Simultaneous observation of linear systems

Algebraic approaches for the design of simultaneous observers for linear systems

“…By letting N = 5, and computing the sets G k, 5 , k = 1, 2, 3, as in (9), one obtains the following polynomials of ℝ[y e, 5 ]:…”

“…Although the design of a state observer for a single continuoustime system can be carried out by using classical techniques, such as the Luenberger observer [7] and the Kalman filter [8], the design of a single observer capable of estimating the state of multiple systems is much more challenging. This problem was firstly introduced in [9] and can be summarised as follows: given a set of plants, design a single observer that can estimate the state of each of these plants. This problem is particularly interesting when one aims at obtaining a reliable estimate of the state of a plant affected by known perturbations that arise from sensor or actuator faults [10].…”

“…Several techniques have been proposed in the literature to deal with such a problem. For instance, it has been addressed in [9], by using coprime factorisation technique, in [11], by using evolutionary strategies, in [12], by using a parametrisation in terms of a stable inverse and a stable null space, and in [13], by using a state-space characterisation. Furthermore, in [14,15], common functional observers have been proposed for two linear systems with unknown inputs.…”

“…The main difference between the tools given in this paper and the ones given in [9,[11][12][13] is that the former, by employing techniques borrowed from algebraic geometry, provides an exact certificate for the simultaneous observability (or lack thereof) of a set of linear continuous-time systems. Namely, given a set of systems, the proposed method allows one to determine a closedform expression for the simultaneous inverse of the observability map of such systems, if any, which can be directly used to design a simultaneous observer.…”

Observer parameterization for simultaneous observation

“…When there are disturbances in given plants, the description for the set of plants in (1) and (2) becomes …”