LMI Conditions for Stability and State-Feedback H∞ Control of Discrete-Time Multimode Multidimensional Systems

This letter deals with stability and state feedback control of discrete-time Multi-Mode Multi-Dimensional (
$M^{3}D$
) linear systems. The
$M^{3}D$
switch dynamics are modeled through a state mapping describing the mode transitions. This
$M^{3}D$
model framework then allows to consider poly-quadratic Lyapunov functions to obtain Linear Matrix Inequalities conditions for stability proof and for the synthesis of state-feedback controllers under
$\mathcal {H}_{\infty }$
performance. A numerical example illustrates the improvement of the controller synthesis conditions here introduced for discrete-time
$M^{3}D$
systems over independent point-wise mode solutions.