Robust state-feedback controllability of linear systems to a hyperplane in a class of bounded controls
The authors deal with the controlled dynamic system \[ x'=A(t)x+b(t)u+c(t)v,\, x(0)=x_0,0\leq t\leq t_f, \] where \(x\in\mathbb R^n\) is a state vector; \(u, v \in \mathbb R^1\) are the control and the disturbance, respectively; \(t_f\) is a prescribed final time. The matrix function \(A(t)\) and the vector functions \(b(t),c(t)\) are continuously differentiable. It is assumed that the admissible controls satisfy the constraint \(u(\cdot)\in U\), where \(U\) is the set of measurable bounded functions on \([0,t_f)\). Two types of constraints on the disturbance are assumed: \(v(\cdot)\in V_i,i = 1,2\), where \(V_1\) is the set of measurable bounded functions on \([0,t_f)\), \(V_2 = L^2(0,t_f)\). The target set is the hyperplane \(\mathcal D=\{x\in\mathbb R^n\mid d^Tx=0\},\) where \(d\in\mathbb R^n\) is a prescribed nonzero vector. The problem investigated is the existence of a state feedback control with measurable bounded time realizations transferring the system to a given hyperplane (a target set) from any initial position in a prescribed time for any admissible input uncertainty realization. Necessary and sufficient conditions for the existence of such a control are derived based on the explicit construction of this control by using an auxiliary zero-sum linear-quadratic differential game. Examples illustrating the theoretical results are presented.
scientific article; zbMATH DE number 2154325
scientific article