Faramarz Mossayebi

Adaptive estimation and synchronization of chaotic systems

Abstract In this Letter the recently introduced method to synchronize deterministic chaotic systems is generalized by means of concepts in adaptive control.

CONTROL OF CHUA'S CIRCUIT

This paper considers the control of a polynomial variant of the original Chuas circuit. Both state space techniques and input-output techniques are presented. It is shown that standard control theory approaches can easily accommodate a chaotic system. Furthermore, it is shown that a harmonic balance approach could predict the period doubling phenomenon and onset of the double scroll chaos, as well as providing a control approach.

A CLASSICAL APPROACH TO CONTROLLING THE LORENZ EQUATIONS

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.

SYSTEM IDENTIFICATION AND MODEL-BASED CONTROL OF A CHAOTIC SYSTEM

In this paper the control of a hyper2chaotic system is considered to show the role of system identification techniques in developing a model for effective control of highly complex systems. An indirect adaptive control scheme is considered and it is shown that simple prediction models which cannot possibly represent the dynamics of the chaotic system lead to stable control. Furthermore, it is shown that higher dimensional prediction models which more closely represent the chaotic process dynamics lead to controlled systems with sparse and disjoint basins of attraction for the desired steady state solution. The use of highly nonlinear models also results in a complex pattern of convergence to the desired state.

Dynamical complexity arising in the adaptive control of a chaotic system

Abstract In this Letter unique properties of an indirect adaptive controller designed to drive a chaotic logistic system to a steady state are presented and compared to published results.

Classical control of a chaotic system

It is shown that classical control techniques can be used for the control of chaotic systems. In particular, it is demonstrated that a model for a fluid loop can be stabilized at a given operating point in the chaotic regime. It is also shown that in the stable regime, output feedback can be used to drive the system into chaos.<<ETX>>

INDIRECT ADAPTIVE CONTROL OF A CHAOTIC SYSTEM

Abstract The adaptive control of a simple chaotic system to a steady reference is investigated. An indirect adaptive control scheme with least-mean-squares (LMS) parameter estimation yields a fractal basin boundary between stable and unstable control. This boundary appears self-similar, indicative of a fractal structure, and has a fractal dimension of 2.61. Adaptive control using a least-squares (LS) estimator appears globally stable with the rate of adaptation a complex function of the distance from the desired set point. The effect of control parameters, noise and a step-change perturbation of the system are also reported.

On matrix integrators for real-time simulation

A matrix integration method is generalized to systems with zero eigenvalues. It is shown that the regression coefficients of the integrator can be determined without explicitly computing the inverse of the system Jacobian. This is done by transforming the original system into a new system whose Jacobian is in block upper triangular form. A numerical example is included for illustrative purposes. >

A fundamental theorem for the model reduction of nonlinear systems

A simple, but fundamental, theorem is given on the extent to which a nonlinear system model can have its order reduced. Essentially, the result is that the order, or the dimension of the state space representation, cannot be reduced to, or below, the dimension of the systems attractor. Several examples are given to illustrate this point. The result is especially applicable to higher order systems such as the infinite dimensional systems arising from the modeling of distributed parameter systems.

Chaos in an adaptive controller

Abstract A specific example is used to show that some bounded instability in adaptive control systems can be understood as a chaotic phemonenon displayed by the closed-loop system. For the given example, it is shown that the identification algorithm used has a direct effect on the convergence of the adaptive system.