105 105 8 Proportional and derivative control 8 1 8 2 8 3 8 4 8 5 Why derivative control Mixing the two methods of control Optimizing the combination Handling inertia Summary 95 96 98 99 103 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 feedback system to make the actual position be the desired position In other 105 2009 09 29 13 11 30 UTC rev b19331f50bbd 105 106 106 8 2 Mixing the two methods of control 96 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 motor controller C R M R R 1 R S R R sensor 1 Since the motor has the functional R 1 R let s put a discrete time derivative 1 R into the controller to remove the 1 R in the motor s denominator 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 integrating systems 8 2 Mixing the two methods of control Proportional control uses as the controller Derivative control uses 1 R as the controller The linear mixture of the two methods is C R 1 R controller C R 1 R 1 motor M R R 1 R S R R 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 106 2009 09 29 13 11 30 UTC rev b19331f50bbd 106 107 107 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 R 1 R R R 1 R 1 R 1 1 R Clear the fractions to get F R whatever 1 R 1 R R2 The whatever indicates that we don t care what is in the numerator It can contribute only zeros whereas what we worry about are the poles The poles arise from the denominator so to avoid doing irrelevant algebra and to avoid cluttering up the expressions we do not even compute the numerator as long as we know that the fractions are cleared The denominator is 1 R R2 R3 This cubic polynomial produces three poles Before studying their locations a daunting task with a cubic do an extreme cases check Take the limit 0 to turn off derivative control The system should turn into the pure proportional control system from the previous chapter It does The denominator becomes 1 R R2 which is the denominator from Section 7 2 As the proportional gain increases from 0 to the poles which begin at 0 and 1 move inward collide at 1 2 when 1 4 then split upward and downward to infinity Here is the root locus of this limiting case of 0 with only proportional control 107 2009 09 29 13 11 30 UTC rev b19331f50bbd 107 108 108 8 3 Optimizing the combination 98 8 3 Optimizing the combination We would like to make the whole system as stable as possible in the sense that the least stable pole is as close to the origin as possible The root locus for the general combination has three branches one for each pole whereas the limiting case of proportional control has only two poles and two branches Worse the root locus for the general combination is generated by two parameters the gains of the proportional and the derivative portions whereas in the limiting case it is generated by only one parameter The general analysis seems difficult Surprisingly the extra parameter rescues us from painful mathematics To see how look at the coefficients in the cubic 1 R R2 R3 The factored form is 1 p1 R 1 p2 R 1 p3 R 1 p1 p2 p3 R p1 p2 p1 p3 p2 p3 R2 p1 p2 p3 R3 z z z 1 So the first constraint is p1 p2 p3 1 showing that the center of gravity of the poles is 1 3 That condition is independent of and So the most stable system has a triple pole at 1 3 if that arrangement is possible To see why that arrangement is the most stable imagine starting from it Now move one pole inward along the real axis to increase its stability To preserve the invariant p1 p2 p3 1 at least one of the other poles must move outward and become less stable Thus it is best not to move any pole away from the triple cluster so it is the most stable arrangement Exercise 42 Where does the preceding argument require that the center of gravity be independent of and If the triple pole arrangement is impossible then the preceding argument which assumed its existence does not work And we need lots of work to find the best arrangement of poles 108 2009 09 29 13 11 30 UTC rev b19331f50bbd 108 109 109 8 Proportional and derivative control 99 Fortunately the triple pole is possible thanks to the extra parameter Having freedom to choose and we can set the R2 coefficient independently from the R3 coefficient which is So using and as separate dials we can make any cubic whose poles are centered on 1 3 Let s set those dials by propagating constraints With p1 p2 p3 1 3 the product p1 p2 p3 1 27 So the gain of the derivative controller is 1 27 The last constraint is that p1 p2 p1 p3 p2 p3 3 9 1 3 So 1 3 With 1 27 this equation requires that the gain of the proportional controller be 8 27 The best controller is then 8 1 1 R C R 1 R 1 27 27 3 9 Exercise 43 What is the pole zero plot of the forward path C R M R This controller has …
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