Abstract. The problem of robust stability for a class of networked control systems (NCSs) based on state observer is studied in this paper. Considering.
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INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 3, Number 4, Pages 594–603
ROBUST STABILITY FOR A CLASS OF NETWORKED CONTROL SYSTEMS BASED ON STATE OBSERVER ZHAN-ZHI QIU1 , QING-LING ZHANG2 2, XUE-FENG ZHANG2 2 AND LI YANG3 Abstract. The problem of robust stability for a class of networked control systems (NCSs) based on state observer is studied in this paper. Considering time-driven sensors, event-driven actuators and the controller, and the uncertain time-delay less than and equal to one sampling period, an augmented mathematical model for the NCSs based on state observer is developed. The conditions of the robust stabilization for the NCSs are derived by using linear matrix inequality formulations. Furthermore, the designs for the robust control law and the state observer are presented. A numerical simulation example shows that the analysis method and the results are valid and feasible. Key Words. networked control systems; robust controller design; states observer; linear matrix inequality; uncertain time-delay
1. Introduction Since the 21st century, the rapid development of the technologies on communication has promoted networked control systems (NCSs) in wide applications ranging from equipment manufacture to electricity production, communication, and aircrafts. So control methodologies in networked control system, increasingly, have been the important topics for scholars and experts. There are many domestic and foreign research achievements of NCSs, such as modeling and stability analysis of systems, robust and H∞ control etc.. But most of them are based on the fact that the states of the controlled objects are measurable. In practical NCSs, when the nodes distributed geographically communicate via shared network, not only because of the limits of wideband and communication mechanism there exists communication delay, but also due to environmental or economic factors it is usually difficult that the states of controlled object are completely measurable , instead, partial information of the states is obtained. So some scholars have studied the problem of state estimation, however, there is few report on the control of NCSs based on observer. This paper is concerned with NCSs with time-driven sensors, event-driven actuators and the controller, considering the uncertain time-delay not more than a sampling period, and robust controller design for NCSs based on state observer is studied. The sufficient condition of existence of robust control law is provided, moreover, a law of robust control and a design method for state observer are presented. Finally, numerical examples and simulation results via Matlab are given to prove the effectiveness of the method. Received by the editors September 1, 2006 and, in revised form, March 22, 2007. 2000 Mathematics Subject Classification. 35R35, 49J40, 60G40. This research was supported by the Natural Science Foundation of Liaoning province, China (2050770). 594
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2. Modelling for NCSs A typical networked control system is shown in Figure 1, where y is sensor measurement output, and u is control input. The sensor is chosen time-driven, and it samples data to output in a period of T. Then the data is transferred to the controller after the network-induced delay τsc , and the controller deals with the data after receiving it. The control signal outputted is transferred to the actuator after the network-induced delay τca . The actuator restores the control signal to a piecewise continuous control input u through a zero-order hold. Suppose there don’t exist disorders of the sequence and data packet dropout when the signal is transmitted in the network., and the processing disturbance and measurement noises are not considered; the delay of the device such as the controller is neglected. k k τk denotes the network-induced delay of the sampling period τk = τsc + τca in the whole closed-loop, thus k, here τk is uncertain or unknown, but it is bounded, and the condition of τk ≤ T is satisfied.
Figure 1. The model of NCSs In an NCS, the model of the plant can be described as ½ ˙ x(t) = Ax(t) + Bu(t) (1) y(t) = Cx(t) Where x(t) ∈ Rn , u(t) ∈ Rr and y(t) ∈ Rm are the state vector, control input vector and measurement output vector, respectively, A, B and C are constant matrices with compatible dimensions. Input u(t) is piecewise continuous in a period, in view of τk ∈ [0, T ], so u(t) is given by ½ (2)
u(t) =
u(k − 1), tk < t ≤ tk + τk u(k), tk + τk < t ≤ tk + T
Then, the discrete model of NCS is described as (3)
x(k + 1) = Ad x(k) + Bd0 (τk )u(k) + Bd1 (τk )u(k − 1)
(4)
y(k) = Cx(k) R RT T −τ where Ad = eAT , Bd0 (τk ) = 0 k eAT Bdt, Bd1 (τk ) = T −τk eAT Bdt According to the eigenvalues of matrix A, the discrete state equation (3) can be transformed into the model with uncertainty in the following
(5) x(k + 1)
=
Ad x(k) + (B0 + DF (τk )E)u(k) + (B1 − DF (τk )E)u(k − 1)
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Z. QIU, Q. ZHANG, X. ZHANG AND L. YANG
Where B0 , B1 , D and E are constant matrices, F T (τk )F (τk ) ≤ I, F (τk ) is timevarying in terms of τk , τk ∈ [0, T ]. ˆ1, λ ˆ2, · · · , λ ˆ n , correspondWhen matrix A has the different non-zero eigenvalues λ ˆ ing eigenvector matrix is denoted by Λ = [Λ1 , Λ2 , · · · , Λn ], so B0 = Λdiag(− λˆ1 , − λˆ1 , · · · , − λˆ1 )Λ−1 B, E = Λ−1 B, 1
2
ˆ
n
ˆ
ˆ
B1 = Λdiag( λˆ1 eλ1 T , λˆ1 eλ2 T , · · · , λˆ1 eλn T )Λ−1 B, 1
2
ˆ
ˆ
n
ˆ
F (τk ) = diag(eλ1 (T −τk −a1 ) , eλ2 (T −τk −a2 ) , · · · , eλn (T −τk −an ) ), where a1 , a2 , · · · , an ˆ ˆ are selected in the condition of eλi (T −τk −ai ) < 1, i = 1, 2, · · · , n, D = Λdiag( λˆ1 eλ1 a1 ˆ 2 a2 1 λ ,··· ˆ2 e λ
1
ˆ
, λˆ1 eλn an ). n When the eigenvalue of the matrix A equals zero or there are the same r as eigenvalues of the matrix A, the values of the matrices B0 , B1 and D need to be changed correspondingly. Because the sensor can’t measure all the states of the plant, we utilize state observer to estimate the states of the plant in order to realize the state feedback. Since there is a delay τk ∈ [0, T ] in the closed-loop, in order to make the estimation error as small as possible, we utilize the model of the observer which has the function of delay compensation. Consider the plant (5) and delay τk ∈ [0, T ], under the assumption that Ad is nonsingular, (Ad , C) is observable, and then the model of the observer is described as (6)
x ˆ(k + 1) = Ad x ˆ(k) + B0 u(k) + B1 u(k − 1) + L[Y (k) − CA−1 d (hatx(k) − B0 u(k − 1) − B1 u(k − 2))]
Where x ˆ(k) ∈ Rn is the output, L is the observer gain with the appropriate dimension. Because of the delay, at the k interval, the observer input is (7)
y(k) = Cx(k − 1)
In terms of (5), we obtain (8) x(k − 1) = A−1 d [x(k) − (B0 + DF E)u(k − 1) − (B1 − DF E)u(k − 2)] Substituting (7) and (8) into (6), then the state observer is given by (9)
−1 x ˆ(k + 1) = LCA−1 x(k) + B0 u(k) d x(k) + (Ad − LCAd )ˆ −1 +(B1 − LCA−1 d DF E)u(k − 1) + LCAd DF Eu(k − 2)
Define the estimation error as (10)
e(k) = x(k) − x ˆ(k)
Feedback controller is given by memoryless model (11)
u(k) = K x ˆ(k)
Combining (5), (9), (10) and (11), the observer-based model of the closed-loop NCS is (12)
x(k + 1) = [Ad + (B0 + DF E)K]x(k) − (B0 + DF E)Ke(k) +(B1 − DF E)Kx(k − 1) − (B1 − DF E)Ke(k − 1)
ROBUST STABILITY FOR A CLASS OF NETWORKED CONTROL SYSTEMS
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Estimation error equation is e(k + 1) = DF EKx(k) + (Ad − LCA−1 d − DF EK)e(k)
(13)
−1 +(LCA−1 d − 1)DF EKx(k − 1) − (LCAd − 1)DF EKe(k − 1) −1 −LCA−1 d DF EKx(k − 2) + LCAd DF EKe(k − 2)
¯0 = −(B0 + DF E)K, A¯1 = Notations are given by A¯0 = Ad + (B0 + DF E)K, B ¯ ˜ ˜ ˜ (B1 −DF E)K, B1 = −(B1 −DF E)K, A0 = DF EK, A1 = (LCA−1 d −1)DF EK, B1 = −1 −1 −1 ˜2 = LCA DF EK, B ˜0 = Ad − DEF K − LCAd DF EK, A˜2 = −LCAd DF EK, B d DEF K −LCA−1 .And then the observer-based augmented model of the closed-loop d NCS is · (14)
x(k + 1) e(k + 1)
¸
· =
A¯0 A˜0
¯0 B ˜ B0
¸·
x(k) e(k)
¸
· + · +
A¯1 A˜1
¯1 B ˜1 B
0 A˜2
0 ˜2 B
¸· ¸·
x(k − 1) e(k − 1) x(k − 2) e(k − 2)
¸ ¸
Definition 1. If the NCS (5) with uncertain delay is asymptotically stable under the observer-based control law (11), the control law (11) is regarded as the observer-based robust control law. Lemma1.Real matrices W, M, N and F (k) , where F (k) is symmetric, and satisfies F T t(k)F (k) ≤ I , we have W + M F (k)N + N T F T (k)M T < 0 If and only if there exists a positive scalar ε > 0 such that W + εM M + ε−1 N N T < 0 Lemma2. (Schur compensation lemma) Given the symmetric matrix A, the symmetric positive definite matrix C, and ¸matrix B , then A + B T CB < 0 if and only · ¸ · −1 T −C B A B if < 0 or < 0. B −C −1 BT A
3. Robust controller design Theorem 1. Observer-based NCS (14) is robust asymptotically stable if there exist symmetric positive definite matrices Pn , n = 1, 2, · · · , 6 and scalars εi > 0, i = 1, 2, · · · , 4 such that
(15)
P3 + P5 − P1 0 0 0 0 0 Ad + B0 K 0 EK EK 0 0
∗ P4 + P6 − P2 0 0 0 0 −B0 K Ad −EK −EK −CA−1 d 0
∗ ∗ −P3 0 0 0 B1 K 0 −EK −EK 0 EK
∗ ∗ ∗ −P4 0 0 −B1 K 0 EK EK 0 −EK
∗ ∗ ∗ ∗ −P5 0 0 0 0 0 0 −EK
∗ ∗ ∗ ∗ ∗ −P6 0 0 0 0 0 EK
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Z. QIU, Q. ZHANG, X. ZHANG AND L. YANG
∗ ∗ ∗ ∗ ∗ ∗ N1 0 0 0 0 0
∗ ∗ ∗ ∗ ∗ ∗ ∗ N2 0 0 0 0
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −ε1 I 0 0 0
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −ε2 I 0 0
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ N3 0
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −ε4 I
