Computational Model Of A Fractional-Order Chemical Reactor System And Its Control Using Adaptive Sliding Mode Control

Dynamics of chemical reactor systems are found with highly nonlinear behavior. Computational modeling of a fractional-order chemical reactor system and investigating nonlinear dynamical changes and its control are the main focus of this research work. Chaos theory is a blooming fertile field in recent years, which is used widely to quantify nonlinear behaviors such as quasi-oscillations, bi-stability and bifurcation. The work starts from deriving state-space model of the system with first-order differential equations. There are six equilibrium points and the Jacobian matrix is derived for investigating the stability of the equilibrium points. Eigenvalues of each equilibrium point are calculated. Based on the sign of the real part of the eigenvalues and the existence of imaginary part, we found two equilibrium points behave as saddle spirals and the remaining four equilibrium points are saddle nodes. The stability of the system for different parameter values is investigated and presented. The influence of parameters in the system dynamics is discussed and significant parameter values are highlighted for further study. We considered Caputo’s definition for formulating the fractional-order (FO) model of the system based on the advantages highlighted in various literatures. The stable and unstable regions are portrayed with parameter variations. The results clarified that the analysis can be refined using fractional-order treatment of chaotic systems. We proceeded with our investigation towards obtaining different oscillations, particularly chaotic oscillations. The challenges lie in finding the proper fractional order to handle the system. We showed the bifurcation diagram for a range of fractional-order values and clarified the transitions from periodic oscillations to chaotic behavior and period-doubling bifurcations. The phase portraits are presented to show the limit cycle oscillations for fractional-order 0.95, period-doubling during 0.98, and chaotic oscillations for higher values. We proceeded with our investigation with fractional-order as 0.99. Bifurcation plots for parameter variation are obtained. Chaotic regions, periodic oscillations, period-halving and period-doubling are observed and the influences are discussed. We emphasize the intricate properties which are not addressed during the integer-order treatment of the system and nail the importance of fractional-order treatment. An adaptive sliding mode (ASM) controller is derived and implemented to control the system precisely. The effectiveness is shown by providing simulation results of the system with parameter estimation and controlled state time history plots. The work can be extended to verify the simulated results with equivalent electronic circuits.