We will consider a practical example that demonstrate the ideas of transfer function methods.
Example: Temperature Control
For the tank, fluid and heater process (that we studied earlier in this course) shown below, let
the energy supplied by the heater = 20V Watt, where V is the input voltage,
the thermal capacitance of the fluid C = 60 Watt.s/ oC, and
the thermal resistance of tank R = 0.2 oC/ Watt.
We would like to control the temperature of fluid. It is required that the temperature tracks given desired constant reference signals. A suitable solution for such a problem is the use of PI controllers. In order to design a control system for the above process, we need to select a sensor for temperature measurement. We will choose a thermocouple that converts every 10 oC to 1 volt. This means that the measurement will be equal to 0.1T , where T is the fluid temperature. Let us apply the following steps in the design of a PI controller for the above process.
1. Modeling: Recall that the differential equation of the above thermal process is obtained using the heat balance equation and is written as (state space representation)
Note that we choose the state variable to be the temperature difference between the fluid temperature and ambient temperature. The reason for this is to obtain a linear state space representation for the system. By adding the value Ta to x we obtain T. For the purpose of simplifying the discussion, we will let Ta = 0, that is, x = T and y =x =T.
The transfer function of the process from the input voltage to the temperature is
If the initial temperature and the input to the process are known one can solve for the time response of the temperature using the transfer function and the state space representation of the system. Note that when we deal with transfer functions, the initial condition is assumed to be equal to zero.
2. PI (proportional plus integral) control: We will now study the effect of connecting the above process to a PI controller in a closed loop (feedback) system. The sensor will be the thermocouple above. The block diagram of the closed loop system is drawn below
The transfer function of the closed loop system is found as
Note that the denominator of the transfer function depends on the controller parameters kp and ki that the control engineer selects based on certain requirements. The roots of the denominator are called poles. Negative poles produce stable system and imaginary roots cause oscillations in the response of the system. The response of the system depends on the poles. These poles depend in turn on the controller parameters. We conclude that it is important to select kp and ki carefully to design of a good control system.
As an exercise let us find the response of the system when the proportional gain is equal to 5, the integral gain is equal to 0 and the reference signal is equal to 5 volts. Note that in this case the controller becomes proportional since there is no integral action. We have
As time goes to infinity the output temperature goes to 33.3. The measured temperature goes to 0.1*33.3 = 3.33 < r, where r = 5 (reference signal). Thus, tracking is not achieved with a P (proportional) controller. Let us test the tracking ability of the closed loop system with PI controller when the input reference signal is constant (r = a). We will use the final value theorem as discussed in the previous lecture.
The PI controller provides the tracking ability.
The denominator of the closed loop system transfer function T (s) is
The poles of the system are the roots of the denominator and are equal to
The poles are negative for all positive values of kp and ki . Therefore, the use of a PI controller produces a stable closed loop system. If the integral gain is equal to zero, one pole will be equal to zero and according to our stability definition, the system will be unstable.
Imaginary poles (roots) will occur if
For example if kp = 5, then oscillations will occur if
In summary, stability, tracking ability, and system response requirements are important factors in determining the controller parameters.