Method of Quasi-Optimal Synthesis Using Invariants

This paper studies the problem of synthesis of terminal control of a dynamic system. It is shown that to solve the problem the convolution product of objective functional and Gaussian constraint can be used as expanded functional. The use of needle variation allows to get a required condition of the objective functional minimum and to reduce the optimization problem to the boundary problem, which can be solved in closed form. The quasi-optimal solution to the problem of optimal speed of operation is obtained. The modes of operation with choosing the control law parameters were studied.


Introduction
The research results show [1]- [5] that the convolution product of the objective functional and the action integral gives the control structure accurate to a synthesis function. The function can be built using the stationarity conditions of the energy invariants which can be used to determine the switching surface. This allows to find that solutions to the extreme problem which satisfy to variational principle (the dynamics base) and provide the stability of the controlled motion in accordance with A.M. Lyapunov's theorem.
In this work it is shown that the application of combined-maximum principle methodology for finding a required condition for the expanded functional minimum as a convolution product of the objective criterion and the Gaussian constraint [4] provides a synthesis of quasioptimal controls. In contrast to known results this leads to a solvable boundary problem and doesn't require to build the synthesis function.
The aim of this investigation is synthesis of terminal control using criterion of operation speed using Gaussian constraint.

Definition of the synthesis problem
According to the Gauss principle at every time t the dynamic system moves in such a way that constraint [ where s m is the material point mass; s q -is the coordinate of the material point relative to a static Cartesian coordinate system; s Q is the resultant force applied to the material point; n is a number of degrees of freedom of the dynamic system. From (2) the Appell equations [5] follow in the form: where G is the Gibbs function, and the time derivative is denoted by two dots. Let the dynamics of the studied system satisfies (1) and is described by equations (3).
It is required to find the restricted possible forces  where a convex function ) (q F is of constant sign and is continuous along with its partial derivatives in the whole domain, and 1 0 ,t t is the time of start and finish of the control accordingly [2,6].

Method of synthesis using invariants
The searching for the required condition for the objective functional minimum (4) is performed by the Lagrange undetermined multipliers method. Let us consider the expanded functional [4] Let the arbitrary generalized force is determined by expression: is the given small finite time interval; 0 ! 't . Then the full variation of the functional has the following form: The relations on the trajectory ends are the transversality conditions: the varied force and the minimizing W W the following equation is true: > @ > @ > @, The developing of this equation should be performed separately for each specific case of synthesis problem.

The results of mathematical simulation
Let dynamic system (3) has the form: It is required to synthesize the law of optimal control of the dynamic system (19), transferring it from initial state to the phase space point 0 , 0 fulfilling the condition of minimum of the objective functional min; The estimation of efficiency of the suggested solution is performed on the basis of comparison with the quasioptimal law of "soft" terminal control [7,8]: The results of the mathematical simulation are shown in the Fig, 1, where the number 1 denotes the phase trajectory of the system (19) with the right part (22), and the number 2 denotes the phase trajectory of the system (19) with the right part (24). Fig, 2 presents the structure of the controlling generalized forces. There are the following notations: 1the control (22), 2the control (24). It can be seen that singularity (24) at the end time leads to a sharp increase of control force in opposite to (22).
The results of modeling are illustrated on Fig. 3. The solution based on the Pontryagin's maximum principle allow getting these results. This confirms the validity of the developed method.

Conclusion
The solution (22) unlike the known solution (24) [7] did not contain singularities at the finite time. As a result, the discontinuity of the control generalized force at the final time is absent. This allows to use it in practice without additional transformation.