Terminal Stabilization for Dynamic Systems with Output Variable Constraints

Abstract

This paper presents a novel approach to finite-time stabilization of dynamic systems, differing from classical nonsmooth solutions. The problem of finite-time control for linear dynamic plants is addressed, including the case with external disturbances, under strictly predefined constraints on the output signal trajectories starting from the initial time instant. The solution is presented in an order of increasing complexity, beginning with the scalar unperturbed case and proceeding to the general case of an arbitrary-order linear system with unknown bounded external disturbances. The imposed constraints can be motivated either by technical requirements on the plant behavior or empirically, according to preferred transient performance specifications. It is shown that in the controller synthesis procedure, one can define the form of output constraints so that the finite-time stabilization condition for the controlled variable is satisfied using bounded control input. The plant state is assumed to be known, with initial conditions either exactly known or belonging to a known bounded set; controllability and observability conditions are fulfilled. The proposed method is based on the idea of applying a functional transformation to the output signal, which allows reformulating the original constrained problem into an unconstrained stability problem with respect to a new variable. It is proven that such a transformation exists and that its inverse ensures the solution of the original problem while maintaining bounded control signals. The resulting control algorithm is compared with known results in finite-time control via computer simulations. It is demonstrated that the proposed method provides comparable regulation performance in terms of control effort, while offering greater flexibility in choosing closed-loop system trajectories.