Page - (000085) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

10.2 Regulation 81 2 4 6 8 0.005 0.010 0.015 2 4 6 8 0.01 0.02 0.03 0.04 0.05 0.06 2 4 6 8 0.05 0.10 0.15 Fig. 10.1 Response to an impulse perturbation by a nonlinear system and a linearized approxi- mation, shown as the deviation fromequilibrium.The nonlinear response in blue arises from the system inEq.10.1.The linearized response ingoldarises from the system inEq.10.2.Thepanels fromleft to right showincreasingmagnitudesof theDiracdelta impulseperturbationat timezero, withtheimpulseweightedby0.1,0.1 √ 10,1,respectively.Largerimpulsescausegreaterdeviations from the equilibriumpoint. The greater the deviation from the equilibrium, the less accurate the linearizedapproximationof thedynamics This section applies the linear state feedback regulation approach. I used that approach inapreviouschapter, inwhich thecost function inEq.9.3, repeatedhere, J = ∫ T 0 ( u′Ru+x′Qx)dt, balancesthetradeoffbetweenthecostofcontrolinputsandthecostofstatedeviation fromequilibrium.Themodel iswrittensothat theequilibriumstatesarex∗ =0.We obtain the optimal state feedback by applying themethods described in the prior chapter (seealso the supplementalMathematicacode). Consider the linear approximation in Eq.10.2. That system has one input, for whichweletR=1andscale thestatecostsaccordingly.Foreachstate,assumethat thecost isρ2, so that the integrandof thecostbecomesu2+ρ2(x21 +x22 ) . We can calculate the feedback input that minimizes the cost for the linearized approximation.Using the optimal feedback,we can formaclosed-loop system for both the linearizedsystemand theoriginalnonlinear system. Figure10.2 shows the response to an impulse perturbation for the closed-loop systems. In eachpanel, thenonlinear (blue) and linear (gold) responses are similar, showing that thedesign for the linear systemworkswell for thenonlinear system. The panels from left to right show a decreasing cost weighting on the inputs relative to the states.As the relative input costs become less heavilyweighted, the optimal feedback uses stronger inputs to regulate the response, driving the system back toequilibriummorequickly. Minimizing a cost function by state feedbackmay lead to systems that become unstable with respect to variations in themodel dynamics. Previous chapters dis- cussed alternative robust techniques, including integral control and combinations ofH2 andH∞ methods.Wemayapply those alternativemethods to the linearized approximationinEq.10.2.Thelinearizedsystemcorrespondstothetransferfunction P(s)= n/4 s2+(1+γ)s+γ .