Allen R. Stubberud

NONLINEAR FILTERING BY APPROXIMATION OF THE A POSTERIORI DENSITY

The problem of estimating from noisy measurement data the state of a dynamical system described by non-linear difference equations is considered. The measurement data have a non-linear relation with the state and are assumed to be available at discrete instants of time. A Bayesian approach to the problem is suggested in which the density function for the state conditioned upon the available measurement data is computed recursively. The evolution of the a posteriori density function cannot be described in a closed form for most systems; the class of linear systems with additive, white gaussian noise provides the major exception. Thus, the problem of non-linear filtering can be viewed as essentially a problem of approximating this density function. For linear systems with additive, white gaussian noise, the a posteriori density is gaussian. The results for linear systems are frequently applied to non-linear systems by introducing linear perturbation theory. Then, the linear equations and gaussian a posterio...

Optimal filtering for Gauss—Markov noise†

The optimal continuous-filtering problem for the caso of linear dynamics, linear measurements, and gaussian whito disturbance and measurement noise has been Solved by Kalman and Buey. In this study, their rosults are generalized for the caso where measurement noise is a Gauss—Markov process, but without the technique of state augmentation, as has already been done. Proof is presented that the optimal continuous-filtering problem can be solved by simply replacing the observation vector with a derived observation vector and an initial condition. When the derived observation vector is used, coloured noise is eliminated and only the standard Kalman filter problem, easily solvable, remains.

Recursive Filtering for Systems with Small but Non-negligible Non-linearities†

Abstract : Linear estimation theory has been applied extensively to nonlinear systems by assuming that perturbations from a reference solution can be described by linear equations. As long as the second order (and higher) terms in the perturbation equations are negligible, linear estimation techniques have been found to yield satisfactory response. Many examples have been encountered in which the linear theory is not satisfactory, however, and it is to this situation that attention is directed here. Time-discrete systems in which the second order effects are small but nonnegligible are considered. Recursion relations for the conditional mean and covariance are developed. While these relations yield approximations to the true values of these moments, they are superior to the approximations provided by applying linear theory to a nonlinear system. Some results for a simple system are presented in which the response from linear and nonlinear filters is compared.

Reduced order Kalman filter

The order of the Kalman filter equations for a wide class of aerospace navigation problems is reduced, yielding an optimal sequential linear filter with a substantial decrease in computer requirements. A theorem is proved generalizing the Kalman filter to handle step-wise correlated noise. An illustrative example is then presented in which computer computation time and storage requirements are reduced by more than half with negligible increase in programming complexity.

On Final Value Control

Final value control of linear plants is discussed. An expression for the closed loop fundamental matrix in terms of the open loop transition matrix is presented. It is shown that for scalar minimum energy final value control the singularities in the feedback loops are inversely proportional to the order of the derivative being fed back. That is, the position loop has an ηth order singularity, the velocity loop an η—lth order singularity, etc. A partially closed loop control is developed that allows the highest-order feedback singularity to be set between η and zero. Those results are illustrated by examples.

Synthesis of Computationally Efficient Sequential Linear Estimators

The Kalman sequential linear estimation theory, although not always utilized because the number of computations required for many systems of practical importance becomes prohibitive, allows straight-forward synthesis of optimal estimators for many complex systems. Some systems designers have chosen to ignore variables and by such a reduction in system dimension have been able to economize with regard to the number of computations. The purpose of this paper is to demonstrate a method which allows economy of computation by partitioning the system state vector; the variables to be eliminated are placed in one subsystem and the remaining variables in one or more additional subsystems. The resultant system is computationally more efficient if some variables are eliminated. This is so because the remaining states have been partitioned into two or more subsystems. The number of computations for a subsystem varies approximately as the cube of the dimension of its state vector. By operating on several subsystems of lesser dimension than that of the unpartitioned system, the number of computations is decreased; performance will deteriorate. The method for determining the partitioning tends to keep this deterioration under control; it is illustrated by application to a marine-type inertial navigation system.

Application of the mathematical theory of sequential sampling to gamma scanning in nuclear medicine

Abstract To minimize the time for performing a gamma scan in nuclear medicine, optimum use must be made of the statistical data from the scan to control the time of observation at any given point. In the work described, the sequential sampling technique has been applied to the problem of proportioning observation time on the basis of observed counts. In the limit of both very high and very low count rates, advantagesis taken of the rapid decision time afforded by the good statistics of high rates and the a priori rejection of areas exhibiting low rates. Areas of intermediate activity require longer times to obtain a statistically significant estimate of activity. In this scanning method, accumulated counts from a stationary sensor are compared with two (linearly) time- varying thresholds: upon crossing either threshold an intensity estimate is made (which may be binary) and the scan continued to the next point. This procedure is well known in sequential hypothesis testing and the theory developed is pertinent to the gamma scanning application. An analysis of the system as compared to a fixed observation- time-in-scanning has been made. For a two-level detection system, with a given error probability and a given ratio of decay rate for a high-activity region to that for a low-activity region, it has been found that the average observation time for a variable threshold scanning is approximately half that for fixed-time scanning.

Identification via Non-linear Filtering

ABSTRACT Non-linear filtering problems are being recognized more arid more as significant. Generally, an exact solution is not available but must be approximated. However, this paper gives the exact non-linear filter associated with identifying a scalar stochastic dynamical system, dx(t)/dt=ax(t)+ξ(t), disturbed by gaussian white noise ξ(t) when the system state x(t) is observed in an additive gaussian white noise environment, i.e. z(t)=x(t)+η(t) is observed over the interval of time 0≤t≤T<∞. The plant parameter, a, is assumed to have the prior probability density PA .(a). The solution, is obtained by giving the conditional probability density functional p(a\z(t), 0≤t≤T<∞). The minimum mean-square-error estimate, that is, the Bayes estimate or conditional expectation, is given along with the minimum-mean-square-error. Limiting cases are described. This approach is a variation on the eigenfunction expansion schemes used in the stochastic signals in noise-detection problems by Helstrom and others. This appr...