This paper will establish the importance and significance of studying the fractional-order control of nonlinear dynamical systems. The foundation and the sources related to this research scope is going to be set. Then, the paper incorporates a brief overview on how this study is performed and p resent the organization of this study. The present work investigates the effectiveness of the physical-fractional and biological-genetic operators to develop an Optimal Form of Fract ional-order PID Controller (O2Fo-PIDC). The newly developed Fo-PIDC with optimal structure and parameters can, also, imp rove the performances required in the modeling and control of modern manufacturing-industrial process (MIP). The synthesis methodology of the proposed O2Fo-PIDC can be viewed as a mu lti-level design approach. The hierarchical Multiobject ive genetic algorith m (M GA), adopted in this work, can be visualized as a comb ination of structural and parametric genes of a controller orchestrated in a h ierarch ical fashion. Then, it is applied to select an optimal structure and knowledge base of the developed fractional controller to satisfy the various design specification contradictories (simplicity, accuracy, stability and robustness).

In this paper, we develop a fractional-order time-delayed feedback control (Fo-TDFC) approach combined with an optimal knowledge base to stabilize and synchronize a large class of fractional-order chaotic systems (FoCS). The proposed control laws are formulated based on the Lyapunov stability theory and the multiobjective optimization process of the closed-loop controlled system. The design and optimization of the Fo-TDFC laws are theoretically rigorous and represent a powerful tool to control and synchronize FoCS. Finally, numerical simulation results are provided to illustrate the effectiveness of the proposed method by considering the fractional-order energy resources demand-supply chaotic system as an illustrative example.

The concept of fractional-order control (F-oC) is exploited in this paper to synchronize fractional-order dynamical systems. The addressed systems in this paper reflect the real physical phenomena characterized by the complicated relationship between supply and demand for energy resources in the Shanghai area. Thus, we provide the developed fractional energy resource attractor and the simulation results regarding synchronization under the proposed control law of the same fractional energy resource attractor. Note that most of the synchronization methods achieved excellent performance when dealing with complex continuous systems; however, no method addressed the synchronization problem of fractional-order energy resource systems based on the F-oC and modern optimization techniques, to the best of our knowledge. By designing the finite-time control theory, the finite-time full synchronization of two identical fractional-order energy resources demand-supply hyperchaotic systems (F-oERDSHSs) is investigated due to its performance. The advanced prediction-based fractional-order control law (Pb-FoCL) is established for finite-time synchronization of F-oERDSHSs. The design procedure becomes a multiobjective optimization problem of the knowledge base of the developed controller while satisfying the desired performance requirements. The Finite-Time Stability (F-TS) of the control-loop system is proved by using the finite-time Lyapunov stability theory. Furthermore, the Improved Artificial Hummingbird Algorithm (I-AHA) is used to find an optimal knowledge base of Pb-FoCL while achieving the design constraints. Simulation results are provided to verify the efficiency of the proposed control strategy.