An extended class of stabilizable uncertain systems

Abstract

The system dx dt = A(·)x + B(·)u, u = S ∗ (·)x, where A(·) ∈ Rn×n, B(·) ∈ Rn×m, S(·) ∈ Rn×m is considered. Elements of matrixes A(·), B(·), S(·) are uniformly bounded and they are functionals of arbitrary origin. It is supposed that among elements placed on right from principal diagonal k elements αiljl (·) (l ∈ 1, k) have constant sign and each of these elements is only one in his linene and his column. Other elements placed on right from principal diagonal are sufficiently small. It is also supposed that m = n − k and elements βij(·) of matrix B(·) have the following property inf (·) |βiss(·)| = β0 > 0 for is ∈ 1, n \{i1, . . . , ik}. Other elements of matrix B(·) are zeroes. Positive definite matrix H = {hij} of a special form is constructed. Positive numbers hii = hi are placed on principal diagonal, hiljl = hjlil = −0, 5 hilhjl sign αiljl (·). Other elements of matrix H are zeroes. Investigating the derivation of Lyapunov function V (x) = x∗H−1x we have such hi (i ∈ 1, n) and λi 0 (i ∈ 1, n) that for S(·) = H −1ΛB(·), Λ = diag (λ1, . . . , λn) the system is globally exponentially stable. The control is robust in relation to elements of matrix A(·). Refs 4.