The Proportional-Integral-Derivative (PID) controllers are the most widely used at industry. Based on the estimation of Prof. Åström and Hägglund, 95 of the control loops at industry are PID controllers and mainly PI. The tuning procedure is really important, where the controllers parameters must be set to reach a desired closed-loop specifications. Thus, it is essential to understand the effect of the PID controller componentes on the feedback loop.
Proporcional (P) control: A proportional controller gives a control signal that is proportional to the feedback error. It acts as an amplifier of gain . Its action is represented by:
where is the proportional gain, that is the quantity multiplying the error signal to obtain the control signal . Considering zero initial conditions, the transfer function of a proportional controller can be obtain as follows:
In general, when the setpoint of a system with proportional control is changed, the error cannot be removed completely and an error in steady state is obtained, usually called offset. This error consists in the permanent deviation of the controlled variable with respect to its reference at the steady state. This is one of the main drawbacks of the proportional controllers. The steady-state error can be reduced by increasing the gain . Moreover, by increasing faster responses are obtained, although this can also lead to high control signal values that are not implementable in practice. As noticed above, the proportional controller does not augment the system order.
Integral (I) control: In a controller with integral action, the control signal is modified in a velocity proportional to the error signal; that is, if the error signal is large, the control signal is increased very fast; and if it is small, the control signal is increased slowly. If zero initial conditions are considered, this controller can be expressed as:
and the transfer function of the integral controller (which is of type 1) is the following:
where is called integral gain. By the original definition of the integral term, these systems have “memory”, since the output at the time instant will depend on the past behaviour from 0 to . If a steady-state situation is considered with and , the following expression must be fulfilled:
that it is only possible if or . Therefore, it can be deduced that with the integral action the error will be zero at the steady state.
The main properties of the integral controller are:
Proportional-Integral (PI) control: As discussed above, the proportional action leads to steady-state errors. On the other hand, although the integral action removes the steady-state error, it can provoke oscillatory behaviors or very slow responses. Therefore, both actions are combined obtaining a Proportional-Integral (PI) controller, which includes the advantages of these two elements. This controller can be represented by:
where is the integral time, which represents in this case the time required by the integral action to provide a signal equal to that given by the proportional action.
The transfer function of the PI controller is given by:
With a PI controller is possible to remove the steady-state error, but faster than with a I controller. When is increased in the PI controller, faster responses are obtained, but oscillations can result in the closed-loop response.
Derivative (D) control: In the derivative control, the controller output is proportional to ratio of the error signal with respect to time:
where the parameter is the derivative gain. The transfer function of the derivative controller is given by .
This type of controllers does not react against error signals in steady state because its derivative is zero. For that reason, this controller is not used alone, but it is combined with some of the other control actions. Furthermore, it is a non-causal element and in practice it is usually implemented by including a derivative filter.
The effect of the derivative control action is to anticipate to changes in the error and to give a faster response for those changes. This faster response of the derivative controller allows the system accounting the time delay in fast processes and to be stable in a short period of time (specially when the error changes constantly).
Proportional-Derivative (PD) control: When a proportional and derivative controllers are combined, the following transfer function is obtained:
The controller output can change when the error varies constantly. When the setpoint is changed in the control loop, the control signal changes quickly at the beginning because of the derivative term, and afterwards a gradual change is provided because of the proportional action. The derivative time represents the time for which the derivative action reacts in advance with respect to the proportional action. In theory, the derivative action contributes to stabilize the system by softening the oscillations in the response.
A PD controller can be expressed as:
where can be interpreted as the prediction error at the time instant by using a lineal extrapolation.
Notice that this controller is type 0 and thus it is not able to remove steady-state errors.
Proportional-Integral-Derivative (PID) control: When the three actions are combined, a PID controller is obtained. This controllers contribute by removing the steady-state errors and reducing the oscillations in the closed-loop response (when the different parameters are properly tuned). It is represented by the following equation:
and thus the transfer function is given by:
It is the most general control and it is likely the most used controller in industry. It allows an optimal use of the three control actions exploiting all their properties. It can be considered as a proportional controller that includes an integral action to remove the steady-state error and a derivative action to improve the stability and increase the response velocity. In summary:
Figure 3.10 shows the Nyquist diagram, system output and system input (control signal) of a first-order system with delay controlled with a PID controller when the reference is a unit step.