Designing Stable Inverters and State Observers for Switched Linear Systems with Unknown Inputs

We present a method for estimating the inputs and states in discrete-time switched linear systems with unknown inputs. We first investigate the problem of system invertibility, which reconstructs the unknown inputs based on knowledge of the output of the system, the switching sequence, and the initial system state. We then relax the assumption on the knowledge of the initial system state, and construct observers that asymptotically estimate the state. Our design, which considers a general class of switched linear observers that switch modes based on the known (but arbitrary) switching sequence, shows that system invertibility is necessary in order to construct state observers. Furthermore, some portion of the observer gain must be used to recover the unknown inputs, and the remaining freedom must be used to ensure stability. The state of the observer is then used to asymptotically estimate the unknown inputs (i.e., it forms the dynamic portion of a stable inverter for the given switched system)