Daniel W. Berns

Brief Paper: Feedback Control of Limit Cycle Amplitudes from A Frequency Domain Approach

In this paper, a problem in feedback control of limit cycle amplitudes is studied. A graphical approach for this bifurcation control problem is developed by means of higher-order harmonic balance approximations for both amplitude and frequency of the system oscillatory outputs. The approach starts with the familiar Hopf bifurcation mechanism, and employs the second, fourth, and sixth-order harmonic balance approximations to generate a sequence of graphical tests for convergence analysis of the system oscillatory outputs. This sequential graphical testing leads to accurate approximations of limit cycles of small amplitudes in the system outputs. Degenerate Hopf bifurcation theory is used to formulate an appropriate control objective of capturing small-amplitude limit cycles, which can avoid reaching unstable equilibria or other undesirable limit sets. A rich cubic planar model is presented for illustration of the proposed control method.

Predicting period-doubling bifurcations and multiple oscillations in nonlinear time-delayed feedback systems

In this work, a graphical approach is developed from an engineering frequency-domain approach enabling prediction of period-doubling bifurcations (PDBs) starting from a small neighborhood of Hopf bifurcation points useful for analysis of multiple oscillations of periodic solutions for time-delayed feedback systems. The proposed algorithm employs higher order harmonic-balance approximations (HBAs) for the predicted periodic solutions of the time-delayed systems. As compared to the same study of feedback systems without time delays, the HBAs used in the new algorithm include only some simple modifications. Two examples are used to verify the graphical algorithm for prediction: one is the well-known time-delayed Chuas circuit (TDCC) and the other is a time-delayed neural-network model.

Feedback control of limit cycle amplitudes

A bifurcation control problem of modifying the amplitudes of limit cycles via feedback is studied. A graphical approach reminiscent to the familiar describing function method is developed for validating the harmonic balance approximations of both the amplitude and the frequency of the system oscillatory outputs, starting from the Hopf bifurcation mechanism. The second, fourth, and sixth-order harmonic balance approximations provide a sequential graphical testing for the convergence of the oscillatory outputs, thereby yielding an accurate approximation of the desired limit cycles of small amplitudes. The knowledge of degenerate Hopf bifurcations and the associate Poincare normal forms are useful for formulating the control objective: to capture small-amplitude oscillatory system outputs and to avoid unstable equilibria or other complicated limit sets. A power system example is included for illustration.

Brief Detecting period-doubling bifurcation: an approximate monodromy matrix approach

A quasi-analytical approach is developed for detecting period-doubling bifurcation emerging near a Hopf bifurcation point. The new algorithm employs higher-order Harmonic Balance Approximations (HBAs) to compute the monodromy matrix, useful for the study of limit cycle bifurcations. Prediction of the period-doubling bifurcation is accomplished very accurately by using this algorithm, along with a detailed approximation error analysis, without using numerical integration of the dynamical system. An example is given to illustrate the results.

CONTROLLING OSCILLATION AMPLITUDES VIA FEEDBACK

This paper studies feedback control of limit cycle amplitudes in nonlinear systems. A graphical approach to bifurcation control is first briefly introduced, followed by the derivation of a nonlinear control law. Then, a suitable implementation of the design is developed via an approximation of the time-derivative term in the nonlinear state feedback controller. A classical model is finally simulated for illustration of the proposed control method.

Bifurcation Control in Feedback Systems

In this chapter the mathematical tools from bifurcation theory are used within the framework of feedback control systems. The first part deals with a simple example where the amplitude of limit cycles and the appearance of period-doubling bifurcations are controlled using a method derived from the frequency domain approach. In the second part, bifurcation theory is used to analyze the dynamical behavior of an inverted pendulum with saturated control. The main objective is to find appropriate values of the controller parameters to achieve the stabilization of the pendulum at the inverted position and, at the same time, to obtain the largest basin of attraction.

A quasi-analytical method for period-doubling bifurcation

Prediction of period-doubling bifurcation is accomplished very accurately by using higher-order Harmonic Balance Approximations (HBAs) and quasi-analytical monodromy-matrix evaluation. Approximation error analysis is carried out for the computation. An accurate detection of first period-doubling bifurcation in Chuas circuit is demonstrated.

On the detection of period doubling bifurcations in nonlinear feedback systems

In this work the detection of period doubling bifurcations is obtained using a frequency domain approach. The original approach has been applied efficiently in approximating periodic solutions from Hopf bifurcations while in this paper the emphasis is in the prediction of the first period doubling sequence. An application using higher-order harmonic balance approximations (HBA) is shown for the illustration of the technique.

On cyclic fold bifurcations in nonlinear systems

Approximations to recover the branch of multiple periodic solutions starting from a Hopf bifurcation under the variation of the main bifurcation parameter are proposed. This task is accomplished with a normal form method and, simultaneously, by using a frequency domain approach. The classical Sibirskiis example and Chuas circuit are shown for illustration of the main results.

Predicting period-doubling bifurcations in nonlinear time-delayed feedback systems

A graphical approach is developed in this paper for detecting the period-doubling bifurcation emerging near the Hopf bifurcation point of a time-delayed feedback system. The new algorithm employs higher-order harmonic balance approximations (HBAs) for estimating the predicted periodic solutions of the system. Prediction of the period-doubling bifurcation is accomplished using a type of distortion index based on some information about the higher-order harmonics. The time-delayed Chuas circuit is used as an example for illustration.