Switched systems with changing derivation order: Stability and applications

This paper deals with systems that can switch their structure, including the differentiation order. It is shown that there are several non-equivalent cases for them, which all coincide when the derivation order is not switched but fixed at 1. For each of these cases, (asymptotic) stability results are obtained in this paper. This is accomplished by generalizing Common Lyapunov Functions (CLF) and Multiple Lyapunov Functions (MLF) methods, the latter when applied to fractional switching systems (FSS) in the resetting. Several examples are presented to illustrate that such Lyapunov functions exist for linear and nonlinear switched order systems. It is shown that the resetting fractional switching can be easily implemented by standard software. Finally, applications in adaptive integer-order problems are made by exploiting features of both fractional and integer-order systems.