Feedback Control over Packet Dropping Network Links

Transcription

1 Control & Automation, July 7-9, 007, Athens - Greece T36-00 Feedback Control over Packet Dropping Network Links Haitao Mo and Christoforos N. Hadjicostis Coordinated Science Laboratory and Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign, Urbana, IL, USA Abstract This paper studies stabilization schemes for a discrete-time control systems in which the feedback loop includes a network link that may suffer packet drops. We model the packet dropping network link as an erasure channel and study state/output feedback stabilization schemes for linear systems of arbitrary dimension. We use the mean square deviation of the system state from zero (MSDZ) as our measure of performance and discuss how to obtain the gain matrix that asymptotically minimizes this quantity. To evaluate the potential of this control scheme, we then focus on the scalar case and compare it against an alternative control scheme that first uses Kalman filtering (with intermittent observations) to estimate the system state and then applies state feedback based on this state estimate. We provide both analytical and empirical comparisons and conclude that, under certain conditions, the two control schemes can have comparable performance; however, the proposed (state or output) feedback strategy is considerably simpler to implement. I. INTRODUCTION Networked control systems (NCSs) [1, [, i.e., closedloop feedback control systems that rely on an external, separate communication network to transmit measurement information (from sensors to controllers) and command signals (from controllers to actuators) have been receiving increasing attention over the last few years due to the broad capabilities and ubiquity of modern networks. In NCSs, packet drops and packet delays are important factors because they can greatly influence the behavior of the underlying NCS. Many researchers have studied the impact of packet drops on the performance of a control scheme; some of them (e.g., [3, [4, [5, [6, [7) investigate schemes in which straight state/measurement feedback (that is subject to packet drops and/or noise) is directly applied; others (e.g., [8, [10, [11, [1) study schemes that first perform state estimation (based on the packetdropped noise-impaired measurement) and then apply state feedback based on the estimated state. This paper focuses on a scheme that uses direct (packet-dropping and/or noise-impaired) state/measurement feedback and extends a previously proposed stabilization scheme for scalar discrete-time (DT) LTI dynamic systems in [3 to higher-dimensional systems. To analyze a given higherdimensional NCS, we use the mean square deviation of the system state from zero (MSDZ) as our measure of performance. More specifically, we consider the system to be stable if its MSDZ (i.e., the trace of the covariance matrix of the state of the closed loop NCS) remains finite as time goes to infinity. We compare the performance of the proposed stabilization scheme against an alternative (but computationally involved) control strategy which first uses Kalman filtering (with intermittent observations) to estimate the system state and then applies state feedback based on that estimated state. For the case of a scalar system, we show analytically and empirically that our proposed control scheme achieves an MSDZ which, under certain conditions, is comparable to the one achieved by this more involved control strategy. Therefore, our proposed scheme can be viewed as a suboptimal (and less computationally intensive) alternative to the scheme that uses Kalman filtering. II. PROBLEM DESCRIPTION A networked control system (NCS) typically consists of a nominal system (plant) to be controlled, a controller, and a network that transmits measurements (from the sensors to the controller) and control signals (from the controller to the actuators in the plant). In this paper, we assume that the network is not a dedicated network (e.g., the Internet) and make the following simplifying assumptions: (i) The unstable nominal plant (system) to be stabilized via feedback is a discrete-time (DT) linear time-invariant (LTI) system. (ii) We ignore the network link from the controller to the actuators and model the successful arrival (i.e., the arrival within one time step) of sensor measurements (which are sent through the network link to the controller) as a Bernoulli random process: at each discrete time step, all measurements are stored in a single packet which is sent over the network link and can be lost with probability p; transmissions (and thus packet losses) of different packets are assumed to be independent. (iii) We consider two kinds of noise: driving noise, which is added to the system input, and measurement noise, which is added to the system output. Both noises are modeled by zero-mean uncorrelated white random processes. The system (plant) we consider can be described as +1 =A + BU k, Y k =C + M k, where R n is the system state, U k R m is the system input, and Y k R l is the system output corrupted by (quantization or other) noise M k (see Fig. 1). At each discrete time step, a quantized measurement Y k of the system output is obtained and then sent over the network so that the quantized measurement vector that the controller receives can be described as Ŷ k = ε k Y k, (1) where ε k is an i.i.d. Bernoulli random process that captures the successful arrival of a packet through the network. More specifically, at each time step k, ε k is 1

2 Control & Automation, July 7-9, 007, Athens - Greece T36-00 N k U k Xk+1 = A + BU k CXk M k p Ω k Output Feedback Controller Y k cy k Packet dropped with Prob. p Packet Dropping Network Link Fig.. 1-p State 1 State p Success 1-p I.i.d. Bernoulli packet drops represented by a Markov model. Drop Fig. 1. DT LTI network control system with feedback controller. with probability 1 p (this means that the transmission of the data packet is successful) or 0 with probability p (this means that the transmission of the data packet fails). 1 After the packet gets through the network, the output feedback Ω k is calculated as Ω k = KŶk, () where K is the feedback gain matrix to be determined. The output feedback plus driving noise N k form the input signal U k = KŶk + N k which is fed back to the system. We assume that the driving noise N k and the quantization (or measurement) noise M k are zero-mean uncorrelated white processes with covariance matrices Q N and Q M of appropriate dimensions. The notation that is used for the remainder of the paper: (i) A B denotes the Kronecker product; (ii) ρ(a) is the spectral radius of a square matrix A; (iii) tr(a) is the trace of matrix A; (iv)cs(a) is the vector formed by stacking the columns of matrix A into a single column vector (i.e., cs(a) =[a T 1 a T... a T n T where a 1,..., a n are the columns of matrix A). III. STABILITY AND OPTIMAL FEEDBACK CONTROL Lemma 1: Let N k and M k be zero-mean uncorrelated white noise processes with covariance matrices Q N and Q M of appropriate dimensions and let X 0 be a zeromean random vector (with covariance Σ X0 ) independent from N k and M k. Define H = pa A +(1 p)(a + BKC) (A + BKC) where K is the output feedback gain matrix. Under the feedback control scheme described in the previous section, the state vector is a zero-mean random vector with covariance Σ Xk given recursively by Σ Xk =(1 p)(a+bkc)σ Xk 1 (A+BKC) T +BQ N B T +(1 p)bkq M K T B T +paσ Xk 1 A T, (3) and MSDZ Xk = tr(σ Xk ) given by MSDZ Xk =cs(i n) T {H k cs(σ X0 )+[ P k 1 i=0 Hi (B B)cs(Q N ) +(1 p)[ P k 1 i=0 Hi (BK BK)cs(Q M ) }. (4) Proof: The proof is omitted due to space limitations. Note that analysis similar to Lemma 1 can also be found in [5. Our goal is to make the MSDZ of the system 1 Technically, p captures the probability that the packet with the measurements is not delivered by the network within the time that elapses between two consecutive time steps. state finite as k goes to infinity for arbitrary (finite-valued) Σ X0, Q N and Q M. Clearly, if K can be chosen so as ρ(h) < 1 then this goal is achieved. The necessity of ρ(h) < 1 is not as clear because cs(σ X0 ) does not necessarily span the whole space (although necessity is claimed in [5, no proof is provided). Lemma below uses results from [13 to establish the necessity of ρ(h) < 1. Lemma : ρ(h) < 1 is a necessary and sufficient condition to make the MSDZ of the system state finite as k goes to infinity for arbitrary (finite-valued) Σ X0, Q N and Q M. Proof: The sufficiency of ρ(h) < 1 follows from Eqn. (4): if ρ(h) < 1, lim k H k =0and i=0 Hi = (I H) 1. To prove necessity we use the results of [13 which analyze the more general case of DT Markovian jump linear systems. We can model i.i.d. Bernoulli packet drops by the Markov Chain shown in Fig.. Let ( ) (1 p)[a L = A (1 p)[a A, p[a A p[a A where A = A + BKC. If we use Proposition 4 of [13, we establish that ρ(l) < 1 is a necessary condition to make the system asymptotically second moment stable for arbitrary (finite-valued) Σ X0, Q N and Q M (this is equivalent to having a finite MSDZ of the system state as k goes to infinity). We now show that ρ(l) < 1 ρ(h) < 1. Letλ and t =[t T 1 t T T be an eigenvalue and the corresponding eigenvector of L, where t 1 and t are both of dimension n 1. FromLt = λt, wehave (1 p)[a A (t 1 + t )=λt 1, (5) p[a A(t 1 + t )=λt, (6) The addition of the left and right parts of Eqns. (5) and (6) leads to H(t 1 +t )=λ(t 1 +t ) which means eigenvalues of L are also eigenvalues of H if t 1 +t 0.Ift 1 +t =0, from Eqns. (5) and (6), we know λ = 0 and it is an eigenvalue of L but may or may not be an eigenvalue of H. Suppose now that there is some other eigenvalue λ 0of H and we have Ht = λt where t is the eigenvector to λ. Since t and λ 0 are known, we can plug them into Eqns. (5) and (6) and solve for t 1 = (1 p)[a A t λ and t = p[a At λ. Since t = t 1 + t 0, and since Lt = λt from Eqns. (5) and (6), we see that λ is an eigenvalue of L. In conclusion, all nonzero eigenvalues of H are eigenvalues of L. Therefore, ρ(l) < 1 ρ(h) < 1 and the proof is complete. Next, we discuss how to choose K to minimize the MSDZ of as k goes to infinity. Lemma 3: The quantity MSDZ X (the limit of MSDZ Xk as k goes to infinity) is finite (given by

3 Control & Automation, July 7-9, 007, Athens - Greece T36-00 MSDZ X = cs(i n ) T {[(I H) 1 (B B)cs(Q N )+ (1 p)[(i H) 1 (BK BK)cs(Q M )}) and is locally minimal as a function of the feedback gain matrix K only if K satisfies the following three conditions: (i) K is such that ρ(h) < 1. (ii) K is a solution to the following equation array (composed of ml equations) where K ij is the (ith,jth) element of matrix K R m l : (MSDZ X ) =0, 1 i m, 1 j l. (7) K ij We refer to these equations as the minimizing equations for the remainder of this paper. (iii) The symmetric matrix = MSDZ X K MSDZ X K ml K MSDZ X K 11 K ml. MSDZ X K ml is positive definite. Proof: Since MSDZ X is finite, it means ρ(h) < 1 from Lemma (). Also MSDZ X being locally minimal implies conditions (ii) and (iii) for K must be satisfied. At this point the proof is complete. Lemma 3 provides us with a method to find the feedback gain K that results in the global minimal MSDZ X for direct state/measurement feedback strategy (although for measurement feedback strategy, such global minimizing K may not always exist (see discussion later)). We see that we can find the global minimizing K by: (i) solving equation array (7) (possibly getting several solutions for K), (ii) checking whether > 0 and ρ(h) < 1, and (iii) if multiple local minimizing K exist, picking the one which results in the minimal MSDZ X. We would like to point out here that the first step can be quite complex, particularly when the dimension of the system is high. Note, however, that the computation for obtaining the optimal feedback gain is done completely offline and no computation needs to be done online. Therefore, this strategy might be amenable for real-time control applications. Also note that, for some cases, the direct measurement feedback gain K may not be obtainable because of the limitations in the degrees of freedom posed on K (see [9). IV. KALMAN ESTIMATION FOR STATE FEEDBACK To gain an understanding of the performance of our proposed strategy, we focus on a scalar system and analyze an alternative feedback control strategy that uses Kalman filtering (under intermittent observations) to estimate the system state and then implements state feedback using this estimated state. We compare the two strategies analytically and then use simulations to verify our results. In the following discussion, we quickly review how to estimate the system state using Kalman filtering under intermittent observations; more details can be found in [10, [14 and [15. We first introduce some notation: k and +1 k are respectively the Linear Minimum Mean Square Error (LMMSE) estimates of the system N k U k +1 = A + BU k CXk M k Ω k State Feedback Controller Fig. 3. b k Kalman Filter with Intermittent Observation Y k Packet dropped with prob. p Packet Dropping Network Link Feedback control scheme using Kalman filtering. state and +1 given Y k and ξ k, defined respectively as the output sequence Y k =[Y 0,..., Y k T and the packet dropping sequence ξ k =[ε 0,..., ε k T. The associated error covariances for k and +1 k are denoted by P k k and P k+1 k, i.e., P k k E[( k )( k ) T Y k,ξ k, (8) P k+1 k E[(+1 +1 k )(+1 +1 k ) T Y k,ξ k. Note that the definitions of k and +1 k are different from those in [10 since {X 0, N 0, N 1,..., M 0, M 1,...} are not necessarily Gaussian here; however, in the sense of LMMSE, the Kalman filtering estimation used here is equivalent to that in [10. The feedback control system using Kalman filtering is shown in Fig. 3. For convenience of our analysis, we write the plant state evolution as +1 =A + BΩ k + BN k. (9) Unlike the feedback strategy discussed in the previous section, the feedback Ω k of the control system in Fig. 3 emulates state feedback by using the estimated system state, i.e. Ω k = K k, (10) where K is a (new) feedback gain matrix to be determined. The system state is estimated as follows (see [10): +1 k =A k + BΩ k, (11) P k+1 k =AP k k A T + BQ N B T, (1) G k+1 =P k+1 k C T (CP k+1 k C T + Q M ) 1, (13) +1 k+1 = +1 k + ε k+1 G k+1 (Y k+1 C +1 k ), P k+1 k+1 =P k+1 k ε k+1 G k+1 CP k+1 k, (14) X 0 1 =E[X 0,P 0 1 =Σ X0. (15) The separation principle still holds in the scenario in Fig. 3: the covariance matrix P k k is independent of the feedback control input Ω k = K k as seen from Eqns. (11) (14) (the feedback control input Ω k only affects k but not P k k ). Thus, the feedback gain K can be chosen separately from the state estimator (to minimize the MSDZ X of the feedback system in Fig. 3) without affecting the optimality of state estimation. Note that our discussion so far is applicable to both scalar and higher-dimensional systems. We now focus on the case of scalar systems and, for clarity of presentation, we replace A, B, C, Q N, Q M, Σ Xk with (scalars) α, β, γ, σn, σ M, σ, keeping the rest of the notation

4 Control & Automation, July 7-9, 007, Athens - Greece T36-00 unchanged. Note that the MSDZ of the system state is now the variance of the system state σ. Without loss of generality, we assume γ =1in our analysis. Lemma 4: Consider the setup of Fig. 3 with an underlying scalar system as described above. Assume that N k and M k are zero-mean uncorrelated white noise processes and X 0 is a zero mean random variable (independent from N k, M k ) with variance σ X 0. Then, the state is a zeromean random variable with variance that can be obtained recursively as σx k =(α + βk) σx k 1 + β σn+ (α (α + βk) )P k 1 k 1, (16) where P k 1 k 1 = E[P k 1 k 1 with the expectation taken over all possible sequences ξ k. Proof: It is easy to show that E[ =0since N k, M k and X 0 are zero-mean. From the properties of Linear Minimum Mean Square Error (LMMSE) estimators, we easily obtain E[( k ) k Y k,ξ k = 0 which implies E[Xk k b Y k,ξ k =E[ Xk k b Y k,ξ k =E[ X b k k Yk,ξ k. (17) We can then use the definition of P k k and the fact that E[ Y k,ξ k =E[ k Y k,ξ k =0to arrive at E[ X k k Y k,ξ k =E[X k Y k,ξ k P k k. (18) If we take the expectation of the equations above over all possible output sequences Y k and all packet dropping sequences ξ k, we get E[ Xk k =E[ k =E[ k, (19) E[ X k k =E[ P k k, (0) where P k k = E[P k k is actually also the expectation of P k k taken only over all possible sequences ξ k (since from Eqns. (1) (14), we know that P k k does not depend on the output sequences Y k ). From Fig. 3, we have +1 = α + βn k + βk k and, by applying Eqns. (19) and (0), and utilizing the fact that N k and M k are white and uncorrelated, we arrive at E[Xk+1 =α E[Xk+β σn +(βk) E[ k + αβke[ Xk k +αβke[ k =(α + βk) E[Xk+β σn +(α (α + βk) )P k k. At this point, the proof is complete. Lemma 5: Consider the setup of Fig. 3 for a scalar system as described above and let P k k = E[P k k denote the expected variance of the estimation error at time k where the expectation is taken over all possible sequences ξ k. Under the same assumptions as in Lemma 4, P k k is finite (bounded) as k goes to infinity if and only if pα < 1. (1) Furthermore, if this condition is satisfied, a lower bound of P k k as k goes to infinity is given by P e 1 pβ 4 σn 4 = + σ M σ N β 1 pα β σn +. () σ M Proof: From Eqns. (1) (14), we can derive that P k k =α P k 1 k 1 +β σ N ε (α P k 1 k 1 +β σ N ) k α P k 1 k 1 +β σ N +σ M and, if we take expectations on both sides, we obtain P k k =α P k 1 k 1 + β σn [ (α P k 1 k 1 + β σ N (1 p)e ) α P k 1 k 1 + β σn + σ M which after some manipulation leads to P k k =(pα ) k P (pα ) k [ 1 pα pβ σn +(1 p)σm { [ (1 p)σm 4 1 E α P k 1 k 1 + β σn + σ M [ + pα 1 E α P k k + β σn + σ M [ } (pα ) k 1 1 E α P β σn +. σ M (3) 1 1 Since 0 <E[ α P i i +β σn +σ M β σn +σ M k 1, Eqn. (3) leads to and for 0 i P k k (pα ) k P (pα ) k 1 pα pβ 4 σ 4 N + β σ M σ N β σ N + σ M (4) P k k (pα ) k P (pα ) k 1 pα [pβ σn +(1 p)σm. (5) Since P 0 0 = σ M +pσ X 0 σ σx +σ X 0 M 0 0 (from Eqns. (14) and (15)), we can easily derive from Eqns. (4) and (5) that a necessary and sufficient condition to have a finite P k k as k goes to infinity is that pα < 1 (note that in Eqns. (4) and (5), the terms 1 (pα ) k [pβ σn +(1 p)σ M and 1 (pα ) k 1 pα pβ 4 σ 4 N +β σ M σ N β σn +σ M 1 pα are always positive). As k goes to infinity, if Eqn. (1) is satisfied, the lower bound on the expected variance of the estimation error in Eqn. (4) can be simplified to Eqn. () and the proof is complete. Remark 6: The necessary and sufficient condition in Lemma 5 for the expected variance of the estimation error to be finite (bounded) as time goes to infinity for the scalar system is similar to that in [10, but our proof is different and the lower bound of Eqn. () is tighter than that in [10. By taking the expectation of both sides of Eqn. (1), we have P k+1 k = α P k k + β σn and by letting k go to infinity and applying the lower bound in Eqn. (), we can obtain P e = α P e + β σn = β σ N 1 pα + (1 p)α β σ M σ N (β σn +σ M )(1 pα ), which is a lower bound of P k k 1 as k

5 Control & Automation, July 7-9, 007, Athens - Greece T36-00 goes to infinity. Note that the corresponding lower bound in [10 for P k k 1 as k goes to infinity is S = β σ N 1 pα and is not as tight as P. e Definition 7: Define the sequence {V k } by V k+1 = g p (V k ), V 0 = P 0 0 where the function g p (V k ) is the modified algebraic Riccati equation g p (V k )=α V k + β σn (α V k + β σn (1 p) ) α V k + β σn +. σ M (6) We can follow the techniques and procedures in [10 to obtain an upper bound for P k k as k goes to infinity. More specifically, when Eqn. (1) is satisfied, we can make use of sequence {V k } to obtain an upper bound for P k k P k k V k (7) and an upper bound for P k k as k goes to infinity from Ṽ = g p (Ṽ ) (the fact that {V k} is a convergent sequence can be shown following the techniques in [10). In our case, Ṽ turns out to be Ṽ = b b 4ac (8) a where a = α 4 p α, b =α β σn p + α σm β σn σm, c = β4 σn 4 p + σ M β σn. Therefore, the expected variance P k k of the estimation error for Kalman filtering (under intermittent observations) as k goes to infinity has lower and upper bounds given by P e in Eqn. () and Ṽ in Eqn. (8). Another property of sequence {V k } is established in the following lemma. Lemma 8: If Eqn. (1) is satisfied and V 0 in the sequence {V k } is large enough, then V 0 is the largest among {V k } for any k. Proof: First we can easily show that function g p (V ) is an increasing function of V, i.e., if X < Y, then g p (X) <g p (Y ). We compare V 1 with V 0 by calculating the difference V 1 V 0= (pα 1)α V 0 +[(pα 1)β σ N +σ M (α 1)V 0 +pβ 4 σ N 4 +β σ N σ M α V 0 +β σ N. +σ M Since Eqn. (1) is satisfied, the term (pα 1)α V 0 in the above equation is negative and, for large enough V 0,this first term dominates and we have V 1 <V 0. By applying the property above, we have V = g p (V 1 ) <g p (V 0 )=V 1. If we iterate this procedure, we obtain V 0 >V 1 >V >... > 0 which means that V 0 is the largest among {V k } for any k and the proof is complete. We are now ready to discuss how to choose the feedback gain K for the closed-loop scalar system of Fig. 3, so as to minimize the expected MSDZ (i.e., the expected variance) of as k goes to infinity. First, we obtain lower and upper bounds for the quantity Λ k =P k 1 k 1 +(α + βk) P k k (α + βk) (k 1) P 0 0. (9) The proof of the following lemma is omitted due to space limitations. Lemma 9: If Eqn. (1) is satisfied, for any ɛ>0, there exists some k 0 and V e such that when k>k 0 (P e ɛ) 1 (α+βk)(k k 0 1) 1 (α+βk) +(α+βk) (k 1) τ 1 (.) Λ k [1 (α+βk)(k k 0 1) ( V e +ɛ) 1 (α+βk) + [(α+βk)(k 1 k 0 ) (α+βk) k Ve 1 (α+βk), (30) where τ 1 (.) is a constant that depends on k 0. Theorem 10: Consider the closed-loop scalar system of Fig. 3 described earlier under the assumptions of Lemma 4. If Eqn. (1) is satisfied, the MSDZ (variance) of, denoted by σx k (with the expectation taken over all possible sequences ξ k ), is finite (bounded) for arbitrary σx 0 as k goes to infinity if and only if (α + βk) < 1. The lower and upper bounds of the resulting σx k as k goes to infinity are given by σx 1 lk = 1 (α + βk) β σn P e +[α (α + βk) 1 (α + βk), (31) σx 1 uk = 1 (α + βk) β σn +[α (α + βk) Ṽ 1 (α + βk), (3) where P e and Ṽ are defined in Eqns. () and (8). Proof: The proof is omitted due to space limitations. Remark 11: In the closed-loop scalar system of Fig. 3, if Eqn. (1) is satisfied and (α+βk) < 1, both the lower and upper bounds of MSDZ Xk, (i.e., the lower and upper bounds of σx k ) as time goes to infinity are minimized when the feedback gain is set to K minkal = α/β. (i) For the closed-loop scalar system of Fig. 3, by plugging K minkal = α β into Eqns. (31) and (3), we obtain the following lower and upper bounds for the minimal σx k as k goes to infinity: σx lkmin =β σn + α P, e (33) σ X ukmin =β σ N + α Ṽ. (34) (ii) For the scalar closed-loop feedback system in Fig. 1, we obtain the minimal MSDZ (i.e., the variance) of the system state to be σx min = β σn +(1 p)k min σ M 1 α p (α + βk min ) (1 p). (35) [The optimal feedback gain K min for the proposed methodology is given by K min = b 1 + b 1 4a 1c 1, (36) a 1 where a 1 = σm (1 p)αβ, b 1 = σm (1 α )+σn β, c 1 = σn αβ (see [3). The expressions of σx and ukmin σ X min can easily be verified to be equal. Moreover, if there is no measurement noise M k, we can easily check that σx = lkmin σx = ukmin σ X min ; as expected, in this special scenario, the two strategies perform exactly the same. Clearly, the use of the control strategy in Fig. 3 is

6 Control & Automation, July 7-9, 007, Athens - Greece T36-00 Ratio of Ave. 1 MSDZ Diff Fig. 4. Simulation results comparing the two feedback control strategies in terms of their MSDZ performance difference for various σn and σ M in the range [0, 30. The highest (blue) curve is for σn =1; the next highest (red) curve is for σn = 10; the other (green) curve is for σn = 100; the lowest (black) curve is for σ N = σ M guaranteed to perform at least as well as the strategy in Fig. 1; this, however, comes at the cost of higher computational complexity. Example 1: In our simulations, we consider a scalar networked control system with packet dropping probability p =0.1and system (plant) parameters α =.5, β =1.5. The measurement and driving noises M k and N k are zero-mean uncorrelated white Gaussian noise processes with variance σm and σ N respectively. Fig. 4 plots simulation results for these two control strategies for several different combinations of σn and σm ; for each combination of σ N and σ M,werunsimulation experiments based on 3000 independent trials (each running a simulation of the system for 1000 time steps). Each point in every curve is the average normalized difference over all time steps between the MSDZ achieved by the two control strategies defined as σ (sim) σ (kal) σ (kal) k where σ (sim) and σ (kal) are the empirical MSDZs at time k, achieved respectively by the proposed strategy and the strategy that uses Kalman filtering. Fig. 4 clearly shows that the empirical performance using the proposed strategy is comparable to the strategy that uses Kalman filtering to estimate the system state (especially) if σm is relatively small. The average performance difference between the two strategies correlates positively with σm and negatively with σ N. More specifically, with fixed σn, the average performance difference is approximately a linear function of σm (after an initial transient). When there is no measurement noise (i.e., σm = 0) or when σ N is large compared to σ M (i.e., when the driving noise dominates), there is no difference or very small difference between the two strategies., V. CONCLUSIONS AND FUTURE RESEARCH In this paper we have studied stabilization schemes in NCSs. More specifically, we have extended a previously proposed stabilization scheme in [3 to handle higherdimensional dynamic systems and we have clarified the performance of this proposed control scheme by comparing against other ways of extending state feedback strategies. We have established that, under certain conditions, the performance of our proposed control scheme is comparable to the performance of a strategy that uses Kalman filtering to estimate the system state and applies state feedback based on this estimated state. 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