**4.1 Smith predictive control structure**

The Smith predictive controller is based in the internal model controller architecture that uses the physical model presented in section II, as illustrated in figure 8. It uses two physical direct models one with time delay for the prediction loop and another with out the time delay for the internal model control structure.

Fig. 8. SPC constituent blocks.

The Smith predictive control structure has a special configuration, because the systems has two inputs with two deferent time delays so it uses two direct models, one model with time delay for compensate its negative effect and another with out time delay needed for the internal model control structure.

Adaptable PID Versus Smith Predictive Control Applied to an Electric Water Heater System 155

For small water flows there is another problem with the multiplicity of the time delay and its resolution. With a sampling period of 1 second it is more difficult to use factional time

> Algorithm MSE Test Set SPC 3,56

The physical model includes à priori knowledge of the real system and has the advantage of been interpretable. This characteristic facilitates the implementation and simplicity the

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

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

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

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

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

For future work some improvements should be made as the enlargement of the resolution of

Tompson M. L. and Kramer M. A., (1994). Modelling chemical processes using prior

Pottman M., Pearson R. K., (1998). Block-Oriented NARMAX Models with Output

Eskinat E., Johnson S. H. and Luyben W., (1991). Use of Hammerstein Models in Identification of Non-Linear Systems, *AIChE Journal*, 1991, vol. 37(2), pp. 255-268.

knowledge and neural networks, *A. I. Ch. E. Journal*, 1994, vol. 40(8), pp. 1328-1340.

the used water flow and the redefinition of the time delay function.

Multiplicities, *AIChE Journal*, 1998, vol. 44(1), pp. 131-140.

The final MSE evaluation control criterion achieved with the SPC is presented in table 2.

delays that happen in reality. This makes the control results a bit aggressive.

Table 2. Mean square errors of the control results.

Smith predictive control algorithm.

well in cold water temperature variations.

**5. Conclusions** 

model accuracy.

characteristics.

reduce resources.

**6. References** 

showed in table 1 and 2.

The SPC separates the time delay of the plant from time delay of the model, so it is possible to predict the *Δt(k)*, *d(k) ) steps* earlier, avoiding the negative effect of the time-delay in the 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 Karim, 2002), (Tan & Cauwenberghe, 1999).

The physical inverse model is mathematically calculated based in the physical direct model presented in section 2 used with out time delay.

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

### **4.2 Smith predictive control results**

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 system is very similar in every working point.

Fig. 9. SPC control results.

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.

For small water flows there is another problem with the multiplicity of the time delay and its resolution. With a sampling period of 1 second it is more difficult to use factional time delays that happen in reality. This makes the control results a bit aggressive.

The final MSE evaluation control criterion achieved with the SPC is presented in table 2.


Table 2. Mean square errors of the control results.

The physical model includes à priori knowledge of the real system and has the advantage of been interpretable. This characteristic facilitates the implementation and simplicity the Smith predictive control algorithm.
