Jorge L. Moiola

BIFURCATION CONTROL: THEORIES, METHODS, AND APPLICATIONS

Bifurcation control deals with modification of bifurcation characteristics of a parameterized nonlinear system by a designed control input. Typical bifurcation control objectives include delaying t...

Hopf Bifurcation Analysis: A Frequency Domain Approach

The Hopf bifurcation theorem continuation of bifurcation curves on the parameter plane degenerate bifurcations in the space of system parameters high-order Hopf bifurcation formulas Hopf bifurcation in nonlinear systems with time delays birth of multiple limit cycles appendix.

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.

An overview of bifurcation, chaos and nonlinear dynamics in control systems

Abstract This article offers a brief review of the fundamental concepts of bifurcation and chaos in nonlinear dynamical and control systems. Both the time-domain and frequency-domain versions of the classical Hopf bifurcation theory are studied in detail. Generalized (or degenerate) Hopf bifurcation is also discussed. Theoretical analysis and potential applications of the bifurcation theory in power systems are introduced. Meanwhile, chaos and the route to chaos from period-doubling bifurcations are described. In particular, chaos and bifurcations in feedback control systems and adaptive control systems are addressed. Because a nonlinear control system is by nature a very complex nonautonomous dynamical system due essentially to the involving of the control input, understanding and utilizing the rich dynamics of nonlinear control systems have an important impact in the modern technology. It calls for new effort and endeavor devoted to this scientific and engineering challenge.

BIFURCATIONS AND HOPF DEGENERACIES IN NONLINEAR FEEDBACK SYSTEMS WITH TIME DELAY

An application of the well-developed frequency-domain approach to detect oscillations in nonlinear feedback systems with time delay is presented. The method depends on an early proof of the Hopf bifurcation theorem known as the Graphical Hopf Theorem (GHT). Several nondegeneracy conditions are included to apply the GHT in nonlinear systems with time delay. The singular conditions corresponding to degeneracies, which include static and dynamic bifurcations, as well as some special cases of degenerate Hopf bifurcations and multiple crossings, are also discussed. Two Single-Input Single-Output (SISO) feedback systems with odd nonlinearities are presented as examples to show that the proposed technique and a standard simulation method have very good agreement in the results, yet the GHT is much simpler in calculation. The first one shows an application of the GHT under classical Hopf conditions while the second emphasizes the presence of degenerate Hopf bifurcations and multiple crossings. For both examples, and others which have appeared recently in the literature, a considerable simplification of the formulas for recovering periodic solutions is also provided in this paper.

Bifurcation Analysis on a Multimachine Power System Model

In this article bifurcation analysis of the 9 bus power system model corresponding to the Western Systems Coordinating Council is performed. In order to use standard continuation packages like MATCONT, a full ordinary differential equations model, including the corresponding dynamics of the control loops and the transmission lines, is derived. Different loading conditions are studied by using the load demands as bifurcation parameters. For variations of one of the loads, it is shown that the equilibrium point undergoes Hopf and saddle-node bifurcations. Furthermore, the bifurcation analysis varying two loads simultaneously reveals the existence of a pair of double Hopf and a zero-Hopf bifurcations, acting as organizing centers of the dynamics. Finally, a power system stabilizer has been added in order to modify the location of a Hopf bifurcation curve.

FREQUENCY DOMAIN APPROACH TO COMPUTATION AND ANALYSIS OF BIFURCATIONS AND LIMIT CYCLES: A TUTORIAL

This paper introduces a frequency domain approach together with some techniques and methodologies for the computation and analysis of bifurcations and limit cycles arising in nonlinear dynamical systems. The frequency domain approach discussed in this paper originates from the classical feedback control systems theory, which has been proven to be successful and efficient for the computation and analysis of regular as well as singular bifurcations and stable as well as unstable limit cycles. While describing these techniques and methods, two representative yet distinct applications of the approach are studied in detail: The graphical analysis of multiple parametric bifurcation curves and the numerical computation of multiple limit cycles. Compared to the classical time domain methods, both the advantages and the limitations of the frequency domain approach are analyzed and discussed. It is believed that this frequency domain approach to the study of nonlinear dynamics has great potential and promising future in both theory and applications.

Computations of limit cycles via higher-order harmonic balance approximation

The detection of limit cycles arising from Hopf bifurcation phenomena by applying the harmonic balance method with different higher-order approximations is discussed. The results are presented using a graphical procedure that indicates clearly how the predictions of amplitude and frequency of a periodic solution can be improved by using higher and higher order approximations. Complete and explicit formulas for eighth-order harmonic balance approximation are provided. >

Hopf bifurcation for maps: a frequency-domain approach

The application of the graphical Hopf theorem (GHT) as a tool for detecting invariant cycles in maps is presented. The invariant cycle emerging from the bifurcation is approximated using an analogous version of the GHT for continuous-time systems. This technique is formulated in the so-called frequency domain and it involves the use of the Nyquist stability criterion and the harmonic balance method. Some examples are included for illustration.

Degenerate Hopf bifurcations via feedback system theory: Higher-order harmonic balance

Abstract In this work we present some higher-order Hopf bifurcation formulas to analyze the rich dynamic behavior of chemical systems under the failure of some hypotheses of the classical Hopf theorem. The treatment is done entirely in an equivalent formulation in the so-called “frequency domain”. This approach gives a natural interpretation about some Hopf degeneracies involving limit points on the periodic branch and multiple periodic solutions. Moreover, we calculate the expression for the second curvature coefficient, extending previous results using this formulation. This coefficient is equivalent to one of the higher-order coefficients in the Hopf bifurcation normal form used previously by other researchers.