ABSTRACT
We studied the effects of using fractional order proportional, integral, and derivative (PID) controllers in a closed-loop mathematical model of deep brain stimulation. The objective of the controller was to dampen oscillations from a neural network model of Parkinson's disease. We varied intrinsic parameters, such as the gain of the controller, and extrinsic variables, such as the excitability of the network. We found that in most cases, fractional order components increased the robustness of the model multi-fold to changes in the gains of the controller. Similarly, the controller could be set to a fixed set of gains and remain stable to a much larger range, than for the classical PID case, of changes in synaptic weights that otherwise would cause oscillatory activity. The increase in robustness is a consequence of the properties of fractional order derivatives that provide an intrinsic memory trace of past activity, which works as a negative feedback system. Fractional order PID controllers could provide a platform to develop stand-alone closed-loop deep brain stimulation systems.
ABSTRACT
In the present work we propose a fractional state observer with constant gain to estimate the periodical force exerted on a mechanical system by measuring only its displacement. The state observer is designed from both the Fourier series that approximates the periodical force and the equations of the damped harmonic oscillator that represents the behavior of the system. Specifically, the reconstruction of the force is carried out from the estimates of the series coefficients, which in fact are part of the dynamical system that composes the observer. Adams-Bashforth-Moulton method is used to compute the fractional derivatives of the observer in the Liouville-Caputo sense. Experiments based on real data are presented to show the advantages of using a fractional observer in the reconstruction of forces.
