Ilia Rapoport

A Crame/spl acute/r-Rao-type estimation lower bound for systems with measurement faults

A Crame/spl acute/r-Rao-type lower bound is presented for systems with measurements prone to discretely-distributed faults, which are a class of hybrid systems. Lower bounds for both the state and the Markovian interruption variables (fault indicators) of the system are derived, using the recently presented sequential version of the Crame/spl acute/r-Rao lower bound (CRLB) for general nonlinear systems. Because of the hybrid nature of the systems addressed, the CRLB cannot be directly applied due to violation of its associated regularity conditions. To facilitate the calculation of the lower bound, the hybrid system is first approximated by a system in which the discrete distribution of the fault indicators is replaced by an approximating continuous one. The lower bound is then obtained via a limiting process applied to the approximating system. The results presented herein facilitate a relatively simple calculation of a nontrivial lower bound for the state vector of systems with fault-prone measurements. The CRLB-type lower bound for the interruption process variables turns out to be trivially zero, however, a nontrivial, non-CRLB-type bound for these variables has been recently presented elsewhere by the authors. The utility and applicability of the proposed lower bound are demonstrated via a numerical example involving a simple global positioning system (GPS)-aided navigation system, where the GPS measurements are fault-prone due to their sensitivity to multipath errors.

A new estimation error lower bound for interruption indicators in systems with uncertain measurements

Optimal mean-square error estimators of systems with interrupted measurements are infinite dimensional, because these systems belong to the class of hybrid systems. This renders the calculation of a lower bound for the estimation error of the interruption process in these systems of particular interest. Recently it has been shown that a Crame/spl acute/r-Rao-type lower bound on the interruption process estimation error is trivially zero. In the present work, a nonzero lower bound for a class of systems with Markovian interruption variables is proposed. Derivable using the well-known Weiss-Weinstein bound, this lower bound can be easily evaluated using a simple recursive algorithm. The proposed lower bound is shown to depend on a measure of the interruption chain transitional determinism, the measurement noise sensitivity to interruption process switchings, and a measure of the systems state estimability. In some cases, identified in this correspondence, the proposed bound is tight. The use of the lower bound is illustrated via a simple numerical example.

Weiss–Weinstein Lower Bounds for Markovian Systems. Part 1: Theory

Being essentially free from regularity conditions, the Weiss-Weinstein estimation error lower bound can be applied to a larger class of systems than the well-known Crameacuter-Rao lower bound. Thus, this bound is of special interest in applications involving hybrid systems, i.e., systems with both continuously and discretely distributed parameters, which can represent, in practice, fault-prone systems. However, the requirement to know explicitly the joint distribution of the estimated parameters with all the measurements makes the application of the Weiss-Weinstein lower bound to Markovian dynamic systems cumbersome. A sequential algorithm for the computation of the Crameacuter-Rao lower bound for such systems has been recently reported in the literature. Along with the marginal state distribution, the algorithm makes use of the transitional distribution of the Markovian state process and the distribution of the measurements at each time step conditioned on the appropriate states, both easily obtainable from the system equations. A similar technique is employed herein to develop sequential Weiss-Weinstein lower bounds for a class of Markovian dynamic systems. In particular, it is shown that in systems satisfying the Crameacuter-Rao lower bound regularity conditions, the sequential Weiss-Weinstein lower bound derived herein reduces, for a judicious choice of its parameters, to the sequential Crameacuter-Rao lower bound

Weiss–Weinstein Lower Bounds for Markovian Systems. Part 2: Applications to Fault-Tolerant Filtering

Characterized by sudden structural changes, fault-prone systems are modeled using the framework of systems with switching parameters or hybrid systems. Since a closed-form mean-square optimal filtering algorithm for this class of systems does not exist, it is of particular interest to derive a lower bound on the state estimation error covariance. The well known Crameacuter-Rao bound is not applicable to fault-prone systems because of the discrete distribution of the fault indicators, which violates the regularity conditions associated with this bound. On the other hand, the Weiss-Weinstein lower bound is essentially free from regularity conditions. Moreover, a sequential version of the Weiss-Weinstein bound, suitable for Markovian dynamic systems, is presented by the authors in a companion paper. In the present paper, this sequential version is applied to several classes of fault-prone dynamic systems. The resulting bounds can be used to examine fault detectability and identifiability in these systems. Moreover, it is shown that several recently reported lower bounds for fault-prone systems are special cases of, or closely related to, the sequential version of the Weiss-Weinstein lower bound

Recursive Weiss-Weinstein lower bounds for discrete-time nonlinear filtering

Being essentially free from regularity conditions, the Weiss-Weinstein lower bound can be applied to a larger class of systems than the well-known Cramer-Rao lower bound. Thus, this bound is of special interest in applications involving hybrid systems, i.e., systems with both continuously and discretely-distributed parameters, which can represent in practice fault-prone systems. However, the requirement to know explicitly the joint distribution of the estimated parameters with all the measurements renders the application of the Weiss-Weinstein lower bound to Markovian dynamic systems impractical. A new algorithm is presented in this paper for the recursive computation of the Weiss-Weinstein lower bound for a wide class of Markovian dynamic systems. The algorithm makes use of the transitional distribution of the Markovian state process, and the distribution of the measurements at each time step conditioned on the appropriate states, both easily obtainable from the system equations. For systems satisfying the Cramer-Rao lower bound regularity conditions, and for a particular choice of its parameters, it is shown that the recursive Weiss-Weinstein lower bound reduces to the recently introduced recursive Cramer-Rao lower bound. Moreover, it is shown that several recently reported lower bounds, derived for systems with fault-prone measurements, are special cases of the proposed recursive Weiss-Weinstein lower bound.

Efficient fault tolerant estimation using the IMM methodology

Space systems are characterized by a low-intensity process noise resulting from uncertain forces and moments. In many cases, their scalar measurement channels can be assumed to be independent, with one-dimensional internal dynamics. The nominal operation of these systems can be severely damaged by faults in the sensors. A natural method that can be used to yield fault tolerant estimates of such systems is the interacting multiple model (IMM) filtering algorithm, which is known to provide very accurate results. However, having been derived for a general class of systems with switching parameters, the IMM filter does not utilize the independence of the measurement errors in different channels, nor does it exploit the fact that the process noise is of low intensity. Thus, the implementation of the IMM in this case is computationally expensive. A new estimation technique is proposed herein, that explicitly utilizes the aforementioned properties. In the resulting estimation scheme separate measurement channels are handled separately, thus reducing the computational complexity. It is shown that, whereas the IMM complexity is exponential in the number of fault-prone measurements, the complexity of the proposed technique is polynomial. A simulation study involving spacecraft attitude estimation is carried out. This study shows that the proposed technique closely approximates the full-blown IMM algorithm, while requiring only a modest fraction of the computational cost.

Optimal filtering in the presence of faulty measurement biases

The problem of fault tolerant estimation with application to a system subjected to faulty measurement biases is addressed. A new estimation approach, called fault adaptive filtering, is proposed. The proposed estimation scheme consists of a single minimum variance state estimator, which takes into account fault probabilities, combined with a fault estimator, which provides estimates of these probabilities. Unlike the multiple model approach this method does not require a large number of parallel filters. On the other hand, unlike separate estimation and fault detection techniques, the fault adaptive filter is able to utilize partial information about the existence of faults. A numerical example of spacecraft attitude determination is presented, which demonstrates the performance of the proposed approach.

A Cramer-Rao type lower bound for the estimation error of systems with measurement faults

A Cramer-Rao type lower bound for a class of systems with faulty measurements is presented. Lower bounds for both the state and the Markovian interruption variables of the system are derived, based on the recently presented sequential version of the Cramer-Rao lower bound (CRLB) for general nonlinear systems. To facilitate the calculation of the lower bound for this class of systems, the discrete distribution of the fault indicators is approximated by a continuous one and the lower bound is obtained via a limiting process applied to the approximating system. The results presented in this paper facilitate a relatively simple calculation of a nontrivial lower bound for the state vector of systems with faulty measurements. The CRLB-type lower bound for the interruption process variables is trivially zero, however, a non-trivial, non-CRLB-type bound for these variables has been recently presented elsewhere by the authors.

Fault-Tolerant Particle Filtering by Using Interacting Multiple Model-Based Rao-Blackwellization

The problem of fault-tolerant particle filtering of a highly nonlinear system with fault-prone scalar measurement channels is addressed, in which each measurement channel is characterized by an additive measurement error generated by a linear scalar hybrid system. Particle filtering is an emerging method that exploits the recent advances in computer technology by using simulation-based techniques to represent probability density functions in nonlinear, non-Gaussian systems. Because, in the system under investigation, the overall state vector includes both the main states of the system and the parameters of the measurement channels, its size can be prohibitively large for efficient application of ordinary particle filtering, due to the required number of particles. The RaoBlackwellization technique is adopted, allowing to estimate just the main system states using a reduced-size set of particles. The parameters of each measurement channel are estimated by a separate interacting multiple model scalar filter. A numerical example is presented, where four fault-prone magnetometers are used to estimate both the attitude of a spacecraft and the fault parameters of the measurement channels. The new state estimator is compared with unscented Kalman filtering-based techniques. The results demonstrate the superiority of the proposed algorithm in terms of estimation accuracy.

Fault Tolerant Particle Filtering Using IMM-Based Rao-Blackwellization

Fault tolerant particle flltering of a system with nonlinear dynamics and fault-prone scalar measurement channels is addressed, where each measurement channel is characterized by an additive measurement error generated by a linear scalar hybrid system. Particle flltering is an emerging method, which exploits the recent advances in computer technology by using simulation-based techniques to represent probability density functions in nonlinear, non-Gaussian systems. Since, in the system under investigation, the overall state vector includes both the main states of the system and the parameters of the measurement channels, its size can be prohibitively large for e‐cient application of ordinary particle flltering, due to the required number of particles. The Rao-Blackwellization technique is adopted herein, allowing to estimate just the main system states using a reduced-size set of particles. The parameters of each measurement channel are estimated by a separate IMM scalar fllter. A numerical example is presented, where four fault-prone rate gyros are used to estimate the angular velocity of a spacecraft and the fault parameters of the measurement channels. The new state estimator is compared with two implementations of the ordinary particle fllter, which difier by the number of particles used to represent the state distribution. The results demonstrate the superiority of the proposed algorithm over the ordinary particle fllters in terms of computational e‐ciency, which translates, in this case, to better accuracy and stability.