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Unformatted text preview: Chapter 9 Reference Input Tracking: feedforward control 9.1 Introduction In previous lectures, we have discussed control designs based on actual or observer estimated state feedback, eigenvalue (pole) assignments, as well as controllers based on internal model principle (IMP) for disturbance rejection. The control laws have in most part been derived to drive the output or the state to 0, (hopefully) despite disturbances etc. Very often in applications, however, we desire the output to track a reference input, i.e. if r ( t ) is the reference input, we would like the output y ( t ) r ( t ) as t . In these notes, we first show how the control techniques can be modified to include reference input. The three methods are: 1. Internal model controller 2. Incorporating reference in state space control system 3. Feedforward control design, especially for non-minimum phase systems. The second half of the notes will discuss a special case of feedforward controller for vibration suppression - known as input shaping. 9.2 Internal model controller (IMC) Using IMP for reference tracking has already been discussed. Here, we will just illustrate it with an example. If the reference input can be generated using a set of autonomous linear differential equation, e.g. d n r dt n + a n 1 d n 1 r dt n 1 + ... + a 1 dr dt + a = 0 , then, taking Laplace transform, we can define a reference generating polynomial r ( s ) just like that the way we did for disturbances in class, so that r ( s ) R ( s ) = I c ( s ) = n summationdisplay i =0 g i r ( i ) (0) = terms corresponding to initial conditions . 149 150 c circlecopyrt Perry Y.Li Go(s) C(s) R(s) Di(s) Do(s) Y(s) Figure 9.1: Internal Model Control for reference tracking where r ( i ) denotes the i-th time derivative of r . The controller design can now proceed based on Fig. 9.1 as if the reference is a disturbance. So, according to the IMP, we need the controller C ( s ) to incorporate the reference generating polynomial r ( s ) into its denominator: C ( s ) = P ( s ) r ( s ) L ( s ) where P ( s ) and L ( s ) are chosen so that the closed loop system is stable. Example: Let the plant be G ( s ) = 5 / ( s + 1). Suppose that the reference to be tracked is r ( t ) = 2 sin (3 t ) Since R ( s ) = s/ ( s 2 + 9), the reference generating polynomial is r ( s ) = ( s 2 + 9). From previous notes, we know that in order to do arbitrary pole assignment, we need the order of the closed loop system to be at least 2 n 1 + q = 3 where n = 1 is the order of plant, q = 2 is the order of the r ( s ). Let the controller be C ( s ) = b 2 s 2 + b 1 s + b s 2 + 9 . and the 2 n 1 + q order closed loop characteristic polynomial is: A cl ( s ) = ( s + 1)( s 2 + 9) + ( b 2 s 2 + b 1 s + b ) 5 . (9.1) If we decide to place all the poles at 2, we need A cl ( s ) = ( s + 2) 2 = s 3 + 6 s 2 + 12 s + 8....
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- Fall '08