For an example of the real application of thermal response compensation, consider the system shown in Figure 4-9.

This system has been characterized to have a time constant, τ, of roughly 9.1s, and has a sample time of around 150ms. If the thermal response compensation algorithm as described in Section 4.2 is applied to this system, the results are as shown below in Figure 4-10. As expected, the response time is noticeably improved, however the minor temperature noise in the system is amplified by the compensation algorithm into noise on the order of ±5 °C.

Introducing a threshold value for the algorithm to ignore the effects of noise can help to correct for this error. When the variance of the portion of the signal being analyzed is higher than the threshold, the predictive algorithm should be active. When it is not, the predictive algorithm should be disabled to avoid the signal being overwhelmed by noise.

As a rule of thumb, the threshold value should be between 1-2 standard deviations of the noise magnitude, in order to prevent the algorithm from turning on and off at incorrect times. With the wrong threshold value, the predictive algorithm will initiate and end too quickly, and cause discontinuities in the response signal. Figure 4-11 shows that adding a threshold trigger does help reduce compensation from error in noise during steady-state operation, but does not eliminate the effects of noise entirely.

When ambient conditions at the steady-state conditions have real variation, it can contribute noise outside the threshold value considered, and cause occasional erroneous spikes. For any single dataset, the threshold value can be changed iteratively to try and find the optimum value, but this is not practical for a real implementation where the algorithm must provide good results under multiple conditions.

A better option is to smooth the spikes in the values of T_gradient by taking the slope estimation over a greater number of samples. Now, calculating T_gradient should be done as so:

Where k is a designer selected range of samples being considered. In general, the value of k should be between 2, and the number of temperature samples taken in a single time constant τ. Figure 4-12 through Figure 4-14 show the compensated responses with k =10, 50, and 100 for this system. In general, higher k values will contribute to greater accuracy and smoothing of the response signal, but will increase computation time and memory requirements. At very high k-values the compensation will begin to make response time appear slower as well. Designers should use the span of k to make a tradeoff between system temperature performance, and processing time as their application allows.