OPTIMAL CONTROL SYSTEMS ~,.~t~.~ " .u&:iuut -: ~ aIMlJ f~t jS~ J '1 z : ~ OJ"':' lrAr If I a : ~ i!J1i Electrical Engineering Textbook Series Richard C Dorf, Series Editor University of California, Davis Forthcoming and Published Titles Applied Vector Analysis Matiur Rahman and Isaac Mulolani Continuous Signals and Systems with MATLAB Taan EIAli and Mohammad A Karim Discrete Signals and Systems with MATLAB Taan EIAIi Electromagnetics Edward J Rothwell and Michael J Cloud Optimal Control Systems Desineni Subbaram Naidu OPTIMAL CONTROL SYSTEMS Desineni Subbaram Naidu Idaho State Universitv Pocatello Idaho USA o CRC PRESS Boca Raton London New York Washington, D.C Cover photo: Terminal phase (using fuel-optimal control) of the lunar landing of the Apollo 11 mission Courtesy of NASA TJ "l13 N1 b'~ [...]... Fuel -Optimal Control 7.20 Switching Curve for a Double Integral Fuel -Optimal Control System 7.21 Phase-Plane Trajectories for u(t) = 0 7.22 Fuel -Optimal Control Sequences 7.23 E-Fuel -Optimal Control 7.24 Optimal Control as Dead-Zone Function 7.25 Normal Fuel -Optimal Control System 7.26 Singular Fuel -Optimal Control. .. Optimality 6.3.2 Optimal Control Using Dynamic Programming 6.3.3 Optimal Control of Discrete-Time Systems 6.3.4 Optimal Control of Continuous-Time Systems 6.4 The Hamilton-Jacobi-Bellman Equation 6.5 LQR System Using H-J-B Equation " 6.6 Notes and Discussion 249 249 252 256 259 261 261 266 272 275 277 283 288 7 Constrained Optimal Control Systems 293 7.1 Constrained Optimal. .. Closed-Loop Optimal Control System for Example 3.1 3.5 Optimal States for Example 3.1 3.6 Optimal Control for Example 3.1 3.7 Interpretation of the Constant Matrix P 3.8 Implementation of the Closed-Loop Optimal Control: Infinite Final Time 3.9 Closed-Loop Optimal Control System 3.10 Optimal States for Example 3.2 3.11 Optimal Control. .. Closed-Loop Optimal Controller for Linear Discrete-Time Regulator 5.3 Riccati Coefficients for Example 5.3 5.4 Optimal Control and States for Example 5.3 5.5 Optimal Control and States for Example 5.3 5.6 Closed-Loop Optimal Control for Discrete-Time Steady-State Regulator System 5.7 Implementation of Optimal Control for Example 5.4 5.8 Implementation of Optimal. .. Discrete-Time Optimal Control Systems 191 5.1 Variational Calculus for Discrete-Time Systems 191 5.1.1 Extremization of a Functional 192 5.1.2 Functional with Terminal Cost 197 5.2 Discrete-Time Optimal Control Systems 199 5.2.1 Fixed-Final State and Open-Loop Optimal Control 203 5.2.2 Free-Final State and Open-Loop Optimal Control 207... Coefficients 91(t) and 92(t) for Example 5.6 Optimal States for Example 5.6 Optimal Control for Example 5.6 Closed-Loop Discrete-Time Optimal Control System 6.1 (a) An Optimal Control Function Constrained by a Boundary (b) A Control Variation for Which -8u(t) Is Not Admissible Illustration of Constrained (Admissible) Controls Optimal Path from A to B A Multistage... Open-Loop Optimal Controller (OLOC) and (b) Closed-Loop Optimal Controller (CLOC) 125 126 127 127 133 Implementation of the Optimal Tracking System Riccati Coefficients for Example 4.1 Coefficients 91(t) and 92(t) for Example 4.1 Optimal States for Example 4.1 Optimal Control for Example 4.1 Riccati Coefficients for Example 4.2 Coefficients 91(t) and 92(t) for Example 4.2 Optimal Control. .. Different Types of Systems: (a) Fixed-Final Time and Fixed-Final State System, (b) Free-Final Time and FixedFinal State System, (c) Fixed-Final Time and Free-Final State System, (d) Free-Final Time and Free-Final State System Optimal Controller for Example 2.12 Optimal Control and States for Example 2.12 Optimal Control and States for Example 2.13 Optimal Control and... Dynamic Programming Framework of Optimal State Feedback Control Optimal Path from A to B 6.2 6.3 6.4 6.5 6.6 6.7 6.8 232 233 239 240 241 241 242 243 254 260 261 262 263 265 271 290 7.1 Signum Function 299 7.2 Time -Optimal Control 299 7.3 Normal Time -Optimal Control System 300 7.4 Singular Time -Optimal Control System 301 7.5 Open-Loop... Control Law 315 7.3 Fuel -Optimal Control Systems 315 7.3.1 Fuel -Optimal Control of a Double Integral System 316 7.3.2 Problem Formulation and Statement 319 7.3.3 Problem Solution 319 7.4 Minimum-Fuel System: LTI System 328 7.4.1 Problem Statement 328 7.4.2 Problem Solution 329 7.4.3 SIMULINK© Implementation of Control Law 333 7.5 Energy -Optimal Control ... Systems: (a) Fixed-Final Time and Fixed-Final State System, (b) Free-Final Time and FixedFinal State System, (c) Fixed-Final Time and Free-Final State System, (d) Free-Final Time and Free-Final State... 6.3.3 Optimal Control of Discrete-Time Systems 6.3.4 Optimal Control of Continuous-Time Systems 6.4 The Hamilton-Jacobi-Bellman Equation 6.5 LQR System Using H-J-B Equation " 6.6 Notes and Discussion... Time-Optimal Control 299 7.3 Normal Time-Optimal Control System 300 7.4 Singular Time-Optimal Control System 301 7.5 Open-Loop Structure for Time-Optimal Control System 304 7.6 Closed-Loop