154 research outputs found
Aspects of Bifurcation Theory for Piecewise-Smooth, Continuous Systems
Systems that are not smooth can undergo bifurcations that are forbidden in
smooth systems. We review some of the phenomena that can occur for
piecewise-smooth, continuous maps and flows when a fixed point or an
equilibrium collides with a surface on which the system is not smooth. Much of
our understanding of these cases relies on a reduction to piecewise linearity
near the border-collision. We also review a number of codimension-two
bifurcations in which nonlinearity is important.Comment: pdfLaTeX, 9 figure
Design of the Impulsive Goodwin's Oscillator in 1-cycle
This paper presents a systematic approach to design a hybrid oscillator that admits an orbitally stable periodic solution of a certain type with pre-defined parameters. The parsimonious structure of the Impulsive Goodwin's oscillator (IGO) is selected for the implementation due to its well-researched rich nonlinear dynamics. The IGO is a feedback interconnection of a positive third-order continuous-time LTI system and a nonlinear frequency and amplitude impulsive modulator. A design algorithm based on solving a bilinear matrix inequality is proposed yielding the slope values of the modulation functions that guarantee stability of the fixed point defining the designed periodic solution. Further, assuming Hill function parameterizaton of the pulse-modulated feedback, the parameters of those rendering the desired stationary properties are calculated. The character of perturbed solutions in vicinity of the fixed point is controlled through localization of the multipliers. The proposed design approach is illustrated by a numerical example. Bifurcation analysis of the resulting oscillator is performed to explore the nonlinear phenomena in vicinity of the designed dynamics
Impulsive feedback control for dosing applications
This paper addresses a design procedure of pulse-modulated feedback control
solving a dosing problem defined for implementation in a manual mode. Discrete
dosing, as a control strategy, is characterized by exerting control action on
the plant in impulsive manner at certain time instants. Dosing applications
appear primarily in chemical industry and medicine where the control signal
constitutes a sequence of (chemically or pharmacologically) active substance
quantities (doses) administered to achieve a desired result. When the doses and
the instants of their administration are adjusted as functions of some measured
variable, a feedback control loop exhibiting nonlinear dynamics arises. The
impulsive character of the interaction between the controller and the plant
makes the resulting closed-loop system non-smooth. Limitations of the control
law with respect to control goals are discussed. An application of the approach
at hand to neuromuscular blockade in closed-loop anesthesia is considered in a
numerical example
Transitions from phase-locked dynamics to chaos in a piecewise-linear map
Recent work has shown that torus formation in piecewise-smooth maps can take place through a special type of border-collision bifurcation in which a pair of complex conjugate multipliers for a stable cycle abruptly jump out of the unit circle. Transitions from an ergodic to a resonant torus take place via border-collision fold bifurcations. We examine the transition to chaos through torus destruction in such maps. Considering a piecewise-linear normal-form map we show that this transition, by virtue of the interplay of border-collision bifurcations with period-doubling and homoclinic bifurcations, can involve mechanisms that differ qualitatively from those described by Afraimovich and Shilnikov
Output corridor control via design of impulsive Goodwin's oscillator
In the Impulsive Goodwin's oscillator (IGO), a continuous positive linear
time-invariant (LTI) plant is controlled by an amplitude- and
frequency-modulated feedback into an oscillating solution. Self-sustained
oscillations in the IGO model have been extensively used to portray periodic
rhythms in endocrine systems, whereas the potential of the concept as a
controller design approach still remains mainly unexplored. This paper proposes
an algorithm to design the feedback of the IGO so that the output of the
continuous plant is kept (at stationary conditions) within a pre-defined
corridor, i.e. within a bounded interval of values. The presented framework
covers single-input single-output LTI plants as well as positive Wiener and
Hammerstein models that often appear in process and biomedical control. A
potential application of the developed impulsive control approach to a minimal
Wiener model of pharmacokinetics and pharmacodynamics of a muscle relaxant used
in general anesthesia is discussed
Impulsive Goodwin’s Oscillator Model of Endocrine Regulation: Local Feedback Leads to Multistability
The impulsive Goodwin’s oscillator (IGO) is a hybrid model that captures complex dynamics arising in continuous systems controlled by pulse-modulated (event-based) feedback. Being conceived to describe pulsatile endocrine regulation, it has also found applications in e.g. pharmacokinetics. The original version of the IGO assumes the continuous part of the model to be a chain of first-order blocks. This paper explores the nonlinear phenomena arising due to the introduction of a local continuous feedback as suggested by the endocrine applications. The effects caused by a nonlinear feedback law parameterized by a Hill function are compared to those arising due to a simpler and previously treated case of affine feedback law. The hybrid dynamics of the IGO are reduced to a (discrete) Poincaré map governing the propagation of the model’s continuous states through the firing instants of the impulsive feedback. Bifurcation analysis of the map reveals in particular that both the local Hill function and affine feedback can lead to multistability, which phenomenon has not been observed in the usual IGO model
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