TWO SCHEMES OF DATA DROPOUT COMPENSATION FOR LQG CONTROL OF NETWORKED CONTROL SYSTEMS

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1 Asian Journal of Control, Vol. 7, No., pp , January 05 Published online November 04 in Wiley Online Library (wileyonlinelibrary.com) DOI: 0.00/asjc.045 WO SCHEMES OF DAA DROPOU COMPENSAION FOR LQG CONROL OF NEWORKED CONROL SYSEMS Jinfeng Gao, Han Wu, and Minyue Fu ABSRAC his paper investigates the LQG control problem for networked control systems (NCSs) with packet losses, where the packet losses are considered to appear in both the sensor-to-controller channel and controller-to-actuator channel. Bernoulli random processes are used to describe the packet losses in the two channels. wo simple compensation schemes are explored for state estimation with missing measurements in which the input of the plant is set to zero if a packet is lost, and the holdinput strategy, in which the previous input is used with packet dropout. he optimal static controller gains and the critical loss probabilities for the two schemes are presented, and their performances are compared in terms of numerical simulations. he conclusion is that neither of the two schemes can be claimed to be superior to the other, as the stability regions of the two strategies are reversely complemented to each other whether for the scalar or vector example. Key Words: LQG, packet dropout, networked control systems, Kalman filter, optimal control. I. INRODUCION Feedback control systems wherein the control loops are closed through a real-time network are called networked control systems (NCSs) []. When sensors, actuators, and controllers are connected with information over a real-time network medium, data packet loss often occurs [], especially in a wireless NCS. he reasons for packet dropout are due to communication noise, interference, or congestion both from sensors to controllers (S/C) and from controllers to actuators (C/A). Many researchers in past decades have analyzed state estimator and filter design under lossy links without packet loss compensator in S/C [3 5]. here are several works that studied these problems with packet loss in both S/C and C/A [3,6,8 0,9], which are modeled by Bernoulli processes []. Nevertheless, Sinopoli et al. [3] doesn tconsider optimal control. he optimal controllers are presented in [6,8] using the state feedback method. Han et al. designed a piecewise state feedback controller for optimal H performance [7]. Zhao et al. [3] provides a design procedure for constructing a controller with the maximum possible H consensus performance region in multi-agent systems. Since the packet losses in the control loop often render the closed loop system unstable, Schenato uses an intermittent Kalman filter Manuscript received December 3, 03; revised May 5, 04; accepted July 3, 04. J. Gao (corresponding author, jfgao@zstu.edu.cn) and H. Wu are with the Institute of Automation, Zhejiang Sci-ech University, Hangzhou, 3008, China M. Fu is with School of Electrical Engineering and Computer Science, University of Newcastle, NSW, 308, Australia his work is supported by National Natural Science Foundation of China under Grant ( , 60377), Science and echnology Department of Zhejiang Province under Grant (04C308), Educational Commission of Zhejiang Province under Grant (Y069) and 5 alent Project of Zhejiang Sci-ech University. to handle packet losses to ensure the statistical convergence properties of the estimation error covariance [3]. Furthermore, Sahebsara et al. deal with these packet losses with H filtering to control the convergence of the estimation error covariance [7] leading to an optimal H performance [,3]. here are some other methods, for example, Zhao et al. [4] presents distributed finite-time tracking control for multi-agent systems under a time-invariant communication topology via an observer-based approach. Wang et al. [5] proposes a one-step prediction-based packet dropout compensation method, and the NCS is modeled as a discrete switched system with parametric uncertainties [6]. Overall, most works in the literature consider two different schemes for handling the problem of packet loss: one is zero-input strategy, where the actuator input to the plant is set to zero when the control packet from the controller to the actuator is lost [8,6,7] with no computational resources in actuators, as shown in Fig.. he other is hold-input strategy, where the latest control input stored in the actuator buffer is used when a packet loss occurs [,4,8], as shown in Fig.. However, there are few studies in the literature that simultaneously discuss the zero-input scheme and hold-input scheme except [6] and [], where Schenato only considers the LQ-like performance in [6]. And Zhang et al. present a generalized compensation scheme which involves the above two schemes to design the optimal linear filters for networked systems with communication constraints [], where controller design is not considered. Motivated by these considerations, the purpose of this work is to design the LQG controller for NCSs with packet losses compensation by using a state estimator. he packet dropouts are considered for the case with both S/C and C/A channels, which are

2 56 Asian Journal of Control, Vol. 7, No., pp , January 05 Fig.. Diagram of NCS with packet loss: zero-input strategy. Fig.. Diagram of NCS with packet loss: hold-input strategy. modeled as two independent Bernoulli processes. We will explore two simple compensation schemes: zero-input strategy, where the input to the plant is set to zero if a packet is lost, and hold-input strategy, where the previous stored value in the buffer is used if the packet fails to transmit. he optimal static gains and the critical loss probabilities are calculated using linear matrix inequalities (LMIs) for the two compensation schemes. Moreover, the performances of these two schemes are compared using numerical cases. Coincidentally, the stability regions of the two strategies relatively complement each other for both scalar feedback and vector feedback examples. Notations. hroughout this paper, R denotes the set of real numbers, R n denotes the n-dimensional Euclidean space, and A n m refers to the set of all n m real matrices. A represents the transpose of the matrix A, while A - denotes the inverse of A. For real symmetric matrices X and Y, the notation X Y (respectively, X > Y) means that the matrix X Y is positive semi-definite, (respectively, positive-definite). I is the identity matrix with appropriate dimensions. II. PROBLEM FORMULAION Consider the following discrete-time linear dynamic system x kþ ¼ Ax k þ Bu a k þ w k () u a k ¼ ν ku c k () y k ¼ γ k Cx k þ v k (3) where u a k is the control input, uc k is the controller output, (x 0, w k, v k ) are Gaussian, uncorrelated, white noises, with means ðx 0 ; 0; 0Þ and covariances (P 0, Q, R), respectively, and Q 0, R 0. w k is independent of w s for s t. (γ k, ν k ) are i.i.d Bernoulli random variables with Prob[γ k =0]=γ and Prob [ν k =0]=ν. We assume that the full state estimator^x kjk is available to a remote controller that adopts the linear feedback rule u c k ¼ L^x ( kjk ¼ L z^x kjk ; when zero-input strategy applied L h^x kjk ; when hold-input strategy applied where L is the controller gain matrix. he subscripts z and h in gains L z and L h indicate the zero-input and the hold-input strategy, respectively. he links between in both S/C and C/A are lossy, and the stochastic binary variables (γ k, ν k ) {0, } describe the packet dropouts in both S/C and C/A, respectively. We consider two control compensation strategies. In the zero-input strategy, the closed loop system is described by (4) x kþ ¼ Ax k þ Bν k u c k þ w k y k ¼ γ k Cx k þ v k (4)

3 J. Gao et al.: wo Schemes of Data Dropout Compensation for LQG Control of Networked Control Systems 57 In the case of the hold-input strategy shown in Fig., the closed loop dynamics system is expressed as follows x kþ ¼ Ax k þ B ν k u c k þ ð ν kþu a k þ wk (5) y k ¼ γ k Cx k þ v k Define the following information sets Γ k {y k, γ k, ν k }, where y k =(y k, y k,, y ), γ k =(γ k, γ k,, γ ), and ν k =(ν k, ν k,, ν ). he following cost function is considered J N u N x 0 ; P 0 ¼ E x N W Nx N þ N x k W kx k þ ν k u k U ku k ju N ; x 0 ; P 0 k¼0 where u N =(u N, u N,, u ). By applying the Kalman filter, the state estimate is given by ^x kjk Ε½x k jγ k Š e kjk x k ^x kjk h i P kjk Ε e kjk e kjk jγ k he following facts are required in the derivation of the estimator. Lemma. [8]. he following facts are true (a) E ðx k ^x k Þ^x k jγ k ¼ E ekjk^x k jγ k ¼ 0; (b) (c) E ^x k Sx kjγ k ¼ ^x k Sx k þ trace SP kjk ; S 0; EEgx ½ ½ ð kþ ÞjΓ kþ (6) (7) ŠjΓ k Š ¼ E½gx ð kþ ÞjΓ k Š; gðþ: III. ESIMAOR DESIGN According to system (4), equations for the optimal estimator are derived by using arguments similar to those used in Standard Kalman filtering, and it follows that ^x kþjk AE½x k jγ k Šþν k Bu c k (8) e kþjk x kþ ^x kþjk ¼ Ae kjk þ w k (9) h i P kþjk E e kþjk e kþjk jγ k ¼ AP kjk A þ Q (0) where the independence of the w k and Г k, and the requirement that u k is a deterministic function of Г k, are used. As y k +, γ k +, w k and Г k are independent, the correction step is given by ^x kþjkþ ¼ ^x kþjk þ γ kþ K kþ y kþ C^x kþjk () e kþjkþ ¼ I γ kþ K kþ C ekþjk γ kþ K kþ v kþ () P kþjkþ ¼ P kþjk γ kþ K kþ CP kþjk (3) K kþ ¼ P kþjk C CP kþjk C þ R (4) For hold-input strategy, we derive the equations for the optimal estimator using similar arguments to zero-input strategy. he innovation step of state is given by ^x kþjk ða þ ν k BLÞ^x kjk þ ð ν k ÞBu a k (5) hus, the innovation step of hold-input strategy is obtained by (5), (9), and (0). he correction step is the same as for the zero-input strategy () (4). And we get ^x kþjkþ ¼ ^x kþjk þ γ kþ K kþ y kþ C^x kþjk ¼ A ν k BL γ kþ K kþ CA ^xkjk þγ kþ K kþ CAx k þ ð ν k ÞBu a k þγ kþ K kþ v kþ þ γ kþ K kþ Cw k By using a modified Kalman filter formulation, it is easy to infer P kþjk ¼ AP kjk A þq γ k AP kjk C CP kjk C CPkjk þ R A (6) he error covariance matrices P k + k are the same through the two strategies. Note that (3) indicates that the N error covariance matrices P kjk are stochastic since they k¼0 depend on the sequence {γ k }. Moreover, as the matrix P k + k + is a nonlinear function of the previous covariance P k k, the accurate forecast of these matrices cannot be computed directly. Nevertheless, they can be bounded by computable deterministic quantities, from which we can derive the following lemma. Lemma 3.. he expected error covariance matrix E [P k k ] satisfied the following bounds [3] ep kjk E P kjk ^P kjk ; k 0 (7)

4 58 Asian Journal of Control, Vol. 7, No., pp , January 05 where the matrices ^P kjk and ep kjk can be computed as follows ^P kþjk ¼ A^P kjk A þq γa^p kjk C C^P kjk C þ R C^P kjk A (8) ^P kjk ¼ ^P kjk γ^p kjk C C^P kjk C þ R C^P kjk (9) ep kþjk ¼ ð γþaep kjk A þ Q (0) ep kjk ¼ ð γþep kjk () where the initial conditions are ^P 0j0 ¼ ep 0j0 ¼ P 0. Proof. he argument is based on the observation that the matrices P k + k and P k k are concave and monotonic functions of P k k. he proof is the same as [3] and is thus omitted. he above results can be summarized as follows. heorem 3.. Consider the system () (3) and the problem of minimizing the cost function (6) within the class of admissible policies u k = f(γ k ), where Г k is the information available under two strategies as shown in Figs and. hus the following results hold. (a) he optimal estimator, given by (8) (5), is independent of the control input u k. (b) he optimal estimator gain K k is time-varying and stochastic since it depends on the past observation loss sequence f g k i¼. γ i While the standard LQG optimal regulator always stabilizes the original system, in the case of observation losses, the stability can be lost if the arrival probabilities γ are below a certain threshold. his observation comes from the Modified Riccati Algebraic Equation (MARE), P k + = Π(P k, A, C, Q, R, γ), as described in (6). he results about the MARE are summarized in the following theorem. heorem 3.3. Assume that A; Q = Þ is controllable, (A, C) is detectable, and A is unstable. Consider the MARE as defined in (6), then the following results hold. (a) he MARE has a unique strictly positive definite solution P when γ > γ c, where γ c is the critical loss probability. (b) he critical probability γ c satisfies the following analytical bounds γ m γ c γ M ; γ m max i λ u i ðaþ ; γ M Π i λ u i ðaþ where λ u i ðaþ are the unstable eigenvalues of A. In particular, γ c = γ m if C is square and invertible, and γ c = γ M if C is rank one. he proof of this theorem can be found in [3]. he proof γ c = γ m when C is square and invertible can be found in [0], and the proof γ c = γ M if C is rank one in []. IV. OPIMAL CONROL Derivation of the optimal feedback control law and the corresponding value for the objective function will follow the dynamic programming approach based on the cost-togo iterative procedure. Define the optimal value function V k (x k ) as follows V N ðx N Þ E x N W Nx N jγ k ; V k ðx k Þ min uk E½x k W kx k þ ν k u k U ku k þv kþ ðx kþ ÞjΓ k Š; k ¼ N ; ; () Now make the following computations, which we use to derive the optimal LQG controller E x kþ Sx kþjγ k ¼ E x k A SAx k jγ k þ νu k B SBu k þνu k B SA^x kjk þ traceðsqþ h i h i E e kjk e kjkjγ k ¼ trace E e kjk e kjk jγ (3) k ¼ trace P kjk ; 0 where both the independence of ν k, w k, x k, and the zero mean property of w k are exploited. 4. Zero-input strategy Under zero-input strategy the following theorem holds. heorem 4.. he value function V k (x k )defined in () for the system dynamics of () (3) can be written as V k ðx k Þ ¼ E x k S kx k jγ k þ ck ; k ¼ N; ; 0 (4) where the matrix S k and the scalar c k can be computed recursively as follows S k ¼ A S kþ A þ W k νa S kþ BB S kþ B þ U k B S kþ A (5) c k ¼ trace A S kþ A þ W k S k þe½c kþ jγ k Š Pkjk þ trace ð Skþ QÞ (6)

5 J. Gao et al.: wo Schemes of Data Dropout Compensation for LQG Control of Networked Control Systems 59 with initial values S N = W N, c N = 0. Moreover, the optimal control input is given by u k ¼ B B S kþ B þ U k S kþ A^x kjk ¼ L Z^x kjk (7) Proof. he proof employs an induction argument. he claim is clearly true for k = N with the choice of parameters S N = W N, c N = 0. Suppose now that the claim is true for k +, i.e. V kþ ðx kþ Þ ¼ E x kþ S kþx kþ jγ kþ þ ckþ. he value function at time step k is the following V k ðx k Þ ¼ min uk E x k W kx k þ ν k u k U ku k þ V kþ ðx kþ ÞjΓ k ¼ min uk E x k W kx k þ ν k u k U ku k jγ k þe Ex kþ S kþx kþ þ c kþ jγ kþ jγk ¼ min uk E½x k W kx k þ ν k u k U ku k þ x kþ S kþx kþ þc kþ jγ k Š ¼ E x k W kx k þ x k A W kþ Ax k jγ k þ trace ð Skþ QÞ þe½c kþ jγ k Š þν min uk u k U ð k þ B S kþ BÞu k þ u k B S kþ A^x kjk (8) herefore, the claim given by (4) is also satisfied for time step k for all x k if and only if (5) and (6) are satisfied. heorem 4.. Consider the modified Riccati equation, which is defined in (5). Assuming A; W is controllable, (A, B) is detectable, and A is unstable, then the following hold. (a) he MARE has a unique strictly positive definite solution S if and only if ν > ν c, where ν c is the critical loss probability. (b) he critical probability ν c satisfies the following analytical bounds ν m ν c ν M ; ν m max i λ u i ðaþ ; ν M ¼ Π i λ u i ðaþ where λ u i ðaþare the unstable eigenvalues of A. In particular, ν c = ν m if B is invertible, andν c = ν M if B is rank one. (c) he critical probability can be numerically computed via the solution for the following quasi-convex LMIs optimization problem ν c ¼ argmin ν Ψ ν ðy; ZÞ > 0; 0 Y I: 3 pffiffi pffiffi pffiffiffiffiffiffiffiffiffiffi Y Y ν ZU ν YA þ ZB νya Y W Ψ ν ðy; ZÞ ¼ pffiffi ν U ZH 0 I 0 0 pffiffi 6 ν AY þ BZ 4 ð Þ 0 0 Y pffiffiffiffiffiffiffiffiffiffi νay Y where we use Lemma.(c) to get the third equality, and (3) to obtain the last equality. he value function is a quadratic function of the input, therefore, the minimizer can be simply obtained by solving V k / u k = 0, which gives (7). he optimal feedback controller is, thus, a simple linear function of the estimated state. If we substitute the minimizer back into () and use Lemma.(b) we get V k ðx k Þ ¼ E x k W kx k þ x k A W kþ Ax k νx k A S kþ BU ð k þ B S kþ BÞ B S kþ Ax k jγ k þtraceðs kþ QÞþE½c kþ jγ k Š þνtrace A S kþ BU ð k þ B S kþ BÞ B S kþ A^x kjk (d) If ν > ν c, then lim k S k = S for all initial conditions S 0 0,where S k + = Π(S k, A, B, W, U, ν). Proof of the previous theorems can be found in [8]. Using the above results, we can prove the following theorem for the infinite horizon optimal LQG under zeroinput strategy. heorem 4.3. Consider the same system in the previous theorem, let W N = W k = W and U k = U. Moreover, let (A, B) and (A, Q / ) be controllable, and let (A, C) and (A, W / ) be observable. Assume that ν > ν c and γ > γ c, where ν c and γ c are defined in heorem 4. and heorem 3.3, respectively. hen the following results are gained.

6 60 Asian Journal of Control, Vol. 7, No., pp , January 05 (a) he infinite horizon optimal controller gain is constant lim L k ¼ L ¼ B B S B þ U S A: (9) k (b) he matrices are the positive definite solutions of the following equation S ¼ A S A þ W νa S BB B S B þ U S A: Proof. (a) Since ν > ν c, from heorem 4.(d) it follows that lim k S k = S. herefore, (9) follows from (7). (b) (5) can be written in the terms of MARE as S k + = Π(S k, A, B, W, U, ν), thus, as ν > ν c from heorem 4.(d) it follows that lim k S k + =lim k S k = S. 4. Hold-input strategy Under hold-input strategy the following lemma holds true: Lemma 4.4. he value function V k (x k )defined in () for the system dynamics of () (3) can be written as V k ðx k Þ ¼ E x k S kx k jγ k þ ck ; k ¼ N; ; 0 (30) where the matrix S k and the scalar c k can be computed recursively as follows S k ¼ A S kþ A þ W k ð νþa S kþ BB B S kþ B þ U k S kþ A c k ¼ trace A S kþ A þ W k S k Pkjk þtraceðs kþ QÞþE½c kþ jγ k Š þν u a k Uk þ B S kþ B u a k (3) (3) with initial values S N = W N, c N = 0. Furthermore the optimal control input is given by u k ¼ B B S kþ B þ U k S kþ A^x kjk ¼ L h^x kjk (33) Proof. he proof uses the familiar way in (8),therefore it is omitted. he value function at time step k is the following V k ðx k Þ ¼ min u c E x k k W kx k þ ν k u a k U ku a k þ V kþðx kþ ÞjΓ k ¼ E½x k W kx k þ x k A S kþ Ax k þð ν k Þ u a kð U k þ B S kþ BÞu a k jγ kš þtraceðs kþ QÞ þe½c kþ jγ k þu c k B S kþ A^x kjk Þ Šþð νþmin u c ðuc k k ðu k þ B S kþ B Þu c k (34) where u c k is independent of u a k and x k. By solving V k = u c k ¼ 0, we get (3) and the same controller gain L h with that of zero-input strategy. If we substitute the minimizer back into (33) we get V k ðx k Þ ¼ E x k ðw k þ A S kþ A ð νþa S kþ BðU k þb S kþ BÞB S kþ AÞx k ÞþtraceðS kþ QÞ þe½c kþ jγ k Šþtrace W k þ A S kþ A S k þνu a kð U k þ B S kþ BÞu a k Pkjk herefore, the claim given by (30) is satisfied for time step k for all x k when (3) and (3) are satisfied. heorem 4.5. Consider the modified Riccati equation defined in (3). Assuming A; W = Þ is controllable, (A, B) is detectable, and A is unstable, then the following hold. (a) he MARE has a unique strictly positive definite solution S if and only if ν < ν c, where ν c is the critical loss probability. (b) he critical probability ν c satisfies the following analytical bounds ν m ν c ν M ; ν m Π i λ u i ðaþ ; ν M ¼ max i λ u i ðaþ where λ u i ðaþare the unstable eigenvalues of A. In particular, ν c = ν M if B is invertible, and ν c = ν m if B is rank one. (c) he critical probability can be numerically computed via the solution of the following quasi-convex LMIs optimization problem ν c ¼ argmax ν Ψ ν ðy; ZÞ > 0; 0 Y I: 3 pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi Y Y νzu ν YA þ ZB pffiffi ν YA Y W Ψ ν ðy; ZÞ ¼ pffiffiffiffiffiffiffiffiffiffi νuz 0 I 0 0 pffiffiffiffiffiffiffiffiffiffi 6 4 νðay þ BZ Þ 0 0 Y pffiffi ν AY Y

7 J. Gao et al.: wo Schemes of Data Dropout Compensation for LQG Control of Networked Control Systems 6 (d) If ν < ν c, then lim k S k = S for all initial conditions S 0 0,where S k + = Π(S k, A, B, W, U, ν). he proof is similar to heorem 4.. From the results above, we can prove the following theorem for the infinite horizon optimal LQG under holdinput strategy. heorem 4.6. Consider the same system in the previous theorem, let W N = W k = W and U k = U. Moreover, (A, B) and (A, Q / ) are controllable, (A, C) and (A, W / ) are observable. Assume that ν < ν c and γ > γ c, where ν c and γ c are defined in heorem 4. and heorem 3.3, respectively. hen we have the following results. (a) he infinite horizon optimal controller gain is constant lim L k ¼ L ¼ B B S B þ U S A: k (b) he matrices are the positive definite solutions of the following equation S ¼ A S A þ W ð νþa S BB B S B þ U S A he proof is similar to heorem 4.3, thus it is omitted. Remark. From heorem 4. and Lemma 4.4, the matrices S in the different compensator strategy are different. Also the critical loss probability bounds ν c are not the same. hese results will be further illustrated in the following numerical examples in Section V. V. NUMERICAL EXAMPLES In this section, we present some cases to compare the performance under zero-input and hold-input control architectures. Example. Consider a scalar unstable system () (3) with parameters A =., B = C =, W = U = x 0 =, and no process and measurement noise, i.e. R = Q =0. From heorem 3.3, we get γ c = , so we assume γ = 0.4 > γ c to ensure the closed loop system is stable only if the critical loss probability ν c satisfies the conditions of the two strategies, respectively. After calculating from the above theorems, the critical loss probability ν cz = for the zero-input strategy and ν ch = 0.69 for the hold-input strategy. Suppose ν = 0.4, which meets the stability conditions for the two strategies simultaneously, then in zeroinput strategy, the matrix S =.386 and the optimal controller gain L = However, the matrix S =.53 and the optimal controller gain L = 0.74 in hold-input strategy. And the figures of relations between loss probability ν c and weight U, and performance S are presented in Figs 3 and 4, respectively. Remark. Regions in Fig. 3 indicate the best performing strategy in the space (ν, U) for hold-input and zero-input control case. he stability region for zero-input is ν (ν cz, ], where ν cz = 0.306, for hold-input ν [0, ν ch ), ν ch = Compared with the result in Schenato s paper [6], the difference is that U [0, 0] if ν cz (0.306, ] when the networked system stabilizes using zero-input strategy, and also U [0, 0] if ν cz [0, 0.69) using holdinput strategy. Stability regions for the two strategies are block diagrams; they have a common stability area (0.306, 0.69). While in [6], the stability packet loss probability regions are the same as [0, 0.694), whether in the hold-input or in the zero-input strategy and the regions are divided into the form of a curve according of U. Remark 3. Fig. 4 shows a plot of the transition to instability in the scalar case, the dashed-dotted line shows the U Zero-input Hold-input Fig. 3. Regions of loss probability ν c and weight U. S 0 cz ch c Zero-input Hold-input Fig. 4. ransition to instability in the scalar case.

8 6 Asian Journal of Control, Vol. 7, No., pp , January 05 asymptotic value of the zero-input strategy, the solid line for the asymptotic value of the hold-input strategy, and the dashed line shows the asymptote. Compared with Sinopoli s result [4], in which the transition curve is concave, in this paper it is convex. Example. Consider another unstable vector system () (3) with parametersa ¼ :5 0,B ¼,W ¼ 0, : 0 U =.5,C ¼ ½ Š, R =.5,Q =0*I. We get 0.36 γ c 0.47 calculated by heorem 3.3, let γ = 0.5 > γ c to guarantee the stability of the closed loop system when the critical loss probability ν c satisfies the conditions of the two control architectures, respectively. In zero-input strategy, the calculations are ν m = 0.36, ν M = 0.48, and ν cz = 0.47 by heorem 4., but 0.58 ν c 0.64 and ν ch = 0.58 from heorem 4.5 for hold-input strategy. o meet the diverse stability conditions, we assume ν = 0.5; coincidentally, the optimal controller gain is equal, L ¼ ½0:748 0:383 Š, and the matrix :058 0:3505 S ¼. 0:3505 :6465 We can intuitively compare the optimality regions from the above examples. Interestingly, the sum of the critical packet loss probabilities of the two strategies is coincidentally. For small packet loss probability rates ν if ν ν ch the hold-input strategy provides a better performance than the zero-input one, but for packet loss probability ν cz ν, zero-input strategy has a better expression. However, for ν ch ν ν cz the two strategies will cause the system to create overshooting or oscillations, which means it is not possible to judge which is superior by packet loss compensator using the state estimation. VI. CONCLUSION his paper considers LQG controller design based on zero-input strategy and hold-input strategy for NCSs where the control packets are subject to loss. he packet dropouts are modeled by Bernoulli processes, which consider both S/C and C/A channels. he packet losses compensator uses the state estimator for NCSs. We study the zero-input strategy in which the input of the actuator is set to zero if a packet is lost and the hold-input strategy in which the previous value is used with packet dropout. he optimal static gains and the critical loss probabilities for the two schemes are obtained, and their performances on numerical cases are compared. Neither of the two schemes can be claimed superior to the other using the method in this paper, while the stability regions of the two strategies are relatively complementary to each other for either a simple scalar system or vector one. Nevertheless, the algorithms in this paper can be used to compute which scheme performs better once the packet loss probability and systems parameters are known. REFERENCES. Zhang, W., M. S. Branicky, and S. M. Phillips, Stability of networked control systems, IEEE Control. Syst. Mag., Vol., No., pp (00).. You, K. and L. Xie, Survey of recent progress in networked control systems, Acta Autom. Sin., Vol. 39, No., pp. 0 7 (03). 3. Sinopoli, B., L. Schenato, M. Franceschetti et al. Kalman filtering with intermittent observations, IEEE rans. Autom. Control, Vol. 49, No. 9, pp (004). 4. Nilsson, J., Real-time control systems with delays, Lund Institute of echnology, Sweden (998). 5. Schenato, L., Optimal estimation in networked control systems subject to random delay and packet drop, IEEE rans. Autom. Control, Vol.53,No.5, pp (008). 6. Schenato, L., o zero or to hold control inputs with lossy links? IEEE rans. Autom. Control, Vol. 54, No. 5, pp (009). 7. Han, F., G. Feng, Y. Wang et al. A novel dropout compensation scheme for control of networked S fuzzy dynamic systems, Fuzzy Sets Syst., Vol.35,pp.44 6 (04). 8. Schenato, L., B. Sinopoli, M. Franceschetti et al. Foundations of control and estimation over lossy networks, Proc. IEEE, Vol. 95, No., pp (007). 9. Sahebsara, M.,. Chen, and S. L. Shah, Optimal H filtering in networked control systems with multiple packet dropouts, Syst. Control Lett., Vol. 57, No. 9, pp (008). 0. Qiu, L., Q. Luo, F. Gong et al. Stability and stabilization of networked control systems with random time delays and packet dropouts, J. Franklin Inst.-Eng. App. Math., Vol. 350, No. 7, pp (03).. Niu, Y.,. Jia, X. Wang et al. Output-feedback control design for NCSs subject to quantization and dropout, Inf. Sci., Vol. 79, No., pp (009).. Yang, F. and Q. Han, H control for networked systems with multiple packet dropouts, Inf. Sci., Vol. 5, pp (03). 3. Jiang, S. and H. Fang, H static output feedback control for nonlinear networked control systems with time delays and packet dropouts, ISA rans., Vol. 5, No., pp. 5 (03). 4. Zhao, Y., Z. Duan, G. Wen et al. Distributed finitetime tracking control for multi-agent systems: An

9 J. Gao et al.: wo Schemes of Data Dropout Compensation for LQG Control of Networked Control Systems 63 observer-based approach, Syst. Control Lett., Vol. 6, No., pp. 8 (03). 5. Wang, Y., Q. Han, and X. Yu, One step predictionbased packet dropout compensation for networked control systems, American Control Conference, San Francisco, CA, USA, pp (0). 6. Wang,Y.,Q.Chen,W.Fanet al. Guaranteed cost control of networked control systems with data-packet dropout, Control heory Appl., Vol. 4, No., pp (007). 7. Imer, O. C., S. Yuksel, and. Basar, Optimal control of LI systems over unreliable communication links, Automatica, Vol. 4, No. 9, pp (006). 8. Mo, H. and C. N. Hadjicostis, Feedback control over packet dropping network links, Mediterranean Conference on Control and Automation, Athens Greece, pp. 6 (007). 9. Ling, Q. and M. D. Lemmon, Optimal dropout compensation in networked control systems, IEEE Conference on Decision and Control, Maui, Hawaii USA, pp (003). 0. Katayama,., On the matrix Riccati equation for linear systems with random gain, IEEE rans. Autom. Control, Vol., No. 5, pp (976).. Elia, N., Remote stabilization over fading channels, Syst. Control Lett., Vol. 54, No. 3, pp (005).. Zhang,W.,L.Yu,and G.Feng, Optimal linear estimation for networked systems with communication constraints, Automatica, Vol. 47, No. 9, pp (0). 3. Zhao, Y., Z. Duan, G. Wen et al. Distributed H consensus of multi-agent systems: A performance region-based approach, Int. J. Control., Vol. 85, No. 3, pp (0). Jinfeng Gao received her bachelor s degree in Automation from Hebei Institute of Science and echnology in 000, master degree and Ph.D. degree in Control Science and Engineering from Zhejiang University of echnology and Zhejiang University in 003 and 008, respectively. Now she is an associate professor at Zhejiang Sci-ech University. Her main research interests include time-delay systems, networked control, and multi-agent systems. Han Wu received her bachelor s degree in Automation from Henan Polytechnic University in 0 and master degree in Control heory and Control Engineering from Zhejiang Sci-ech University in 04. Now she is a General Manager Assistant at Zhejiang Hangzhou Bolian Zhixin echnology Co.,LD of China. Her main research interests include descriptor system and networked control. Minyue Fu received his Bachelor s Degree in Electrical Engineering from the University of Science and echnology of China, Hefei, China, in 98, and M.S. and Ph.D. degrees in Electrical Engineering from the University of Wisconsin-Madison in 983 and 987, respectively. From 987 to 989, he served as an Assistant Professor in the Department of Electrical and Computer Engineering, Wayne State University, Detroit, Michigan. He joined the Department of Electrical and Computer Engineering, the University of Newcastle, Australia, in 989 and was promoted to a Chair Professor in Electrical Engineering in 00. He has served as the Head of Department for Electrical and Computer Engineering and Head of School of Electrical Engineering and Computer Science over a period of 7 years. In addition, he was a Visiting Associate Professor at University of Iowa in , a Visiting Professor at Nanyang echnological University, Singapore, 00, and Visiting Professor at okyo University in 003. He has held a Chang Jiang Visiting Professorship at Shandong University, a visiting Professorship at South China University of echnology, and a Qian-ren Professorship at Zhejiang University in China. He was elected to a Fellow of IEEE in late 003. His current research projects include networked control systems, multi-agent systems, smart electricity networks and super-precision positioning control systems. He has been an Associate Editor for the IEEE ransactions on Automatic Control, IEEE ransactions on Signal Processing, Automatica, and Journal of Optimization and Engineering.