Trajectory Planning for Linear Control Systems with Generalized Splines

Tra jectoryPlanningforLinearControlSystemswithGeneralizedSplinesMagnusEgerstedtyDivision ofOptimization andSystems TheoryRoyal Institute ofTechnologySE -10044Sto ckholm, Swedenmagnuse@math.kth. seClyde F. MartinDepartment ofMathematicsexas Th UniversitLubb o ck,79409Texasmartin@math.ttu. eduKeywords:Tra jectoryplanning,optimal control,gen-eralized splinesAbstractWhen planning tra jectoriesfor linear control systems, ademand that arisesnaturally in, for instance,air traccontrol, noise contaminated data interp olation, and plan-ning for switched control systems, is that the curve inter-p olate throughgiven p ointsorintervals aten times.In this article, we address this by showing how standardoptimal control techniques,togetherwith mathematicalprogramming, solve the problem and provide a theoreti-cal framework for pro ducing a set of curves called gener-alized smo othing splines.1Intro ductionIn this article, we lo ok at the problem of nding the con-trol that drives a given linear control system b etween pre-de ned states, while interp olating through given p oints,orintervals.Ourresultingcurvewill minimize afunc-tional, and it turnsoutthat thecurvewill b eso calledgeneralizedspline,whosecharacteristicsdep endonthedynamics ofthecontrolsystem[5]. So,by establishinga framework for constructing optimal tra jectories, basedon controltheory,weprovideasystematicy tocon-struct all of the classical splines, and even other typ es ofcurves that are p otentially useful.This typ e of interp olation problem needs to b e solvedfor a numb er of di erent reasons.One is that when doingtra jectory planning for switched control systems, we needto b e able to generate curves that pass through prede nedstates at given times where the switching is supp osed totake place.In other cases, such as air trac control, thesame problem needs to b e solved, since in these cases, weneedtob eablesp ecifythep osition thatsystemwill b e in at a sequence of times.However, in most situ-ations, it is not really crucial that we pass through theseThis work was sp onsored in part by the Swedish Foundationfor Strategic Research through its Centre for Autonomous Systemsat KTH.yThis work was supp orted in part by NASA Grants NAG 2-902and NAG 2-899 and by NSF Grant DMS 9628558.p oints exactly, but rather that we go reasonably close tothem,whileminimizingthecostfunctional.Thisisadesiredprop ertyfor twoapparentreasons.Firstofall,a small deviation from thepresp eci edp oint can resultin a signi cant decreasein the cost, and secondly, whenthe data thatwe work with isnoise contaminated, it isnot even desirable to interp olate through these p oints ex-actly.Inspiredby [4], wecan incorp oratetheseasp ectsinto our prop osed metho d, and thus pro duce somethingthat normally is called smo othing splines.In Section2, westart by discussingsome facts ab outlinear control systems,and showhow theycanb eusedto arrive at an optimal control problem, suitable for pro-ducing thecurvesdiscussedab ove.We thenpro ceedtoactually solving the problem of pro ducing the generalizedsmo othing splines, using standardoptimal control tech-niques together with mathematical programming, reduc-ingtheproblemintoaconvex,easy-to-solvequadraticprogramming problem.We then, in Section 4, concludebydiscussingsomeprop ertiesofthegeneratedcurves,such as di erentiability and numerical eciency.2OptimalControlforLinearSystemsIn this article, we study linear single input, single outputcontrol systems on the form_x(t)=Ax) +buy(t)=cTx;(1)wherex2Rn,u; y, and (A; b; c) are constant matri-ces of compatible dimensions.Now, the problem that wewant to solve is the following - Given intervals and times([ai; b]; t). Howdoeconstructcurvthatpassesthrough each of these intervals [ai; b] at the sp eci ed timeti, 01< t2< : : : < tmT, while minimizingminuJ= min12ZT0u2(t)dt:(2)It is a well known fact [1], [5] that the problem of inter-p olating the output of the system through given, distinctp oints, while minimizing (2), has a solution if the system(1) is b oth controllable and observable.