## Section: Application Domains

### Adaptive & parametric robust control $-$ collaboration with Safran Electronics & Defense

We have developed a collaboration with *Safran Electronics & Defense*
(Massy Palaiseau) and Rouillier (Ouragan , Inria Paris) on a *parametric robust control theory* based on computer
algebra methods (symbolic-numeric methods), as well as its
applications to the robust stabilization of certain mechanical
systems (e.g. gyrostabilized systems, two mass-spring-damper system,
stabilized mirrors).

For low-dimensional systems of ODEs, this approach aims to determine closed-form solutions for robust controllers and for the robustness margins in terms of the model parameters (e.g. mass, length, inertia, mode) [12], [98], [100]. The main applications of these results are twofold: the feasibility of an industrial project can be simplified by speeding up the computation of robust controllers and robust margins for systems with rapidly changing architecture parameters, and avoiding usual time-consuming optimization techniques. Secondly, adaptive and embeddable schemes for robust controllers can be proposed and tested while coupling our approach with real-time parameter estimation methods such as the ones developed in the GAIA team. For more details, see [12].

Preliminary works in the direction have opened
a great variety of questions such as the explicit search for positive
definite solutions of algebraic or differential Riccati equations (i.e. polynomial or differential systems)
with model parameters, the reduction of these
equations, and of the parameters based on symmetries, the
development, of efficient tools for plotting high degree curves and surfaces showing the robustness margins
in terms of the model parameters (collaboration with Moroz (GAMBLE , Inria Nancy)), the use of a certified
numeric Newton-Puiseux algorithm for the design of robust
controllers, etc. [12],
[98], [100]. These results require the use of a
large spectrum of computer algebra methods such as linear algebra
with parameters, polynomial systems with parameters, ordinary
differential systems with parameters, symmetries and reduction,
rational parametrizations, discriminant varieties, semi-algebraic
sets, critical point methods, real root isolation methods, etc. We
shall further develop the parametric robust control in collaboration
with *Safran Electronics & Defense*.

In connection with the above results, parameter estimation methods will be studied to develop *adaptive robust controllers* for
gyrostabilized systems. Indeed, combining explicit characterizations
of robust controllers in terms of the model parameters with
time-to-time estimations of these model parameters (which can change
with the system production, the heat, the wear, etc.), the robust
controllers can then be automatically tuned to conserve their
robustness performances [12], [99].

Finally, as explained in [11], [99],
constant and distributed delays naturally appear in *Safran E & D* systems (e.g. gyrostabilized systems using visual trackers, stabilized mirror models). Extensions of the above problems and results will be studied for differential time-delay systems based on robust control techniques for infinite-dimensional systems (see, e.g., [54] and the references therein) and its algebraic extension to include model parameters.