Design of linear observers for a class of linear hybrid systems

This paper deals with the design of linear observers for a class of linear hybrid systems. Such systems are composed of continuous-time and digital substates and possess, in general, coupling dynamics between both substates. Two observer prototypes based on the prediction error are proposed. The first is based on the observation of an extended discrete system at sampling instants. The discrete extended state is composed of the sampled values of the continuous substate at sampling instants and the digital substate. The estimation of the continuous substate in between sampling instants is made by using the plant parametrization and the sampled prediction error at the preceding sampling instant. The continuous-time state estimates are reinitialized at each new sampling instant by taking values from the corresponding components of the discretized substate of the observer of the auxiliary discrete extended system. The second observer prototype estimates the continuous-time substate for all time from initial conditions which are taken only when the estimation algorithm starts. The observer involves feedback information of the current observation error, that of the preceding sampling instant and that associated with the estimation of the discrete system. As a result, the discontinuities of the estimated substate at sampling instants, which typically occur with the use of the first prototype, are not present in this observation scheme. The exponential convergence to zero of both prediction and observation errors may be ensured under observability and detectability assumptions Furthermore, prescribed pole placement of the state estimation error is achieved under observability of the discrete extended plant. Also, prescribed pole placement of the combined dynamics of the extended plant and observation error can be obtained. For that purpose, the extended hybrid plant is assumed to be controllable when the linear control input is generated from measurements of the state observation. in both observation prototypes.