Stabilization of some class of uncertain systems

Abstract

Consider the problem of stabilizing control u synthesis for systems dx dt = Ax + Bu, where A ∈ Rn×n,B ∈ Rn×m. Elements i,j(·) of matrix A are uniformly bounded functionals of arbitrary nature. In case of continuous system matrix B elements are also continuous and uniformly bounded functionals. In case of pulse-modulated system matrix B elements are differentiable uniformly bounded functions of time. Assumed that k isolated uniformly bounded elements il,jl (·) exist over the main diagonal of matrix A(·) such as inf (·) | il ,jl (·)| ≥ − > 0, l ∈ 1, k. Gk is a set of isolated elements existing in the system. J1 is a set of indexes of rows of matrix A(·), where isolated elements are located and J2 is a set of indexes of rows of matrix A(·), where isolated elements do not exist. Assumed that other elements located over the main diagonal with indexes from J1 are sufficiently small. sup (·) | i,j (·)| < , i,j /∈ Gk, i ∈ J1, j > i. Other elements located over the main diagonal are uniformly bounded. In continuous case u = S(·)x, in pulse-modulated case u = (t), where vector components are outputs of synchronized pulse elements. Using construction of special quadratic Lyapunov function one can determine S(·), such that for continuosed case closed loop system is globally exponentially stable. In pulsemodulated case such output pulses are synthesised so that system is globally asymptotically stable.