Volterra Series Expansion for Bilinear Systems in a Nonlinear Feedback Loop

1981 ◽  
Vol 103 (2) ◽  
pp. 84-88
Author(s):  
A. Bertuzzi ◽  
C. Bruni ◽  
A. Gandolfi ◽  
A. Germani
Keyword(s):  
Series Expansion ◽  
Existence And Uniqueness ◽  
Feedback Loop ◽  
Volterra Series ◽  
Approximation Theorem ◽  
Bilinear Systems ◽  
Nonlinear Feedback ◽  
Input Output ◽  
Distributed Bilinear Systems ◽  
Weierstrass Approximation Theorem

In this paper the class of distributed bilinear systems in a nonlinear feedback loop is considered. First of all the existence and uniqueness of solution is stated when input and state belong to L2 spaces. Then the existence of the Volterra series expansion for the input-output map is proved for inputs in any sphere around the origin; the proof is based on a new version of the Weierstrass approximation theorem.

Quasi-Periodic Orbits in the Five-Dimensional Nondissipative Lorenz Model: The Role of the Extended Nonlinear Feedback Loop

2018 ◽  
Vol 28 (06) ◽  
pp. 1850072 ◽  
Author(s):  
Sara Faghih-Naini ◽  
Bo-Wen Shen
Keyword(s):  
Periodic Solution ◽  
Feedback Loop ◽  
Three Dimensional ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Restoring Force ◽  
Oscillatory Solutions ◽  
Nonlinear Temperature

A recent study suggested that the nonlinear feedback loop (NFL) of the three-dimensional nondissipative Lorenz model (3D-NLM) serves as a nonlinear restoring force by producing nonlinear oscillatory solutions as well as linear periodic solutions near a nontrivial critical point. This study discusses the role of the extension of the NFL in producing quasi-periodic trajectories using a five-dimensional nondissipative Lorenz model (5D-NLM). An analytical solution to the locally linear 5D-NLM is first obtained to illustrate the association of the extended NFL and two incommensurate frequencies whose ratio is irrational, yielding a quasi-periodic solution. The quasi-periodic solution trajectory moves endlessly on a torus but never intersects itself. While the NFL of the 3D-NLM consists of a pair of downscaling and upscaling processes, the extended NFL within the 5D-NLM additionally introduces two new pairs of downscaling and upscaling processes that are enabled by two high wavenumber modes. One pair of downscaling and upscaling processes provides a two-way interaction between the original (primary) Fourier modes of the 3D-NLM and the newly-added (secondary) Fourier modes of the 5D-NLM. The other pair of downscaling and upscaling processes involves interactions amongst the secondary modes. By comparing the numerical simulations using one- and two-way interactions, we illustrate that the two-way interaction is crucial for producing the quasi-periodic solution. A follow-up study using a 7D nondissipative LM shows that a further extension of NFL, which may appear throughout the spatial mode-mode interactions rooted in the nonlinear temperature advection, is capable of producing one more incommensurate frequency.

Combinatorial Approximations of Volterra Series by Bilinear Systems

1991 ◽  
pp. 304-315 ◽  
Author(s):  
Françoise Lamnabhi-Lagarrigue ◽  
Pierre Leroux ◽  
Xavier G. Viennot
Keyword(s):  
Volterra Series ◽  
Bilinear Systems

On Convergence of Volterra Series Expansion of a Class of Nonlinear Systems

2017 ◽  
Vol 19 (3) ◽  
pp. 1089-1102 ◽  
Author(s):  
Xingjian Jing ◽  
Zhenlong Xiao
Keyword(s):  
Nonlinear Systems ◽  
Series Expansion ◽  
Volterra Series

scholarly journals Slow oscillations in blood pressure via a nonlinear feedback model

2001 ◽  
Vol 280 (4) ◽  
pp. R1105-R1115 ◽  
Author(s):  
John V. Ringwood ◽  
Simon C. Malpas
Keyword(s):  
Blood Pressure ◽  
Feedback Loop ◽  
Sympathetic Nerve Activity ◽  
Linear Feedback ◽  
Scaling Effects ◽  
Nonlinear Feedback ◽  
Nerve Activity ◽  
Feedback Model ◽  
The Central Nervous System ◽  
Proportional Derivative Controller

Blood pressure is well established to contain a potential oscillation between 0.1 and 0.4 Hz, which is proposed to reflect resonant feedback in the baroreflex loop. A linear feedback model, comprising delay and lag terms for the vasculature, and a linear proportional derivative controller have been proposed to account for the 0.4-Hz oscillation in blood pressure in rats. However, although this model can produce oscillations at the required frequency, some strict relationships between the controller and vasculature parameters must be true for the oscillations to be stable. We developed a nonlinear model, containing an amplitude-limiting nonlinearity that allows for similar oscillations under a very mild set of assumptions. Models constructed from arterial pressure and sympathetic nerve activity recordings obtained from conscious rabbits under resting conditions suggest that the nonlinearity in the feedback loop is not contained within the vasculature, but rather is confined to the central nervous system. The advantage of the model is that it provides for sustained stable oscillations under a wide variety of situations even where gain at various points along the feedback loop may be altered, a situation that is not possible with a linear feedback model. Our model shows how variations in some of the nonlinearity characteristics can account for growth or decay in the oscillations and situations where the oscillations can disappear altogether. Such variations are shown to accord well with observed experimental data. Additionally, using a nonlinear feedback model, it is straightforward to show that the variation in frequency of the oscillations in blood pressure in rats (0.4 Hz), rabbits (0.3 Hz), and humans (0.1 Hz) is primarily due to scaling effects of conduction times between species.

Nonlinear models of the first synapse in the light-adapted fly retina

1995 ◽  
Vol 74 (6) ◽  
pp. 2538-2547 ◽  
Author(s):  
M. Juusola ◽  
M. Weckstrom ◽  
R. O. Uusitalo ◽  
M. J. Korenberg ◽  
A. S. French
Keyword(s):  
Membrane Potential ◽  
Synaptic Input ◽  
Light Adaptation ◽  
Volterra Series ◽  
Dynamic Properties ◽  
Second Order ◽  
Synaptic Function ◽  
Input Output ◽  
Light Stimuli ◽  
Synaptic Gain

1. Randomly modulated light stimuli were used to characterize the nonlinear dynamic properties of the synapse between photoreceptors and large monopolar neurons (LMC) in the fly retina. Membrane potential fluctuations produced by constant variance contrast stimuli were recorded at eight different levels of background light intensity. 2. Representation of the photoreceptor-LMC input-output data in the form of traditional characteristic curves indicated that synaptic gain was reduced by light adaptation. However, this representation did not include the time-dependent properties of the synaptic function, which are known to be nonlinear. Therefore nonlinear systems analysis was used to characterize the synapse. 3. The responses of photoreceptors and LMCs to random light fluctuations were characterized by second-order Volterra series, with kernel estimation by the parallel cascade method. Photoreceptor responses were approximately linear, but LMC responses were clearly nonlinear. 4. Synaptic input-output relationships were measured by passing the light stimuli to LMCs through the measured photoreceptor characteristics to obtain an estimate of the synaptic input. The resulting nonlinear synaptic functions were well characterized by second-order Volterra series. They could not be modeled by a linear-nonlinear-linear cascade but were better approximated by a nonlinear-linear-nonlinear cascade. 5. These results support two possible structural models of the synapse, the first having two parallel paths for signal flow between the photoreceptor and LMC, and the second having two distinct nonlinear operations, occurring before and after chemical transmission. 6. The two models were cach used to calculate the synaptic gain to a brief change in photoreceptor membrane potential. Both models predicted that synaptic gain is reduced by light adaptation.

Sign in / Sign up

Export Citation Format

Share Document