Fixed-order sampled-data estimation

Abstract

For the Kalman filter-type sampled-data estimation problem utilizing an averaging AID device, the equivalent discrete-time problem is shown to be of increased order. The fixed-structure optimal projection approach for reduced-order, discrete-time estimation is applied to the equivalent discrete-time problem in order to characterize reduced-order estimators. NomenclatureIn O"sO, r X r identity matrix, r x s zero matrix, and r x r zero matrix ( ) T, tr transpose, trace[, IR, IR' x S expected value, real numbers, r x s real matrices 11, I, 11" q positive integers, I ,;; lie ,;; 11 + 1x, y, X e , Ye II, I, lie' q-dimensional vectors A, C II X 11, I x II matrices A e, Be, C e , De lie X II" lie X I, q X lie, q X I matrices W" W 2 II, I-dimensional, zero-mean, continuous-time white noise processes V, II X II non-negative-definite intensity of w,R q x q positive-definite matrix L q x II matrix I, k 1 E [0, 00), discrete-time index, 1,2,3, ... IntroductionOwing to the advances in digital computers, discrete-time filtering and control of continuous-time systems have been developed and used in numerous applications. In the present paper we consider a Kalman filter-type, sampled-data, problem. It is well known that the optimal discrete-time estimates of the dynamic states of a continuous-time model are given by a discrete-time Kalman filter, which is based on an equivalent discrete-time model. Thus, we derive here an equivalent discrete-time problem and apply the fixed-structure approach developed by Bernstein et al. (1986 b), andHaddad (1987), to obtain reduced-order filters.

Sampled-data filtering with error covariance assignment

“…Consider the following linear time-invariant continuous stochastic system _ x(t) = Ax(t) + w(t) (1) with continuous-time measurements y(t) = Cx(t) + v(t) (2) where x n-dimensional state vector; y m-dimensional measured output vector; w(t) zero mean Gaussian white noise process with covariance W > 0; v(t) zero mean Gaussian white noise process with covariance V > 0; where w(t) and v(t) are uncorrelated.…”

“…The purpose of this correspondence is to generalize the ECA theory to sampled-data systems, i.e., design discrete-time Kalman filters for a continuous-time system such that the sampled-data estimation covariance can be assigned to a prespecified value. We develop here an equivalent presentstate dependent discrete-time model (see, e.g., [2], [6], and [7]) to obtain the desired sampled-data filters. Note that the dual problem for sampled-data feedback controller design was initially studied in [1] with state covariance assignment and was further investigated for a class of uncertain systems in [11].…”