Linear Optimal Estimation Problems in Systems with Actuator Faults

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1 14th International Conference on Information usion Chicago, Illinois, USA, July 5-8, 11 Optimal Estimation Problems in Systems with Actuator aults Daniel Sigalov echnion Israel Institute of echnology Program for Applied Mathematics Haifa 3, Israel Yaakov Oshman echnion Israel Institute of echnology Department of Aerospace Engineering Haifa 3, Israel Abstract We consider estimating the state of a dynamic system subject to actuator faults. he discretely-valued fault mechanism renders the system hybrid, and results in anomalous changes in the dynamics equation that may be interpreted as random accelerations. wo closely related problem formulations are considered. In the first formulation multiple models are used to describe the system s behavior: one model stands for the nominal, fault-free actuator condition, all other models correspond to various actuator fault conditions, and the system can freely assume any model at any time. In the second formulation the abnormal mode is described by a single dynamical model, and the system can switch between the nominal and anomalous conditions a bounded number of times, with the bound assumed known. In both formulations, the minimum mean squared error MMSE) optimal state estimator requires a polynomially growing number of primitive Kalman filters, and is, thus, computationally infeasible. We derive sequential, linear MMSE-optimal state estimation algorithms for both problem formulations. Depending only on the first two moments of the random quantities of the problem, linear optimal filters are robust with respect to the actual driving noise distributions, in the sense that they achieve the smallest worst-case estimation error of all other nonlinear) filters. Although derived assuming seemingly different problem formulations, both filters share essentially the same structure, thus exposing a certain duality between the underlying problems. he performance of both estimators is demonstrated in a simulation study, where they are compared to the interacting multiple model filter. Keywords: Multiple model estimation, target tracking, hybrid systems, fault detection and isolation. I. INRODUCION ault detection and isolation DI) is a task of prime importance in mission-critical systems, that are characterized by tough reliability requirements. Autonomous aerospace systems, e.g., missiles and unmanned airborne vehicles UAVs), are usually required to implement self-diagnostic tools that automatically detect faults, thus increasing efficiency and reducing risks. In addition, implementation of fault-tolerant systems is desired, such that the system s performance would not be severely affected by an undetected component fault. Avoiding explicit fault isolation and ensuing reconfiguration, fault-tolerant estimation and control schemes rely on assumed statistical models for the possible faults, and take them into account when estimating the overall system s state that may include the operational condition of the fault-prone components). his approach is taken in this paper, where we focus on fault-tolerant state estimation in the presence of random actuator faults. In compliance with [1 [3, we model actuator faults as additive terms in the state dynamics equation. However, instead of being unknown deterministic inputs, these are taken to be random) process noise terms with different statistical properties covariances) indicating anomalous actuator conditions. In addition, we allow the faults to explicitly alter the plant dynamics, such that, under a fault occurrence, the state transition matrix might change to one of its predefined non-nominal values. We consider two different problem formulations. In the first, we consider systems with intermittent faults modeled as an independent process. At any time an actuator may remain in its nominal, fault-free condition, or switch to one of possible faulty conditions. he second problem differs by the fault generation mechanism. Instead of allowing an arbitrary number of transitions between nominal and faulty actuator conditions, we assume that the number of such transitions is bounded by a known integer. As long as the number of transitions has not exceeded this bound, faults keep being generated in an independent manner, similarly to the first formulation. When the imal number of transitions has occurred, the system freezes in its current faulty or faultfree) mode. As will be discussed in the sequel, neither of these problem formulations may be represented as a special case of the other. Both problems considered here may be viewed as special cases of the hybrid system framework since the continuous uncertainty associated with the state variable is accompanied with the discrete one associated with the fault generation mechanism. It is well known [4 that under limited computational resources, the optimal state estimator in the minimum mean-squared error MMSE) sense does not exist for general hybrid systems. hus, when dealing with state estimation in hybrid systems, suboptimal approaches are usually considered. hese may be divided into two categories. he first category comprises efficient suboptimal estimators, with the nonlinear GPB [5 and [6 filters being the most popular and efficient ones. hese are designed to handle cases where the discrete uncertainty is governed by a Markov process, which may easily be adopted to our first problem formulation. or ISI 117

2 the case of a bounded number of mode transitions, such as the present paper s second problem formulation, an alternative to, recently proposed by the authors, may be utilized [7. Although, in practice, these filters demonstrate excellent performance in terms of the mean-squared error MSE) criterion, to the best of the authors knowledge, no theoretical performance guarantees are known to exist. Moreover, they explicitly assume that the system noises are Gaussian. he second category consists of estimators that are MMSEoptimal within the narrower subclass of linear filters. Among this class, Nahi [8 discussed the problem of independently evolving faults in the system sensors and derived a linear optimal estimator. His work was further generalized in [9 and [1 who derived linear filters for systems in which fault indicators in the measurement equation could be correlated. Costa [11 derived a linear optimal filter for general hybrid systems. However, for the case of actuator faults, the resulting filter requires state augmentation, increasing computational complexity and sacrificing the filter s desired sequential structure. Moreover, Costa s method does not apply for bounded cases, such as the second problem considered herein. Usually, linear filters do not outperform nonlinear, heuristic filters that belong to the first category. Nevertheless, their availability is important, because they possess a certain robustness property that is generally not shared by nonlinear filters. hus, linear estimators depend on the first two moments of the random quantities involved. Hence, as long as these moments are kept constant, these algorithms are indifferent to the specific noise distributions. Consequently, linear filters attain the smallest worst-case MSE, in comparison to any other nonlinear) filter. his makes their MSE a valuable bound when assessing the performance of other algorithms. In this paper we develop linear optimal estimators for the two problem formulations mentioned above. We show that in spite of the difference between the problems, both filters possess a similar structure, indicating conceptual similarities that are not obvious prior to the actual derivation. he remainder of this paper is organized as follows. In section II we formulate the problem at hand. he linear optimal estimators for both problem formulations are derived in Sections III and IV, respectively, and discussed in Section V. A simulation study, comparing both algorithms, is presented in section VI. Concluding remarks are made in section VII. II. PROBLEM ORMULAION Consider the following standard state-space representation of a stochastic dynamical system: x k+1 A k x k + w k 1) z k H k x k + v k. ) Here {w k } and {v k } are mutually independent, zero-mean white sequences with covariance matrices {Q k } and {R k }, respectively, independent of x, which is assumed to have zero mean and covariance P. he system 1) ) is specified by four matrix sequences {A k }, {H k }, {Q k }, and {R k }. At time k the set M k {A k, H k, Q k, R k } comprises the mode of the system. Different values of the mode correspond to, for example, different flight regimes of an aircraft, e.g., maneuvering/nonmaneuvering, or nominal/faulty conditions of an actuator. In this work we focus on modes that explicitly affect the dynamics equation 1). hus, the mode variable comprises the dynamics transition matrix A k and the covariance matrix of the process noise, Q k, namely, M k {A k, Q k }. We consider the case where, at time k, the mode M k may assume one of r possible values, m 1 through m r. We designate the mode m 1 as the nominal one corresponding to the fault-free actuator operation. he second set of modes, {m,...,m r }, represents various actuator fault conditions, that manifest themselves in sudden, unexpected accelerations. Even in the presence of faults, the state of the system x k may be estimated optimally in the mean-square sense using a standard Kalman filter, provided that the mode sequence evolves in time in a deterministic manner, namely, the exact value of M k is known for each k. he system mode M k is allowed to evolve according to some stochastic mechanism. wo such mechanisms are considered. In the first one, the value of the mode at time k, M k, is chosen according to some probability distribution defined over the set of r modes independently of the modes at other times. In the second mechanism, it is assumed that the total number of modes is r, and the total number of mode transitions is upper bounded by some known integer r 1. As long as the system has not performed r mode transitions, it alternates between the modes in an independent manner. Once r mode switches have been performed, the system remains in the most recent mode forever. here is a clear conceptual difference between the two transition laws considered herein. he conditional independence of the modes in the second formulation prevents the mode variables from being unconditionally independent. hus, it appears that solving the first problem does not provide a solution for the second one. he optimal MMSE estimate of the state x k based on the measurement sequence Z k {z,..., z k } is given by the conditional expectation E [x k Z k. he optimal filter for computing ˆx k E [x k Z k requires the use of primitive) Ks, in a number that grows exponentially in time, in the unbounded case i.e., when the number of mode transitions is unbounded) [4, and polynomially in time and exponentially in r, in the bounded case [1. hese observations call for a suboptimal solution approach. In both cases, our goal is to obtain linear optimal algorithms requiring moderate computational resources, that are capable of tracking the state x k using the measurements Z k {z 1,..., z k }. We thus set out to obtain the linear optimal estimator in the following sequential form: ˆx k+1 1 k)ˆx k + k)z k+1. 3) 1 Notice the difference between r and r. he same letter is used due to the similar role the quantities play in the derivations. 1173

3 Notice that, a-priori, it is not clear that the linear optimal estimators can be cast in the sought for sequential form 3). Our ensuing derivations show that, indeed, this can be done. Before proceeding with the actual derivations, we state the following well known theorem to justify the potential benefits in using linear optimal filters. heorem 1. Let P {p x, z)} be a family of probability distributions, all having the same first two moments. Let ˆx gz) be any estimator of x based on the observation z, and let ˆx L g L z) be the linear optimal estimator of x based on z. Denote by MSEg; p) the mean squared error of gz) corresponding to the distribution p, and let p arg MSEg; p) p P hen, MSEg; p ) MSEg L ; p ). Proof: By definition of p, MSEg; p ) MSEg; N). But, by linearity of the optimal filter in the Gaussian case, MSEg; N) MSEg L ; N). Recognizing that MSEg L ; p ) MSEg L ; N) then yields the result. heorem 1 tells us that the linear optimal estimator is also optimal in the mini sense [13, i.e., its worst-case MSE is not larger than that of any other estimator over the set of distributions having the same first two moments). In other words, for any nonlinear estimator, there exists a mismatching distribution of the process and measurement noises, for which it cannot outperform the linear optimal filter. III. UNBOUNDED NUMBER O MODE RANSIIONS Consider the following dynamical system x k+1 A k x k + w k 4) z k Hx k + v k, 5) where M k {A k, Q k } {m 1,..., m r } and r is a known integer. he mode transition dynamics is independent, such that, for i, j 1,...,r. P {M k+1 m i M k m j } P {M k+1 m i } p i, 6) where {p i } are known probabilities summing up to 1. With a slight abuse of notation, subscripts of A and Q will be used to denote different modes and not the time index, namely, m i {A i, Q i }. he first result of this paper is summarized in the following theorem. heorem. he linear optimal estimator has the sequential form 3) where 1 k) and k) are matrix coefficients that are given by: 1 k) I k)h) A i p i, 7) k) S k+1 [ H S k+1 A i p i V k A i p i V k ) A i p i H 1 A i p i )H + R, 8) where S k E [ x k x k and Vk E [ˆx kˆx k are given by S k+1 Ai S k A ) i + Q i pi, 9) V k+1 1 V k 1 + HS k+1 H + R) 1) ) + H A i p i V k V k A i p i )H. Proof: We prove the theorem constructively by showing that the expressions 7) and 8) for the coefficient matrices of the estimator 3) constitute sufficient conditions for the estimator s optimality in the linear MMSE LMMSE) sense. We do this by examining the following orthogonality conditions, that are known to be necessary and sufficient conditions for LMMSE optimality [14 E [ x k+1 ˆx k+1 )zj, j,...,k ) We proceed in three steps, omitting, for brevity, the explicit dependence of 1 and on k. Consider, first, the orthogonality conditions 11) for j,...,k. Substituting the estimator s equation 3) yields E [ x k+1 zj [ E 1ˆx k + z k+1 )zj E [ x k+1 zj [ E 1ˆx k + Hx k+1 + v k+1 ))zj E [ x k+1 zj 1 E [ˆx k zj HE [ x k+1 zj I H)E [ x k+1 zj 1 E [ˆx k zj, 1) where the third transition follows from the independence of v k+1 and z j for j < k + 1. Conditioning on the mode at time k, 1) becomes I H) E [ A i x k + w k )zj M k m i pi 1 E [ˆx k zj I H) A i E [ x k zj pi 1 E [ˆx k zj, 13) where we have utilized the facts that 1) the mode sequence is i.i.d. and, therefore, M k is independent of {x k, w k, z j }, and ) the sequences z j and w k are independent for j,...,k. Now, E [ x k zj E [ˆxk zj because of the optimality of ˆxk, rendering 7) a sufficient condition for LMMSE optimality. Consider, next, the last orthogonality condition in 11): E [ x k+1 ˆx k+1 ) z k+1. 14) Substituting 5) and 3) in 14), and utilizing the independence of v k+1 and x k+1, yields E [ x k+1 Hx k+1 + v k+1 ) E [ 1ˆx k + z k+1 )Hx k+1 + v k+1 ) E [ x k+1 x k+1 H 1 E [ˆx k x k+1 H E [ z k+1 x k+1 H E [ z k+1 vk+1 S k+1 H 1 E [ˆx k x k+1 H HS k+1 H R, 15) 1174

4 where we have defined S k E [ x k x k. Conditioning on Mk gives the expectation of the right hand side of 15) as E [ˆx k x k+1 E [ˆxk x r k A i p i E [ˆx kˆx r k A i p i, 16) where the last transition follows from orthogonality satisfied by ˆx k. Defining V k E [ˆx kˆx k and plugging into 15) yields the second relation between 1 and : I H)S k+1 H 1 V k A i p i R. 17) Substituting 7) in 17) and solving for yields 8). It remains to provide recursive schemes for the computation of the matrices S k and V k. We have S k+1 E [ x k+1 x k+1 M k m i pi [ Ai E x k + wk) i Ai x k + wk) i Mk i p i Ai S k A ) i + Q i pi. 18) V k+1 E [ 1ˆx k + z k+1 ) 1ˆx k + z k+1 ) 1 E [ˆx kˆx k 1 + E [ z k+1 z k+1 + E [ z k+1ˆx k E [ˆx k z k+1 1 V k 1 + HS k+1 H + R) + E [ z k+1ˆx k E [ˆx k z k+1. 19) It remains to compute E [ z k+1ˆx k E [ˆxk zk+1). o this end we condition once again on M k : E [ z k+1ˆx [ k HE xk+1ˆx k H E [ x k+1ˆx k M k m i pi H A i E [ x kˆx k pi H A i p i )V k, ) which completes the proof of the theorem. IV. BOUNDED NUMBER O MODE RANSIIONS or the second problem of this paper we next consider the case where the mode variable may assume one of two predefined values, namely, M k {A k, Q k } {m 1, m }. Under this formulation, one of the modes may stand for, e.g., a nominal, fault-free operation of the system, whereas the second mode describes a faulty, abnormal operation. he state space representation of the considered system is identical to the one used in the previous section. We assume that the total number of mode transitions from m 1 to m and vice versa) is bounded by a predefined known integer r. Mathematically, the mode transition law is described as follows. Let {N k } be a sequence of random variables assuming the values,...,r, such that N k l if l mode transitions have occurred by time k. hen, the mode at time k + 1 is chosen according to following stochastic rule { } P M k+1 m i Nk l, {M k } k k, {N k } k <k { p i, l < r, i {1, }, 1) δ i,r mod )+1, l r where {p i }, i {1, } are known probabilities and δ i,j is Kronecker s delta. hus, before r switches have occurred, the system switches independently between the two models. After r transitions, the most recent mode remains to be the system s active mode. or simplicity, and without loss of generality, we assume that the initial model is the nominal one, that is, P {M m 1 } 1. his assumption can be relaxed at the cost of a more cumbersome derivation. As before, we are looking for the linear optimal estimator of x k+1, given Z k {z 1,...,z k+1 }. he second result is summarized in the following theorem. heorem 3. or the second problem considered in this paper, the linear optimal estimator has the sequential form 3) with matrix coefficients 1 k) and k) given by: 1 k) I k)h) A l q l k), ) k) [ H S k+1 S k+1 A l q l V k A l q l V k A l q l )H 3) A l q l )H + R 1. Here A l A l mod )+1, and q l k) P {N k l}, that is computed, for 1 l k, as follows: and q l k) p l 1) mod ) q l 1 k 1) { p + l mod )+1 q l k 1), l < r q l k 1), l r, p 1 q k 1), l, k > q l k) 1, l k, l > k. he second-order moment matrix of x k, S k, satisfies S k+1 4) 5) Al S k A l + Q l ) ql k), 6) where Q l Q l mod )+1, and V k is the second-order moment matrix of ˆx k, that is computed via the recursive equation V k+1 1 V k 1 + HS k+1 H + R) 7) r ) r ) + H A l q l k) V k V k A l q l k) H. l1 l1 1175

5 Proof: Similarly to heorem, the constructive proof is based on enforcing the necessary and sufficient) orthogonality conditions on the recursive estimator 3). or notation simplicity we drop the explicit dependence of 1 and on k. Consider, first, the orthogonality conditions for j,..., k: E [ x k+1 ˆx k+1 )zj. 8) ollowing along the lines of the first step in the proof of heorem we obtain I H)E [ x k+1 zj 1 E [ˆx k zj. 9) Using the smoothing property of the conditional expectation on the first term on the RHS of 9) by conditioning on the number of mode transitions by time k, N k, yields I H) E [ x k+1 zj 1 E [ˆx k zj I H) E [ x k+1 zj N k l P {N k l} Nk l q l k) 1 E [ˆx k zj. 3) Now, under the event {N k l}, x k+1 is determined by the model l mod )+1 and, upon substituting the expression for ˆx k+1 from 3), 3) reads I H) E [ A l x k + wk)z l j Nk l q l 1 E [ˆx k zj I H) A l E [ x k zj ql 1 E [ˆx k zj, 31) where the conditioning on {N k i} has been omitted since, due to the conditional independence of the modes, N k is independent of x k, wk l, and z j. urther, note that wk l that should be interpreted as the process noise of the model corresponding to l mode transitions), and v k+1, are independent of z j. Using the orthogonality condition E [ˆx [ k zj E xk zj in 31) thus yields that ) is a sufficient condition for the fulfillment of the first k + 1 orthogonality conditions 8). Consider, next, the last orthogonality condition: E [ x k+1 ˆx k+1 )zk+1 E [ x k+1 zk+1 E [ˆxk+1 zk+1. 3) Substituting 1) and the conjectured form 3) into 3) yields E [ x k+1 Hx k+1 + v k+1 ) E [ 1ˆx k + z k+1 )Hx k+1 + v k+1 ) E [ x k+1 x k+1 H 1 E [ˆx k x k+1 H E [ z k+1 x k+1 H E [ z k+1 vk+1 S k+1 H 1 E [ˆx k x k+1 H HS k+1 H R I H)S k+1 H 1 E [ˆx k x k+1 H R, 33) where, as before, S k E [ x k x k. he remaining expectation on the RHS of 33) is computed by conditioning on N k E [ˆx r k x k+1 E [ˆx k x k+1 E [ˆx k x k Nk l q l E [ˆx k A l x k + w l k) Nk l q l r E [ˆx k x k A l ql A l q l E [ˆx kˆx k r A l q l, 34) where the last transition follows from the orthogonality obeyed by the linear optimal) estimate ˆx k. Substituting 34) and ) into 33) and solving for yields 3). inally, we derive the recursions for the second-order moment matrices. We have S k+1 r r E [ x k+1 x k+1 Nk l q l [ Al E x k + wk) l Al x k + wk) l Nk l q l Al S k A ) l + Q l ql. 35) V k+1 E [ 1ˆx k + z k+1 ) 1ˆx k + z k+1 ) 1 E [ˆx kˆx k 1 + E [ z k+1 z k+1 + E [ z k+1ˆx k E [ˆx k z k+1 1 V k 1 + HS k+1 H + R) + E [ z k+1ˆx k E [ˆx k z k+1. 36) o compute E [ z k+1ˆx k E [ˆxk zk+1) on the RHS of 36) we condition on N k : E [ z k+1ˆx [ k HE xk+1ˆx k H E [ x k+1ˆx k Nk l q l H l1 r H E [ A l x k + wk)ˆx l k ql l1 l1 thus completing the proof. A l E [ x kˆx k ql H V. DISCUSSION r l1 A l q l )V k 37) hough solving seemingly very different problems, it is readily seen that the linear optimal filters of Sections III and IV have strikingly similar structures. In fact, both estimators implement Kalman filter-like procedures, that use what can be called average transition matrices: r A ip i in the case 1176

6 of Section III, and r l1 A lq l in the case of Section IV. Moreover, the computational requirements of both filters are very similar, requiring the inversion of a matrix similar to the Kalman filter s innovation covariance matrix) that has the same dimensions in both cases. Indeed, this phenomenon exposes a fundamental duality between the two problems, that relates to the definition and role of the system modes in both cases. In the first case, there are r explicit dynamical modes, m 1,...,m r, each defined by a particular operational model the dynamic system follows in that mode. In the case with bounded number of transitions, there are just two such physical modes, but the duality exposed by the structural similarity of the linear optimal filters tells us that the system can be viewed as effectively having r temporal modes, each corresponding to a point in time when a physical transition has occurred before the imal number of transitions has been reached. In passing, we note that the notion of temporal multiple model is not new and was previously introduced in [15, where a nonlinear, heuristic filter was derived. We also note that the linear optimal filter of Section III could also be derived as a special case of the idea presented in [11. he present derivation, as opposed to the one in [11, does not require state augmentation, resulting in a more efficient algorithm. VI. NUMERICAL SUDY In this section we demonstrate the performance of the algorithms devised in Sections III and IV via a simple example, consisting of state estimation in a scalar stochastic dynamical system. he performance is compared to that of the stateof-the-art nonlinear algorithm [6 which, in the special case of independent mode dynamics, reduces to the GPB algorithm [16. We consider three numerical examples. In the first, we test a hybrid system driven by Gaussian noises, that alternates an unbounded number of times between nominal and faulty conditions. In the second, we change the distribution of the measurement noise from Gaussian to Laplacian, and demonstrate the robustness of the linear optimal filter and its advantage, in this regard, over the Gaussian distributionassuming. inally, we test the performance of the filter of Section IV in an example with a bounded number of mode transitions, and compare it to that of the unbounded filter and the nonlinear. he system considered is described by the state space model 1) and ), where x k is a one-dimensional state representing the target position at time k, initialized by x, P, and {w k } are i.i.d. zero mean Gaussian random variables with variances {q k }. he mode affects only the variance of the process noise, such that M k q k { σ1, σ}, where σ 1 represents the nominal variance, and σ represents the faulty case. he measurement noise sequence {v k } is zero mean with constant variance σv. In all examples in the sequel the following common parameters were used σ 1.1, σ, σ v 1. 38) A. Example 1: Unbounded number of faults, Gaussian case In this example the number of the faults is unbounded, and P { q k σ 1} 1 P { qk σ } p, 39) with p.95. he measurement noise is Gaussian. A single realization of the state sequence, accompanied by the measurements and estimates of the linear optimal and algorithms, as well as by the MMSE-optimal estimate of a genie-based filter, possessing perfect knowledge of the values of {q k }, is presented in ig. 1a). Squared position errors of the filters, averaged over 1, independent Monte Carlo runs, are presented in ig. 1b). he performance of the linear optimal algorithm is slightly inferior to that of the nonlinear. It is, however, much more efficient, since, as may be readily observed from the filter equations in Section III, its computational requirements are of a single K, as opposed to the that maintains two Ks in parallel and has an additional overhead due to the mixing stage. B. Example : Unbounded number of faults, Laplacian case he setting of this example is identical to that of Example 1, except that now the measurement noise has a Laplace distribution, retaining the same first and second-order moments as the Gaussian noise of Example 1. A sample realization of the state sequence, the measurements, and the corresponding estimated trajectories are presented in ig. a). Squared position errors of the filters, averaged over 1, independent Monte Carlo runs, are presented in ig. b). his example is a simple demonstration of the robustness of linear optimal filters, discussed in heorem 1. Since all second-order statistics are unaltered, the linear optimal filter generates essentially the same MSEs as in the Gaussian case of Example 1. he, on the other hand, is sensitive to the measurement noise distribution, and thus yields higher position errors. he somewhat inferior performance of the is best visualized via the sample run depicted in ig. a). As opposed to the Gaussian case, due to the heavy tails of the Laplace distribution, high-valued measurements are not as unlikely, as can be seen at, e.g., k 14, 61, 77. Operating under the Gaussian assumption, the filter interprets the correspondingly large innovations as originating from the faulty model, thus increasing the weight of the K corresponding to σ. his results in the strong spikes observable at the times of the seemingly abnormal measurements. Although the recovery from this mistaken interpretation of the measurements is rapid, the resulting overall MSE is increased. C. Example 3: Bounded Number of aults, Gaussian case In this example the number of transitions between the nominal and faulty modes is bounded by r 6. he measurement noise is Gaussian, and the conditional probabilities of the modes are P { q k σ1 N k < } 1 P { q k σ N k < } p, 4) 1177

7 6 4 arget Measurements ime a) Sample trajectory, measurements and estimates ime b) Average squared estimation error, Monte Carlo igure 1: Estimation performance, Example 1 Gaussian measurement noise) arget Measurements ime a) Sample trajectory, measurements and estimates Bounded ime b) Average squared estimation error, Monte Carlo igure : Estimation performance, Example Laplacian measurement noise). where, as before, p.95 and the rest of the parameters remain unchanged. In ig. 3 we compare the algorithm derived in Section IV to the linear optimal filter for the unbounded case and the algorithm, where the latter algorithms are implemented with the same mode probabilities as in the previous examples. In addition, we present, for reference, the squared errors of the MMSE-optimal genie-based K. All results are averaged over 1, independent Monte Carlo runs. ig. 3a) depicts the errors for the case when the filter for the bounded case assumes that the number of mode transitions is bounded by r 6. In other words, it operates under the nominal conditions it was designed for. he errors achieved by this filter coincide with those of the linear estimator for the unbounded case in the first third of the simulation interval. Both are slightly inferior to the nonlinear. In the second part of the interval, whereas the linear filter for the unbounded case remains inferior to the in compliance with the results of Example 1), the filter for the bounded case achieves significantly better errors, almost coinciding with the geniebased K. his is due to the fact that, after the imal number of transitions has been exhausted, a single K yields optimal results. In contradistinction, filters that do not take this into account, keep maintaining irrelevant Ks in the case of the ) or irrelevant terms in the filter gain in the case of the unbounded linear filter), thus increasing the MSE. In ig. 3b) we present an example demonstrating the 1178

8 .35.3 Bounded.35.3 Bounded ime ime a) Nominal r b) Underestimated r igure 3: Squared position estimation errors, Example 3 bounded number of transitions). robustness of the linear filter against imprecise knowledge of r. Whereas the true number of transition is 6, the filter uses as the bound on the number of transitions. Consequently, the errors generated at the beginning are higher than those of the nominal case, and are even worse than those of the linear filter for the unbounded case. However, the 6 transitions are eventually exhausted, and the filter s errors converge again to the optimal ones, resulting in a superior performance over the alternatives. It should be mentioned that, while being robust with respect to the exact value of r, the filter is not robust with respect to the parity of r. When such a mismatch occurs, the performance degrades severely. his is not very surprising, since, in the case of such a mismatch, the filter effectively operates, after the imal number of faults is exhausted, under an incorrect model. In contradistinction, the linear filter for the unbounded case and the utilize, to some extent, both dynamical models. VII. CONCLUDING REMARKS We have addressed the problem of state estimation in dynamical systems subject to random actuator faults. wo variants of the problem were considered, corresponding to cases with unbounded and deterministically bounded total number of mode transitions. Multiple number of modes are allowed in the first unbounded) case, whereas the bounded case permits just two system modes. or both problems we have derived sequential, linear optimal state estimation algorithms, that are known to be robust with respect to mismatch of the actual noise distributions to the assumed ones. hough originating from different problems, both filters share essentially the same structure, and may be viewed as dual to each other. he performance of both filters has been demonstrated through a numerical simulation study. REERENCES [1 R. V. Beard, ailure Accomodation in Systems hrough Self- Reorganization. PhD thesis, Dept. of Aeronautics and Astronautics, Massachusetts Inst. of echnology, eb [ W. Chung and J. Speyer, A Game heoretic ault Detection ilter, IEEE rans. on Automatic Control, vol. 43, no., pp , [3 R. Chen and J. Speyer, Sensor and Actuator ault Reconstruction, Journal of Guidance, Control, and Dynamics, vol. 7, no., pp , 4. [4 Y. Bar-Shalom, X. Li, and. Kirubarajan, Estimation with Applications to racking and Navigation. New York : Wiley, 1. [5 G. Ackerson and K. u, On state estimation in switching environments, IEEE rans. on Automatic Control, vol. 15, no. 1, pp. 1 17, 197. [6 H. Blom and Y. Bar-Shalom, he interacting multiple model algorithm for systems with Markovian switching coefficients, IEEE rans. on Automatic Control, vol. 33, no. 8, pp , [7 D. Sigalov and Y. Oshman, State estimation in hybrid systems with a bounded number of mode transitions, in Proc. usion 1, 13th International Conference on Information usion, 1. [8 N. Nahi, Optimal recursive estimation with uncertain observation, IEEE rans. on Information heory, vol. I-15, pp , July [9 Y. Sawaragi,. Katayama, and S. ujishige, Adaptive estimation for a linear system with interrupted observation, IEEE rans. on Automatic Control, vol. 18, no., pp , [1 M. Hadidi and S. Schwartz, recursive state estimators under uncertain observations, IEEE rans. on Automatic Control, vol. AC- 4, pp , December [11 O. Costa, minimum mean square error estimation for discretetime Markovian jump linear systems, IEEE rans. on Automatic Control, vol. 39, no. 8, pp , [1 D. Sigalov and Y. Oshman, Efficient tracking for hybrid systems with a bounded number of mode switches, Proc. 5th Israel Annual Conf. on Aerospace Sciences, ebruary 1. [13. Michaeli and Y. C. Eldar, Hidden relationships: Bayesian estimation with partial knowledge, IEEE ransactions on Signal Processing, 11. to appear. [14 A. Jazwinski, Stochastic Processes and iltering heory. Academic Press, 197. [15 G. Hexner, H. Weiss, and S. Dror, emporal Multiple Model Estimator for a Maneuvering arget, in Proc. of AIAA, GNC, 8. [16 B. D. Anderson and J. B. Moore, Optimal iltering. Prentice-Hall Information and System Sciences Series, Englewood Cliffs, New Jersey: Prentice-Hall,