Identification of linear stochastic systems through projection filters

Abstract

A novel method is presented for identifying a state-space model and a state estimator for linear stochastic systems from input and output data. The method is primarily based on the relationship between the state-space model and the finite difference model of linear stochastic systems derived through projection filters. It is proved that least-squares identification of a finite difference model converges to the model derived from the projection filters. System pulse response samples are computed from the coefficients of the finite difference model. In estimating the corresponding state estimator gain, a z-domain method is used. First the deterministic component of the output is subtracted out, and then the state estimator gain is obtained by whitening the remaining signal. An experimental example is used to illustrate the feasibility of the method. YSTEM identification, sometimes also called system modeling, deals with the problem of building a mathematical model for a dynamic system based on its input/output data. This technique is important in many disciplines such as economics, communication, and system dynamics and control.1 The mathematical model allows researchers to understand more about the properties of the system, so that they can explain, predict, or control the behaviors of the system. Recently, a method has been introduced in Refs. 2 and 3 to iden- tify a state-space model from a finite difference model. The differ- ence model, called autoregressive with exogeneous input (ARX), is derived through Kalman filter theories. However, the method re- quires to use an ARX model of large order, which causes intensive computation in the embedded least-squares operation. In Ref. 4 a method is derived to obtain a state-space model from input/output data using the notion of state observers. This approach can use an ARX model with an order much smaller than that derived through the Kalman filter, but the derivation is based on a deterministic ap- proach. In Ref. 5, it has been proved that, as the order of the ARX model increase to infinity, the observer identification converges to the Kalman filter identification. However, for a stochastic system and an ARX model of a small order, to what the least-squares iden- tification of the ARX model will converge in a stochastic sense is not clear. This paper addresses the above-mentioned problems using a stochastic approach. The approach is primarily based on the re- lationship between the state-space model and the finite difference model via the projection filter.3 First, an ARX model is chosen, and then the ordinary least squares is used to estimate the coefficient matrices. Based on the relationship between the projection filter and the state-space model matrices, the system pulse response samples (i.e., the system Markov parameters) can be calculated from the co- efficients of the identified ARX model. The eigensystem realization algorithm (ERA)6 is used to decompose the Markov parameters into a state-space model. In contrast to the time-domain approaches used in Refs. 2 and 5, a different method is developed in this paper using a z-domain approach to compute the state estimator gain. After identifying a state-space model, the deterministic part of the output is subtracted out. The remaining signal represents the stochastic part. A moving- average (MA) model is then introduced to describe the remaining signal. The MA model is computed by identifying the correspond- ing autoregressive (AR) model first and then inverting it. From the identified MA model, the state estimator gain is then calculated. Finally, identification of a 10-bay structure is used to illustrate the feasibility of the approach.