Letter An Extended Convex Combination Approach for Quadratic $\mathcal{L}_{2}$ Performance Analysis of Switched Uncertain Linear Systems

Dear Editor, This letter is concerned with quadratic
$\mathcal{L}_{2}$
performance (quadratic stability and
$\mathcal{L}_{2}$
gain) for switched uncertain linear systems (SULS) with norm-bounded uncertainties. Assuming that no single uncertain subsystem achieves quadratic
$\mathcal{L}_{2}$
performance
$y$
but a convex combi-nation of the subsystems can make it, we propose a state-dependent switching law such that the SULS achieves the desired quadratic
$L_{2}$
performance. The discussion is extended to the SULS with state feed-back control, and a sufficient condition is proposed to design the state feedback gains and the switching law simultaneously.