## ECE 553: Optimum Control Systems (Spring 2007)

"Since the building of the universe is perfect and is created by the wisdom creator, nothing arises in the universe in which one cannot see the sense of some maximum or minimum." (Leonard Euler)

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.

### LECTURE NOTES

The most recent lecture is posted soon after it is given in class. Previous lectures are also being modified even after their initial posting.
Note: if you notice errors, please communicate them to me (by email or in person) so I can correct them.

### FINAL EXAM QUESTIONS

Schedule: Mon Wed 12:30-1:50pm, room 1109, Siebel Center.

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: Tue 4-5:30pm

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.
Other references will be announced in class later and made available on the web whenever possible. For other reserve materials for ECE 553 in Grainger library, click here.

Assignments and grading policy: Will be explained in class.

Brief course outline:

1. Introduction (1.5 weeks)
The goals of the course; path optimization vs. point optimization; basic facts from finite-dimensional optimization.

2. Calculus of variations (3 weeks)
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 (5 weeks)
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 (2.5 weeks)
Sufficient conditions for optimality; viscocity solutions of the HJB equation.

5. LQR problems (1 week)

6. Other topics (1.5 weeks)