ARMA identification

In view of recent results on the asymptotic behavior of the prediction error covariance for a state variable system (see Ref. 1), an identification scheme for autoregressive moving average (ARMA) processes is proposed. The coefficients of thed-step predictor determine asymptotically the system momentsU ₀,...,U _d–1. These moments are also nonlinear functions of the coefficients of the successive 1-step predictors. Here, we estimate the state variable parameters by the following scheme. First, we use the Burg technique (see Ref. 2) to find the estimates of the coefficients of the successive 1-step predictors. Second, we compute the moments by substitution of the estimates provided by the Burg technique for the coefficients in the nonlinear functions relating the moments with the 1-step predictor coefficients. Finally, the Hankel matrix of moment estimates is used to determine the coefficients of the characteristic polynomial of the state transition matrix (see Refs. 3 and 4).A number of examples for the state variable systems corresponding to ARMA(2, 1) processes are given which show the efficiency of this technique when the zeros and poles are separated. Some of these examples are also studied with an alternative technique (see Ref. 5) which exploits the linear dependence between successive 1-step predictors and the coefficients of the transfer function numerator and denominator polynomials.In this paper, the problems of order determination are not considered; we assumed the order of the underlying system. We remark that the Burg algorithm is a robust statistical procedure. With the notable exception of Ref. 6 that uses canonical correlation methods, most identification procedures in control are based on a deterministic analysis and consequently are quite sensitive to errors. In general, spectral identification based on the windowing of data lacks the resolving power of the Burg technique, which is a super resolution method.This work was supported by NATO Research Grant No. 585/83, by University of Nice, by Thomson CSF-DTAS, by Instituto Nacional de Investigação Científica, and by CIRIT (Comissió Interdepartmental de Recerca i Innovació Technológica de Catalunya). The work of the third author was also partially supported by Army Research Office Contract DAAG-29-84-k-005.Simple ARMA(2, 1) Basic language analysis programs to construct random data were written by the second author and Dr. K. D. Senne, MIT Lincoln Laboratory. Lack of stability of the direct estimation was observed at TRW with the help of Dr. G. Butler. Analysis programs in FORTRAN for ARMA(p, q) were written and debugged at CAPS by the fifth author. The research was helped by access to VAXs at Thomson CSF-DTAS, Valbonne, France, at CAPS, Instituto Superior Técnico, and at Universitat Politécnica de Catalunya. In particular, the authors explicitly acknowledge Thomson CSF-DTAS and Dr. H. Gautier for extending the use of their facilities to the authors from September 1983 until June 1984, when the examples presented were simulated.on leave from University of Southern California, Los Angeles, California.formerly visiting at Massachusetts Institute of Technology and Laboratory for Information and Decision Systems, while on leave from Instituto Superior Técnico, Lisbon, Portugal.