**5. Conclusions**

154 Frontiers of Model Predictive Control

The SPC separates the time delay of the plant from time delay of the model, so it is possible

control results. The time delay is a known function that depends of the water flow *wf(k)*. The incorrect prediction of the time delay may lead to aggressive control if the time delay is under estimated or conservative control if the time delay is over estimated (Tan & Nazmul

The physical inverse model is mathematically calculated based in the physical direct model

The low pass filter used in the error feedback loop is a digital first order filter used to filter the feedback error and indirectly to filter the control signal *f(p(k))*. The time delay function is a function of the water flow, which is explained in section 2 and expressed in equation 4.

To test the SPC based in the physical model it was used the same reference signals *r(t)* and

The SPC results are shown in figure 9. As it was predicted from previews work the results are very good in reference and in water flow changes. The behaviour of the closed loop

It can be seen that for small water flows the resolution of the measure is small that makes the control signal a bit aggressive but it does not affect the output hot water temperature.

*d(k) ) steps* earlier, avoiding the negative effect of the time-delay in the

to predict the *Δt(k)*,

Karim, 2002), (Tan & Cauwenberghe, 1999).

**4.2 Smith predictive control results** 

Fig. 9. SPC control results.

system is very similar in every working point.

presented in section 2 used with out time delay.

water flow *wf(t)* used to test the adaptive PID controller.

For comparing the two control algorithms, APID and SPC, the reference signals were applied in controlling the system and the respective mean square errors were calculated as showed in table 1 and 2.

This work present and validate the physical model of the electric water heater. This model was based in the model of a gas water heater because of the similarities of both processes.

The MSE of the validation test is very small which validate the physical electric water heater model accuracy.

Finally, the proposed APID and SPC controllers were successful applied in the electric water heater system. It is verify that the SPC achieved much better results than the adaptive proportional integral derivative controller did as it was expected because of the system characteristics.

The best control structure for varying first order systems with varying large time delay is the Smith predictive controller based in physical model of the system as presented in this work. The SPC controller proposed in opposition to the APID controller reacts also very well in cold water temperature variations.

This controller is mathematically simple and easily implemented in a microcontroller with reduce resources.

For future work some improvements should be made as the enlargement of the resolution of the used water flow and the redefinition of the time delay function.
