Modeling continuous-time processes via input-to-state filters

Abstract

A direct algorithm to estimate continuous-time ARMA (CARMA) models is proposed in this paper. In this approach, we first pass the observed data through an input-to-state filter and compute the state covariance matrix. The properties of the state covariance matrix are then exploited to estimate the half-spectrum of the observed data at a set of user-defined points on the right-half plane. Finally, the continuous-time parameters are obtained from the half-spectrum estimates by solving an analytic interpolation problem with a positive real constraint. As shown by simulations, the proposed algorithm delivers much more reliable estimates than indirect modeling approaches, which rely on estimating an intermediate discrete-time model.

Introduction

The identification of continuous-time stochastic processes is a fundamental research issue which has received considerable interest recently. Since many natural processes are continuous-time, it is of interest in many practical applications to identify a continuous-time model instead of a discrete-time model (Larsson, 2004). Although the signal is in continuous-time, in practice one works with sampled data. One popular approach is to identify a discrete-time system from uniformly sampled data, as shown in Larsson (2004) and references therein. Subsequently, the estimated discrete-time model is converted back to a continuous-time model via a nonlinear transformation (Söderström, 1991). This approach will be referred to as the indirect approach. Apart from the obvious difficulty of solving nonlinear equations, this approach also suffers from several other setbacks: (i) at a fast sampling rate, the poles and the zeros of the associated discrete-time system cluster close to the point $1+\mathrm{i}0$ in the complex plane, leading to a numerically ill-conditioned identification problem; (ii) the continuous-time parameters can be very sensitive to the sampled data. These issues have been discussed in detail in Fan, Söderström, Mossberg, Carlsson, and Zhou (1999), Fan, Söderström, and Zhou (1999), Larsson and Mossberg (2003), Larsson and Söderström (2002), Söderström (1991) and Söderström and Mossberg (2000). A second approach is to identify the continuous-time parameters directly. The basic idea here is to replace the differentiation operator with the delta operator (Feuer and Goodwin, 1996, Goodwin et al., 1992). Several methods have been developed using this approach for autoregressive models, see Fan, Söderström, Mossberg et al. (1999), Fan, Söderström, and Zhou (1999), Larsson and Söderström (2002) and Söderström and Mossberg (2000). This approach is advantageous in many cases as it is computationally efficient and avoids nonlinear transformations if the underlying model is autoregressive. This technique also benefits from nonuniform sampling (Larsson, 2004, Larsson and Söderström, 2002). However, it is not well understood how we can extend this technique for an ARMA model since the mapping from the continuous-time zeros to the equivalent discrete-time zeros is complicated. It is also not known how to guarantee the positivity of the estimated spectrum. Therefore, for ARMA models, theonly available time domain approach seems to be the indirect method described above (Larsson and Mossberg, 2003, Söderström, 1991).

If we consider a discrete-time process, we can ensure the stability of the estimated autoregressive model in a fairly simple way (Ljung, 1999, Makhoul, 1975, Söderström and Stoica, 1989). But unlike the discrete-time counterpart, the mapping from the lagged covariance estimates to the system parameters for a continuous-time system is more complicated. Hence, the standard discrete-time algorithms cannot be extended directly. In this paper, we propose a direct approach for modeling continuous-time stochastic processes. Specifically, we provide an estimation algorithm with the following properties: (i) it is computationally efficient; (ii) the stability of the estimated model is guaranteed; (iii) it can handle irregularly sampled data; (iv) it is possible to circumvent the problems associated with the sampling zeros for an ARMA model.

Our approach uses the framework of input-to-state filtering (Georgiou, 2001, Georgiou, 2002a, Georgiou, 2002b) where we first estimate the half-spectrum and its derivatives evaluated at some pre-specified points in the right-half plane. This is achieved using a linear operation on the covariance matrix of the output of an input-to-state filter. Subsequently, we present an approach for estimating a stable rational model from the estimates of the half-spectrum. In this step, we apply linear interpolation combined with a regularization step similar to that proposed in Mari, Stoica, and McKelvey (2000) and Stoica, McKelvey, and Mari (2000). If the resulting model is unstable then a recent spectral zero assignment algorithm (Byrnes, Georgiou, & Lindquist, 2001), (see also Georgiou, 1999), is used to compute a stable model. The approach can also be used when the data are irregularly sampled. We provide additional insights into the estimation of the half-spectrum, and discuss the numerical and statistical issues involved in the estimation of such statistics. We also carry out an asymptotic second-order statistical analysis. The proposed algorithm is tested using numerical simulations.

In the framework of input-to-state filtering it is also possible to model the spectrum in terms of the generalized orthogonal basis functions (Heuberger, Van den Hof, & Wahlberg, 2005). This approach has been explored in Blomqvist and Fanizza (2003).