Relay-based identification with a quantizer

The procedure for the proposed CLID scheme based on W ang et al. (1997b) is as following:

• The process was run as normal (with quantization interval very small, for example 0.005, equivalent to 12 bit A/D). The standard deviation o f process output (7^is calculated.

• A mid-rise quantizer is used and a specific quantization interval (in the range of greater than 4 O'y) is chosen together with the hysteresis width and paramteters for the lowpass

Butterworth filter. The process is run with this quantization interval. The quantizer output and the filter output are logged.

The relay-based identification method (Wang et a l, 1997b) is used directly.

The controller dynamics and the filter dynamics are separated (by complex dividing) from the result identified from the third step in order to get the process estimate.

5 .3 C o m p u ta tio n m eth o d s

5.3.1 M odel accu racy m easure

For assessment of the accuracy of the estimate, the identification error in this thesis is measured by the worst-case error.

E R R % = xlOO% f = 1,2, " M (5.13)

In (5.13), G (7’ty.) and G{jCO^)diiQ, the actual and the estimated process frequency response respectively. Only the part of the Nyquist curve from zero frequency to the crossover frequency 69^^ where the argument is -180° was considered since this part is the most important for system identification and controller design.

5.3.2 N o n lin ea rity test m ethod Principle for nonlinearitv test

The principle o f non-linearity test is based on the following three main elements: Null hypothesis, surrogate data and nonlinear prediction (Kantz and Schreiber, 1995).

An appropriate null hypothesis for many nonlinear dynamics tests is that the time trend arises from a linear dynamic system. In order to establish the significance o f a test against this null, one can generate many realizations of the Null, and estimate the significance.

The Null o f ‘linear dynamic system’ is not specific. One way to make a specific Null is to set the mean and variance to the same values as those of the original data. Surrogate data is random data generated to have the same mean, variance, and autocorrelation function as the original data. In other words, surrogate data has the same power spectrum as the original data, but the phase relationships that arise from nonlinearity are randomised.

M atlab codes to test nonlinearitv

The above principle is implemented with Matlab codes provided by the author’s supervisor (Thornhill, N.F., Shah, S.L., and Huang, B., 2(X)1). In the algorithm, surrogate data sets that have the same power spectrum and auto-covariance as the test data set are created first. A suitable sub-set o f the data is found with a good end-to-end match. Then the surrogates are calculated. 50 surrogate data sets are used for the test in this thesis. The non-linear zero prediction error of the surrogates is compared with the prediction error for the test data. If the test data were generated by a non-linear process, they are generally more predictable than the surrogates and the error is significantly smaller. "Significant" means lying more than two standard deviations from the mean error o f the surrogates.

The result o f the algorithm gives the nonlinearity index. It is the prediction error o f the test data minus the mean error of the surrogates divided by 2 c r . If the data are much more predictable than the surrogates then the value of nonlinearity index is bigger than 1. If the data are roughly as predictable as the surrogates then the value o f the nonlinearity index is typically less than 1.

5 .4 D iscrete tra n sfe r fu n c tio n m o d el sim u la tio n m eth o d

In order to demonstrate the theoretical analysis and the methods discussed, a second order ARMAX model presented in Huang and Shah (1997) was chosen as a simulation example. The purpose o f this section is to emphasize the validation o f the left side o f Figure 4-9 (or Figure 4-15, i.e. when the quantization interval is small) and to emphasize the procedures for further recommendations on the selection of the quantization interval.

5.4.1 S im ulation ex am ple

A second order ARMAX model presented in (Huang and Shah, 1997) was chosen to validate the theoretical analysis and the methods discussed. The transfer function was:

-1 1 n o „ - i . n 1 o -2

■■.■■03403.

0.2417, -

1-0.8,- .

0.12,-

1 - 0 . 7 8 5 9 9 ''+ 0 . 3 6 7 9 9 '" 1 - 0 . 7 8 5 9 9 ''+ 0 . 3 6 7 9 9 '"

A unit feedback control law is implemented in this simulation (Figure 5-5). The white noise

z" - 0 .7 8 5 9 Z + 0.3679

z^-0.8z + 0.12

z" - 0 .7 8 5 9 Z + 0.3679 0.3403Z + 0.2417

F igure 5-5 Simulation fo r CLID with a quantizer f o r discrete transfer function model

5.4.2 C orrelation and signal-to-noise ratio test

Simulation in this subsection with Figure 5-5 is used for the purpose o f exploring the characteristics o f the quantizer excitation under different quantization intervals. In this simulation, a white noise with (7^=1 is applied and fixed throughout the tests. The standard deviation of the process output is =1.373 when the quantization interval is 0.005 (the same as Case 3 of the following subsection 5.4.4). There are altogether 30 tests ranging from qi=0.10, qi=0.20, qi=0.30 until qi=3.00. The results will be presented in subsection 6.1.1.