Introduction
This is a virtual experiment. It is performed on the computer. We study temperature control of a fluid contained in a tank. Manual (human) and automatic control are available. The basic operation of the automatic PI (proportional plus integral) controller is explored. The user can set the controller parameters and thus study their effect on the system’s performance.
Process (Plant): Fluid, Tank and Heater
The system (or process) that we will consider consists of a tank that contains fluid. The tank sits on a heater which supplies energy to the fluid. The energy supplied by the heater depends on the input voltage to the heater. Please refer to Figure 1 for a sketch of this thermal system.
Figure 1: Fluid, tank and heater thermal system.
In this application we are interested in regulating the fluid temperature. The system can be represented by a block diagram as shown in Figure 2. The input to the system is the voltage to the heater, whereas the output of the system is the fluid temperature.
Figure 2: Block diagram of the process.
Let us develop a mathematical model (differential equation) for this system which describes its dynamics. The different signals and components needed to derive a mathematical model for the thermal system are shown in Figure 3.
We first identify the input and output.
Input: voltage to heater, V, output: fluid temperature, T.
Figure 3: Diagram of the thermal system signals and components.
We apply the heat balance equation to obtain the differential equation of the system:
Energy supplied by the heater = Energy stored by fluid + Energy lost to the surrounding environment by conduction
where, k (=20) is a constant provided by the heater manufacturer, V is the voltage to the heater, C is thermal capacitance of the fluid , T is the fluid temperature, Ta (=20 oC) is the ambient temperature (environment), R is the thermal resistance of the tank wall (heat conduction) and t is the time. Equation (1) is the differential equation that describes the dynamics of our thermal system.
Important note: If the input voltage to the system is negative, the energy supplied by the heater will be equal to zero. This means that the effect of providing a negative voltage is the same as the effect of providing zero voltage. The heater can only supply heat to the fluid. It cannot extract energy from the fluid. This is considered as a limitation on the system.
Temperature Control (feedback or closed loop system):
In order to control the fluid temperature of the fluid, heater and tank process we need to have a controller. The controller takes the difference between the desired (or reference) and measured temperatures and provides the voltage to the heater such that this difference becomes as close as possible to zero. The temperature measurement is obtained by connecting a sensor to the process. This measurement is fed back and compared with the desired temperature. The difference between the desired and measured temperatures is called the error. Figure 4 shows the block diagram of the closed loop system.
Figure 4: Block diagram of the closed loop system.
The objective of the controller is make the measured temperature follow (or track) the desired temperature. In other words, the controller tries to take the error to zero. If the error is positive, the desired temperature is larger than the measured temperature. The controller increases the voltage to the heater which results in a higher measured temperature and closer error to zero. If the error is negative, the desired temperature is less than the measured temperature. The controller reduces the voltage to the heater which results in a lower measured temperature and closer error to zero.
We will explore two types of controllers:
1. Human (Manual Control): see Figure 5
In this case the voltage to the heater is adjusted via a knob. The human uses his hand to change the knob’s position. He/She watches the temperature gauge (sensor) such as a thermometer and adjusts the voltage so the that the difference between the desired and measured temperatures is as close as possible to zero. Note that the sensor is the combination of human’s eyes and temperature gauge. The controller is the human’s brain.
Figure 5: Human or manual control of temperature.
2. Proportional plus Integral or PI (Automatic Control): see Figure 6
In this case the voltage to the heater is adjusted directly by an electronic controller. The controller is proportional plus integral, PI. The input to the controller is the error e. The output of the controller is the voltage V to the heater. The voltage is given as
The controller parameters KP and KI are the proportional and integral gains, respectively. The sensor is a thermocouple that converts every 10 oC to 1 Volt. The PI controller drives the error to zero. Proportional control alone does not drive the error to zero.
Figure 6: PI automatic control of temperature.