Abstract
In this paper we consider the problem of recursive identification of ARMA processes. This recursive procedure is parameterized by a weight-matrix acting on the stochastic gradient. The optimal weight-matrix will be defined using a risk-sensitive identification criterion. First the cost function will be expressed using LEQG-theory. Then, applying stochastic realization theory and the bounded real lemma we derive alternative expressions for the cost function. We prove among others, that the LQG functional of a properly augmented system gives the LEQG cost function of the original system. Furthermore, we point out that this cost function can be interpreted as mutual information between two stochastic processes. The optimal weight-matrix will be computed first as the optimum of a multi-dimensional constrained minimization, then a direct approach for solving the optimization problem will be presented. Finally, we briefly indicate that the results above can be extended to multivariate stochastic systems.
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10.6 Bibliography
A. Benveniste, M. Metivier, and P. Priouret. Adaptive algorithms and stochastic approximations. Springer-Verlag, Berlin, 1990.
A. Bensoussan and J.H. van Schuppen. Optimal control of partially observable stochastic systems with an exponential-of-integral performance index. SIAM J. Control Optim., 23:599–613, 1985.
L. Gerencsér. Rate of convergence of recursive estimators. SIAM J. Control and Optimization, 30(5):1200–1227, 1992.
L. Gerencsér. A representation theorem for the error of recursive estimators. In Proc. of the 31st IEEE Conference on Decision and Control, Tucson, pages 2251–2256, 1992.
L. Gerencsér. A representation theorem for the error of recursive estimators. Submitted to SIAM J. Control and Optimization, 2000.
D. Mustafa and K. Glover, Minimum Entropy H ∞ Control. Lecture Notes in Control and Information Sciences, Vol. 146. Springer Verlag, 1990.
T. Söderström and P. Stoica. System identification. Prentice Hall, 1989.
A. Stoorvogel and J.H. van Schuppen. System identification with information theoretic criteria. Report BS-R9513, CWI, Amsterdam, 1995.
J. C. Willems, Least squares stationary optimal control and the algebraic Riccati equation, IEEE Transactions of Automat. Contr. 18(6):621–634, 1971.
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Gerencsér, L., Michaletzky, G. (2003). Risk Sensitive Identification of ARMA Processes. In: Rantzer, A., Byrnes, C.I. (eds) Directions in Mathematical Systems Theory and Optimization. Lecture Notes in Control and Information Sciences, vol 286. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36106-5_10
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DOI: https://doi.org/10.1007/3-540-36106-5_10
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