Optimal Control by Anderson B.D.O., Moore J.B.

By Anderson B.D.O., Moore J.B.

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Main points of the section. Complete controllability or stabilizability of the state space equations ensure existence of optimal linear quadratic indices and associated Riccati equation solutions in the infinite-time case. 1-1. have the form Consider the time-invariant problem, and suppose F and G ‘=[~’ d ‘=[:’1 so that the pair [F, G] is not completely controllable and [ Fll, Gl] is completely controllable; see Appendix B. Suppose that the solution P(t) of the Riccati equation associated with the finite-time regulator problem is partitioned as 44 The Standard Regulator Problem—n Chap.

16 The Standard Regulator Problem—1 Chap. 2-6) is but a first-order partial differential equation with one dependent variable, V*, and two independent variables, x(t) and t, because f, ~, and D are known functions of their arguments. 2-6) is very simply derived. ), and, accordingly, the minimum value of this performance index with respect to u(“) is also m (x (T)). 2-7) may also be referred to as the Hamilton–Jacobi equation, and constitute a true partial differential equation. , “) does not exist—then the whole procedure is invalidated and the Hamilton–Jacobi approach cannot be used in tackling the optimization problem.

3. We shall establish existence of V*(X (t), t). 4. We shall find the optimal control. To carry out this program, it is necessary to make the following temporary assumption. 3-1 Assume that F(t), G(t), R(t), and Q(~) have entries that are continuously differentiable. This assumption is removed in Prob. 3-2. We note that some treatments of the regulator problem assume a priori the form x‘ (t) P(t)x (t) for the optimal performance index. It is therefore interesting to observe a simple derivation of this form, most of which appears in [12].

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