ajout TP2 424

424-Systeme_Non_Lineaires/TP2/Grue_L.mdl.r2009b

@@ -0,0 +1,882 @@
+Model {
+  Name			  "Grue_L"
+  Version		  7.4
+  MdlSubVersion		  0
+  GraphicalInterface {
+    NumRootInports	    0
+    NumRootOutports	    0
+    ParameterArgumentNames  ""
+    ComputedModelVersion    "1.70"
+    NumModelReferences	    0
+    NumTestPointedSignals   0
+  }
+  SavedCharacterEncoding  "ISO-8859-1"
+  SaveDefaultBlockParams  on
+  ScopeRefreshTime	  0.035000
+  OverrideScopeRefreshTime on
+  DisableAllScopes	  off
+  DataTypeOverride	  "UseLocalSettings"
+  MinMaxOverflowLogging	  "UseLocalSettings"
+  MinMaxOverflowArchiveMode "Overwrite"
+  MaxMDLFileLineLength	  120
+  Created		  "Fri Mar 23 18:37:01 2012"
+  Creator		  "abbasturki"
+  UpdateHistory		  "UpdateHistoryNever"
+  ModifiedByFormat	  "%<Auto>"
+  LastModifiedBy	  "abbasturki"
+  ModifiedDateFormat	  "%<Auto>"
+  LastModifiedDate	  "Thu Mar 24 18:48:48 2016"
+  RTWModifiedTimeStamp	  380746096
+  ModelVersionFormat	  "1.%<AutoIncrement:70>"
+  ConfigurationManager	  "None"
+  SampleTimeColors	  off
+  SampleTimeAnnotations	  off
+  LibraryLinkDisplay	  "none"
+  WideLines		  off
+  ShowLineDimensions	  off
+  ShowPortDataTypes	  off
+  ShowLoopsOnError	  on
+  IgnoreBidirectionalLines off
+  ShowStorageClass	  off
+  ShowTestPointIcons	  on
+  ShowSignalResolutionIcons on
+  ShowViewerIcons	  on
+  SortedOrder		  off
+  ExecutionContextIcon	  off
+  ShowLinearizationAnnotations on
+  BlockNameDataTip	  off
+  BlockParametersDataTip  off
+  BlockDescriptionStringDataTip	off
+  ToolBar		  on
+  StatusBar		  on
+  BrowserShowLibraryLinks off
+  BrowserLookUnderMasks	  off
+  SimulationMode	  "normal"
+  LinearizationMsg	  "none"
+  Profile		  off
+  ParamWorkspaceSource	  "MATLABWorkspace"
+  AccelSystemTargetFile	  "accel.tlc"
+  AccelTemplateMakefile	  "accel_default_tmf"
+  AccelMakeCommand	  "make_rtw"
+  TryForcingSFcnDF	  off
+  RecordCoverage	  off
+  CovPath		  "/"
+  CovSaveName		  "covdata"
+  CovMetricSettings	  "dw"
+  CovNameIncrementing	  off
+  CovHtmlReporting	  on
+  CovForceBlockReductionOff on
+  covSaveCumulativeToWorkspaceVar on
+  CovSaveSingleToWorkspaceVar on
+  CovCumulativeVarName	  "covCumulativeData"
+  CovCumulativeReport	  off
+  CovReportOnPause	  on
+  CovModelRefEnable	  "Off"
+  CovExternalEMLEnable	  off
+  ExtModeBatchMode	  off
+  ExtModeEnableFloating	  on
+  ExtModeTrigType	  "manual"
+  ExtModeTrigMode	  "normal"
+  ExtModeTrigPort	  "1"
+  ExtModeTrigElement	  "any"
+  ExtModeTrigDuration	  1000
+  ExtModeTrigDurationFloating "auto"
+  ExtModeTrigHoldOff	  0
+  ExtModeTrigDelay	  0
+  ExtModeTrigDirection	  "rising"
+  ExtModeTrigLevel	  0
+  ExtModeArchiveMode	  "off"
+  ExtModeAutoIncOneShot	  off
+  ExtModeIncDirWhenArm	  off
+  ExtModeAddSuffixToVar	  off
+  ExtModeWriteAllDataToWs off
+  ExtModeArmWhenConnect	  on
+  ExtModeSkipDownloadWhenConnect off
+  ExtModeLogAll		  on
+  ExtModeAutoUpdateStatusClock on
+  BufferReuse		  on
+  ShowModelReferenceBlockVersion off
+  ShowModelReferenceBlockIO off
+  Array {
+    Type		    "Handle"
+    Dimension		    1
+    Simulink.ConfigSet {
+      $ObjectID		      1
+      Version		      "1.6.0"
+      Array {
+	Type			"Handle"
+	Dimension		8
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+	  Version		  "1.6.0"
+	  StartTime		  "0.0"
+	  StopTime		  "20"
+	  AbsTol		  "auto"
+	  FixedStep		  "auto"
+	  InitialStep		  "auto"
+	  MaxNumMinSteps	  "-1"
+	  MaxOrder		  5
+	  ZcThreshold		  "auto"
+	  ConsecutiveZCsStepRelTol "10*128*eps"
+	  MaxConsecutiveZCs	  "1000"
+	  ExtrapolationOrder	  4
+	  NumberNewtonIterations  1
+	  MaxStep		  "auto"
+	  MinStep		  "auto"
+	  MaxConsecutiveMinStep	  "1"
+	  RelTol		  "1e-3"
+	  SolverMode		  "Auto"
+	  Solver		  "ode45"
+	  SolverName		  "ode45"
+	  ShapePreserveControl	  "DisableAll"
+	  ZeroCrossControl	  "UseLocalSettings"
+	  ZeroCrossAlgorithm	  "Nonadaptive"
+	  AlgebraicLoopSolver	  "TrustRegion"
+	  SolverResetMethod	  "Fast"
+	  PositivePriorityOrder	  off
+	  AutoInsertRateTranBlk	  off
+	  SampleTimeConstraint	  "Unconstrained"
+	  InsertRTBMode		  "Whenever possible"
+	}
+	Simulink.DataIOCC {
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+	  Decimation		  "1"
+	  ExternalInput		  "[t, u]"
+	  FinalStateName	  "xFinal"
+	  InitialState		  "xInitial"
+	  LimitDataPoints	  on
+	  MaxDataPoints		  "1000"
+	  LoadExternalInput	  off
+	  LoadInitialState	  off
+	  SaveFinalState	  off
+	  SaveCompleteFinalSimState off
+	  SaveFormat		  "Array"
+	  SaveOutput		  on
+	  SaveState		  off
+	  SignalLogging		  on
+	  InspectSignalLogs	  off
+	  SaveTime		  on
+	  ReturnWorkspaceOutputs  off
+	  StateSaveName		  "xout"
+	  TimeSaveName		  "tout"
+	  OutputSaveName	  "yout"
+	  SignalLoggingName	  "logsout"
+	  OutputOption		  "RefineOutputTimes"
+	  OutputTimes		  "[]"
+	  ReturnWorkspaceOutputsName "out"
+	  Refine		  "1"
+	}
+	Simulink.OptimizationCC {
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+	  Version		  "1.6.0"
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+	    Type		    "Cell"
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+	    Cell		    "BooleansAsBitfields"
+	    Cell		    "PassReuseOutputArgsAs"
+	    Cell		    "PassReuseOutputArgsThreshold"
+	    Cell		    "ZeroExternalMemoryAtStartup"
+	    Cell		    "ZeroInternalMemoryAtStartup"
+	    Cell		    "OptimizeModelRefInitCode"
+	    Cell		    "NoFixptDivByZeroProtection"
+	    PropName		    "DisabledProps"
+	  }
+	  BlockReduction	  on
+	  BooleanDataType	  on
+	  ConditionallyExecuteInputs on
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+	  PassReuseOutputArgsAs	  "Structure reference"
+	  ExpressionDepthLimit	  2147483647
+	  FoldNonRolledExpr	  on
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+	  RollThreshold		  5
+	  SystemCodeInlineAuto	  off
+	  StateBitsets		  off
+	  DataBitsets		  off
+	  UseTempVars		  off
+	  ZeroExternalMemoryAtStartup on
+	  ZeroInternalMemoryAtStartup on
+	  InitFltsAndDblsToZero	  off
+	  NoFixptDivByZeroProtection off
+	  EfficientFloat2IntCast  off
+	  EfficientMapNaN2IntZero on
+	  OptimizeModelRefInitCode off
+	  LifeSpan		  "inf"
+	  BufferReusableBoundary  on
+	  SimCompilerOptimization "Off"
+	  AccelVerboseBuild	  off
+	}
+	Simulink.DebuggingCC {
+	  $ObjectID		  5
+	  Version		  "1.6.0"
+	  RTPrefix		  "error"
+	  ConsistencyChecking	  "none"
+	  ArrayBoundsChecking	  "none"
+	  SignalInfNanChecking	  "none"
+	  SignalRangeChecking	  "none"
+	  ReadBeforeWriteMsg	  "UseLocalSettings"
+	  WriteAfterWriteMsg	  "UseLocalSettings"
+	  WriteAfterReadMsg	  "UseLocalSettings"
+	  AlgebraicLoopMsg	  "warning"
+	  ArtificialAlgebraicLoopMsg "warning"
+	  SaveWithDisabledLinksMsg "warning"
+	  SaveWithParameterizedLinksMsg	"warning"
+	  CheckSSInitialOutputMsg on
+	  UnderspecifiedInitializationDetection	"Classic"
+	  MergeDetectMultiDrivingBlocksExec "none"
+	  CheckExecutionContextPreStartOutputMsg off
+	  CheckExecutionContextRuntimeOutputMsg	off
+	  SignalResolutionControl "UseLocalSettings"
+	  BlockPriorityViolationMsg "warning"
+	  MinStepSizeMsg	  "warning"
+	  TimeAdjustmentMsg	  "none"
+	  MaxConsecutiveZCsMsg	  "error"
+	  SolverPrmCheckMsg	  "warning"
+	  InheritedTsInSrcMsg	  "warning"
+	  DiscreteInheritContinuousMsg "warning"
+	  MultiTaskDSMMsg	  "error"
+	  MultiTaskCondExecSysMsg "error"
+	  MultiTaskRateTransMsg	  "error"
+	  SingleTaskRateTransMsg  "none"
+	  TasksWithSamePriorityMsg "warning"
+	  SigSpecEnsureSampleTimeMsg "warning"
+	  CheckMatrixSingularityMsg "none"
+	  IntegerOverflowMsg	  "warning"
+	  Int32ToFloatConvMsg	  "warning"
+	  ParameterDowncastMsg	  "error"
+	  ParameterOverflowMsg	  "error"
+	  ParameterUnderflowMsg	  "none"
+	  ParameterPrecisionLossMsg "warning"
+	  ParameterTunabilityLossMsg "warning"
+	  FixptConstUnderflowMsg  "none"
+	  FixptConstOverflowMsg	  "none"
+	  FixptConstPrecisionLossMsg "none"
+	  UnderSpecifiedDataTypeMsg "none"
+	  UnnecessaryDatatypeConvMsg "none"
+	  VectorMatrixConversionMsg "none"
+	  InvalidFcnCallConnMsg	  "error"
+	  FcnCallInpInsideContextMsg "Use local settings"
+	  SignalLabelMismatchMsg  "none"
+	  UnconnectedInputMsg	  "warning"
+	  UnconnectedOutputMsg	  "warning"
+	  UnconnectedLineMsg	  "warning"
+	  SFcnCompatibilityMsg	  "none"
+	  UniqueDataStoreMsg	  "none"
+	  BusObjectLabelMismatch  "warning"
+	  RootOutportRequireBusObject "warning"
+	  AssertControl		  "UseLocalSettings"
+	  EnableOverflowDetection off
+	  ModelReferenceIOMsg	  "none"
+	  ModelReferenceVersionMismatchMessage "none"
+	  ModelReferenceIOMismatchMessage "none"
+	  ModelReferenceCSMismatchMessage "none"
+	  UnknownTsInhSupMsg	  "warning"
+	  ModelReferenceDataLoggingMessage "warning"
+	  ModelReferenceSymbolNameMessage "warning"
+	  ModelReferenceExtraNoncontSigs "error"
+	  StateNameClashWarn	  "warning"
+	  SimStateInterfaceChecksumMismatchMsg "warning"
+	  StrictBusMsg		  "Warning"
+	  LoggingUnavailableSignals "error"
+	  BlockIODiagnostic	  "none"
+	}
+	Simulink.HardwareCC {
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+	  Version		  "1.6.0"
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+	  ProdBitPerShort	  16
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+	  ProdEndianess		  "Unspecified"
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+	  ProdShiftRightIntArith  on
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+	  TargetPreprocMaxBitsUint 32
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+	  ProdEqTarget		  on
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+	  UpdateModelReferenceTargets "IfOutOfDateOrStructuralChange"
+	  CheckModelReferenceTargetMessage "error"
+	  ModelReferenceNumInstancesAllowed "Multi"
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+	  ModelReferenceMinAlgLoopOccurrences off
+	}
+	Simulink.SFSimCC {
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+	  SimUseLocalCustomCode	  off
+	  SimBuildMode		  "sf_incremental_build"
+	}
+	Simulink.RTWCC {
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+	  Version		  "1.6.0"
+	  Array {
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+	    Cell		    "IncludeHyperlinkInReport"
+	    Cell		    "GenerateTraceInfo"
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+	    Cell		    "GenerateTraceReportSl"
+	    Cell		    "GenerateTraceReportSf"
+	    Cell		    "GenerateTraceReportEml"
+	    PropName		    "DisabledProps"
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+	  SystemTargetFile	  "grt.tlc"
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+	  RTWCompilerOptimization "Off"
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+		Cell			"PurelyIntegerCode"
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+	  BlockType		  Inport
+	  Name			  "Delta F"
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+	  OutScaling		  "2^0"
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+	Block {
+	  BlockType		  Inport
+	  Name			  "Delta C"
+	  SID			  284
+	  Position		  [25, 148, 55, 162]
+	  Port			  "2"
+	  IconDisplay		  "Port number"
+	  OutDataType		  "fixdt(1, 16)"
+	  OutScaling		  "2^0"
+	}
+	Block {
+	  BlockType		  Demux
+	  Name			  "Demux1"
+	  SID			  236
+	  Ports			  [1, 6]
+	  Position		  [295, 28, 300, 227]
+	  ShowName		  off
+	  Outputs		  "6"
+	  DisplayOption		  "bar"
+	}
+	Block {
+	  BlockType		  Mux
+	  Name			  "Mux1"
+	  SID			  233
+	  Ports			  [2, 1]
+	  Position		  [95, 81, 100, 179]
+	  ShowName		  off
+	  Inputs		  "2"
+	  DisplayOption		  "bar"
+	}
+	Block {
+	  BlockType		  StateSpace
+	  Name			  "Syst. linéaire"
+	  SID			  232
+	  Position		  [160, 113, 220, 147]
+	  A			  "[0 0 0 1 0 0 ; 0 0 0 0 1 0 ; 0 0 0 0 0 1 ; 0 0 m*g/Mc -Cd/Mc 0 0 ; 0 0 0 0 -Cr/(b^2*(J/b^2+m)) 0 ; 0 0 -g/R*"
+	  "(1+m/Mc) Cd/(Mc*R) 0 0]"
+	  B			  "[0 0 ; 0 0 ; 0 0 ; 1/Mc 0 ; 0 -1/(b*(J/b^2+m)) ; -1/(R*Mc) 0 ];"
+	  C			  "eye(6)"
+	  D			  "zeros(6,2)"
+	}
+	Block {
+	  BlockType		  Outport
+	  Name			  "Delta d"
+	  SID			  279
+	  Position		  [480, 33, 510, 47]
+	  IconDisplay		  "Port number"
+	  OutDataType		  "fixdt(1, 16)"
+	  OutScaling		  "2^0"
+	}
+	Block {
+	  BlockType		  Outport
+	  Name			  "Delta r"
+	  SID			  280
+	  Position		  [445, 68, 475, 82]
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+	  IconDisplay		  "Port number"
+	  OutDataType		  "fixdt(1, 16)"
+	  OutScaling		  "2^0"
+	}
+	Block {
+	  BlockType		  Outport
+	  Name			  "Delta theta"
+	  SID			  282
+	  Position		  [415, 103, 445, 117]
+	  Port			  "3"
+	  IconDisplay		  "Port number"
+	  OutDataType		  "fixdt(1, 16)"
+	  OutScaling		  "2^0"
+	}
+	Block {
+	  BlockType		  Outport
+	  Name			  "Delta d'"
+	  SID			  283
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+	  IconDisplay		  "Port number"
+	  OutDataType		  "fixdt(1, 16)"
+	  OutScaling		  "2^0"
+	}
+	Block {
+	  BlockType		  Outport
+	  Name			  "Delta r'"
+	  SID			  285
+	  Position		  [350, 173, 380, 187]
+	  Port			  "5"
+	  IconDisplay		  "Port number"
+	  OutDataType		  "fixdt(1, 16)"
+	  OutScaling		  "2^0"
+	}
+	Block {
+	  BlockType		  Outport
+	  Name			  "Delta theta'"
+	  SID			  286
+	  Position		  [320, 208, 350, 222]
+	  Port			  "6"
+	  IconDisplay		  "Port number"
+	  OutDataType		  "fixdt(1, 16)"
+	  OutScaling		  "2^0"
+	}
+	Line {
+	  SrcBlock		  "Mux1"
+	  SrcPort		  1
+	  DstBlock		  "Syst. linéaire"
+	  DstPort		  1
+	}
+	Line {
+	  SrcBlock		  "Delta F"
+	  SrcPort		  1
+	  DstBlock		  "Mux1"
+	  DstPort		  1
+	}
+	Line {
+	  SrcBlock		  "Syst. linéaire"
+	  SrcPort		  1
+	  DstBlock		  "Demux1"
+	  DstPort		  1
+	}
+	Line {
+	  SrcBlock		  "Delta C"
+	  SrcPort		  1
+	  DstBlock		  "Mux1"
+	  DstPort		  2
+	}
+	Line {
+	  SrcBlock		  "Demux1"
+	  SrcPort		  1
+	  DstBlock		  "Delta d"
+	  DstPort		  1
+	}
+	Line {
+	  SrcBlock		  "Demux1"
+	  SrcPort		  2
+	  DstBlock		  "Delta r"
+	  DstPort		  1
+	}
+	Line {
+	  SrcBlock		  "Demux1"
+	  SrcPort		  3
+	  DstBlock		  "Delta theta"
+	  DstPort		  1
+	}
+	Line {
+	  SrcBlock		  "Demux1"
+	  SrcPort		  4
+	  DstBlock		  "Delta d'"
+	  DstPort		  1
+	}
+	Line {
+	  SrcBlock		  "Demux1"
+	  SrcPort		  5
+	  DstBlock		  "Delta r'"
+	  DstPort		  1
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+	  SrcPort		  6
+	  DstBlock		  "Delta theta'"
+	  DstPort		  1
+	}
+      }
+    }
+  }
+}

424-Systeme_Non_Lineaires/TP2/Ini_Grue.m

@@ -0,0 +1,120 @@
+% Ce script permet de d<EFBFBD>clarer les valeurs des param<EFBFBD>tres du mod<EFBFBD>le
+% Il sert aussi <EFBFBD> d<EFBFBD>clarer et <EFBFBD> calculer les diff<EFBFBD>rentes variables 
+% de commande tout au long du TP (Exemples : Point de fonctionnement,
+% trajectoire, commande <EFBFBD> grand gain, ...
+clear, clc,
+
+
+
+%% Param<EFBFBD>tres : 
+m  = 500;  %kg
+Mc = 5000; %kg
+g  = 10;   %m/s^2
+J  = 50;   %kg.m^2
+b  = 0.4;  %m
+Cd = 20;   %kg/s
+Cr = 20;   %kg.m^2/s
+
+%% Point de fonctionnement 
+R = 10;
+D = 0;
+C0 = m*g*b; 
+F = 0;
+ 
+%% Trajectoire
+Rini = R;
+Dini = 0;
+Rfin = 5;
+Dfin = 20;
+zh = 1;
+dt = 10;
+%% Manip 1 
+A = [0 0 0 1 0 0 ; 
+     0 0 0 0 1 0 ; 
+     0 0 0 0 0 1 ; 
+     0 0 m*g/Mc -Cd/Mc 0 0 ; 
+     0 0 0 0 -Cr/(b^2*(J/b^2+m)) 0 ; 
+     0 0 -g/R*(1+m/Mc) Cd/(Mc*R) 0 0];
+ 
+B = [0 0 ; 
+     0 0 ;
+     0 0 ;
+     1/Mc 0 ;
+     0 -1/(b*(J/b^2+m)) ;
+     -1/(R*Mc) 0 ];
+C= eye(6);
+Com = [B A*B A^2*B A^3*B A^4*B A^5*B];
+rank(Com)
+
+%% Manip 2 
+
+vprA = damp(eig(A)); % valeur propres de A 
+omega0 = vprA(2);
+xi = 0.5
+i=complex(0,1);
+p1 = omega0*(-xi+ i *sqrt(1-xi^2));
+p2 = omega0*(-xi- i *sqrt(1-xi^2));
+p = [-2 -2.5 -3 -4 p1 p2];
+
+
+K = place(A,B,p)
+C_1 =C(1,:);
+B_1 =B(:,1);
+
+eta = -1/(C_1*(A-B*K)^-1*B_1);
+
+%%
+sys = ss(A-B*K,B(:,2),C(2,:),D)
+H = tf(sys)
+bode(H)
+% margin(H)
+% Avec le correcteur intégral :
+Ti = 5e-4
+CI =  tf(1,[-Ti 0]); % négatif pour avoir une phase >180
+%bode(H,CI*H,'grid')
+fig =figure();
+margin(CI*H)
+grid on;
+saveas(fig,"manip_5marge.png");
+
+%% pretty figure
+% plot(simout)
+% legend("d","r","\theta","d d/dt","dr/dt","d\theta/dt");
+% grid on;
+%% Commande grand gain
+Deltat= 10
+Dini = 0;
+Rini = 10;
+Dfin = 20;
+Rfin = 5;
+ed = omega0/10;
+er = ed;
+
+coeff_phi = [6/Deltat^5 -15/Deltat^4 10/Deltat^3 0 0 0]
+
+
+%%
+close all;
+fig= figure()
+plot(Dc)
+hold on
+plot(simout.Time, simout.Data(:,1));
+xlabel('temps (s)');
+grid on;
+title('Vérification de la planification de trajectoire');
+legend('d_c','d');
+
+fig2= figure()
+plot(Rc)
+hold on
+plot(simout.Time, simout.Data(:,2));
+xlabel('temps (s)');
+grid on;
+title('Vérification de la planification de trajectoire');
+legend('r_c','r');
+
+
+
+
+
+

424-Systeme_Non_Lineaires/TP2/TP2.tex

@@ -0,0 +1,518 @@
+\documentclass[10pt,a4paper,notitlepage]{article}
+\usepackage[utf8]{inputenc}
+\usepackage[french]{babel}
+\usepackage[T1]{fontenc}
+\usepackage{mathtools}
+\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
+\usepackage{graphicx}
+\usepackage{float}
+\usepackage{subcaption}
+\usepackage{tikz}
+
+\usepackage{siunitx}
+\usepackage{numprint}
+
+\addto{\captionsfrench}{\renewcommand{\abstractname}{Introduction}}
+
+\author{Pierre-Antoine Comby}
+\title{TP1 : Commande d'un bras à liaison flexible par bouclage linéarisant}
+
+\input{/home/pac/Scripts/Raccourcis.tex}
+\renewcommand{\R}{\mathbb{R}}
+\renewcommand{\Z}{\mathbb{Z}}
+\renewcommand{\vec}{\overrightarrow}
+\usepackage{minted}
+\begin{document}
+\maketitle
+\section{Modélisation}
+\paragraph{Prépa.1}
+\begin{itemize}
+\item On applique le PFD au chariot \{1\}de masse $M_c$:
+\begin{equation}
+  M_c\vec{a_{\{1\}}} = \vec{F}+\vec{T}-C_d\vec{v_{\{1\}}}+M_c\vec{g}
+\end{equation}
+Soit en projetant:
+\begin{equation}
+  M_c \ddot{d} = F+T\sin(\theta)-C_d\dot{d}
+\end{equation}
+\item On fait de meme pour la masse pendulaire \{2\} $m$:
+\begin{equation}
+  m \vec{a_{\{2\}}} = -\vec{T} +m \vec{g}
+\end{equation}
+et il vient directement:
+\begin{equation}
+  \begin{cases}
+    m\ddot{x} = -T\sin(\theta)\\
+    m\ddot{z} = -T\cos(\theta)+mg
+  \end{cases}
+\end{equation}
+\item et un théorème du moment cinétique sur l'arbre donne (avec $\alpha$ angle du tambour par rapport au chariot):
+  \begin{equation}
+    J\ddot{\alpha} = -C +bT-C_r \dot{\alpha}
+  \end{equation}
+  soit :
+  \begin{equation}
+    J \frac{\ddot{r}}{b} = -C+bT -C_r \frac{\dot{r}}{b}
+  \end{equation}
+\end{itemize}
+\paragraph{Prépa.2}
+Comme $b\ll r$ on a :
+\begin{equation}
+  \begin{cases}
+    x = r\sin\theta+d\\
+    z = r\cos\theta
+  \end{cases}
+\end{equation}
+soit :
+\begin{equation}
+  \begin{cases}
+    \dot{x} = \dot{r}\sin\theta + r\dot{\theta}\cos\theta +\dot{d}\\
+    \dot{z} = \dot{r}\cos\theta -r\dot{\theta}\sin\theta
+  \end{cases}
+  \quad\text{ et }\quad
+  \begin{cases}
+    \ddot{x} = \ddot{r}\sin\theta+2\dot{r}\dot{\theta}\cos\theta+r\ddot{\theta}\cos\theta- r\dot{\theta}^2\sin\theta+\ddot{d}\\
+    \ddot{z} = \ddot{r}\cos\theta-2\dot{r}\dot{\theta}\sin\theta-r\ddot{\theta}\sin\theta-r\dot{\theta}^2\cos\theta
+  \end{cases}
+\end{equation}
+\paragraph{Prépa.3} en remplacant dans le système (1) donné dans l'énoncé on trouve:
+\begin{equation}
+  \begin{cases}
+    m(
+    \ddot{r} \sin\theta+2\dot{r}\dot{\theta}\cos\theta+r\ddot{\theta}\cos\theta-r\dot{\theta}^2\sin\theta+\ddot{d}
+    ) &=-T\sin\theta\\
+    m(
+    \ddot{r}\cos{\theta}-2\dot{r}\dot{\theta}\sin\theta-r\ddot{\theta}\sin\theta-r\dot{\theta}^2\cos\theta
+    ) &= - T\cos\theta+ mg
+  \end{cases}
+\end{equation}
+A partir des identités trigonométrique on a :
+\begin{equation}
+  \begin{cases}
+    m\ddot{r} &= - m\ddot{d}\sin\theta+mr\dot{\theta}^2 -T + mg \cos{\theta}\\
+    r \ddot{\theta} &= -\ddot{d}\cos\theta-2\dot{r}\dot{\theta}-g\sin\theta 
+  \end{cases}
+\end{equation}
+\paragraph{Prépa.4}
+Des équations précédentes on en déduit le modèle d'état( en remplacant l'expression de $T$ par celle déduite de la prépa.3):
+\begin{equation}
+  \begin{cases}
+    M_c\ddot{d} &= F +T \sin\theta-C_d\dot{d}\\
+    r\ddot{\theta} &= -\ddot{d}\cos\theta-2\dot{r}\dot{\theta}-g\sin(\theta)\\
+    (m+\frac{J}{b^2})\ddot{r} = -\frac{C}{b}-\frac{C_r\dot{r}}{b^2}-m\ddot{d}\sin\theta-mr\dot{\theta}^2+mg\cos{\theta}
+  \end{cases}
+\end{equation}
+
+\paragraph{Prépa.5}
+On linéarise autour de $d=D$,$r=R$,$\theta=0$, $T=mg$,$F=0$,$C=mbg$:
+\begin{equation}
+    \begin{cases}
+M_{c}(\Delta d+D) &=(\Delta F)+T \sin (\Delta \theta)-C_{d}(\Delta d+D) \\(\Delta r+R)(\Delta \theta) &=-(\Delta d+D) \cos (\Delta \theta)-2(\Delta r+R)(\Delta \theta)-g \sin (\Delta \theta) \\\left(m+\frac{J}{b^{2}}\right)(\Delta r+R) &=-\frac{\Delta C+m g b}{b}-\frac{C_{r}(\Delta r+R)}{b^{2}}-m(\Delta d+D) \sin (\Delta \theta)-m(\Delta r+R)(\Delta \theta)^{2}+m g \cos (\Delta \theta)
+\end{cases}
+\end{equation}
+En faisant une approximation au 1er ordre on a:
+\begin{equation}
+  \begin{cases}
+    M_{c} \Delta d &=\Delta F+m g \Delta \theta-C_{d} \Delta d \\
+    R \Delta \theta+\Delta d &=-g \Delta \theta \\
+    \left(m+\frac{J}{b^{2}}\right) \ddot{\Delta r} &=-\frac{\Delta C+m g b}{b}-\frac{C_{r} \Delta r}{b^{2}}-m g \Delta \theta
+  \end{cases}
+  \implies
+  \begin{cases} M_{c} \ddot{\Delta} d &=\Delta F+m g \Delta \theta-C_{d} \dot{\Delta} d \\
+    R \Delta \theta+\Delta d &=-g \Delta \theta \\
+    \left(m+\frac{J}{b^{2}}\right) \ddot{\Delta r} &=-\frac{\Delta C}{b}-\frac{C_{r} \Delta r}{b^{2}}
+  \end{cases}
+\end{equation}
+On en deduit le modèle d'état linéarisé:
+\[
+  \deriv{t}\vect{\Delta d \\ \Delta r \\ \Delta\theta \\ \dot{\Delta d}\\ \dot{\Delta r}\\ \dot{\Delta \theta}} = 
+  \left[ \begin{array}{cccccc}
+           {0} & {0} & {0} & {1} & {0} & {0} \\
+           {0} & {0} & {0} & {0} & {1} & {0} \\
+           {0} & {0} & {0} & {0} & {0} & {1} \\
+           {0} & {0} & {\frac{m g}{M_{c}}} & {-\frac{C_{d}}{M_{c}}} & {0} & {0} \\
+           {0} & {0} & {0} & {0} & {-\frac{C_{r}}{J+m b^{2}}} & {0} \\
+           {0} & {0} & {-\left(1+\frac{m}{M c}\right) \frac{g}{R}} & {\frac{C_{d}}{R M_{c}}} & {0} & {0}
+         \end{array}\right] \cdot
+       \left[ \begin{array}{c}{\Delta d} \\ {\Delta r} \\ {\Delta \theta} \\ {\Delta d} \\ {\Delta r} \\ {\Delta \theta}\end{array}\right]+
+       \left[ \begin{array}{ccc}{0} & {0} \\ {0} & {0} \\ {0} & {0} \\ {\frac{1}{M_{c}}} & {0} \\ {0} & {-\frac{1}{\frac{J}{b}+m b^{2}}} \\ {-\frac{1}{R M_{c}}} & {0}\end{array}\right]
+       \cdot \left[ \begin{array}{c}{\Delta F} \\ {\Delta C}\end{array}\right]
+\]
+
+\section{Commande linéaire}
+\paragraph{Manip.1}
+Dans le cas linéaire on construit la matrce de Kallman :
+\[
+  \mathcal{C} = \vect{B & AB & A^2B & A^5B }
+\]
+avec la fonction matlab \texttt{rank(Com)} on vérifie que le système est bien commandable.
+\begin{minted}{matlab}
+  A = [0 0 0 1 0 0 ; 
+     0 0 0 0 1 0 ; 
+     0 0 0 0 0 1 ; 
+     0 0 m*g/Mc -Cd/Mc 0 0 ; 
+     0 0 0 0 -Cr/(b^2*(J/b^2+m)) 0 ; 
+     0 0 -g/R*(1+m/Mc) Cd/(Mc*R) 0 0];
+B = [0 0 ; 
+     0 0 ;
+     0 0 ;
+     1/Mc 0 ;
+     0 -1/(b*(J/b^2+m)) ;
+     -1/(R*Mc) 0 ];
+Com = [B A*B A^2*B A^3*B A^4*B A^5*B];
+rank(Com)
+\end{minted}
+On a un rang de 6, le système linéaire est bien commandable.
+\paragraph{Manip.2}
+avec la fonction \texttt{damp(eig(A))} on trouve les valeurs propres et constantes de temps de la matrice d'état
+\begin{table}[H]
+  \centering
+  \begin{tabular}[t]{llll}
+  \hline
+  {\bf Pole}                 & {\bf Damping}   & {\bf Time Constant}  &            \\
+\hline
+                       &           & {(rad/TimeUnit)} & {(TimeUnit)} \\   
+  
+  0.00e+00             & -1.00e+00 & 0.00e+00       & Inf        \\
+ -1.82e-04 + 1.48e+00i & 1.23e-04  & 1.48e+00       & 5.50e+03   \\ 
+ -1.82e-04 - 1.48e+00i & 1.23e-04  & 1.48e+00       & 5.50e+03   \\ 
+ -3.64e-03             & 1.00e+00  & 3.64e-03       & 2.75e+02   \\ 
+  0.00e+00             & -1.00e+00 & 0.00e+00       & Inf        \\
+  -1.54e-01             & 1.00e+00  & 1.54e-01       & 6.50e+00   \\
+  \hline
+\end{tabular}
+
+\caption{Valeur propre et pulsation caractéristique}
+\end{table}
+On a donc la pulsation propre du système : $\omega_0$= \SI{1.48}{rad/s}. On choisi alors d'imposer les poles suivant, (d'après le cahier des charges) en ammortissant le pole double conjugué principal:
+\[
+  p_{1,2} = \omega_0(\xi\pm j \sqrt{1-\xi^2}) = 1.48(0.5\pm \sqrt{1-0.5^2})
+\]
+\begin{minted}{matlab}
+  
+vprA = damp(eig(A)); % valeur propres de A 
+omega0 = vprA(2);
+xi = 0.5
+i=complex(0,1);
+p1 = omega0*(xi+ i *sqrt(1-xi^2));
+p2 = omega0*(xi- i *sqrt(1-xi^2));
+p = [-2 -2.5 -3 -4 p1 p2];
+
+K = place(A,B,p)
+
+\end{minted}
+\[K = 10^{5}\cdot
+\begin{bmatrix}
+  
+    0.4288 & 0.1360  & 2.2614  & 0.0215 & 0.0506  & -0.9539 \\
+   -0.0035 & -0.0261 & -0.0026 & 0.0019 & -0.0182 & 0.0195  \\
+
+\end{bmatrix}\]
+\paragraph{Manip.3}
+Pour le terme de précommande, si l'on veux un gain statique unitaire, il faut\footnote{cf UE 421}:
+\[
+  \eta =\frac{-1}{C_1(A-BK)^{-1}B_1} = 4.1039. 10^{4}
+\]
+Avec $C_1$ et  $B_1$ lignes et colonnes des matrices $C$ et $B$ propre à $\Delta d$
+\paragraph{Manip.4} L'ajout d'une composante intégrale va rendre notre système précis. On construit alors la fonction de transfert grace à MATLAB:
+\begin{minted}{matlab}
+  sys = ss( A-B*K,B(:,2),C(2,:), 0 );
+  H = tf(sys);
+  bode(H);
+\end{minted}
+On obtient le diagramme de bode suivant:
+\begin{figure}[ht]
+  \centering
+  \includegraphics[width=0.9\textwidth]{manip_5_bode.png}
+  \caption{Diagramme de bode système en boucle ouverte}
+  \label{fig:label}
+\end{figure}
+\paragraph{Manip.5}
+La marge de phase est nulle, on met en place un correcteur intégrale pur qui apporte stabilité et précission au système:
+
+\begin{minted}{matlab}
+Ti = 4e-4
+CI =  tf(1,[-Ti 0]); % négatif pour avoir une phase >180
+%bode(H,CI*H,'grid')
+figure();
+margin(CI*H)  
+\end{minted}
+\begin{figure}[H]
+  \centering
+  \includegraphics[width=0.7\textwidth]{manip_5marge.png}
+  \caption{Marge de Phase et Gain pour $T_i=4.10^{-4}$}
+  \label{fig:margin}
+\end{figure}
+sur la figure\ref{fig:margin} on relève une marge de phase de 50$^o$ , le cahier des charge est respecté.
+
+\begin{figure}[ht]
+  \centering
+  \includegraphics[width=0.9\textwidth]{bouclage_modele_L.png}
+  \caption{Modèle Linéaire compléter par le correcteru par retour d'état.}
+  \label{fig:label}
+\end{figure}
+
+\paragraph{Manip.6}
+Avec une commande $\Delta d_c = 20$m on obtient la figure suivante:
+\begin{figure}[ht]
+  \centering
+  \includegraphics[width=0.7\textwidth]{manip6_20.png}
+  \caption{sortie du système pour une commande de 20m}
+  \label{fig:gain20}
+\end{figure}
+
+On remarque que le système n'est plus précis, il faut recaler le gain, ici manuellement, pour obtenir un système précis.
+On atteint les limites du modèle linéaire, pour des commandes plus grande ou plus complexe il va falloir tenir compte des non-linéarités du système.
+
+\paragraph{Manip.7}
+On applique le correcteur au modèle non linéaire, en ayant recentrer le
+retourd'état autour de son point de fonctionnement.
+On obtient une sortie chaotique, mais douce. comme on peux le voir sur la figure \ref{fig:dc_var} 
+\paragraph{Manip.8}
+Un changement de consigne apporte un comportement différents :
+\begin{figure}[H]
+  \centering 
+\begin{subfigure}{0.5\textwidth}
+  \centering
+  \includegraphics[width=\textwidth]{NL_correcL10.png}
+  \caption{$d_c=10$}
+  \label{fig:label}
+\end{subfigure}%
+\begin{subfigure}{0.5\textwidth}
+  \centering
+  \includegraphics[width=\linewidth]{NL_correcL100.png}
+  \caption{$d_c=100$}
+  \label{fig:label}
+\end{subfigure}\\
+\begin{subfigure}{0.5\textwidth}
+  \centering
+  \includegraphics[width=\linewidth]{NL_correcL1.png}
+  \caption{$d_c=1$}
+  \label{fig:label}
+\end{subfigure}
+\caption{différentes sorties pour des consignes différentes}
+\label{fig:dc_var}
+\end{figure}
+
+
+
+
+
+
+\section{Commande non linéaire hiérarchisante}
+\subsection{commande à grand gain}
+\paragraph{Prépa.6} On planifie la trajectoire avec:
+\begin{equation}
+  \begin{aligned} D_{c}(t) &=(1-\phi(t)) D_{i n i}+\phi(t) D_{f i n} \\
+    R_{c}(t) &=(1-\phi(t)) R_{i n i}+\phi(t) R_{f i n}
+  \end{aligned}
+\end{equation}
+Avec les conditions initiales et finales sur les positions
+$\phi(0)=0 \quad$ et $\quad \phi(\Delta t)=1$
+et celles sur la vitesse et l'accélération
+$\forall i=1,2\left.\frac{d^{i} \phi(t)}{d t^{i}}\right|_{t=0}=0 \quad$ et $\quad\left.\frac{d^{i} \phi(t)}{d t^{i}}\right|_{t=\Delta t}=0$
+On a 6 conditions à respecter, donc on choisit un polynôme de degré 5:
+\[
+  \phi(t) = a_5 t^5 + a_4 t^4 +a_3 t^3 + a_2 t^2+a_1 t+ a_0
+\]
+
+Les conditions initiales imposent  $a_0=a_1=a_2 = 0$ on a donc le polynome : $ \phi(t) = a_5 t^5 + a_4 t^4 +a_3 t^3$.
+Alors :
+\begin{equation}
+  \begin{aligned} \alpha(\Delta t)=1 & \Rightarrow \quad a_5 \Delta t^{5} \quad+a_4 \Delta t^{4} \quad+a_3 \Delta t^{3}=1 \\
+    \left.\frac{d \alpha(t)}{d t}\right|_{t=\Delta t}=0 & \Rightarrow 5 a_5 \Delta t^{4} \quad+4 a_4 \Delta t^{3}+3 a_3 \Delta t^{2}=0 \\
+    \left.\frac{d^{2} \alpha(t)}{d t^{2}}\right|_{t=\Delta t}=0 & \Rightarrow 20 a_5 \Delta t^{4}+12 a_4 \Delta t^{3}+6 a_3 \Delta t^{2}=0
+  \end{aligned}
+\end{equation}
+En résolvant le système on obtient:
+\[
+  a_5 = \frac{6}{\Delta t^5} ,\quad a_4 = \frac{-15}{\Delta t^4}, \quad a_3 = \frac{10}{\Delta t^3}
+\]
+
+\paragraph{Prépa.7} à partir de l'équation (2.4) de l'énoncé on a :
+
+\begin{equation}
+  \frac{1}{\omega_{0}^{2}}=\frac{R}{g} \Rightarrow \omega_{0}=\sqrt{\frac{g}{R}}
+\end{equation}
+
+\paragraph{Prépa.8} on a les commandes :
+\begin{equation}
+  F=\frac{M_{c}}{\epsilon_{d}}\left(\dot{D}_{c}-\dot{d}\right) \text{ et } C=-\frac{J / b+m b}{\epsilon_{r}}\left(\dot{R}_{c}-\dot{r}\right)
+\end{equation}
+soit le modèle :
+\begin{equation}
+  \dot{x}=\left[ \begin{array}{c}{\Delta d} \\ {\Delta r} \\ {\frac{m g}{M_{c}} \Delta \theta-\frac{C_{d}}{M_{c}} \Delta d+\frac{1}{\epsilon_{d}}\left(\dot{D}_{c}-\dot{d}\right)} \\ {\frac{-C_{r}}{J+m b^{2}}+\frac{F_{c}-\dot{r}}{\epsilon_{r}}} \\ {\Delta \theta}\end{array}\right]
+\end{equation}
+D'où :
+\begin{equation}
+  \begin{aligned}
+    \ddot{\Delta d} &=-\left(\frac{C_{d}}{M_{c}}+\frac{1}{\epsilon_{d}}\right) \Delta d+\frac{\dot{D}_{c}}{\epsilon_{d}}+\frac{m g}{M_{c}} \Delta \theta \\
+    \ddot{\Delta r} &=-\left(\frac{C_{r}}{J+m b^{2}}+\frac{1}{\epsilon_{r}}\right) \Delta r+\frac{\dot{R}_{c}}{\epsilon_{r}} \\
+    \text { On peut poser } \tau_{d} &=\frac{1}{\frac{C_{d}}{M_{c}}+\frac{1}{\epsilon_{d}}} \\
+    \text { et } \tau_{r} &=\frac{1}{\frac{C_{r}}{J+m b^{2}}+\frac{1}{\epsilon_{r}}}
+  \end{aligned}
+\end{equation}
+En considérant $\epsilon_d \ll 1 $ et $\epsilon_r <<1 $ on a $\tau_d \simeq \epsilon_d$ et $\tau_r \simeq \epsilon_r$
+
+\paragraph{Prépa.9}
+On doit prendre  $\epsilon_d$ et $\epsilon_r$ suffisamment petit devant la pulsation du sytème $\omega_0$, pour avoir une commande qui puisse  compenser suffisament vite les oscillations du système.
+\paragraph{Manip.9}
+On réalise la commande suivante :
+\begin{figure}[ht]
+  \centering
+  \includegraphics[width=0.9\textwidth]{boucleNL.png}
+  \caption{Élaboration de la commande pour la porsuite de trajectoire}
+  \label{fig:label}
+\end{figure}
+
+\begin{figure}[ht]
+  \centering
+  \begin{subfigure}{0.5\textwidth}
+    \centering
+    \includegraphics[width=\linewidth]{traj_plan_d}
+    \caption{poursuite sur $d$}
+    \label{fig:label}
+  \end{subfigure}%
+\begin{subfigure}{0.5\textwidth}
+  \centering
+  \includegraphics[width=\linewidth]{traj_plan_r}
+  \caption{Poursuite sur $r$}
+  \label{fig:label}
+\end{subfigure}
+\caption{Commande en poursuite de trajectoire}
+\label{fig:pours_traj}
+\end{figure}
+sur la figure \ref{fig:pours_traj} on remarque que la poursuite est plutot bien respecté pour $d$, mais celle sur $r$ conduit à un écart final non nul.
+\paragraph{Manip.10}
+
+
+\paragraph{Manip.11}
+\paragraph{Manip.12}
+\paragraph{Manip.13}
+
+\subsection{Platitude}
+\paragraph{Prépa.10}
+Pour montrer que le systeme est plat, ayant pour sortie plates $(x, z),$ il faut montrer que les états,$d, r, \theta$ et les commandes $F, C$ ne dépendent que des sorties $x, z$ et de leurs derivées.
+\begin{equation}
+  \left\{
+    \begin{aligned}
+      x &=r \sin \theta+d \\
+      m \ddot{x} &=-T \sin \theta
+    \end{aligned}
+  \right. \Rightarrow d=x+r \frac{m \ddot{x}}{T}
+\end{equation}
+
+Or on a:
+\begin{equation}
+  \begin{cases}
+    m \ddot{x}=-T \sin \theta \\
+    m \ddot{z}=-T \cos \theta+m g
+  \end{cases} \Rightarrow T=m \sqrt{\ddot{x}^{2}+(g-\ddot{z})^{2}}
+\end{equation}
+\begin{equation}
+  \begin{cases}
+    x=r \sin \theta+d \\
+    z=r \cos \theta
+  \end{cases} \Rightarrow r^{2}=(x-d)^{2}+z^{2}
+\end{equation}
+Soit :
+\begin{equation}
+  \begin{cases}
+    x-d &=-r \frac{m \ddot{x}}{T} \\
+    r^{2} &=(x-d)^{2}+z^{2}
+  \end{cases} \Rightarrow r^{2}=\frac{z^{2}}{1-\frac{\ddot{x}^{2}}{\vec{x}^{2}+(g-\ddot{z})^{2}}}
+\end{equation}
+
+Or $\theta = \arccos{\frac{z}{r}}$ on a donc :
+
+\begin{equation}
+r=\sqrt{\frac{z^{2}}{1-\frac{\ddot{x}^{2}}{\ddot{x}^{2}+(g-\ddot{z})^{2}}},}, d=x+\sqrt{\frac{z^{2}}{(g-\ddot{z})^{2}}} \ddot{x}, \quad \text { et } \theta=\arccos \sqrt{1-\frac{\ddot{x}^{2}}{\ddot{x}^{2}+(g-\ddot{z})^{2}}}
+\end{equation}
+Les variables d'états de dépendent donc que des sorties et de leurs dérivées, de même pour la commande 
+\begin{equation}
+  \begin{aligned}
+    C &=b T-C_{r} \frac{\dot{r}}{b}-J \frac{\ddot{r}}{b} \\
+    F &=M_{c} \ddot{d}-T \sin \theta+C_{d} \dot{d}
+  \end{aligned}
+\end{equation}
+Le système est plat ,avec le sorties  plates $x,z$
+\paragraph{Prépa.11}
+Les conditions imposées sont :
+\begin{equation}
+  \begin{cases}
+    d(t=0) = D_{ini} \\
+    r(t=0) = R_{ini} \\
+    \theta(t=0) = 0
+  \end{cases} \quad\text{ et }\quad
+  \begin{cases}
+    d(t=\Delta t ) = D_{fin} \\
+    r(t=\Delta t ) = R_{fin} \\
+    \theta(t=\Delta t ) = 0
+  \end{cases}
+\end{equation}
+Ce qui se transpose aux coordonnées:
+\begin{equation}
+  \begin{cases}
+    x(t=0) = D_{ini}\\
+    z(t=0) = R_{ini} \\
+  \end{cases} \quad\text{ et }\quad
+  \begin{cases}
+    x(t=\Delta t) = D_{fin}\\
+    z(t=\Delta t) = R_{fin} \\
+  \end{cases}
+\end{equation}
+
+On pose donc :
+\begin{equation}
+  \begin{aligned}
+    x_{c}(t) &=(1-\alpha(t)) D_{i n i}+\alpha(t) D_{f i n} \\
+    z_{c}(t) &=(1-\alpha(t)) R_{i n i}+\alpha(t) R_{f i n}
+  \end{aligned}
+\end{equation}
+De manière analogue à la préparation 6 on a les conditions initiales et finales sur les positions  et leur dérivées:
+\[
+  \alpha(0) = 0 ,\quad \alpha(\Delta t) =1, \quad \forall i = 1,2,3\left. \deriv[^i\alpha(t)]{t^i}\right|_{t=0}= 0 \text{ et } \left.\deriv[^i\alpha(t)]{t^i}\right|_{t=\Delta t}= 0
+\]
+Pour satisfaire les 8 conditions on choisit un polynome d'ordre 7 :
+\begin{equation}
+  \alpha(t)=a_{7} t^{7}+a_{6} t^{6}+a_{5} t^{5}+a_{4} t^{4}+a_{3} t^{3}+a_{2} t^{2}+a_{1} t+a_{0}
+\end{equation}
+Les conditions initiales imposent $a_0=a_1=a_2=a_3 =0$ et les conditions finales donnent :
+\begin{equation}
+  \begin{array}{lllllll}
+    \alpha(\Delta t)                              & =1  & a_7\Delta t^7    & + a_6 \Delta t^6  & + a_5 \Delta t^5 & + a_4 \Delta t^4 & =1 \\
+    \frac{d \alpha(t)}{d t}|t=\Delta t            & =0  & 7a_7\Delta t^6   & + 6a_6 \Delta t^5 & 5a_5\Delta t^4   & 4a_4 \Delta t^3  & =0 \\
+    \frac{d^{2} \alpha(t)}{d t^{2}}|_{t=\Delta t} & =0  & 42a_7\Delta t^5  & +30a_6 \Delta t^4 & 20a_5 t^3   & 12a_4\Delta t^2            & =0 \\
+    \frac{d^{3} \alpha(t)}{d t^{3}} |_{t=0}  & t=\Delta & 210a_z\Delta t^4 & +120a_6\Delta t^3 & 60a_5 \Delta t^2 & 24a_4\Delta t            & =0
+  \end{array}
+\end{equation}
+On a donc :
+\begin{equation}
+  a_{7}=\frac{-20}{\Delta t^{7}}, \quad a_{6}=\frac{70}{\Delta t^{6}}, \quad a_{5}=\frac{-84}{\Delta t^{5}}, \quad a_{4}=\frac{35}{\Delta t^{4}}
+\end{equation}
+
+\paragraph{Prépa.12}
+
+On impose une trajectoire parabolique ainsi,au niveau niveau de l'obstacle :$x_c=x_H \implies z_c=z_H$. On pose donc :
+\[
+  z_c = a(x_c-x_h)^2+z_h
+\]
+En évaluant cette expression à la position initiale on a:
+\[
+  a = \frac{R_{ini}-z_H}{(D_{ini}-x_H)^2}
+\]
+Pour déterminer $x_H$ on utilise la position finale et on a:
+
+\begin{equation}
+  x_{H}=\frac{D_{i n i}+\sqrt{\frac{R_{i n i}-z_{H}}{R_{i n i}-z_{H}}} D_{f i n}}%
+{1+\sqrt{\frac{R_{i n i}-z_{H}}{R_{f i n}-z_{H}}}}
+\end{equation}
+
+
+
+\end{document}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End:

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