State feedback control for nonlinear NCSs with time-delay of partly unknown transition probabilities

State feedback control for nonlinear networked control systems with time-delay of partly unknown transition probabilities

Y an-Feng Wang, Zu-Xin Li, Hui-Ying Chen

School of Engineering, Huzhou University, Huzhou, Zhejiang 313000, China

Abstract:The problem of state feedback control for a class of nonlinear

networked control systems with time-delay is discussed in this paper. The

time-delay is modeled as a finite state Markov chain of which transition

probabilities are partly unknown. The closed-loop system model is obtained

by means of state augmentation. A sufficient condition is given which

guarantees the stochastic stability of the closed-loop system in the form of

linear matrix inequalities and the maximum bound of the nonlinearity is also

obtained. Finally, a simulation example is used to show the validity of the

proposed method.

Keywords: state feedback; nonlinear; transition probability matrix; linear

matrix inequality

1.Introduction

Networked control systems (NCSs) are feedback control systems with control loops closed via digital communication channels [1]. As is well known, NCSs are widely used in many fields such as remote surgery, mobile sensor networks, and unmanned aerial vehicles especially in a number of emerging engineering applications such as arrays of micro actuators, and even social-economical systems [2-4]. Compared with the point-to-point wiring, the use of communication channels can reduce the costs of cables, simplify the installation and maintenance of the system. However, the insertion of the network will make system analysis and synthesis more challenging [5-9]. Network-induced delays can degrade the performance of control systems designed without considering them, and even destabilize the system [10-12].

Up to now, the effect of time-delay on the stability and performance of NCSs has received great attention [13, 14]. The time-delay may be constant, time-varying and in most cases, random [15-17]. For the random cases, the time-delays are commonly modeled as white sequences taking on values of zero and one with certain probability [18, 19] or as finite state Markov chains [20-22]. In the real networks the current time-delays are usually related with the previous delays, so the Markov chain model is

proper than the white sequence model to describe the time-delay.

In [23, 24], the random delays are modeled as Markov chains such that the closed-loop system is a jump linear system. In [25-27], the random time-delays are modeled as Markov chains, and output feedback controller design methods for NCSs with random network-induced delays are proposed. In [28] the mixed

H/H control

2

issue of NCSs with random time delays has been investigated based on Markov jump linear systems method. In the above literature [23-28], all part of the elements in the transition probabilities matrix of the time-delays are assumed to be completely known. However, in almost all kinds of networks, all part of the elements in the desired transition probabilities matrix are hard or costly to obtain. In [29-31], the stability issue of discrete-time networked control systems with random Markov delays and uncertain transition probabilities is studied. And the controlled plants in [23-31] are assumed to be linear. However almost all the practical controlled pants are nonlinear; it is significant and necessary to further study nonlinear NCSs with partly unknown transition probabilities. To the best of the authors’ knowledge, up to now, very limited efforts have been devoted to studying the system with uncertain transition probability matrices for nonlinear NCSs, which motivates our investigation.

This paper studies the state feedback problem of a class of nonlinear NCSs. The time-delay is modeled as a finite state Markov chain of which transition probabilities are partly unknown. The closed-loop system model is obtained by means of state augmentation and the mode-dependent state feedback controller is designed which guarantees the stochastic stability of the closed-loop system. The state feedback controller is designed which guarantees the stability of the resulting closed-loop systems. The maximum bound of nonlinear disturbance is also obtained.

Notations: The superscript T stands for the transpose of a matrix.n R and n m

R denote the n-dimensional Euclidean space and the set of all n m real matrices, respectively. Diag{…} denotes the block diagonal matrix. I and 0 stands for unit matrix and zero matrix with appropriate dimensions respectively. * denotes the entry of matrices implied by symmetry.

2.Problem Formulation

Consider the NCS setup in Figure 1, in which the controlled plant is a nonlinear time-invariant discrete-time system, and0

is the bounded and random

k

time-delay from the senor to the controller which takes value in{1,2,,}d and

the transition probability matrix of

k

is

[

]ij

.

Fig.1. NCSs with random delay

That is,k

jump from mode i to j with probability

ij

which is defined

by

1

Pr(|

)ij

k k

j i , where

0ij

,,i j ,

1

1d

ij

j .

Likewise the transition probability matrix is defined by

11121212223132341

42

4d d d d

.

The set contains d modes of k , and the transition probabilities of the jumping

process in this paper are considered to be partly accessed, i.e., some elements in

matrix

are unknown. For example, for the time-delay

k

with 4 modes, the

transition probabilities matrix

may be as:

12

2331

33

44

?

?

??

????

??

?

.

where “?” represents the inaccessible elements. For notational clarity,i , we

denote

i i k

uk

with

{:is known}i k

ij

j

{:

is unknown}i uk

ij

j .

(1)

Moreover, if i k

, it is further described as

12

={,}i i i i k

k k k ,1

d , wher

e i k

represents the th known element

with the index i k in the th i row of the matrix

.

And

i uk

is described as

12={,}

i i i

i uk

d k k k , wher

e i

d k represents th

e ()th

d

unknown element with the index ()th d in the th i row of the matrix

.

Assume that the model of the plant is a nonlinear time-invariant discrete-time system as follows

（(1)

()

()

,())p p x k

A x k

B u k f k x k

(2)

where ()n x k R is state vector and ()m u k R is the control input,n n p A R and

n m p B R are all real constant matrices,（,())f k x k is nonlinear disturbance, satisfying the following Lipschitz condition[32]:

（（2,()),())

()()T T

T f k x k f k x k x k H Hx k (3)

Where H is a real constant matrix,0is the bound of the nonlinear disturbance.

It is noticed that when the controller is designed, the exact k

is known. Hence,

the controller gain can be designed depending on

k

, that is

()()()k k

u k K x k , (4)

()

(),{,

,0}x t t t

d

Consequently, the closed-loop system from (2) and (4) can be expressed as

（(1)

()

()()

,())p p k k

x k

A x k

B K x k

f k x k (5)

At sampling time k , if we augment the state-variable as

()[()(1)()]T T

T T X k x k x k x k

d

the closed-loop system (5) can be written as

(1)

(()())()

(,())k k X k

A

BK E X k DF k X k , (6)

(0)[(0)(1)()]T T

T T X d .

where

(2

)(2)

000000000000

p

n d n d A I I A R

I , (2

)000

p n d m

B B

R , (2

)000

n d n

I D

R ,

(2)

000n

n d G I R ,(,())

(,())F k X k f k GX k ,(2)

()

000n

n d k E I R

has all elements being zeros except for the (1)th k

block being identity. It can be

seen that the closed-loop system (6) is a jump linear system with d different modes.

To proceed, we will need the following lemmas. Lemma1 (see [2]) Let 1T ,2

n

n

T R be symmetric matrices. The conditions on

10,0T x T x

x such that 20T x T x hold if there exist a scalar

such

that 1

2

0T T .

Lemma 2

(see [33]) For given scalars 0i

and matrices 0i

P ,

1

1

1

N

N N i

i

i i i i i P P

always holds.

Lemma3 (see [19]) The matrix ()T E i is of full-array rank, then there exist two orthogonal matrices (2

)(2)

n d n d i U R and g

g

i

V R , such that ()0

i

T

T

T i i E i U V ,

where

12=(,

,

)i g

diag , where

i

(1,2,

,)i

g are nonzero singular values of T i E . If

matrix (2

)(2)

()n d n d F i R has the following structure

120

()

0i T

i i i F F i U U F , (7) there exists a nonsingular matrix g

g

i

X R such that ()()()T

T i F i E i E i X , where

1

0g

g

i F R ,(2

)(2)

2

0n d g n d g i F R .

Throughout this paper, we use the following definition:

Definition 1(see [21]) The system (6) is stochastically stable if for every finite

(0)X and initial mode

, there exists a finite matrix 0W

such that the

following holds

2

00

00

{

()

|,

}

T k E X k X X

WX . (8)

Our objective for this paper is to find the controller gain ()k K such that the closed-loop system (5) is stochastically stable. In the following, if we know that

k

i ,()k K and ()k E for this paper is denoted as ()K i and ()E i , respectively.

3. Controller Design

Theorem 1 Consider system (6) with partly unknown transition probabilities (1). If there exists matrices ()

0F i ,()Y i and a scalar 0 such that 13142324

33

44

()0()*-0**0

00***0**

*

*

T T

F i F i

G

H I

I

(9)

where

1

2

13

(()

()())i i i

T

ik ik ik AF i BY i E i I

I

I ,

14

(()

()())T

i i i AF i BY i E i v I

v I

v I

,

1223i i i ik ik ik D D D , 24i i i v D v D v D ,

3312{()()()i i i diag F k F k F k ,

4412{()()

()i i i d diag F k F k F k , 120()0i T

i i i F F i U U F , 1

i

k i ij j ,

21/,

holds for all i , there exists a mode-dependent controller of the form (4) such that the resulting system (6) is stochastically stable. Furthermore, if the LMI (9) has a solution, an admissible controller is given by

111()()T i i i i i K i Y i V F V . (10)

Proof.

For the closed-loop system (6), consider the quadratic function which is given by ((),)()()()T k V X k k X k P X k . We have

{((),)}E V X k k （+1{(+1)()(1)|),}T k k

E X k P X k X k i ()()()T k X k P X k =()T k L (11)

where

1111(()())()()*()d T ij j d k T ij j L A BK i E i P j D L D P j D

, 111(()())()(()())()d T ij j L A

BK i E i P j A BK i E i P i , ()(,()T T T X k F k X k .

If ()0k L , it can be proofed that the closed-loop system (6) is stochastically stable

which is similar to that of [19], and is omitted here.

From (3) one can get

2

0*T T T G H HG I . (12)

According to lemma 1, if there exists a scalar 0such that

2

11

1

1

(()())()0*

()d

T T

T

ij

j d

T

ij

j L G H HG A BK i E i P j D

D

P j D

I

,

(13)

the inequality (11) subject to (12) holds. Assume 0, inequality (13) can be written

as

2

11

1

1

(()())()0*

()d

T T

T

ij

j d

T

ij

j L G H HG A BK i E i P j D

D

P j D I

,

(14)

where

11

1

(()())

()(()())

()d

T

ij

j L A

BK i E i P j A BK i E i P i ,

()()/P j P j , ()

()/P i P i .

According to lemma 2, one can get

1

()(

)

(

)d

i i ij

k

i

uk

j P j P P (15)

where

(

)

()i k

i k

ij

j

P P j , (

)

()i uk

i uk

j

P P j ,

1

i k

i

ij

j

.

By Schur complement, one can obtain

1314

2324

33

44

()000**0

00***0*

*

*

*

T T

P i G H I I

(16)

where

1

2

13

(()())i i i

T

ik ik ik A

BK i E i I

I

I ,

14

(()())T

i i i A

BK i E i v I

v I

v I

,

11133

12{()()

()i i

i diag P k P k P k , 11144

12{()()

()i i i d diag P k P k P k .

Performing a congruence transformation to (16) by 1{()}diag P i I I I I

, and let 1()

()P i F i ,1111

2{()()()

i i

i diag P k P k P k 12{()()

()}i i

i diag F k F k F k ,

11112{()()()i i i d diag P k P k P k 12{()()()i i i d diag F k F k F k , we can get

13142324

33

44

()0()*-0**0

00***

0**

*

*

T T

F i F i

G

H

I I

, (17)

where

1

2

13

()(()())i i i

T

ik ik ik F i A

BK i E i I

I

I ,

14

()(()())T

i i i F i A

BK i E i v I

v I

v I

.

For the matrix ()T E i of full-array rank, there always exist two orthogonal matrices

(2

)(2)

n d n d i

U R and g

g

i

V R such that 0

i

T

T

T i i i E U V , where

12

=(,

,

)

i

g

diag ,

i

(1,2,

,)i g are nonzero singular values of T i E . Assume that the matrix ()F i has the

following structure

120

()

0i T i

i

i F F i U

U F , (18) according to lemma 3, there exists matrix g g i X R such that ()()()T

T i F i E i E i X ,

setting ()()T T i X K i Y i , we can obtain (9) from (17). Since ()()()T

T i F i E i E i X , we can

get

12

00i T

i i F U F 0

i

i

T T

T i i i i V U V X ,

that is 11T i i

i

i i i X V F V , which implies that 111()

()T i

i

i i i K i Y i V F V , which completes

the proof.

Remark 1: when all the elements in the transition probabilities matrix of the time-delays are known, then

0i

, which is a special case.

Remark 2: We introduce some

structure restrictions on the matrix variables ()F i which enable us to present the existence conditions of the controller in terms of LMIs. Nevertheless, such restriction will introduce some extents of conservatism into the controller design.

The maximum bound of nonlinear disturbance max

1/

can be found by solving

the following optimization problem

min

s.t.(9) and ()

0F i ,()

0Y i ,

0.

4. Numerical Examples

In this section, one numerical example will be given to show the validity and

potential of our developed theoretical results. Consider the nonlinear time-invariant discrete-time plant with the following data,

0.20.5

0.670.6

p

A ,0.1

0.2

p

B , （1223sin(())

,())0.7()cos(())

x k f k x k x k x k .

Assume the time-delay k

from the sensor to the controller takes values

from

{1,2,3,4}. Consider the following three transition probabilities matrices

1

?0.20.3??

??0.7

0.5?0.3?

?0.6?

?

,

2

?0.20.3??

0.1?0.7

0.5?0.3??0.60.2

?

,

3

0.40.20.30.1?0.1?0.7

0.50.10.30.1?0.60.2?

.

Our purpose is to design a mode-dependent controller of the form (3) such that the close-loop system (5) is stochastically stable with the partly unknown transition probabilities. Let H in (3) to be unit matrix, by solving (9) in Theorem 1, the controller gain under three different transition probabilities matrices is solved as follows, respectively.

1=0.20010.0638K ,2=0.1157

0.0090K , 3

0.0102

0.0058K ,4=0.0006

0.0001K .

1=0.21560.0664K ,2=

0.13610.0189K ,

3=0.00560.0135K ,4=0.0017

0.0016K . 1=0.20300.0564K ,2=0.08340.0229K , 3

0.0086

0.0301K ,4=0.0022

0.0015K .

The maximum bound of nonlinear disturbance max

under the three transition

probability matrices is abstained as table 1.

Table 1

max

under the transition probability matrix

1

,

2

and

3

.

transition matrix

1

2

3

max

0.0523 0.1142 0.1876

From the table above, we can see that the more elements of transition probability matrix are known, the larger max is.

Figure 2 shows one simulation run of the Markov jump delays under the transition probability matrix 3. Figure 3 shows the state response of the closed-loop system using the obtained controller under the transition probability matrix 3

when (0)

2

1T

x .

k

t i m e -d e l a y

Fig.2. Step of delay

k

k

x (k )

Fig.3. States of the closed-loop system

It can be seen that the closed-loop system with the partly unknown transition probabilities under the designed controller is stochastically stable. 5. Conclusions

In this paper, the state feedback stabilization problem has been considered for a kind of nonlinear networked control systems with random communication delays. A Markov chain is employed to describe the random communication time-delay of which transition probabilities are partly unknown. The closed-loop system is established through the state augmentation technique and the state feedback controller is designed which guarantees the stability of the resulting closed-loop systems. It is shown that the controller design problem under consideration is solvable if a set of LMIs is feasible. Simulation results show the validity of the proposed method.

Acknowledgements

This work was supported by the National Natural Science Foundation of China under grant No. 61174029.

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