Stability analysis and controller design for networked control systems

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Yongxian Jin ; Yi Di , vol 6 Issue 10, pp 49-54, October 2018

Stability analysis and controller design for networked control systems

Yongxian Jin

¹

，Yi Di

²

1

College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua Zhejiang, 321004, China

2

Graphic information center, Zhejiang Normal University, Jinhua Zhejiang, 321004, China [email protected]

ABSTRACT

This work is devoted to design the stabilizing controller and stability analysis in the case of networked real-time control systems (NCSs). By considering the relationship between the network-induced delay and its bound, an improved stability criterion for NCSs is proposed in the derivative of Lyapunov function. A stabilizing state feedback controller is applied to generate the next control information for each subsystem using delayed sensing data in a free-weighting LMI formulation.

Moreover the system control design focuses on the network-induced delays which may deteriorate the closed-loop performances and even destabilize the closed-loop system in real-time control. Simulation examples show the interest of the proposed approach. Keywords: NCSs; network-induced delay; LMI;

delay-dependent

1 INTRODUCTION

Feedback control systems that are closed over a real-time communication network have more and more popular and hardware devices for networks have become cheaper and the internet has been more and more common. So a new area of research activity called „networked real-time control systems‟ (NCSs) has received more attention in recent years. Although NCSs have several advantages such as re-configurability, low installation cost and easy maintenance, the use of communication networks makes it necessary to deal with networked-induced delays.

These delays may be unknown and time-varying, and may deteriorate the closed-loop performances and even destabilize the closed-loop system ^[1-9].

As for network-induced delays, it has been dealt with by the control theory community in most research work. In an NCS, the network-induced delays include computing time of network nodes and network transmission time from sensor to controller and from controller to actuator which occurs because a variety of information is exchanging among devices connected the shared network. The performance and stability of control scheme strongly relies on the respect of the specified sampling times and network-induced delays. Recently, delay-dependent criteria on the MADB for the stability of NCSs with network-induced delays has attracted much attention^[11-14]. The main methods reported are based on four fixed model transformations^[15]. Among them, the descriptor system approach combined with the Park‟s and Moo et.al.‟s inequalities ^[16-18]. A free-weighting matrix approach^[19]is reported to cover the results using Moo et.al.‟s inequality and the descriptor system approach ^[20-21]. As for NCSs, the work in [9] establishes some stability conditions under an assumption that the network-induced delay is less than the sampling period for continuous-time systems based on sampling-date method. Some methods to calculate the MADB for NCSs using Moon et.dl.‟s inequality for both discrete-time and continuous-time plants are proposed in

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© 2018, IJournals All Rights Reserved www.ijournals.in

Page 50 [22].

This paper deals with stability analysis and controller design in the presence of the network-induced delays.

our objective is to keep stability and good performances for NCSs in the presence of timing uncertainties as communication delays. Thus it can be useful to consider more dynamic solutions.

The outline of the paper is the following. In section 2, stability analysis and stabilizing controller design are presented. In section 3, simulation example presents the system design for discrete systems with time delays.

Section 4 ends with some concluding remarks.

2 STABILITY ANALYSES AND

STABILIZING CONTROLLER DESIGN

We are interested here in real-time control of systems with communications delays (see fig 1). There are mainly two sources of delays from the network-induced delays in an NCS: device delay and transmission delay.

The computing delay includes the time delay at the plant node(P)



_sou and the controller nodes (C)



_des. The transmission delay includes the time delay from sensor(S) to controller(C)



_sc ^and ^from controller(C) to actuator(A)



_ca.The total time delay can be expressed as follows

sou des sc ca

        

（1）

2.1 Delay-dependent NCS model

In this section, we will review and improve some results on the modeling of NCSs. We need the following assumptions：

Assumption 1: In an NCS, the total time delay



is less than one sampling period

h

.

Assumption 2: In an NCS, the NCS uses the way of single-packet transmission. The data packet loss and noise effects on the NCS are not considered.

Assumption 3: In an NCS, the sensor is assumed to be

time-driven, whereas the controller and actuator are event-driven, and the controller is time-invariant.

The discrete system equations can be written as:

( 1) ( ) ( ) ( )

( ) 0, ( 0)

x k Ax k A x k Bu k

x k k



    

     



（2）

where, state vector

x k  ( )

ⁿ ;control input vector

u k  ( )

^m ;

A

,

A

_ ,

B

are real constant matrices with appropriate dimensions.

The total time delay



satisfies:

1 2

    

（3）

where,



₁and



₂are minimum and maximum time delay respectively and less than one sampling time.

where

11 2 1

1 1 2 11

( 1) ( ) (

)

^T ^T

Q P A I A

I P N N X

 



       

  

12 2 1 2 12

PA

_

N

T

N  X

    

22 2 2 2 22

Q N N

T

 X

     

H   P 

2

Z

2.2 Controller design

Our object is to design state feedback controller for the NCSs.

( ) ( )

u k  Kx k

(4) shared network

C S

A P

C S

A P

system 1 system n

………

…

Fig. 1 NCS Control Structure

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Page 51 where,

u k  ( ) 0 k   (  , 0)

Lemma 1 Given constant

matrices

M

,

P

and

Q

,where

P  P

^T and

T

0 Q  Q 

,then,

P MQ M 

^¹ ^T

 0

if only if

T

0 P M

M Q

 

  

 

^,or

0 Q M

T

M P

 

  

 

（5）

Theorem 1 Given



_i

 0

(

i

=1,2), if there exist constant matrices

P

,

Q

,

Z

,

X

,

N

1 ,

N

₂ ,where

P  P

^T

 0

,

Q  Q

^T

 0

,

T

0 Z  Z 

, ¹¹ ¹²

22

* 0 X X X

X



    

 

^,

N

¹and

N

₂have appropriate dimensions, hold,

11 12

22

( )

* 0

* *

T

A I H A H

H



    

 

     

  

 

（6）

11 12 1

22 2

* 0

* *

X X N

Z

 

 

   

 

 



（7）

then, the system(2) with the time-delay constraint (3) is stable when

u k  ( ) 0

.

Proof We assume

y k ( )  x k (   1) x k ( )

,

then

x k (   1) x k ( )  y k ( )

.

So,

1

( ) ( ) ( ) 0

k

l k

x k x k y l



^

 

    

（8）

Choose a Lyapunov functional candidates as:

1 2 3

( ) ( ) ( ) ( ) V k  V k  V k  V k

1

( )

^T

( ) ( )

V k  x k Px k

2

0 1

2

1 1

( ) ( ) ( )

k T

l k

V k y l Zy l

  



    

  

1

2

1 1

3

1 1

( ) ( ) ( )

k T

l k

V k x l Qx l



  

  

    

 

where

P  P

^T

 0

,

Q  Q

^T

 0

and

Z  Z

^T

 0

. Let

 V k ( )  V k (   1) V k ( )

,then

1

( ) ( 1) ( 1) ( ) ( )

2 ( ) ( ) ( ) ( )

T T

V k x k Px k x k Px k x k Py k y k Py k

    



2

1

2 2

1 2

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

k

T T

l k k

T T

l k

V k y k Zy k y l Zy l

y k Zy k y l Zy l





 



 

  

 



1

2

3 2 1

2 1

( ) ( 1) ( ) ( ) ( ) ( ) ( 1)

( ) ( ) ( ) ( )

T

k T

l k

T T

V k x k Qx k

x l Qx l

x k Qx k x k Qx k



 



 

    

  

  



We use equation(8) and any matrices

N

_i (i=1,2), then the following the equation holds:

1 2

1

2[ ( ) ( ) ]

( ) ( ) ( ) 0

T T

k

l k

x k N x k N

x k x k y l



^

 

  

    

 





 （9）

On the other hand, as for any matrix

11 12

22

* 0 X X

X X

 

   

 

, the following equation holds:

2

1 1

1 1 1 1

1

2 1 1 1 1

( ) ( ) ( ) ( )

k k

T T

l k l k

k

T T

l k

k X k k X k

 



   

    

 

   



 



 

 



（10）

where,



₁

( ) k  [ x k

^T

( ) x k

^T

(   )]

, add the left sides of equation(9)and equation(10) into

 V k ( )

, hence we have

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Page 52

1

1 1 2 2

( ) ( ) ( ) ( ) ( )

k

T T

l k

V k k k k k



 

^

 

 

    

^where

2

( , ) k l [

1^T

( ) k y l

^T

( )]

^T

  

and

11 12

12 22

( )

( ) ( )

( )

T

T T T

d

A I HA A I H A I

A H A I A HA



 

  

    

   

    

 



So, an NCS is stable if only if

  0

^and

  0

^.

According to Lemma 1, we know that

  0

is equal to

  0

.

Theorem 2 Given



_i

 0

(

i

=1,2), if there exist constant matrices

L

,

W

,

R

,

Y

,

M

1 ,

M

₂ ,

V

,where

L  L

^T

 0

,

W  W

^T

 0

,

T

0 R  R 

^, ¹¹ ¹²

22

* 0 Y Y Y

Y



    

 

^,

M

¹,

M

₂and

V

have appropriate dimensions, holds:

11 12 13 2 13

22 23 2

2

* 0

* * 0

* * *

LA

T

L R



    

  



 

 

       

  

 

(11)

11 12 1

22 2

1

* 0

* * Y Y M

Y Y M

LR L

^

 

 

   

 

 



(12)

then the system(2) with the time-delay constraint (3) can be controlled and

K  VL

^¹.

where

11 2 1

1 1 2 11

( 1)

^T

2

T T T

W AL LA L

BV V B M M Y

  



      

   

12 2 1 2 12

A L

_

M

T

M Y

     

13

L A I ( )

^T

V B

^T ^T

   

22 2 2 2 22

W M M

T

Y

      

Proof According to the proof of theorem 1, the LMI(6)

is changed by using lemma 1.

11 12 2

22 2

2

( ) ( )

* 0

* * *

T

T T

d

A I P A I Z A P A Z

Z



     

 

     

  

 

 

(13)

then, the NCS(8) under the control of the memoryless state feedback controller and its state equation is

( 1) ( ) ( ) ( )

x k   A BK x k   A x k

_

 

(14)

Let

A    A BK

then equation(13) and equation(14) are

11 12 2

22 2

2

( ) ( )

* 0

* * 0

* * *

T T

T

A BK I P A BK I Z

A P A Z

P

Z

 

  

 



     

 

       

  

 



(15) where

11 2 1

1 1 2 11

( 1) ( )

( )

^T ^T

Q P A BK I A BK I P N N X

  



      

    

12 2 1 2 12

PA

_

N

T

N X

     

22 2 2 2 22

Q N N

T

X

      

( 1) ( ) ( )

x k   A k   A k

_

 

(16)

So, we can use the method of proof for Theorem 1, then

1 1 1 1 1 1

1 1

{ , , , } { , ,

, } 0

diag P P P Z diag P P P Z

     

 







(17)

1 1 1 1 1

1

{ , , } ( , ,

} 0

diag P P P X diag P P P

    



 





(18)

where

L  P

^¹ ,

W  P QP

^¹ ^¹ ^，

1 1 1 1

{ , } { , }

Y  diag P

^

P

^

  X diag P

^

P

^

R  Z

^1,

M

₁

 P N P

^¹ ₁ ^¹,7, and

V  KP

^¹.

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Page 53 We can draw the final conclusion by calculating the

equation (17) and equation (18).

3 SIMUALTION EXAMPLE

In this section, one example is given to show effectiveness of the proposed methods. They are solved by the LMIs from Theorem 1 and Theorem 2.

Consider the system state equation(2).

When

u k  ( ) 0

, the system matrices are denoted by

0.8 0 0 0.97

A  

  

 

^,

0.1 0 0.1 0.1 A

_

  

      

When

 

₁



₂,



is constant. If we use the methods in[23, 24], we can get the maximum value of



₂,which are 12 and 16 respectively. However, its value is 16 used the method proposed in this paper.

As for network-induced delay



,we use the methods in[23, 24]and in this paper respectively. The results as follows

From the table, we know that the results used the method in this paper is less conservative than that used the method of Fridman et.al.

4 CONCLUSIONS

In this paper, we analyze the stability of NCSs through an improved stability criterion that proposed in the derivative of Lyapunov function and consider the relationship between the network-induced delay which is time- varying and its bound. A stabilizing state feedback

controller is applied in each subsystem of NCSs in a free-weighting LMI formulation. Simulation example shows the effectiveness of the method.

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