Principles of System Identification

25

Advanced Topics in SISO Identification

The objective of this chapter is to present a brief overview of certain advanced topics in

identification of SISO systems. In particular, we review topics of identification of linear time-

varying systems using recursive and wavelet-based methods, non-linear system identification

(both under open-loop conditions) and closed-loop identification.

The theory and practical guidelines for identification laid down in the previous parts of this text

have been in the framework of open-loop linear time-invariant systems. The general situation is,

however, that the system is time-varying and/or non-linear and/or under closed-loop conditions. In

this respect, the set of open-loop LTI systems is certainly a restricted class. However, as we shall

shortly learn, the core principles and guidelines for the identification of aforementioned systems

are similar to those of LTI systems. Moreover, many of the algorithms use the LTI system model

as the basic building block. Needless to add, the form of optimization problem, complexity and

dimensionality of the associated methods can be significantly different from those of LTI systems.

While acknowledging the numerous deviations from open-loop LTI scenario, we shall restrict

ourselves in this chapter to three prominent classes, namely, linear time-varying, non-linear and

closed-loop systems.

25.1 IDENTIFICATION OF LINEAR TIME-VARYING SYSTEMS

There are several compelling reasons in practice to respect the time-varying nature of a system, par-

ticularly when the time scales of changes in process become comparable to that of the observation

time scale. The time-varying nature is induced largely by “rapidly” changing physical properties of

the system and/or wear and tear of the actuators. For example, in continuous operation of heat ex-

changers, it is common to observe either a build-up or erosion of material leading to what is known

as fouling. This causes significant changes in the heat transfer coefficient of the material with the

passage of time. A time-invariant model built on initial operating data would obviously be unable

to capture the changing system characteristics.

In this chapter, we shall restrict ourselves, as earlier, to linear time-varying (LTV) systems. One

of the main differences between LTV and LTI systems is that the elementary responses (impulse,

step) change with time. Moreover, the FRF is also a function of time. The non-parametric model in

(17.18) for instance would now be

y[k] =

∞

X

n=0

g[n, k]u[k − n] +

∞

X

m=1

h[n, k]e[k − n] + e[k] (25.1)

A class of parametric descriptions on the other hand make use of those of LTI systems but with

time-varying coefficients. Thus, the parametric model in (17.25) would now take the form

y[k] = G(q

−1

,θ

G

[k])u[k] + H(q

−1

,θ

H

[k])e[k] (25.2)

Systems that are described by (25.2) are also known as the linear parameter-varying (LPV) systems.

A popular one is the TVARX process, for instance

y[k] = −

n

a

X

i=1

a

i

[k]y[k −i] +

n

b

X

j=1

b

j

[k]u[k − n

j

] + e[k] (25.3)

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