# CDS 110b: Linear Quadratic Regulators

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.

• 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.

Q: What do you mean by penalizing something, from $Q_{x}\geq 0$ "penalizes" state error?
According to the form of the quadratic cost function $J$ , there are three quadratic terms such as $x^{T}Q_{x}x$ , $u^{T}Q_{u}u$ , and $x(T)^{T}P_{1}x(T)$ . When $Q_{x}\geq 0$ 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 $P_{1}$ represent how important $X$ , $U$ , and $X(T)$ are in the designer's concerns.