Some methods of construction of a common Lyapunov solution to a finite set of complex systems

Stabilization criteria of a class of switched systems

“…Rearranging the above inequality, it follows that (Ã A [22] + A [21] (−Q) −1 CA [12] ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ [21] (−Q) −1 CA [12] ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟…”

“…with à p ∶= A p [22] + A p [21] (−Q p ) −1 CA p [12] ,B p ∶= A p [21] (−Q p ) −1 A p T [21] , andC p ∶= A p T [12] C (−Q p ) −1 CA p [12] . Under this condition, the switched system S = (,   ,   ) is stabilizable by partial reset of order z.…”

“…Let A ∈ R n×n be a Metzler two-block matrix for which all blocks A [i ] ∈ R z×z , i, = 1, 2, are square and C ≻ 0 is a positive Lyapunov solution for the block (1,1) satisfying (5). Considerà ∶= A [22] + A [21] (−Q) −1 CA [12] .…”

“…1.Ã + A [21] is Hurwitz; 2. Q − A T [12] CQ −1 CA [12] is a negative definite matrix; then A is two-block diagonally stable with respect to diag(C, D), being D a diagonal Lyapunov solution that satisfies…”

“…This approach is based on quadratic stability that is still relevant, attracting the attention of several researchers (see, for instance, Arcak and Góra). Although not easy to tackle, several attempts have been made, in order to provide conditions less restrictive than the conditions established for the problem of existence of common Lyapunov solution, which allows to stabilize a switched system.…”