IMT Institute for Advanced Studies, Lucca Lucca, Italy Embedded Model Predictive Control: Finite Precision Arithmetic and Aerospace Applications PhDPrograminComputerScienceandEngineering XXVII Cycle by Alberto Guiggiani 2015 ThedissertationofAlbertoGuiggianihasbeenapproved. ProgramCoordinator:Prof.AlbertoBemporad, IMTInstituteforAdvancedStudies,Lucca(IT). Advisor:Prof.AlbertoBemporad, IMTInstituteforAdvancedStudies,Lucca(IT). Co-advisor:Prof.PanagiotisPatrinos, IMTInstituteforAdvancedStudies,Lucca(IT). Co-advisor:Prof.IlyaKolmanovsky, UniversityofMichigan,AnnArbor,MI(US). ThedissertationofAlbertoGuiggiani hasbeenreviewedby: Prof.IonNecoara, UniversityPolitehnicaofBucharest(RO). Dr.SamirBennani, EuropeanSpaceAgency(ESA). IMT Institute for Advanced Studies, Lucca 2015 allamiafamiglia Table of Contents Page ListofFigures xiii ListofTables xvii Acknowledgements xix VitaandPublications xxi Abstract xxv NotationandAbbreviations xxvii 1 Introduction 1 1.1 ModelPredictiveControl . . . . . . . . . . . . . . . 2 1.1.1 LinearTrackingFormulation . . . . . . . . . 5 1.1.2 Extensions . . . . . . . . . . . . . . . . . . . . 8 1.2 QuadraticProgrammingforModelPredictiveControl 12 1.2.1 LagrangianDuality . . . . . . . . . . . . . . . 14 1.2.2 Methods . . . . . . . . . . . . . . . . . . . . . 15 vii 1.2.3 FromModelPredictiveControltoQuadratic Programming . . . . . . . . . . . . . . . . . . 21 1.3 Fixed-PointComputations . . . . . . . . . . . . . . . 24 1.3.1 OverflowErrors . . . . . . . . . . . . . . . . . 27 1.3.2 Round-offErrors . . . . . . . . . . . . . . . . 28 1.3.3 ErrorsduetoMathematicalOperations . . . 29 1.4 EmbeddedModelPredictiveControl . . . . . . . . . 30 1.5 Offset-FreeModelPredictiveControl . . . . . . . . . 33 1.5.1 Methods for Offset-Free Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . 33 1.5.2 Simulation . . . . . . . . . . . . . . . . . . . . 41 1.6 ModelPredictiveControlforAerospaceApplications 46 1.7 MotivationandContribution . . . . . . . . . . . . . 48 2 Gradient Projection Methods in Finite Precision Arith- metic 57 2.1 InexactGradientProjection . . . . . . . . . . . . . . 58 2.2 InexactDualGradientProjection . . . . . . . . . . . 64 2.2.1 ModifiedPrimal-DualPair . . . . . . . . . . 66 2.2.2 InexactOracle . . . . . . . . . . . . . . . . . . 68 2.2.3 PrimalConvergenceRates . . . . . . . . . . . 70 2.2.4 OptimalChoiceof α forFixedOracleErrors (cid:15) , (cid:15) . . . . . . . . . . . . . . . . . . . . . . . 76 z ξ 2.2.5 BoundoftheNumberofIterations . . . . . . 76 2.2.6 MaximumAdmissibleOracleErrors (cid:15) , (cid:15) . 78 z ξ 2.3 Fixed-PointDualGradientProjectionforQuadratic Programs . . . . . . . . . . . . . . . . . . . . . . . . . 81 viii 2.3.1 Fixed-pointImplementation . . . . . . . . . . 82 2.3.2 GuidelinesfortheNumberofFractionalBits 83 2.3.3 GuidelinesfortheNumberofIntegerBits . . 85 2.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . 87 2.4.1 SampleEvolutions . . . . . . . . . . . . . . . 87 2.4.2 InfeasibilityandSuboptimalityBounds . . . 90 2.4.3 TargetInfeasibility . . . . . . . . . . . . . . . 92 2.4.4 BoundsonIterationCount . . . . . . . . . . 95 2.4.5 MassesSeriallyConnectedExample . . . . . 97 3 ProximalNewtonMethodsinFinitePrecisionArithmetic101 3.1 ProblemSetup . . . . . . . . . . . . . . . . . . . . . . 102 3.2 ProximalNewtonAlgorithm . . . . . . . . . . . . . 104 3.3 Fixed-PointProximalNewtonAlgorithm . . . . . . 107 3.3.1 Round-offErrorAnalysis . . . . . . . . . . . 107 3.3.2 AvoidingOverflowErrors . . . . . . . . . . . 108 3.4 OptimizationoftheAlgorithm . . . . . . . . . . . . 110 3.4.1 Preconditioning . . . . . . . . . . . . . . . . . 110 3.4.2 Division-freeComputations . . . . . . . . . . 113 3.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . 115 3.5.1 ComputationalComplexity . . . . . . . . . . 115 3.5.2 SolutionAccuracy . . . . . . . . . . . . . . . 119 3.5.3 ControlofaF16AircraftExample . . . . . . 121 4 ExperimentalTests 125 4.1 EmbeddedOptimizationonARMCortex . . . . . . 126 4.1.1 TheARMCortex-M3ProcessingUnit . . . . 126 ix 4.1.2 GradientProjectionMethodsonARMCortex 127 4.1.3 ProximalNewtonMethodsonARMCortex . 130 4.2 EmbeddedOptimizationonFPGA . . . . . . . . . . 134 4.2.1 IntroductiontoFPGADevices . . . . . . . . 134 4.2.2 Fixed-PointDualGradientProjectiononFPGA138 5 AerospaceApplications 147 5.1 SpacecraftNonlinearModel . . . . . . . . . . . . . . 148 5.2 ControlObjective . . . . . . . . . . . . . . . . . . . . 150 5.3 ControlModel . . . . . . . . . . . . . . . . . . . . . . 152 5.4 MPCFormulation . . . . . . . . . . . . . . . . . . . . 154 5.5 ComputationalComplexity . . . . . . . . . . . . . . 161 5.6 Simulations . . . . . . . . . . . . . . . . . . . . . . . 163 5.6.1 SinusoidalReferencesTracking . . . . . . . . 163 5.6.2 Rest-to-RestOrientationManeuver . . . . . . 166 5.6.3 Fixed-PointAccuracy . . . . . . . . . . . . . . 168 5.7 ReactionWheelsDesaturationbyGravityGradients 170 5.7.1 Background . . . . . . . . . . . . . . . . . . . 170 5.7.2 NonlinearModel . . . . . . . . . . . . . . . . 170 5.7.3 ControlModel . . . . . . . . . . . . . . . . . 172 5.7.4 SimulationResults . . . . . . . . . . . . . . . 173 5.7.5 ComparisonwithLQR . . . . . . . . . . . . . 176 5.8 ReactionWheelsDesaturationbyMagneticMoments 178 5.8.1 Background . . . . . . . . . . . . . . . . . . . 178 5.8.2 NonlinearModel . . . . . . . . . . . . . . . . 179 5.8.3 ControlModel . . . . . . . . . . . . . . . . . 180 5.8.4 SimulationResults . . . . . . . . . . . . . . . 182 x