Design of observers with commutation-dependent gains for linear switching systems

The problem of constructing state observers is addressed for a class of discrete-time systems that may switch among different modes taken from a finite set. Sufficient conditions for the exponential stability of the estimation error dynamics are derived that can be expressed in terms of Linear Matrix Inequalities (LMIs), thus guaranteeing the numerical tractability. The observer gains are designed such as to take into account the commutation from one system mode to the next one. Due to the increase in the degrees of freedom available in the synthesis of the observer, the proposed approach turns out to be much less conservative if compared with "state of the art" observers. Finally, under suitable observability assumptions, the stability of the estimation error dynamics is ensured even when the system mode is not known and the system and measurement equations are affected by disturbances.