Stabilization of chain of integrators with arbitrary order in finite-time

Abstract

A control algorithm for finite-time stabilization of a chain of integrators with arbitrary order is introduced. The method is based on Implicit Lyapunov Function (ILF) approach with applying properties of homogeneous systems. Scheme of control parameter tuning is presented in Linear Matrix Inequality (LMI) form. The method is simple in implementation and does not assume any additional computational on-line procedures that is an improvement with respect to [8], [11]. The theoretical results are supported by numerical simulations.

On finite‐time robust stabilization via nonlinear state feedback

“…, and parameter c u should satisfy inequality (15). Finally, the settling-time function can be estimated as follows:…”

“…This paper addresses the problem of a control design for the finite‐time stabilization of a chain of integrators. This paper represents an extension and simplification of the results in the works of Polyakov et al and Zimenko et al The developed finite‐time control law does not depend on the implicitly defined Lyapunov function that allows to avoid any additional online procedures. The result is obtained by using the method of ILF with applying the properties of homogeneous systems.…”

“…Since, in addition to solving the system of LMIs (11)-(13), (24), and (25), it is necessary to satisfy inequality (15), particular attention should be paid to selecting the values of the parameters min and max . Taking into account that according to (22), d(t, x) = 0 for = 1, it is preferable to choose 0 < min < 1 and max > 1 to satisfy inequality (14).…”

“…Taking into account that according to (22), d(t, x) = 0 for = 1, it is preferable to choose 0 < min < 1 and max > 1 to satisfy inequality (14). On the other hand, to fulfill inequality (15), it is necessary to solve inequalities (11)-(14) with a sufficiently large value of max min . Thus, to select the parameters, the following strategy can be used along with Algorithm 1: starting with max = min = 1, it is necessary to solve the system of inequalities (11)-(13), (24), and (25) using Algorithm 1 with slightly increasing the parameter max and/or decreasing min until the inequality c 2 max ≤ c 1 min is satisfied.…”

“…Thus, to select the parameters, the following strategy can be used along with Algorithm 1: starting with max = min = 1, it is necessary to solve the system of inequalities (11)-(13), (24), and (25) using Algorithm 1 with slightly increasing the parameter max and/or decreasing min until the inequality c 2 max ≤ c 1 min is satisfied. Then, parameter c u should be chosen according to (15).…”