# Course Websites

## ECE 553 - Optimum Control Systems

### Last offered Spring 2022

#### Official Description

Theoretical and algorithmic foundations of deterministic optimal control theory, including calculus of variations, maximum principle, and principle of optimality; the Linear-Quadratic-Gaussian design; differential games and H-infinity optimal control design. Course Information: Prerequisite: ECE 313 and ECE 515.

Control Systems

#### Description

Theoretical and algorithmic foundations of optimal control theory; deterministic dynamical systems described in the continuous time; the Linear-Quadratic-Gaussian design; differential games, H∞-optimal control design.

#### Topics

• Introduction: formulation of optimal control problems; parameter optimization versus path optimization; local and global optima; general conditions on existence and uniqueness; some useful results finite-dimensional optimization
• Calculus of variations: Euler-Lagrange equation and the associated transversality conditions; path optimization subject to equality and inequality constraints; differences between weak and strong extrema; second-order conditions for extrema
• Minimum principle and Hamilton-Jacobi theory: Pontryagin's minimum principle; optimal control with state and control constraints; time-optimal control; singular solutions; Hamilton-Jacobi-Bellman equation, and relationship with dynamic programming
• Linear quadratic problems: basic finite-time and infinite-time state regulator (review of material covered in ECE 415); spectral factorization, robustness, frequency weightings; tracking and disturbance rejection; the Kalman filter and duality; the linear-quadratic-Gaussian (LQG) design
• Perturbational and computational methods: near-optimal designs; gradient methods; numerical methods based on the second variation
• Differential games: solution concepts for zero-sum and nonzero-sum games; general theorems on existence and uniqueness; explicit solutions to linear-quadratic games
• H[infinity]-optimal control design: relationships with zero-sum differential games; optimum or near-optimum designs under different information patterns

#### Detailed Description and Outline

Topics:

• Introduction: formulation of optimal control problems; parameter optimization versus path optimization; local and global optima; general conditions on existence and uniqueness; some useful results finite-dimensional optimization
• Calculus of variations: Euler-Lagrange equation and the associated transversality conditions; path optimization subject to equality and inequality constraints; differences between weak and strong extrema; second-order conditions for extrema
• Minimum principle and Hamilton-Jacobi theory: Pontryagin's minimum principle; optimal control with state and control constraints; time-optimal control; singular solutions; Hamilton-Jacobi-Bellman equation, and relationship with dynamic programming
• Linear quadratic problems: basic finite-time and infinite-time state regulator (review of material covered in ECE 415); spectral factorization, robustness, frequency weightings; tracking and disturbance rejection; the Kalman filter and duality; the linear-quadratic-Gaussian (LQG) design
• Perturbational and computational methods: near-optimal designs; gradient methods; numerical methods based on the second variation
• Differential games: solution concepts for zero-sum and nonzero-sum games; general theorems on existence and uniqueness; explicit solutions to linear-quadratic games
• H[infinity]-optimal control design: relationships with zero-sum differential games; optimum or near-optimum designs under different information patterns

#### Texts

Main:
D. Liberzon, Calculus of Variations and Optimal Control Theory: A Concise Introduction, Princeton Univ Press, December 2011.

Recommended:
T. Basar and P. Bernhard, H∞-Optimal Control and Related Minimax Design Problems, Birkhäuser, 1995.

TitleSectionCRNTypeHoursTimesDaysLocationInstructor
Optimum Control SystemsN33998DIS41400 - 1520 T R  4070 Electrical & Computer Eng Bldg Daniel M Liberzon