Section: New Results

Interval Observer

We made several progresses in the domain of the construction of state estimators called interval observers.

1) In [16] , we have shown how interval observers can be constructed for nonlinear (and not Lipschitz) systems possessing a special triangular system. These systems are not cooperative and not globally Lipschitz and have a rather general structure which may result from a change of coordinates or an output injection. Besides, under additional assumptions, input to state stability (ISS) properties are derived. We illustrated the constructions by designing a framer and an ISS interval observer for two models of bioreactors.

2) The contributions [17] and [18] present major results for the design of interval observers for discete-time systems. In [18] , coordinate transformations which change an arbitrary linear discrete-time system into a positive one and general nonlinear designs of interval observers for nonlinear systems (satisfying a restricitive stability assumption) are proposed. In [17] , it is explained how two classical Luenberger observers can be used, (even in the absence of the positivity property of the studied system or the error equations) as interval observer, provided two appropriate outputs, which compose the lower and the upper bound of the interval observer, are selected.

3) In [33] , we present a new type of interval observers for nonlinear systems that are affine in the unmeasured part of the state.They are composed of two copies of classical observers and upper and lower bounds which are designed by taking advantage of positivity properties of the error equations when written in appropriate coordinates.