Optimal Control of Mechanical Oscillations by Prof. Dr. Agnessa Kovaleva (auth.)

By Prof. Dr. Agnessa Kovaleva (auth.)

This e-book bargains with elements of the optimum regulate of oscillatory platforms: the initiation of optimum oscillatory regimes and chances of the regulate for random disturbances. the most content material of the ebook relies on a few assertions of the optimum regulate thought nad the disturbance conception. Informations invaluable for the knowledge are given within the appendix. Al the theoretical propositions are illustrated through examples with designated mechanical context. because the publication as a rule comprises mathematical foundations, purposes aren't constrained to the given examples.

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Optimal Control of Mechanical Oscillations

This ebook offers with elements of the optimum keep an eye on of oscillatory structures: the initiation of optimum oscillatory regimes and probabilities of the keep an eye on for random disturbances. the most content material of the ebook relies on a few assertions of the optimum keep an eye on conception nad the disturbance thought. Informations useful for the certainty are given within the appendix.

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In the same way, during the stop at the left x-(r-/2)=-A, the intermediate elements are transferred from the state y- (r- /2) into the state y+ (0) ;t: y- (r- /2). Thus, at the start of each half-interval limiter all the points of the system will be in the state, corresponding to the initial conditions of the asymmetric periodic movement to be found. Such an approach is also possible in the case when the actuator makes not simply one but several stops with change of dynamic characteristics of the system during the cycle.

And of multiplier If x(t) E R" p(t). is ann-dimensional vector satisfying the equation 28 1 Optimal Periodic Control. , then obtaining the scalar product of left and right parts of Eq. 63) in the form g0 = [ {P'(t)x(t)- [ P'(s)K(s-t)f(s,t,x(t~u(t~a)ds}dt. 73) hold; but Eq. 72) reduces to the system of n equations. If the constraint equations contain the unknown parameter a which is determined by the optimality conditions, then the variation of the functional (I. 76) where & is an admissible variation of the parameter a.

161) 44 1 Optimal Periodic Control. The Integral Equations Method Eq. 163) Let us analyze an example of control for the simplest positioning system (Fig. 8). The control excitation u2 (t), 2 U , is applied to the element 2 which is lu 15 connected by the elastic link with the actuator 1. 1. 8 Here m~> m2 are masses of elements, c~> c2 are stiffness of elastic links, x~> x 2 are coordinates of mass centers of the elements with respect to the equilibrium state. 5 Problems of Optimal Displacement for Linear Systems and leaving the limiters: i- 1 = 0 for x 1 =±6..

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