Predictor Performance

As a benchmark against which to assess the predictions generated by the adaptive algorithm the optimal predictor parameters for predictors up to ten steps ahead were calculated using the recursive Diophantine equation (3.140). As a measure of prediction quality the normalized mean square prediction error at k steps ahead is defined as

[y(i+k/J) -y(i+k)]2

e(t+Jc) = (4.21)

52 yil+k) 2

i = 0

where the signal y(k) has been adjusted to have zero mean. The normalized mean square errors time history from the "optimal" predictors are shown in Figure 4.3.

The self tuning predictor was initially implemented using Peterka's square root algorithm, with a predictor based on

A R M A (5,5) model. The resulting normalised mean square errors are shown in Figure 4.4 for the PM signal. It is encouraging that the predictor performance for all prediction distances converges

close to the performance of the optimal predictor.

Evolution of the parameter estimates for the one step ahead predictor (10 parameters) is shown in Figure 4.5. The first five estimates always correspond to . . .o?5. and the remaining to

...02, where I will depend on the prediction distance k as I = 5+k-l. The parameters continue to vary long after the overall performance of the predictor has settled. In Figure 4.4 the mean square error of the 1-step ahead predictor has settled at about 500 seconds but the parameter plot shows variations well up to 1000 seconds and smaller but continual variations in some of the estimates associated with lagged variables in the data vector. Figure 4.6 shows the parameter estimates of the 3 -step predictor and in Figure 4.7 the estimates associated with the 7-step predictor. It is clear from the plots that with increasing prediction distances the variability of the estimates increases and for a prolonged duration. However this variability does not appear to adversely affect the predictor performance. The root cause of the variability is not due to any adaptive aspect of the algorithm since the iterated forgetting factor used in these tests rapidly approaches unity. Furthermore with the persistently exciting noise input it is expected that the estimator gain falls rapidly as the trace of the covariance matrix Figure 4.3 9 illustrates. The estimator essentially "falls asleep". The variations can therefore only be attributed to the underlying signal/noise ratio which decreases rapidly with prediction distance. This is a characteristic behaviour of the self tuning predictor. The PM signal with various predictions superimposed is shown in Figures 4.8 and 4.9.

4.5.1 Tests on Residuals

Recall from (3.103) that the self-tuning predictor works by making certain auto correlations and cross correlations zero. Or

equivalently the self-tuning predictor may be considered as a minimum variance controller for the fictitious system (Figure

2.2) with dead time k. The minimum variance viewpoint can be exploited to demonstrate predictor performance.

The output of the control system, Figure 2.3, is the prediction error e(t) and this from (3.118) is a moving average process of order k-1 where k is the prediction distance.

e(t) = E{z~^) w{t) (4.22)

For a moving average process of order k-1 it is known (Box and Jenkins 1976) that the autocorrelations up to lags k-1 are non zero and zero beyond k-1. This is the equivalent result from Theorem I . Hence the performance of the predictions at various distances k can be "diagnosed" by examining a plot of the autocorrelations. If the autocorrelations are indeed zero at lags greater than k then the predictor has achieved minimum variance. Plots of the autocorrelations from the predictor errors for

k=l,..10 and the optimal predictor are shown in Figures 4.10-

4.12. The plots demonstrate that the predictor has to a significant degree achieved minimum variance. It is anomalous that the errors from the optimal predictor exhibit significant autocorrelation (in comparison to the 2a) limits. This could perhaps be attributable to the random noise generating mechanism. It is interesting to note in passing that the above idea can be adapted as a tool to diagnose any single input single output regulator, since minimum variance control is the best possible control in the sense that no other controller can have lower variance. For a regulator loop with dead time k (assumed known) achieving minimum variance control the autocorrelation of the errors will be moving average of order k-1. The same autocorrelation test can then be applied to determine if the loop is minimum variance. Under minimum variance control the output will have variance

Oy — ( 1 + 01 + 02 + . . . 0^_i ) • 23 j Where is the standard deviation of the driving noise.

If significant autocorrelations exist beyond lag k then by fitting a moving average model,which will be of order greater than k-I, to the error sequence the theoretical achievable minimum variance can be determined as follows.

Fit the model

y(t) = (l + 0iZ“^+0 2Z"^+. . .0;,_^z^"^ + 0j^z”^+. . . w( t)

The variance is therefore

Oy = ( 1 + 01 + 02 + . . . ^k+f^ * 25 j

the terms up to lag k-1 will be identical since the assumed dead time is k and no control can have any effect on these terms, they are dependent solely on the disturbance process. Terms at lag k and beyond can be influenced by control ; minimum variance makes these zero. The theoretical minimum variance achievable can be found by simply truncating terms in (4.25), or the ratio

'’1 = 1 + ___ (e^+. ■ -etf) (4.26) ^Vmv (1 + 01 + 02 + . • ■Ok-l)

can be found. The analysis can be used to give a bound on achievable performance and improvements possible. If control loop performance is unsatisfactory but the analysis shows it to be minimum variance then the designer must change the structure of the system to reduce time delay or minimise the effect of disturbances perhaps through the use of feedforward.