A nonlinear black box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics. There has been considerable recent interest in this area with structures based on neural networks, radial basis networks, wavelet networks, hinging hyperplanes, as well as wavelet transform based methods and models based on fuzzy sets and fuzzy rules. This paper describes all these approaches in a common framework, from a user's perspective. It focuses on what are the common features in the di erent approaches, the choices that have to be made and what considerations are relevant for a successful system identi cation application of these techniques. It is pointed out that the nonlinear structures can be seen as a concatenation of a mapping from observed data to a regression vector and a nonlinear mapping from the regressor space to the output space. These mappings are discussed separately. The latter mapping is usually formed as a basis function expansion. The basis functions are typically formed from one simple scalar function which is modi ed in terms of scale and location. The expansion from the scalar argument to the regressor space is achieved by a radial or a ridge type approach. Basic techniques for estimating the parameters in the structures are criterion minimization, as well as two step procedures, where rst the relevant basis functions are determined, using data, and then a linear least squares step to determine the coordinates of the function approximation. A particular problem is to deal with the large number of potentially necessary parameters. This is handled by making the number of \used" parameters considerably less than the number of \o ered" parameters, by regularization, shrinking, pruning or regressor selection. A more mathematically comprehensive treatment i s g i v en in a companion paper (Juditsky et al., 1995).

Recur"sive algonithms, whene r.andom obsenvations enter^ are studied in a fainly general frrarraewot'k. An ínpontant featr¡ne is that the obsenvations rnay depenci on pneviotts rrout¡:utÊ't of the algonithm. The considered class of algonithms containsr e 9, stochastic apprrrxirnation algonít¡rrns, necur.sive identification algonithns arid algonithr"ns fon adaptíve cont:rol of linear. systerns. The assunption (2) seems to be appnopniate fon nrany applícations. The sãnìe results at¡ those belcn^¡ can be obtaj¡red also fon non linear dyr.amics. 9(t) = e(t;ç(t-t) rx(t-1),e(t)) and tåe proofs fon this cðse âre grrren in t?1.

a b s t r a c tMost of the currently used techniques for linear system identification are based on classical estimation paradigms coming from mathematical statistics. In particular, maximum likelihood and prediction error methods represent the mainstream approaches to identification of linear dynamic systems, with a long history of theoretical and algorithmic contributions. Parallel to this, in the machine learning community alternative techniques have been developed. Until recently, there has been little contact between these two worlds. The first aim of this survey is to make accessible to the control community the key mathematical tools and concepts as well as the computational aspects underpinning these learning techniques. In particular, we focus on kernel-based regularization and its connections with reproducing kernel Hilbert spaces and Bayesian estimation of Gaussian processes. The second aim is to demonstrate that learning techniques tailored to the specific features of dynamic systems may outperform conventional parametric approaches for identification of stable linear systems.

Absrmcr-me extended galman filter is an approximate Nter for nonlinear systems, based on first-order linearization. Its rrse for the wit parameter and state estimation problem for linear systems with uuknom parameters is well known and widely spread. H q e a convergence of this method is given. It is shown that in general, the estimates may be biased or divergent and the c a w for this are displayed. Some common special cases where convergence is gnaranteed are also given. The analysis gives insight i n @ the convergence mechanisms and it is shown that with a modification of the algorithm, global convergence results can be obtained for a general case. The scheme can then be mterpreted as mrudmization of the likelihood function for the estimation problem, or as a reun-sive prediction error algorithm.

The sections in this article are The Problem Background and Literature Outline Displaying the Basic Ideas: Arx Models and the Linear Least Squares Method Model Structures I: Linear Models Model Structures Ii: Nonlinear Black‐Box Models General Parameter Estimation Techniques Special Estimation Techniques for Linear Black‐Box Models Data Quality Model Validation and Model Selection Back to Data: The Practical Side of Identification