linear quadratic dynamic programming

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1、Chapter5LinearQuadraticDynamicProgramming5.1.IntroductionThischapterdescribestheclassofdynamicprogrammingproblemsinwhichthereturnfunctionisquadraticandthetransitionfunctionislinear.Thisspecificationleadstothewidelyusedoptimallinearregulatorproblem,forwhichtheBellm

2、anequationcanbesolvedquicklyusinglinearalgebra.Weconsiderthespecialcaseinwhichthereturnfunctionandtransitionfunctionarebothtimeinvariant,thoughthemathematicsisalmostidenticalwhentheyarepermittedtobedeterministicfunctionsoftime.Linearquadraticdynamicprogramminghas

3、twousesforus.Afirstistostudyoptimumandequilibriumproblemsarisingforlinearrationalexpectationsmodels.Herethedynamicdecisionproblemsnaturallytaketheformofanoptimallinearregulator.Asecondistousealinearquadraticdynamicprogramtoapproximateonethatisnotlinearquadratic.La

4、terinthechapter,wetellhowtheKalmanfilteringproblemfromchap-ter2relatestothelinear-quadraticdynamicprogrammingproblem.Suitablyreinterpreted,formulasthatsolvetheoptimallinearregulatoraretheKalmanfilter.–127–128LinearQuadraticDynamicProgramming5.2.Theoptimallinearregu

5、latorproblemTheundiscountedoptimallinearregulatorproblemistomaximizeoverchoiceof{u}∞thecriteriontt=0∞−{xtRxt+utQut},(5.2.1)t=0subjecttoxt+1=Axt+But,x0given.Herextisan(n×1)vectorofstatevariables,utisa(k×1)vectorofcontrols,Risapositivesemidefinitesymmetricmatrix,

6、Qisapositivedefinitesymmetricmatrix,Aisan(n×n)matrix,andBisan(n×k)matrix.Weguessthatthevaluefunctionisquadratic,V(x)=−xPx,wherePisapositivesemidefinitesymmetricmatrix.Usingthetransitionlawtoeliminatenextperiod’sstate,theBellmanequa-tionbecomes−xPx=max{−xRx−uQu−

7、(Ax+Bu)P(Ax+Bu)}.(5.2.2)uThefirst-ordernecessaryconditionforthemaximumproblemontherightsideofequation(5.2.2)is1(Q+BPB)u=−BPAx,(5.2.3)whichimpliesthefeedbackruleforu:−1u=−(Q+BPB)BPAx(5.2.4)oru=−Fx,where−1F=(Q+BPB)BPA.(5.2.5)Substitutingtheoptimizer(5.2.4)int

8、otherightsideofequation(5.2.2)andrearranginggivesPA−APB(Q+BPB)−1BPA.(5.2.6)P=R+AEquation(5.2.6)iscalledthealgebraicmatrixRiccatiequation.Itexpressesthematr