CDS 110b: Linear Quadratic Regulators

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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.

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 \(Q_x \geq 0\) "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.

Zhipu Jin,13 Jan 03