The corresponding author, Tel: +61 7 4930 9270; Fax: +61 74930 9729 X. Zhang and X. Jiang are with School of Automation, Hangzhou Dianzi University, Hangzhou 310018 Zhejiang, China. E-mail: jiangxf@hdu.edu.cn The research work was partially supported by the research start-up foundationof Hangzhou Dianzi University, People's Republic of China under GrantKYS041507020 and the national Natural Science Foundation of People's Republic of China under Grant 60774058.

An improved network-based controller design method

Xinxin Zhang School of Automation

Hangzhou Dianzi University Hangzhou 310018, Zhejiang, China

Stars108197@yahoo.com.cn

Xiefu Jiang School of Automation

Hangzhou Dianzi University Hangzhou 310018, Zhejiang, China

jiangxf@hdu.edu.cn

Abstract— This paper is concerned with the design problem of network-based controller for systems with both the network-induced delays and data packet dropouts. A new Lyapunov-Krasovskii functional candidate is proposed to drive a new stabilization criterion. The sufficient condition on the existence of the controller is obtained in terms of a matrix inequality. And a feasible solution of the proposed matrix inequality is obtained by an iterative algorithm. No parameter needs to be selected in advance.

Keywords- Lyapunov-Krasovskii functional; networked control systems (NCSs); stability; linear matrix inequality (LMI)

I. INTRODUCTION

Networked control systems (NCSs) are feedback control systems wherein the control loops are closed through a real-time network. Recently, much attention has been paid to the study of stability analysis and control design of NCSs due to their low cost, reduced weight and power requirements, simple installation and maintenance, and high reliability. NCSs have been adopted in many application areas. Examples include manufacturing plants, automobiles, aircraft, and spacecraft. However, the insertion of a real-time network introduces time delays due to time-sharing of the communication media. The existence of a network-induced delay can degrade the performance of an NCS, and can even destabilize the system. Therefore it is of great significance to consider stability analysis and controller design of NCSs.

Recently, more attention has been paid on the controller design of the NCSs. For example, in [1], the authors designed the memory feedback controller employed a predictor-based delay compensation method. However, the effect of controller-to-actuator delays was neglected, and no method was given to estimate the maximum allowable value of the network-induced delay that guarantees the stabilizability of the NCSs. A dynamic observer-based H controller was designed for discrete-time networked systems with random communication delays without considering any data packet dropouts [2]. In [3], the authors have investigated robust H control for uncertain NCSs by considering both time-varying network-induced delays and data packet dropouts simultaneously. However, some parameters need to be selected in advance for the controller design which leads to a more conservative result.

In this paper, a network-based controller design method is proposed based on new Lyapunov-Krasovskii functional, and

the new sufficient condition on the existence of the controller is proposed in term of a matrix inequality. In order to find a less conservative approach to network-based controller design and avoid selecting any parameter in advance, an iterative algorithm is proposed.

II. PROBLEM STATEMENT

Consider the following system controlled through a network

00 )(

)()()(xtx

tButAxtx (1)

where ( ) nx t R is the state vector, ( ) nu t R is the input vector, A and B are constant matrices of appropriate dimensions.

Suppose that all the systems' states are available for a state feedback control. If we consider the effect of the network-induced delay in addition to the controller proceeding delay and data packet dropout, the closed-loop system can be modeled as [4]

1 1( ) ( ) ( ), [ , )k k k k kx t Ax t BKx i h t i h i h (2)

where 1,2,...,k h is the sampling period and K is a controller gain to be determined. The time-delay k denotes the time from the instant ki h when sensor nodes sample sensor data from a plant to the instant when actuators transfer data to the plant. ( 1,2,...)ki k are some integers and

1 2 3{ , , , } {0,1, 2, }i i i .Obviously,

1 1 0 01[ , ) [ , ), 0k k k kki h i h t t .In this paper,

( ) 0u t is assumed before the first control signal reaches the plant. If 1k ki i , then the new data packet reaches the plant before the old one. At this time, the old data packet should be discarded and its successive data packet used instead. Therefore, it is necessary to find an appropriate network scheduling method that can discard the old data packet when the new one reaches the plant before the old one. In the following discussion, we assume that 1 , 1, 2,3,...k ki i k .

Throughout this paper, the following assumptions are needed.

Assumption 1 The sensor is clock-driven, the controller and actuator are event-driven.

978-1-4244-2251-7/08/$25.00 ©2008 IEEE

Assumption 2 There exist two constants 0m and 0such that

1 1( ) ,, 1, 2,3,...

k k k

k m

i i hk

III. MAIN RESULTS

We now state and establish the following proposition that gives a sufficient condition on the existence of the network-based controller for the system (1).

Proposition 1 For some given scalars m , , the closed-loop system (2) is asymptotically stable if there exist matrices

0TX X , 1 2

2 3

0T

Q QQ Q

, 4 5

5 6

0T

Q QQ Q

, 1 1 0TR R ,

2 2 0TR R , 0TS S , Y of appropriate dimensions such that

1 2

2 3

0T (3)

where

11 13 15

13 33 21

2 44

15 55 5

5 66

0 0( ) 2 0 0

0 0 00 0 0

0 0 00 0 0

T

T

T

T

T

BYBY S S S

QS Q

QS Q

2

2 2

2 20 0 00 0 00 0 00 0 0

T T Tm

T T T T T Tm

XA XA XA

Y B Y B Y B,

1 1 13 1 2

11 1 4 1 2

13 2 1 15 5 2 33 3 1 1

44 3 55 6 4 2 66 6

{ , , },,

,

, , ,

, , ,

m

T

diag XR X XR X XS X

AX XA Q Q R R

Q R Q R Q Q R

Q S Q Q R Q S

Moreover, the controller gain is 1K YX .

Proof: Choose a Lyapunov-Krasovskii functional candidate as:

1 2

2 2 3

( ) ( ) ( )

( ) ( )

( ) ( )2 2

m

Tt

T

t

Tm mt

V x x t Px t

x s x sQ Qds

Q Qx s x s

4 5

2 5 6

0 0

1 22 2

( ) ( )

( ) ( )2 2

( ) ( ) ( ) ( )2 2

( ) ( )

m

m

T

t

Tt

t tT Tmt s t s

t T

t s

x s x sQ Qds

Q Qx s x s

ds x R x d ds x R x d

ds x Sx d

(4)

where 0P , 1 2

2 3

0T

Q QQ Q

, 4 5

5 6

0T

Q QQ Q

, 1 0R , 2 0R ,

and 0S are symmetric positive-definite matrices of appropriate dimensions, and m . Taking the derivative of ( )tV x with respect to t along the trajectory of the system (2) yields

1 2 1 2

2 3 2 3

4 5

5 6

( ) 2 ( ) [ ( ) ( )]

( ) ( ) ( ) ( )2 2

( ) ( ) ( ) ( )2 2( ) ( )

( )2

Tt k

TTm m

T Tm m

m m

T

T

V x x t P Ax t BKx i h

x t x tQ Q Q Qx t x tQ Q Q Qx t x t x t x t

x t x tQ QQ Qx t

4 5

5 6

1 22 2

( ) ( )2 2

( ) ( ) ( )2

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 2

m

m

T

T

t t tT T T Tmt t t

Q Qx t x tQ Qx t x t x t

x t x t x s R x s ds x s R x s ds x s Sx s ds

for 1 1[ , )k k k kt i h i h , 1, 2,...,k where

2 2 21 2

1 ( 4 )4 mR R S . Use Lemma 1 in [5] we can get

12

1 1

1 1

( ) ( )2

( ) ( )

( ) ( )2 2

m

t Tmt

T

m m

x s R x s ds

x t x tR RR Rx t x t

22

2 2

2 2

( ) ( )2

( ) ( )

( ) ( )2 2

t T

t

T

x s R x s ds

x t x tR RR Rx t x t

( ) ( )

( ) ( ) ( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

m

k m

k

t T

t

i h tT T

t i h

Tk k

m m

Tk k

x s Sx s ds

x s Sx s ds x s Sx s ds

x i h x i hS Sx t x tS S

x i h S S x i hx t S S x t

Then, we have

1 1( ) ( ) ( ), [ , ), 1,2,...Tt k k k kV x t t t i h i h k

where

( ) ( ) ( ) ( ) ( ) ( ) ( ) ,2 2

T T T T T T Tmk mt x t x i h x t x t x t x t

11 12 13 5 2

12 22

13 33 2

2 44

5 2 55 5

5 66

0 00 0

0 0 00 0 0

0 0 00 0 0

T

T

T

T

T

Q RS SQ

S QQ R Q

S Q

with

11 1 4 1 2

12 13 2 1

22 33 3 1 1

44 3 55 6 4 2 66 6

,

, ,

( ) 2 , ,, , .

T T

T

T

PA A P Q Q R R A APBK A BK Q R

BK BK S Q Q RQ S Q Q R Q S

If

0, (5)

then there exists a 0 such that ( ) ( ) ( )TtV x x t x t for

1 1[ , ), 1, 2,...k k k kt i h i h k . Similar to the proof of [4], theclosed-loop system (2) is asymptotically stable. Pre- and postmultiplying both sides of (5) with { , , , , , }diag X X X X X Xand its transpose, respectively, and introducing

1X P , Y KX , ( 1,...,6)i iQ XQ X i ,

1 1R XR X , 2 2R XR X , S XSX , then using Schur complement yields (9). This completes the proof.

Remark 1 Xu and Lam in [7, 8] have provided a delay-dependent stability criterion which involves the least number of unknown variables to be determined; hence it is mathematically least complex and computationally most efficient compared with some LMI-based delay-dependent stability criteria obtained in recent years. According to the stability criterion in [7, 8], the system

( ) ( ) ( ),( ) ( ), [ ,0]x t Ax t Bx t hx t t t h

(6)

where the scalar 0h is a constant delay of the system (6), is asymptotically stable, if there exist some matrices 0P ,

0Q , 0Z , Y , and W such that

11 12

12 22 0.0

0

T

T T

T T

hY hA ZhW hB Z

hY hW hZhZA hZB Z

(7)

where

11 12 22, ,T T T TPA A P Y Y Q PB Y W Q W W

(7) is equivalent to

1 2 0,0 0 0 0

T T

T T

T T

PA A P Q PB hA RB P Q hB RhRA hRB R

Y Y Y IW R W W I

where 1R h Z . It is clear that

0 0

TT T

T T

PA A P Q PB hA R I IB P Q hB R I R IhRA hRB R

.

Equality holds if and only if 0 0

Y IW I R . So the solvability

of (7) is equivalent to LMI

0

T T

T T

PA A P Q R PB R hA RB P R Q R hB RhRA hRB R

. (8)

It is clear that LMI (8) involves 21.5n + 1.5n unknown variables while LMI (7) involves 23.5n +1.5n unknown variables [8]. To the best of our knowledge, LMI (8) is a mathematically least complex and computationally most efficient condition since it involves the least number of variables while providing an equivalent stability condition for time-delay system (6).

Remark 2 Similar to Proposition 1, the system (6) is asymptotically stable if there exist matrices

0P , 11 12

12 22

0T

Q QQ Q

and 0R of appropriate dimensions

such that the following LMI holds:

11 12 2

12 22 11 12

12 22 2

2 2

00.

0

Th

T

T T Th

h h

Q R PB A RQ R Q Q R QB P Q Q B RRA RB R

(9)

where 11 11 .TPA A P Q R

If there exist 0, 0, 0P Q R such that LMI (8) holds, then it is easy to prove that there exist some matrices 0P ,

2 20

2 2Q R R

R Q Rand 1

4 0R such that LMI (9) holds, i.e.

the solution of LMI (9) covers the solution of LMI (8). Therefore, LMI (9) can provide a less conservative result than

LMI (8) due to the introduction of 12Q and 22Q in (9). And LMI (9) can provide a less conservative result, too.

Remark 3 It is clear that (9) is not an LMI, it can not be solved directly by MATLAB LMI Toolbox. In order to solve this problem, similar to [4], we propose the following algorithm. First, we need to difine three variables 1 2,T T and 3Tsuch that

1 1 11 1 2 2 30, 0, 0XR X T XR X T XS X T

which are equivalent to 1 1 1 1 1 1

1 2 31 1 1 1 1 1

1 2

0, 0, 0T X T X T XX R X R X S

Then, introducing some new variables 11 1T T , 1

2 2T T ,1 1 1 1 1

3 3 1 1 2 2, , , ,T T X X R R R R S S , above matrix inequalities can be represented as

1 2 3

1 2

0, 0, 0T X T X T XX R X R X S

.

Now, using the cone complementary linearization algorithm in [6], a feasible solution of the nonlinear matrix inequality (9) can be found by the following minimization problem.

1 1 2 2 3 3 1 1 2 2

1 2 1 2

2 3 1 2

13

1

32 1

32 1

2

Minimize tr ( )

0, 0, 0,

0, 0, 0,subject to

0, 0, 0,

T

XX TT T T T T R R R R SS

T X T XX R X R

T IX IT XI TI XX S

T IT I R II TI T I R

R

2

0, 0I S I

I R I S

where 3 1 2 3{ , , }diag T T T . The aforementioned minimization problem can be solved by the following iterative algorithm.

Algorithm 1:

For two given constants 0, 0m ,find a feasible solution under the LMI conditions in (3)

0 0 10 20 30 40 50 60 10 20 0 10 20

0 10 20 30 10 20 30 0

( , , , , , , , , , , , , ,

, , , , , , , ).

X X Q Q Q Q Q Q R R S R RS T T T T T T Y

set 0k . If there are none, exit.

Solve the following LMIs problem with a feasible solution

1 2 3 4 5 6 1 2 1 2 1 2 3

1 2 3

( , , , , , , , , , , , , , , , , ,, , , ).X X Q Q Q Q Q Q R R S R R S T T TT T T Y

1 1 1 1 2 2 2 2

3 3 3 3 1 1 1 1 2 2 2 2

Minimize tr (

)k k k k k k

k k k k k k k k

X X X X T T T T T T T T

T T T T R R R R R R R R S S S S subject to LMIs in (3).

1 1 1, 1 1 2, 1 2 3, 1 3

1, 1 1 2, 1 2 3, 1 3 1, 1 1 2, 1 2

1 1, 1 1 2, 1 2 1 1

Set , , , , ,

, , , , ,

, , , , .

k k k k k

k k k k k

k k k k k

X X X X T T T T T T

T T T T T T R R R R

S S R R R R S S Y Y

If the condition (9) is satisfied, then the controller gain is designed as 1K YX and exit. If the condition (9) is not satisfied within a specified number of iterations, then say “no solution'” and exit. Otherwise, set 1k k and return to Step 2).

IV. CONCLUSIONS

The method of stability analysis and controller design for linear NCSs with both the network-induced delay and the data packet dropout has been proposed based on a new Lyapunov-Krasovskii functional. No slack variable is introduced, which means the obtained results is mathematically least complex.

REFERENCES

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[3] D. Yue, Q.-L. Han, and J. Lam, “Network-based robust H control of systems with uncertainty, ”Automatica, vol.41, no.6, 2005, pp. 999-1007.

[4] X. Jiang and Q.-L. Han, “A new H stabilization criterion for networked control system, ” IEEE Transactions on Automatic Control, vol. 53, no. 4, 2008, pp. 1025-1032.

[5] Q.-L. Han, “Absolute stability of time-delay systems with sector-bounded nonlinearity,” Automatica, vol. 41, no. 12, 2005, pp. 2171-2176.

[6] L.E. Ghaoui, F. Oustry, and M. AitRami, “A Cone Complementarity Linearization Algorithm for Static Output-feedback and Related Problems,” IEEE Transactions on Automatic Control, vol. 42, no. 8, 1997, pp. 1171--1176.

[7] S. Xu and J. Lam, “Improved delay-dependent stability criteria for time-delay systems,” IEEE Transactions on Automatic Control, vol. 50, no.3, 2005,pp. 384-387.

[8] S. Xu and J. Lam, “On equivalence and efficiency of certain stability criteria for time-delay systems,” IEEE Transactions on Automatic Control, vol. 52, no.1, 2007, pp. 95-101.