Wyniki wyszukiwania - BazTech

1

On a neural control close to time-optimal of a non-linear and discontinuous object

Hejmo W., Majewska K.

Systems Science

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2002

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Vol. 28, nr 2

51-58

EN

The paper deals with the time-optimal problem of the controlled object the dynamics of which is given by the mapping: x = y, y = f (x) + u, |u| [left angle bracket]or= 1, with selected discontinuous function of motion resistance f. For any target state the time optimal trajectory may be formed by the solution induced by the "bang-bang" control function with one, two or three switching operations. The switching curve, generally discontinuous, cannot be defined in open, algebraic form. The unique one way of its shape formation is the numerical approach. Moreover, in the time-optimal process the phenomenon of nonunique solution may appear, the trajectories of which reach the target state along a totally different way in the same minimum time. We are able to determine the value of the time optimal control function belonging to each state in the state plane, thus, feedback system may be theoretically constructed. Unfortunately, because of non-algebraic formulas determining the states in which the switching operations should be executed and, moreover, because a singular phenomenon of non-unique time optimal solutions may appear, the standard concept of regular closed-loop system synthesis becomes inappropriate. In order to synthesize the engineering closed-loop control structure, there is suggested an idea of the controller of the three-layer feed-forward back propagation network. The construction and the training of the neural network have been used with Levenberg-Marquardt method. A comparison between theoretical minimum time of target reaching and time of target reaching in the system created in accordance with the neural system proposed is made.

2

Niejednoznaczne rozwiązania czasooptymalne w pewnym zamkniętym układzie sterowania

Hejmo W., Majewska K.

Czasopismo Techniczne. Elektrotechnika

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2000

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R. 97, z. 4-E

1--11

EN

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.

3

On non-unique time-optimal control of a planar discontinuous dynamic object.

Hejmo W., Majewska K.

Systems Science

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1999

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Vol. 25, nr 3

37-55

EN

The purpose of this paper is to present the solution of time-optimal problem of the controlled object, the dynamics of which is given by x=y, y=f(x)+u, where /u/<1, and motion resistance function f(x)=0 if x<0, f(x)=-A if x>0, 0,A<1. That model describes dynamics of of industrial devices called position mechanisms. It has been shown that in the formula defining resistance functionf(x) there exists a value Ab that plays an essential role in time-optimal structure formation. Namely, if AAb we will examine the following two singular phenomena. The first phenomenon appears if the target state z1=(0,0). Then, there exists the switching curve, dividing the state plane into two sets, however, only one in branch is formed by the time-optimal solution. None of the solutions form the second branch of the switching curve. Thus, the time-optimal process is generated by bang-bang control with none, one or two switching operations. The second singular phenomenon appears if the target state z1=(x1,0), x1>0. Then, there exists a set of starting states from which there start two trajectories reaching he target in the same minimal time. It appeared that only two starting points from which there start the non-unique trajectories, may be defined in algebraic open form. The next starting points may be calculated in numerical way only. There have been shown some examples of numerical solutions. Finally, several suggestions as to practical applications have been given, too.