31 research outputs found
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
Border Collision Route to Quasiperiodicity: Numerical Investigation and Experimental Confirmation
Numerical studies of higher-dimensional piecewise-smooth systems have recently shown how a torus can arise from a periodic cycle through a special type of border-collision bifurcation. The present article investigates this new route to quasiperiodicity in the two-dimensional piecewise-linear normal form map. We have obtained the chart of the dynamical modes for this map and showed that border-collision bifurcations can lead to the birth of a stable closed invariant curve associated with quasiperiodic or periodic dynamics. In the parameter regions leading to the existence of an invariant closed curve, there may be transitions between an ergodic torus and a resonance torus, but the mechanism of creation for the resonance tongues is distinctly different from that observed in smooth maps. The transition from a stable focus point to a resonance torus may lead directly to a new focus of higher periodicity, e.g., a period-5 focus. This article also contains a discussion of torus destruction via a homoclinic bifurcation in the piecewise-linear normal map. Using a dc-dc converter with two-level control as an example, we report the first experimental verification of the direct transition to quasiperiodicity through a border-collision bifurcation
Hidden attractors in fundamental problems and engineering models
Recently a concept of self-excited and hidden attractors was suggested: an
attractor is called a self-excited attractor if its basin of attraction
overlaps with neighborhood of an equilibrium, otherwise it is called a hidden
attractor. For example, hidden attractors are attractors in systems with no
equilibria or with only one stable equilibrium (a special case of
multistability and coexistence of attractors). While coexisting self-excited
attractors can be found using the standard computational procedure, there is no
standard way of predicting the existence or coexistence of hidden attractors in
a system. In this plenary survey lecture the concept of self-excited and hidden
attractors is discussed, and various corresponding examples of self-excited and
hidden attractors are considered