# chap7 - 8 Proportional and derivative control 8.1 Why...

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Unformatted text preview: 8 Proportional and derivative control 8.1 Why derivative control 95 8.2 Mixing the two methods of control 96 8.3 Optimizing the combination 98 8.4 Handling inertia 99 8.5 Summary 103 The goals of this chapter are: • to introduce derivative control; and • to study the combination of proportional and derivative control for taming systems with integration or inertia. The controllers in the previous chapter had the same form: The control signal was a multiple of the error signal. This method cannot easily control an integrating system, such as the motor positioning a rod even without inertia. If the system has inertia, the limits of proportional control become even more apparent. This chapter introduces an alternative: derivative control. 8.1 Why derivative control An alternative to proportional control is derivative control. It is motivated by the integration inherent in the motor system. We would like the feed- back system to make the actual position be the desired position. In other 96 8.2 Mixing the two methods of control words, it should copy the input signal to the output signal. We would even settle for a bit of delay on top of the copying. This arrangement is shown in the following block diagram: + C ( R ) =? M ( R ) = R 1 − R S ( R ) = R − 1 controller motor sensor Since the motor has the functional R / ( 1 − R ) , let’s put a discrete-time de- rivative 1 − R into the controller to remove the 1 − R in the motor’s de- nominator. With this derivative control , the forward-path cascade of the controller and motor contains only powers of R . Although this method is too fragile to use alone, it is a useful idea. Pure derivative control is fragile because it uses pole–zero cancellation. This cancellation is mathematically plausible but, for the reasons explained in lecture, it produces unwanted offsets in the output. However, derivative control is still useful. As we will find, in combination with proportional control, it helps to stabilize in- tegrating systems. 8.2 Mixing the two methods of control Proportional control uses β as the controller. Derivative control uses γ ( 1 − R ) as the controller. The linear mixture of the two methods is C ( R ) = β + γ ( 1 − R ) . + C ( R ) = β + γ ( 1 − R ) M ( R ) = R 1 − R S ( R ) = R − 1 controller motor sensor Let F ( R ) be the functional for the entire feedback system. Its numerator is the forward path C ( R ) M ( R ) . Its denominator is 1 − L ( R ) , where L ( R ) is the loop functional or loop gain that results from going once around the feedback loop. Here the loop functional is 8 Proportional and derivative control 97 L ( R ) = − C ( R ) M ( R ) S ( R ) . Don’t forget the contribution of the inverting (gain = − 1 ) element! So the overall system functional is F ( R ) = ( β + γ ( 1 − R )) R 1 − R 1 + ( β + γ ( 1 − R )) R 1 − R R ....
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chap7 - 8 Proportional and derivative control 8.1 Why...

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