ECE 553

OPTIMUM CONTROL SYSTEMS

This graduate-level course is on the theoretical and algorithmic foundations of optimal control theory, and will deal primarily with deterministic dynamical systems described in the continuous time. In addition, some aspects of stochastic control systems, and in particular the Linear-Quadratic-Gaussian (LQG) theory, will be covered. Furthermore, extensions of the single-criterion dynamic optimization theory to zero-sum and nonzero-sum differential games will be discussed, and the theory of H-infinity optimal control along with its extensions to nonlinear systems will be thoroughly studied. The course builds on a first-level graduate control course, such as ECE 515, and also requires some probability background at the level of ECE 413 (or better ECE 534). Some familiarity with the basics of finite-dimensional optimization would be helpful, but this is not absolutely necessary since the necessary tools on this topic will be developed throughout the course.

SPRING 2006 OFFERING

Instructor : Professor Tamer Basar

Office : 356 CSL (Phone: 3-3607)

Email : tbasar@control.csl.uiuc.edu

Required Text (for the second half of the course): Tamer Basar and Pierre Bernhard, H-infinity Optimal Control and Related Minimax Design Problems, 2nd edition, Birkhauser, 1995.

Recomended Text 1 (for the first half of the course): I.M. Gelfand and S.V. Fomin, Calculus of Variations, Dover, 2000. (This is paperback version of the original edition by Prentice Hall, dated 1963.)

Recomended Text 2 (for the first half of the course): Dimitri P. Bertsekas, Dynamic Programming and Optimal Control, Volume I, 3rd edition, Athena Scientific, 2005.

All three books are on reserve in the Grainger Library.

Meeting times : Mondays and Wednesdays, 12:30 - 1:55 p.m. in Rm. 106 B3 Engineering Hall

COURSE OUTLINE

I. Introduction

  1. Formulation of optimal control problems
  2. Parameter optimization versus path optimization
  3. Local and global optima; general conditions on existence and uniqueness
  4. Some useful results from finite-dimensional optimization

II. The Calculus of Variations

  1. The Euler-Lagrange equation and the associated transversality conditions
  2. Path optimization subject to equality and inequality constraints
  3. Differences between weak and strong extrema
  4. Second-order conditions for extrema

III. The Minimum Principle and Hamilton-Jacobi Theory

  1. Pontryagin's minimum principle
  2. Optimal control with state and control constraints
  3. Time-optimal control
  4. Singular solutions
  5. Hamilton-Jacobi-Bellman (HJB) equation, and relationship with dynamic programming
  6. Viscosity solutions to the HJB equation

IV. Linear Quadratic Problems

  1. Basic finite-time and infinite-time state regulator (review of material covered in ECE 515)
  2. Riccati equation and its properties (review of ECE 515 material)
  3. Tracking and disturbance rejection
  4. The Kalman filter and duality
  5. The Linear-Quadratic-Gaussian (LQG) design

V. H-infinity Optimal Control Designs

  1. Notions of sensitivity and complementary sensitivity in the control of uncertain plants, and relationship with H-infinity control
  2. Relationship between H-infinity optimal control and zero-sum differential games; some relevant results from zero-sum differential games
  3. Optimal or near-optimal designs under perfect state measurements
  4. Designs under imperfect state and sampled-data measurements
  5. Nonlinear systems

VI. Perturbational and Computational Methods

  1. Near-optimal designs
  2. Gradient methods
  3. Numerical methods based on the second variation

VII. Singularly Perturbed Systems (as time permits)

  1. Time-scale separation in singularly perturbed differential equations
  2. Near-optimal control on fast and slow time scales
  3. Robustness of near-optimal designs
  4. H-infinity optimal control on two time scales

VIII. Nonzero-Sum Differential Games (as time permits)

  1. Solution concepts for nonzero-sum games
  2. Nash equilibria; general theorems on existence and uniqueness; computational issues for static games
  3. The significance of information patterns in differential games; the role of representations of strategies
  4. Linear-quadratic differential games; applications

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