ECE 553
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ECE 553: Optimum Control Systems (Spring 2005)

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*"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.

**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*.

Other references will be announced in class later. Many other reserve materials
for ECE 453/553 are available in Grainger.
**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