EEE 587
Optimal Control Theory
Spring 2014
TTh 12:00-1:15 SCOB 302

Instructor: Kostas Tsakalis, GWC 358, 965-1467, tsakalis
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Office Hours: see schedule or by appointment

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

Course Outline:

  1. Introduction, Review of state-space concepts, Linear Algebra review, Performance measures of control systems
  2. Numerical optimization fundamentals, (also see Optimization Notes handout ) Solution of two-point boundary value problems
  3. Calculus of variations, Pontryagin's minimum principle
  4. Applications: The linear quadratic regulator
  5. Applications: Minimum time problems, Minimum energy problems
  6. Dynamic Programming, Hamilton-Jacobi-Bellman Equation
  7. 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


More Optimal Control Problems