In this section, we shall investigate the effects of integral and derivative control actions on the system performance. Here we shall consider only simple systems so that the effects of integral and derivative control actions on system performance can be clearly seen.
Integral Control Action. In the proportional control of a plant whose transfer function does not possess an integrator 1/s, there is a steady-state error, or offset, in the response to a step input. Such an offset can be eliminated if the integral control action is included in the controller. The main function of the integral action is to ensure that the exit of the process matches the set point in steady state.
In the integral control of a plant, the control signal, the output signal from the controller, at any instant, is the area under the actuating error signal curve up to that instant.
The control signal u(t) can have a nonzero value when the actuating error signal e(t) is zero, as shown in Figure 5-39(a). This is impossible in the case of the proportional controller since a nonzero control signal requires a nonzero actuating error signal.
(A nonzero actuating error signal at steady state means that there is an offset.) Figure
5-39(b) shows the curve e(t) versus t and the corresponding curve u(t) versus t when the
controller is of the proportional type.
Note that integral control action, while removing offset or steady-state error, may lead
to oscillatory response of slowly decreasing amplitude or even increasing amplitude,
both of which are usually undesirable.
Proportional Control of Systems. We shall show that the proportional control of a system without an integrator will result in a steady-state error with a step input. We shall then show that such an error can be eliminated if integral control action is included in the controller.
Consider the system shown in Figure 5-40. Let us obtain the steady-state error in the
unit-step response of this system. Define
the error E(s) is given by
For the unit-step input R(s) = 1/s, we have:
The steady-state error is
Such a system without an integrator in the feedforward path always has a steady-state error in the step response. Such a steady-state error is called an offset. Figure 5-41 shows the unit-step response and the offset.
Integral Control of Systems. Consider the system shown in Figure 5-42. The controller is an integral controller. The closed-loop transfer function of the system is:
Since the system is stable, the steady-state error for the unit-step response can be obtained by applying the final-value theorem, as follows:
Integral control of the system thus eliminates the steady-state error in the response to the step input. This is an important improvement over the proportional control alone, which gives offset.
Derivative Control Action. Derivative control action, when added to a proportional controller, provides a means of obtaining a controller with high sensitivity. An advantage of using derivative control action is that it responds to the rate of change of the actuating error and can produce a significant correction before the magnitude of the actuating error becomes too large. Derivative control thus anticipates the actuating error, initiates an early corrective action, and tends to increase the stability of the system.
Although derivative control does not affect the steady-state error directly, it adds damping to the system and thus permits the use of a larger value of the gain K, which will result in an improvement in the steady-state accuracy. Because derivative control operates on the rate of change of the actuating error and not the actuating error itself, this mode is never used alone. It is always used in combination with proportional or proportional-plus-integral control action.
Proportional Control of Systems with Inertia Load. Before we discuss the effect
of derivative control action on system performance, we shall consider the proportional
control of an inertia load.
Consider the system shown in Figure 5-46(a). The closed-loop transfer function is obtained as:
The characteristic equation is:
Since the roots of the characteristic equation are imaginary, the response to a unit-step input continues to oscillate indefinitely, as shown in Figure 5-46(b). Control systems exhibiting such response characteristics are not desirable. We shall see that the addition of derivative control will stabilize the system.
Proportional-Plus-Derivative Control of a System with Inertia Load. Let us modify the proportional controller to a proportional-plus-derivative controller whose transfer function is Kp(1+Tds). The torque developed by the controller is proportional to Kp(e+Tde’). Derivative control is essentially anticipatory, measures the instantaneous error velocity, and predicts the large overshoot ahead of time and produces an appropriate counteraction before too large an overshoot occurs.
Consider the system shown in Figure 5-47(a).
The closed-loop transfer function is given by:
The characteristic equation is:
Now it has two roots with negative real parts for positive values of J, Kp, and Td. Thus
derivative control introduces a damping effect. A typical response curve c(t) to a unit step
input is shown in Figure 5-47(b). Clearly, the response curve shows a marked improvement over the original response curve shown in Figure 5-46(b).
Proportional-Plus-Derivative Control of Second-Order Systems. A compromise between acceptable transient-response behavior and acceptable steady-state behavior may be achieved by use of proportional-plus-derivative control action. Consider the system shown in Figure 5-48.
The closed-loop transfer function is:
The steady-state error for a unit-ramp input is:
The characteristic equation is:
The effective damping coefficient of this system is thus B + Kd rather than B. Since the
damping ratio of this system is:
it is possible to make both the steady-state error Ess for a ramp input and the maximum
overshoot for a step input small by making B small, Kp large, and Kd large enough so that
is between 0.4 and 0.7.
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