After studying this chapter you will be able to

• describe the output-error model-estimation problem;

• parameterize the system matrices of a MIMO LTI state-space model of fixed and known order such that all stable models of that order are presented;

• formulate the estimation of the parameters of a given system parameterization as a nonlinear optimization problem;

• numerically solve a nonlinear optimization problem using gradient-type algorithms;

• evaluate the accuracy of the obtained parameter estimates via their asymptotic variance under the assumption that the signal-generating system belongs to the class of parameterized state-space models; and

• describe two ways for dealing with a nonwhite noise acting on the output of an LTI system when estimating its parameters.

Introduction

After the treatment of the Kalman filter in Chapter 5 and the estimation of the frequency-response function (FRF) in Chapter 6, we move another step forward in our exploration of how to retrieve information about linear time-invariant (LTI) systems from input and output measurements. The step forward is taken by analyzing how we can estimate (part of) the system matrices of the signal-generating model from acquired input and output data. We first tackle this problem as a complicated estimation problem by attempting to estimate both the state vector and the system matrices. Later on, in Chapter 9, we outline the so-called subspace identification methods that solve such problems by means of linear least-squares problems.

Nonparametric models such as the FRF could also be obtained via the simple least-squares method or the computationally more attractive fast Fourier transform.