Well-posedness analysis of switch-driven piecewise affine systems

The well-posedness problem (existence and uniqueness of solutions) of a class of multi-modal piecewise affine systems is addressed, where binary-switches individually act under autonomous switching. First, a new transition rule on the discrete state, called the switch-based transition rule, is introduced and some relations with the mode-based transition rule are discussed. Next, a sufficient condition for such a multi-modal system to be well-posed for all external inputs is derived in terms of well-posedness of its subsystems of lower complexity "bimodal systems". Finally, an easily checkable condition for the bimodal system to be well-posed for all external inputs is given, which consequently allows us to algebraically determine well-posedness of the multi-modal systems in question.