ESE 543 Control System Design Using State Space MethodsClass OutlineTopicsTopics (cont)Topics (cont)Topics (cont)Grading PolicyLecture 1Robust and Adaptive Control ChallengesX-45A, X-45C at Edwards AFBPhantom EyeBasic Control TheoryControl Theory BasicsControl System DesignBasic Control TheoryControl Theory BasicsClosed Loop Feedback ControlPlant ModelsOpen Loop vs Closed LoopMathematical FoundationClassical Control TheoryLaplace Transform ReviewTransfer FunctionsBlock DiagramsControl System Block DiagramsClosed Loop Transfer FunctionError Transfer FunctionRoot LocusExampleRoot LocusTypical Transfer FunctionsFeedback ControlPID ControlP ControlPI ControlPD ControlPID ControlIntegrator MgmtSteady State Tracking and Controller TypeSteady State ErrorSystem TypeSystem TypeSystem TypeSystem TypeSystem TypeSystem Type SummaryFrequency Domain Analysis and Stability MarginsIntroductionStability of SISO SystemsStability of SISO SystemsThe Frequency ResponseThe Frequency ResponseThe Argument PrincipleNyquist Stability TheoryNyquist Theory (cont)Nyquist CriterionLoop gain Crossover Frequency Phase Crossover FrequencyClassical Frequency Response MethodsLecture 2State Space ModelingExampleExampleSolving First Order Differential EquationsSolving First Order Differential EquationsVector and Matrix AlgebraFirst Order Differential EquationLinear SystemsWhat is control system design about?Formulating the Control Design Problem Slide Number 70System Modeling and AnalysisSuspended Ball ModelSuspended Ball ModelSuspended Ball ModelSuspended Ball ModelLinearization of Nonlinear SystemsLinearization of Nonlinear SystemsReview of Transfer Functions and Transfer Function Matrices Important Transfer FunctionsControl Design DilemmaMIMO System Transfer Function MatricesSISO vs. MIMOAerodynamicsAircraft DynamicsAircraft DynamicsAircraft DynamicsAircraft DynamicsAccel Transfer FunctionLinear Pitch ModelAerodynamic DataExample: Pitch AutopilotState Space SystemsAll Integrator DiagramState Space MatricesIntegrator Diagram in Parallel FormMinimal RealizationsTransfer Functions From State Space Models Transfer Functions/State Space ModelsUncertainty In Dynamic ModelsUncertainty In Dynamic ModelsClassical Control Building BlocksLinear Vector SpacesLinear Vector SpacesOperatorsDefinitions and TheoremsDefinitions and TheoremsDefinitions and TheoremsSimilarity TransformationsReviewExampleInner ProductNormsGram-Schmidt ProcessExampleExampleEigenvalues, EigenvectorsEigenvalues, EigenvectorsEigenvalues, EigenvectorsSimilarity TransformationsEigenvalue FactsEigenvalues and EigenvectorsEig and EigV of Missile ModelDefiniteness of MatricesMatrix NormsMatrix Norms (cont)Singular Values, Singular VectorsSingular Values, Singular Vectors (cont)Singular Values, Singular Vectors (cont)Useful Properties of Singular ValuesHow Do We Compute Singular ValuesFactESE 543 Control System Design Using State Space MethodsK.A. Wise 2 Class Outline Text: Modern Control Theory, W. L. Brogan Text: Robust and Adaptive Control with Aerospace Applications, Lavretsky and Wise 1. Review of classical methods 2. System modeling and analysis 3. System structural properties 4. State Feedback design 5. Output feedback design 5. Multivariable frequency domain analysis 6. Robustness Theory 7. Optimal Control Alt: Optimal Control, Anderson and MooreK.A. Wise 3 Topics 1. System modeling and analysis Linearization State space systems Linear vector spaces Basis vectors, Gram-Schimdt orthogonalization Transformations, mappings, canonical forms Range space, null space, rank Eigenvalues, eigenvectors Singular values, singular vectors State transition matrix Φ Solution to state equationsK.A. Wise 4 Topics (cont) 2. System structural properties Controllability Observability Duality Degree of controllability, observability Poles, zeros, MIMO transmission zeros Stability Lyapunov stability Regions of asymptotic stabilityK.A. Wise 5 Topics (cont) 3. Feedback design Pole placement Stabilizability Observer feedback Detectability Pole placement using observer feedback Linear Quadratic Regulator Theory Robust Servomechanism Projective Control TheoryK.A. Wise 6 Topics (cont) 4. Multivariable frequency domain analysis Multivariable Nyquist theorem Relationship between SISO and MIMO stability analysis Singular value frequency response analysis 5. Optimal Control Linear quadratic regulator 6. Robustness Theory Uncertainty modeling Small gain theoremK.A. Wise 7 Grading Policy Homework 30% Computer Projects 20% Midterm Exam 25% Final Exam 25% Boeing Phone: 314-232-4549 Cell: 636-866-9162 Email: [email protected] [email protected] MatlabLecture 1K.A. Wise 9 Robust and Adaptive Control Challenges 4th Generation Escape System X-45A J-UCAS • Nonlinear Aero • Large Uncertainties • Nonlinear Control Effectors • Limited Actuation X-36 Tailless Agility Research Aircraft Unstable In Multiple Axes, Non-minimum Phase Vulture II Solar EagleK.A. Wise 10 X-45A, X-45C at Edwards AFBPhantom Eye K.A. Wise 11Basic Control TheoryK.A. Wise 13 Control Theory Basics • Differential Equations, Dynamics, Dynamical Systems • Linearization • Stability • Feedback Control • Time Domain, Frequency Domain • Methods For Compensator DesignK.A. Wise 14 Control System Design Control Design Problem: Design control inputs to produce satisfactory output response in the presence of disturbances and plant uncertainties. wyPuOutput Variables Control Inputs Disturbances Plant Model • • • • • • • • •K.A. Wise 15 Basic Control Theory System Model: Derive and or model all relevant dynamics. Analytical or experimental Open Loop Poles and Zeros: Form transfer functions or state space models. Determine open loop properties (poles and zeros). Determine control design requirements. Close Loop Poles: Design compensator to achieve desired pole locations. Analysis Of The Closed Loop System: Analyze system response in time domain (simulation). Determine stability robustness properties in frequency domain (Bode, Nyquist). Steps to a good design Done Implementation Signal Constraints Robustness Sensitivity Disturbance Rejection Transient Behavior Steady State Accuracy StabilityK.A. Wise 16 Control Theory Basics 1st Principles ODE’s Linear Nonlinear Physics Dynamics Infinite Dimensional n-Dimensional Classical Modern Analysis Feedback Linearization Backstepping Adaptive Control