Shi Bai

Complicated dynamics of prototype continuous-line adaptive control system

The authors investigate the possible dynamics that a prototype model reference adaptive control system can experience when using the so-called sigma -modification adaption law described by P.I. Ioannou and P.V. Kokotovic (1984). Limiting their attention to first-order plants with a single unknown pole and an external disturbance, they verify analytically the bifurcations that an example adaptive system has previously been shown to exhibit based on computer simulations. The parameter-space-to-bifurcation diagram is constructed summarizing all possible behaviors of the class of prototype systems under consideration. It is shown that a certain range of constant-plus-sinusoidal disturbances generate chaos. For a certain range of constant disturbances, it is shown that a sinusoidal reference input signal generates chaos as well. In both cases, chaos arises form transversal homoclinic orbits. The use of the parameter space as a design tool for the proper implementation of the scheme is demonstrated. >

Disturbance-generated bifurcations in a simple adaptive system: Simulation evidence

Abstract We present complete characterization based on computer simulation of the possible dynamics exhibited by an adaptive control system where the (first order) plant with a single unknown pole has constant disturbance at the input. The adaptive system is taken from the current adaptive control systems literature. Here we show that the system in fact undergoes bifurcations and exhibits rich dynamics. As the constant disturbance is varied, the system undergoes saddle-node bifurcation, (subcritical) Hopf bifurcation, and a saddle connection bifurcation. The last bifurcation means that the system acquires a homoclinic orbit for a specific disturbance. While the first two bifurcations are local, the presence of the homoclinic orbit has the potential, when periodically disturbed, to generate nonlocal complicated behavior. This latter behavior is often referred to as ‘horseshoe chaos’.

A feedback neural network with supervised learning

A model for continuous-time dynamic feedback neural networks with supervised learning ability is introduced. Modifications were introduced to conventional models to guarantee analytically that a given desired vector, and its negative, are indeed stored in the network as asymptotically stable equilibrium points. The modifications entail that the output signal of a neuron is multiplied by the square of its associated weight to supply the signal to an input of another neuron. A simulation of the complete dynamics is presented for a prototype one neuron with self-feedback and supervised learning. The simulation illustrates the (supervised) learning capability of the network

A new feedback neural network with supervised learning

A model is introduced for continuous-time dynamic feedback neural networks with supervised learning ability. Modifications are introduced to conventional models to guarantee precisely that a given desired vector, and its negative, are indeed stored in the network as asymptotically stable equilibrium points. The modifications entail that the output signal of a neuron is multiplied by the square of its associated weight to supply the signal to an input of another neuron. A simulation of the complete dynamics is then presented for a prototype one neuron with self-feedback and supervised learning; the simulation illustrates the (supervised) learning capability of the network.

Chaotic dynamics even in the highly damped swing equations of power systems

The authors present a simulation of the equations of a two-machine swing, each component of which has nonzero damping. The simulation confirms the nonempty transversal intersection of the stable and unstable manifolds of a saddle point and its replica. This intersection means that a homoclinic orbit exists in the dynamics. This in turn implies the existence of chaotic dynamics in the two-machine system.<<ETX>>

Disturbance-generated bifurcation in a prototype adaptive system with e/sub 1/-modification law

The e/sub 1/-modification law has been proposed by K.S. Narendra and A.M. Annaswamy (ibid., vol.32, no.2, pp.134-5, 1987) for robust model reference adaptive control in the presence of bounded disturbances. Using this law, it is shown by the present authors that, in the case of regulation, a prototype adaptive control system with a single unknown pole and an unknown constant disturbance at its input possesses multiple equilibria and undergoes various bifurcations. Simulations and local analysis give a complete characterization of the possible dynamics exhibited by the disturbance adaptive system. As the constant disturbance is varied, saddle-node bifurcation, (subcritical) Hopf bifurcation, and saddle-connection bifurcation occur in that order. Consequently, in this case, the e/sub 1/-modification law is qualitatively equivalent to the sigma -modification law. >

Dynamics of feedback neural nets with unsupervised learning

The dynamics of feedback artificial neural networks (ANNs) with unsupervised learning is examined in order to describe their inherent genetic characteristics. A simple prototype continuous-time feedback ANN with unsupervised learning is considered and its inherent characteristics are described. Based on the choice of the genetic parameters, the prototype can learn only one class, only two classes, or only three classes. To which class an initial external datum would converge, however, would also depend on the initial value of the weight. Some of the conclusions are extended in order to describe qualitatively the characteristics of an arbitrarily large feedback ANN with unsupervised learning

Analysis of disturbance-generated bifurcation in an adaptive system with e/sub 1/-modification law

A modified adaptation law, the so-called e/sub 1/-modification law, has been proposed by K.S. Narendra and A.M. Annaswamy (1987) for robust model-reference adaptive control in the presence of bounded disturbances. Using this law, the authors show analytically that in the case of regulation, a prototype adaptive control system, with a single unknown pole and an unknown constant disturbance at its input, possesses multiple equilibria and undergoes various bifurcations. The analysis begins by transforming the augmented system equations so that there are only two independent parameters. Then explicit expressions for the equilibria and their bifurcations are given, followed by the proof of the existence of the Hopf bifurcation. As the constant disturbance is varied, saddle-node bifurcation, (subcritical) Hopf bifurcation, and saddle-connection bifurcation occur in that order. Consequently, in this case, the e/sub 1/-modification law is qualitatively equivalent to the sigma -modification law.<<ETX>>

A Robust Adaptive Controller For Linear Time-Invariant Plants With External Disturbances

We present a direct adaptive control algorithm which is robust with respect to external plant disturbances. In the presence of constant disturbances, or time-dependent disturbances which eventually approach a constant, this algorithm ensures that the tracking error approaches zero asymptotically. For general time-varying bounded distances, the (norm of the) tracking error will be asymptotically reduced to a sufficiently small number. Moreover, in the absence of disturbances, the algorithm guarantees zero tracking error.