CDS 110b: Linear Quadratic Regulators

WARNING: This page is for a previous year. See current course homepage to find most recent page available.

Course Home

L7-2: Sensitivity

L8-1: Robust Stability

L9-1: Robust Perf

Schedule

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 Outline

1. Derivation of the LQR regulator
2. Choosing LQR weights
3. Incorporating a reference trajectory
4. Integral feedback
5. Design example

Lecture Materials

• Lecture Presentation (MP3)
• Lecture notes on LQR control
• Homework 2

References and Further Reading

• Friedland, Ch 9 - this is the assigned reading for this lecture. The derivation of the LQR controller is done differently, so it gives an alternate approach.
• 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.

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 $J$, there are three quadratic terms such as $x^{T}Qx$, $u^{T}Ru$, and $x(T)^{T}P_{1}x(T)$. When $Q\geq 0$ 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 $P_{1}$ represent how important $X$, $U$, and $X(T)$ are in the designer's concerns.

Zhipu Jin,13 Jan 03

Q: What do you mean by the terms damping ratio and natural frequency in 1c and 1d on HW #2?

The position control system is a second-order system. So for any control law, there would be some natural frequency and damping ratio for the closed loop system (Recall from CDS110a that these terms are defined for any second order system; for a revision of these concepts, you can take a look at ocw.mit.edu/NR/rdonlyres/Mathematics/18-03Spring2004/ B76E6F4F-7B05-4DA0-A5A5-03FA4ACCB6B2/0/sup_13.pdf) All you have to do in 1c and 1d is to find out the value of these terms for the control law from 1b.

Vijay Gupta, 17 Jan 05

Q: How are closed loop and open loop defined (for a state space system)

The basic definitions for open and closed loop systems are given in AM05 (Chapter 1). For state space systems, we follow the same basic idea: a closed loop system is one in which the output of the system affects its input (through the "controller"). Thus if we have a state feedback compensator $u=Kx$, then the dynamics of the closed loop system are given by

${\dot {x}}=(A-BK)x$

The open loop dynamics are given by

${\dot {x}}=Ax$

The eigenvalues of the closed and open loop systems are given by the eigenvalues of the respective system matrices.