The BRADLEY DEPARTMENT of ELECTRICAL and COMPUTER ENGINEERING

# ECE 6744 Linear Control Theory | ECE | Virginia Tech

### Course Information

#### Description

Advanced introduction to the theory of optimal control of time-varying and time-invariant linear systems; Solutions to the linear-quadratic regulator, optimal filtering, and linear-quadratic-gaussian problems; Robustness analysis and techniques to enhance robustness of controllers.

#### Why take this course?

This course provides a rigorous introduction to linear optimal control systems. The emphasis is on being able to derive the fundamental properties of these systems as well as to provide an introduction to important theoretical tools such as dynamic programming and least-squares optimization in function spaces.

### Learning Objectives

• Derive the solutions to the Linear-Quadratic Regulator problem and the Linear-Quadratic-Gaussian problem.
• Use linear optimal control techniques to solve problems such as disturbance rejection and tracking.
• Test the robustness of linear control systems to unstructured uncertainty and implement methods to improve robustness.
• Derive the basic equations of the Kalman filter and describe the important stochastic properties of this filter.

### Course Topics

#### Percentage of Course

1. Introduction and Review 10%
2. Linear Quadratic Regulator (LQR) 25%
a. Derivation from Dynamic Programming Theory %
b. Derivation from Least Squares Theory %
c. Penalty Matrix Selection %
d. Application to Disturbance Rejection and Tracking %
3. Robustness 15%
a. Singular Values and the Multivariable Nyquist Test %
b. Gain and Phase Margin Properties of LQR %
c. General Uncertainty Bounds %
4. Kalman Filtering 10%
a. Stochastic Dynamical Systems %
b. Derivation as Linear, Minimum-Variance Estimator %
c. Properties %
5. Linear-Quadratic-Gaussian Control 25%
a. Stochastic Dynamic Programming %
b. Derivation of LQR with Additive or Multiplicative Noise %
c. Separation Principle %
d. Loss of Robustness and Loop-Transfer Recovery %
e. Approaches to Robustness with Structured Uncertainty %
6. Advanced Topics 15%