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lqnotesv6 - Bassam Bamieh Notes on Optimal Control and...

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Bassam Bamieh Notes on Optimal Control and Linear Quadratic Problems (V. 1/22/03) c ° Bassam Bamieh, 2001

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Preface The main purpose of these notes is to present a treatment of Linear Quadratic (LQ) problems with possibly indeFnite quadratic functionals. The necessary conditions are stated as stationarity conditions for a quadratic objective sub- ject to linear dynamical constraints. With this setup, it is possible to de- rive systematically conditions for a variety of problems such as the Linear Quadratic Regulator, H norm computation and state feedback synthesis, the Bounded Real Lemma, and the Positive Real Lemma. The key point is that for any of these problems, the correct Riccati equation can be simply derived from the necessary conditions without guessing. Suﬃciency can then be veriFed by completing-the-squares arguments. Perhaps the distinction between this treatment and others of the same subject is that a Riccati equation is used as a necessary condition for a family of LQ problems, and as a suﬃcient condition for a single LQ problem. This is accomplished by a careful treatment of the Two Point Boundary Value Prob- lems (TPBVPs) resulting from application of the minimum principle to LQ problems. The connection between linear TPBVPs and Riccati equations is illustrated. We also show how this is a special case of the connection between more general TPBVPs and a certain partial di±erential equation through the procedure of invariant imbedding. These notes begin with some tutorial material on di±erentiation in R n and Hilbert spaces, the calculus of variations, Lagrange multipliers, and the minimum principle of optimal control. Readers who are familiar with this material may skip directly to the chapters on Invariant Imbedding (4) and LQ problems (5).

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Table of Contents 1 Diferentiability and the calculus oF variations ............ 1 1.1 Diferentiation in R n .................................... 1 1.2 Calculus oF variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Minima, maxima and saddle points . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Exercises. ............................................. 15 2 Optimization and constrained stationarity ................ 17 2.1 Constrained stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Global minimizers and convexity . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 Exercises. 26 3 Optimal control and the Minimum Principle .............. 29 4 Invariant Imbedding ...................................... 35 4.1 Linear Two Point Boundary Value Problems and Riccati Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5 Linear Quadratic Problems ............................... 43 5.1 Stationarity oF quadratic Forms subject to linear constraints . . 44 5.2 Some properties oF the diferential Riccati equation . . . . . . . . . 47 5.3 The Linear Quadratic Regulator (LQR) . . . . . . . . . . . . . . . . . . . 50 5.4 The bounded real lemma and H norm computation . . . . . . . 54 5.5 H state Feedback design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.6 The positive real lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

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1 Diferentiability and the calculus oF variations We begin with some preliminaries on diferentiation oF Functions in R n . Read- ers Familiar with this material should still skim over this section as the con- cepts are presented in a manner that is that immediately generalizes to the in±nite dimensional case and the calculus oF variations.

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lqnotesv6 - Bassam Bamieh Notes on Optimal Control and...

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