Download e-book for kindle: Optimal Control with Engineering Applications by Hans P. Geering Ph.D. (auth.)

By Hans P. Geering Ph.D. (auth.)

ISBN-10: 3540694374

ISBN-13: 9783540694373

ISBN-10: 3540694382

ISBN-13: 9783540694380

Because the theoretical a part of the publication relies at the calculus of diversifications, the exposition is especially obvious and calls for often a trivial mathematical history. in terms of open-loop optimum regulate, this ends up in Pontryagin’s minimal precept and, in terms of closed-loop optimum keep an eye on, to the Hamilton-Jacobi-Bellman thought which exploits the main of optimality.

Many optimum regulate difficulties are solved thoroughly within the physique of the textual content. in addition, the entire workout difficulties which seem on the ends of the chapters are sketched within the appendix.

The publication additionally covers a few fabric that isn't frequently present in optimum keep an eye on textual content books, particularly, optimum keep an eye on issues of non-scalar-valued functionality standards (with purposes to optimum filtering) and Lukes’ approach to approximatively-optimal keep an eye on design.

Furthermore, a quick creation to differential online game idea is given. This ends up in the Nash-Pontryagin Minimax precept and to the Hamilton-Jacobi-Nash thought. the cause of together with this subject lies within the vital connection among the differential video game concept and the H-control thought for the layout of sturdy controllers.

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Extra resources for Optimal Control with Engineering Applications

Example text

According to the philosophy of the Lagrange multiplier method, this inequality must hold for arbitrary combinations of the mutually independent variations δtb , and δx(t), δu(t), δλ(t) at any time t ∈ (ta , tb ), and δλa , δx(ta ), and δx(tb ). Therefore, this inequality must be satisfied for a few very specially chosen combinations of these variations as well, namely where only one single variation is nontrivial and all of the others vanish. The consequence is that all of the factors multiplying a differential must vanish.

Since we are working our way backwards in time, we can write: λo2 (t) = λo2 (11) + λo1 (11 − t) , where, of course, the values of λo1 and λo2 (11) are unknown yet. ) to be conditions: ⎧ o λ2 (11) < 0 ⎪ ⎪ ⎪ o o ⎪ ⎪ ⎨ λ2 (11) + λ1 ≤ 0 λo2 (t) = λo2 (11) + λo1 − 2µo0 ⎪ ⎪ ⎪ λo (11) + 10λo1 − 2µo0 ≥ 0 ⎪ ⎪ ⎩ 2o λ2 (11) + 11λo1 − 2µo0 > 0 optimal we need the following for t = 11 for t = 10+ for t = 10− , µoo ≥ 0 for t = 1 for t = 0 . Thus, at the moment, we have three unknowns: λo1 , λo2 (11), and µo0 .

In this case, the optimal control law is: uo (λo2 (t)) = −amax sign{λo2 (t)} . • In all other cases, we have λ00 = 1. • If tb > tb,min , there are cases where λo2 (t) ≡ −1 or λo2 (t) ≡ +1 is necessary. Then, there are infinitely many optimal solutions. They are characterized by the fact that only acceleration or deceleration occurs in order to arrive at the specified terminal state in time. 34 2 Optimal Control • In the remaining (more general) cases with tb > tb,min , we obtain the following optimal control law with co1 = 0: ⎧ ⎪ ⎨ +amax uo (λo2 (t)) = 0 ⎪ ⎩ −amax for λo2 (t) < −1 for λo2 (t) ∈ [−1, 1] for λo2 (t) > +1 .

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Optimal Control with Engineering Applications by Hans P. Geering Ph.D. (auth.)


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