Chaos control for the Lorenz system (original) (raw)
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20th Iranian Conference on Electrical Engineering (ICEE2012), 2012
The objective of this paper is to design a finite-time controller for Lorenz system in discrete-time. First, a discrete model is derived through the Taylor series expansion. In the next step, a discrete-time terminal sliding mode controller (DTSMC) is developed to reach a finite-time and high precision control. The stability analysis of DTSMC is presented in the presence of external disturbance and model uncertainties. Numerical simulations of Lorenz system are shown and compared with a discrete-time sliding mode control (DSMC) to illustrate the effectiveness of the proposed control scheme.
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Chaos control means to design a controller that is able to mitigating or eliminating the chaos behavior of nonlinear systems that experiencing such phenomenon. In this paper, a nonlinear Sliding-Mode Controller (SMC) is presented. Two nonlinear chaotic systems are chosen to be our case study in this paper, the well known Chua's circuit and Lorenz system. The study shows the effectiveness of the designed nonlinear Sliding-Mode Controller.
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This work presents a robust chaotic control strategy for the Lorenz chaos via backstepping design. Backstepping technique is a systematic tool of control law design to provide Lyapunov stability. The concept of extended system is used such that a continuous sliding mode control (SMC) effort is generated using backstepping scheme. In the proposed control algorithm, an adaptation law is applied to estimate the system parameter and the SMC offers the robustness to model uncertainties and external disturbances so that the asymptotical convergence of tracking error can be achieved. Regarding the SMC, an equivalent control algorithm is chosen based on the selection of Lyapunov stability criterion during backstepping approach. The converging rate of error state is relative to the corresponding dynamics of sliding surface. Numerical simulations demonstrate its advantages to a regulation problem and an orbit tracking problem of the Lorenz chaos.
Adaptive Fuzzy Sliding Mode Controller Design for Lorenz System
2009 International Workshop on Chaos-Fractals Theories and Applications, 2009
This paper presents an adaptive fuzzy sliding mode control (AFSMC) scheme for chaos control of Lorenz system. In this scheme, the reaching law required to drive the system states of Lorenz system to the sliding surface is inferred by an adaptive technique and a set of fuzzy logic rules based upon the output of a sliding mode controller (SMC). The feasibility and effectiveness of the AFSMC scheme are demonstrated via a numerical simulation. The numerical results demonstrate the ability of AFSMC scheme to suppress the chaotic Lorenz system and reveal that the control signal is chatter free.
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The Lorenz chaotic system is based on a nonlinear behavior and this causes the system to be unstable. Therefore, two different controller models were developed and named as the adaptive pole placement and sliding mode control (SMC) methods for the establishment of continuous time nonlinear Lorenz chaotic system. In order to achieve this, an improved controller structure was developed first theoretically for both the controller methods and then tested practically using the numerical samples. During the establishment of adaptive pole placement method for the Lorenz chaotic system, various stages were applied. The nonlinear chaotic system was also linearized by means of Taylor Series expansion processes. In addition, the feedback matrix of the adaptive pole placement method was determined using linear Jacobian matrix. The chaotic system reached an equilibrium point by using both the SMC and adaptive pole placement methods; however the simulation results of the SMC had better success th...
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This paper presents a novel fuzzy sliding mode approach to control Lorenz chaotic system without chattering problem. A boundary layer and a moving fuzzy surface with an algorithm to remove chattering are used in the proposed control law. Two folds present advantages of the proposed approach: 1) it has a flexibility to define the control law without cancelling useful nonlinearities; 2) the system performance is robust against parametric uncertainties. Consequently, the goals of stabilization of chaotic motion and tracking a reference signal are achieved without chattering. Theoretical analysis and numerical simulations illustrate the effectiveness of the proposed control approach. Moreover, the proposed method is superior to backstepping method in overcoming parametric uncertainties with less control effort.
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In this paper we analyze some dynamical properties of a chaotic Lorenz system driven by a control input. These properties are the input-state and the input-output feedback linearizability, the stability of the zero dynamics, and the phase minimality of the system. We show that the controlled Lorenz system is feedback equivalent to a controllable linear system. We also show that the zero dynamics are asymptotically stable when the output is an arbitrary state. These facts allow designing control laws such that the closed-loop system has asymptotically stable equilibrium points with dynamic behavior free from chaotic transients. The controllers are robust in the sense that the closed-loop system is stable and non chaotic around a nominal set of parameter values. The results also show that the proposed controllers give better responses compared to linear algorithms obtained from standard linearizatiou techniques, and exhibit a good performance even when the control input is bounded.
Waves in Random and Complex Media, 2023
By replacing a quadratic nonlinear term in Lü system with a piecewise linear signum (PWL) function, a new simplified three-dimensional piecewise continuous autonomous system (a modified Lü system) is introduced. The qualitative properties of the modified Lü system are studied. Based on these properties, the feedback control law is applied to suppress chaos to one of the three equilibria. Several different synchronized methods, such as the active control, one way coupling by active control, and the adaptive active control are applied to achieve the state synchronization of two identical modified Lü systems. These results show that after the simplification, the modified Lü system can still keep the basic and typical nonlinear phenomena. Compared with the original Lü system, the modified Lü system has a lot of advantages, by which the modified Lü system can be more easily implemented by theoretical analysis, and more practicable made by secret communications.