CDS 110b: Linear Quadratic Regulators

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CDS 110b

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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 "penalizes" state error?

According to the form of the quadratic cost function J, there are three quadratic terms such as x^TQ_xx, u^TQ_uu, and x(T)^TP₁x(T). When and if Q_x 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_x can keep x in small value regions. This is what the "penalizing" means.

So in the optimal control design, the relative values of Q_x, Q_u, and P₁ represent how important X, U, and X(T) are in the designer's concerns.

Zhipu Jin,13 Jan 03