Extended superstability in control theory

The notion of superstability that was recently used to tackle various problems of robustness and the linear control theory was generalized to attain higher flexibility. For the continuous and discrete cases, a class of matrices $E$ was introduced for which the superstability condition is satisfied after the diagonal transformation. Systems with these matrices have piecewise-linear Lyapunov function $V(x)=\max\limits_{i}|x_{i}/d_{i}|$. Problems such as verification of the membership $\widetilde{A}\subset E$ for the interval matrices, existence of a feedback $K$ such tha $A+BK\in E$, the best componentwise estimation, and disturbance attenuation were all of them reduced to the easily solvable linear programming problems. Efficient numerical methods were proposed to solve the arising linear inequalities.