LQG Control For MIMO Systems Over multiple TCP-like Erasure Channels

Transcription

1 LQG Control For MIMO Systems Over multiple CP-like Erasure Channels E. Garone, B. Sinopoli, A. Goldsmith and A. Casavola Abstract his paper is concerned with control applications over lossy data networks. Sensor data is transmitted to an estimation-control unit over a network and control commands are issued to subsystems over the same network. Sensor and control packets may be randomly lost according to a Bernoulli process. In this contet, the discrete-time Linear Quadratic Gaussian LQG optimal control problem is considered. In 1, a complete analysis was carried out for the case that sensor measurements are delivered to the estimator into a single data packet, as well as control inputs. Here, a nontrivial generalization for MIMO systems is presented under the assumption that each sensor and each actuator echange data with the control unit in an independent way by using its own packet no aggregation. In such a framework, it will be shown that the separation principle still holds in the case packets are acknowledged by the receiver i.e. the CP-like case. Moreover, it will be pointed out that the optimal LQG control is a linear function of the state that eplicitly depends on the loss probabilities of the actuator channels. Such a dependence is not present in the single channel case considered in 1. In the infinite horizon case, stability conditions on the packet arrival probabilities will be provided in terms of Linear Matri Inequalities LMIs. I. INRODUCION he term Cyber-Physical Systems CPS refers to the embedding of information and telecommunication technologies into physical systems, with the purpose of monitoring and controlling them at a fine temporal and spatial scale. Several industries, such as automotive, process control, energy and built environments will be able to monitor and improve their performance while being safer, more robust and energy efficient. Closing the loop around CPS raises numerous issues. Unlike the classical control paradigm, where components, i.e. sensors, controllers and E. Garone and A. Casavola are with the Dipartimento di Elettronica, Informatica e Sistemistica, Università degli Studi della Calabria, Via Pietro Bucci, Cubo 42-C, Rende CS, 87036, Italy {egarone,casavola}@deis.unical.it B. Sinopoli is with the Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA brunos@ece.cmu.edu A. Goldsmith is with the Department of Electrical Engineering, Stanford University, Stanford, CA 15213, USA andrea@ee.stanford.edu

2 actuators, are often co-located and therefore connected via dedicated channels, e.g. serial cables, spatially distributed systems will make use of general purpose networks to communicate and echange information. As a consequence the communication introduces uncertainty as data may get delayed or lost in the communication. Networked Control Systems NCS is the discipline that studies the interaction between control and communication, providing analysis and design tools to address this new challenge see 2,3,4,5,6,7,8. In this paper we provide a complete generalization of the Linear Quadratic Gaussian LQG optimal control problem where sensors and actuators are connected to the estimator-controller pair via analog erasure channels in the so-called CP-like case, i.e. the case where packet acknowledgements are available 9. Previous results on this topic 10, 11, 12, 13 refer to the case where only a single channel eists between both sensors and controller and between the latter and the actuators. hey have shown the eistence of a critical values for the parameters of the Bernoulli arrival processes, ν and γ, modeling the arrival rate for the control and sensor channels respectively, outside which a transition to instability occurs and the optimal controller fails to stabilize the system. In particular, in the CP-like case, the critical arrival probabilities for the control and observation channel are independent of each other. his is another consequence of the fact that the separation principle holds under the assumption the acknowledgment is available. A more involved situation regards the UDP-like case as shown in 1. In this paper we focus on the general CP-like case where multiple communication channels are interposed between the sensors, the controller and the actuators. Accordingly, in full generality, multiple independent Bernoulli processes are considered, with parameters γ i and ν j, that govern packet losses between the sensors and the estimation-control unit, and between the latter and the actuation points respectively see Figure 1. herefore, contrarily to the previous work, the system can eperience partial observation and control loss. In 14, this case was considered for the estimation problem eclusively and with two observation channels. We will show that the separation principle holds for the general case of multiple channels. Also the optimal estimator is linear as well as the optimal controller. Differently from the single channel case, the optimal control inputs will depend eplicitly upon the arrival rates of their respective channels. We will solve the LQG problem for both the finite and infinite horizon cases. For the latter, we provide implicit stability conditions. he remainder of this paper is organized as follows. Section 2 provides the problem formulation. In Section 3, we solve the estimation problem. In Section 4, we consider the control

3 synthesis problem for both finite and infinite horizon. Finally, Section 5 draws conclusions and outlines the agenda for future work. II. PROBLEM FORMULAION Consider a linear system whose state equations are k+1 = A k + Bu a k + w k, y m k = C k + v k, 1 where k IR n is the state vector, u a k IRm the input vector applied to the plant at time k Z + and yk m IR p the output measured by the sensors. he white noise processes w k IR n and v k IR p are Gaussian, uncorrelated with zero mean and covariances Q IR n n and R IR p p respectively. As reported in Figure 1, the input signals are remotely computed by a control u a k N u k k System A,B,C m y k NEWORK Control Unit y k y k m k Fig. 1. System Diagram unit that communicates the control action to the actuators through unreliable communication channels. Here, it is assumed that at each sampling time the control unit sends to each actuator j = 1,..., m a packet containing the desired command u j,k IR. Any packet can be lost according to an i.i.d. Bernoulli random binary variable ν j,k, j = 1,.., m, such that ν j,k = 1 if the j-th packet is successfully transmitted to the j-th actuator at time k. he probability of successful transmission of the j-th packet is ν j. By defining u k = u 1,k,..., u m,k as the aggregate vector of the commands generated by the control unit, we can write the input applied to the plant at time k as u a k = N k + I m m N k u l, 2 where N k = diag {ν 1,k,..., ν m,k } and u l IR m is the locally generated control signal to be applied to the actuators in the case that all packets sent to the actuators are lost. As highlighted in 15, different choices are possible. In this paper we will consider the case u l = 0,..., 0.

4 Similarly to what happens at the actuator, the remote controller receives the output measurements from the sensors via unreliable channels. Again, we assume that the controller receives a packet from each sensor. Packets can be lost according to an i.i.d. Bernoulli random binary process γ i,k, i = 1,..., p, with probability of successful transmission γ i. herefore, at each sampling instant k, the vector y k IR p k represents the measurements available to the controller at time k y k = Γ k yk m, 3 where p k, 0 p k p, is the number of sensor packets successfully arrived at destination at time k and Γ k IR pk p a row selection matri defined as the collection of p k rows Γ k = e i, 4 i γ i,k =1 where e i denotes the i-th vector of the standard basis of IR p. Notice that Γ k = I p p will correspond to the case where all packets are delivered. On the contrary, it is the null matri Γ k IR 0 p when no sensor packet are available at the controller side at time k. Remark 1 - Most of the specific literature on control over erasure channel see 4-16 relies on the assumption packets dropout can be modeled as Bernoulli processes. Even though it may appear simplistic, such an assumption is not unrealistic in a large class of networked control systems. For instance, in those applications where the time constants are sufficiently high, the temporal window between two subsequent transmissions will be large enough to assume their arrival probabilities uncorrelated, as clearly motivated in 16. he overall system, from the controller viewpoint, can be rewritten as k+1 = A k + BN k + w k, y k = Γ k C k + v k, 5 where Γ k, N k, v k, w k are stochastic variables. Finally it is important to clearly define the Information Set I k available to the controller at time k. In fact, while it is reasonable to include the delivered measurements y τ, τ = 0,..., k, the corresponding matrices Γ τ, τ = 0,..., k, and the past input vectors u τ, τ = 0,..., k 1, some considerations about the possible inclusion of N τ, τ = 0,..., k 1 are in order. From the controller viewpoint, the knowledge of N k 1 at time k corresponds to knowing which actuator packets were successfully delivered to their respective actuators at the previous

5 time instant. Such a knowledge may be obtained via an acknowledgment mechanism, that has to be carefully characterized. In the NCS literature, e.g. see 9, it is usual to consider the following three cases: the UDP-Like case see 13, in which N k 1 is unknown, the case in which the knowledge of portions of N k 1 at time k depends on stochastic variables known as the Quasi CP-Like case see 17, and finally the CP-Like see 10 case, in which the matri N k 1 is assumed to be perfectly known at time k. In this paper, we focus on the CP-Like case only, whose corresponding Information Set becomes: I k = {y τ, Γ τ, N τ 1, u τ 1, 0 τ = 1,.., k}. 6 Remark 2 - For the sake of clarity, it is worth noting that the terms UDP-Like, CP-Like and Quasi CP-Like, usually adopted in the NCS literature, do not indicate specific communication protocols but only refer to the particular acknowledgement model used. he aim of this paper is to analyze the LQG control for the presented networked system under the further assumption that the unknown initial state 0 is Gaussian, with known epected value 0 and covariance P 0. More formally, let us introduce the following cost function N 1 J N u N 1, 0, P 0 = E NW N N + k W k k + u a k U k u a k un 1, 0, P 0, 7 k=0 where W k IR n n, k = 0,..., N and U τ IR m m, k = 0,..., N 1 are proper weighting matrices and u N 1 denotes the set of input vectors, k = 0,..., N 1 to be applied from time 0 through N 1. he control design problem for both the finite and infinite horizon cases will involve computing the optimal control input sequence u = f k I k that minimizes the cost functional 7, i.e. u N 1 = argmin =f k I k J N u N 1, 0, P 0. 8 Remark 3 - Notice that in the above CP-like problem setting, we have assumed that at time k the controller has a complete and deterministic knowledge of the actuator packet arrivals up to time k 1. his could be unrealistic, unless dedicated acknowledgment channels are available, because the acknowledgment mechanism may be using the same unreliable network. However, the CP-like case is still relevant not only for its theoretical importance as a lower bound for the case the acknowledgment is stochastic but also because, in some cases, it reasonably approimates realistic situations. In fact, usually, as depicted in Figure 2, while the packets

6 containing measurements and commands have to be received within short temporal portions of the sampling time, the acknowledgment packets have a much longer admissible temporal windows, due to the fact that the knowledge about N k 1 will be needed only at time k. his longer time window, coupled with the less demanding bandwidth requirements for the acknowledgment packets an ACK is a binary information, allows re-sending-until-received policies to be effective in making negligible the probability to lose acknowledgment packets. Sampling ime k k+1 Measures Receinving ime Control Unit Computational ime Actuator Signals Receinving ime Acknowledgment Receinving ime Fig. 2. Sampling time and transmission windows. III. OPIMAL ESIMAION In this section we will compute the optimal state estimator for a multichannel system under deterministic acknowledgment. o this end, let us define the following variables: ˆ k k E k I k, 9 e k k k ˆ k k, 10 P k k E e k e k Ik. 11 By eploiting the independence of w k, the one-step ahead predictions of the above quantities can be easily obtained: ˆ k+1 k = E k+1 I k, N k, = AE k I k + BN k = Aˆ k k + BN k, 12 e k+1 k = k+1 ˆ k+1 k = Ae k k + w k, 13 P k+1 k = e k+1 k e k+1 k = AP k k A + Q. 14 Please note that the eact knowledge of N k plays a fundamental role in the above predictions. he latter equations have been used in 14 as the innovation step for the optimal Kalman filter formulation in the case of possibly partial observation losses in the case of two sensors. However in that paper, the correction step is described through a combinatorial epression that takes into

7 account all possible packets arrival configurations. his is overcome by using the matri Γ k defined in 4 that allows to formulate the following more effective and intuitive correction step: ˆ k+1 k+1 = ˆ k+1 k + K k+1 yk+1 Γ k+1 C ˆ k+1 k, 15 K k+1 = P k+1 k C Γ k+1 Γk+1 CPk+1 k C +R Γ k+1 1, 16 P k+1 k+1 = P k+1 k K k+1 Γ k+1 CP k+1 k. 17 It is worth noting that in such formulation, the Kalman filter for networked systems can be seen as a Kalman filter for a time-varying system, with output matri C k = Γ k C, with C k R p k n. he convergence properties of the aforementioned estimator will be presented in Section V. IV. FINIE HORIZON LQG CONROL In this section we analyze the Finite Horizon LQG control design problem for system 5. In order to proceed systematically, we need first to recall some preliminary results presented in 10: Lemma 1 - he following equalities hold true: E e k kˆ k k I k = 0 18 E k S k I k = ˆ k Sˆ k + trace SP k k, S 0 19 E E f k+1 I k+1 I k = E f k+1 I k, f. 20 E e k k e k k I k = trace E e k k e k k I k = trace Pk k. 21 Moreover, it is important to understand what the epected value of k S k represents. By eploiting the independence of N k, w k, k, the zero-mean property of w k and by taking advantage of Lemma 1 we obtain: E k+1 S k+1 I k = E A k + BN k + w k S A k + BN k + w k I k = = E k A SA k + 2u k N kb SA k + u k N k B SBN k I k + E w k Sw k I k = = E k A SA k I k + u k Ψ N, B SB + 2u k NB SAˆ k k + trace SQ, where N = diag{ ν 1,..., ν m } and Ψ N, X is defined as Ψ N, X = ν i 1 ν i N I XN I. 23 I 2 I i I i/ I N I denotes a diagonal matri defined on the inde set I I = {1,..., m} such that its diagonal elements are 1, if i I N I ii = 0, if i / I 22

8 and 2 I is the set of all the possible subsets of I. In order to derive the optimal control law and the corresponding value for the objective function we follow a dynamic programming approach based on a cost-to-go iterative procedure 18. o this end, let us define the following optimal value function V k k V N N = E NW N N I N, 24 V k k = min E k W k k + N k U k N k + V k+1 k+1 I k, 25 where k = N 1,..., 1, 0 and such that J N u N 1, 0, P 0 = V0 0. We can prove that Lemma 2 - he value function V k k defined in for system 5 along with the Information Set 6 can be written as V k k = E k S k k I k + ck k = N,..., 0, 26 where the matri S k and the scalar c k can be computed by means of the follow matri iterations: S k = W k + A S k+1 A A S k+1 B N Ψ N, U k + B S k+1 B 1 NB S k+1 A, 27 c k = E c k+1 I k + trace S k+1 Q + trace A S k+1 A + W k S k Pk k, 28 with initial values S N = W N and c N = 0. Moreover, the optimal control input is given by = Ψ N, U k + B S k+1 B 1 NB S k+1 A k k = L k k k. 29 Proof - he proof employs an induction argument. he claim is clearly true for k = N with parameter S N = W N and c N = 0. Suppose now that the claim is true for k + 1: V k+1 k+1 = E k+1s k+1 +1 I k+1 + c k+1. he cost at time k is then V k k = min E k W k k + u k N ku k N k + V k+1 k+1 I k = = min E k W k k + u k N ku k N k I k + E E k+1 S k+1 k+1 + c k+1 I k+1 Ik. By eploiting equation 20, the latter becomes V k k = min E k W k k + u k N ku k N k I k + E k+1 S k+1 k+1 + c k+1 I k = = min E k W k k + u k N ku k N k I k + E k+1 S k+1 k+1 I k + E ck+1 I k. As already seen, the second term of this equation can be rewritten as 22. hen, the value function becomes V k k = min E k W k k + u k N ku k N k I k + E k A S k+1 A k I k + +u k Ψ N, B S k+1 B + 2u k NB S k+1 Aˆ k k + trace S k+1 Q + E c k+1 I k.

9 Finally, by using the linearity of the epectation operator, by epanding E u k N ku k N k I k as in 22 and by taking advantage of the linearity of Ψ we obtain: V k k = min E k Wk + A S k+1 A k + c k+1 Ik + u k Ψ N, U k +u k Ψ N, B S k+1 B + 2u NB k S k+1 Aˆ k k + trace S k+1 Q = = min E k Wk + A S k+1 A k + c k+1 Ik + u k Ψ N, U k + B S k+1 B +2u k NB S k+1 Aˆ k k + trace S k+1 Q. Because the value function is a quadratic function of the input, the minimizer can be obtained by solving V k / = 0: V k k = 2 NB S k+1 A k k + 2Ψ N, U k + B S k+1 B = 0, which yields equation 29. he optimal control law is thus a linear function of the state estimate. If we substitute such a minimizer back into the cost V k k, we obtain the following epression: V k k = E k Wk + A S k+1 A k + c k+1 Ik + trace Sk+1 Q + ˆ k k A S k+1 B N Ψ N, U k + B S k+1 B 1 NB S k+1 Aˆ k k. Finally, using 19 of Lemma 1, we can rewrite V k k as V k k = E k Wk + A S k+1 A k + c k+1 Ik + trace Sk+1 Q + E k A S k+1 B N Ψ N, U k + B S k+1 B 1 NB S k+1 A k Ik + +trace A S k+1 B N Ψ N, U k + B S k+1 B 1 NB S k+1 AP k k = = E k W k +A S k+1 A A S k+1 B N Ψ N, U k + B S k+1 B 1 NB S k+1 A k Ik + +c k+1 I k +traces k+1 Q+trace A S k+1 B N Ψ N, U k +B S k+1 B 1 NB S k+1 AP k k. Using this last equation we can easily show that equation 26 is satisfied also at each step k for all k if and only if the matrices S k and the scalars c k satisfy equations 27 and 28 respectively. Note that in 28, the last term of the above equation is written in a more compact form by using the definition of S k in 27. As a consequence and because J N u N 1, 0, P 0 = V0 0, it follows that the value of the cost function at the optimum is JN = 0 S 0 0 +traces 0 P 0 + N 1 traces k+1 Q+ N 1 trace A S k+1 A+W k S k EΓ Pk k k=0 where E Γ Pk k denotes the fact that the epectation is calculated with respect to the arrival sequence {Γ k }. It is worth remarking that the error covariance matrices { } N P k k are stochastic k=0 k=0 30

10 because their evolution follows the stochastic Riccati equation 17. heir epected values cannot be computed analytically, as the Riccati is nonlinear, but can only be bounded. For more details on this subject please refer to 19 and 14. he following heorem summarizes the results for the finite horizon LQG control for physically distributed networked systems in the CP-like case: heorem Finite Horizon LQG under CP - Consider system 5 and the problem of minimizing the cost function 26 under policy = fi k, where I k is the information available in the CP like case. hen: he optimal controller is a linear function of the estimated system state 29, where the matri S k can be computed iteratively using 27. he separation principle holds true because the optimal estimator is independent of the control input and the optimal control law depends on the data only via ˆ k k he optimal state estimator is given by and the optimal cost value is 30. Remark 4 - It is interesting to note that, as epected, in the multichannel case the optimal control depends on the arrival rates of every single control channel. V. INFINIE HORIZON LQG CONROL he infinite horizon LQG can be obtained by taking the limit for N of the previous equations under the additional hypothesis W N = W k = W, U k = U, A, B and A, Q 1/2 controllable and A, C and A, W 1/2 observable. However, as eplained, the matrices {P k k } depend nonlinearly on the specific realization of the matri stochastic process {Γ k }. herefore, the epected covariance matrices E Γ P k k and the minimal cost J N cannot be computed analytically and do not seem to have limit. Moreover, it is important to understand that, unlike the standard LQG regulator, in the case of observation and control packet losses, the stability can be lost if the arrival probabilities are below a certain threshold. In order to analyze this behavior we need to study the convergence of the following Modified Algebraic Riccati Equations MAREs for the controller and the estimator respectively, as their convergenge is a sufficient condition for boundedness of the cost function 1: S k+1 = Π c Sk, A, B, U, W, N, 31 P k+1 = Π o Pk, A, C, Q, R, Γ, 32

11 where the nonlinear functions Π c,π o are defined as follows: Π c S, A, B, U, W, N = W + A SA A SB N Ψ N, U +B SB 1 NB SA, 33 Π o P, A, C, Q, R, Γ = AP A + Q + 34 AP C γ i 1 γ i Γ I ΓI CP C + R Γ 1 I ΓI CPA, I 2 I i I i/ I and Γ I IR card{i} p is a row selection matri defined as the collection of a number of rows equal to the cardinality of the set I I, i.e. card{i}, such that Γ k = e i being e i the i-th vector of the standard basis of IR p. o study the MAREs convergence we will introduce the following auiliary operators: F c ϕ c K, S = ν i 1 ν i I SFI c I 2 I i I i/ I F o ϕ o K,..., K I, P = γ i 1 γ i I P FI o I 2 I i I i/ I with FI c = A + KN I B, VI c = W + KN I UNI K, = A + K I Γ I C, V o = Q + K I Γ I RΓ I KI, F o I I i I, + VI c, 35 + VI o, 36 and where,..., I, denotes the enumeration of any I 2 I. By taking advantage of the above definitions and by following a similar argument as in 19 and 1, it is possible to state the convergence conditions for the MAREs. Such results are here reported without proofs for the sake of brevity. Details can be found in 20. heorem Convergence of MAREs - If there eists a pair K c, S such that S > ϕ c K c, S, S > 0 37 then, for any initial condition S 0 > 0, the MARE 31 converges and the limit is independent of the initial condition, i.e. lim t S t = S where S is the unique positive-semidefinite fied point of the MARE. Similarly, if there eists a 2 p + 1-tuple such that P > ϕ o K,..., K I, P, P > 0 38 then, for any initial condition P 0 > 0, the MARE 32 converges to P, the latter being the unique positive-semidefinite fied point of the MARE. Lemma 3 - Conditions 37 and 38 are equivalent to the solution of the following two LMIs

12 Ψ c Y, Z = Y Y η Y A + ZN B η ZN U 1/ W Y I Y > 0. η I Y A + ZN I B η I ZN I U 1/ >0, Y 0 I 39 where Ψ o Y, Z,..., Z I = Y Y λ Y A + Z Γ C λ Z Γ R 1/ Q Y I Y > 0. λ I Y A + Z I Γ I C λ I Z I Γ I R 1/ >0, Y 0 I η I = ν i 1 ν i, λ I = i 1 γ i. 41 i I i/ I i I γ i/ I heorem Infinite Horizon LQG under CP - Consider system 5 under the following additional hypothesis: W N 40 = W k = W and U k = U. Moreover, let A, B and A, Q 1/2 be controllable and A, C and A, W 1/2 observable. hen, if the arrival probabilities N,Γ are such that MAREs converge, the infinite horizon optimal controller gain is constant and can be computed by L = Ψ N, U + B S B 1 NB S A. 42 he infinite horizon optimal estimator gain K k, given by equation 16, is time-varying and depends on the realization of the observation arrival process {Γ j } k j=1. Moreover the infinite horizon cost cannot be computed, as it depends upon the realization of the observation arrival process, but can be deterministically bounded

13 VI. EXAMPLE A. Eample 1 - An unstable batch reactor he goal of this eample is to show how the proposed methodology can be used to compute optimal control laws and determine the stability regions for real applications. Consider a batch reactor. his is a large system where components, i.e. sensors, controllers and actuators cannot be co-located and therefore a networked architecture may be desirable. We want to assess the impact of connecting these components via lossy channels. Let us consider, in particular, the unstable batch reactor given in 21. he plant can be modeled as a 2-input-2-output system. he linearized continuous time process model presented in 21 has been discretized with a sampling time τ = 0.1sec, yielding the following discrete-time state space representation k+1 = k y k = k with k IR 4, IR 2, y k IR 2. he regions of guaranteed convergence for the observation and control MAREs w.r.t. the arrival rates γ 1, γ 2, ν 1, ν 2 are shown in Fig. 3a and Fig. 3b, respectively, for W = I 4 4, U = I 2 2, Q = 2I 4 4, R = 0.001I Output Packet Arrival Probabilities and Riccati Equation Convergence 0.5 Control Packet Arrival Probabilities and Riccati Equation Convergence Stable 0.3 Stable Eγ Eν Unstable Unstable Eγ Eν 1 a b Fig. 3. Eample 1 - Observation MARE Left and Control MARE Right regions of guaranteed convergence. Note that both MAREs need to be stable to guarantee boundedness of the LQG cost.

14 B. Eample 2 - A simple stable networked system he goal of this second eample is to give evidence of the dependence of the control law from the packet arrival probabilities. his phenomenon is present eclusively when more than one channel is used. o this end, consider the following demonstrative stable networked system with two inputs. Such a model can be used to describe a water reservoir whose level is controlled by two valves. k+1 = 0.5 k In order to concentrate on the control law only, let us suppose that the eact value of the state is provided at each instant k. In Fig. 4a, the values of the optimal control law gains L = L 1, L 2 are shown for different arrival rates ν 1, ν 2. It is possible to note that the control law typically privileges the most reliable channel. his peculiar property can be better noticed in Fig. 4b, where Fig. 4a is projected on the hyperplane ν 1 = Constant Gain vs Arrival Rate L1 L Constant Gain vs Arrival Rate ν 1 = L L ν ν ν 2 L1 L2 a b Fig. 4. Eample 2 - Left Control Gain vs. Arrival Rates Right Control Gain vs. Arrival Rates, Projection on ν 1 = 0.5 VII. CONCLUSION AND FUURE WORK Motivated by applications where control is performed over a communication network, this paper etends previous results on optimal control over lossy networks to the case where both observation and control packets travel across a multichannel network. As a consequence, partial observation and control input losses may occur. We have assumed Bernoulli with acknowledgement of receipt of control packets always available to the controller. First, we have computed the optimal estimator for this case. hen, we have proved that the optimal LQG control is a linear function of the state, showing that the separation principle still holds true. Finally, we have computed the optimal controller for both finite and infinite horizon cases, providing sufficient

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