Realization of the paradigm of prescribed control of a nonlinear object as the problem on maximization of adequacy - Trunov, Alexander (Eastern-European Journal of Enterprise Technologies, 2016)
Here we propose a new realization of the paradigm of prescribed control on the example of a vector model in the form of nonlinear non-stationary system of differential equations of the first order, for which a control system is synthesized. The estimation of efficiency is introduced as a continuous, three times differentiated, dimensionless function. Dimensionless objective function and the function of Lagrange with the inequalities constraints is formed. The problem of minimization of square deviation is posed for them as the problem of maximization of one of the criteria of adequacy – accuracy. It is written in a formalized way in the form of a system of nonlinear algebraic equations. For construction of its solution, a recurrent approximation of objective function is used, which made it possible to complement the system with new equations and to find, in a general form, expressions of auxiliary vectors independent on the number of components of the vectors of strategies and the Lagrange multipliers. Two additional equations are brought out, which realize maximization of adequacy by two additional criteria: depth and completeness. All this taken together made it possible to complement the system with new nonlinear equations, which, in turn, increased substantially the number of permissible inequalities constraints. This approach considers the properties of a non-stationary nonlinear model, which describes the object independent of the selection of the purpose of control. As a result of the solution of the problem, it was possible to connect the idea about the adequate, prescribed behavior of model and the properties of a nonlinear object, to obtain a differential equation that describes optimal controlling action, which, in this case, maximizes adequacy. It was demonstrated that in a particular case, under condition of independence of the rate of change in controlling influence on its magnitude, the obtained results coincide with expressions of the method of speed gradient. The estimations of the norm of error in the vector of strategies are presented depending on the properties of object, error in the function of efficiency and synthesized law of controlling action for the prescribed law of the object’s functioning. The set and solved problem demonstrated possibilities to set, with the aid of cybernetic methods, a new type of problems on adequate control. The latter is especially relevant in connection with application of integrated computer technologies. The obtained solutions without artificial assumptions make it possible to synthesize controlling action for the conditions that change over time both in the properties of object and in constraints.