Interval methods in the problems of optimal control
Authors: Skibitskiy N. V. 
Published: May 2022

The problem of optimal control of a linear dynamic object in conditions of incomplete information about the initial data is considered. The approaches based on various models of the uncertainty description are analyzed. It is shown that the use of the approach based on a probabilistic model for describing uncertainty is advisable when the uncertainty is associated only with randomness, the description of other sources of uncertainty within this model is rather difficult and the formal application of the regression analysis provides the results that are far from true. Though the fuzzy model is suitable for describing a wide range of the uncertainty sources, application of the model faces methodological difficulties when comparing and ranking fuzzy numbers and smoothing fuzzy data. In this regard, it seems promising to use an approach based on an interval model which allows description of a wide class of uncertain and inaccurate initial data. To unify control algorithms for the systems described by equations of state of different types with interval-given parameters, we developed an algorithm of equivalent transformations that provides the transition to special forms of representing the state matrix while maintaining preservation of all the dynamic properties of the original system. The problems of constructing the range of values of the roots of the matrix of the system and its description by an approximation in the form of an interval vector are solved to ensure the implementation of the algorithm. An approach is proposed to solve the problems of terminal control and maximum performance, when the uncertainty of the initial data is described by the interval model and the apparatus of interval analysis is used to solve the problem. It is shown that in this case with the direct use of the classical formulation of the optimal control problem without taking the uncertainty into account, there is no single optimal control which guarantees the exact transfer of the object to the required final state for any value of the parameters from a given range of their possible values. Therefore, in the presence of the interval uncertainty of the initial data, the solution of the problem can’t be obtained in the sense in which it is understood with precisely known parameters and the approach to the formulation of the control problem itself should be revised in order to determine in the future a solution that ensures guaranteed accuracy of the system translation. In this regard, we propose to formulate the control problem in conditions of the interval uncertainty as a problem of determining the set of control actions that guarantee the solution with the accuracy set up to the interval, on the set of initial data known up to the interval. Using an example of the problem with an inaccurately known initial state, it is shown that if the set of possible initial states of an object belongs to a n-dimensional rectangular parallelepiped, then when implementing a control on an object calculated for any initial state from a given set, the set of final states is convex and represents a n-dimensional parallelepiped. To construct the parallelepiped, it is sufficient to determine the coordinates of the vertices corresponding to the vertices of the abovementioned n-dimensional rectangular parallelepiped. We propose a system of inequalities which determines the condition for the membership of a set of finite states of the object obtained upon implementation of control for any possible value of the initial state from a given set to the required set of finite states. On the basis of this system of inequalities, conditions are formulated that provide a priori determination of the solvability of the problem.