Estimating Nonlinear Systems in a Neighborhood of LTI-approximants

The estimation of Linear Time Invariant (LTI) models is a standard procedure in System Identification. Any real-life system will however be nonlinear and time-varying, and the estimated model will converge to the LTI second order equivalent (LTI-SOE) of the true system. In this paper we consider some aspects of this convergence and the distance between the true system and its LTI-SOE. We show that there may be cases where even the slightest nonlinearity may cause big differences in the LTI-SOE. We also show a result that gives conditions that guarantee that the LTISOE is close to “the natural” LTI approximant. Finally, an upper bound on the distance between the LTI-SOE of a nonlinear FIR system with a white input signal and the linear part of the system is derived. 1 LTI Model Identification To estimate Linear Time Invariant (LTI) models from observed data is a standard tool in systems and control, see e.g. [1]. A brief summary of the basic procedure is as follows: A general LTI-model of a dynamical system can always be described as y(t) = G(q, θ)u(t) +H(q, θ)e(t) (1) Here, q is the shift operator, and G and H are the transfer matrices from the measured input u and the noise source e, which is modeled as white noise (sequence of independent random variables). For notational convenience we will from now on only consider Single-Input-Single-Output systems, but the theory is the same in the multi-variable case. The transfer functions are parameterized by a finite-dimensional parameter vector θ, and this parameterization can be quite arbitrary. For black-box models, it is common to parameterize G and H in terms of the coefficients of numerator and denominator polynomials, perhaps constraining G and H to have the same denominators. This leads to well established model classes, known under names like ARX, ARMAX, OE, BJ, etc. The model parameterizations could also correspond to state-space models in discrete or continuous time. Whatever the parameterization, the problem is to estimate the parameters in (1) based on observed input-output sequences {y(t), u(t), t = 1, 2, . . . , N}.