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Intelligent Information Management Systems and Technologies (2011)

Volume 7, No.1

17--24

Dual Random Model of Increasing Life Insurance

Wang Li-Yan1, 2 Zhang Mao-Jun3 Yang De-Li2

1. College of Information Engineering, Dalian University, Dalian, 116622, P. R. China 2. School of Management, Dalian University of Technology, Dalian, 116024, P. R. China 3. Department of Applied Mathematics, Dalian University of Technology, Dalian, 116024, P. R. China

ABSTRACT

This paper discusses modeling and control of networked control systems with uncertain short time delay and data packet dropout, the uncertain time delay system was modeled as an uncertain discrete liner system by appropriate transformation. This paper further analyzes the stability of the system using the relevant knowledge of Lyapunov stability theory and linear matrix inequalities (LMI), and gives sufficient conditions and a controller for system stability. Key words: networked control systems(NCS), time delay, data packet dropout, LMI, stability.

1. Introduction In networked control systems (NCS), there are essentially two more kinds of delays than traditional control systems: Communication delay between the sensor and the controller τ sc and communication delay between the controller and the actuator τ ca . The control delay for NCS, in principle, equals the sum of these delays. We define τ = τ sc + τ ca as the total time delay. If the time delay of NCS is a certain distribution character random quantity, the NCS can be modeled as a stochastic system, we can use the method of stochastic control theory to analyze and control the system [1]. If the time delay is time varying or uncertain, we can use the method of robust control, adaptive control or intelligent control to analyze and control the system [2] [3]. Moreover, we can transform time varying delay into fixed delay, then we can use the theory and method of deterministic system [4]. Time delay also can be modeled as independent random delay or random delay using markov chain theory [5].

? International Society for Scientific Inventions? ISSN 1551-2606 (Print) 1551-2614 (Online)

18

Control of a class of Networked Control Systems with time delay and data packet dropout

Using a Lyapunov-based method and the knowledge of linear matrix inequalities (LMI) and on the bases of research achievements up to now, this paper models an asynchronous dynamical system, and analyzes the sufficient conditions and control law for system exponential stability.

2. Problem statement In order to model the system, we make the following assumptions about the network control system: (1) The senor is time-driven, the output of the controlled object is sampled periodically, the sampling period is T (T>0); the controller and actuator are event-driven, data is processed as soon as it arrives. (2) Data packet may lose in communication and probability of data packet dropout is r . (3) The communication delay τ is uncertain randomly varying, and τ is less than the sampling period, i.e. τ ∈ [0, T ] . (4) The process time of controller, which is much less than T, is negligible. Assume that the controlled object in NCS is a linear time invariant (LTI) system, considering the noice of system, it is of the form

& ? x (t ) = Ax (t ) + Bu (t ) + v(t ) ? ? y (t ) = Cx(t ) + w(t )

(1)

where x (t ) ∈ R n , u (t ) ∈ R m , y (t ) ∈ R k , A , B and C are known matrices of compatible dimensions. v (t ) and w(t ) are uncorrelated white noise with zero mean. According to [6], discretizing (1) at the sampling instant kT taking into account the time delay, we can obtain

? x(k + 1) = Ad x(k ) + Γ0u (k ) + Γ1u (k ? 1) + v(k ) ? ? y (k ) = Cx(k ) + w(k )

T ?τ k

(2)

where Ad = e AT ， Γ 0 =

∫

0

e As dsB ， Γ1 = ∫

T

T ?τ k

e As dsB . Because of the time delay τ k ∈ [0, T ] ，

we can find τ min and τ max satisfying

τ k ∈ [τ min ,τ max ] ， 0 ≤ τ min < τ max ≤ T .

Define τ = (τ min + τ max ) / 2 , B0 =

∫

T ?τ

0

e As dsB , B1 = ∫

T

T ?τ

e As dsB , F = ∫

T ?τ k

T ?τ

e As ds , D = I ,

E0 = B , E1 = ? B , (2) can be rewritten as

Deng Jun-hui Ge Zhao-qiang

19

? x(k + 1) = Ad x(k ) + ( B0 + ?B0 )u (k ) + ( B1 + ?B1 )u (k ? 1) + v(k ) ? ? y (k ) = Cx(k ) + w(k )

(3)

where ?B0 = DFE0 , ?B1 = DFE1 , and F = (3) can be denoted by:

∫

T ?τ k

T ?τ

e As dsB is an uncertain matrix.

? xk +1 = Ad xk + ( B0 + ?B0 )uk + ( B1 + ?B1 )uk ?1 + vk , ? ? yk = Cxk + wk .

(4)

Because of data packet dropout, our system can be described as a switch system as Figure 1, the switch S is off means data packet dropout.

Controlled Object

xk

S

Controller

xk

Figure 1: the structure of the system Suppose the control law is uk = K x k ： S is on: we have x k = xk , (4) can be written as

? xk +1 = Ad xk + ( B0 + ?B0 ) K x k + ( B1 + ?B1 ) K x k ?1 + vk , ? ? ? x k = xk . ?

S is off: we have x k = x k ?1 ，(4) can be written as

(5)

? xk +1 = Ad xk + ( B0 + B1 ) K x k ?1 + vk , ? ? ? x k = x k ?1 , ?

T define zk = [ xk +1 , x k ]T , (5) and (6) can be rewritten as T

(6)

20

Control of a class of Networked Control Systems with time delay and data packet dropout

zk +1 = Φ s zk ， s = 1, 2 ,

where

(7)

? A + ( B0 + ?B0 ) K Φ1 = ? d I ?

? S11 ?*

( B1 + ?B1 ) K ? ? Ad ? , Φ2 = ? 0 0 ? ?

( B0 + B1 ) K ? ?. I ?

Lemma 1 [7] Let S = ?

S12 ? be a symmetric matrix, where “*” means the symmetric S 22 ? ?

portion of the matrix, the follows are equivalent:

T ? ? T (1) S < 0 ; (2) S11 < 0 , S 22 ? S12 S111S12 < 0 ; (3) S 22 < 0 , S11 ? S12 S 221S12 < 0 .

Lemma 2 [8] For the asynchronous dynamical system (7), if we have a Lyapunov function

1 V ( xk ) : R n → R + and α1 , α 2 >0 satisfying α1rα 2? r > α > 1 and V ( xk +1 ) ? α s?2V ( xk ) ≤ 0,

s = 1, 2 , then the system (7) is an exponential stable system, α is the decay rate of the system.

Lemma 3 [9] Y,D and E are known matrixes of compatible dimensions, and Y is a symmetric matrix, then Y + DFE + E T F T DT < 0 for all F satisfying F T F ≤ I , if and only if there exists a scalar ε > 0 such that Y + ε DDT + ε ?1 E T E < 0 .

3.

Main Results

According to [10], using the similar method, we can obtain following theorem. Theorem 1 For the given system (7), without white noise of the system, assume that the data packet dropout rate is r , then the sufficient conditions for exponential stability are: there exists matrix P = P > 0 , Q = Q T > 0 and positive scalars α1 , α 2 , λ , ε satisfying

T

1 α1rα 2? r > 1 ,

(8)

?ΓT PΓ 0 + Q ? α1?2 P ? ΓT PΓ1 0 0 ? ? < 0, T * Γ1 PΓ1 ? α1?2Q ? ?

(9)

Deng Jun-hui Ge Zhao-qiang

21

T ? Ad PAd ? α1?2 P ? * ?

? ? <0, K ( B + B ) P( B0 + B1 ) K + Q ? α Q ?

T T 0 T 1 ?2 1

T Ad P( B0 + B1 ) K

(10)

where Γ 0 = Ad + ( B0 + ?B0 ) K , Γ1 = ( B1 + ?B1 ) K . Proof (i)

T Define the Lyapunov function is Vk = xk Pxk + x k ?1 P x k ?1 . T

Assume that the switch S is on, we have that system is of the form (5), we can obtain

T T

T T Vk +1 ? α1?2Vk = xk +1 Pxk +1 + x k Qx k ? α1?2 ( xk Pxk + x k ?1Qx k ?1 )

T T = (Γ 0 xk + Γ1 x k ?1 )T P(Γ 0 xk + Γ1 x k ?1 ) + xk Qxk ?α1?2 ( xk Pxk + x k ?1Qx k ?1 ) T T T T = xk (ΓT PΓ 0 + Q ? α1?2 P) xk + xk ΓT PΓ1 x k ?1 + x k ?1Γ1 PΓ 0 xk + x k ?1 (Γ1 PΓ1 ? α1?2Q) x k ?1 0 0 T T

T

? T = ? xk ?

T ?2 ? ? xk ? T ΓT PΓ1 0 ? ?Γ PΓ + Q ? α1 P x k ?1 ? ? 0 0 ?, ?? T ?? Γ1 PΓ1 ? α1?2Q ? ? x k ?1 ? *

according to (9), we have: Vk +1 ? α1?2Vk < 0 . (ii) Assume that the switch S is off, we have that system is of the form (6), we can obtain

T T

T ? T ? Vk +1 ? α 2 2Vk = xk +1 Pxk +1 + x k Qx k ? α 2 2 ( xk Pxk + x k ?1Qx k ?1 )

? T = ( Ad xk + ( B0 + B1 ) K x k ?1 )T P ( Ad xk + ( B0 + B1 ) K x k ?1 ) + x k ?1Qx k ?1 ?α 2 2 ( xk Pxk + x k ?1Qx k ?1 ) T T T T ? T = xk ( Ad PAd ? α 2 2 P ) xk + xk Ad P( B0 + B1 ) K x k ?1 + x k ?1 K T ( B0 + B1T ) PAd xk T + x k ?1 ( K T ( B0 + B1T ) P( B0 + B1 ) K + Q ? α1?2Q) x k ?1 T ?2 ? ? Ad PAd ? α1 P ?? ?? * T ? ? xk ? Ad P( B0 + B1 ) K ?, ?? T K T ( B0 + B1T ) P ( B0 + B1 ) K + Q ? α1?2Q ? ? x k ?1 ? T T T T

T = ? xk ? ?

x

T k ?1

? according to (10), we have: Vk +1 ? α 2 2Vk < 0 .

From (i) and (ii), for the given system (7), we have: Vk +1 ? α s?2Vk < 0, s = 1, 2 .

22

Control of a class of Networked Control Systems with time delay and data packet dropout

Using Lemma 2, the given system (7) is an exponential stable system. Corollary 1 For the given system (4), without white noise of the system, assume that the data packet dropout rate is r , then the sufficient conditions for exponential stability of the given system which has the state feedback control law uk = K x k are: there exists matrix X = X T , Y ,

Z = Z T > 0 and positive scalars α1 , α 2 , λ , ε satisfying

1 α1rα 2? r > 1 ,

(11)

? Z ? α1?2 X ? 0 ? ? * ? * ?

0 ?α1?2 Z * *

T T X T Ad + Y T B0 Y T B1T

? X + λ 2ε 0

0

? Z ? α2 2Z

Y T BT ? ? ?Y T BT ? < 0, 0 ? ? ?ε I ?

(12)

? ? ?α 2 2 X ? ? 0 ? * ?

*

T X T Ad ? T ? Y T Bd ? < 0 , ?X ? ?

(13)

and the state feedback control law can be expressed as uk = YX ?1 xk . Proof Using the theorem above, we only need to prove that (9) can deduce (12) and (10) can deduce (13). Using Lemma 1, inequality (9) can be expressed as

?Q ? α1?2 P 0 ΓT ? 0 ? ?2 T ? 0 α1 Q Γ1 ? < 0 , ? ? * * ? P ?1 ? ? ?

using (3), the inequality above can be rewritten as

?Q ? α1?2 P 0 ? ?2 0 α1 Q ? ? * * ?

T T Ad + K T B0 ? ?0 ? ? BK ? ? K T BT ? ? ? ? ? ? ? ? K T B1T ? + ?0 ? F ? ? BK ? + ? ? K T BT ? F T ? P ?1 ? ? I ? ? 0 ? ? 0 ? ? ? ? ? ? ? ? T

?0 ? ? ? , (14) ?0 ? < 0 ?I ? ? ?

T

define σ max ( F ) as the maximum singular value of matrix F , let

Deng Jun-hui Ge Zhao-qiang

23

λ = (eσ

max

( F )(T ?τ min )

? eσ max ( F )(T ?τ ) ) / σ max ( F ) , we get

min ,τ max

σ max ( F ) ≤ maxτ ∈[τ

k

]

∫

T ?τ k

T ?τ

e As ds ≤ max(

2

∫

T ?τ max

T ?τ

e As ds ,

2

∫

T ?τ min

T ?τ

e As ds )

2

≤∫

T ?τ min

T ?τ

e

A 2s

ds ≤ ∫

T ?τ min

T ?τ

eσ max ( A) s ds (eσ max ( F )(T ?τ min ) ? eσ max ( F )(T ?τ ) ) / σ max ( F ) = λ ,

(15)

using Lemma 3, we get

?Q ? α1?2 P 0 ? ?2 0 α1 Q ? ? * * ?

using Lemma 1 again, we can obtain

T T Ad + K T B0 ? ?0 ? ?0 ? ? BK ? 1? ? T T 2 ? ?? ? K B1 ? + ελ ?0 ? ?0 ? + ? ? BK ? ? ε ?I ? ?I ? ? 0 ? ? P ?1 ? ? ?? ? ? ? ? T

T

? K T BT ? ? T T? ? ? K B ? < 0 , (16) ? 0 ? ? ?

?Q ? α1?2 P 0 ? ?2 0 α1 Q ? ? * * ? * * ?

T T Ad + K T B0

K T B1T ? P ?1 + ελ 2 0

K T BT ? ? ? K T BT ? < 0, 0 ? ? ?ε I ?

(17)

let X = P ?1 , Z = P ?1QP ?1 , Y = KX , multiply the both sides of (17) by diag ( X , X , I , I ) , we can get inequality (12), it means that (9) can deduce (12). Similarly, (10) can deduce (13). Corollary is established. 4. Conclusion

This paper studies a Lyapunov-based method for modelling and controller design for a networked control system with short time delay and data packet dropout, and the uncertain time delay system was modeled as an uncertain discrete liner system by appropriate transformation. Using the knowledge of Lyapunov stability theory and linear matrix inequalities (LMI), this paper further analyzes the stability of the system and gives sufficient conditions and a controller for system stability. The conclusion of this paper is specific to NCS with short time delay, and it has certain theoretical value and practical value. There needs further study of much complex system of NCS with long time delay.

24

Control of a class of Networked Control Systems with time delay and data packet dropout

References

1. 2. 3. 4. 5. 6. Nillsson J.,Bernhardsson B.,Wittenmark B., Stochastic analysis and control of real-time systems with random time delays, Automatica, 1998, 34(1), 57-64. Dong Y., Han Q. L., Lam J., Networked-based robust H∞ control of systems with uncertainty, Automatica, 2005, 41, 999-1007. Andrey V. S., Analysis and synthesis of networked control systems: Topological entropy, observability, robustness and optimal control, Automatica, 2006, 42, 51-62. Wang Suqing, Control and scheduling co-design method of networked control systems, Nanjing University of Science and Technology, 2006. Zhang L. Q., Yang S., A new method for stabilization of networked control systems with random delays, IEEE Transactions on Automatica Control, 2005, 50(8), 1177-1181. Guangming Xie, Long Wang, Stabilization of Networked Control Systems with Time-Varying Network-Induced Delay, 43rd IEEE Conference on Decision and Control, 2004. 7. 8. Yu Li, Robust control – Linear matrix inequalities method, Tsinghua University Press, 2002. Hassibi A., Boyd S. P., How J. P., Control of Asynchronous Dynamical Systems with Rate Constraints on Events, Proe. of the IEEE Conference on Control and Decision, 1999, 1345-1351. 9. Kosmidou O. I., Generalized Riccati equations associated with guaranteed cost control: An overview of solutions and features, Applied Mathematics and Computation, 2007 (2), 511-520. 10. Wei Zhang, Michael S. Branicky, Stephen M. Phillips, Stability of Networked Control Systems, IEEE Control Systems Magazine, February 2001, 84-97.

ACKNOWLEDGEMENT This work is supported by the Nature Science Foundation of Inner Mongolian in China, Grant No.200607010113.

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