Kenneth Hsu

Identification of Structured Nonlinear Systems

This paper is concerned with the identification of static nonlinear components in a complex interconnected system. These nonlinear components are treated nonparametrically, in the sense that no natural parameterization is assumed to be available.

Filter design from data: direct vs. two-step approaches

Consider a nonlinear system with input u and outputs y, z. Assume that ut and yt are measured for all times t and that zt is measured only for t les Tm, but it is of interest to know zt for t > Tm. Such situation may arise when the sensor measuring z fails and it is important to recover this variable, e.g., for feedback control. Another case arises when the sensor measuring z is too complex and costly to be used, except for an initial set of experiments. Assuming that z is observable from the couple (u, y), the standard approach consists of a two-step procedure: identify a model and then design an observer/Kalman filter based on the identified model. Observing that an estimator of z t, t > Tm is a system using (ut, y t) as inputs and producing as output an estimate of zt , the idea of directly identifying an estimator model from the available noisy data (utildet, ytildet) and ztildet in the time interval (0, Tm) is investigated in this paper. The two-step procedure is proved to perform, in the case of exact modeling, no better than the direct approach. In the presence of modeling errors, the directly identified filter is proved to be anyway the minimum variance estimator, among the selected approximating filter class. A similar result is not assured by the two-step design, whose performance deterioration due to modeling errors may be significantly larger. Another relevant point is that minimum variance filters for nonlinear systems are in general difficult to derive and/or to implement, and widely used approximate solutions, such as extended Kalman filters, quite often exhibit poor performance. On the contrary, the recent progresses in nonlinear identification methods may allow the direct filter identification. An example related to the Lorenz attractor is presented to demonstrate the effectiveness of the presented approach

Input design for structured nonlinear system identification

This paper is concerned with the input design problem for a class of structured nonlinear models. This class contains models described by an interconnection of known linear dynamic systems and unknown static nonlinearities. Many widely used model structures are included in this class. The model class considered naturally accommodates a priori knowledge in terms of signal interconnections. Under certain structural conditions, the identification problem for this model class reduces to standard least squares. We treat the input design problem in this situation. An expression for the expected estimate variance is derived. A method for synthesizing an informative input sequence that minimizes an upper bound on this variance is developed. This reduces to a convex optimization problem. Features of the solution include parameterization of the expected estimate variance by the input distribution, and a graph-based method for input generation.

Parametric identification of structured nonlinear systems

In this paper, identification of structured nonlinear systems is considered. Using linear fractional transformations (LFT), the a priori information regarding the structural interconnection is systematically exploited. A parametric approach to the identification problem is investigated, where it is assumed that the linear part of the interconnection is given and the input to the nonlinear part is measurable. An algorithm for the identification of the nonlinear part is proposed. The uniqueness properties of the estimate provided by the algorithm are examined. It is shown that the estimate converges asymptotically to its true value under a certain persistence of excitation condition. Two simulated examples and a real-data example are presented to show the effectiveness of the proposed algorithm.

Brief paper: The filter design from data (FD2) problem: Nonlinear Set Membership approach

In the paper, we consider the problem of designing a filter that, operating on noisy measurements of input u and output y of a dynamical system, gives estimates (possibly optimal in some sense) of some other variable of interest z. A large body of literature exists, which investigates this problem assuming that the system equations relating u, y and z are known. However, in most practical situations, the system equations are not (completely) known, but a data set composed of noisy measurements of u, y and z is available. In such situations, a two-step procedure is typically adopted: a model is identified from the set of measured data, and the filter is designed on the basis of the identified model. In this paper, we propose an alternative solution, which uses the available data set of measured u, y and z not for the identification of system dynamics, but for the direct design of the filter. Such a direct design is investigated within the Nonlinear Set Membership framework. In the case of full observability, an almost optimal filter is derived, where optimality refers to minimizing a worst-case estimation error. In the case of partial observability, conditions are given for which the direct design is guaranteed to give bounded estimation error. Three examples are presented, related to the Lorenz chaotic system the first two, and to an automotive application the third one.

Parametric and nonparametric curve fitting

We are concerned with convergence issues in the identification of a static nonlinear function. Our investigation focuses on properties of the input signal that ensure convergence of the estimate. Both parametric and nonparametric approaches are considered. In the parametric case, we offer sufficient conditions under which the estimated parameters converge to their true values almost surely. For the nonparametric case, we offer necessary and sufficient conditions under which the estimated function converges almost surely to the true nonlinearity.

Nonparametric methods for the identification of linear parameter varying systems

In this paper, we consider the identification of linear parameter varying (LPV) systems. By taking a nonparametric approach, we do not impose a priori parameterizations on the model structure, nor is it assumed that a natural parameterization is suggested from an analytical understanding of the underlying process. In this case, it is shown that the LPV identification problem reduces to a closely related problem in nonlinear system identification. By referring to previous results, we offer a dispersion-based cost criterion and a sufficient condition on the exogenous parameter under which asymptotic convergence of the identified LPV system is assured almost surely.

Identification of Nonlinear Maps in Interconnected Systems

We offer a systematic algorithm for the identification of static nonlinear maps in interconnected systems. The class of systems considered are those consisting of linear time-invariant systems and static nonlinear functions. Under the conditions that the linear dynamics are known and the inputs to the nonlinearities are measurable, the identification problem is reduced to a least squares problem. Notions of identifiability and persistence of excitation are introduced to demonstrate convergence of the estimate to the true nonlinear function under the L1-norm.

A KERNEL BASED APPROACH TO STRUCTURED NONLINEAR SYSTEM IDENTIFICATION PART II: CONVERGENCE AND CONSISTENCY

Abstract In (Hsu et al. , 2005c), an algorithm for the identification of structured nonlinear systems was proposed and its computational properties were explored. In this paper, we continue the investigation and formalize notions of identifiability and persistence of excitation. Conditions under which the estimated nonlinearity converges uniformly to the true nonlinearity are developed for a class of kernel based dispersion functions.

An LFT approach to parameter estimation

In this paper we consider a unified framework for parameter estimation problems. Under this framework, the unknown parameters appear in a linear fractional transformation (LFT). A key advantage of the LFT problem formulation is that it allows us to efficiently compute gradients, Hessians, and Gauss-Newton directions for general parameter estimation problems without resorting to inefficient finite-difference approximations. The generality of this approach also allows us to consider issues such as identifiability, persistence of excitation, and convergence for a large class of model structures under a single unified framework.