu et al., 2014), which leads to some new results. Compared with quadratic Lyapunov functions, the results based on linear Lyapunov functions are easily to analyze. Additionally, there are many different performances such as HN performance (Li et al., 2010), H2 performance, L2 performance, L1 performance (Sun et al., 2012), and so on. It is noticed that some frequently used performance measures such as HN norm are based on the L2 signal space (Ebihara et al., 2014; Fornasini and Valcher 2012; Li et al., 2010). In some situations, these performance measures induced by L2 signal are not very natural to describe the features of practical positive systems. In addition, 1 -norm provides a more useful description for positive systems because it represents the sum of the values of the components, which is more appropriate, for example, if the values represent the amount of material. For positive Markovian jump system, there is further room for investigation. To mention a few, the transition rates in Bolzern et al. (2014) and Zhang et al. (2014b) are completely known, which may lead to some conservativeness. To the best of our knowledge, most of the existing results based on positive Markovian jump systems are based on nominal system without taking time delay, finite-time control and exogenous disturbance into account. As time delay, finite-time control and exogenous disturbance are frequently encountered in practice, it is necessary and significant to further consider positive Markovian jump system with those kinds of phenomena. When taking those kinds of phenomena into account, the problem of choosing an appropriate mode-dependent co-positive Lyapunov function candidate and how to reduce some conservativeness of Lyapunov function will be more complicated and challenging. However, until now, no relevant work that considers this kind of system has been published, which motivates our investigation. In this paper, finite-time L1 control for positive Markovian jump systems with constant time delay and partly known transition rates will be investigated. The main contributions of this paper are threefold: (i) By constructing appropriate copositive type Lyapunov function, sufficient conditions for finite-time boundedness of the underlying system are proposed; (ii) Based on the finite-time boundedness, L1-gain performance is analyzed; (iii) Based on the obtained results, s state feedback controller is designed such that the closedloop Markovian jump system with constant time delay and partly known transition rates is positive and L1 finite-time bounded. Notation: In this paper, A < ( 4 0,  ,  ) means that all entries of matrix A are nonnegative (non-positive, positive, negative); A  B(A < B) means that A  B  0(A  B < 0); R (R + ) is the set of all real (positive real) numbers; Rn(Rnþ ) in n-dimensional real (positive) vector space; The vector 1-norm P is denoted by jjxjj1 = nk = 1 jxk j, where xk is the kth element of x 2 RÐn; Given v: R!Rn, the L1-norm is defined by ‘ jjvjjL1 = 0 jjvjj1 dt; L1 ½0, + ‘) is the space of absolute integrable vector-valued functions onÐ [0, + N), i.e., we say x: ‘ [0, + N) !Rn is in L1[0, + N) if 0 jjx(t)jj1 dt\‘; Matrix A is said to be a Metzler matrix if its off-diagonal elements are all nonnegative real numbers; Symbol E{} represents the

mathematical expectation; In denotes identity matrix and 1n means the all-ones vector in Rn.

Problem statement and preliminaries Consider the following positive Markovian jump system with constant time delay: x_ (t) = A(gt )x(t) + Ad (gt )x(t  t) + G(gt )w(t), ð1Þ

z(t) = C(gt )x(t) + D(gt )w(t), x(t + u) = u(u), 8u 2 ½t, 0,

where x(t) 2 Rn is the state vector; w(t) 2 Rl is the disturbance input which belongs to Ln1 ½0, + ‘); z(t) 2 Rq is the controlled out; u(u) is a vector-valued initial continuous function which is defined on interval [2t,0];{gt, t  0} is a time-homogeneous stochastic Markovian process with right continuous trajectories and takes values in a finite set S = {1,2,.,N} with transition rate matrix P = {pij}, i, j 2 S. The transition rate from mode i at time t to mode j at time t + Dt is given by:  Pfgt + Dt = jjgt = ig =

pij Dt + o(Dt), 1 + pij Dt + o(Dt),

i 6¼ j, i = j,

where Dt  0, lim (o(Dt)=Dt) = 0 and pij  0, for i6¼j and P Dt!0 N j = 1, i6¼j pij =  pii . Throughout the paper, the transition rates are built to be partly known, that means there are only some elements to be i obtained Sini matrix P = {pij}. For 8i 2 S, the set S represents S i = Ski Suk , with D

Ski ¼ fj : pij is known, for j 2 Sg, D

i Suk ¼ fj : pij is unknown, for j 2 Sg

And if Si6¼Ø, it is further given by D

Ski ¼ fk1i , k2i , . . . , kmi g, 1  m  N , where kmi 2 S means the mth known transition rate of Ski in the ith row of the matrix P. For simplicity, when gt = i, the system matrices of the ith mode are denoted as Ai, Adi, Gi, Ci and Di. Assumption 1. The disturbance input w(t) 2 Rl satisfies ÐT 0

kw(t)k1 dt\d, d  0

ð2Þ

Definition 1. (Kaczorek, 2002). System (1) is said to be positive if, for any initial condition u(u) < 0, u 2 [2t, 0], the corresponding trajectory x(t) < 0 and z(t) < 0. Lemma 1. (Kaczorek, 2002). System (1) is said to be positive if and only if Ai, for all i 2 S, are Metzler matrices and Adi < 0, Gi < 0, Ci < 0, Di < 0. Lemma 2. (Zhang et al., 2014b). Matrix A is a Metzler matrix if and only if there exists a positive constant e such that A + eI < 0.

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Definition 2. (finite-time stability). For given time constant T and vectors q . y . 0, positive system (1) (w(t) = 0) is said to be finite-time stable with respect to (q, y, T, gt), if Efsupt  t  0 ½xT (t)qg  1 ) EfxT (t)y g\1, 8t 2 ½0, T .

In this paper, our aim is to design the state feedback controller (3) under which the closed-loop Markovian jump system (5) is positive and L1 finite-time bounded.

Remark 1. From Definition 2, we can see that the concept of finite-time stability is different from the one of Lyapunov asymptotic stability. A Lyapunov stable system may not be finite-time stable because its states may exceed the prescribed bounds during the fixed interval time.

Remark 2. From Lemma 1, the closed-loop Markovian jump system (5) is positive if and only if Ai + BiKi, 8i 2 S are Metzler matrices and Adi < 0, Gi < 0, Ci < 0, Di < 0.

Definition 3. (finite-time boundedness). For given time constant T and vectors q . y . 0, positive system (1) is said to be finite-time bounded with respect to (q, y, T, d, gt), where w(t) satisfies (2), if Efsupt  t  0 ½xT (t)qg  1 ) EfxT (t)y g\1, 8t 2 ½0, T .

This section will focus on the problem of finite-time boundedness, L1 finite-time boundedness and L1 controller design.

Definition 4. (L1 finite-time boundedness). For given time constant T, positive constant g and vectors q . y . 0, positive system (1) is said to be L1 finite-time bounded with respect to (q, y, T, d, gt), if the following conditions are satisfied:

In this subsection, we turn the focus on the problem of finitetime boundedness for positive Markovian jump system (1). The following theorem gives a sufficient condition of the finite-time boundedness for the corresponding system.

(i) Positive system (1) is finite-time bounded with respect to (q, y, T, d, gt);

Theorem 1. For given time constant T, positive constant l and vectors q . y . 0, if there exist vectors ni, si, s 2 Rnþ , r1i, r2i 2 Rn and positive constants j1, j2, j3, j4, such that the following inequalities hold,

(ii) Under zero initial condition (u(u) = 0, 8u 2 ½t, 0), system (1) satisfies ð T E 0

 jjz(t)jj1 dt  gE

ð T 0

Main results

Finite-time boundedness analysis

1 ~ )  lj1 elT (j2 + tj3 + t 2 j4 + (1  elT )d w 2

 jjw(t)jj1 dt ,

j1 y  ni  j2 q, si  j3 q, s1  j4 q when w(t) satisfies (2). Definition 5. Considering V (x(t), i) as the Lyapunov function for the system (1), we define the weak infinitesimal operator as follows:

ATi ni +

X

pij (nj  r1i ) + si + ts  lni  0

ð6Þ

ð7Þ ð8Þ

j2Ski

ATdi ni  s  0

1 GV (x(t), i) = lim ½EfV (x(t + Dt), g(t + Dt))j Dt!0 Dt

X

x(t), g(t) = i)g  V (x(t), g(t) = i):

ð9Þ

pij (sj  r2i )  s 4 0

ð10Þ

j2Ski

In this paper, the state feedback controller is designed as follows:

i nj  r1i 4 0, sj  r2i 4 0, j 2 Suk , j 6¼ i

ð11Þ

i ,j=i nj  r1i < 0, sj  r2i < 0, j 2 Suk

ð12Þ

ð3Þ

u(t) = Ki x(t)

where Ki are the controller gain matrices. Then, the control synthesis problem will be investigated for the following positive Markovian jump system x_ (t) = Ai x(t) + Bi u(t) + Adi x(t  t) + Gi w(t), z(t) = Ci x(t) + Di w(t), x(t + u) = u(u), 8u 2 ½t, 0,

ð4Þ

Furthermore, with the state feedback controller (3), the closed-loop Markovian jump system is denoted as: x_ (t) = (Ai + Bi Ki )x(t) + Adi x(t  t) + Gi w(t), z(t) = Ci x(t) + Di w(t), x(t + u) = u(u), 8u 2 ½t, 0,

ð5Þ

then, the positive system (1) with constant time delay and partly known transition rates is finite-time bounded with ~ = maxp, i2L 3 S (~ respect to (q, y, T, d, gt), where w wip ), ~ ip is the pth element of the vector L = {1,2,.,l}, w ~ i = GiT qj2 . w Proof. For the positive Markovian jump system (1), choose the co-positive Lyapunov function candidate as follows V (x(t), i) = xT (t)ni +

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ðt tt

xT (s)si ds +

ð0 ðt t

xT (s)sdsdu

t+u

ð13Þ

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where ni, si, s 2 Rnþ and

N P

elt EfV (x(t), i)g  EfV (x0 , t0 )g\

pij sj 4 s.

EfV (x(t), i)g\elt EfV (x0 , t0 )g ðt 1 ~ wi ds  elt EfV (x0 , t0 )g + elt (1  elt )d w + elt els wT (s)~ l 0 ð20Þ

pij nj ) + wT (t)GiT ni + xT (t  t)ATdi ni

j=1

+ xT (t)si  xT (t  t)si +

ðt

xT (s)

tt

+ txT (t)s 

ðt tt

Due to

j=1

pij sj ds

j=1

Considering equality (13) and Definition 2 yields N X

pij nj + si + ts)

j=1

+w N P

N X

xT (s)sds  xT (t)(ATi ni + T

(t)GiT ni

pij r1i =

N P j=1

T

+ x (t 

ð19Þ

which shows

GV (x(t), i) = xT (t)(ATi ni +

els wT (s)~ wi ds

0

j=1

According to Definition 5, it can be shown that

N X

ðt

t)(ATdi ni

EfV (x(t), i)g  j1 xT (t)y, EfV (x0 , t0 )g  j2 xT (0)q  j2 + tj3 supt  u  0 (xT (u)q)

 si )

ð14Þ

pij r2i = 0 for a set of vectors r1i and

1 + t 2 j4 supt  u  0 (xT (u)) 2

ð21Þ

Combining (20)–(21) yields

r2i, we have GV (x(t), i)  xT (t)(ATi ni +

N X

  elT 1 ~) (j + tj3 + t 2 j4 + (1  elt )d w E xT (t)y  2 lj1 2

pij nj

j=1



N X

=x

T

Substituting (6) into (22), we have T

pij r1i + si + ts) + w

j=1

(t)(ATi ni

+

X

(t)GiT ni

pij (nj  r1i ) +

j2Ski

X

  E xT (t)y \1 pij (nj  r1i )

i j2Suk

+ si + ts) + wT (t)GiT ni + xT (t  t)(ATdi ni  s),

ð15Þ

and N X

pij sj 

j=1

X

N X

pij r2i  s =

j2Ski

X

pij (sj  r2i )  s 4 0

ð16Þ

i j2Suk

Note that pii \ 0 for all j = i and pij  0 for all j6¼i, if i 2 Ski , inequalities (7)–(11) imply that GV (x(t), i)\lxT (t)ni + wT (t)GiT ni \lxT (t)ni ðt ð0 ðt +l xT (s)si ds + l xT (s)s t

tt

= lV (x(t), i) + w

According to the Definition 3, we can see that the positive system (1) with partly known transition rates is finite-time bounded with respect to (q, y, T, d, gt). The proof is completed.

t+u

ð17Þ

(t)GiT j2 q:

i On the other hand, if i 2 Suk , inequalities (7–12) also imply that (17) holds. ~ i = GiT qj2 and multiplying inequality (17) by Denoting w e2lt yield:

wi G(elt V (x(t), i))\elt wT (t)~

ð18Þ

Applying Dynkin’s formula to inequality (18) yields

Remark 4. An appropriate mode-dependent co-positive type Lyapunov function (13) is constructed, and parameter in integral term is mode-dependent, which may reduce some conservativeness. When w(t) = 0, the result on finite-time stability can be obtained.

+ wT (t)GiT j2 q T

ð23Þ

Remark 3. The free-connection weighting vectors ri 2 Rn are proposed to study the stability of Markovian jump system through considering the relationship among the transition rates of subsystems, which increase the choice room for vector parameter and reduce the conservativeness of systems.

j=1

pij (sj  r2i ) +

ð22Þ

Corollary 1. For given time constant T, positive constant l and vectors q . y . 0, if there exist vectors ni, si, s 2 Rnþ , r1i, r2i 2 Rn and positive constants j1, j2, j3, j4, such that inequalities (7)–(12) and the following inequality hold, 1 elT (j2 + tj3 + t 2 j4 )  lj1 2

ð24Þ

then, the positive system (1) (w(t) = 0) with constant time delay and partly known transition rates is finite-time stable with respect to (q, y, T, gt).

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Remark 5. The concept of finite-time boundedness implies the one of finite-time stability. It is easy to see that finite-time stability can be regarded as a special class of finite-time boundedness by setting w(t) = 0. i Remark 6. If Suk = ˘, then system (1) becomes Markovian switching system with completely known transition rates. Also if Ski = ˘, namely, the transition rates are completely unknown, then the problem is changed into the finite-time control for switched time-delayed linear system under arbitrary switching.

Corollary 2. For given time constant T, positive constant l and vectors q . y . 0, if there exist vectors ni, si, s 2 Rnþ , r1i, r2i 2 Rn and positive constants j1, j2, j3, j4, such that inequalities (6)–(7), (9) and the following inequality hold, ATi ni +

N X

pij nj + si + ts  lni  0,

j=1

N X

GV (x(t), i)\lxT (t)ni  kz(t)kL1 + gkw(t)kL1 ðt ð0 ðt \lxT (t)ni + l xT (s)si ds + l xT (s)sdsdu t

tt

t+u

ð28Þ

 kz(t)kL1 + gkw(t)kL1 = lV (x(t), i)  kz(t)kL1 + gkw(t)kL1 Multiplying inequality (28) by e2lt yields G(elt V (x(t), i))\elt (  kz(t)kL1 + gkw(t)kL1 )

ð29Þ

Under zero initial condition, using Dynkin’s formula yields elt V (x(t), i)\

ðt

els (  kz(s)k1 + g kw(s)k1 )ds

ð30Þ

 ð T  els jjz(s)jj1 ds  gE els jjw(s)jj1 ds

ð31Þ

0

and ð T

pij sj  s 4 0,

j=1

E 0

then, the positive system (1) (w(t) = 0) with constant time delay and complete known transition rates is finite-time stable with respect to (q, y, T, gt).

0

which further implies that ð T E 0

 ð T  jjz(t)jj1 dt  gelT E jjw(t)jj1 dt

ð32Þ

0

L1 finite-time boundedness analysis

~ = gelT . with the L1-gain performance index g The proof is completed.

In this subsection, we will consider the problem of L1 finitetime boundedness analysis of positive Markovian jump system (1).

L1 controller design In this subsection, we will consider the problem of L1 controller design of positive Markovian jump system (1).

Theorem 2. For given time constant T, positive constants l, g and vectors q . y . 0, if there exist vectors ni, si, s 2 Rnþ , r1i, r2i 2 Rn and positive constants j1, j2, j3, j4, such that inequalities (7), (9)–(12) and the following inequalities hold, 1 elT (j2 + tj3 + t 2 j4 + (1  elT )dg)  lj1 2 ATi ni

+

X

pij (nj  r1i ) + si + ts 

lni + CiT 1

ð25Þ

Theorem 3. For given time constant T, positive constants l, g and vectors q . y . 0, if there exist vectors ni, si, s 2 Rnþ , ki, r1i, r2i 2 Rn and positive constants j1, j2, j3, j4, ei, such that inequalities (7), (9)–(12), (25), (27) and the following inequalities hold, ATi ni + ki +

0

j2Ski

ð26Þ GiT ni

+ DTi 1

 g1  0

P j2Ski

pij (nj  r1i )

+ si + ts  lni + CiT 1  0

ð33Þ

n~iT BTi ni Ai + Bi n~i kTi + ei I < 0

ð34Þ

ð27Þ

then, the positive system (1) with constant time delay and partly known transition rates is L1 finite-time bounded with respect to (q, y, T, d, gt). Proof. It is clear that inequality (8) holds if inequality (26) ~ i = g in inequality (6) of Theorem 1, we satisfies. Letting w derive that system (1) is finite-time bounded with respect to (q, y, T, d, gt). For positive system (1) with Lyapunov function candidate (13), inequalities (7), (9)–(12) and (25)–(27) imply

where Bi n~i  0, n~i 2 Rm are a set of given vectors, then the closed-loop system (5) with constant time delay and partly known transition rates is positive and L1 finite-time bounded with respect to (q, y, T, d, gt). Moreover, the state feedback controller can be given as u(t) = Ki x(t) =

1 n~i kTi x(t) n~iT BTi ni

ð35Þ

Proof. Firstly, we prove the positivity of system (5). Dividing two sides of (34) by n~iT BTi ni yields

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Transactions of the Institute of Measurement and Control Ai +

1 n~iT BTi ni

Bi n~i kTi +

1

ei I n~iT BTi ni