Remark

The updated interval of each above system is listed Table 3-1.

System Ⅰ System Ⅱ System Ⅲ

Original poles ±1.5i −2.57, 8.57 0.526 0.6023i± ,−6.37, −2.6814 New poles − , 21 − − , 21 − − , 21 − , 4− ,−7

Updated interval 0.27 sec 0.07 sec 0.08 sec

From system Ⅰand system Ⅱ, we can obtain that if the original system is more unstable (the poles is more positive), then in general the updated interval is shorter. From system Ⅲ, we can get Lemma 4 also suit to use in high order system.

Table 3-1 List of the updated interval.

Figure 3-3-3(c) Control input in system Ⅲ.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0 0.5 1 1.5 2 2.5 3 3.5 4

u(t)=-Kx(tk)

Time(s)

u(t)=-Kx(t) u(t)=-Kx(tk)

Chapter4

Networked Control Systems with Disturbance

4.1 System Model

4.1.1 Normal Control Systems

Now we model the NCSs for system with disturbance. If there is no network, the sketch map of this system can be shown as in Fig. 4-1-1.

Consider the system

1 2

( ) ( ) ( ) ( )

x t = Ax t +B w t +B u t , (4-1) where ( )x t ∈ℜ is the state of the system, ( )ⁿ u t ∈ℜ is the control signal, ( )^m w t ∈R^p is the disturbances satisfying w t( ) ≤ρ (ρ is a positive scalar), and A , B , ₁ B are known ₂ matrices with proper dimensions. Suppose the original system ( ( )x t = Ax t( )) has eigenvalues in the right-half-plane. We must design a state feedback controller such that the system to be

Figure 4-1-1 System x(t) = Ax(t)+ B w(t)+ B u(t) , where _{1 2} u(t) = -Kx(t) . 1

s

A

K

B

2

^{x t} ( ) ^{x t} ( )

−

B

1

( )

w t

( )

x t

stable. For convenience, we define A= −A B K₂ , which is stable, and the controlled system

The sketch map of this system is shown in Fig. 4-1-1. Now, we want to find the condition of the system (4-2) to be stable via Theorem1 and Theorem 2. Let ( ( ))V x t =x t Px t^T( ) ( ), where the system (4-2) converges to neighborhood of origin.

4.1.2 Networked Control Systems

When we connect the feedback channel of sensors to the network, the system can be given as in Fig. 4-1-2. Then, ( )u t = −Kx t( )_k as t∈

[

t t_k, _k+1

)

. The closed-loop system

Let ( ( ))V x t =x t Px t^T( ) ( ) be a Lyapunov function of the networked control system, where

It means that the system (4-4) converges to neighborhood of origin, if

2 1

( ) 2 ( ) 2 0

x t PB K e t ρ B P

− + + < . (4-6) Figure 4-1-2 System x(t) = Ax(t) + B w(t) + B u(t)_{1 2} , where u(t) = -Kx(t ) , _k

H denotes a zero-order-hold stage.

1

4.2 Transmission Error Upper Bound

From now on, the state is not real time feedback, and it would be interesting to find out how much the error is created. The system we consider here is (4-4), and then we define

( ) ( ) ( )_k

e t =x t −x t where t_k ≤ <t t_k+1. We call e t( ) as “transmission error”. Furthermore, we will derive the upper bound of transmission error in Lemma 5, and we call it as

‘transmission error upper bound’ shown in Fig. 4-2-2.

( ) x t

( )0

x t

( )1

x t

( )2

x t

( )3

x t

( )4

x t

( )5

x t

0 1 2 3 4 5 6

t t t t t t t t

Figure 4-2-1 Signification of x(t) and x(t ) . _k

( ) e t

0 1 2 3 4 5 6

t t t t t t t t

The Norm of actual transmission error Transmission error upper bound

Figure 4-2-2 Signification of transmission error and transmission error upper bound.

Lemma 5.

(Transmission Error Upper Bound of Networked Control Systems with Disturbances) The system is x t( )=Ax t( )+B w t₁ ( )+B u t₂ ( ), where w t( ) ≤ρ, and the transmission error, Taking the integral on both sides, we have

( )

∫

e t dt=

∫

Ae v( )+Ax t( )_k +B w v1 ( )  dv^.

4.3 Simulation

4.3.1 Validating Lemma 5

There are two systems chosen in this section for verifying Lemma 5.

System .Ⅳ

Consider the following system:

1 2

( ) ( ) ( ) ( )

x t = Ax t +B w t +B u t ,

where x t( )∈ℜ^{2 1×} is the state of the system, ( )u t ∈ is the control signal, and R 1 5

3 7

A − 

=  

 , ₁ 1 B  11

=  

 , ₂ 1 B  1 

=  − , 0 (0) 0

x  

=  

 , and τ =0.1s.

The poles of matrix A are 8.6 and -2.6. The eigenvalues of A A BK= − are 1− ,

− with choosing ( )2 u t = −Kx t( ), where ^{K = −}

[

^{4.2 13.2−}

]

^.

Suppose the disturbance ^{w t( ) sin 20=}

( )

^{t , as t∈}

[

^{0sec, 2sec}

]

is shown in Fig. 4-3-1(a). The system state ( )x t is shown in Fig. 4-3-1(b). The upper bound of e t( ) and actual e t( ) is shown in Fig. 4-3-1(c).

Figure 4-3-1(a) Noise of system Ⅳ.

Figure 4-3-1(b) States of system Ⅳ under controlled.

System .Ⅴ

Consider the following system:

x t( )= Ax t( )+B w t₁ ( )+B u t₂ ( ),

0 1 2 3 4 5 6 7 8 9 10 -10

-8 -6 -4 -2 0 2 4 6 8 10

Noise

Time(s)

Figure 4-3-2(a) Noise of system Ⅴ.

Figure 4-3-2(b) States of system Ⅴ under controlled.

0 1 2 3 4 5 6 7 8 9 10

-8 -6 -4 -2 0 2

State 1

0 1 2 3 4 5 6 7 8 9 10

-5 0 5 10 15 20

State 2

Time(s)

4.3.2 Remark

We take a view of Lemma 6: ( ) ^{( 1)} ( ) ₁

A

k

e t e A x t B

A

τ ^{− } ρ^

≤  + . When the value of

τ is small, according to the Taylor Expansion, the term e^Aτ can be rewritten as 1+ Aτ +..., and then the upper bound is decided mainly by A x t( )_k + B₁ ρ. However, the upper bound will become extreme large if τ is large. At this time, the bound will be useless.

Figure 4-3-2(c) Error upper bound of system Ⅴ as w(t) = 10sin 20t

( )

.

0 1 2 3 4 5 6 7 8 9 10

0 5 10 15 20 25 30 35 40

Error

Time(s)

Error upper bound

( ) e t

4.4 H

_∞

Control

4.4.1 Basic H

_∞

Control Concept

^[13]

For linear system, the system is defined as follows ,

1 2 then we can get the results that

0^∞z t z t dt^T( ) ( )

Consequently, by setting, ( ) ( ) 0

2 12 1 12 11 12 12

Replacing to (4-7), we have

1 2 1 1

We obtain the conclusion that only if there exists a positive symmetric matrix P satisfying 1_{2 1 1 1 1 2 2}

In this section, we derive the condition such that ²

0^∞z t z t dt^T( ) ( ) ≤γ 0^∞w t w t dt^T( ) ( )

Lemma 6. ( H_∞ Control of Networked Control Systems) Therefore, (4-11) can be simplied as

e t PB B P x t^T( ) 2 2^T

(

( )−x t( )_k

)

≤x t Qx t^T( ) ( ). (4-12)

(4-11) holds. It means that ²

0^∞z t z t dt^T( ) ( ) ≤γ 0^∞w t w t dt^T( ) ( )

∫ ∫

^.

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 dt^T( ) ( ) ≤100 0^∞w t w t dt^T( ) ( )

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 dt^T( ) ( ) ≤100 0^∞w t w t dt^T( ) ( )

∫ ∫

^as

[ ]

( ) 2^T ( )_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.

Reference

[1] G. C. Walsh, H. Ye, and L. Bushnell, ”Stability analysis of networked control systems,”

in Proc. Amer. Contr., pp.2876-2880, San Diego, CA., June 1999.

[2] G. C. Walsh, H. Ye and L. Bushnell, ”Stability analysis of networked control systems,”

IEEE Trans. Contr. Sys. Tech., vol.10, pp. 438-446, May 2002.

[3] W. Zhang, “Stability analysis of network control systems,” Partial fulfillment of the requirements for the degree of Doctor of Philosophy, Department of Electrical Engineering and Computer Science, Case Western Reserve University, Aug., 2001.

[4] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilal, LMI Control Toolbox, Math Works.

[5] G.. C. Walsh and H. Ye, ”Scheduling of networked control systems,” IEEE Control System Magazine, pp. 57-65, February 2001.

[6] M. S. Branicky and S. M. Philips, W. Zhang, ”Scheduling and feedback co-design for networked control systems,” in Proc. of the 41^st IEEE Conference on Decision and Control Las Vegas, Nevada USA, pp.1211-1217, Dec. 2002.

[7] C.A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties, Academic Press, Onc., New York, NY, 1975.

[8] C. T. Chen, Linear System Theory and Design, 3^rd Ed., Oxford University Press, 1999.

[9] W. Zhang, M. S. Branicky, and S. M. Philips, ”Stability of networked control systems,”

IEEE Control System Magazine, pp. 84-98, Feb. 2001.

[10] P. J. Antsaklis and A. N. Michel, Linear Systems, McGraw-Hill, 1997.

[11] L. Xie, J. M. Zhang, and S. Q. Wang, ”Stability analysis of networked control systems,”

in Proceedings of the 1^st International Conference on Machine Learning and Cybernetics, pp.757-841, Beijing, Nov. 2002.

[12] C. Scherer and S. Weiland, Lecture Notes DISC Course on Linear Matrix Inequalities in Control, version 2.0, April 1999.

[13] W. S. Levine and R. T. Reichert, “An introduction to H_∞ control system design,” in Proc. Decision and Control, pp. 2966-2974, Dec. 1990.

[14] J. Jacques, E. Slotine, and W. P. Li, “Applied nonlinear control”, published by Pearson Education, Oct. 1990.

[15] R. S. Raji, “Smart networks for control,” IEEE Spectrum, pp.49-55, June 1994.

[16] A.S. Tanenbaum, Computer Networks, 3^rd Ed., Englewood Cliffs, NJ: Prentice-Hall,

1996.

[17] A. Ray, ”Performance evaluation of medium access control protocols for distributed digital avionics,” ASME J. Dynamic Sys., Mea. Contr., vol. 103, no. 4, pp. 370-377, Dec.

1987.

[18] A. Ray, “Distributed data communication networks for real-time process control,” Chem.

Eng. Comm., vol. 65, pp.139-154, Mar. 1988.

[19] J. Nilsson, B. Bernhardsson, and B. Wittenmark, “Stochastic analysis of control of real-time systems with random time delays,” Automatica, vol. 34, no. 1, pp. 57-64, Jan.

1998.

[20] J. Nilsson, “Real-time control systems with delays,” PhD. dissertation, Dept. Automatic Control, Lund Institute of Technology, Lund, Sweden, Jan. 1998.

[21] W. S. Wong and R. W. Brockett. “Systems with finite bandwidth constraints-part Ⅱ:

Stabilization with limited information feedback,” IEEE Trans. Automatic Control, vol.

42, no. 5, pp. 1049-1052, 1999.

[22] R. W. Brockeet. “Stabilization of motor networks,” in Proc. 34^th IEEE CDC, pp.

1484-1488, 1995.

[23] D. Hristu and K. Morgansen, “Limited communication control,” Systems and Control Letters, July 1999.

[24] U. Ozguner, H. Goktag, and H. Chan, “Automotive suspension control through a computer communication network,” in Proc. IEEE Int. Conference on Control Applications, pp.895-900, 1992.

[25] D. Radford, “Spread-spectrum data leap through ac power wiring,” IEEE Spectrum, pp. 48-53, Nov. 1996.

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

在文檔中 網路控制系統的穩定性分析 (頁 33-0)