Page - (000077) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

9.1 RegulationExample 71 x = Ax + Bu yC xd K u u* . Fig.9.1 Statefeedbackmodelofregulation.Theprocessandoutputdescribethestateequationsin Eq.2.6.Thecontrol input signal,u∗ =Kx, is obtainedbyminimizing the cost function inEq.9.3 toderive theoptimal stategains.Adisturbance,d, is added to the input signal equivalent to theerror,e= y−r,becausethereferenceinput isr =0.Acostbased onu2 ande2matches theearlier cost function inEq.8.1. Inthiscase,IweightthecostsforeachstateequallybylettingQ=ρ2I2, inwhich In is theidentitymatrixofdimensionn,andρ is thecostweightingforstatesrelative to inputs.With thosedefinitions, thecostbecomes J = ∫ T 0 [ u2+ρ2(x21 +x22)]dt, inwhichx21 +x22measuresthedistanceofthestatevectorfromthetargetequilibrium ofzero. We obtain the gain matrix for state feedback models, K, by solving a matrix Riccati equation. Introductory texts on control theory derive theRiccati equation. For our purposes,we can simply use a software package, such asMathematica, to obtain the solution forparticular problems.See the supplemental software code for anexample. Figure9.2 shows the response of the state feedback system in Fig.9.1with the Riccati solution for the feedback gain values,K.Within each panel, the different curvesshowdifferentvaluesofρ, theratioofthestatecostsforx relativetotheinput costs foru. In thefigure, the blue curves showρ=1/4,whichpenalizes the input costs four timesmore than the state costs. In that case, thecontrol inputs tend tobe costlyandweaker, allowing the statevalues tobe larger. At the other extreme, the green curves showρ=4. That value penalizes states moreheavilyandallowsgreatercontrol inputvalues.The larger inputcontrolsdrive the statesback towardzeromuchmorequickly.Thefigure captionprovidesdetails about eachpanel. In thisexample, theunderlyingequationsfor thedynamicsdonotvarywithtime. Time-invariant dynamics correspond to constant values in the statematrices,A,B, andC.Atime-invariantsystemtypically leads toconstantvalues in theoptimalgain matrix,K, obtainedbysolving theRiccati equation. TheRiccatisolutionalsoworkswhenthosecoefficientmatriceshavetime-varying values, leading to time-varying control inputs in the optimal gainmatrix,K. The general approach can also be extended to nonlinear systems.However, theRiccati equation isnot sufficient to solvenonlinearproblems.