In this example, the non-linear model developed in Section 8.5 is used for preliminary evaluation of the temperature controller. Here, the dynamic component’s structure and parameters estimates are identified from simulated data, where the fan voltage is assumed to be 2.5 VDC and the heater voltage steps between 2.5 and 3.5 VDC. As in equation9.1, the dynamic component is a second order model with two delay terms, and the parameter estimates are b1 = 0.7769, b2 =−0.7570, a1 =−1.6491,
and a2 = 0.6556. The coefficients of the polynomials that find the static function’s
coefficients i.e. ymax, θ, x0 and c, are the same with the ones used in table 8.2. The
structure of the controller is similar to the one shown in Fig. 9.1, however the control input goes through the non-linear static function defined by equation 8.10. The control simulation set-point yd(k) is also formulated by means of a second non-linear
static function, which is the same as equation 8.10 again. Fig.9.2 shows the block diagram that was used to run the simulation. The PIP control gains were calculated on the basis of trial and error pole placement. For this simulation, one pole was selected close to the origin i.e p1=0.9744 and the second one was selected to be
further away, p2=0.85. Subsequently, the proportional and output feedback gains are
f0=0.1232, and f1=-0.0534; the input feedback filter is g1=-0.9744, and the integral
gain is kI=0.0043.
Figure 9.2: PIP controller implemented on a 2-dimensional Hammerstein model.
the voltage to the fan input is fixed at 2.5 VDC, while the voltage to the heater input is initially zero, stepped up in turn to 2.5 VDC; then stepped up again to 3.5 VDC, and finally stepped down back to 2.5 VDC. These signals are depicted at the lower plot of Fig. 9.3 by a dashed blue and a dashed green trace, respectively. Passing those two signals through equation 8.10yields the command input to the controller, which is represented by a black trace at the upper plot of Fig. 9.3. The controlled temperature response (upper plot of Fig. 9.3, dashed red trace) shows that controller works well, while the control input (lower plot of Fig. 9.3, red trace) shows that for every step change of the set-point yd(k), it follows and eventually
settles at a voltage, the value of which is the same as the heater input previously used to formulate the set-point.
For another scenario (Fig. 9.4), the command input is changed mainly due to the fan input changing. For example, at the 120th sample, the voltage to the fan is increased to 3.5 VDC, while the heater voltage remains at 2.5 VDC. The latter combination yields a lower temperature set-point. This results in a slower response as can been seen by the control input voltage, but the set-point is still tracked.
Finally, for a fixed fan setting and various step changes applied in the heater input (thus the set-point is only determined by the latter), this controller is compared with a PIP controller that uses a second order linear TF model. Its parameter estimates are b1 = 0.2855, b2 = −0.2786, a1 = −1.8737, and a2 = 0.8754. The following
feedback gains f0=0.5722, f1=-0.4212, g1=-0.8177, kI=0.0125 are also calculated
50 100 150 200 250 300 0 5 10 Temperature ( ° C) 50 100 150 200 250 300
One minute samples 0 1 2 3 4 Volts DC
Figure 9.3: PIP control simulation on the non-linear thermal model. The upper plot shows the command input and temperature response with a black and a red trace, respectively. The lower plot depicts the control input with a red trace, whilst the fan and heater signals that eventually formulate the command input are represented by dashed blue and green traces, respectively.
PIP controller that uses the linear model is depicted in green colour. The first two plots show drastic step changes to the set-point; although the control with the linear model shows a consistent response, the control with the non-linear model is slower and overshoots in the upper plot; and it is slightly slower in the middle plot. On the contrary, for smaller set-point changes, the responses are almost identical; as seen in the lower plot of Fig. 9.5.
These simulations show that the responses are satisfactory when the set-points are near the operating point of the controller but somewhat slower for instances where the fan input is changed or drastic steps occur in the heater input. If a faster controller were to be used, with both poles closer to the origin, the simulation response would be highly oscillatory and unstable for certain step changes in the set-points that do not fall within its operating point. Hence, an SDP-PIP control approach may be more suitable in this regard.
50 100 150 200 250 300 0 2 4 6 Temperature ( ° C) 50 100 150 200 250 300
One minute samples 0 1 2 3 4 Volts DC
Figure 9.4: PIP control simulation on the non-linear thermal model. In this case, it is the fan that induces a change in the command input. The upper plot shows the command input and temperature response with a black and a red trace, respectively. The lower plot depicts the control input with a red trace, whilst the fan and heater signals that eventually formulate the command input are represented by dashed blue and green traces, respectively.
50 100 150 200 250 6 8 10 50 100 150 200 250 300 0 5 10 Temperature ( ° C) 20 40 60 80 100 120 140 160 180
One minute samples 4
5 6
Figure 9.5: PIP control simulation on the non-linear thermal model (dashed red trace), compared with the PIP control simulation on the linear model, for three different set-point change scenarios.