A stability theorem with applications to adaptive control

Abstract

305importance of (7)-(10).All of these "residual-based" methods suffer from the drawback of requiring the computation of prediction error signals at each intermdate stage. As noted in [A, the use of "fast algorithms" can reduce this burden. However, if recursive least squares methods are used, as suggested by Morf and Vieira [16], the minimum phase property is not guaranteed.The estimation method presented here can be viewed as an efficient approximate residual-based technique. From the alternative form of the estimate given in (36) and the definitions (12)-(18), we see that instead of using N -j prediction error terms to determine the jth partial correlation estimate, N -p terms are used. By keeping the number of terms the same for each estimate, the efficiency of the Cholesky decomposition may be exploited. Indeed, for long data sequences and modest order autoregressive fits, there will be virtually no difference between the estimates obtained here and those of [A. For limited data, simulations will be required to compare the performance of the various methods.Finally, we turn to a small generalization that may be of interest in spectrum analysis applications. It is somewhat troubling that our estimation procedure is not invariant with respect to reversing the time index. This situation may be remedied by replacing the normal equations (15) with the normal equations associated with the problem of minimizing f J T + @ i d . Then Cholesky factorization may be applied just as in the previous case. This forces the estimates to be fit both forwards and backwards over the data and is invariant to time reversal of the data. This modification requires no more computation than the method based on (1 5), since the matrix "Yp 9 : is simply replaced by its persymmetrized (symmetrized about the NE-SW diagonal) version. Methods of obtaining time-reversal invariance for residual-based estimates are described in [A and [SI.One problem not yet mentioned, but discussed in some detail by Jones [SI, is the order selection problem. Determination of an appropriate value of p is somewhat more difficult in the multivariate case than in the scalar case. Further research and experience in practical applications w i l l be needed to provide good order selection techniques. REFERENCES canonical factorization of a spectral density matrix," Biometrikn, vol. 50, pp. P. Whittle, "On the fitting of multivariate autoregressions, and the approximate 129-134, 1963. R A. Wiggins and E. A. Robinson, "Recursive solution to the multichannel filtering problem," , "Identification and autoregressive spectrum estimation," IEEE Tram J. P. Burg, "Maximum entropy spectral analysis," Ph.D. dissertation, Dep. Gecphysics. Stanford Univ, Stanford, CA, May 1975. A. H. Nuttall, "Multivariate linear predictive spectral analysis employing weighted forward and backward averaging: A generalization of Burg's algorithm," Naval Underwater Syst Center Tech. Rep. 5501, Oct. 1976. 0. N. Strand, "Multichannel complex maximum entropy (autoregressive) spectral analysis," IEEE...

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