An Introduction to System Modeling and Control

1 Introduction 1 1.1 Aircraft 1 1.2 Quadrotors 7 1.3 Inverted Pendulum 11 1.4 Magnetic Levitation 12 1.5 General Control Problem 14 2 Laplace Transforms 15 2.1 Laplace TransformProperties 17 2.2 Partial Fraction Expansion 21 2.3 Poles and Zeros 31 2.4 Poles and Partial Fractions 32 Appendix: Exponential Function 35 Problems 38 3 Differential Equations and Stability 45 3.1 Differential Equations 45 3.2 PhasorMethod of Solution 48 3.3 Final Value Theorem 52 3.4 Stable Transfer Functions 56 3.5 Routh-Hurwitz Stability Test 59 3.5.1 Special Case - A Row of the Routh Array has all Zeros* 65 3.5.2 Special Case - Zero in First Column, but the Row is Not Identically Zero* 68 Problems 71 4 Mass-Spring-Damper Systems 81 4.1 MechanicalWork 81 4.2 ModelingMass-Spring-Damper Systems 82 4.3 Simulation 88 Problems 92 5 Rigid Body Rotational Dynamics 103 5.1 Moment of Inertia 103 5.2 Newton's Law of RotationalMotion 104 5.3 Gears 111 5.3.1 Algebraic Relationships Between Two Gears 112 5.3.2 Dynamic Relationships Between Two Gears 112 5.4 Rolling Cylinder* 117 Problems 125 6 The Physics of the DC Motor 139 6.1 Magnetic Force 139 6.2 Single-LoopMotor 141 6.2.1 Torque Production 141 6.2.2 Wound Field DCMotor 143 6.2.3 Commutation of the Single-LoopMotor 143 6.3 Faraday's Law 145 6.3.1 The Surface Element Vector S 146 6.3.2 Interpreting the Sign of 147 6.3.3 Back Emf in a Linear DCMachine 147 6.3.4 Back Emf in the Single-LoopMotor 149 6.3.5 Self-Induced Emf in the Single-LoopMotor 150 6.4 Dynamic Equations of the DCMotor 152 6.5 Optical EncoderModel 154 6.6 Tachometer for a DCMachine* 157 6.6.1 Tachometer for the Linear DCMachine 157 6.6.2 Tachometer for the Single-Loop DCMotor 157 6.7 TheMultiloop DCMotor* 159 6.7.1 Increased Torque Production 159 6.7.2 Commutation of the Armature Current 159 Problems 163 7 BlockDiagrams 173 7.1 Block Diagramfor a DCMotor 173 7.2 Block DiagramReduction 175 Problems 185 8 System Responses 191 8.1 First-Order SystemResponse 191 8.2 Second-Order SystemResponse 193 8.2.1 Transient Response and Closed-Loop Poles 194 8.2.2 Peak Time and Percent Overshoot 198 8.2.3 Settling Time 200 8.2.4 Rise Time 202 8.2.5 Summary of 202 8.2.6 Choosing the Gain of a Proportional Controller 202 8.3 Second-Order Systems with Zeros 205 8.4 Third-Order Systems 210 Appendix - Root LocusMatlab File 211 Problems 212 9 Tracking and Disturbance Rejection 221 9.1 Servomechanism 221 9.2 Control of a DC ServoMotor 226 9.2.1 Tracking 226 9.2.2 Disturbance Rejection 231 9.2.3 Summary of the PI Controller for a DC Servo 234 9.2.4 Proportional plus Integral plus Derivative Control 234 9.3 Theory of Tracking and Disturbance Rejection 238 9.4 InternalModel Principle 242 9.5 Design Example: PI-D Control of Aircraft Pitch 244 9.6 Model Uncertainty and Feedback* 250 Problems 258 10 Pole Placement, 2 DOF Controllers, and Internal Stability 271 10.1 Output Pole Placement 271 10.1.1 DisturbanceModel 276 10.1.2 Effect of the Initial Conditions on the Control Design 278 10.2 Two Degrees of FreedomControllers 283 10.3 Internal Stability 292 10.3.1 Unstable Pole-Zero Cancellation Inside the Loop (Bad) 295 10.3.2 Unstable Pole-Zero Cancellation Outside the Loop (Good) 298 10.4 Design Example: 2 DOF Control of Aircraft Pitch 300 10.5 Design Example: Satellite with Solar Panels (Collocated Case) 303 Appendix: Output Pole Placement 306 Appendix:Multinomial Expansions 310 Appendix: Overshoot 311 Appendix: Unstable Pole-Zero Cancellation 315 Appendix: Undershoot 317 Problems 320 11 Frequency Response Methods 339 11.1 Bode Diagrams 339 11.1.1 Simple Examples 343 11.1.2 More Bode DiagramExamples 345 11.2 Nyquist Theory 359 11.2.1 Principle of the Argument 359 11.2.2 Nyquist Test for Stability 368 11.3 Relative Stability: Gain and Phase Margins 377 11.4 Closed-Loop Bandwidth 383 11.5 Lead and Lag Compensation 387 11.6 Double Integrator Control via Lead-Lag Compensation 392 11.7 Inverted Pendulum with Output 399 Appendix: Bode and Nyquist Plots inMatlab 401 Problems 402 12 Root Locus 419 12.1 Angle Condition and Root Locus Rules 420 12.2 Asymptotes and Their Intercept 427 12.3 Angles of Departure 434 12.4 Effect of Open-Loop Poles on the Root Locus 450 12.5 Effect of Open-Loop Zeros on the Root Locus 451 12.6 Breakaway Points and the Root Locus 452 12.7 Design Example: Satellite with Solar Panels (Noncollocated) 453 Problems 458 13 Inverted Pendulum, Magnetic Levitation, and Cart on a Track 467 13.1 Inverted Pendulum 467 13.1.1 Mathematical Model of the Inverted Pendulum 467 13.1.2 Linear ApproximateModel 470 13.1.3 Transfer FunctionModel 470 13.1.4 Inverted PendulumControl Using Nested Feedback Loops 472 13.2 Linearization of NonlinearModels 475 13.3 Magnetic Levitation 478 13.3.1 Conservation of Energy 479 13.3.2 StatespaceModel 480 13.3.3 Linearization About an Equilibrium Point 481 13.3.4 Transfer FunctionModel 483 13.4 Cart on a Track System 483 13.4.1 Mechanical Equations 484 13.4.2 Electrical Equations 485 13.4.3 Equations ofMotion and Block Diagram 486 Problems 488 14 State Variables 501 14.1 Statespace Form 501 14.2 Transfer Function to Statespace 503 14.2.1 Control Canonical Form 505 14.3 Laplace Transformof the Statespace Equations 513 14.4 Fundamental Matrix Phi 516 14.4.1 Exponential Matrix 517 14.5 Solution of the Statespace Equation* 520 14.5.1 Scalar Case 521 14.5.2 Matrix Case 522 14.6 Discretization of a StatespaceModel* 523 Problems 525 15 State Feedback 529 15.1 Two Examples 529 15.2 General State Feedback Trajectory Tracking 537 15.3 Matrix Inverses and the Cayley-Hamilton Theorem 538 15.3.1 Matrix Inverse 538 15.3.2 Cayley-Hamilton Theorem 541 15.4 Stabilization and State Feedback 543 15.5 State Feedback and Disturbance Rejection 547 15.6 Similarity Transformations 551 15.7 Pole Placement 555 15.7.1 State Feedback Does Not Change the SystemZeros 559 15.8 Asymptotic Tracking of Equilibrium Points 560 15.9 Tracking Step Inputs via State Feedback 562 15.10 Inverted Pendulum on an Inclined Track* 569 15.11 Feedback Linearization Control* 574 Appendix: Disturbance Rejection in the Statespace 579 Problems 581 16 State Estimators and Parameter Identification 595 16.1 State Estimators 595 16.1.1 General Procedure for State Estimation 600 16.1.2 Separation Principle 608 16.2 State Feedback and State Estimation in the Laplace Domain* 610 16.3 Multi-Output Observer Design for the Inverted Pendulum* 613 16.4 Properties ofMatrix Transpose and Inverse 615 16.5 Duality* 617 16.6 Parameter Identification 619 Problems 626 17 Robustness and Sensitivity of Feedback 641 17.1 Inverted Pendulum with Output 641 17.2 Inverted Pendulum with Output 655 17.3 Inverted Pendulumwith State Feedback 657 17.4 Inverted Pendulumwith an Integrator and State Feedback 661 17.5 Inverted Pendulumwith State Feedback via State Estimation 663 Problems 666 References 671 Index 675