System Identification

Zadeh (1962) gave the following definition of the identification problem

"Identification is the determination ,on the basis of input and output, of a system within a specified class of systems,to which the system under test is equivalent."

An identification problem is then characterised by three quantities: a class of systems S e , input signals u, and an equivalence which is usually a criterion of performance or loss function. The loss function is a functional of the process output y and the model output y^

F=F{y,yJ (3.1)

Thus two models and are equivalent if the associated loss function for each model is the same.

with

equivalence defined by a loss function the identification problem is essentially the minimisation problem of finding

The choice of f and u and the criterion will depend on the purpose of the identification and any priori knowledge. With the abstract definition of Zadeh there is a large choice in how the identification problem is formulated. If for example the identification problem is imbedded in a stochastic framework and ^ is defined as a parameter class f = (Sg) where 0 is a parameter vector, the problem reduces to one of parameter estimation. Such parametric formulations will be of considerable interest.

The solution to the identification problem raises interesting questions concerning the existence,uniqueness and accuracy of the solution and how these are affected by ^ and u, (Goodwin and Payne 1977) . In general the choice of ^,u and the criterion can often be resolved if the purpose of the identification is made explicit . Detailed discussions of system identification may be found in numerous articles and books (Ljung 1987, Norton 1988). The emphasis in this chapter will be on the parameter estimation of linear discrete time stochastic systems.

Non parametric representations such as spectral densities,impulse response,covariance functions will be useful as intermediate steps in determining parametric representations of for example disturbances.

The parametric state space and ARMAX models are a natural choice for developing a prediction algorithm because they are flexible and able to describe the behaviour of many time series (Box and Jenkins 1976) . Furthermore ARMAX type models are ideally suited for digital implementation. In order to obtain unique solutions and construct efficient algorithms it is important to find parsimonious representations of the system that is representations that contain the smallest number of parameters. As has been demonstrated in Chapter 2 the ARMAX models are parsimonious representations since they correspond directly to the canonical state-space form. The parsimonious representation crystallizes all knowledge of the system dynamics and leads to a nice solution to the prediction problem.

Possible objections to an ARMAX representation are that nonlinear dynamics cannot be described and in any case the ARMAX structure is too simplistic to be a complete mathematical description of the signal generating process. It could also be argued that the discrete nature of model will cause loss of information. These objections can to a large degree be dispelled since as Box (1976) asserts all models are incorrect and should be judged by there usefulness in the proposed application. By including nonlinear regressors ARMAX models can to a certain extent be generalised to capture nonlinearities. Rigorous extensions to nonlinear systems can be effected by the Nonlinear NARMAX model

(Leontaritis and Billings 1985).

In numerous practical applications (Astrom and Wittenmark 1984) a sampling interval of approximately one tenth the dominant time constant gives negligible problems due to "loss of information".

3.2.1 Class of Input Signals

In system identification the model is evaluated from observing input and output sequences of the system. The accuracy and quality of the identified model depends fundamentally on the input signal u. In particular the input needs to be persistent or roughly speaking it must be varied sufficiently so that all important dynamic modes of the system are excited. The resulting data will then provide all the necessary information about the input-output properties of the system. For controller development the input signal should be chosen to vary the output about the normal operating region.

An input sequence u (k) is said to be persistently exciting of order n (Ljung 1987) if the following limits exist.

Ü = u(k) (3.3)

1 ^

V

iu{k)-u) (uik+l) -u) (3,4)

^k=o

and the matrix A defined by

A = { a^j = R(i-j) } i, j = 1,2, . . .n (3.5)

is positive definite.

These conditions effectively specify the number of independent modes or components present in the input signal. The "necessary" number to elucidate an accurate model naturally depends on the order of the m o d e l . A sequence is persistently exciting of order n if no selection of every n successive samples of the sequence is linearly dependent. The condition ensures that the least squares solution is nonsingular. The frequency domain interpretation of a persistently exciting signal is that the spectrum of the signal #(&) is non zero at n or more frequencies. It has been shown for example by Astrom and Bohlin (1966) that in order to get consistent parameter estimates using least squares or maximum likelihood requires persistent excitation.

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