Page - (000079) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

9.2 TrackingExample 73 withα=a+b=10.1andβ =ab=1.Thestate-spacemodel isgiven inEq.9.2, expressed inmatrix form in Eq.9.4. The state-spacemodel describes the process outputover time, y(t),whichweabbreviate as y. Here, I describe a state-space design of tracking control for this process. For this example, Iuse the tracking referencesignal inEq.8.3, ignoringhigh-frequency noise (κ2 =0). The reference signal is the sum of low-frequency (ω0 =0.1) and mid-frequency(ω1 =1)sinewaves.Thetransfer functionfor thereferencesignal is R(s)= ω0 s2+ω20 + ω1 s2+ω21 . In state-space form, the reference signal,r(t), is Ar = ⎛ ⎜⎜⎝ 0 1 0 0 0 0 1 0 0 0 0 1 −ω20ω21 0−ω20−ω21 0 ⎞ ⎟⎟⎠ Br = ( 0 0 0 1 )T Cr = ( ω20ω1+ω0ω21 0ω0+ω1 0 ) . Wecan transforma tracking problem into a regulator problemand then use the methods from the previous chapter (Anderson andMoore 1989). In the regulator problem,weminimizedacombinationofthesquaredinputsandstates.Foratracking problem,weusetheerror,e= y−r, insteadofthestatevalues,andexpressthecost as J = ∫ T 0 ( u′Ru+e2)dt. (9.5) Wecancombine the state-spaceexpressions for y andr intoa single state-space modelwith output e. That combinedmodel allowsus to apply the regulator theory to solve the trackingproblemwith state feedback. Thecombinedmodel for the trackingproblemis At = ( A 0 0 Ar ) Bt = ( B 0 0 Br ) Ct = ( C−Cr ) , whichhasoutputdeterminedbyCtase= y−r (AndersonandMoore1989).Inthis form,wecanapplytheregulator theorytofindtheoptimalstatefeedbackmatrix,K,