PDF e-Pub

## Section: New Results

### Observers

Participants : Frederic Mazenc, Michael Malisoff [LSU] , Saeed Ahmed [Inria] , Ali Zemouche [CRAN] , Rajesh Rajamani [University of Minneapolis, USA] , Maruthi Akella [University of Texas, USA] .

We produced several works which pertain to the case where only a part of the state variables can be measured.

In the paper [58], we adopted a technique based on the indroduction of several observers in cascade (such a cascade is called 'sequential observer') for a class of time-varying linear systems in which the inputs and outputs containing sampling and arbitrarily long delays. The observers are of a continuous-discrete type. We used the observers to design controllers that ensure a strong robustness property with respect to uncertainties in the system and the output, under delays and sampling. A fundamental aspect of the approach is that it produces the observers and controllers without distributed terms. We have assessed the performance of the control laws through two examples, which inlcude a DC motor model that illustrates the utility of the work in engineering applications.

In two papers, we developped the theory of the finite time observers. In [53], we study a class of linear continuous-time time-varying systems with piecewise continuous disturbances and piecewise constant outputs. Under a classical assumption of observability, we designed a new type of observers to estimate the solutions of the system in a predetermined finite time. In contrast to the well-established finite time observer design techniques which estimate the system state using a continuous output, our proposed observer applies when only piecewise constant measurements are available. In [54], we construct finite-time reduced order observers for a broad family of nonlinear time-varying continuous-time systems. The motivation for this is the fact that in practice the time-varying aspect of a system may be an obstacle to the design of full-order finite-time observers, but not for the design of reduced order ones. We illustrated our results using a tracking problem for nonholonomic systems in chained form.

Two of our works present construction of asymptotic observers without delay. The paper [61] solves an ${H}_{\infty }$ observer design problem for a class of descriptor nonlinear systems. The method we established is theoretical and can be applied to many automatic control design problems such as unknown input estimation problem, which plays an important role in control systems, namely for diagnosis and fault tolerant control. The design relies on the Linear Matrix Inequality condition (LMI) technique. We applied our resut to a model of a flexible joint robot system.

The work [16] is dedicated to the design of a smooth six-degree-of-freedom observer to estimate the incorporating linear and angular velocity, called dual angular velocity, for a rigid body. The approach is based on the dual-quaternion description and we proved that the estimation errors exhibit asymptotic convergence. Furthermore, to achieve tracking control objective, we combined the proposed observer with an independently designed proportional-derivative-like feedback control law (using full-state feedback), and a special Lyapunov "strictification" process is employed to ensure a separation property between the observer and the controller. We performed numerical simulations for a prototypical spacecraft hovering mission application.