By Anderson B.D.O., Moore J.B.
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Extra info for Optimal Control
3-12) define P(t) for all t s T, and therefore the index V*(X (t), t)= x ‘(t) P(t)x (t) exists for all ts T. In the course of deriving the Riccati equation, The optimal control. 3-9)in Eq. 3-10). 3-10), P(t) is unknown. The product 2P (t)x (t)represents dV */ax, and z(r) is to be regarded as being defined by independent variables x (t)(actually absent from the functional form for z), dV*/dx, and t. 3-1 1). 3-13). Notice, too, that Eq. 3-13) is a linear feedback law, as promised. 3-1, and, accordingly, we may summarize the results as follows.
It might be thought curious that R(t) is not positive definite, since in the corresponding continuous time performance index, the corresponding matrix is positive definite. In the latter case, the presence of the matrix rules out the possibility of using an infinitely large control to take the state to zero in an infinitely short time. In the discrete time case, it is not possible to take the state to zero in an infinitely short time, and the possibility of an infinitely large control occurring does not even arise.
This new function ii (“, “) has two important properties. 2-9) V(X(t), u(”), t) = ~~l(x(~), u(7), T) d~ + m(x(T)) 1 That is, to achieve the optimal performance index V* (x (t), t), the optimal control to implement at time t is d (x (t), t). ) over [to, t). To some, this result will be intuitively clear, being in fact one restatement of the Principle of Optimality, to the effect that a control policy, optimal over an interval [to, T], is Optimal over all subintervals [t, T]. 2-4). ). ) regarded as the sequential use of u~O,,,and u~,T], and with the assumption that u~O,,)is applied until time t, evidently Sec.