On State Space Realizability of Bilinear Systems Described by Higher Order Difference Equations ¨ Ulle Kotta Institute of Cybernetics at Tallinn Technical University Akadeemia tee 21, 12618 Tallinn, Estonia kotta@cc.ioc.ee Sven N˜ omm Institute of Cybernetics at Tallinn Technical University Institut de Recherche en Communications et Cybernetique de Nantes 1 rue de la No¨ e, BP 92101, France sven@cc.ioc.ee Alan S. I. Zinober Department of Applied Mathematics The University of Sheffield Sheffield S10 2TN, UK a.zinober@sheffield.ac.uk Abstract— This paper studies the realizability property of bilinear input-output (i/o) models in the classical state space form. Constraints on the parameters of the bilinear i/o model are suggested that lead to re- alizable models. The complete list of 2nd and 3rd order realizable input- output bilinear models together with the corresponding state equations is given. In the general case some subclasses of realizable bilinear models together with their state-space realizations are presented, including the diagonal bilinear model and the special subclass of the superdiagonal bilinear model. I. INTRODUCTION In many situations the higher order nonlinear input-output (i/o) difference equation y(t + n)= f (y(t),...,y(t + n − 1),u(t),...,u(t + n − 1)) (1) is obtained from experimental data using identification procedures or neural networks. It is known that an arbitrarily structured empir- ical model (1) does not necessarily have a state space realization. Hereinafter, if we speak about realizability, we mean finding the minimal, i. e. accessible and observable realization. Using such a model is highly undesirable for further analysis and control design, since practically all existing control theory for nonlinear systems is based now upon a state space description. Choosing a model structure that leads to state space realization and hence to a tractable control problem, is therefore an important task. It has been noted by various authors that the “black-box” empirical modelling problem is generally not unique and a large collection of candidate models can be obtained. Because the model structure must be chosen a priori, it can be chosen for convenience in subsequent application steps. So why not choose a model structure that leads to a realizable model? The class of models (1) is an extremely large, and in practice, it is usual to focus on various structurally defined subsets of this class. The selection of a subclass of models (1) amounts to specifying a general form for the function f (·) in equation (1). The papers [1], [2] and [3] study realizability of NN-based models, polynomial models and associative models, respectively. A bilinear i/o equation y(t + n) = n  i=1 ai y(t + n − i)+ n  i=1 bi u(t + n − i) + n  i=1 n  j=1 cij y(t + n − i)u(t + n − j ) (2) is a simple non-linear extension of a linear system. It can be used to approximate a non-linear system in cases where a linear model is not sufficiently accurate. During the last thirty years much research has been directed towards the class of bilinear systems. The main reason for the interest in bilinear systems is the fact that many relevant physical systems have been satisfactorily modeled by means of bilinear processes [4]. Moreover, though there is no systematic approach for building nonlinear dynamic models (1), the i/o bilinear representation (2) has the advantage that it is well- suited to parameter estimation, and [5] describes a least-squares procedure for estimating the parameters ai , bi , cij directly from i/o data. In this paper the class of bilinear i/o equations (2) is analyzed with respect to realizability in the classical state space form. It is important to note that the results presented in this paper assume that the parameters are specified independently which is the case in most identification procedures. In cases where external constraints imply relationships between these parameters, there may be additional realizable strucures besides those shown in this paper. The general necessary and sufficient realizability conditions [6] in terms of integrability of certain system-related sequence of subspaces of one-forms, though transparent and inherently simple, yield little insight regarding which strucures of model (1) or (2) imply realizability. In [7] a general subclass of (1), namely an i/o model of the general form y(t)= n-s  i=1 φi (y(t − i),...,y(t − i − s),u(t − i − s)) has been proved to have the state space realization. More specif- ically, it is obvious as a conclusion of the above result that all diagonal i/o bilinear models exhibit a bilinear state space realization. The objective of this paper is to study further the realizability property of the subclass of i/o bilinear models and to suggest constraints on the parameters a i , bi , cij of the bilinear model (2) that can lead to realizable models. More precisely, we will demonstrate that if certain combinations of coefficients in (2) are zero, then the system is realizable. Our analysis is based on necessary and sufficient algebraic realizability conditions for equation (1) [6]. We will provide the complete list of 2nd and 3rd order realizable i/o bilinear models together with their state space equations. In the general case, i. e. when n> 3, we provide several practical realizable subclasses of (2), again together with the corresponding state equations. II. ALGEBRAIC FRAMEWORK Consider a discrete-time single-input single-output nonlinear sys- tem Σ described by the input-output difference equation (1), where u ∈ IR is the scalar input variable, y ∈ Y ⊂ IR is the scalar output variable, n is nonnegative integer, f is a real analytic function defined on Y n × IR n . We will associate with the system Σ an extended state-space system Σe with input v(t)= u(t + n) and Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003 FrP02-6 0-7803-7924-1/03/$17.00 ©2003 IEEE 5685