Asymptotically stable hybrid observer in the problem of multiprogram stabilization

For the first time, the problem of mulriprogram stabilization was formulated by V. I. Zubov in 1991. Representation of rights parts of systems of differential equations with a given finite family of solutions is offered, and also the problem of control synthesis that realizes the group of program motions and provides its asymptotical stability is considered. Multiprogram control is constructed as the Hermite interpolation polynomial whose nodes are program motions and whose values are corresponding program controls. According to the approach, multiprogram stabilizing controls with incomplete feedback are constructed. Different types of continuous observers of the system state that is closed with multiprogram control are offered. The realization of this approach requires the construction of the so-called hybrid state observers. In the present paper, the method of construction of hybrid multiprogram control with incomplete feedback is given. The theorem of sufficient conditions of an asymptotically stable hybrid observer is proved. The proof is constructive. It is base on the Lyapunov method and includes an algorithm of the mentioned observer construction.