Singular Switched Systems in Discrete Time: Solvability, Observability, and Reachability Notions

Discrete-time singular (switched) systems, also known as
(switched) difference-algebraic equations and discrete-time (switched)
descriptor systems, have in general three solvability issues:
inconsistent initial values, nonexistence or
nonuniqueness of solutions, and noncausalities, which are generally
not desired in applications. To deal with those issues, new
solvability notions are proposed in the study, and the corresponding
necessary and sufficient conditions have been derived with the help of
(strictly) index-1 notions. Furthermore, surrogate (switched)
systems--ordinary (switched) systems that have equivalent
behavior--have also been established for solvable systems. By
utilizing those surrogate systems, fundamental analysis including
observability, determinability, reachability, and controllability has also been
characterized for singular linear (switched) systems. The solvability
study has been extended to singular nonlinear (switched) systems, and
moreover, Lyapunov and incremental stability analyses have been
derived via single and switched Lyapunov function approaches.