Continuous-time subspace system identification using generalized orthonormal basis functions

This paper proposes a new subspace identification algorithm for continuous-time systems using generalized orthonormal basis functions. It is shown that a generalized orthonormal basis induces the transformation of continuoustime stochastic systems into discrete-time stochastic systems, and that the transformed noises have the ergodicity properties. With these basic observations, the standard subspace identification methods such as the MOESP algorithm are applied to estimate the system matrices. I. I NTRODUCTION Identification of the dynamical system model from inputoutput data is a fundamental problem of the system control theory. Subspace system identification methods, studied extensively in [17], [18], [20], [21], are powerful tools for modeling multi-input multi-output linear dynamical systems. In most of these studies, the system is assumed to be discrete-time. When we need continuous-time system models, what we usually do is to first carry out discretetime system identification using sampled input-output data and then to transform the system back into continuous-time models. There are a couple of reasons why the study on continuous-time system identification is necessary. Firstly, in many applications continuous-time models are preferred because the characteristics of the system is more visible from model parameters such as time-constants and bandwidth. Secondly, if we first identify discrete-time systems, then the transformation into continuous-model is sometimes non unique. For example, we need a matrix logarithm to transform the discrete-time system matrix into the continuous-time system matrix. Continuous-time system identification based on subspace method are studied in [3], [6], [8], [12], [15], [19]. In [19], a method to estimate the system matrices from the frequency response data is discussed. In [8], approximate derivative is exploited to construct algebraic input-output equation. In [6], [15], another way to approximate derivative, or so called δ-operator, is used to obtain a similar conclusion. Ohsumi et. al. proposed to use the distribution-based approach [12]. The approach is to use the derivative in the sense of distribution. Haverkampet. al. used a Laguerre filter, and applied a discrete-time subspace system identification method [3]. The class of generalized orthonormal bases include the Laguerre basis and the Kautz basis [5] as special subclasses. The Laguerre basis and the Kautz basis [9] are applied to study system identification and system approximation in [10], [11], [22], [23]. In these studies, the system response The authors are with the Department of Computer–Controlled Mecahnical Systems, Osaka University, 2–1 Yamadaoka, Suita 565 0871, Japan. is expanded by the Laguerre (or Kautz) basis, and the coefficients of the expansion are determined by the least square method. This line of argument is extended in [16] to include generalized orthonormal bases. This paper uses generalized orthonormal bases to define the transformation of the continuous-time stochastic system into the discrete-time stochastic system, and applies a discrete-time subspace system identification method. Unlike the result in [16], the idea is to transform the system using generalized orthonormal bases. Though the Laguerre basis is a special class of generalized orthonormal bases, the method in this paper differs from that in [3]: (i) The Laguerre basis is used to define the transformation of the systems in this paper, whereas it is used as a filter in [3]. (ii) The generalized orthonormal basis have more degree of freedom than the Laguerre basis. As for (i), it turns out that the identified discrete-time system is causal, and thus it avoids the singularity of the transformation formulae encountered in [3] when the pole of the Laguerre filter coincides with a pole of the system. As for (ii), the extension is by no means trivial because the bilinear transformation on complex plane is not readily available to the case of generalized orthogonal basis functions. We shall come back to these points later in the main sections. This paper is organized as follows. In Section II, we review basic notions of signal and system spaces, and generalized orthonormal bases. In Section III, using a similar ideas as lifting technique considered in [13], we show that a generalized orthonormal basis induces the transformation of stochastic systems. The fact that a generalized orthonormal basis induces the transformation of deterministic systems was first pointed out in [4], where they call it the Hambo transform. In Section IV, we show that the transformed discrete-time signals satisfy the ergodicity properties. Then from the algebraic input-output equation, we conclude that the standard MOESP type algorithms are applicable to estimate the system matrices. A numerical example shows differences of the proposed method and the method in [3].