The effect of erroneous models on the Kalman filter response

Abstract

The optimal filtering equations, as derived by K h a n [I], 121, require the specification of a number of models for a given application. This paper concerns itself with the effect of errors in the assumed models on the filter response. The types of errors considered are those in the covariance of the initial state vector, the covariance of the stochastic inputs to the system, and the covariance of the uncorrelated measurement noise.Presented here is a derivation of a recursive equation for the actual covariance matrix of the estimation error when the filter design is based upon erroneous models. The derived equation can also be used to obtain the covariance matrix of the estimation error when the optimal filter gains are approximated by simple functions of time to be used in a real-time 6ltering application.A numerical example illustrates the use of the derived equations. IETRODUCTIONThis paper is concerned with the determination of the performance of the suboptimal filter obtained when a Kalman [l ] filter design is based upon erroneous noise models and a priori statistics. The performance is specified by the covariance matrix of the suboptimal estimation error.A recursive equation is derived for this matrix which makes possible a quantitative evaluation of the statistical quality of the estimate obtained. SYSTEM MODELThe system model used is as follows:(1) y(tk) ~ll(tk)?.(fJ:) + .L'(lk) ( 4 where 1) r(tt) is the n vector of states at time tk E ( x ( t o ) ) = 0 and E ( x ( t o ) x ' ( f o ) ) = P*(to) 2) + ( f k + l ; a ) is the transition matrix ( n x s ) of the system 3) u(tk) is an n vector of stochastic inputs such that E{,t&)) = 0 and E(zt(t&'(tj)} = St.jQ(tk) 4) y ( f k ) is an m vector of measurements taken a t time t k 5) M(tJ:) is an n t x n matrix (Measurement Sensitivity Matrix) 6) o(tk) is an m vector of additive measurement noise with the following properties: OPTIMAL FILTERThe optimal filter can be represented as a linear dynamic system by the following system of equations [2]:x*(4+1 I fk) = $*(tJ:+l; tk)Z*(tk I fm-1) 4-R * ( t k ! > ' ( t k ) (3) $*(tk+1; tk) = @(tk+1; tk) -R * ( t k ) X ( t J J (4) R*(tk) = q t i + 1 ; a)P*(tl.).l~'(k)[-~~(tn)P*(fk)l~~'(fe) + R(td]-I (5) P * ( f . 4 = $*(ta+l; k)P*(k)+'O~+l; t d + Q(h4 (6)where 1) x * ( t n + l ! t t ) is the optimal estimate of the state vector x ( f t + d , having observed y(to), y(tl), . . , y ( h ) 2) $ * ( f k + l ; ts) is the transition matrix ( n X n ) of the filter 3) R * ( t b )is an n x m matrix of optimal gains 4) P * ( t k + l ) is the covariance matrix of the estimation error. ManuscriptThe author is with Bell Telephone Laboratories. Inc., Whippany. N. J. ANALYTIC RESULTSNow assume that the actual gains employed in the filter design deviate from the optimal gains, i.e., that R ( t k ) = R*(tk) + 6K*(tk) (7) and $ ( k + 1 ; tk) = +*(tk+1; tn) -a K * ( t k ) i ) K ( t k ) .(8)In order to determine the effect of these errors we ask the following question. How close do we come to the optimum by using the following filter equation ?&...

A semi-definite programming approach for robust tracking

“…The Basic KF is sensitive to modeling errors and implementation errors, as described early on by [12,23]. The filter can be made more robust by using such heuristics as adding process noise and fading memory of "old" measurements.…”

“…ChoosingS = S, the normalized vector χ resides in the uncertainty setΩ, defined by (12), with m = 3S. Moreover, matrices A T and T , are replaced by their normalized versionẼ andẼ z such that δ t =Ẽχ and z t =Ẽ z χ, andẼ = K T,S C+D.…”

Unknown Input Estimation using the Optimal FIR Smoother

“…Kalman filter can give optimal solution, but the estimates of each unknown input depends on the initial information and other unknown inputs. Moreover, since Kalman filter use all of the information from initiation to the current time, it has some potential problems due to its structure [6,7]. Kalman filter may diverge for system with initial state uncertainty, modeling errors, and numerical errors.…”

Process and Measurement Noise Estimation for Kalman Filtering

“…An important result, apparently first published by Son and Anderson [1], is the fact that the information needed to determine K is encoded in the output signals and thus, given sufficiently long data sequences K can be estimated from the measurements without the need to know or estimate the Q, R matrices. Perhaps the most widely quoted strategies to carry out the estimation of K are due to Mehra [2] who built on the work by Heffes [3] and the subsequent paper by Carew and Bellanger [4], both techniques being iterative in nature. A noteworthy contribution from this early work is the contributions by Neethling and Young [5], who suggested some computational adjustments that could be used to improve accuracy.…”