Numerical solution of positive control problem via linear programming
The following linear time-invariant control system \(x_{t+1} = Ax_ t + Bu_ t\), \(t = 0, 1, 2, \dots\), is considered, where \(x_ t \in \mathbb{R}^ m\), \(u_ t \in \mathbb{R}^ p_ +\) \((\mathbb{R}^ p_ +\) is the set of the vectors with nonnegative coordinates), and \(A\) and \(B\) are matrices of suitable dimensions. The reachability problem is to find a function \(u\) such that \(x_ 0 = 0\) and \(x_ t = x\). An iterative procedure (based on the simplex method) for determining the minimal number of time steps, assuring the reachability of the terminal point \(x\), is proposed. This procedure is used to obtain explicitly a corresponding control. The case of perturbations of the terminal point that leaves it inside the reachability cone is considered and two numerical examples are presented.
scientific article; zbMATH DE number 720550
scientific article