Fractional order adaptive control for systems of locally nonlinearizable nonlinearities

In this paper electromagnetic servo valve controlled differential hydraulic cylinders are considered as paradigms of non-linear systems that have locally nonlinearizable nonlinearities. One of them is of hydrodynamic origin: the flow resistance of the system is proportional to the square root of certain pressure difference that has infinite derivative at zero. The other nonlinearity is caused by the discontinuous behavior of the friction and adhesion forces between the piston and the cylinder at zero piston velocity. Such a behavior is difficult to control by the traditional PID controllers. Furthermore, uncertainties and variation of the hydrodynamic parameters in general make it unrealistic to develop an accurate model for such systems. Brocker and Lemmen proposed two different control approaches for the differential hydraulic cylinders based on the disturbance rejection, and on the partial flatness principles, respectively. In each case it was necessary to measure the external disturbance force and its time-derivative as well as to know the exact model of the system. Later on Tar et al. proposed an alternative adaptive approach that does not require to measure the disturbance force and to know the exact parameters of the cylinder. This method rejected to use time-derivatives because of the presence of friction, and, as a consequence it resulted in a very hectic transient phase of learning. In this paper an alternative approach is presented that combines this approach with the use of calculated time-derivatives that are "rejected" by adoptively varying the order of the derivation applied. In this way the harsh initial transients can be evaded. The operation of the method is presented by simulations.