**1. Introduction**

144 Frontiers of Model Predictive Control

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Industry control processes presents many challenging problems, including non-linear or variable linear dynamic behaviour, variable time delay that means time varying parameters. One of the alternatives to handle with time delay systems is to use prediction technique to compensate the negative influence of the time delay. Smith predictor control (SPC) is one of the simplest and most often used strategies to compensate time delay systems. In this algorithm it is important to choose the right model representation of the linear/non-linear system. The model should be accurate and robust for all working points, with a simple mathematical and transparent representation that makes it interpretable.

This work is based in a previews study made in modelling and controlling a gas water heater system. The problem was to control the output water temperature even with water flow, cold water temperature and desired hot water temperature changes. To succeed in this mission one non-linear model based Smith predictive controller was implemented. The main study was to identify the best and simple model of the gas water heater system.

It has been shown that many variable industry linear and non-linear processes are effectively modelled with neural and neuro-fuzzy models like the chemical processes (Tompson & Kramer, 1994). Hammerstein and Wiener models like pH-neutralization, heat exchangers and distillation columns (Pottman & Pearson, 1992), (Eskinat et al., 1991). And hybrid models like heating and cooling processes, fermentation (Psichogios & Ungar, 1992), solid drying processes (Cubillos et al., 1996) and continues stirred tank reactor (CSTR) (Abonyi et al., 2002).

In this previews work there were explored this three different modelling types: neuro-fuzzy (Vieira & Mota, 2003), Hammerstein (Vieira & Mota, 2004) and hybrid (Vieira & Mota, 2005) and (Vieira & Mota, 2004a) models that reflex the evolution of the knowledge about the first principles of the system. These kinds of models were used because the system had a nonlinear actuator, time varying linear parameters and varying dead time systems. For dead time systems some other sophisticated solutions appear like in (Hao, Zouaoui, et al., 2011)

Adaptable PID Versus Smith Predictive Control Applied to an Electric Water Heater System 147

The electric water heater is a multiple input single output (MISO) system. The controlled output water temperature will be called hot water temperature (*hwt(t)*). This variable depends of the cold water temperature (*cwt(t)*), water flow (*wf(t)*), power (*p(t)*) and of the electric water heater dynamics. The hot and cold water temperature difference is called delta

The electric water heater is physically composed by an electric resistance, a permutation chamber and several sensors used for control and security of the system as shown on figure 2. Operating range of the *hwt(t)* is from 20 to 50ºC. Operating range of the *cwt(t)* is from 5 to 25ºC. Operating range of the *wf(t)* is from 0,5 to 2,5 litters / minute. Operating range of the

The applied energy in to the heating resistance is controlled using 100 alternated voltage cycles (one second). In each sample, the applied number of cycles is proportional to the

Figure 3 shows one photo of the electric water heater and the micro-controller board.

Fig. 1. System main blocks.

water temperature (*Δt(t)*).

**2.1 Electric water heater description** 

*p(t)* is from 0 to 100% of the available power.

delivery energy to the heating element.

Fig. 2. Schematic of the electric water heater: sensors and actuator.

that used a neuro-fuzzy compensator based in Smith predictive control to achieved better results. Or other solutions for unknown dead time delays like (Dong-Na, Guo, et al., 2008) that use gray predictive adaptive Smith-PID control because the dead time variation is unknown. There is an interesting solution to control processes with variable time delay using EPSAC (Extended Prediction Self-Adaptive Control) (Sbarciog, Keyser, et al., 2008) that could be used in this systems because the delay variations is caused by fluid transportation.

At the beginning there was no knowledge about the physical model and there were used black and grey box model approaches. Finally, the physical model was found and a much simple adaptive model was achieved (the physical model white box modelling).

This chapter presents two different control algorithms to control the output water temperature in an electric water heater system. The first approach is the adaptive proportional integral derivative controller and second is the Smith predictive controller based on the physical model of the system. From the previews work it is known that the first control approach is not the best algorithm to use in this system, it was used just because it has a simple mathematical structure and serves to compare results with the Smith predictive controller results. The Smith predictive controller has a much more complex mathematical structure because it uses three internal physical models (one inverse and two directs) and deals with the variable time delay of the system. The knowledge of the physical model permits varying the linear parameters correctly in time and gives an interpretable model that facilitate its integration on any control schemes.

This chapter starts, in section 2, with a full description of the implemented system to control the electric water heater, including a detailed description of the heater and its physical equations allowing the reader to have a comprehension of the control problems that will be explained in later sections.

Section 3 and 4, describes the two control algorithms presented: the adaptive proportional integral derivative control structure and the Smith predictive control based in the physical models of the heater. These sections show the control results using the two approaches applied in to a domestic electric water heater system. Finally, in section 5, the conclusions are presented.
