H∞ CONTROLLER DESIGN FOR NETWORKED
CONTROL SYSTEM WITH SHORT UNCERTAIN TIMEDELAY
Author*a, Authora, Authorb
aDepartment
of Mathematics and Physics
North China Electric Power University, Beijing, 102206, PRC
bDepartment
of Automation
North China Electric Power University, Beijing, 102206, PRC
ABSTRACT: For the NCS with disturbance, we point out a sufficient condition ensuring the system
meeting prescribed H∞ performance γ and a state feedback controller ensuring the system meeting
optimal performance index. Numerical examples are given to illustrate the effectiveness of our results
via TrueTime toolbox.
KEY WORDS: networked control system (NCS); time-delay; asymptotical stability; state feedback;
H∞ performance
1.
INTRODUCTION
In modern industrial control systems, sensors, controllers and plants are often connected over a network medium,
which are called networked control systems (NCSs). NCS has many advantages, such as low cost, and simple
installation, thus NCS has been becoming increasingly important in practical applications (Zhang W. (2001)).
However, there are several drawbacks in NCS such as packet dropout and network-induced delay (Dang X.D.
(2008), Walsh G.C. (2002)) and so on. The delay can degrade the performance of NCS and even cause the system
unstable. Therefore, it is necessary to study the stability of NCS with network-induced delay (Richard J.P. (2003)).
2.
NETWORKED CONTROL SYSTEMS MODELING
Consider a continuous time linear time-invariant plant described by:
{
𝑥̇ (𝑡) = 𝐴𝑥(𝑡) + 𝐵1 𝑢(𝑡) + 𝐵2 𝑤(𝑡)
𝑦(𝑡) = 𝐶1 𝑥(𝑡) + 𝐶2 𝑤(𝑡)
( 1)
where x(t), u(t), w(t), y(t) are the state vector of plant, input vector of plant, disturbance of plant and the
output of plant respectively. A, B1 , B2 , C1 , C2 in equation (1) are known constant matrices with compatible
dimensions. Discretize the continuous plant into the following model during the interval [kh, kh+h], where k is a
non-negative integer.
3.
STATE FEEDBACK CONTROLLER DESIGN FOR NCS WITHOUT DISTURBANCE
Consider the asymptotical stability of NCS without disturbance.
Theorem 3.1 Consider a continuous time linear time-invariant plant described by (2.1) with state
feedback controller u(t) = K x(t), the sampling period h is fixed and known, the overall delay in the
system satisfies dk ∈ [0, h]. The closed-loop system is asymptotically stable if there exist symmetric positive
definite matrices P, Q, feedback gain matrix K , and a scalar α > 0, such that the following matrix
inequality holds.
4.
STATE FEEDBACK CONTROLLER DESIGN FOR NCS WITH DISTURBANCE
Figure 1 shows a schematic of the feedback controller design while the results are found in Table 1.
Figure 1 Diagram of feedback controller design.
Description
Case
1
Initial guesses
n
tr
C10
C01
1000
1
-0.5
g
C10
C01
Final values
n
tr
0.5
0.2
118
0.41
-0.38
0.12
g
0.23
10
1
-0.5
0.5
0.2
75
0.17
3
100
10
-0.5
0.5
0.2
97
1.78
-0.49
0.66
0.09
4
100
0.1
-0.5
0.5
0.2
100
1.00
-0.50
0.50
0.20
5
100
1
-1
0.5
0.2
99
0.90
-0.60
0.64
0.25
6
100
1
-0.25
0.5
0.2
102
1.09
-0.42
0.37
0.15
7
100
1
-0.5
5
0.2
97
1.30
-0.76
1.06
0.22
8
100
1
-0.5
0.05
0.2
109
0.61
-0.39
0.20
0.21
9
100
1
-0.5
0.5
0.4
102
0.54
-0.60
0.48
0.30
10
100
1
-0.5
0.5
0.1
99
1.30
-0.47
0.53
0.15
all high
11
1000
10
-1
5
0.4
101
0.05
-1.38
1.00
0.46
all low
Table
12
10
0.1
-0.25
0.05
0.1
0.98
0.00
13
90
the
10
10
-1
5
0.4
91
15.67
-1.69
2.94
0.00
14
1000
0.1
-1
5
0.4
80
26.51
-2.18
2.20
0.03
15
1000
10
-0.25
5
0.4
100
1.00
-0.50
0.50
0.20
10
-1
0.05
0.4
241
0.00
-5.E+09
2.E+07
0.27
10
-1
5
0.1
97
1.34
-1.15
1.87
0.26
5.
1 low, 4 highs
1 low or high at a time
2
1 Comparison between initial guesses and
CONCLUSION
16
1000
17
1000
-8.11E+09 1.36E+11
0.01solutions.
-0.30
final
0.00
2 lows, 3 highs
18
10
0.1
0.4
76
0.03
-6565 been
5.E+05
0.00 as a discrete time system with
The networked
control
system-1 with 5uncertain
time-delay
has
modelled
19
10
10
-0.25
5
0.4
96
10.34
-0.95
1.15
0.00 without disturbance which make
uncertain parameters. A state feedback controller has been designed for NCS
20
10
10
-1
0.05
0.4
96
7.56
-1.61
3.92
0.01
the system 21asympotically
stable.
For
NCS
with
disturbance,
a controller
has been presented which stabilizes
10
10
-1
5
0.1
89
16.88
-1.89
2.45
0.00
the closed-loop
system
with
prescribed
and
optimal
performance
index.
The
numerical examples in the end of
22
1000
0.1
-0.25
5
0.4
100
1.03
-0.62
0.67
0.24
23
1000
0.1
-1
0.05
0.4
216
0.00
-1999
92.24
0.27
this paper have illustrated the effectiveness of our results.
24
1000
0.1
-1
5
0.1
80
26.00
-1.74
1.50
0.03
25
1000
10
-0.25
0.05
0.4
103
1.12
-0.39
0.33
0.13
26
1000
REFERENCES
10
-0.25
5
0.1
100
1.00
-0.50
0.50
0.20
0.27
27
1000
10
-1
0.05
0.1
117
0.32
-0.42
0.13
28
10
0.1
-0.25
5
0.4
101
0.59
-0.01
76.39
31
10
10
-0.25
0.05
0.4
76
0.26
3 lows, 2 highs
0.84
Zhang W, Branicky MS, Phillips SM (2001) Stability of networked control
system.
29
10
0.1
-1
0.05
0.4
97
1.03
-12.40
35.53
0.30
Magazine,30 21, 84-99.
10
0.1
-1
5
0.1
91
14.46 -241.24
5.E+02
0.00
-4997.55 73931.58
IEEE Control Systems
0.00
Gahinet P, Nemirovski
A10(1995)
LMI5 Control
Toolbox
Users-0.72
Guide. 1.06
USA:MathWorks.
32
10
-0.25
0.1
97
5.69
0.00
33
4 lows, 1 high
10
0.05
0.1
114
11.08
-0.55
0.30
0.00
Dang XD (2008) 10
Stability
and-1 controller
design
of networked
control
system
with time-varying delays and data
34
1000
0.1
-0.25
0.05
0.4
216
0.00
-0.49
0.02
0.17
packet dropout.
Control
and
Decision
Conference,
Shandong,
1376-1379.
35
1000
0.1
-0.25
5
0.1
86
25.85
-1.24
0.97
0.00
36
1000
0.1
-1
0.05
0.1
129
0.19
-0.44
0.08
0.27
37
1000
10
-0.25
0.05
0.1
225
0.00
-0.24
0.00
0.05
38
10
0.1
-0.25
0.05
0.4
76
0.50
-1949
2.E+04
0.00
39
10
0.1
-0.25
5
0.1
77
0.29
-5819.84
9.E+04
0.00
40
10
0.1
-1
0.05
0.1
0
905.02
-15.35
0.59
0.00
41
10
10
-0.25
0.05
0.1
101
1.85
-0.36
0.43
0.00
42
1000
0.1
-0.25
0.05
0.1
248
0.00
-0.22
0.00
0.09