Simultaneous Input and State Estimation with a Delay

Transcription

1 15 IEEE 5th Annual Conference on Decision an Control (CDC) December 15-18, 15. Osaa, Japan Simultaneous Input an State Estimation with a Delay Sze Zheng Yong a Minghui Zhu b Emilio Frazzoli a Abstract In this paper, we present recursive algorithms for linear iscrete-time stochastic systems that simultaneously estimate the states an unnown inputs in an unbiase minimumvariance sense with a elay. By allowing potential elays in state estimation, the stricter assumptions in a previous wor [1] can be relaxe. Moreover, we show that a system property nown as strong etectability plays a ey role in the existence an stability of the asymptotic estimator with a elay we propose. I. INTRODUCTION For linear iscrete-time stochastic systems with nown inputs, the Kalman filter optimally extracts information about a variable of interest from noisy measurements. However, these inputs that may represent unnown external rivers, instrument faults or attac signals are often not accessible. This problem of simultaneous state an input estimation is foun across many isciplines an applications, from the real-time estimation of mean areal precipitation uring a storm [] to input estimation in physiological an transportation systems [1], [] to fault etection an iagnosis []. Literature review. While state estimation for linear stochastic systems with unnown inputs have been wiely stuie uner various assumptions [], [5] [7], the problem of concurrently obtaining minimum-variance unbiase estimates for both the states an the unnown inputs has receive less attention. Initial research was focuse on particular classes of linear systems with unnown inputs [8] [1], an more recently, less restrictive estimators of both state an unnown input have been propose in [1], [1], [1]. However, these estimators are restricte to estimating the states an unnown inputs at the same time step (i.e., without elay) an thus only apply to a limite class of systems. On the other han, current results for linear eterministic systems [15], [16] suggest that state an input estimation is possible for a broaer class of systems if elays are allowe. Such a filter with a elay for stochastic systems has been recently propose in [17], but only for systems without irect feethrough an with an emphasis on unbiaseness but not the optimality of the input estimates. Contributions. We consier simultaneous input an state estimation with a elay (i.e., the estimation of inputs an states up to time step from the measurements up to time step + L for some integer L ) with less restrictive assumptions on the system than currently assume in the literature, an hence for a broaer class of systems. We a S.Z Yong an E. Frazzoli are with the Laboratory for Information an Decision Systems, Massachusetts Institute of Technology, Cambrige, MA, USA ( szyong@mit.eu, frazzoli@mit.eu). b M. Zhu is with the Department of Electrical Engineering, Pennsylvania State University, University Par, PA, USA ( muz16@psu.eu). propose recursive algorithms that are optimal in the unbiase minimum-variance sense for these systems, along with necessary an sufficient conitions for the existence of stable estimators. Finally, we relate the stability an existence of our estimators to strong etectability of the system. II. PROBLEM STATEMENT Consier the linear time-invariant iscrete-time system x +1 = Ax + Bu + G + w (1) y = Cx + Du + H + v where x R n is the state vector at time, u R m is a nown input vector, R p is an unnown input vector, an y R l is the measurement vector. The process noise w R n an the measurement noise v R l are assume to be mutually uncorrelate, zero-mean, white ranom signals with nown an boune covariance matrices, Q = E[w w ] an R = E[v v ], respectively for all. The matrices A, B, C, D, G an H are also nown. Note that no assumption is mae about H either being the zero matrix (no irect feethrough), or having full column ran when there is irect feethrough. Without loss of generality, we assume that n l 1, l p an m, that the current time variable r is strictly nonnegative, an that x is inepenent of v an w for all an r[g H ] = p, where r(m) enotes the ran of matrix M. Given that filtering an smoothing algorithms are typically initiate by an initial state estimate (biase or otherwise), it maes sense to thin of an estimator with a elay as an estimator that can uniquely provie state an input estimates at all times after a possible initial elay (cf. the notion of invertibility in Definition ). In aition, we wish for an estimator whose the state an input estimates are asymptotically unbiase with more observations. More formally, we efine the esire asymptotic estimator with a elay as follows: Definition 1 (Asymptotic/Stable Estimation with a Delay). For any initial state x R n an sequence of unnown input { j } j N in R p, an asymptotic estimator with a elay L (i) uniquely estimates the state ˆx an the unnown inputs { i } i= for all from observations of outputs up to time step + L, i.e., {y i } +L i=, an (ii) provies asymptotically unbiase estimates, i.e., E[ˆx x ] an E[ ˆ ] as. The estimator esign problem can be state as follows: Given a linear iscrete-time stochastic system with unnown inputs (1), esign an asymptotic/stable estimator with a possible elay L (cf. Definition 1) that optimally estimates system states an unnown inputs in the unbiase minimumvariance sense /15/$1. 15 IEEE 68

2 III. PRELIMINARY MATERIAL A. Linear System Properties In this section, we provie efinitions of several linear system properties, which we shall relate to the estimator existence an stability in Section IV. Without loss of generality, we assume that w = an v = 1, an that B = D = (since u is nown). Definition (Invertibility ). The system (1) is sai to be invertible if, given the initial state x, there exists a nonnegative integer L such that the unnown inputs { i } i= (an thus the state x ) can be uniquely recovere from the outputs up to time step + L, enote {y i } +L i=. Definition (Strong observability). The system (1) is sai to be strongly observable if there exists a nonnegative integer L such that x can be uniquely recovere from the outputs up to time step L, enote {y i } L i=, for any initial state x an any sequence of unnown inputs { i } L i=. Equivalently, the system (1) is strongly observable if y = implies x = for any sequence of unnown inputs { i } i N. Definition (Strong etectability). The system (1) is sai to be strongly etectable if y = implies x as for any initial state x an input sequence { i } i N. Next, we consier the Rosenbroc system matrix, also nown as the matrix pencil, R S (z) of system (1): [ ] zi A G R S (z) :=. () C H Definition 5 (Invariant Zeros). The invariant zeros z of the system matrix R S (z) in () are efine as the finite values of z for which the matrix R S (z) rops ran, i.e., r(r S (z)) < nran(r S ), where nran(r S ) enotes the normal ran (maximum ran over z C) of R S (z). A useful characterization of invertibility, strong observability an strong etectability base on the invariant zeros (see, e.g., [1], [] [] for proofs) are as follows: Theorem 1 (Invertibility [], [1]). The system (1) is invertible if an only if r(r S (z)) = n + p for at least one z C. Theorem (Strong observability []). The system (1) is strongly observable if an only if r(r S (z)) = n+p, z C. Theorem (Strong etectability [1]). The system (1) is strongly etectable if an only if r(r S (z)) = n + p, z 1 The analysis can be extene to the case with non-zero w an v by applying the Gauss-Marov theorem [18, Theorem.1.1], an y an x can be replace by E[y ] an E[x ]. This simplification also provies a connection to the system properties of eterministic systems. Note the slightly ifferent efinition compare to [19] in which { i } i= is to be uniquely etermine from {y i } +L i=. C, z 1. From the above theorems, we observe that strong observability implies both invertibility an strong etectability, while strong etectability implies invertibility, i.e., strong observability strong etectability invertibility. Moreover, comparing the above conitions to the PBH test, we observe that strong observability/etectability implies that (A, C) is observable/etectable. B. System Transformation 1) Initial System Transformation: To eal with the potentially ran eficient H, we first carry out a transformation of the system, as is one in our previous wor [1], to ecouple the output equation into two components, one with a full ran irect feethrough matrix an the other without irect feethrough. Let p H := r(h). Using singular value ecomposition, we rewrite the matrix H as H = [ U 1 U ] [ Σ ] [ ] V 1 V where Σ R p H p H is a iagonal matrix of full ran, U 1 R l p H, U R l (l ph), V 1 R p p H, V R p (p ph), an U := [ ] [ ] U 1 U an V := V1 V are unitary matrices. Note that when H =, Σ, U 1 an V 1 are empty matrices while U an V are arbitrary unitary matrices. Then, we efine two orthogonal components of the unnown input given by () 1, = V 1,, = V. () Since V is unitary, = V 1 1, + V, an the system (1) can be rewritten as x +1 = Ax + Bu + G 1 1, + G, + w (5) y = Cx + Du + H 1 1, + v, (6) where G 1 := GV 1, G := GV an H 1 := HV 1 = U 1 Σ. Next, we ecouple the output y using a nonsingular transformation [ ] [ ] [ ] T1 IpH U1 T = = RU (U RU ) U 1 (7) T I (l ph ) to obtain z 1, R p H an z, R l p H given by z 1, = T 1 y = C 1 x + D 1 u + Σ 1, + v 1, z, = T y = C x + D u + v, (8) where C 1 := T 1 C, C := T C = U C, D 1 := T 1 D, D := T D = U D, v 1, := T 1 v an v, := T v = U v. This transform is also chosen such that the measurement noise terms for the ecouple outputs are uncorrelate. The covariances of v 1, an v, are: R 1 := E[v 1, v 1, ] = T 1RT 1, R := E[v, v, ] = T RT = U RU, R 1,(,i) := E[v 1, v,i ] = T 1E[v v i ]T =,, i. Since the initial state, process an measurement noise are assume to be uncorrelate, it can be verifie that v 1, an v, are also uncorrelate with the initial state x an process We aopt the convention that the inverse of an empty matrix is also an empty matrix an assume that operations with empty matrices are possible. U 69

3 noise w. ) Further Transformations: As we have seen in [1], that Σ has full ran enables us to estimate 1, without elay. On the other han, by substituting (5) into (8) to obtain z, = C Ax + C Bu + C G 1 1, +C G, + D u + v,, we observe that, can be estimate with one-step elay if I () := C G has full column ran, i.e., p I () := r(c G ) = p p H, which is the origin for the ran conition that is provie in [1] for the existence of an MVU filter. We emphasize that this ran conition is not only sufficient but also necessary for obtaining a state estimate of x without elay ue to the influence of, on x in (5). We now aress the question of whether a potential elay in estimating the state x from observations {y i } +L i= for some integer L 1 woul relax the requirement that C G be full ran. That is, we consier the scenario when p I () := r(c G ) < p p H. In this case, we again use singular value ecomposition to rewrite the matrix C G as C G = [ ] [ ] [ ] Σ U U V V where Σ R p I () p I () is a iagonal matrix of full ran, U R (l p H) p I (), U R (l p H) (l p H p I () ), V R (p p H) p I (), V R (p p H) (p p H p I () [ ] ), an U () := U U an V () := [ ] V V are unitary matrices. As before, if p I () =, then Σ, U an V are empty matrices an U an V are arbitrary unitary matrices. We then further ecompose, into two orthogonal components:, = V,,, = V,. (9) Since V () is unitary,, can be reconstructe from, an, using, = V, + V,. The system ynamics in (5) can also be rewritten as x +1 = Ax +Bu +G 1 1, +G V, +G V, +w = Ax +Bu +G 1 1, +G, +G, +w (1) where G := G V an G := G V. Next, we again ecouple the output z, using a nonsingular transformation [ ] [ T () T = = U IpI () R U (U R U ) ][ ] U (11) T I (l ph p I () ) to obtain z, R p I () an z, R l p H p I () given by U z, = T z, = C x + D u + v, z, = T z, = C x + D u + v, (1) where C := T C, C := T C = U C, D := T D, D := T D = U D, v, := T v, an v, := T v, = U v,. With this transform, v, an v, are uncorrelate, with covariances given by: R := E[v, v, ] = T R T, R := E[v, v, ] = T R T = U R U, R,(,i) := E[v, v,i ] = T E[v, v,i ]T =,, i. Moreover, v, an v, are uncorrelate with the initial state x an process noise w. Next, from (1) an (1), an simplifying, we have z, = C Ax + C Bu + C G 1 1, + Σ, +C w + D u + v, z, = C Ax + C Bu + C G 1 1, + C w +D u + v, (1) Since 1, can be rewritten from (8) as 1, = Σ (z 1, C 1 x D 1 u v 1, ), we observe that if Σ has full ran, we can uniquely estimate, (with one-step elay). On the other han,, cannot be estimate from z, or z,, but may instea be estimate with two-step elay, i.e., from z,+1 = C Ax + C Bu + C G 1 Σ (z 1, C 1 x D 1 u v 1, ) + C w + D u +1 + v,+1 = C Âx + C Bu + C G 1 Σ (z 1, D 1 u v 1, ) + C w + D u +1 + v,+1 = C ÂAx + C ÂBu + C ÂG 1 1, +C ÂG, + C ÂG, + C Âw +C Bu + C G 1 Σ (z 1, D 1 u v 1, ) +C w + D u +1 + v,+1 (1) where  := A G 1Σ C 1. Thus, we see that if I (1) := C ÂG has full column ran,, (an thus, ) can be uniquely etermine with two-step elay, an the state x can be estimate with one-step elay, i.e., L = 1. Otherwise, further ecomposition proceures as above can be repeate until such a full column ran matrix I (L) is obtaine. Remar 1. Further ecomposition proceures woul involve the repetition of all steps in Section III-B.. That is, for any elay L, we recursively use the singular value ecomposition of I (L) = [ ] [ ] [ ] Σ U L+1 U L+1 V L+1 L+ VL+ to further ecompose L, into L+1, := VL+1 L, an L+, := VL+ L,, as well as z L, into z L+1, := T L+1 z L, an z L+, := T L+ z L, with [ ] T L+1 TL+ = T (L), where T (L) is obtaine similar to (11). In the process, we recursively efine, among others, C L+ := T L+ C L+, C L+ := T L+ C L+, G L+ := G L+ V L+ an G L+ := G L+ V L+. It can be shown that the L-elay invertibility matrices form a sequence {I (L) } L L= as follows: C G, C ÂG, C 6 ÂÂ(1) G 6, C 8 ÂÂ(1)  () G 8, C 1 ÂÂ(1)  ()  () G 1,..., C L+ ÂÂ(1)... Â(L) G L+,..., C L+ ÂÂ(1)... Â(L) G L+, (15) where we have efine Â(1) := (I G Σ C ) an  (L) := (I G L+1 Σ L+1 CL+1ÂÂ(1)... Â(L ) )Â(L) for all L =,..., L 1. We enote as L the maximum number of elay steps beyon which input estimation (given x ) is no longer possible. That is, if, after a elay of L time steps, is not uniquely etermine with given x, then an thus x cannot be uniquely obtaine with The elay L can be foun a priori as the inex for the first matrix in the sequence in (15) with full ran. 7

4 any aitional elay, i.e., with L > L. We next characterize a (conservative) upper boun on this maximum elay. Lemma 1. An upper boun on the maximum elay is given by L u = (n 1)(p p H ). Proof. Note that we repeat the process of further ecomposing L, into L+1, an (L+1), base on the ran of I (L) only if I (L) is ran eficient (incluing having zero ran). In the case that I (L) has ran zero, we observe from the above construction of I (L) that C L+ = C L+, G L+ = G L+ an Â(L) = Â(L ). By the Cayley-Hamilton theorem, if the matrices {I (L) } corresponing to n 1 consecutive elays have ran zero, then any further elay cannot increase the ran of the next I (L) such that an x can be uniquely etermine given x. Next, if I (L) has nonzero but eficient ran, the ecomposition leas to a reuction of the imension of the resulting non-empty (L+1), an thus the number of columns of I (L) by at least 1. Since we starte with p p H columns of I (), this reuction in number of columns can tae place at most p p H times. Therefore, combining the two worst case scenarios gives us L u = (n)(p p H ). Remar. The ability to uniquely etermine the unnown inputs with a elay given a previous state is equivalent to the efinition of invertibility in Definition. Thus, we can compare the upper boun obtaine in Lemma 1 as L u = n(p p H ) p + p H with the upper boun on the inherent elay for invertibility systems given in [19] as n p + p H. Thus, L u is a more conservative upper boun except when p = p H or p = p H + 1. IV. ALGORITHMS FOR SIMULTANEOUS INPUT AND STATE ESTIMATION WITH A DELAY A. Existence Conition for Estimation with a Delay We first visit the question of when an asymptotic estimator with a elay as efine in Definition 1 exists. The proof of the following claims will be provie in Section V-B. Lemma (Unique Estimates with a Delay). Given any any initial estimate ˆx (biase or otherwise), the state an unnown inputs x an can be uniquely estimate for all with a elay L if an only if the system (1) is invertible. Lemma (Asymptotic Unbiaseness). Given any initial state estimate ˆx (biase or otherwise), the estimate biases with a elay L exponentially ten to zero if the pairs (Ã(L), C (L) ) an (Ã(L), ( Q (L) ) 1 ) are etectable an stabilizable, respectively, with Ã(L), C(L) an Q (L) as efine below in Remar. Theorem (Existence). An asymptotic estimator with a elay L (base on Definition 1) exists if: (i) the system (1) is invertible, an (ii) the pairs (Ã(L), C (L) ) an (Ã(L), ( Q (L) ) 1 ) are etectable an stabilizable, respectively. Remar. We now from [1] that à () = (I G M C ) + G M C, C() = C an Q () = (I G M C )(G 1 M 1 R 1 M 1 G 1 +Q)(I G M C ). For L 1, the matrices Ã(L), C (L) an Q (L) can be obtaine irectly with further ecompositions (cf. Remar 1) an with the proceure outline in Section V-B. It can be verifie that we obtain a sequence { C (L) } L L= given by C, C Â, C 6 ÂÂ(1), C 8 ÂÂ(1)  (), C 1 ÂÂ(1)  ()  (),..., C L+ ÂÂ(1)... Â(L),..., C L+ ÂÂ(1)... Â(L), an Ã(L) is of the form of (I G L+ ML+ C(L) )Â(L) + G L+ ML+ C(L), whereas the escription of Q (L) is much more involve an hence, for brevity, only aitionally given for L = 1 as Q(1) = (I G M C Â)(I G M C )(G 1 M 1 R 1 M1 G 1 + Q)(I G M C ) (I G M C Â) + (I G M C Â)G M R M G (I G M C Â) + G M C QC M G, with M L+1 = Σ L+1 an M L+ = I (L). B. Estimation Algorithms As shown in [1], if r(c G ) = p p H, then state estimates can be obtaine without elay (i.e., L = ). For the sae of brevity, the reaer is referre to [1] for etails of the filter algorithm, its erivation an properties. Notably, the stability conition for the filter is etectability an stabilizability of (Ã(), C () ) an (Ã(), ( Q () ) 1 ), respectively (as is given in Remar ), an strong etectability is a necessary conition for the existence of a stabilizing solution (i.e., convergence of the error covariance to steay-state). As escribe in Section III-B., if p I () := r(c G ) < p p H but p I (1) := r(c ÂG ) = p p H p I (), then state estimates can be obtaine with one step elay. In this case, given measurements up to time step, we consier the following three-step recursive filter 5 : Unnown Input Estimation: ˆ 1, = M 1, (z 1, C 1ˆx +1 D 1 u ) (16) ˆ, = M, (z, C ˆx D u ) (17) ˆ, = M, (z,+1 D u +1 C ˆx C Bu C ÂG ˆ, C G 1 Σ (z 1, D 1 u )) (18) ˆ = V 1 ˆ1, + V V ˆ, + V V ˆ, (19) Time Upate: ˆx = Aˆx + Bu + G 1 ˆ1, () ˆx +1 = ˆx + G ˆ, + G ˆ, (1) Measurement Upate: ˆx +1 = ˆx +1 + L (z,+1 C ˆx +1 C Bu C G 1 Σ (z 1, D 1 u ) D u +1 ) () where ˆx, ˆ1,, ˆ,, ˆ, an ˆ enote the optimal estimates of x, 1,,,,, an. The matrices L, M 1,, M, an M, (with appropriate imensions) are filter gains that are chosen to minimize the state an input error covariances. Note that for the 5 To initialize the filter, arbitrary initial values of ˆx 1, P x 1 an ˆ 1, can be use since we will show that the filter is exponentially stable in Theorem 6. If y an u are available, we can fin the minimum variance unbiase initial estimates given in the initialization of Algorithm 1 using the linear minimum-variance-unbiase estimator [18]. 71

5 measurement upate in (), we only use a component of the measurement given by z,+1. There is no loss of generality in iscaring the rest because it can be verifie as in [1] (in which only z, is use) that the inclusion of z 1,+1 an z,+1 will result in a biase state estimate. Algorithm 1 summarizes the filter with elay L = 1. Similar to ULISE [1] (with L = ), this filter possesses some nice properties, given by the following theorems. Its erivation an proofs will be provie in Section V. Theorem 5 (Optimality). Let the initial state estimate ˆx 1 be unbiase. If r(c ÂG ) = p p H p I (), then the filter algorithm given in Algorithm 1 provies the unbiase, best linear estimate in the mean square sense of the unnown input an the minimum-variance unbiase estimate of states. Theorem 6 (Stability). Let r(c ÂG ) = p p H p I (). Then, that (Ã(1), C (1) ) is etectable is sufficient for the bouneness of the error covariance. Furthermore, if (Ã(1), ( Q (1) ) 1 ) is stabilizable, the filter is exponentially stable (i.e., its expecte estimate errors ecay exponentially). Furthermore, the following proposition shows that the invariant zeros of system (1) are the poles of the input an state filter with elay L = 1, specifically of the state error ynamics E[ x +1 ] (see proof in Section V-C): Proposition 1. All invariant zeros of the system (1) are eigenvalues of the state matrix (Ã(1) Ã(1) L C(1) ) of the propagate state error ynamics E[ x +1 ]. Proposition 1 has the implication that the invariant zeros of the system (1) cannot be stabilize by any choice of filter gain L. Thus, the invariant zeros of the system (1) must be stable such that the input an state filter algorithm in Algorithm 1 is stable by Theorem 6. In other wors, the strong etectability of the system 1 is necessary for the stability of the filter with elay L = 1. Moreover, it is observe in simulation (cf. Section VI) that the converse of Proposition 1 oes not hol. In that example, there are less invariant zeros than eigenvalues of (Ã(1) Ã(1) L C(1) ). For L L, we can repeat the ecomposition proceure (cf. Remar 1) an corresponingly construct asymptotic filtering algorithms as outline in this section, provie that the system is invertible. The same optimality an stability properties as Theorems 5 an 6, as well as Proposition 1 can be also verifie. However, the escription of these cases woul require much more notations an hence, for conciseness, is eferre to a later publication. Remar. Strong etectability of the system (1) is necessary for a stable filter because strongly unetectable moes of the system cannot be stabilize by any choice of filter gain (Proposition 1). Since strong etectability implies invertibility (Section III-A), an invertibility is necessary an sufficient for obtaining unique estimates (Lemma ), we conclue that strong etectability is a ey system property for the existence of a stable asymptotic estimator (Theorem ). Remar 5. Input an state smoothing with a elay L (i.e., the estimation of x :N L an :N L from the observa- Algorithm 1 Filtering with a Delay (L = 1) 1: Initialize: P x 1 = P x = (C R C) ; ˆx 1 = E[x ] = P 1C x R (z, Du);  = A G 1 Σ C 1; ˆQ = G 1Σ R 1Σ G 1 +Q; R = C G 1Σ R 1Σ G 1 C +R ; ˆ 1, = Σ (z 1, C 1ˆx 1 D 1u ); P1, = Σ (C 1P 1C x 1 + R 1)Σ ; P1, x = P 1C x 1 Σ ; : for = 1 to N 1 o Estimation of,,, an : ˆx = Aˆx + Bu + G 1 ˆ1, ; : ˆ, = Σ (z, C ˆx D u ); 5: P = ÂP  x + ˆQ; 6: R, = C P C + R ; 7: P, = Σ R, Σ ; 8: P, x = P A x C Σ P1,G x 1 C Σ ; 9: P1, = Σ C 1P A x C Σ P 1,G 1 C Σ ; 1: R, = C  P  C +C G 1Σ R 1Σ G 1 C +C QC +R + C Â(AP, x + G 1P1, + G P, QC Σ )G  C +(C Â(AP,+G x 1P1, +G P, QC Σ )G  C ) ; 11: P, = (G  C R 1: M, = P,G  C, CÂG) ; R, ; 1: ˆ, = M, (z,+1 C ˆx C ÂG ˆ,, C Bu D u +1 C G 1Σ (z 1, D 1u )); 1: P, x = (P A x + P1,G x 1 + P,G x )  C M,; 15: P 1, = (P x 1,A + P 1,G 1 + P 1,G )  C M,; 16: P, = (P x,a + P1,G 1 + P,G )  C M, + M C QÂC M,; 17: P1, = P1,V + P1,V ; 18: P, = [ ] [ ] [ ] P, V V P, V ; P, P, V 19: ˆ = V 1 [ ˆ1, + V V ˆ, ] + V V ˆ, ; : P P = V 1, P1, V ; 1: P x = P x Time upate P1, 1,V P,,V V 1 + P x + P x,v V ; : ˆx +1 = ˆx + G ˆ, + G ˆ, ; : R w = G Σ CQ GM,C Â(I G Σ C)Q; : P x +1 = [ ] P x P1, x P, x A G 1 G P1 x P1, P1, A G P, x P1, P, 1 G +Q + R w + R w ; Measurement upate 5: R, = C ÂP +1 x C + R C ÂG M, R 6: L = (P x +1 C R M,G  C ; G M, R ) R, ; 7: ˆx +1 = ˆx +1 + L (z,+1 C ˆx +1 C Bu C G 1Σ z 1, + C G 1Σ D 1u D u +1 ); 8: P x +1 = (I L C Â)P +1(I x L C Â) + L R L (I L C Â)G M, R L L R M,G (I L C Â) ; Estimation of 1, 9: R1, = C 1P +1C x 1 + R 1; : P1, = Σ R1, Σ ; 1: ˆ1, = Σ (z 1, C 1ˆx +1 D 1u ); : P1, x = P +1C x 1 Σ ; : en for tions in a fixe time interval given by y :N an u :N where L N 1) is also possible with the two-pass approach of 7

6 [1], with the filter in the previous section (cf. Algorithm 1 for L = 1) as the forwar pass. Since the bacwar pass in [1] is agnostic to whether the filtere estimates are obtaine with or without elay, the smoothing algorithm remains the same for all time steps for which filtere estimates can be obtaine with a elay L. Moreover, if follows that the smoothe estimates are also unbiase an achieve minimum mean square error an maximum lielihoo [1]. V. ANALYSIS For the analysis of the results provie in the previous section, let x +1 := x ˆx +1, x +1 := x ˆx +1, := ˆ, P +1 x := E[ x +1 x x +1 ], P +1 := E[ x +1 x +1 ] an P := E[ ], as well as i, := i, ˆ i,, P i, := E[ i, i, ], P x i, = (P x i, ) := E[ x +1 i, ] for i = 1,,,, an P ij, = (P ij, ) := E[ i, j, ], for i, j = 1,,,, i < j. We begin with the erivation of the filter with elay L = 1, which by esign maintains the unbiaseness of the filter an minimizes variance of the estimate errors, thus proving Theorem 5. Then, we erive the stability conitions for the filter in Theorem 6 by means of fining an equivalent system without unnown inputs. Since Lemma is straightforwar to verify an Lemma follows from Theorem 6 for L = 1 an by extension for all L when erive with the same proceure, Theorem hols. Finally, we prove the claim of Proposition 1 that the invariant zeros of the system are poles of the filter regarless of the choice of the filter gain L. A. Filter Derivation with L = 1 (Proof of Theorem 5) The following lemma shows the unbiaseness of the state an unnown input estimates is preserve for all time steps. Lemma. Let ˆx 1 = ˆx 1 be unbiase, then the estimates, ˆ, ˆx +1 an ˆx +1, are unbiase for all, if an only if M 1, Σ = I, M, Σ = I an M, C ÂG = I. Proof. From (8), (1), (1), (16), (17) an (18), we have ˆ 1, = M 1, (C 1 x +1 + Σ 1, + v 1, ) () ˆ, = M, (C (A x + G 1 1, + w ) +v, + Σ, ) ˆ, = M, (C Â(A x + G 1 1, +G, + w ) C G 1 Σ v 1, +C w + v,+1 + C ÂG, ). () (5) On the other han, from () an (1), the error in the propagate state estimate can be obtaine as: x +1 = A x + G 1 1, + G, +G, + w. (6) Then, from (1) an (), the upate state estimate error is x +1 = (I L C Â) x +1 L v,, (7) where v, := v, C G 1 Σ v 1,. It can be easily shown by inuction (hence omitte for brevity) that M 1, Σ = I, M, Σ = I an M, C ÂG = I, are necessary an sufficient for unbiaseness of ˆ, ˆx +1 an ˆx +1. We continue the proof of Theorem 5 in three subsections, one for each step of the three-step recursive filter. 1) Unnown Input Estimation: To obtain an optimal estimate of ˆ using (19), we estimate all components of the unnown input as the best linear unbiase estimates (BLUE). This means that the expecte input estimate is unbiase, i.e., E[ ˆ 1, ] = 1,, E[ ˆ, ] =,, E[ ˆ, ] =, an E[ ˆ ] =, as was shown in Lemma, an that the mean square error of the estimate is the lowest possible. Theorem 7. Suppose ˆx 1 = ˆx 1 are unbiase. Then (16), (17) an (18) provie the best linear input estimate (BLUE) with M 1,, M, an M, given by M 1, = Σ, M, = Σ, M, = (G Â C R, CÂG ) G Â C while the input error covariance matrices are P1, = Σ R1, Σ, P, = (G Â C P, = Σ R, C ÂG ), R, Σ, with R 1,, R, an R, efine in Algorithm 1. R,, (8) (9) Proof. We wish to choose M 1,, M, an M, such that Lemma hols, resulting in input estimate errors given by 1, = M 1, e 1,,, = M, e,, (), = M, e,, where e 1, := C 1 x +1 + v 1,, e, := C (A x + G 1 1, + w ) + v, an e, := C Â(A x + G 1 1, + G, + w ) C G 1 Σ v 1, + v,+1. Then, with R i, := E[e i, e i, ] for i = 1,,, we apply the well nown generalize least squares (GLS) estimation approach (see, e.g., [18, Theorem.1.1]) to obtain the optimal M 1,, M, an M, given by (8), such that the estimates have minimum variance, i.e., are best linear unbiase estimates (BLUE). The corresponing covariance matrices are P1, = E[ 1, 1, ] = Σ R1, Σ, R, Σ P, = E[,, ] = Σ P, = E[,, ] = (G Â C Next, we note the following equality:, R, C ÂG ). tr(e[ ]) = tr(p 1,) + tr(p,) + tr(p,). Since the unbiase estimates of ˆ 1, an ˆ, are unique (albeit at ifferent time steps) because Σ an Σ are invertible, we have min tr(e[ ]) = tr(e[ 1, 1, ]) + tr(e[,, ]) + min tr(e[,, ]), from which the unbiase estimate ˆ has minimum variance when ˆ 1,, ˆ, an ˆ, have minimum variances. ) Time Upate: The time upate is given by () an (1), an the error in the propagate state estimate by (6) an its covariance matrix can be compute as P x = A G 1 G P x P x P x 1, P, x 1, P 1, P 1, P, x P 1, P, A G 1 G +Q + Rw +R w, where R w:= G M, C Â(I G Σ C )Q G Σ C Q. 7

7 ) Measurement Upate: In the measurement upate step, the measurement z,+1 is use to upate the propagate estimate of ˆx x +1 an P +1. Next, the covariance matrix of the upate state error is compute as P x +1 = (I L C Â)P +1 x (I L C Â) + L R L + (I L C Â)G M, R L + L R M, G (I L C Â) := P x + L R, L L S S L (1) where R := E[v, v, ] = C G 1 Σ R 1 Σ G 1 C + R, R, := C ÂP +1 x C + R C ÂG M, R R M, G ÂC an S := P +1 x C G M, R. Theorem 8. Suppose ˆx 1 = ˆx 1 are unbiase. Then, the minimum-variance unbiase state estimator is obtaine with the gain matrix L given by L = (P +1 x C G M, R )( R,). () Proof. To obtain L, we minimize tr(p +1 x ) by using the conventional metho of ifferential calculus. In aition, it can be verifie that the (cross-)covariances P, x, P 1,, P, x, P 1,, P,, P, x, P, an P1, are as given in Algorithm 1. B. Stability Conition (Proof of Theorem 6) To obtain the stability conition for the asymptotic filter with elay L = 1, we begin by fining an equivalent system without unnown inputs. From (7), we have x +1 = x +1 L (C  x +1 +v,). Then, substituting () into (6) an the above equation, an rearranging, we obtain x +1 = A (1) x + w (1) L (C ÂA (1) x +C Âw (1) + v, ), () where A (1) = (I G M, C Â)Â(1), w = (I G M, C Â)[(I G Σ C )(G 1 Σ v 1, + w ) G Σ v,] G M, C w G M, v, an v, := v, C G 1 Σ v 1,. Note that the state estimate error ynamics above is the same for a Kalman filter [] for a linear system without unnown inputs: x e +1 = A (1) x e + w(1) ; ye = C Âxe + v,. Since the objective for both systems is the same, i.e., to obtain an unbiase minimum-variance filter, they are equivalent systems from the perspective of optimal filtering. Furthermore, since the noise terms of this equivalent system are correlate, i.e., E[w (1) v, ] = G M, R, we further transform the system into one without correlate noise (see, e.g., [1]): x e +1 = Ā (1) xe + ū(1) + w (1) ; ye = CÂxe + v,, with Ā (1) = A (1) + G M, C Â, ū (1) = G M, y e is a nown input an w (1) = w (1) + G M, v,. The noise terms w (1) an v, are now uncorrelate with covariances Q(1) := E[ w (1) w (1) ], R, an E[ w (1) v, ] =. Finally, if we substitute M, by M, := (C ÂG ), we obtain the stability conition given in Lemma from stanar results of the stability of Kalman filtering (see [1] for the justification of the substitution). C. Connection between Strong Detectability an Stability (Proof of Proposition 1) First, we note that the following ientity hols [ ] [ ] zi A G zi  G r p C H H = r C [ ] [ ] [ ] I zi  G = r zi  G = r C I C C  C G [ ] I = r zi  G [ ] [ ] T () C  U () Σ I V () V () = r I G Σ zi  G G I C Â Σ I C  zi Â(1) G [ ] = r C Â Σ zi  = r (1) G + p C  I () C  where the first equality is obtaine from [1]. Thus, the invariant zeros of system [(1) are all z C] for which the system matrix R (1) zi S (z) :=  (1) G rops ran. C  Let z be any invariant zero of R (1) S. Then, there exists [ ν µ ] (1) such that R S (z) [ ν µ ] =, i.e., (zi Â(1) )ν G µ =, () C Âν =. (5) Premultiplying () with (I G M, C Â) an applying (5) as well as the fact that M, C ÂG = I, we have = (I G M, C Â)(zI Â(1) )ν +(I G M, C Â)G µ = (zi Ã(1) )ν = (zi Ã(1) )ν + Ã(1) L C Âν = (zi (Ã(1) Ã(1) L C Â))ν. If ν =, then from (), G µ =, which implies that µ =, which is a contraiction. Hence, ν an the eterminant of zi (Ã(1) Ã(1) L C Â) is zero, i.e., any invariant zero of the system matrix R (1) S (z) is also an eigenvalue of the error ynamics of E[ x +1 ] = (Ã(1) Ã(1) L C Â)E[ x ]. VI. ILLUSTRATIVE EXAMPLE In this example, we consier the state estimation an fault ientification problem when the system ynamics is plague by faults,, that influence the system ynamics an the outputs through G an H, as well as zero-mean Gaussian white noises. Specifically, the linear iscrete-time problem we consier is base on the system given in [1], [7]: A = ; G = 1 ; H = 1 1 ;.1 B = 5 1; C = I 5; D = 5 1; Q = ; R =

8 x1 x 6 x1 ˆx f 1 ˆx s 1 1 x ˆx f ˆx s x ˆx f ˆx s Time, Time, Time, 6 1 x ˆx f ˆx s x5 ˆx f 5 ˆx s 5 1 ˆf 1 ˆs 1 5 Time, 1 5 Time, 1 5 Time, 1 x x5 5 6 ˆf ˆs 1 ˆ f ˆf ˆs 5 ˆ s Time, Time, Time, Fig. 1: Actual states x 1, x, x, x, x 5, unnown inputs 1,,, an their filtere ( f ) an smoothe ( s ) estimates. trace(p x ) 1 filter 1 1 smoother Time, trace(p ) 1 x 1 1 filter 1 1 smoother Time, Fig. : Trace of estimate error covariance of states, tr(p x ), an unnown inputs, tr(p ). The unnown inputs use in this example are { 1, 5 7,1 =, otherwise { 1 ( 1), 1 8, = 7, otherwise, 5 59, 6 69, 7 79, =, , , , otherwise, = sin(.1 + ),. The invariant zeros of the system matrix R S (z) are {.7,.7}. Thus, this system is strongly etectable. Since r(c G ) = an r(c ÂG ) = 1, the states an unnown inputs can be estimate with elay L = 1. We observe from Figure 1 an that the propose algorithm is able to estimate the system states an unnown inputs. For the sae of comparison, we have inclue smoothe estimates (cf. Remar 5), which shows a lower error covariance, as expecte. Moreover, with the steay-state L obtaine in the simulation, we fin the eigenvalues of (Ã(1) Ã(1) L C(1) ) to be {.7,.7,,,.98}. Hence, as is preicte in Proposition 1, all invariant zeros of the system are eigenvalues of the filter. VII. CONCLUSION We presente recursive algorithms that simultaneously estimate the states an unnown inputs in an unbiase minimum-variance sense with a possible elay. The stricter requirement to ensure estimation without elay is relaxe an an asymptotic estimator is evelope for this broaer class of systems. Notably, strong etectability is ientifie as a ey system property that ictates the existence an stability of an input an state estimator with a elay. ACKNOWLEDGMENTS This wor was supporte by the National Science Founation, grant #1918. M. Zhu is partially supporte by ARO W911NF (MURI) an NSF CNS REFERENCES [1] S.Z. Yong, M. Zhu, an E. 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