Linear Optimal Estimation Problems in Systems with Actuator Faults

Linear Optimal Estimation Problems in Systems with Actuator Faults Daniel Sigalov

Yaakov Oshman

Technion – Israel Institute of Technology Program for Applied Mathematics Haifa 32000, Israel Email: [email protected]

Technion – Israel Institute of Technology Department of Aerospace Engineering Haifa 32000, Israel Email: [email protected]

Abstract—We consider estimating the state of a dynamic system subject to actuator faults. The discretely-valued fault mechanism renders the system hybrid, and results in anomalous changes in the dynamics equation that may be interpreted as random accelerations. Two 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. The 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. I NTRODUCTION Fault detection and isolation (FDI) 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). This 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. The 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 maximal 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. Thus, when dealing with state estimation in hybrid systems, suboptimal approaches are usually considered. These may be divided into two categories. The first category comprises efficient suboptimal estimators, with the nonlinear GPB [5] and IMM [6] filters being the most popular and efficient ones. These 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. For

the case of a bounded number of mode transitions, such as the present paper’s second problem formulation, an alternative to IMM, 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. The 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 [10] 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. Thus, 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. This 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. The remainder of this paper is organized as follows. In section II we formulate the problem at hand. The 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. P ROBLEM F ORMULATION Consider the following standard state-space representation of a stochastic dynamical system: xk+1 = Ak xk + wk zk = Hk xk + vk .

(1) (2)

Here {wk } and {vk } are mutually independent, zero-mean white sequences with covariance matrices {Qk } and {Rk }, respectively, independent of x0 , which is assumed to have zero mean and covariance P0 . The system (1)–(2) is specified by four matrix sequences {Ak }, {Hk }, {Qk }, and {Rk }. At time k the set Mk ,

{Ak , Hk , Qk , Rk } 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). Thus, the mode variable comprises the dynamics transition matrix Ak and the covariance matrix of the process noise, Qk , namely, Mk , {Ak , Qk }. We consider the case where, at time k, the mode Mk may assume one of r possible values, m1 through mr . We designate the mode m1 as the nominal one corresponding to the fault-free actuator operation. The second set of modes, {m2 , . . . , mr }, represents various actuator fault conditions, that manifest themselves in sudden, unexpected accelerations. Even in the presence of faults, the state of the system xk 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 Mk is known for each k. The system mode Mk is allowed to evolve according to some stochastic mechanism. Two such mechanisms are considered. In the first one, the value of the mode at time k, Mk , 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 = 2, and the total number of mode transitions is upper bounded by some known integer rmax 1 . As long as the system has not performed rmax mode transitions, it alternates between the modes in an independent manner. Once rmax mode switches have been performed, the system remains in the most recent mode forever. There is a clear conceptual difference between the two transition laws considered herein. The conditional independence of the modes in the second formulation prevents the mode variables from being unconditionally independent. Thus, it appears that solving the first problem does not provide a solution for the second one. The optimal MMSE estimate of the state xk based on the measurement sequence Zk , {z0 , ..., zk } is given by the conditional expectation E [xk | Zk ]. The optimal filter for computing x ˆk = E [xk | Zk ] requires the use of (primitive) KFs, 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 rmax , in the bounded case [12]. These 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 xk using the measurements Zk , {z1 , ..., zk }. We thus set out to obtain the linear optimal estimator in the following sequential form: x ˆk+1 = F1 (k)ˆ xk + F2 (k)zk+1 .

(3)

1 Notice the difference between r max and r. The same letter is used due to the similar role the quantities play in the derivations.

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. Theorem 1. Let P = {p (x, z)} be a family of probability distributions, all having the same first two moments. Let x ˆ= g(z) be any estimator of x based on the observation z, and let x ˆL = gL (z) be the linear optimal estimator of x based on z. Denote by MSE(g; p) the mean squared error of g(z) corresponding to the distribution p, and let p⋆ , arg max MSE(g; p) p∈P

⋆

Then, MSE(g; p ) ≥ MSE(gL ; p⋆ ). Proof: By definition of p⋆ , MSE(g; p⋆ ) ≥ MSE(g; N ). But, by linearity of the optimal filter in the Gaussian case, MSE(g; N ) ≥ MSE(gL ; N ). Recognizing that MSE(gL ; p⋆ ) = MSE(gL ; N ) then yields the result. Theorem 1 tells us that the linear optimal estimator is also optimal in the minimax 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. U NBOUNDED N UMBER OF M ODE T RANSITIONS Consider the following dynamical system xk+1 = Ak xk + wk zk = Hxk + vk ,

(4) (5)

where Mk ≡ {Ak , Qk } ∈ {m1 , ..., mr } and r is a known integer. The mode transition dynamics is independent, such that, for i, j = 1, . . . , r. P {Mk+1 = mi | Mk = mj } = P {Mk+1 = mi } = pi , (6) where {pi } 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, mi = {Ai , Qi }. The first result of this paper is summarized in the following theorem. Theorem 2. The linear optimal estimator has the sequential form (3) where F1 (k) and F2 (k) are matrix coefficients that are given by: r X F1 (k) = (I − F2 (k)H) Ai pi , (7) i=1

F2 (k) =

Sk+1 −

r X

A i p i Vk

i=1

"

× H

Sk+1 −

r X i=1

A i p i Vk

r X

ATi pi

i=1

r X i=1

ATi pi

!

!

HT T

H +R

#−1

, (8)

    ˆk x ˆTk are given by where Sk , E xk xTk and Vk , E x Sk+1 =

r X i=1


Ai Sk ATi + Qi pi ,

(9)

(10) Vk+1 = F1 Vk F1T + F2 (HSk+1 H T + R)F2T ! ! r r X X + F2 H Ai pi Vk F1T + F1 Vk ATi pi H T F2T . i=1

i=1

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 (xk+1 − x ˆk+1 ) zjT = 0, ∀ j = 0, . . . , k + 1. (11)

We proceed in three steps, omitting, for brevity, the explicit dependence of F1 and F2 on k. Consider, first, the orthogonality conditions (11) for j = 0, . . . , k. Substituting the estimator’s equation (3) yields     ˆk + F2 zk+1 )zjT 0 = E xk+1 zjT − E (F1 x     ˆk + F2 (Hxk+1 + vk+1 ))zjT = E xk+1 zjT − E (F1 x       ˆk zjT − F2 HE xk+1 zjT = E xk+1 zjT − F1 E x     (12) ˆk zjT , = (I − F2 H)E xk+1 zjT − F1 E x

where the third transition follows from the independence of vk+1 and zj for j < k + 1. Conditioning on the mode at time k, (12) becomes r X   0 = (I − F2 H) E (Ai xk + wk )zjT Mk = mi pi i=1   − F1 E x ˆk zjT r X     ˆk zjT , = (I − F2 H) Ai E xk zjT pi − F1 E x

(13)

i=1

where we have utilized the facts that 1) the mode sequence is i.i.d. and, therefore, Mk is independent of {xk , wk , zj }, and 2) the sequences  zj and  wTk are independent for j = 0, . . . , k.  ˆk , ˆk zj because of the optimality of x Now, E xk zjT = E x rendering (7) a sufficient condition for LMMSE optimality. Consider, next, the last orthogonality condition in (11):   T = 0. (14) E (xk+1 − x ˆk+1 ) zk+1

Substituting (5) and (3) in (14), and utilizing the independence of vk+1 and xk+1 , yields   0 = E xk+1 (Hxk+1 + vk+1 )T   − E (F1 x ˆk + F2 zk+1 )(Hxk+1 + vk+1 )T     ˆk xTk+1 H T = E xk+1 xTk+1 H T − F1 E x     T − F2 E zk+1 xTk+1 H T − F2 E zk+1 vk+1   = Sk+1 H T − F1 E x ˆk xTk+1 H T − F2 HSk+1 H T − F2 R, (15)

  where we have defined Sk , E xk xTk . Conditioning on Mk gives the expectation of the right hand side of (15) as r r X  X    ATi pi , ˆk x ˆTk ATi pi = E x ˆk xTk E x ˆk xTk+1 = E x i=1

i=1

(16)

where the last transitionfollows  from orthogonality satisfied by x ˆk . Defining Vk , E x ˆk x ˆTk and plugging into (15) yields the second relation between F1 and F2 : T

(I − F2 H)Sk+1 H − F1 Vk

r X

ATi pi

− F2 R = 0.

(17)

i=1

Substituting (7) in (17) and solving for F2 yields (8). It remains to provide recursive schemes for the computation of the matrices Sk and Vk . We have Sk+1 =

r X i=1

=

r X i=1

=

r X i=1

  E xk+1 xTk+1 Mk = mi pi

E

h

Ai xk + wki




Ai xk + wki


Ai Sk ATi + Qi pi .

i
T Mk = i pi

(18)

i h Vk+1 = E (F1 x ˆk + F2 zk+1 ) (F1 x ˆk + F2 zk+1 )T  T    T F2 = F1 E x ˆk x ˆTk F1T + F2 E zk+1 zk+1  T    T ˆk zk+1 T F2T + F2 E zk+1 x ˆ k F1 + F1 E x

Let {Nk } be a sequence of random variables assuming the values 0, . . . , rmax , such that Nk = ℓ if ℓ mode transitions have occurred by time k. Then, the mode at time k + 1 is chosen according to following stochastic rule o n P Mk+1 = mi Nk = ℓ, {Mk′ }k′ ≤k , {Nk′ }k′