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Niejednoznaczne rozwiązania czasooptymalne w pewnym zamkniętym układzie sterowania

Hejmo W., Majewska K.

Czasopismo Techniczne. Elektrotechnika

2000

R. 97, z. 4-E

1--11

The purpose of this paper is to present the solutions of a time-optimal control problem of a position mechanism, in a case when the motion resistance function depends on a position of this mechanism. The dynamics of the controlled object is described by a planar, non-linear and discontinuous differential equation: x=f(x)+u, where |u| is less than or equal to 1, motion resistance function f(x) = 0 if x is less than or equal to 0, f(x) = -A if x is greater than 0 and 0 is less than or equal to A is less than 1. In a case of such defined motion resistance function the two following singular phenomena appears: 1) if the target z[1] = (0,0) and A is greater than A[b], A[b] = 2-2^(1/2) then the switching curve is composed of two branches, but only one of them is formed by the solution of the time-optimal problem. Thus, the closed-loop system executes none, one or two switching operations and any small change in the value of the resistance function requires to change the closed-loop system structure. 2) if the target z[1] = (x[1], 0), x[1] is greater than 0 then there exists the set of states from which two different trajectories reaching the target in the same minimum time start. The switching curve is composed of three branches. One of the branches is induced by a singular set of states and is formed by none of the solutions of time-optimal problem. The paper presents the sets of non-unique states for different values of the motion resistance function and the target z[1] = (1,0) in the graphical form. Finally, some suggestions as to practical application are given.