On Bootstrap Identification Using Stochastic Approximation

A two-stage state and parameter estimation algorithm for linear systems has been developed. Stage 1 uses a stochastic approximation method for state estimation, while stage 2 considers parameter estimation through a linear Kalman filter. These two stages are conpled in a bootstrap manner. The algorithm is computationally much simpler than the usual extended Kalman filter. A fourth-order numerical example has been solved, and results have been compared with those obtained using an extended Kalman filter. 1. ~NTRODUCTION The use of minicomputers and microprocessors for adaptive control of processes has necessitated the need for a computationally simpler identification algorithm. A common technique for state and parameter estimation of the state-variable models of a process has been the extended Kalman filter (e.g., see [I]-[2]). This method is known to have the problems of increased computation, storage requirements, and divergence. Two-stage estimation techniques have recently been employed [3] to overcome these difficulties. In [3], linear Kalman filters have been employed in two stages and no precaution has been taken against divergence in the state estimator because of parameter uncertainty. In the present note. a computationally simpler two-stage estimation algorithm has been developed for estimating states and parameters of a linear system. Stage 1 deals with the state estimation with assumed nominal values of the parameters. A convergence-based stochastic-approximation algorithm [4] has been proposed in stage 1 which also gives savings in computation. In stage 2, a parameter dynamic model and a pseudo-parameter measurement equation are developed and a linear Kalman filter is employed for obtaining the parameter estimates. The parameter-measurement equation contains the state-variable terms that are substituted by their recent estimates available as output of stage 1. These two stages are coupled together in a bootstrap manner [SI. The complete procedure and algorithm are given in the sequel. 11. THE PROCEDURE We consider the following discrete-time model of a linear system: (1) (2) where x(k) is the n-vector state, u(k) is the m-vector input, y(k) is the r-vector output, 8 is the p-vector parameter assumed constant but unknown. The state noise vector w(k) and the measurement noise vector ~(k) are mutually independent and assumed zero mean white Gaussian with constant covariances Q and R, respectively. The parameter model is described by TABLE I B(k+l) = B(k) (3) (4) where (4) is a pseudo-measurement equation obtained by the proper splitting of (2). Matrix C and vector d are functions of state x(k) and input u(k). The ne I) is obtained by adding H(k)w(k) to u(k+ l). Assuming H(k) to be independent of w(k), u,(k+ 1) is a zero mean white Gaussian sequence with a covariance equal to R + H(k)QH(k). Manuscript received December 28, 1976. The authors are with the Department of Electncal Engineenng. lndlan Inst,tute of Technology. New Delhi. India. Parameters Actual. values starting values Estimated Values. Tvro-stage Al m rithm Extended Kalman filter I1=0. 01 R = O . 1 % 0 . 0 1 R=O.