This is a graduate-level course on optimal control systems. It presents a rigorous introduction to the theory of calculus of variations, the minimum principle, and the HJB equation. The course deals mainly with general nonlinear systems, but the linear theory will be examined in detail towards the end.
Schedule: Mon Wed 12:30-13:50, Room 106B3, Engineering Hall.
Prerequisites: ECE 515 (Linear Systems) or equivalent is required. Previous or concurrent enrollment in ECE 528 (Nonlinear Systems) is desirable. Proficiency in mathematical analysis, at the level of Math 447 or a course where analysis is used (such as ECE 490 or 528), is also essential.
Instructor: Daniel Liberzon
Office: 144 CSL
Email: liberzon at uiuc.edu
Office hours:
Texts: (on reserve in Grainger library)
- Calculus of variations: Gel'fand and Fomin, Calculus of Variations. $8 on Amazon.com!
- Minimum principle: Athans and Falb, Optimal Control. Also has useful background material.
- LQR theory: Anderson and Moore, Optimal Control: Linear Quadratic Methods.
Assignments and grading policy: Will be explained in class.
Brief course outline:
1. Introduction
The goals of the course; path optimization vs. point optimization; basic facts
from finite-dimensional optimization.
2. Calculus of variations
Examples of variational problems; Euler-Lagrange equation; Hamiltonial formalism
and Legendre transformation; mechanical interpretation; constraints; second
variation and Legendre's necessary condition; weak and strong extrema; conjugate
points and sufficient
conditions.
3. The minimum principle
Statement of the optimal control problem; variational argument and preview of
the minimum principle; statement and proof of the minimum principle; relation to
Lie brackets; bang-bang and singular optimal controls; Hamiltonian systems
and the minimum principle on
manifolds.
4. Hamilton-Jacobi-Bellman equation
Sufficient conditions for optimality; viscocity solutions of the HJB equation.
5. LQR problems.
6. Other topics