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Procedia Engineering
Procedia Engineering 00 (2011) 000–000 Procedia Engineering 29 (2012) 4192 – 4197 www.elsevier.com/locate/procedia
2012 International Workshop on Information and Electronics Engineering (IWIEE 2012)
Robust H ∞ Control for Networked Systems with Random Packet Dropouts and Time Delays Xiuying Lia, Shuli Suna* a
Electronic Engineering Department of Heilongjiang University, No. 74 Xuefu Road Nangang District, Harbin, 150080, China
Abstract This paper is concerned with the robust H ∞ control problem for a class of networked systems with random transmission delays and packet dropouts. Both the sensor-to-estimator channel and the controller-to-actuator channel are considered. The random one-step transmission delays and packet dropouts are modeled by a Bernoulli distributed stochastic variable. Applied the linear matrix inequality (LMI) approach, an observer-based feedback controller is designed to make the closed-loop networked system to be robustly exponentially stable in the sense of mean square and the prescribed H ∞ disturbance-rejection-attenuation level is also achieved. A simulation example is given to illustrate the proposed method.
© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Harbin University of Science and Technology Open access under CC BY-NC-ND license. Keywords: Robust H ∞ control; linear matrix inequality (LMI); networked system; packet dropout; random communication delay; mean-square stability
1. Introduction Networked control systems (NCSs) are spatially distributed systems in which the sensors, actuators and controllers communicate through the wireless links or communication networks. Due to the unreliable communication media, random time delays and packet dropouts will often occur during the information transmission. But, the NCSs have a lot of advantages compared to the conventional point-to-point system connection, so they are more popular in the engineering practice. With the growing applications, NCSs have gained much attention by the researchers [1-8]. As for the random packet dropouts, the H ∞ and
* Corresponding author. Tel.: +86 136 74686865. E-mail address:
[email protected]
1877-7058 © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. doi:10.1016/j.proeng.2012.01.642
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Xiuying Li and Shuli / Procedia Engineering (2012) 4192 – 4197 Author nameSun / Procedia Engineering 00 29 (2011) 000–000
LQG controllers are designed in [1] and [2], respectively. Yang et al. investigate the control problems for NCSs with random transmission delays in [3,4]. Relevant filtering problems have gained lots of interests for such a system [5,6]. As for the NCSs with both random transmission delays and packet dropouts, there are some results on filtering problems [7,8], but the results on control problems are seldom reported. In this paper, we study the robust H ∞ control problem for NCSs with random transmission delays and packet dropouts, both sensor-to-controller channel and controller-to-actuator channel are considered simultaneously. These random delays and packet dropouts can be modelled by a stochastic variable satisfying Bernoulli distribution. An observer-based controller is designed via an LMI approach such that the closed-loop networked control system is exponentially stable in the sense of mean square, and the prescribed H ∞ disturbance attenuation performance is achieved. 2. Problem Formulation Consider the following discrete-time linear system: ( A + ΔA k )x k + B 2 u k + B1w k ⎧x k +1 = ⎨ z k =+ (C1 ΔCk )x k + D1w k ⎩
(1)
where x k ∈ R n is the state, u k ∈ R p is the control input, z k ∈ R m is the controlled output, w k ∈ R q is the disturbance input belonging to l2 [0, ∞) , A, B1 , B 2 , C1 and D1 are known real constant matrices with appropriate dimensions, ΔA k and ΔCk are parameter uncertainties assumed to be the form of ΔA k = H1Fk E , ΔCk = H 2 Fk E , where H1 , H 2 and E are known real constant matrices of suitable dimensions, and Fk represents an unknown real-valued time-varying matrix satisfying Fk FkT ≤ I . In this paper, we choose the dynamic observer-based control scheme for system (1), and assume that there exist possible one-step transmission delays and random packet dropouts from the sensor to the observer and from the controller to the actuator in the network. Consider the sensor-to-estimator channel first, the measurement received at time k by the observer with random transmission delays and packet dropouts is described by yk = (1 − α k )C2 x k + α k α k −1C2 x k −1 + D2 w k
(2)
where y k ∈ R r , C2 and D2 are known real matrices with appropriate dimensions. α k is a Bernoulli distributed stochastic variable with Pr ob{α k = 1} = α and Pr ob{α k = 0}= 1 − α . It is clear that the probability for a packet from the sensor at instant k to be received by the observer on time is Pr ob{α k = 0}= 1 − α , one unit delay rate is Pr ob{α k= 1, α k −1= 1}= α 2 and packet dropout rate is Pr ob{α= 1, α k −= 0} = α (1 − α ) . Similar model can be implemented for the controller-to-actuator k 1 channel. The dynamic observer-based control scheme for the system (1) is described by ⎧⎪xˆ k +1 =Axˆ k + B 2u k + L(y k − yˆ k ) Observer : ⎨ (1 − α )C2 xˆ k + α 2C2 xˆ k −1 ⎪⎩ yˆ k = uˆ k = -Kxˆ k ⎧ Controller : ⎨ (1 − β k )uˆ k + β k β k −1uˆ k −1 ⎩u k =
(3) (4)
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where xˆ k ∈ R n is the state estimate of the system (1), uˆ k ∈ R p is the designed control input to be transmitted, u k ∈ R p is the controller received by the actuator, and L ∈ R n×r and K ∈ R p×n are the observer and controller gains, respectively. β k is also a Bernoulli distributed random variable and mutually independent of
αk
which satisfies probabilities Pr ob{β k = 1} = β and Pr ob{β k = 0}= 1 − β .
The probability analysis for the packet dropout rate and one-step delay rate is similar with the sensor-toobserver channel. Let the estimation error be e k = x k − xˆ k
(5)
The closed-loop system can be obtained by substituting (2), (3) and (4) into (1) and (5) , which is given in a compact form as follows: ⎡ xk +1 ⎤ ⎡ A 0 ⎢ e ⎥ =⎢ % ⎣ k +1 ⎦ ⎣ A 0
⎡A ⎡A B0 ⎤ ⎡ xk ⎤ B01 ⎤ ⎡ x k ⎤ ⎡ A1 B1 ⎤ ⎡ x k −1 ⎤ B11 ⎤ ⎡ x k −1 ⎤ ⎡ B1 ⎤ + ξ k ⎢ 01 +⎢ + ς k ⎢ 11 ⎥ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥+⎢ ⎥ w k (6) % % % % ⎣ ek ⎦ % % % ⎥⎢ B 0⎦ ⎣ A11 B11 ⎦ ⎣ e k −1 ⎦ ⎣B1 − LD2 ⎦ ⎣ A 01 B01 ⎦ ⎣ e k ⎦ ⎣ A1 B1 ⎦ ⎣ e k −1 ⎦
where ξ k = diag{( β k − β )I, (α k − α )I} , ς k = diag{( β k β k −1 − β 2 )I, (α k α k −1 − α 2 )I} , A 0 = A + ΔA − (1 − β )B 2 K , % = ΔA , B% = A − (1 − α )LC , A = − β 2 B K , B = β 2 B K , A % = 0 , B% = −α 2 LC , B = (1 − β )B K , A 0
2
0
0
2
1
2
1
2
1
1
% = LC , B% = 0 , A = −B K , B = B K , A % = −LC , B% = 0 . A 01 = B 2 K , B01 = −B 2 K , A 01 2 01 11 2 11 2 11 2 11
2
(7)
Our aim is to design the controller (4) for the system (1), such that the closed-loop system (6) guarantees the stochastic stability in the mean-square sense and meets the H ∞ performance constraint. That is, we like to design a controller (4) such that the closed-loop system (6) satisfies the following two performance requirements (Q1) and (Q2). (Q1) The closed-loop system (6) is exponentially mean-square stable. i.e., if with w k = 0 , there exist 2
2
constants ϕ > 0 and τ ∈ (0,1) , such that E{ ηk } ≤ ϕτ k E{ η0 } for all η0 ∈ R n , k ∈ Ι + , where ηk = ⎡⎣ xTk
eTk
T
xTk −1 eTk −1 ⎤⎦ .
(Q2) The closed-loop system (6) satisfies the H ∞ performance constraint. i.e., under the zero-initial condition, the controlled output z k satisfies ∞
∑ E{ z k
2
∞
} < γ 2 ∑ E{ w k } 2
(8)
k 0= k 0 =
for all nonzero w k ,where γ > 0 is a prescribed scalar. 3. Robust H ∞ Controller Design Theorem 1: Suppose that both the controller gain matrix K and the observer gain matrix L are given. The closed-loop system (6) is exponentially mean-square stable if there exist positive definite matrices P1 , P2 , P3 and P4 satisfying
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XiuyingAuthor Li andname Shuli/Sun / Procedia Engineering 29 (2012) 4192 – 4197 Procedia Engineering 00 (2011) 000–000
P3 − P1 ⎡ ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ A + ΔA − λ 2B K 2 2 ⎢ ⎢ ΔA ⎢ ⎢ λ1λ2B 2K ⎢ λ3λ4LC2 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎣
* P4 − P2
A
* *
* *
* *
* *
* *
* *
0 0
* * − P3 0
* −P4
* *
* *
* *
* *
* *
λ22B 2K
−λ14B 2K
λ14B 2K
−P1−1
*
*
*
*
0
−λ14LC2 λ13λ2B 2K
0
−P2−1
*
*
*
0
0
− P1−1
*
*
0
0
0
0
− P2−1
*
λ12λ2B 2K
0
0
0
0
− P1−1
0
0
0
0
0
0
− λ42LC2
−λ1λ2B 2K
−λ13λ2B 2K
0
−λ33λ4LC2 −λ12λ2B 2K −λ32λ4LC2
0 0
* ⎤ * ⎥⎥ * ⎥ ⎥ * ⎥ * ⎥⎥ 0 . The system (6) is robustly exponentially mean-square stable and the H ∞ norm constraint (8) is achieved for all nonzero w k if there exist positive definite matrices P1 , P2 , P3 and P4 , a positive real scalar ε > 0 , and real matrices K and L satisfying (10) . P3 − P1 * * * * * * * ⎡ ⎢ 0 P − P * * * * * * 4 2 ⎢ ⎢ 0 0 −P3 * * * * * ⎢ 0 0 0 − P * * * * 4 ⎢ ⎢ 2 γ 0 0 0 0 − I * * * ⎢ 2 4 4 ⎢ P A − λ 2P B K λ λ λ − − * * P B K P B K P B K P B P 2 1 2 2 1 2 1 1 2 1 1 2 1 1 1 ⎢ 1 ⎢ 0 P2 A − λ42P2LC2 0 −λ34P2LC2 P2B1 − P2LD2 0 −P2 * ⎢ C1 0 0 0 D1 0 0 −I ⎢ ⎢ 0 0 0 0 −λ1λ2P1B2K −λ13λ2P1B2K λ13λ2P1B2K ⎢ λ1λ2P1B2K 3 ⎢ λ λ P LC 0 P LC 0 0 0 0 0 λ λ − 3 4 2 2 ⎢ 3 4 2 2 ⎢ 0 0 0 0 0 0 −λ12λ2P1B2K λ12λ2P1B2K ⎢ 2 ⎢ 0 0 0 0 0 0 0 −λ3 λ4P2LC2 ⎢ 0 0 0 0 0 H1T P1 H1T P2 HT2 ⎢ ⎢ 0 0 0 0 0 0 0 εE ⎣
* *
* *
* *
* *
* *
* *
* *
* *
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
−P1
*
*
*
0
−P2
*
*
0
0
−P1
*
0
0
0
−P2
0 0
0 0
0 0
0 0
* ⎤ * ⎥⎥ * * ⎥ ⎥ * * ⎥ * * ⎥⎥ * * ⎥ ⎥ * * ⎥ ⎥ 0 . The system (6) is exponentially mean-square stable, and the H ∞ norm constraint (8) is achieved for all nonzero w k if there exist positive definite matrices P11 ∈ R p× p ,
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Xiuying Liname and Shuli Sun / Procedia Engineering (2012) 4192 – 4197 Author / Procedia Engineering 00 (2011)29000–000
P22 ∈ R ( n − m )×( n − m ) , and P2 ∈ R n×n , and real matrices M ∈ R p×n and N ∈ R n×r , such that * * * * * * * ⎡ P3 − P1 ⎢ 0 P − P * * * * * * 4 2 ⎢ ⎢ 0 0 −P3 * * * * * ⎢ 0 0 0 −P4 * * * * ⎢ ⎢ 0 0 0 0 −γ 2I * * * ⎢ 2 4 4 ⎢ P A − λ 2B M λ2 B2M λ1 B2M −λ1 B2M P1B1 −P1 * * 2 2 ⎢ 1 ⎢ 0 P2 A − λ42NC2 0 −λ34NC2 P2B1 − ND2 0 −P2 * ⎢ C1 0 0 0 D1 0 0 −I ⎢ ⎢ −λ1λ2B2M −λ13λ2B2M λ13λ2B2M 0 0 0 0 ⎢ λ1λ2B2M 3 ⎢ λ λ NC 0 0 0 0 0 0 −λ3 λ4NC2 ⎢ 3 4 2 ⎢ 0 0 0 0 0 0 −λ12λ2B2M λ12λ2B2M ⎢ 2 ⎢ 0 0 0 0 0 0 0 −λ3 λ4NC2 ⎢ T T T 0 0 0 0 0 H P H P H ⎢ 1 1 1 2 2 ⎢ εE 0 0 0 0 0 0 0 ⎣
* * * *
* * * *
* * * *
* * * *
*
*
*
*
*
*
*
*
* *
* *
* *
* *
−P1
*
*
*
0
−P2
*
*
0
0
−P1
*
0
0
0
−P2
0 0
0 0
0 0
0 0
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ * * ⎥⎥ * * ⎥ ⎥ * * ⎥ ⎥ 0, P22 > 0, P2 > 0, M,N
4. Simulation Research In order to demonstrate the effectiveness of the proposed method, an uninterruptible power system (UPS) will be taken as our example. We consider the UPS with 1KVA. The discrete-time model (1) can be obtained with sampling time 10ms at half-load operating point as follows [9]: ⎡0.9226 −0.6330 0 ⎤ ⎡ 0.5⎤ ⎡1 ⎤ ⎡1 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A = ⎢ 1.0 0 0 ⎥ , B1 = ⎢ 0 ⎥ , B 2 = ⎢ 0 ⎥ , H1 = ⎢⎢0 ⎥⎥ , H 2 = 0.2 , E = [1 0 0] ⎢⎣ 0 ⎢⎣ 0.2 ⎥⎦ ⎢⎣ 0 ⎥⎦ ⎢⎣1 ⎥⎦ 1.0 0 ⎥⎦ C1 = [ 0.1 0 0] , C2 = [ 23.738 20.287 0] , D1 = 0.1 , D2 = 0.2
Our aim is to design an H ∞ controller (4) based on (3), such that the H ∞ performance index is minimized. Let α= β= 0.7 , i.e., the probability for a packet received on time is 0.3, one unit delay rate is 0.49, and the packet dropout rate is 0.21. We assume that the disturbance input is w k = 1 k 2 , and the initial conditions are x0 = [0 0 0]T , xˆ 0 = [0.2 0.2 0]T . By using the MATLAB LMI Toolbox to solve PROBLEM 1, we can obtain that γ min = 0.2279 , and K = [-0.0005 0.0011 -0.0011] , L = [0.0010 0.0020 0.0018]T . The simulation results of the state responses are given in Fig.1, from which we can see that our goal is achieved.
Xiuying Li and Shuli / Procedia Engineering (2012) 4192 – 4197 Author nameSun / Procedia Engineering 00 29 (2011) 000–000
The state response xk
0.3 x1 0.2
x2 x3
0.1 0 -0.1
0
5
10
15
20
25 k/step
30
35
40
45
50
Fig. 1 H ∞ control with γ min = 0.2279 under α= β= 0.7
5. Conclusion In this paper, an observer-based H ∞ control problem has been studied for NCSs with random packet dropouts and communication delays which can be described by a Bernoulli distributed stochastic variable. Both the sensor-to-observer channel and the controller-to-actuator channel are considered. The controller has been designed via an LMI approach to make the closed-loop networked system exponentially meansquare stability and achieve a desired H ∞ disturbance rejection level. Acknowledgements This work was supported by Natural Science Foundation of China under Grant No. NSFC-60874062. References [1] Wang ZD, Yang FW, Ho DWC, Liu XH. Robust H ∞ control for networked systems with random packet losses. IEEE Trans Syst Man and Cyber-Part B: Cybernetics 2007; 37:916-23. [2] Imer OC, Yuksel S, Basar T. Optimal control of LTI systems over unreliable communication links. Automatica 2006; 42:1429-39. [3] Yang FW, Wang ZD, Hung YS, Gani M. H ∞ control for networked systems with random communication delays. IEEE Trans Autom Control 2006; 51:511-8. [4] Lin C, Wang ZD, Yang FW. Observer-based networked control for continuous-time systems with random sensor delays. Automatica 2009;45:578-84. [5] Sun SL. Optimal estimators for systems with finite consecutive packet dropouts. IEEE Signal Processing Letters 2009;16:557-60. [6] Lu X, Xie LH, Zhang HS, Wang W. Robust Kalman filtering for discrete-time systems with measurement delay. IEEE Trans Circuits Syst II, Express Briefs 2007; 54:522-6. [7] Luca S. Optimal estimation in networked control systems subject to random delay and packet drop. IEEE Trans Autom Control 2008;53:1311-17. [8] Sahebsara M, Chen T, Shan SL. Optimal H 2 filtering with random sensor delay, multiple packet dropout and uncertain observations. Int J Control 2007;80:292-301. [9] Rong Y. The design of H ∞ iterative learning control and its application to UPS. M.S. thesis: Fuzhou University, China: Fuzhou; 2003.
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