## Section: New Results

### Interval observer

Participants : Frederic Mazenc, Silviu Niculescu, Thach Ngoc Dinh, Olivier Bernard [Inria - Sophia-Antipolis] , Eric Walter [CNRS - L2S - Supelec] , Michel Kieffer [CNRS - L2S - Supelec] .

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

1) We presented the design of families of interval observers for continuous-time linear systems with a point-wise delay after showing that classical interval observers for systems without delays are not robust with respect to the presence of delays and that, in general, for linear systems with delay, the classical interval observers endowed with a point-wise delay are unstable. We proposed a new type of design of interval observers enabling to circumvent these obstacles. It incorporates distributed delay terms [26] .

2) We considered a family of continuous-time systems that can be transformed through a change of coordinates into triangular systems. By extensively using this property, we constructed interval observers for nonlinear systems which are not cooperative and not globally Lipschitz. For a narrower family of systems, the interval observers possess the Input to State Stability property with respect to the bounds of the uncertainties [42] , [21] .

3) For the first time, we addressed in [44] the problem of constructing interval observers for discrete-time systems. Under a strong assumption, we proposed time-invariant interval observers for a very broad family of systems. In a second step, we have shown that, for any time-invariant exponentially stable discrete-time linear system with additive disturbances, time-varying exponentially stable discrete-time interval observers can be constructed. The latter result relies on the design of time-varying changes of coordinates which transform a linear system into a nonnegative one.

4) We considered continuous-time linear systems with additive disturbances and discrete-time measurements. First, we constructed a standard observer, which converges to the state trajectory of the linear system when the maximum time interval between two consecutive measurements is sufficiently small and there are no disturbances. Second, we constructed interval observers allowing to determine, for any solution, a set that is guaranteed to contain the actual state of the system when bounded disturbances are present [46] .