THE FREQUENCY SAMPLING FILTER MODEL

We begin with the single input, single output (SISO) case and assume that the process to be identified is stable, linear, time invariant and can be represented by the following discrete-time, finite impulse response (FIR) transfer function model  

To derive the frequency sampling filter (FSF) model, we first make use of the inverse Discrete Fourier IVansform (DFT) relationship between the process frequency response and its impulse response, under the assumption that N is an odd number 

This relationship maps a set of discrete-time frequency response coefficients, G ), I = 0, 1, 2. into the set of discrete-time unit impulse 

Interchanging the summations in Equation (4.3) gives the transfer function in its frequency sampling filter model form 

One other point worth making is the relationship between the FSF model parameters and the entire process frequency response. As stated earlier, the parameters of the FSF model are the values of the discrete-time process 

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We assume that the process being identified is stable, linear and time invariant and can be accurately represented by a reduced nth order FSF model. For an arbitrary process input u k) and the measured process output y k), the frequency sampling filter model can be written as... 

The previous chapter introduced the frequency sampling filter model structure in the context of process identification. This chapter pursues the topic further by focusing on the identification of step response models using the... 

We begin by assuming that the unit step response y t) is approximately equal to unity for t >Ts, where Ts is the desired closed-loop settling time. We also assume that the closed-loop system is sampled with an interval At. It is well known that, at the sampling instants, the continuous-time step response is equal to the discrete-time step response. This property is called step response invariance with respect to discretization. Therefore, the results developed in Chapter 5 between the frequency sampling filter model and a discrete-time step response model can be applied. 

This chapter describes two new methods for obtaining frequency response and step response models from processes operating under relay feedback control. Both methods are based on the frequency sampling filter model structure and a recursive least squares estimator. 

For each process output, the p inputs are denoted as ui k), U2 k),..., Up k), the times to steady state for the individual subsystems are given by Ni, N2,- Np, and the reduced orders for each subsystem represented by its own FSF model are chosen to be ni, ri2,..., np. In the matrix representation for this MISO system, the first input ui k) is passed through a set of n frequency sampling filters based on N to form the first n columns in the data matrix, followed by passing the second input U2 k) through a set of U2 frequency sampling filters based on N2 to form the next U2 columns in the data matrix, etc. The parameter vector 9 contains the n firequency response parameters associated with the first subsystem, followed by the n2... 

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