Stabilization of a Class of Uncertain Systems

Abstract

We consider the problem to synthesize a stabilizing control u synthesis for systems \(\frac{{dx}}{{dt}} = Ax + Bu\) where A ∈ ℝ^n×n and B ∈ ℝ^n×m, while the elements α_i,j(·) of the matrix A are uniformly bounded nonanticipatory functionals of arbitrary nature. If the system is continuous, then the elements of the matrix B are continuous and uniformly bounded functionals as well. If the system is pulse-modulated, then the elements of the matrix B are differentiable uniformly bounded functions of time. It is assumed that k isolated uniformly bounded elements \({\alpha _{{i_l},{j_l}}}\left( \cdot \right)\) satisfying the condition \(\mathop {\inf }\limits_{\left( \cdot \right)} \left| {{\alpha _{{i_l},{j_l}}}\left( \cdot \right)} \right|{\alpha _ - } > 0,\quad l \in \overline {1,k}\) are located above the main diagonal of the matrix A(·), where G_k is the set of all isolated elements of the system, J₁ is the set of indices of rows of matrix A(·) containing isolated elements, and J₂ is the set of indices of its rows free of isolated elements. It is assumed that other elements located above the main diagonal are sufficiently small provided that their row indices belong to J₁, i.e., \(\mathop {\sup }\limits_{\left( \cdot \right)} \left| {{\alpha _{i,j}}\left( \cdot \right)} \right| < \delta ,\quad {\alpha _{i,j}} \notin {G_k},\quad i \in {J_1},\quad j > i\). All other elements located above the main diagonal are uniformly bounded. The relation u = S(·)x is satisfied in the continuous case, while the relation u = ξ(t) is satisfied in the pulse-modulated case; here the components of the vector ξ are outputs of synchronous pulse elements. Constructing a special quadratic Lyapunov function, one can determine a matrix S(·) such that the closed system becomes globally exponentially stable in the continuous case. In the pulse-modulated case, input pulses are synthesized such that the system becomes globally asymptotically stable.