Page - (000080) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

74 9 StateFeedback -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 (a) (b) (c) (d) Fig. 9.3 Tracking a reference input signal with state feedback. The blue curve shows the input signal,r(t),whichis thesumoftwosinewaveswithfrequenciesω0 =0.1andω1 =1.Thesystem responds to the inputbyproducinganoutput, y(t).Theoutput isdeterminedby theprocess,P(s), and theoptimal state feedback,K, aspresented in the text.Thegoldcurves showthesystemerror, which is e= y−r, the difference between the output and the reference signal. a Squared input valuesareweightedbyR=wIn,withw=0.1andnas thenumberof inputs to theprocess. In this case,wefixtheinputtotheembeddedreferencesignalinthestate-spacemodeltozeroandhaveone input into theprocess givenbyB inEq.9.4.Theerror curve shows that this systemclosely tracks the low-frequencyreferencesinewavebutdoesnot trackthehigh-frequencyreferencecomponent. bThis case allows feedback inputs into both states of the process, augmenting x˙1 inEq.9.2with a separate input and letting B= I2. Other aspects as in the prior panel. c As in panel a, with w=0.01. Theweaker cost for inputs allows stronger feedback inputs and closer tracking of the high-frequency component of the reference signal, thus shrinking the tracking error in the gold curve.dNearly perfect trackingwithw=0.01 and inputs directly into both process states. See supplementalMathematicacode fordetails about assumptionsandcalculations thatminimizes thecosts, J, inEq.9.5.Figure9.3presentsanexampleandmentions some technical issues in thecaption. Theexample illustrates twokeypoints.First, as therelativecostweightingof the inputs declines, the systemapplies stronger feedback inputs and improves tracking performance. Second,thestateequationsfortheintrinsicprocess,P(s), inEq.9.4provideinput only into the second state of the process, as can be seen in the equation for x˙2 in Eq.9.2.Whenweallowasecond input into the intrinsicprocess,P(s), byallowing feedbackdirectly intoboth x˙1 and x˙2,weobtainmuchbetter trackingperformance, as shown inFig.9.3.