Chapter 4 Networked Control Systems with Disturbance
4.4 H ∞ Control
4.5.1 Validating Lemma 6
4.5 Simulation
For convenience, we call min
max 2 2 out ‘weight’ of the following system.
4.5.1 Validating Lemma 6
System .Ⅵ
Consider the following system:
1 2 obtain the control input signal which makes
0∞z t z t dtT( ) ( ) ≤100 0∞w t w t dtT( ) ( )
Fig. 4-5-1(b) shows the original states of system Ⅵ. Fig. 4-5-1(c) shows the states of controlled system under instantlyⅥ updated control or updated hold control under the
‘weight’=0.3296 of Lemma 6. The control input signal as updated instantly or as
‘weight’=0.3296 is shown in Fig. 4-5-1(d).
The requirement of Lemma 6, e t( ) ≤0.3296 ( )x t , is shown in Fig. 4-5-1(e).
Figure 4-5-1 (b) Original states in system Ⅵ.
0 0.5 1 1.5 2 2.5 3
0 100 200 300 400
State 1
0 0.5 1 1.5 2 2.5 3
-100 0 100 200 300 400
State 2
Time(s)
0 0.5 1 1.5 2 2.5 3
-5 -4 -3 -2 -1 0 1 2 3 4 5
Noise
Time(s)
Figure 4-5-1 (a) Noise of system Ⅵ.
Figure 4-5-1 (c) States of system Ⅵ under controlled.
System .Ⅶ
Consider the following system :
1 2
of them are in the right-half-plane, it is an unstable system Using H∞ control approach, we can obtain the control input signal which makes
0∞z t z t dtT( ) ( ) ≤100 0∞w t w t dtT( ) ( )
∫ ∫
as[ ]
( ) 2T ( )k 26.22 3.8 11.56 10.58 ( )k
u t = −B Px t = − − x t . And then we calculate the value of
‘weight’= min
max 2 2
( ) 0.1027
( T )
Q PB B P λ
λ = .
Fig. 4-5-2(b) shows the original states of system VII. Fig. 4-5-2(c)(d) shows the states of controlled system VII under instantly updated control or updated hold control under the
‘weight’=0.1027 of Lemma 6. The control input signal as updated instantly or as
‘weight’=0.1027 is shown in Fig. 4-5-2(e).
The requirement of Lemma 6, e t( ) ≤0.1027 ( )x t , is shown in Fig. 4-5-2(f).
0 1 2 3 4 5 6
-50 -40 -30 -20 -10 0 10 20 30 40 50
Noise
Figure 4-5-2(b) Original states in system Ⅶ.
Figure 4-5-2(d) State 3,4 of system Ⅶ under controlled.
0 1 2 3 4 5 6
-5 0 5 10 15
State 3
u(t)=-Kx(t) u(t)=-Kx(tk)
0 1 2 3 4 5 6
-10 -5 0 5 10 15
State 4
Time(s)
u(t)=-Kx(t) u(t)=-Kx(tk)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-60 -40 -20 0 20 40 60
control input
u(t)=-Kx(t)
u(t)=-Kx(tk) as weight=0.1027
4.5.2 Remark
System VI System VII
Times of real-time updated 3000 5000
Weight 0.3296 0.1027
Times of updated in Lemma 6 66 152
From Table 4-1, we can conclude that the usage of network is reduced by using the control method provided in Lemma 6.
Table 4-1 List of the updated times.
Chapter5
Conclusions and Future Work
5.1 Conclusions
We investigate the stability of the linear system operating under limited communication.
Using Lyapunov theory, we give a sufficient condition for stability of NCSs. In the system without disturbances, we really and truly get the maximum time interval of state updating which still guarantees the stability of the system. In the system with disturbances, we obtain the upper bound of state error caused by jamming in the network based on the H∞ design. It not only ensures the stability of the controlled system, but also guarantees the closed-loop system satisfying the L2 −gain requirement. From these results, the time of state to be feedback will be reduced, so it minimizes the network usage.
5.2 Future Work
In the future, we hope to derive other corresponding results for NCSs about that time delay, scheduling, dropping packets out etc. Additionally, in this thesis, we only concentrated on the stability requirements. In future work, we can consider some other performance.
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Appendix
The state transition matrix is ( , ) ( )
Proof. (Lemma 2)
The state transition matrix is
( ) ( )
Proof. (Lemma 3) Because of ( )e t =x t( )−x t( )k
( )
e t = Ae t( )+Ax t( )k .
Taking the integral on both sides,
k Taking Norm on both sides, we obtain
k