By Anderson B.D.O., Moore J.B.
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Extra resources for Optimal Control
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 .