Stabilization of dynamic systems providing additional properties of transients
The dynamical system considered in the paper is described by the equation \[ \dot x=Ax+bu\quad (x,b\in\mathbb R^n,\;u\in\mathbb R,\;A\in\mathbb R^{n\times n}, \text{ rank}(b,Ab,\dots ,A^{n-1}b)=n), \] where \(x=x(t)\) is the state vector and \(u=u(t)\) is the scalar control. The problem is to construct bounded stabilization feedbacks that provide additional properties of transient processes: degree of stability, oscillation, absence of overshoot, and monotonicity. The problem is solved by means of optimal control methods. The results are illustrated by an example of stabilization of a dynamic fourth-order system. The study is related to the following investigations: \textit{R. Gabasov}, \textit{F. M. Kirillova} and \textit{O. I. Kostyukova}, Dokl. Akad. Nauk SSSR 320, No. 6, 1298--1299 (1991; Zbl 0769.49028); \textit{R. Gabasov}, \textit{F. M. Kirillova} and \textit{O. I. Kostyukova}, Izv. Ross. Akad. Nauk, Tekh. Kibern. 1992, No. 4, 3--19 (1992; Zbl 0808.93027); \textit{N. V. Balashevich}, \textit{R. Gabasov} and \textit{F. M. Kirillova}, Kibern. Sist. Anal. 1994, No. 1, 66--73 (1994; Zbl 0820.93026); \textit{N. V. Balashevich}, \textit{R. Gabasov} and \textit{F. M. Kirillova}, Avtom. Telemekh. 1994, No. 2, 23--31 (1994; Zbl 0866.49035), where methods for the realization of a feedback-type optimal control and their application to the stabilization problem are described.
scientific article; zbMATH DE number 1978429