## Section: New Results

### Modeling, observation and control: systems modeled by ordinary differential equations

#### Control ideas for optimization

Participant : Pierre-Alexandre Bliman.

This subject has been worked on with A. Bhaya and F. Pazos (UFRJ, Rio de Janeiro, Brazil).
For large size problems, solving the equation Ax = b for x , where A is a symmetric positive definite matrix and b a vector, cannot be done simply by inversion of A .
Rather, one can try to search recursively for the minimum of the quadratic function .
The various gradient methods search at each step along the current gradient (x_{k}) = Ax_{k}-b , where x_{k} is the current estimate.
Among them, the steepest descent amounts to choose the step length which minimizes the residue norm at step k .
The latest, equal to (Rayleigh coefficient), ensures a monotone decrease to zero of the residue norm, but is not especially fast.
Quite surprisingly, the choice made by Barzilai and Borwein in a renowned paper of the delayed expression works remarkably better.

Our work has been to try to find better choices, considering the problem as a control issue. Deadbeat control ideas yield interesting results, while some preliminary cooperative control ideas seem to lead to quite sensible speed up. Further studies are presently under consideration.

#### Robustness Properties of Linear Systems

Participant : Pierre-Alexandre Bliman.

We went on developing with P.L.D. Peres and R.C.L.F. Oliveira (Unicamp, Campinas, Brazil), M.C. de Oliveira (University of California San Diego, USA) and V.F. Montagner (University of Santa Maria, Brazil) tools for robust analysis, robust synthesis and gain-scheduling dedicated to uncertain linear systems subject to parametric uncertainties.

Previous efforts had been concentrated on a representation of the parameters more fitted to the computational techniques. In particular, cartesian products of simplexes (“multi-simplexes”) had been introduced. This formalism has been used now for gain-scheduling purposes.

#### The two-stage algorithm for Hammerstein system identification

Participant : Qinghua Zhang.

A Hammerstein system is composed of a static nonlinearity block followed by a linear dynamic block. Typically, the nonlinearity of such a system is caused by actuator distortions. The Two-Stage Algorithm (TSA) has been widely used and adapted for the identification of Hammerstein systems. It is essentially based on a particular formulation of Hammerstein systems in the form of bilinearly parameterized linear regressions. In collaboration with Jiandong Wang (Peking University, China) and Lennart Ljung (Linköping University, Sweden), we have studied various aspects of the TSA. We first elaborated a constructive method for deriving the TSA, whereas it was initially proposed as a heuristic method in the literature. We then studied the parametrization of the weighting matrix used in the TSA. These results are reported in [59] . We also studied a somewhat contradictory fact about the TSA: though the optimality of the TSA has been established by Bai in 1998 [80] only in the case of some special weighting matrices, the unweighted TSA is usually used in practice. Our investigation shows that the unweighted TSA indeed gives the optimal solution of the weighted nonlinear least-squares problem formulated with a particular weighting matrix. This provides a theoretical justification of the unweighted TSA, and also leads to a generalization of the result to the case of colored noise with noise whitening. These results are reported in [58] , [43] .