A Classical Approach to Controlling the Lorenz Equations

Abstract

This paper presents a classical approach to controlling the Lorenz equations, a well known chaotic system. It is shown that proportional-plus-integral control using an easily measurable state variable gives both good stability and tracking properties well into the normally chaotic region. Here the input signal to the Lorenz equations is the applied heat via the Rayleigh number. This paper demonstrates that widely used classical control approaches can stabilize a chaotic system and have it track input signals in its usually chaotic regime.

Synchronization of a class of delayed neural networks with reaction–diffusion terms

“…Under assumption (H 1 ), drive-response neural networks (17) and (18), with the boundary condition (2) and initial conditions (3), (5), are exponentially synchronized, if there exists a constant v > 1 such that the controller gain matrix M in (8) satisfies…”

“…Exponential synchronization of neural networks has been considered in [1,2]. The synchronization, control and applications of chaotic systems have also been studied, we refer to [3][4][5][6][7][8][9][10][11][12][13][14][15][16][22][23][24] and the references cited therein. In [3], Pecora and Carroll propose the drive-response concept, and use the output of the drive system to control the response system so that the state synchronization is achieved.…”

Bifurcations and Chaos in a Linear Control System with Saturated Input

“…For a survey of some techniques see, for example [Chen & Dong, 1993a, 1993b. The use of conventional, linear control actions to stabilize asymptotically a trajectory of a system in a chaotic regime has been analyzed for different types of nonlinear systems [Chen, 1993;Chen & Dong, 1992, 1993a, 1993b, 1993cChen et al, 1993;Hartley & Mossayebi, 1992;Ogorzalek, 1993;Vincent & Yu, 1991]. Some other nonconventional approaches for stabilizing equilibria or periodic orbits present in chaotic attractors have been developed [Alvarez, 1994a;Genesio et al, 1993;Hubinger et al, 1994;Vincent & Yu, 1991].…”