Stability of uncertain discrete systems

Abstract

The uncertain system

$$ x_{n + 1} = A_n x_n , n = 0,1,2, \ldots , $$

is considered, where the coefficients a _ij (n) of the m×m matrix A _n are functionals of any nature subject to the constraints

$$ \begin{array}{*{20}c} {\left| {a_{i,i} (n)} \right| \leqslant \alpha _ * < 1,} \\ {\left| {a_{i,j} (n)} \right| \leqslant \alpha _0 for j \geqslant i + 1,} \\ {\left| {a_{i,j} (n)} \right| \leqslant \delta for j < i.} \\ \end{array} $$

Such systems include, in particular, switched-type systems, whose matrix A can take values in a given finite set.

By using a special Lyapunov function, a bound δ ≤ δ(α₀,α_*) ensuring the global asymptotic stability of the system is found. In particular, the system is stable if the last inequality is replaced by a _i,j (n) = 0 for j < i.

It is shown that pulse-width modulated systems reduce to the uncertain systems under consideration; moreover, in the case of a pulse-width modulation of the first kind, the coefficients of the matrix A are functions of x(n), and in the case of a modulation of the second kind, they are functionals.