Identification of Stochastic Time-Delay Systems

-A methad is presented to identify time lags present in systems described by linear, stochastic dynamical models nsuaUy enconntered in process control. 'zhe model equation is fii disaetized and converted into a non-time-delayed form: a two-stage estimator is then proposed to identify the model parameten. Manuscript received August 12, 1977. K. K. Biswas is with the Department of Electrical Engineering, Indian hstitute of Technology. New Delhi, India. G. Sin& is with the Department of Electrical Engineering, Motilal Nehru Regional Engineering College, Mahabad. India INTRODUCTION Time lags are generally encountered in industrial processes; e.g., thermal lags or transportation lags. An effective process-control setup is possible only when various process lags have been identified in addition to other system parameters. Recently, Rao and Sivakumar [I] indicated a technique for identification of deterministic time-lag systems, using moment functionals. However, because of the random disturbances present in the processes as well as in the associated instrumentation, it is preferable to use a stochastic model in such situations. The aim of this note is to present a method for identification of time lags in such models. IDENTIFICATION OF S Y ~ TIME LAGS Consider a stochastic dynamical system described by x(t) = F,x(r) + F2x(t-T) + Bu(t) + w(t) (1) where x is n X 1 state vector, T is scalar time lag, Fl and F2 are n X n unknown/known matrices, B is n Xp known input matrix, u is p X1 known input vector, and w is n x 1 disturbance vector. The system output is assumed to have a model y(t) = Hx(t) + o(r) (2) wherey is mX 1 output vector, H is mXn known output matrix, and o is m X 1 measurement noise vector. w(r) and o(r) can be assumed to be white-Gaussian zero-mean noise processes with known covariance. The idenuication of the unknown time lag T and the associated matrices Fl and F2 can be carried out as follows. The first step is to descretize the system equations (1) and (2) with a sampling interval of Ar, to yield model equations of the form ,(k) (3) (4) where k = r/Ar and L = T/ &. The matrices A,, A,, and B, are obtained through first-order Euler's integration process. The next step is to construct an augmented state vector which contains the state x(k) and some delayed states. We define X (k) = [x'(k)x\k -1), • • •, x\k N )]' where N is chosen slightly higher than the unknown value L. Adjoining equations of the type to (3), a higher dimensional state model can be constructed as X(k+\) = AX(k) + Bu y(k) = HX(k) + v( where (5)