On the effects of strong asymmetries on the adaptive controllers based on Robust Fixed Point Transformations

For replacing Lyapunov´s sophisticated “2nd Method” in the design of adaptive controllers a novel approach based on Robust Fixed Point Transformations (RFPT) was proposed that directly concentrates on the designer´s intent instead of forcing global stability. It guarantees convergence only in a bounded basin while iteratively generating the sequence of the appropriate control signals. In the initial phase of this iterative learning considerable fluctuation may occur in the control signal that otherwise may be limited due to phenomenological reasons. While in mechanical systems positive or negative force or torque components can be allowed, in controlling chemical reactions negative ingress rates of pure reactants into a stirring tank reactor phenomenologically cannot be realized. While velocity components may have well interpreted positive or negative values, negative concentrations physically cannot make sense. On this reason the mathematical models of chemical reactions normally containing the products of various powers of the concentrations must be completed with truncation-type nonlinearities that introduce strong asymmetric nonlinearities. In this paper the effects of these phenomena are investigated via computer simulations in the adaptive control of a Classical Mechanical and a chemical system. It was found that in spite of these limitations the adaptive controller can still work at least in certain segments of the whole control section.