Algebraic approaches for the design of simultaneous observers for linear systems

In this paper, algebraic techniques are proposed to design observers capable of estimating the state of multiple linear continuous-time systems. In order to pursue this objective, firstly an algebraic technique is given to compute the set of all the linear inverses of the observability map of a single plant. Such a result is then used to characterize, through algebraic geometry tools, the simultaneous observability of multiple linear systems both in the forced and in the autonomous case. Such a characterization is finally employed to design a single observer that is capable of estimating the state of multiple linear systems.