Stochastic Systems and Recursive Estimation
by László Gerencsér
This article summarizes results achieved in the area of recursive estimation of stochastic systems by the author in the past decade. The main objective of this work was to develop a rigorous foundation of the theory of recursive estimation.
The spectacular success story of stochastic systems in the sixties and seventies had been interrupted by the emergence of worst case approaches to control and identification. Another less obvious reason for the loss of momentum was a series of unresolved technical difficulties. The book of Ljung and Söderström about system identification in 1981 was instrumental in bringing attention to the enormous flexibility of recursive identification. The blueprint of a new theory of recursive estimation has also been given in this book, but it took almost a decade to develop a practically useful rigorous methodology. As it stands now there are two basic approaches in Europe: one based on Markov-processes, presented in the book of Benveniste, Metivier and Priouret, and a second one based on a rigorous development of the ODE (Ordinary Differential Equation) method. The power of the second approach is that it allows the handling of enforced boundedness and is applicable for time-varying systems.
In what follows we provide a short survey of results obtained via the theory of L-mixing processes, and give a glimpse of related works and further potentials. The basic technology behind the modern theory of recursive identification is the notion of exponentially stable processes introduced by Lennart Ljung on the one hand and Jorma Rissanen and Peter E. Caines on the other hand. A systematic investigation and generalization of this notion has lead to the concept of L-mixing processes (Stochastics, 1989). The usefulness of this concept is due to certain invariance properties. Furthermore, we have powerful moment inequalities for weighted integrals, multiple integrals of Volterra-type, and for exponential moments. The new techniques have been used extensively by Andrew Heunis.
On-line prediction error estimators were traditionally derived from off-line estimators using approximations. It was conjectured by Gábor Tusnády that the difference between the two estimators is negligible compared to the standard deviation. This had in fact been proved in 1993 in my paper published in Systems and Control Letters. This paper gives the most precise and convenient characterization of recursive prediction error estimators. The harder part of the analysis is given in my paper, published in SIAM Journal on Control and Optimization, 1992, which provides a rigorous foundation of a practically useful ODE-method.
A major challenge to the theory of system-identification was the emergence of the theory of stochastic complexity developed mainly by Jorma Rissanen. More generally, the study of the interaction of uncertainty and performance became a widely investigated subject. Using the new theory of recursive identification the pathwise performance of an on-line computable adaptive predictor for ARMA-processes was analyzed in my paper in the Journal of Statistical Planning and Inference, 1994.
More recently we have considered the problem of interaction of identification and control. On the technical level, we ask about the effect of statistical uncertainty, due to identification, on control performance. The level of external excitation is an interesting design variable. A key element of the analysis is understanding closed loop recursive identification, and the interaction seems to be unmanageable complex. It turns out that the rigorous theory of recursive estimation provides the appropriate tools to solve this problem. For details see my paper with Zsuzsanna Vágó in Journal of Mathematical Systems, Estimation and Control, 1998. For other aspects of this problem, see the article by József Bokor on SZTAKIs Systems and Control Laboratory in this issue.
The asymptotic theory of recursive identification and stochastic realization theory have also been used to develop a general approach to the design and analysis of risk-sensitive identification methods in a joint work with György Michaletzky and Zsuzsanna Vágó, thus extending the works of Jan H. van Schuppen and Anton Stoorvogel.
Standard recursive estimation methods are generalizations of the classical Robbins- Monroe procedure. However, in direct adaptive control we need Kiefer-Wolfowitz-type procedures, that are designed for function minimization under noise. A remarkable progress in this direction is the development of a randomized version of the latter procedure by James C. Spall. This so-called simultaneous perturbation stochastic approximation (SPSA) method, had been the subject of my paper in the IEEE Transactions on Automatic Control, 1999, where the extension of the ODE method is given. The analysis is also applicable to discrete-variable stochastic optimization problems. The latter is a joint work with Stacy Hill and Zsuzsanna Vágó.
Estimation of Hidden-Markov processes is another recent research interest. The prime example is the estimation of Gaussian ARMA-processes under quantized observation representing low-accuracy sensors (a joint work with Francois LeGland and György Michaletzky). Surprisingly, no computationally viable procedure is known today. Some simpler benchmark problems have been studied in cooperation with Lorenzo Finesso and Ildikó Kmecs.
Further topics in recursive identification, that we considered earlier are: rate of convergence for the LMS method of adaptive filtering; fixed gain recursive estimation; real-time change-detection. Continuous-time Modelling seems to be as attractive as ever (a joint work with Arun Bagchi). An emerging area is the real-time analysis of financial time-series. In particular, the estimation of cointegrated processes is on the agenda. Another pending project is the application of stochastic realization theory to choose appropriate parameterizations for recursive system-identification. Some of the basic problems have been formulated in the book of György Michaletzky, József Bokor and Péter Várlaki, 1998.
László Gerencsér - SZTAKI
Tel : +36 1 4665 644