CDS 110b: Linear Quadratic Regulators

CDS 110b

Schedule

Project

Course Text

This lecture provides a brief derivation of the linear quadratic regulator (LQR) and describes how to design an LQR-based compensator. The use of integral feedback to eliminate steady state error is also described.

• Lecture Presentation
• Homework 3 - due 30 Jan 08
• pvtol_lqr.m - MATLAB script demonstrating LQR design

References and Further Reading

• R. M. Murray, Optimization-Based Control. Preprint, 2008: Chapter 2 - Optimal Control
• Lewis and Syrmos, Section 3.4 - this follows the derivation in the notes above. I am not putting in a scan of this chapter since the course text is available, but you are free to have a look via Google Books.
• Friedland, Ch 9 - the derivation of the LQR controller is done differently, so it gives an alternate approach.

Frequently Asked Questions

Q: What do you mean by penalizing something, from Q>=0 "penalizes" state error?

According to the form of the quadratic cost function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J} , there are three quadratic terms such as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^T Q x} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u^T R u} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(T)^T P_1 x(T)} . When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q \geq 0} and if $Q$ is relative big, the value of $x$ will have bigger contribution to the value of $J$. In order to keep $J$ small, $x$ must be relatively small. So selecting a big $Q$ can keep $x$ in small value regions. This is what the "penalizing" means.

So in the optimal control design, the relative values of $Q$, $R$, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_1} represent how important Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X(T)} are in the designer's concerns.

Zhipu Jin,13 Jan 03