ON OBSERVABILITY CONTROL FOR DIFFERENTIAL EQUATIONS

Abstract

We consider a controlled linear differential equation with constraints as in the author's previous paper. The controller's goal is to displace an initial state of \(x_0\) to a specified final state \(x_T\). An observer, unaware of the system's state vector, attempts to determine \(x_T\) by analyzing the vector \(y(t)\), which is linked to \(x(t)\). Using \(y(t)\), the observer constructs a set of possible values for \(x_T\). When specific constraints are used for the controls (or disturbances, from the observer's opinion), this set becomes an ellipsoid, characterized by a set of differential equations. The controller, in turn, aims to achieve its own objectives while simultaneously generating the most challenging signals for the observer. Unlike the previous article of the author not scalar, but two-criterion control observation problem is considered here. It is solved in functional spaces in two ways, without passing to sampling of a system. The solution boils down to determination of finite-dimensional parameters of optimal control from the system of linear algebraic equations. As the third option the problem can be solved also by sampling, but then the solution turns out piecewise-constant. We explore an example to illustrate these concepts.