Continuous-time identification of SISO systems using Laguerre functions

This paper looks at the problem of estimating the coefficients of a continuous-time transfer function given samples of its input and output data. We first prove that any nth-order continuous-time transfer function can be written as a fraction of the form /spl Sigma//sub k=0//sup n/b~/sub k/L/sub k/(s)//spl Sigma//sub k=0//sup n/a~/sub k/L/sub k/(s), where L/sub k/(s) denotes the continuous-time Laguerre basis functions. Based on this model, we derive an asymptotically consistent parameter estimation scheme that consists of the following two steps: (1) filter both the input and output data by L/sub k/(s), and (2) estimate {a~/sub k/, b~/sub k/} and relate them to the coefficients of the transfer function. For practical implementation, we require the discrete-time approximation of L/sub k/(s) since only sampled data is available. We propose a scheme that is based on higher order Pade approximations, and we prove that this scheme produces discrete-time filters that are approximately orthogonal and, consequently, a well-conditioned numerical problem. Some other features of this new algorithm include the possibility to implement it as either an off-line or a quasi-on-line algorithm and the incorporation of constraints on the transfer function coefficients. A simple example is given to illustrate the properties of the proposed algorithm.