Principles of System Identification

20

Identification of Non-Parametric

Input-Output Models

This chapter is devoted to the methods for estimating non-parametric models presented in

Chapter 17. Estimation of response models, namely, impulse, step and frequency responses,

is discussed.

20.1 RECAP

Before we set out to estimate non-parametric models, it is useful to recap certain relevant points

concerning their uses from previous chapters.

Non-parametric models are usually the preferred starting points in identification for reasons dis-

cussed previously (recall §17.4). They make minimal assumptions about the process and do not

require any significant intervention from the user. In Chapter 17 we studied three important descrip-

tions, namely, impulse, step and frequency response models. The focus is primarily on estimation of

these models, specifically in Sections 20.2, 20.3 and 20.4, respectively.

Chapters 11 and 16 described approaches for construction and estimation of non-parametric noise

models based on spectral representations. In §20.5 we shall study a non-parametric method for

estimating the disturbance spectrum from input-output data, which is essentially an extension of the

non-parametric techniques of §16.5.5 that were developed for estimating the spectral density of a

random process.

One of the reasons for non-parametric descriptions being strong contenders as starter models is

that they provide valuable knowledge on time-delay (dead time), order and process characteristics

that are required for estimating parametric models. The impulse and frequency response estimates

have been the traditional tools for this purpose. Chapter 22, specifically §22.5, is devoted to a pre-

sentation of techniques for delay estimation. The use of IR estimates in order estimation through

Hankel matrices is discussed in Chapter 23.

As with previous chapters, wherever applicable, theory is supported with demonstration on case

studies using the System Identification toolbox of MATLAB. Throughout this chapter the following

are assumed - open-loop conditions, inputs are quasi-stationary and unmeasured disturbances are

stationary.

20.2 IMPULSE RESPONSE ESTIMATION

As was discussed in §17.4.1.1, the FIR model representation is the suitable version of the IIR model

for non-parametric identification. The problem statement is as follows. Estimate the M unknown

coefficients

θ =

f

g[0] g[1] ··· g[M − 1]

g

T

of the FIR model

y[k] =

M−1

X

n=0

g[n]u[k − n] + v[k] (20.1)

from input-output data.

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