Nonlinear adaptive control of adrenal-postpituitary imbalances and identifiability analysis

Adrenal-postpituitary imbalances express pathological evolutions of the nonlinear biological oscillator due to hormonal coupling between adrenocortical hormones and vasopressin. This system, based on agonistic-antagonistic equilibration, can be represented by a nonlinear model to be controlled in the pathological case, in order to reach a physiological state. The modeling introduced by E. Bernard-Weil has already led to efficient therapeutics and can thus be considered realistic. We can therefore use the simulated data given by Bernard-Weil, and although our results on control are obtained by simulation, they are meaningful. The therapy is based on the idea of moving the pathological controlled system from the pathological state to the physiological one. However, it is proved that with a periodic control one is not able to achieve the precise objective. This leads us to introduce the locking concept, which allows system parameters to change and provides the basis for an adaptive and iterative control, here given by a sequence of polynomial correctors. In a few iterations we are now able to find the physiological behavior again. Moreover, as we have to identify the parameters of the considered models and control laws, we have to study their structural identifiability. We can prove, thanks to the work of Vajda and his colleagues, the global identifiability of the uncontrolled 12-parameter model. We also prove the local identifiability of the eight-parameter controllers.