**EEE 587**

**Optimal
Control Theory**

**Spring
2014**

TTh 12:00-1:15 SCOB 302

**Instructor: Kostas Tsakalis, GWC
358, 965-1467, **

This page: http://www.tsakalis.faculty.asu.edu

Office Hours: see
schedule or by appointment

**Textbook:** D. Subbaram Naidu, *Optimal
Control Systems.* CRC Press, ISBN: 0849308925

**Course Outline:**

- Introduction, Review of
state-space concepts, Linear Algebra review, Performance measures of
control systems
- Numerical optimization
fundamentals, (also see Optimization Notes
handout ) Solution of two-point boundary value problems
- Calculus of variations, Pontryagin's minimum principle
- Applications: The linear
quadratic regulator
- Applications: Minimum time
problems, Minimum energy problems
- Dynamic Programming,
Hamilton-Jacobi-Bellman Equation
- Applications to robust control

**Grading:** HW 30%,
Midterms 40% (30% best + 10% worst), Project and/or Final 40% (30% best + 10%
worst, max 100%).

**Important Dates:** **FINAL EXAM Tue May
6, 12:10-2:00**

**HW Assignments**

**HW 1, due 2/13
Problem I. **Find a minimal state space realization for the 2x2 transfer
function matrix {G_ij(s)} where the matrix entries
are given as follows:

G_11 = 1/(s+1), G_12 = (s+3)/(s+1)(s+2), G_21 = 1/(s+1)(s+2), G_22 = (s+5)/(s+1)(s+2)

For the resulting realization, find the Lyapunov
function V = x'Px whose derivative along the
trajectories of the unforced system is -x'x.

Use Simulink to generate a trajectory of the unforced
system for four different initial conditions, evaluate the Lyapunov
function V along each trajectory and plot V(t) as a
function of t. Verify that V is monotonically decreasing.

Hint: 1. Realize each term separately, concatenate,
and apply the Kalman Canonical Decomposition,
Corollary 4.5.5 of the notes, or “mineral” in MATLAB.

2. Solve the Lyapunov equation A'P+PA = -I using the
MATLAB function “lyap”.

**Problem II.** Derive the state equations
for an inverted pendulum and an inverted pendulum on a cart (see linked Modeling Notes, Ch. 0.5, 0.6).

Derive the linearized model. Formulate a performance
index to drive the pendulum to the vertical position with as little energy as
possible.

HW 2, due 2/27

Problems 2.1, 2.6, 2.7, 2.11 from the textbook (SOLUTIONS)

**Problems 1.7, 2.1, 2.4, 3.1 from
the linked collection of problems**

Test 1, Thu
3/6 SOLUTIONS

Material:
Ch. 1,2

**HW 3, due 4/3
Problems 3.8, 3.9, 4.2, 4.3 from the textbook (SOLUTIONS)**

**Problems 3.2, 3.4 from the linked collection of problems**

**Application Note: Observer Design for
Feedback Control with Integrator Augmentation.**

Test 2, Thu
4/17, SOLUTIONS

Material:
Ch. 3,4

**HW 4, due 5/1
Problems 6.3, 7.3, 7.6, 7.8, 7.9, 7.11 from the textbook (SOLUTIONS)**

Project, Due
5/6

Provide a
solution to the computational problems 5.1, 5.2 from the linked Collection of
problems

**====================================
**

**Collection of
Optimal Control Problems**