On the output stabilization problem: constructing a delay feedback for a chain of integrators

A feedback structure stabilizing an $ n $-fold integrator with even $ n $ is proposed. It is well known that for many differential systems, in particular, for models of mechanical systems, stabilization problems can be transformed into a special form containing a chain of integrators as a subsystem. The control constructed for an arbitrary even order of the integrator is a linear combination of coordinates of the delay state with odd indices and depends on three numerical parameters. These parameters satisfy constraints of a simple form and can vary widely depending on the control performance requirements. Examples of more general system structures are given for which the control constructed ensures the asymptotic stability of the equilibrium. The stability properties of cascade delay systems are used to prove the stabilizing properties of the control.