Dynamical analysis of an inverted pendulum with positive position feedback controller approximate uniform solution

Abstract

The inverted pendulum is controlled in this article by using the nonlinear control theory. From classical analytical mechanics, its substructure equation of motion is derived. Because of the inclusion of the restoring forces, the Taylor expansion is employed to facilitate the analysis. An estimated satisfactory periodic solution is obtained with the aid of the modified Homotopy perturbation method. A numerical technique based on the fourth-order Runge–Kutta method is employed to justify the previous solution. On the other hand, a positive position feedback control is developed to dampen the vibrations of an IP system subjected to multi-excitation forces. The multiple time scale perturbation technique of the second order is introduced as a mathematical method to solve a two-degree-of-freedom system that simulates the IP with the PPF at primary and 1:1 internal resonance. The stability of these solutions is checked with the aid of the Routh–Hurwitz criterion. A set of graphs, based on the frequency response equations resulting from the MSPT method, is incorporated. Additionally, a numerical simulation is set up with RK-4 to confirm the overall controlled performance of the studied model. The quality of the solution is confirmed by the match between the approximate solution and the numerical simulation. Numerous other nonlinear systems can be controlled using the provided control method. Illustrations are offered that pertain to implications in design and pedagogy. The linearized stability of IP near the fixed points as well as the phase portraits is depicted for the autonomous and non-autonomous cases. Because of the static stability of the IP, it is found that its instability can be suppressed by the increase of both the generalized force as well as the torsional constant stiffness of the spring. Additionally, the presence of the magnetic field enhances the stability of IP.

Different controllers for suppressing oscillations of a hybrid oscillator via non-perturbative analysis

“…15 . It is shown that from this figure and from both Tables 1 and 2 that the analytical solution is merely reliable with the calculation path as shown earlier 15 , 30 , 42 , 48 .…”

“…Subsequently, we may indicate that the optimal mode of oscillation lessening is . Previous instances were involved 15 , 30 , 42 , 48 .…”

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“…The inverted pendulum 1 , 2 is a well-known and extensively studied control problem that has attracted significant research attention due to its inherent instability and practical applications in various fields. The rotary inverted pendulum (RIP) 3 , 4 , a fascinating and extensively studied control problem, presents unique challenges due to its rotational nature.…”

Comparative Study of Takagi-Sugeno-Kang and Madani Algorithms in Type-1 and Interval Type-2 Fuzzy Control for Self-Balancing Wheelchairs

“…The domain of control systems has long been a cornerstone of engineering, dating to the application of Lagrangian mechanics and Newtonian mechanics to derive the governing equations of various systems [1]- [3]. While these classical mechanics have paved the way for modern control strategies, such as Proportional-Integral-Derivative (PID) controllers, linear quadratic regulators (LQR), fuzzy logic, and neural networks, a shift toward more advanced approaches is evident [4]- [17].…”