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# 6.3 Notes and references for Chapter 6

The linear quadratic regulator problem is a classical subject covered in many textbooks. This chapter has particularly benefited from the expositions in [AF66,AM90,Bro70,KS72]; basic background facts from linear system theory can be found in these books as well. Kalman's original LQR paper [Kal60] is a must-read, as it also contains the first treatment of controllability for linear systems and a few other fundamental concepts and techniques. Among several possible ways of deriving the solution to the finite-horizon LQR problem, we favor the approach followed in [AF66] and [KS72] where the linear state feedback form of the optimal control is established independently of the Riccati differential equation (which is derived later). Section 24 of [Bro70] contains an insightful discussion of Riccati differential equations and their solutions. Various methods for numerical solution of the RDE are described in [AM90, Appendix E], [KS72, Section 3.5], and the references therein. For the infinite-horizon LQR problem, different sources again take different routes to arrive at the main results. In particular, [AM90] gives a purely linear-algebraic proof of closed-loop stability which does not rely on the Lyapunov argument that we used in Section 6.2.2.

Many generalizations of the LQR problem are not discussed here. These include: more general quadratic costs involving cross-terms; tracking problems; the output feedback version of the LQR problem and its relation to optimal state estimation (the Kalman filter). The references [AF66,AM90,KS72] cited earlier can be consulted for detailed information on these topics. In Section 7.3, we will discuss robust control problems which can be considered as extensions of the LQR problem.

Next: 7. Advanced Topics Up: 6. The Linear Quadratic Previous: 6.2.4 Complete result and   Contents   Index
Daniel 2010-12-20