This article provides a graphical parameter tuning method of PI^λ controllers for fractional-order time-delay systems. First, the complete stabilizing region of PI^λ controller in proportional-integral plane, for a fixed A, is determined in terms of a graphical stability criterion applicable to fractional-delay systems. Then, the stabilizing region is maximized analytically with respect to parameter ）, to expect the most various behaviors of the closed-loop systems. Finally, by defining appropriate functions relative to the requirements of gain and phase margins, the curves in the maximized stabilizing region satisfying the pre-specified gain and phase margins are drawn, which releases a flexible parameter tuning procedure. Numerical examples are given to illustrate the design steps.
In the paper, a novel four-wing hyper-chaotic system is proposed and analyzed. A rare dynamic phenomenon is found that this new system with one equilibrium generates a four-wing-hyper-chaotic attractor as parameter varies. The system has rich and complex dynamical behaviors, and it is investigated in terms of Lyapunov exponents, bifurcation diagrams, Poincare maps, frequency spectrum, and numerical simulations. In addition, the theoretical analysis shows that the system undergoes a Hopf bifurcation as one parameter varies, which is illustrated by the numerical simulation. Finally, an analog circuit is designed to implement this hyper-chaotic system.
