Chaos in PID Controlled Nonlinear Systems

Abstract

-Controlling nonlinear systems with linear feedback control methods can lead to chaotic behaviors. Order increase in system dynamics due to integral control and control parameter variations in PID controlled nonlinear systems are studied for possible chaos regions in the closed-loop system dynamics. The Lur'e form of the feedback systems are analyzed with Routh's stability criterion and describing function analysis for chaos prediction. Several novel chaotic systems are generated from second-order nonlinear systems including the simplest continuous-time chaotic system. Analytical and numerical results are provided to verify the existence of the chaotic dynamics.

The Application of Accurate Exponential Solution of a Differential Equation in Optimizing Stability Control of One Class of Chaotic System

“…is obtained from System (3) Based on the above description, if the controller u exists in the form of Equation (9), then System (3) has the following solution:…”

“…where the following results are found: a 1 e λ 1 t = A 1 and a 2 e λ 2 t = A 2 . They are substituted into Equation (9), and the following equation is obtained:…”

“…This research is also of significance for the abovementioned chaos control of one class of system in practice. Now, plentiful methods have been proposed for chaos control, including adaptive control [4], sliding mode control [5,6], fuzzy control [7], the backstepping method [8], PID control [9], optimal control [10], linear feedback control [11], neural network intelligent control [12], and pulse control [13]. Most of the implemented methods are based on the Lyapunov method.…”

A model-based velocity controller for chaotization of flexible joint robot manipulators

“…31,32 Finally, a chaotization problem can be stated as a tracking problem, but the goal is to meet some formal criteria of chaos. 33 Chaotization for mechanical systems can be accomplished by means of several methods including proportional derivative control, 34 saturated proportional control, 35 time-delay feedback control applied to DC motors for industrial mixing, 36 chaotization of brushless DC motor systems, [37][38][39] adaptive techniques for rigid robot manipulators, 40 feedback linearization for mobile robots, 41 proportional integral derivative control, 42 neural networks, 43 path planning generators for autonomous robots, [44][45][46] mechanism synthesis, 47 computing robot kinematics, 48 parameter identification, 49 passive dynamic walking, 50 among others. There are also some studies showing the importance of chaotization in FJR manipulators.…”

Chaotic Encryption Based Data Transmission Using Delta and Delta-Sigma Modulators

“…To encrypt the Δ modulation systems, while the required random bits can be generated from a hardware-based generator (e.g., using thermal noise [24] and radioactive decay [25]) or from softwarebased generators (e.g., linear congruential generators [26]), chaotic systems are very simple to realize and offer a hybrid structure with the features of hardware and software based approaches [27]- [30]. The number of the chaotic systems have been increased over time in the literature, which allows us to benefit from chaotic dynamics for generating efficient chaotic random bits for use in cryptographic applications [31]- [35]. In this work, a chaotic random bit generator is developed and integrated into the Δ and ΔΣ modulators for data encryption.…”