Page - (000086) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

82 10 Nonlinearity 0 2 4 6 8 0.05 0.10 0.15 0 2 4 6 8 0.05 0.10 0.15 0 2 4 6 8 0.05 0.10 0.15 Fig. 10.2 State feedback increases the rate of return to equilibrium for the original nonlinear system (blue) and a linear approximation (gold). Theoptimal state feedbackwas calculated from the linearizedsysteminEq.10.2.Eachpanel shows theclosed-loopresponse toperturbationusing statefeedback.Theperturbationisanimpulsefunctionwithweightingofone,correspondingtothe rightpanelofFig.10.1.Theweightingsof thestatedeviations in thecost functionare, fromleft to right,ρ2 =1,10,100 Robust feedbackbasedon this transfer functionmaybeapplied to theoriginalnon- linear system. 10.3 PiecewiseLinearAnalysisandGainScheduling Linear approximations at a particular operatingpoint providenearly exact descrip- tions of nonlinear dynamics near the operating point.As the systemmoves further fromtheoperatingpoint, the linear approximationbecomes lessaccurate. In somecases, significant divergence from theoperatingpoint causes the quali- tative nature of the nonlinear dynamics to differ from the linear approximation. In other cases, such as inFig.10.1, the qualitative dynamics remain the same, but the quantitative responsesdiffer. The distance froman operating point atwhich the linear approximation breaks down depends on the particular nonlinear system. By considering the region over which the linearapproximationholds,onecanapproximateanonlinear systembya sequenceof linear approximations. Starting froman initial operatingpoint, thefirst linear approximationholdsnear that point.Then, as theapproximationbreaksdownaway fromthe initial operating point, onecanuseanewapproximationaroundasecondoperatingpoint. By repeatedlyupdating the approximation as needed for new regions, the series of linearapproximationsdescribes thenonlinear system.Each linearapproximation holds in its own region or “piece.” That approach leads to the piecewise linear approximationmethod (Rantzer andJohansson2000). Foreachpiece, linearmethodsspecifythedesignoffeedbackcontrol.Theoverall control becomesa sequenceof individual controllers basedon linear analysis,with each particular control regime appliedwhen the system is in the associated oper- ating region.Alternative control in different operating regions is often called gain scheduling.