A New Parametrization of Observers

This paper extends and generalizes recent results on the characterization and parametrization of observers for linear systems in the behavioral framework. We formulate the results in the language of quotient signal modules that was developed by Oberst and first used in the context of observer theory by the first author. The resulting characterization of observers in terms of a generalized internal model principle is both elegant and concise. It includes all such results known to the authors as special cases, including the classical results for linear time-invariant state space systems. Moreover, this new characterization of observers leads to a clean and simple oneto-one parametrization result with only free parameters. This new parametrization allows to decide certain additional observer properties (such as input/output structure or nonintrusiveness) purely by inspection. 1. Problem formulation. We study plants (observed systems) of the form P =  w1 w2 w3  ∈F w1+w2+w3 ; R1 ◦w1 = R2 ◦w2−R3 ◦w3  , where w1 denotes the measured variable, w2 the to be estimated variable and w3 an irrelevant variable, respectively. As usual in the behavioral context [2], the wi should be thought of as vector-valued trajectories of time (vectors with wi entries, each of which is a scalar function of time) and the equation given in terms of operators Ri holds in some function space F . More precisely, let F be a field and let D := F [s] denote the polynomial ring in one indeterminate over F . We interpret D as a ring of operators that act on (scalar) trajectories of time, turning the set F of these into a D-module with operation ◦. For example, we could choose F = R and F = C ∞(R,F) with the operator s = d dt , the usual continuous time derivative. The Ri ∈Dk×wi are then just constant coefficient (higher order, matrix) differential operators and the plant P is a linear time-invariant continuous time system. Another choice would be F = R, F = FN and (s ◦w)(t) = w(t + 1), i.e., s is the left shift operator. This yields standard linear discrete time systems. Many more choices are possible, all we require is that the signal module F is an injective cogenerator in the category of D-modules, cf. [1]. Following the ideas developed by Valcher and co-authors [3, 4], an observer in our context is just another system O = {( w1 ŵ2 ) ∈F w1+w2 ; R̂1 ◦w1 = R̂2 ◦ ŵ2 }