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 MQ1M 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 AT 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 AT 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 AT 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 LAT
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 VL1 .
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 AT 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 P1
,
W P1QP 1
，
Y diag{P1 , P1} X diag{P 1 , P 1}
R Z 1 , M1 P1 N1P1 ,7, and V KP1 . Page 52
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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