Optimal state filter from sequential unknown input reconstruction: Application to distributed state filtering

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1 Received: 29 December 215 Revised: 31 July 217 Accepted: 3 August 217 DOI: 1.12/acs.2828 RESEARCH ARTICLE Optimal state filter from sequential unnown input reconstruction: Application to distributed state filtering Taouba Rhouma 1 Jean-Yves Keller 2 Karim Chabir 1 Dominique Sauter 2 Mohamed Naceur Abdelrim 1 1 Modeling, Analysis and Control of Systems (MACS Laboratory LR 16ES22), National Engineering School of Gabés (ENIG), University of Gabés, Gabés, Tunisia 2 Centre de Recherche en Automatique de Nancy, University of Lorraine, CNRS UMR 739, Vandœuvre-lés-Nancy, France Correspondence Taouba Rhouma, Modeling, Analysis and Control of Systems (MACS Laboratory LR 16ES22), National Engineering School of Gabés (ENIG), University of Gabés, 629 Gabés, Tunisia. taouba.rhouma@gmail.com Summary This paper generalizes Kalman filtering with an intermittent unnown input problem to be left invertible discrete-time stochastic linear systems with zero, one, or more structural delays. Contrary to the state filtering based system inversion where the unnown input vector is reconstructed with a time delay that is equal to the structural delay of the plant, we propose an optimal state filtering by reconstructing some linear combinations of the unnown input vector with a time delay less than the structural delay. Designed under a sequential unnown input decoupling constraint, which has never been previously studied in the literature, all presented filters are very computationally efficient. The proposed state filtering is used to solve the autonomous distributed state filtering problem in large-scale networed control systems when the unnown input vector represents interactions between subsystems and when each subsystem receives intermittent information about the interaction from unreliable networs. The stochastic stability conditions of the extended intermittent unnown input Kalman filter are established when the arrival binary sequence of pacet dropouts follows a random Bernoulli process. KEYWORDS distributed state filtering, intermittent unnown inputs, Kalman filtering, linear systems 1 INTRODUCTION Over the last 3 decades, Kalman filtering has attracted more attention of researchers from different areas. It plays an essential role in system theory (see the wor of Kailath et al 1 for linear systems and the wors of Yin et al 2,3 for nonlinear systems) and state filtering with unnown inputs representing unnown external drives, actuators, or instrument faults. In the wor of Kitanidis, 4 an optimal linear state filter is obtained by minimizing the trace of the state estimation error covariance matrix under the constraint that the state estimation error is decoupled from unnown inputs or obtained from the state filtering of singular systems as in the wor of Darouach et al. 5 Other optimal filters having a structure closer to that of the standard Kalman filter are designed in previous wors The problem studied in previous wors, which is the most closely related to joint input and state estimation, is of great importance to fault-tolerant control when each component of the unnown input vector may represent actuator, sensor, or transmission faults as explained in the wors of Blane et al 17 and Patton et al Copyright 217 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/acs Int J Adapt Control Signal Process. 218;32:

2 RHOUMA ET AL. 99 Recent technological advances are revolutionizing our ability to build massively distributed networed control systems (NCSs). Infrastructures such as power grids, water distribution networs, and transport systems are examples of large-scale cyberphysical systems integrating communication channels, computational algorithms, and physical processes. The most common networ effect can be categorized into additional networ-induced delays or pacet losses. The state-of-the-art in control system designing that taes into account the effects of pacet delay or pacet loss in NCSs was surveyed in the wor of Hespanha et al. 19 In particular, Kalman filtering with intermittent measurements described by nown binary random sequences following Bernoulli or Marovian processes was studied in the wors of Schenato et al, 2 Shi et al, 21 and Sinopoli et al. 22 In the framewor of this paper, we are mainly concerned with the distributed state filtering when each subsystem receives information about its physical interactions with other subsystems over unreliable networs. For each subsystem, a solution to the autonomous distributed state filtering problem consists in solving online the state filtering problem of the discrete-time linear stochastic systems subject to data-driven unnown inputs triggered by data losses. The intermittent unnown input Kalman filter (IIKF) in the wor of Keller and Sauter 23 solves this state filtering problem when data losses are described as nown binary sequences by parameterizing the solution to the intermittent unnown input decoupling constraint from 2 constant-size matrices. These matrices are called the free and the constrained part of the filter gain that are used to minimize the trace of the state and the unnown input estimation error covariance matrix from a 2-stage optimization strategy. When the number of intermittent unnown inputs is equal to the number of measurements, Keller et al 24 showed the duality between Kalman filters with intermittent observations in the wor of Sinopoli et al 22 and with intermittent unnown inputs in the wor of Keller and Sauter. 23 The problem of inverting linear time-invariant systems has been of interest to control engineers for many years. The existence conditions under which a linear discrete-time system with permanent unnown inputs is left invertible with zero, one or more structural delays has been discussed in previous wors or in Appendix A. This paper solves the autonomous distributed state filtering problem when subsystems are assumed to be left invertible. In other words, when the physical interactions between subsystems can be reconstructed without communication between them. Applied on each subsystem, the extended IIKF (EIIKF) presented in this paper will depend on the arrival binary sequence of unnown inputs from a combinatorial logic for systems with zero structural delays or from a sequential logic for systems with structural delays greater than zero. We also showed that the stochastic stability conditions of the EIIKF with random occurrence of unnown inputs may be satisfied even if the system has unstable invariant zeros. In the literature, to the best nowledge of the authors, the computational time for a real-time implementation of the Kalman filter with permanent unnown inputs has never really been studied. However, this problem must be taen into account when the unnown inputs appear sequentially that need to optimize the filter under a sequential unnown input decoupling constraint within one period of time. This article provides a real-time solution to the optimization of the EIIKF from a 2-stage optimization strategy. This technique maes it possible to obtain the state prediction via that of produced by the standard Kalman filter. This is corrected by using a hybrid estimation of the intermittent unnown inputs having the nontrivial property of being zero in their absences. This explains why the EIIKF presented in this paper does not require much more computing time than the standard Kalman filter. This paper is organized as follows. Section 2 presents the problem statement in the area of the distributed large-scale NCS. Sections 3 and 4 design the EIIKF for left invertible linear systems with zero and one delays, respectively. Section 5 extends the obtained results for delays greater than one. Section 6 gives an illustrative example before we provide concluding remars in Section 7. 2 PROBLEM STATEMENT Consider a large-scale cyberphysical system described by the following linear discrete-time stochastic dynamic system: x +1 = Ax +ΓU + w Y = Ex + HU + V, (1) where x R n is the state vector, w R n is the process noise vector, U = [ u T 1, ut ] T,and i, ut N, Y = [ y T 1, yt ] T i, yt N, are the control and measurement vectors with ui, R d i and y i, R m i,andv = [ v T 1, vt ] T i, vt N, with vi, R m i is the measurement noise vector, where i {1,, N} is the set of subsystems.

3 1 RHOUMA ET AL. - Controller 1 Controller i Controller N FIGURE 1 Networ of distributed actuators Matrices E = [ C T 1 ] T, [ ] [ CT C T i N Γ= B1 B i B N,andH = D T 1 ] DT D T i N are of appropriate dimensions. The process and sensor noises w R n and v i, R m i are zero mean uncorrelated Gaussian random sequences with { [ ][ ] } T [ ] w wj W E v i, v = i, j V δ, j W, V i >. (2) i Uncorrelated with w and v i,, the initial state x is a Gaussian random variable with E {x } = x and P = E { (x x )(x x ) T}. The pair (A, C i ) is assumed to be detectable i {1,, N}. System (1) is controlled by a set of N spatially distributed controllers, each of them transmits its local control law to other controllers over unreliable networs. Let u i, be the control law transmitted by the ith controller, ρ i ={, 1} is the acnowledgement signal indicating the status of the reception/delivery (for example, Transmission Control Protocol). Let ρ i = 1whenuc = [ u T i, 1, ut i 1, ut ] T i+1, ut N, R q i is correctly received by the ith controller or ρ i = whenuc is i, lost (see Figure 1). We assume in our analysis that there exists no transmission noise, no quantification error, and m i > q i.fromthehybrid signal d i =ρi uc +(1 i, ρi) d i,where d i represents the unnown vector uc when i, ρi =, the state model of the plant viewed by the ith controller is described by x +1 = Ax + B i u i, + F i d i, + w y i, = C i x + D i u i, + J i d i, + v i, (3) with F i = [ B 1 B i 1 B i+1 B N ] R n,q i (4) J i = [ D 1 D i 1 D i+1 D N ] R m i,q i. (5) Without communication between subsystems, Appendix A recalls the structural conditions under which d i, can be reconstructed with finite structural delays α i from local measurements y i, available until time (see also previous wors ). This paper assumes that Equation 3 is left invertible with structural delays α i = if ran(j i )=q i (6) or when ran(j i ) < q i with structural delays α i = 1if ([ ]) Ji ran = ran(j C i F i J i )+q i. (7) i When J i =, note that Equation 7 becomes ran(c i F i )=q i and recovers the necessary existence condition of the unnown input observers (see the wor of Kobayashi and Naamizo 26 ). We also consider the case of structural delays α i > 1when Equations 6 and 7 do not hold. For simplicity of the notation, we omit the index i and assume that u i, =. The state model (3) can be then rewritten as follows: x +1 = Ax + Fρ u c + F ρ d + w y = Cx + Jρ u c + J ρ d + v, (8)

4 RHOUMA ET AL. 11 where ρ u c Rρq is the intermittent nown input vector and ρ d R ρ q with ρ = 1 ρ is the intermittent unnown input vector. For the state model (8), consider the following n-order linear state filter: x +1 = A x 1 + Fρ u c + K ( y C x 1 Jρ u c ) P +1 =(A K C)P 1 (A K C) T + K VK T + W, (9) where x +1 is the state prediction of covariance P +1 = E{(x +1 x +1 )(x +1 x +1 ) T } based on sequences {y j } and { ρ j } and the hybrid state feedbac gain K R n,m depending on the binary sequence { ρ j }. This paper extends the use of the IIKF presented in the wor of Keller and Sauter 23 to left invertible systems of structural delays α=, α=1orα > 1. If α=, the unbiased minimum-variance (UMV) state prediction so that E { x } +1 = x +1 { ρ j } will depend on the binary sequence from a combinational logic. If α=1, the UMV state prediction so that E { x } +1 = x+1 when ρ = will depend on the binary sequence from a sequential logic. For linear systems with structural delay α > 1, the UMV state prediction will be recovered whenthe unnowninputs are zeroduring α successive periods of time. The sufficient stochastic stability conditions will be established when the arrival sequence of the pacet dropouts follows a random Bernoulli process with λ=pr[ρ = ] where λ is the occurrence rate of the pacet dropouts. For systems having unstable invariant zeros, we will show that the obtained stochastic stability conditions can be satisfied when the unnown input occurrence rate λ =1 λdoes not exceed a critical limit. 3 STATE FILTERING FOR SYSTEMS WITH ZERO STRUCTURAL DELAY For the filter's design, the arrival sequence of the unnown inputs is considered as deterministic. From systems (8) and (9), the state prediction error e +1 = x +1 x +1 propagates as follows: e +1 =(A K C)e 1 + w K v +(F K J) ρ d. (1) With E{e / 1 }=, since x 1 is the UMV prediction of the initial state x,wehavee{e +1/ }= ifandonlyifthe hybrid state feedbac gain K satisfies the following decoupling constraint: ( F K J) ρ =. (11) Theorem 1 presents the EIIKF for left invertible linear discrete-time systems of structural delay α=. Theorem 1. The EIIKF generating the UMV state prediction so that E{ x +1 }=x +1 { ρ j } includes a Kalman filter, ie, x +1 = A x 1 + Fρ u c + K ( y C x 1 Jρ u c ) +μ d (12) P +1 =(A K C)P 1 (A K C) T + K V K T + W +μ Q μ T (13) with K = AP 1 C T (CP 1 C T + V) 1 updated online via μ =(F K J)I ρ with Iρ = I q ρ from the minimum variance unnown input estimate d given by d = G (y C x 1 Jρ u c ) (14) Q = ρ [J T( CP 1 C T + V ) 1 J ] 1 (15) with G = Q J T (CP 1 C T + V) 1 where E{ d }= ρ d. Initialized by x 1 = x and P 1 = P, the EIIKF can also be expressed from L = P 1 C T (CP 1 C T + V) 1 as follows: ( )) x = x 1 + L y C x 1 J (ρ u c + d (16) P =(I L C)P 1 (I L C) T + L V L T + L JQ J T L T (17) x +1 = A x + F(ρ u c + d ) (18) P +1 = AP A T + W + FQ F T FQ J T L T AT A L JQ F T, (19) where x is the UMV state estimation so that E{ x }=x { ρ j }.

5 12 RHOUMA ET AL. Proof. The decoupling constraint (11) can be rewritten as follows: K JI ρ = FIρ. (2) ([ ]) F The existence condition ran J = ran( J) for a solution to Equation 2 when ρ = 1 is always satisfied under ran(j) =q.thegaink parameterized as K = K +μ G (21) is a solution to Equation 2 K if and only if I ρ G is a solution to I ρ G J = I ρ, (22) where K isthefreepartofthegainandi ρ G is the constrained hybrid part of the gain. From the implicit unnown input estimator, ie, d = I ρ G ( y C x 1 Jρ u c ) (23) Q = I ρ G ( CP 1 C T + V )( I ρ G ) T (24) satisfying E{ d }=I ρ d under Equation 22, the global optimization problem, ie, min tr ( ) P +1 under K JI ρ = K FIρ (25) can be equivalently expressed as 2 decoupled optimization problems as follows: min tr ( ) P +1 (26) K min tr(q ) under I ρ G I ρ G J = I ρ (27) with P +1 =(A K C)P 1 (A K C) T + K V K T + W (see the 2-stage optimization strategy developed in the wor of Keller and Sauter 23 ). The unique solution to Equation 26 coincides with the Kalman filter's gain, ie, K = AP 1 C T (CP 1 C T + V) 1, (28) and the unique solution to Equation 27 gives I ρ G = Q J T (CP 1 C T + V) 1. (29) Relations (16) to (19) derive to the lin K = A L between predictive and estimated forms of the Kalman filter's gain. Notice that when the arrival sequence {ρ j } of pacet dropouts follows a random Bernoulli process with λ=pr[ρ = ], the arrival sequence { ρ j } of unnown inputs follows a random Bernoulli process with λ =Pr[ ρ = 1] =1 λand the time evolution of the state prediction error covariance matrix P +1/ is random. Let E{P +1 { ρ j } } be the mathematical expectation of P +1/ taen with respect to { ρ j },and λ c is the critical arrival rate of unnown inputs so that lim E{P +1 { ρ j } } < when λ λ c or lim E{P +1 { ρ j } } when λ > λ c. From Appendix B in which we give an equivalent time-varying state filter decoupled from unnown input estimation, let us define the following matrices A = A, W = W, C =Σ C,andV = V when ρ = anda 1 = A, W 1 = W, C 1 =Σ C,andV 1 = V when ρ = 1. Theorem 2. The system has no unstable invariant zeros, and we have lim E{P +1 { ρ j } } < λ [ 1] when (A 1, C 1 ) is detectable. Otherwise, when (A 1, C 1 ) is undetectable, we get lim {P E +1 { ρ } j < λ if there exists K R n,m,k 1 R n,m q,andy= Y T R n,n with < Y < Isothat } Y T Ψ λ(y, K, K 1 )= p S T p1 S 1 p S p1 S 1 Y Y [ λ c ] (3) >, (31)

6 RHOUMA ET AL. 13 where S i = YA i + K i C i,p = 1 λ, p 1 = λ. { The lower bound λ c of λ c } in Equation 3 is a solution to the linear matrix inequality (LMI) feasibility problem λ c = arg max Ψ λ(y, K, K 1 ) >. λ Proof. The time-varying Riccati difference equation (RDE) in Appendix B can be expressed as a switching standard RDE, ie, 1 [ P +1 = Ai P 1 A T i + W i A i P 1 C T i (C ip 1 C T i + V i ) 1 C i P 1 A T ] i (32) σ i i= with σ = 1 ρ, σ 1 = ρ. A deterministic upper bound so that lim E{P +1 { ρ j } } S is a solution to the modified algebraic RDE (ARDE), ie, S = 1 p i [A i SA T i + W i A i SC T ( i Ci SC T ) ] 1Ci i + V i SA T i = g λ(s), (33) i= where p i is the probability of the events σ i = 1 depending on λ. Let us define the auxiliary function, ie, satisfying X = 1 p i [(A i K i C i )X(A i K i C i ) T + K i V i K T i + W i ]=Φ λ(x) (34) i= g λ(x) Φ λ(x) (K, K 1 ). (35) As explained in the wor of Sinopoli et al, 22 if there exists K, K 1,andZ > suchthatz > Φ λ(x), then there exists a unique stabilizing solution to the modified ARDE (33) and the following statements are equivalent. 1. (K, K 1 ) and Z > suchthatz > Φ λ(z). 2. (K, K 1 ) and < Y < I such that Ψ λ(y, K, K 1 ) >. The above result leads to the lower bound of the { critical value for the} unnown input occurrence rate obtained by solving the LMI feasibility problem λ c = arg max Ψ λ(y, K, K 1 ) >.Wearegoingtoshowthatthereexistsno λ critical value beyond which the prediction error covariance matrix becomes unbounded when the pair (A 1, C 1 ) is detectable (or equivalently, when the plant has no unstable invariant zero). When λ =1, we have p =, p 1 = 1, and lim E{P +1 { ρ j } } S. Weget lim P +1 S since the time evolution of P +1/ is deterministic when ρ = 1, where S is now a solution to the standard ARDE, ie, S = A 1 SA T 1 + W 1 A 1 SC T 1 ( C1 SC T 1 + V 1) 1C1 SA T 1. (36) The standard ARDE (36) has a strong solution (all the modes of (A 1 K 1 C 1 ) with K 1 = A 1 SC T 1 (C 1SC T 1 + V 1) 1 are inside or on the unit circle) if and only if the pair (A 1, C 1 ) is detectable leading to lim P +1 S < when λ =1 and to lim E{P +1 { ρ j } } < λ [ 1], since the mathematical expectation of the random covariance decreases with the arrival rate of the unnown inputs. 4 STATE FILTERING FOR SYSTEMS WITH ONE STRUCTURAL DELAY Consider when α=1the input transformation matrix S computed in Appendix A so that the transformed state prediction error (1) given by e +1 =(A K C)e 1 + w K v +(FS K JS) ρ S 1 d (37) and can be equivalently expressed as e +1 =(A K C)e 1 + w K v + ([ F F 1 ] K [ J ]) [ ρ d ρ d1 ]. (38)

7 14 RHOUMA ET AL. With E(e / 1 )=since x 1 is the UMV prediction of the initial state x,wehavee{e +1 }=F 1 ρ d1 ifandonly if the state feedbac gain K satisfies the following sequential decoupling constraints: (F K J ) ρ = (39) (A K C)F 1 ρ 1 = (4) with ρ 1 = for the binary memory initial state. Theorem 3 presents the EIIKF for left invertible linear discrete-time systems of structural delay α=1. Theorem 3. The EIIKF generating the UMV state prediction so that E{ x +1 }=x +1 when ρ = includes a Kalman filter, ie, x +1 = A x 1 + Fρ u c + ( K y C x 1 Jρ u c ) +μ d (41) P +1 =(A K C)P 1 (A K C) T + K V K T + W +μ Q μ T (42) with K = AP 1 C T (CP 1 C T + V) 1 updated online via μ =(M K N)I ρ,m = [ ] [ ] F AF 1,N = J CF 1,and I ρ = diag ([ ]) ρ ρ 1 from the unnown input estimate d of covariance Q / given by ( d = G y C x 1 Jρ u c ) (43) [ (NI ] ρ T(CP 1 + Q = ) C T + V) 1 NI ρ (44) with G = Q (NI ρ )T (CP 1 C T + V) 1 where E{ d }= [ ρ ( d )T ρ 1 ( d ] 1 1 )T T and where X + in Equation 44 is the unique Moore-Penrose generalized inverse of X. Initialized by x 1 = x, P 1 = P, and ρ 1 =, the EIIKF can be expressed on its estimated form from L = P 1 C T (CP 1 C T + V) 1 as follows: ( ) x = x 1 + L y C x 1 Jρ u c N d (45) P =(I L C)P 1 (I L C) T + L V L T + L NQ N T L T (46) x +1 = A x + Fρ u c + M d (47) P +1 = AP A T + W + MQ M T MQ N T L T AT A L NQ M T, (48) where x is the UMV state estimation so that E{ x }=x { ρ j }. Proof. The decoupling constraints (39) and (4) can be jointly expressed as K MI ρ = NIρ. (49) ([ ]) N The existence condition ran M = ran(n) for a solution to Equation 49 when ρ = 1and ρ 1 = 1 are satisfied, ([ ] ) N and thus, ran M I ρ = ran(ni ρ ) { ρ, ρ 1 } since N is a full column ran matrix, as explained in Appendix A. The gain K can then be parameterized as follows: K = K +μ G (5) with I ρ G constrained to satisfy I ρ G NI ρ = Iρ. (51) From the implicit unnown input estimator, ie, d = I ρ G ( y C x 1 Jρ u c ) (52) Q = I ρ G (CP 1 C T + V) ( I ρ G ) T (53) satisfying E{ d }= [ ρ ( d )T ρ 1 ( d 1 1 )T ] T under Equation 51, the global optimization problem, ie, min K tr(p +1 ) under K MI ρ = NIρ (54)

8 RHOUMA ET AL. 15 can be equivalently expressed as 2 decoupled optimization problems, ie, min tr( P +1 ) (55) K min tr(q ) subject to I ρ G I ρ G NI ρ = Iρ (56) with P +1 =(A K C)P 1 (A K C) T + K V K T + W (see the 2-stage optimization strategy developed in the wor of Keller and Sauter 23 ). The unique solution to Equation 55 coincides with the Kalman filter's gain, ie, K = AP 1 C T (CP 1 C T + V) 1, (57) and the unique solution to Equation 56 gives I ρ G = Q (NI ρ )T (CP 1 C T + V) 1. (58) Relations (45) to (48) derive to the lin K = A L between predictive and estimated forms of the Kalman filter's gain. Under permanent unnown inputs, the filter of Theorem 3 reconstructs the unnown input sequence { d, d 1 }, 1 where d = θ d R q and d1 = θ 1 1d 1 R q 1 represent 2 independent linear combinations of d and d [ ] 1 θ described by θ and θ 1 in S 1 =. Leading to reduce the computational time requirements, the estimation of d 1 θ 1 is avoided, which is the main difference with the joint unnown inputs and the state filtering problem presented in the wor of Yong et al, 16 where d is estimated from measurements y and y +1. From the different binary situations of { ρ, ρ 1 } described by {, }, {1, }, {, 1},and{1, 1}, let us define {σ, σ1, σ2, σ3} where σ i = 1when{ ρ, ρ 1 } is equal to its (i + 1)th situations, otherwise σ i =. From Appendix C, given an equivalent time-varying state filtering decoupled from the unnown input estimation, let us define the following matrices A i = A 1, W i = W 1, C i =Σ 1C,andV i = V 1 when { ρ, ρ 1 } is equal to its (i + 1)th situation. Theorem 4. The system has no unstable invariant zero, and we have lim E{P +1 { ρ j } } < λ [ 1] when (A 3, C 3 ) is detectable. Otherwise, when (A 3, C 3 ) is undetectable, we have lim {P E +1 { ρ } } j < λ [ λ c ] (59) if there exists K R m,n,k 1 R m q,n,k 2 R m q1,n, K 3 R m q,n,and < Y < IwithY= Y T R n,n so that Y p S p1 S 1 p2 S 2 p3 S 3 p S T Y Ψ λ(y, K, K 1, K 2, K 3 )= p1 S T 1 Y p2 S T Y 2 p3 S T Y 3 > (6) with S i = YA i + K i C i,p =(1 λ) 2,p 1 = p 2 = { λ (1 λ), andp 3 = λ 2. The lower } bound λ c of λ c in Equation 59 is a solution to the LMI feasibility problem λ c = arg max Ψ λ(y, K, K 1, K 2, K 3 ) >. λ Proof. The time-varying RDE in Appendix C can be expressed as a switching standard RDE, ie, P +1 = 3 σ i i= [ Ai P 1 A T i + W i A i P 1 C T i (C ip 1 C T i + V i ) 1 C i P 1 A T ] i, (61) and the deterministic upper bound S so that lim E{P +1 { ρ j } } S is solution to the modified ARDE, ie, S = 3 [ p i Ai SA T i + W i A i SC T i (C isc T i + V i ) 1 C i SA T ] i = g λ(s), (62) i= where p i is the probability of the event σ i = 1 depending on λ. Therefore, let us define the auxiliary function as X = 3 p i [(A i K i C i )X(A i K i C i ) T + K i V i K T i + W i ]=Φ λ(x) (63) i=

9 16 RHOUMA ET AL. satisfying g λ(x) Φ λ(x) (K, K 1, K 2, K 3 ). (64) By incrementing the results presented in the wor of Sinopoli et al, 22 there exists a unique stabilizing solution to the modified ARDE (62) if there exists K, K 1, K 2, K 3,andZ > such that Z > Φ λ(x), and then, the following statements are equivalent. 1. (K, K 1, K 2, K 3 ) and Z > suchthatz > Φ λ(z). 2. (K, K 1, K 2, K 3 ) and < Y < I such that Ψ λ(y, K, K 1, K 2, K 3 ) >. The above results lead to the lower bound of { the critical value for the unnown } input occurrence rate obtained by solving the LMI feasibility problem λ c = arg max Ψ λ(y, K, K 1, K 2, K 3 ) >. As in the previous section, we are going λ to show that there exists no critical value beyond which the prediction error covariance matrix becomes unbounded when the pair (A 3, C 3 ) is detectable. When λ =1, we have p = p 1 = p 2 =, p 3 = 1, and lim E{P +1 { ρ j } } S.We get lim P +1 S since the time evolution of P +1/ is deterministic when ρ = 1, where S is now solution to a standard ARDE, ie, S = A 3 SA T 3 + W 3 A 3 SC T ( 3 C3 SC T 3 + V 1C3 3) SA T 3. (65) The standard ARDE (65) has a strong solution if and only if the pair (A 3, C 3 ) is detectable. This leads to lim P +1 S < when λ =1andto lim E{P +1 { ρ j } } < λ [ 1], since the mathematical expectation of the random covariance decreases with the arrival rate of the unnown inputs. 5 STATE FILTERING FOR SYSTEMS WITH STRUCTURAL DELAY GREATER THAN ONE When α > 1, Appendix A computes the input transformation matrix S so that Equation 1 can be equivalently rewritten as follows: ρ d e +1 =(A K C)e 1 + w K v + ([F F i F α ] K [J ]). (66) ρ dα With E(e / 1 )=since x 1 is the UMV prediction of the initial state x, the bias caused by the active unnown inputs is confined on the state prediction error as E{e +1 }= satisfies the following sequential decoupling constraints: i 1 α i=1 j= (F K J ) ρ = (A K C)A i 1 F i ρ i = (A K C)A α 1 F α ρ α = with ρ 1 =,, ρ α = for the binary memory initial state. ρ di A j F i ρ j di j if and only if the state feedbac gain K Theorem 5. The EIIKF generating the UMV state prediction so that E{ x +1 }=x +1 when ρ =,, ρ α 1 = includes a Kalman filter, ie, x +1 = A x 1 + Fρ u c + K ( y C x 1 Jρ u c ) +μ d (68) P +1 =(A K C)P 1 (A K C) T + K V K T + W +μ Q μ T (69) with K = AP 1 C T (CP 1 C T + V) 1 updated online via μ = (M K N)I ρ,m = [ ] F A i F i A α F α, N = [ ] J CA i 1 F i CA α 1 ρ F α,andi = diag ([ ]) ρ ρ i ρ α from the unnown input estimate d of covariance Q / given by d = G (y C x 1 Jρ u c ) (7) (67)

10 RHOUMA ET AL. 17 Q = [ with G = Q (NI ρ )T (CP 1 C T + V) 1, where E{ d }= initialized with x 1 = x,p / 1 = P,and ρ 1 =,, ρ α =. [ (NI ρ) T ( CP 1 C T + V ) ] 1 + ρ NI (71) ρ ( d )T ρ i ( d i i )T ρ α ( d α α )T ] T. The EIIKF is Proof. The decoupling constraints (67) can be jointly expressed as K MI ρ = NI ρ. The existence condition ran ([ ]) N M = ran(n) for a solution to K MI ρ = NIρ when ρ = 1,, ρ α = 1 is satisfied, and thus, ([ ] ) N ran M I ρ = ran(ni ρ ) for a solution to K MI ρ = NI ρ { ρ,, ρ i,, ρ α } since N is a full column ran matrix as explained in Appendix A (when ρ =,, ρ α = 1, the decoupling constraints disappear). The EIIKF with α > 1 derives then to the optimization strategy previously described in Section 3 or 4. Under permanent unnown inputs, the filter of Theorem 5 reconstructs the unnown input sequence { d [, d 1,, d i ],, d α }, 1 i α where d i =θ i id i R q i represents a linear combination of d i described by θ i in S 1 = θ i. Leading to reduce the computational time requirements, the estimation of d α is avoided. This is the fundamental difference with the joint unnown inputs and state filtering problem presented in the wor of Gillijns and De Moor, 15 where d is estimated from measurements y, y +1,, y +α. The stochastic stability conditions should be derived from a time-varying version of the EIIKF depending on matrices of 2 α+1 variable size. The minimum-variance state prediction reaches the UMV state prediction of the whole state vector when the communication between subsystems is restored during α successive periods of time. This is the ey point of our autonomous distributed state filtering strategy. 6 ILLUSTRATIVE EXAMPLE Consider a compartmental model of 1 compartments (see Figure 2) that exchanges mass or energy through mutual interaction. The inputs are provided to compartments, 8, and 9, whereas compartments, 1, 4, 5, 7, and 9 are measured and constitute the outputs. The conservation laws governing the compartment model are given by x +1 = x βx +α(x1 x ) x i +1 = xi βxi +α(xi+1 x i ) α(xi xi 1 ), i = 1..8 x 9 +1 = x9 βx9 α(x9 x8 ), (72) where < β < 1 is the loss coefficient and < α < 1 is the flow coefficient. The above relations can be written as a state model with x = [ x x i x 9 T ] of the transition matrix, ie, 1 α β α α 1 α β α A = R 1,1. (73) α 1 α β Directly affected by energy sources, measurements ỹ, and ỹ 9, are expressed as ỹ, = x +γũ 1, and ỹ 9, = x 9 +γũ 2,, where γ is a nown parameter. FIGURE 2 Compartmental model of 1 compartments

11 18 RHOUMA ET AL. With y 1, = [ ] T, ỹ 9, ỹ 7, ỹ 1, u1, = ũ 1,,andu c = [ ] T, ũ 1, 2, ũ 3, the state model of the plant (8) is obtained with [ ] [ ] [ ] 1 γ C 1 = 1, D 1 =, B 1 = e 1, J 1 =,andf 1 = [ e 1 e ] 9,wheree i is a column vector of the 1 appropriate size with one at the ith position and zero elsewhere. With y 2, = [ ] T, ỹ, ỹ 4, ỹ 5, u 2 = [ ] T,andu ũ 2, ũ c 3, = ũ 2, 1,, the state model of the second subsystem (8) is obtained [ ] [ ] 1 γ with C 2 = 1, D 2 = J 1, B 2 = F 1, J 2 =,andf 2 = B 1. 1 For loss and flow coefficients, we have chosen α=β=, 1andV 1 = V 2 = I 3, W = I 1 for measurement and state noise covariance. The first subsystem is left invertible with structural delay α 1 = 1 and the second with α 2 =. The input transformation used in Theorem 3 is here equal to S = I 2, so we conserve the same filter's order and structure for both cases, ie, α 1 = 1orα 2 =. Both subsystems have no invariant zero and EIIKF's stochastic stability conditions are satisfied, ie, λ [ 1 ] for α 1 = 1andα 2 =. In our simulations, the exchange rate of data u c = [ ] T ũ 1, 2, ũ 3, and u c = ũ 2, 1, between subsystems is fixed at λ=pr[ρ = ] =, 7and λ =Pr[ ρ = 1] =, 3, which represents the arrival rate of random unnown inputs d 1 = [ ] T ũ 2, ũ 3, and d 2 = ũ 1, for the first and the second subsystem, respectively. Figure 3 shows the random evolution of data exchange between the 2 subsystems. First subsystem: The trace of the estimation error covariance matrix and its upper bound, obtained from the EIIKF described in Theorem 3, are plotted in Figure 4. Figure 5 shows the state estimate of the third unmeasured state. The actual and estimated second drive inputs ũ 2, are shown in Figure 6. Second subsystem: The trace of the estimation error covariance matrix and its upper bound, obtained from the IIKF of Theorem 1, are plotted in Figure 7. Figure 8 shows the state estimate of the third unmeasured state. The actual and estimated first drive inputs ũ 1, are shown in Figure FIGURE 3 Random binary sequence ρ [Colour figure can be viewed at wileyonlinelibrary.com] FIGURE 4 First subsystem: trace of the estimation error covariance matrix and its upper bound [Colour figure can be viewed at wileyonlinelibrary.com]

12 RHOUMA ET AL FIGURE 5 First subsystem: third component of the state estimation vector [Colour figure can be viewed at wileyonlinelibrary.com] FIGURE 6 First subsystem: intermittent reconstruction of ũ 2, [Colour figure can be viewed at wileyonlinelibrary.com] FIGURE 7 Second subsystem: trace of the estimation error covariance matrix and its upper bound [Colour figure can be viewed at wileyonlinelibrary.com] FIGURE 8 Second subsystem: third component of the state estimation vector [Colour figure can be viewed at wileyonlinelibrary.com]

13 11 RHOUMA ET AL FIGURE 9 Second subsystem: intermittent reconstruction of ũ 1, [Colour figure can be viewed at wileyonlinelibrary.com] 7 CONCLUSION This paper has extended the use of the Kalman filter with the intermittent unnown inputs for joint input and state estimation in left invertible linear discrete-time systems with zero, one, or more delays. Under sequential unnown input decoupling constraints and from a 2-stage optimization strategy, all presented filters have been designed to be computationally efficient. We have shown that obtained filters can be used to solve online the autonomous distributed state filtering problem in a distributed actuator networ. When the arrival rate of the unnown inputs does not exceed a critical value, the stochastic stability conditions have shown that the conditions for the existence of a stable left inverse can be relaxed. The case where each subsystem receives time-varying delay information about the physical interaction between subsystems with delays less, equal, or greater than the structural delays is currently under consideration by the authors. ORCID Taouba Rhouma REFERENCES 1. Kailath T, Sayed AH, Hassibi B. Linear Estimation. Vol 1. Upper Saddle River, NJ: Prentice Hall; Yin S, Zhu X, Qiu J, Gao H. State estimation in nonlinear system using sequential evolutionary filter. IEEE Trans Ind Electron. 216;63(6): Yin S, Shi P, Yang H. Adaptive fuzzy control of strict-feedbac nonlinear time-delay systems with unmodeled dynamics. IEEE Trans Cybern. 216;46(8): Kitanidis PK. Unbiased minimum-variance linear state estimation. Automatica. 1987;23(6): Darouach M, Zasadzinsi M, Keller J-Y. State estimation for discrete systems with unnown inputs using state estimation of singular systems. Paper presented at: American Control Conference; 1992; 29, ; Chicago, IL, USA. 6. Chen J, Patton RJ. Optimal filtering and robust fault diagnosis of stochastic systems with unnown disturbances. IEE Proc Control Theory Appl. 1996;143(1): Chen J, Patton RJ. Robust Model-Based Fault Diagnosis for Dynamic Systems. Vol 3. Berlin, Heidelberg, Germany: Springer Science & Business Media; Cheng Y, Ye H, Wang Y, Zhou D. Unbiased minimum-variance state estimation for linear systems with unnown input. Automatica. 29;45(2): Darouach M, Zasadzinsi M. Unbiased minimum variance estimation for systems with unnown exogenous inputs. Automatica. 1997;33(4): Darouach M, Zasadzinsi M, Boutayeb M. Extension of minimum variance estimation for systems with unnown inputs. Automatica. 23;39(5): Hou M, Patton RJ. Optimal filtering for systems with unnown inputs. IEEE Trans Autom Control. 1998;43(3): Sundaram S, Hadjicostis CN. Optimal state estimators for linear systems with unnown inputs. Paper presented at: 26 45th IEEE Conference on Decision and Control; 26; ; San Diego, CA, USA. 13. Fang H, Shi Y, Yi J. On stable simultaneous input and state estimation for discrete-time linear systems. Int J Adapt Control Signal Process. 211;25(8): Gillijns S, De Moor B. Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feed through. Automatica. 27;43(5):

14 RHOUMA ET AL Gillijns S, De Moor B. System inversion with application to filtering and smoothing in the presence of unnown inputs. Internal Report. Leuven, Belgium: Katholiee Universiteit Leuven; Yong SZ, Zhu M, Frazzoli E. A unified filter for simultaneous input and state estimation of linear discrete-time stochastic systems. Automatica. 216;63: Blane M, Kinnaert M, Lunze J, Staroswieci M, Schröder J. Diagnosis and Fault-Tolerant Control. Secaucus, NJ: Springer-Verlag New Yor, Inc; Patton RJ, Kambhampati C, Casavola A, Zhang P, Ding S, Sauter D. A generic strategy for fault-tolerance in control systems distributed over a networ. Eur J Control. 27;13(2): Hespanha JP, Naghshtabrizi P, Xu Y. A survey of recent results in networed control systems. Proc IEEE. 27;95(1): Schenato L, Sinopoli B, Franceschetti M, Poolla K, Sastry SS. Foundations of control and estimation over lossy networs. Proc IEEE. 27;95(1): Shi L, Epstein M, Murray RM. Kalman filtering over a pacet-dropping networ: a probabilistic perspective. IEEE Trans Autom Control. 21;55(3): Sinopoli B, Schenato L, Franceschetti M, Poolla K, Jordan M, Sastry SS. Kalman filtering with intermittent observations. IEEE Trans Autom Control. 24;49(9): Keller J-Y, Sauter DD. Kalman filter for discrete-time stochastic linear systems subject to intermittent unnown inputs. IEEE Trans Autom Control. 213;58(7): Keller J-Y, Chabir K, Sauter D. Input reconstruction for networed control systems subject to deception attacs and data losses on control signals. Int J Syst Sci. 216;47(4): Antsalis P. Stable proper nth-order inverses. IEEE Trans Autom Control. 1978;23(6): Kobayashi N, Naamizo T. An observer design for linear systems with unnown inputs. Int J Control. 1982;35(4): Sain MK, Massey JL. Invertibility of linear time-invariant dynamical systems. IEEE Trans Autom Control. 1969;14(2): Silverman LM. Inversion of multivariable linear systems. IEEE Trans Autom Control. 1969;14(3): Keller J-Y. Fault isolation filter design for linear stochastic systems. Automatica. 1999;35(1): Howtocitethisarticle: Rhouma T, Keller J-Y, Chabir K, Sauter D, Abdelrim MN. Optimal state filter from sequential unnown input reconstruction: Application to distributed state filtering. Int J Adapt Control Signal Process. 218;32: APPENDIX A Consider the autonomous noise-free system (3), without data exchange between subsystems, namely, under permanent unnown inputs, rewritten as x +1 = Ax + Fd y = Cx + Jd. (A1) System (A1) is left invertible with structural delay α = {, 1, 2, } if and only if there exists a finite solution to the recursion, ie, α=min {v ran(p ν ) ran(p ν 1 )=q, v =, 1, 2, } (A2) initialized by P 1 =, where J CF J P ν = [D ν D ν 1 D ], D ν =, D ν 1 =,, CA ν 1 F CA ν 2 F D 1 = J, D =. (A3) CF J

15 112 RHOUMA ET AL. When [ α solution to Equation A2 satisfies < α <, the input transformation d = S 1 d = ( d )T ( d 1 )T ( d i )T ( d α)t] T with S R q,q nonsingular can always be computed so that the transformed system, ie, { x+1 = Ax + F d y = Cx + J d F = FS = [F F 1 F i F α ], J = JS = [J ], (A4) satisfies ran(n) =q (A5) with N = [ J CF 1 CA i 1 F i CA α 1 F α ],whereca j 1 F i = j {1,, i 1} for i = 2, 3,, α. Proof. Obtained from Equation A1, the following expression d = M y ν y ν+1 y C CA CA ν 1 d ν x ν = MP ν d d ν+1 (A6) shows that d = d α if and only if M R q,m(ν+1) is the solution to MP ν = [I ]. (A7) ([ ]) P The existence condition ran ν [I ] = ran(p ν ) for a solution to Equation A7, rewritten as ([ ]) Dν P ran ν 1 I = ran(p ν ) or equivalently as ran(p ν ) ran(p ν 1 )=q, shows that the minimum structural delay for the reconstruction of d is given by Equation A2. If Equation A2 gives α=, then the unnown input vector d can be reconstructed without any delay. If Equation A2 gives α=1, then ran(j) =q < q and there exists a permutation matrix S 1p so that [ ] J J = JS1p [ I J + with ran(j )=q and the upper triangular matrix S 1e = J ] so that JS I 1p S 1e = [J ]. Let [ ] F F 1 = FS1 with S 1 = S 1p S 1e. We have leading to ([ ][ ]) ([ ]) J S1 J ran(p 1 )=ran CF J S = ran 1 CF C F 1 J ([ ]) J = ran = ran(p C F 1 J )+ran ([ ]) J C F 1 ran(p 1 ) ran(p )=ran ([ J C F 1 ]) = q. (A8) We conclude from Equation A8 that the transformed system (A4) with d = [ ( d )T ( d 1)T] T = S 1 d, S = S 1, d R q, d1 Rq 1, q1 = q q, F = [F F 1 ], F 1 = F 1, J = [J ] satisfies Equation A5, where N = [J CF 1 ] is a full column ran matrix. If Equation A2 gives α = 2, then ran ([ ]) J C F 1 = q + q 1 < q and there exists a permutation matrix S 2p so that [ ] J C F 1 S2p = [ ] J CF 1 C F 2 with ran ([J CF 1 ]) = q + q 1 and the upper triangular matrix S 2e = [ ] Iq + q 1 [J CF 1 ] + C F 2 so that [ ] J I CF 1 C F 2 S2e = [J CF 1 ].Let [ ] F F 1 F 2 = FS2 with S 2 = S 1 S 2p S 2e.

16 RHOUMA ET AL. 113 We have ran(p 2 )=ran = ran = ran ([ J CF J CAF CF J ][ S2 S 2 S 2 ]) ([ J CF CF 1 J CAF CAF 1 CA F 2 CF CF 1 J ([ J CF 1 J CA F 2 CF 1 J ]) ]) leading to ran(p 2 ) ran(p 1 )=ran ([ ]) J CF 1 CA F 2 = q (A9) after [ some ran manipulations. We conclude from Equation A9 that the transformed system (A4) with d = ( d )T ( d 1 )T ( d 2)T] T = S 1 d, S = S 2, d Rq, d1 Rq 1, d2 Rq 2, q2 = q q q 1, F = [F F 1 F 2 ], F 2 = F 2, J = [J ], d satisfies Equation A5, where N = [J CF 1 CAF 2 ] is a full column ran matrix [ and CF 2 =. The recursive computation of S so that the transformed system (A4) with d = S 1 d = ( d )T ( d 1 )T ( d i )T ( d α)t] T satisfies Equation A5 derives from the same inductive reasoning. Note that, in Equation A4, vector d i and matrix F i may be empty when i < α during the sequential processing. APPENDIX B For measurement noise covariance V = I, the state filtering of Theorem 1 is equivalent to the following time-varying Kalman filter: x +1 = A x 1 + F(ρ u c +Π y )+K (Σ y Σ C x 1 Jρ u c ) (B1) with P +1 =(A K Σ C)P 1 (A K Σ C) T + K V (K ) T + W K = A P 1 C T Σ T (Σ CP 1 C T Σ T + V ) 1 R n,m ρ q Σ =ρ I m + ρ (ΣΣ T ) 1 2 Σ, Π = ρ J + A = A FΠ C, W = W + FΠ Π T FT (B2) (B3) (B4) (B5) V =ρ I m + ρ I m q, (B6) where Σ=β(I J(J) + ) is an arbitrary matrix so that ran(σ) = m q with β R m q,m. [ ] Σ Proof. Let T = Π R m,m be the output transformation satisfying ran(t )=m ρ so that the transformed measurement vector T y generates the time-varying size measurement vector, ie, Σ y =Σ Cx + Jρ u c +Σ v R m ρ q (B7) decoupled from ρ d. The intermittent measurement vector, ie, Π y =Π Cx + ρ d +Π v R ρ q is sensitive to ρ d. Deduced from Equation B7, the expression ρ d =Π y Π Cx Π v substituted in Equation 8 leads to the following free unnown input system: x +1 = A x + F(ρ u c +Π y )+ w (B8) (B9) with w = w FΠ v and where E Σ y =Σ Cx + Jρ u c +Σ v { [ ][ ] } T [ ] w wj W Σ v Σ j v = δ j V,j. (B1) (B11)

17 114 RHOUMA ET AL. For system (B9)-(B1) and Equation B11, consider the following time-varying Kalman filter: x +1 = A x 1 + F(ρ u c +Π y )+K (Σ y Σ C x 1 Jρ u c ) P +1 =(A K Σ C)P 1 (A K Σ C) T + K V (K ) T + W. (B12) (B13) The minimization of tr(p +1/ ) with respect to K leads to the variable-size gain (B3). APPENDIX C For measurement noise covariance V = I, the state filtering of Theorem 3 is equivalent to the following time-varying Kalman filter: x +1 = A 1 x 1 + Fρ u c + F Π y + A F 1Π 1 y + K (Σ 1 y Σ 1 C x 1 Jρ u c ) (C1) with P +1 =(A 1 K Σ 1 C)P 1(A 1 K Σ 1 C)T + K V 1 (K ) T + W 1 K = A 1 P 1C T (Σ 1 )T (Σ 1 CP 1C T (Σ 1 )T + V 1 ) 1 R n,m ρ q ρ 1 q 1 Σ =ρ I m + ρ (Σ Σ T ) 1 2 Σ, Π = ρ (J ) + Σ 1 =ρ 1Σ + ρ ρ 1 (Σ 1 Σ T 1 ) 1 2 Σ 1, Π 1 = ρ 1(Σ CF 1) + A = A F Π C, A1 = A A F 1Π 1 C (C2) (C3) (C4) (C5) (C6) W = W + F Π ΠT FT, W 1 = W + A F 1Π 1 Π1T FT 1 (A )T (C7) V =ρ I m + ρ I m q, V 1 =ρ 1V + ρ ρ 1 I m q, (C8) where Σ =β (I J (J ) + ) with β R m q,m being an arbitrary matrix so that ran(σ )=m q and Σ 1 =β 1 (I Σ CF 1 (Σ CF 1 ) + )Σ with β 1 R m q,m q being an arbitrary matrix so that ran(σ1 )=m q. [ Σ ] Proof. Let T = R m,m be the output transformation satisfying ran(t )=m ρ so that the transformed Π measurement vector T y generates the time-varying size measurement vector, ie, Σ y =Σ Cx + Jρ u c +Σ v R m ρ q decoupled from ρ d and the intermittent measurement vector, ie, Π y =Π Cx + ρ d +Π v R ρ q (C1) sensitive to ρ d. Deduced from Equation C1, the expression ρ d =Πy Π Cx Π v substituted in Equation 8 [ with ( d )T ( d ] T 1 )T = S 1 d and [ ] F F 1 = FS leads to the following system: (C9) x +1 = A x + Fρ u c + F Π y + F 1 ρ d1 + w Σ y =Σ Cx + Jρ u c +Σ v with w = w F Π v,where [ w ] [ ] w T [ E j W Σ v Σ j v = j V For system (C11)-(C12) and Equation C13, consider the following filter: ] δ,j. x +1 = A x 1 + F (ρ u c +Π y )+K (Σ y Σ C x 1 Jρ u c ) (C11) (C12) (C13) (C14) P +1 =(A K Σ C)P 1(A K Σ C)T + K V (K )T + W. (C15) The minimization of tr(p +1/ ) with respect to K under (A K ΣC) ρ 1F 1 = sothate { x } +1 = x+1 + F 1 ρ d1 derived in the wor of Keller 29 leads to the variable-size gain (C3).