Optimal compensation by linear robust control for uncertain systems (Q1290885)

Consider the uncertain discrete-time controlled system \[ x_{k+1} =Ax_k+ Bu_k+ BDv_k(x_k, \sigma_k) \] with \((A,B)\) stabilizable and \(v:{\mathcal R}^n \times {\mathcal R}^s\to {\mathcal R}^n\) is a continuous mapping satisfying \[ \bigl| v_k(x,\sigma) \bigr| \leq\gamma_1| x| +\gamma_2. \] It is required to find a feedback control in order to make the closed loop system uniformly bounded and uniformly ultimately bounded. The following control is proposed \[ u= Kx-\gamma B^TP(A+BK)x. \] Here \(K\) is such that \(A+BK\) is a Hurwitz matrix while \(P>0\) is the unique solution of the Lyapunov equation \[ (A+BK)^T P+P(A+BK) =-Q \] \(Q>0\) being arbitrary. The parameter \(\gamma\geq 0\) is chosen according to an optimization algorithm in order to maximize the size of the ultimate boundedness region. An example taken from macroeconomic control is considered.