280 likes | 367 Views
This study guide delves into the Linear-Quadratic-Gaussian (LQG) problem, exploring approaches like calculus of variation and dynamic programming, alongside the Kalman filter and linear feedback control. It discusses the LQG problem in the context of a linear system dynamics and initial state variables, and provides insights into optimal solutions, cost-to-go analysis, state transitions, and control variables. The guide outlines applications in aerospace, control systems, chemical processes, and guidance technology, offering valuable insights for students and professionals alike.
E N D
Linear-Quadratic-Gaussian Problem Study Guide for ES205 Yu-Chi Ho Jonathan T. Lee Feb. 5, 2001
Outline • Linear-Quadratic Problem • Calculus of Variation Approach • Dynamic Programming Approach • Kalman Filter • Linear Feedback Control
N Linear-Quadratic Problem subject to linearsystem dynamics given the initial state x(0) where x(i) is the state variables at time i u(i) is the control variable at time i A(i) and B(i) are the cost matrices at time i
N Calculus of Variation Approach • Let
N Dynamic Programming Approach • Cost-to-gowithwhereand
N DP Approach (cont.) • Set • We have • LetandThen
N DP Approach (cont.) • By induction, we have the optimal solution to bewhereand with boundary conditionwith
N Static State subject to state transitionwith initial conditionwhereand is noise
N Static State (cont.) • Set • Thus,
N Static State (cont.) • New Estimate = Old Estimate + Confidence Factor Correction Term
N Dynamic State subject to state transition with disturbance (i)where and is noise
N Dynamic State (cont.) • Cost-to-go • Setwhere • We have
N Dynamic System (cont.) • Cost-to-go • Substitute
N Dynamic System (cont.) • Set • Let Then, we have
N Dynamic System (cont.) • Cost-to-go • Substitute
N Dynamic System (cont.) • Set • Let and Then, we have
N With Control subject to state transitionwith disturbance (i)where and is noise
N With Control (cont.) • Cost-to-go • Set • We have
N With Control (cont.) • Cost-to-go • Substitute
N With Control(cont.) • Set • Let Then, we have
N With Control(cont.) • Cost-to-go • Substitute
N With Control(cont.) • Set • Let andThen, we have
N Linear Feedback Control • The predicted statewith initial statebased on the estimator
N Linear Feedback Control (cont.) • Optimal control based on the estimated state:whereandwith boundary conditionwith
N Applications • Apollo program: control of the space craft. • Airplane controllers: autopilot. • Neighboring Optimal Control for Non-Linear Systems • Chemical process controller • Guidance and control
References: • Bryson, Jr., A. E. and Y.-C. Ho, Applied Optimal Control: Optimization, Estimation, and Control, Taylor & Francis, 1975. • Ho, Y.-C., Lecture Notes, Harvard University, 1997.