Stability analysis and controller design for networked control systems

iJournals: International Journal of Software & Hardware Research in Engineering ISSN-2347-4890 Volume 6 Issue 10 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 jyx@zjnu.cn ABSTRACT This work is devoted to design the stabilizing controller

These delays may be unknown and time-varying, and

and stability analysis in the case of networked real-time

may deteriorate the closed-loop performances and even

control systems (NCSs). By considering the relationship

destabilize the closed-loop system [1-9].

between the network-induced delay and its bound, an

As for network-induced delays, it has been dealt

improved stability criterion for NCSs is proposed in the

with by the control theory community in most research

derivative of Lyapunov function. A stabilizing state

work. In an NCS, the network-induced delays include

feedback controller is applied to generate the next

computing time of network nodes and network

control information for each subsystem using delayed

transmission time from sensor to controller and from

sensing data in a free-weighting LMI formulation.

controller to actuator which occurs because a variety of

Moreover the system control design focuses on the

information is exchanging among devices connected the

network-induced delays which may deteriorate the

shared network. The performance and stability of control

closed-loop performances and even destabilize the

scheme strongly relies on the respect of the specified

closed-loop system in real-time control. Simulation

sampling times and network-induced delays. Recently,

examples show the interest of the proposed approach.

delay-dependent criteria on the MADB for the stability

Keywords: NCSs; network-induced delay; LMI;

of NCSs with network-induced delays has attracted

delay-dependent

much attention

[11-14].

The main methods reported are

based on four fixed model transformations[15]. Among

1 INTRODUCTION

them, the descriptor system approach combined with the

Feedback control systems that are closed over a real-time

Park‟s

communication network have more and more popular

free-weighting matrix approach [19] is reported to cover

and hardware devices for networks have become cheaper

the results using Moo et.al.‟s inequality and the

and the internet has been more and more common. So a

descriptor system approach [20-21]. As for NCSs, the work

new area of research activity called „networked real-time

in [9] establishes some stability conditions under an

control systems‟ (NCSs) has received more attention in

assumption that the network-induced delay is less than

recent years. Although NCSs have several advantages

the sampling period for continuous-time systems based

such as re-configurability, low installation cost and easy

on sampling-date method. Some methods to calculate the

maintenance, the use of communication networks makes

MADB for NCSs using Moon et.dl.‟s inequality for both

it necessary to deal with networked-induced delays.

discrete-time and continuous-time plants are proposed in

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

and

Moo

et.al.‟s

inequalities

[16-18].

A

iJournals: International Journal of Software & Hardware Research in Engineering ISSN-2347-4890 Volume 6 Issue 10 October, 2018 [22].

time-driven, whereas the controller and actuator are

This paper deals with stability analysis and controller

event-driven, and

the controller is time-invariant.

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

system n

system 1 A

P

……… …

A

P

C

S

communication delays. Thus it can be useful to consider more dynamic solutions.

shared network

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

C

S

Fig. 1 NCS Control Structure

system design for discrete systems with time delays.

The discrete system equations can be written as:

Section 4 ends with some concluding remarks.

STABILIZING CONTROLLER DESIGN

 x(k  1)  Ax(k )  A x(k   )  Bu (k )  (  k  0)  x(k )  0,

We are interested here in real-time control of systems

（2）

with communications delays (see fig 1). There are

where,

2

STABILITY

ANALYSES

AND

state

vector

x(k ) n ;control

input

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

u(k ) m ; A , A , B are real constant

matrices with appropriate dimensions.

The total time delay  satisfies:

1     2

（3）

where,  1 and  2 are minimum and maximum time delay respectively and less than one sampling time. where

follows

   sou   des   sc   ca

vector

（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

11  ( 2  1  1)Q  P( A  I )  ( A  I )T P  N1  N1T   2 X 11 12  PA  N2T  N1   2 X12 22  Q  N2  N2T   2 X 22

assumptions： Assumption 1: In an NCS, the total time delay  is less than one sampling period h .

H  P  2Z

2.2 Controller design

Assumption 2: In an NCS, the NCS uses the way of

Our object is to design state feedback controller for the

single-packet transmission. The data packet loss and

NCSs.

noise effects on the NCS are not considered.

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

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u(k )  Kx(k )

(4)

Page 50

iJournals: International Journal of Software & Hardware Research in Engineering ISSN-2347-4890 Volume 6 Issue 10 October, 2018 where,

u(k )  0 k  ( ,0)

Lemma

1

matrices M

V2 (k ) 

Given

constant

, P and Q ,where P  PT and

Q  QT  0 ,then, P  MQ1M T  0 if only if

k 1

0

  y   

1 1 

(l ) Zy(l )

2 1 l  k 1

k 1

  x  

V3 (k )

T

T

(l )Qx(l )

2 1 l  k 1

where P  P  0 , Q  Q  0 and Z  Z  0 .

T

P M  Q M T   T   0 ,or  0 M Q  M P 

（5）

Theorem 1 Given  i  0 ( i =1,2), if there exist constant

matrices

P

,

Q ,

Z ,

 X X 12   0 , N1 and N 2 have Z  Z T  0 , X   11 X 22  *

（6）

l k 

  2 yT (k ) Zy (k ) 

y T (l ) Zy (l ) 2

k 1

 y

T

(l ) Zy (l )

l k 

V3 (k )  ( 2  1  1) xT (k )Qx(k )  xT (l )Qx(l )  ( 2  1  1) 2

x (k )Qx(k )  xT (k   )Qx(k   ) T

（7）

then the following the equation holds:

2[ xT (k ) N1  xT (k   ) N 2 ] 

y(k )  x(k  1)  x(k ) ,

（8）

l k 

the other

hand,

as for

any matrix

 X 11 X 12  X    0 , the following equation holds:  * X 22  k 1

k 1

（9）

k 1   x ( k )  x ( k   )  y (l )   0   l  k   

On

 y(l )  0

k 1



We use equation(8) and any matrices N i (i=1,2),

then x(k  1)  x(k )  y(k ) .

So, x(k )  x(k   ) 

V2 (k )   2 yT (k ) Zy (k ) 

l k 

is stable when u (k )  0 .

We assume

2 xT (k ) Py(k )  yT (k ) Py (k )



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

Proof

V1 (k )  xT (k  1) Px(k  1)  xT (k ) Px(k )

k 1

appropriate dimensions, hold,

 X 11 X 12 N1  X   * X 22 N 2   0  * * Z 

Let V (k )  V (k  1)  V (k ) ,then

X ,

N1 , N 2 ,where P  PT  0 , Q  QT  0 ,

 11 12 ( A  I )T H       *  22 AT H   0  * *  H  

T

T



l k 

 1T (k ) X  1 (k )  2

  2 1T (k ) X  1 (k ) 

Choose a Lyapunov functional candidates as:

k 1

 

l k 

T 1

k 1

 

l k 

T 1

(k ) X  1 (k ) （10）

(k ) X  1 (k )

 1 (k )  [ xT (k ) xT (k   )] , add the left

V (k )  V1 (k )  V2 (k )  V3 (k )

where,

V1 (k )  xT (k ) Px(k )

sides of equation(9)and equation(10) into V (k ) , hence we have

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iJournals: International Journal of Software & Hardware Research in Engineering ISSN-2347-4890 Volume 6 Issue 10 October, 2018

V (k )   (k ) 1 (k )  T 1

 2 (k , l )  [ 1T (k )

k 1



l  k 

T 2

(k ) 2 (k ) where

yT (l )]T

and

11  ( A  I )T H ( A  I ) 12  ( A  I ) HA   T  T  22  AT HA 12  Ad H ( A  I )    0 is equal

to   0 .

Theorem 2 Given constant

i  0

(

i =1,2), if there exist W

,

R ,

Y

,

M 1 , M 2 , V ,where L  LT  0 , W  W T  0 ,

x(k  1)  ( A  BK ) x(k )  A x(k   )

11 12 ( A  BK  I )T P  2 ( A  BK  I )T Z    A P  2 AT Z 0   * 22 * *  P 0   * *  *  2 Z (15)

appropriate dimensions, holds:

11  ( 2  1  1)Q  P( A  BK  I ) 

 12  13  2 13    22  23  2 LAT 

0 * L 0  * *   2 R 

Y11 Y12 M 1    Y  * Y22 M 2   0 * * LR 1 L   

where

( A  BK  I )T P  N1  N1T   2 X 11 (11)

12  PA  N2T  N1   2 X12

22  Q  N2  N2T   2 X 22 x(k  1)  A (k )  A (k   ) (12)

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

K  VL1 .

where

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

diag{P 1 , P 1 , P 1 , Z 1}    diag{P 1 , P 1 , P 1 , Z 1}  0 (17)

11  ( 2   1  1)W  AL  LAT  2L  BV  V T BT  M1  M1T   2Y11

diag{P 1 , P 1 , P 1 }  X  diag ( P 1 , P 1 , P 1 }  0

12  A L  M 2T  M1   2Y12

(18)

13  L( A  I )  V B

where

T

T

T

 22  W  M 2  M   2Y22 T 2

Proof

(14)

A  BK then equation(13) and equation(14) are

Y Y12  R  RT  0 , Y   11   0 , M 1 , M 2 and V have * Y22 

 11  *    *  *

(13)

state feedback controller and its state equation is

 Let A

L ,

matrices

11 12 ( A  I )T P  2 ( A  I )Z      *  22 AdT P  2 AT Z   0 * * *   2 Z   

then, the NCS(8) under the control of the memoryless

So, an NCS is stable if only if   0 and   0 .

According to Lemma 1, we know that

is changed by using lemma 1.

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

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L  P1

,

W  P1QP 1

，

Y  diag{P1 , P1}  X  diag{P 1 , P 1}

R  Z 1 , M1  P1 N1P1 ,7, and V  KP1 .

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iJournals: International Journal of Software & Hardware Research in Engineering ISSN-2347-4890 Volume 6 Issue 10 October, 2018 We can draw the final conclusion by calculating the

controller is applied in each subsystem of NCSs in a

equation (17) and equation (18).

free-weighting LMI formulation.

Simulation example

shows the effectiveness of the method.

3 SIMUALTION EXAMPLE In this section, one example is given to show

5 REFERENCES

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