Solving the Terminal Control Problem for a Linear Plant under Interval Uncertainty

Abstract

An approach to solving an optimal control problem subject to a quadratic criterion under the assumption that the initial data are known imperfectly is considered. The uncertainty in the initial data is described by an interval model based on the assumption that the values of certain parameters of the problem truly belong to intervals with the known lower and upper boundaries. Noteworthy is that no probability measure is defined on the interval as a probability density function (as in a stochastic model) or a membership function (as in a fuzzy model). The problem is solved by means of interval analysis techniques. It is shown that an attempt to solve the optimal control problem in its classical formulation without taking into account the uncertainty in its parameters cannot provide a unique optimal control that would guarantee that the plant will be transferred into the required final state with any values of the parameters from the specified interval of their possible values. This means that an attempt to perform control of the system without taking into account the uncertainty may result in the system having been transferred into inadmissible state. Hence, if there is interval uncertainty in the parameters of the problem, one cannot speak about its solution in the sense in which it is understood with precisely known parameters. Therefore, the very approach to the control problem formulation should be revised with a purpose to determine a solution ensuring that the system will be transferred into the desired state with guaranteed accuracy. In this connection, the control problem involving interval uncertainty is proposed to be formulated as a problem of determining the set of control actions guaranteeing its solution with the accuracy specified to the interval in a set of parameters known to within an interval. It is shown using the example of a problem with an imperfectly known initial state that, if the set of possible initial states of a plant belongs to a set represented by an n-dimensional rectangular parallelepiped in the state space, then, when being implemented in a controlled plant calculated for any parameter value from a given set of possible initial states, the set of final states is a convex one and is represented by an n-dimensional parallelepiped. To construct this parallelepiped it is sufficient to determine only the coordinates of its vertices corresponding to the vertices of the n-dimensional rectangular parallelepiped determined above that defines the domain of possible values of the problem’s parameters determined to within an interval. A system of inequalities depending on the plant’s parameters, on the range of its initial states and on the duration of the control period is obtained that determines the condition under which the set of the plant final states belongs to the desired rectangular set. The conditions subject to which the question whether or not the problem is solvable can a priori be answered are formulated proceeding from this system of inequalities.