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Figure 2.2 illustrates the effects of adding integral. It follows from D2.1E that the strength of integral action increases with decreasing integral time Ti. The figure shows that the steady state error disappears when integral action is used. Compare with the discussion of the “magic of integral action” in Section 2.2. The tendency for oscillation also increases with decreasing Ti. The properties of derivative action are illustrated in Figure 2.3.
Figure 2.3 illustrates the effects of adding derivative action. The parameters K and Ti are chosen so that the closed loop system is oscillatory. Damping increases with increasing derivative time, but decreases again when derivative time becomes too large. Recall that derivative action can be interpreted as providing prediction by linear extrapolation over the time Td. Using this interpretation it is easy to understand that derivative action does not help if the prediction time Td is too large. In Figure 2.3 the period of oscillation is about 6 s for the system without derivative Chapter 6. PID Control
Derivative actions cease to be effective when Td is larger than a 1 s (one sixth of the period). Also notice that the period of oscillation increases when derivative time is increased.
A Perspective
There is much more to PID than is revealed by （2.1）. A faithful implementation of the equation will actually not result in a good controller. To obtain a good PID controller it is also necessary to consider。
Figure 2.3
•Noise filtering and high frequency roll off
•Set point weighting and 2 DOF
•Windup
•Tuning
•Computer implementation
In the case of the PID controller these issues emerged organically as the technology developed but they are actually important in the implementation of all controllers. Many of these questions are closely related to fundamental properties of feedback, some of them have been discussed earlier in the book.
III. Filtering and Set Point Weighting
Differentiation is always sensitive to noise. This is clearly seen from the transfer function G(s) =s of a differentiator which goes to infinity for large s. The following example is also illuminating.
where the noise is sinusoidal noise with frequency w. The derivative of the signal is
The signal to noise ratio for the original signal is 1/an but the signal to noise ratio of the differentiated signal is w/an. This ratio can be arbitrarily high if w is large.
In a practical controller with derivative action it is there for necessary to limit the high frequency gain of the derivative term. This can be done by implementing the derivative term as
3.2
instead of D=sTdY. The approximation given by (3.2) can be interpreted as the ideal derivative sTd filtered by a first-order system with the time constant Td/N. The approximation acts as a derivative for low-frequency signal components. The gain, however, is limited to KN. This means that high-frequency measurement noise is amplified at most by a factor KN. Typical values of N are 8 to 20.
Further limitation of the high-frequency gain
The transfer function from measurement y to controller output u of a PID controller with the approximate derivative is
This controller has constant gain
at high frequencies. It follows from the discussion on robustness against process variations in Section that it is highly desirable to roll off the controller gain at high frequencies. This can be achieved by additional
low pass filtering of the control signal by
where Tf is the filter time constant and n is the order of the filter. The choice of Tf is a compromise between filtering capacity and performance. The value of T f can be coupled to the controller time constants in the same way as for the derivative filter above. If the derivative time is used, T f= Td/N is a suitable choice. If the controller is only PI, T f =Ti/N may be suitable.
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