Differential flatness properties and multivariable adaptive fuzzy control of hormonal system dynamics

Abstract

Computer-based control for hormonal (ovarian) system dynamics and for administering in a computationally optimized manner the associated pharmaceutical treatment has been little investigated up to now. Although there has been prior work on the modeling of the ovarian system dynamics and its stability features, there have been hardly any results for applying to the ovarian system exogenous computer-based control. This is due to the complexity, nonlinearity and the high dimensionality of this system. Actually, the ovarian system exhibits nonlinear dynamics which is modeled by a set of thirteen coupled nonlinear differential equations, while receiving only two control inputs. Besides, the fact that the parameters of the model cannot be precisely identified and the fact that any mathematical model for such a biosystem is likely to be subjected to uncertainties and external perturbations, make the application of model-based control schemes for this system be questionable. To overcome this, the paper proposes model-free adaptive fuzzy control based on differential flatness theory for the complex dynamics of the ovarian system. It is proven that the dynamic model of the ovarian system, having as state variables the LH and the FSH hormones and their derivatives, is a differentially flat one. This means that all its state variables and its control inputs can be written as differential functions of the flat output and its derivatives. By exploiting differential flatness properties the system's dynamic model is written in the multivariable linear canonical (Brunovsky) form, for which the design of a state feedback controller becomes possible. After this transformation, the new control inputs of the system contain unknown nonlinear parts, which are identified with the use of neurofuzzy approximators. Lyapunov stability analysis shows that H-infinity tracking performance is achieved for the feedback control loop and this assures improved robustness to the aforementioned model uncertainty as well as to external perturbations.