EEE 587

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”.

(SOLUTIONS)

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.

(SOLUTIONS)

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