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Second Order systems II

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In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
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In an open-loop system, such as a basic thermostat, the poles of the transfer function influence the system's response but do not determine its stability. However, when feedback is introduced to form a closed-loop system, such as an advanced thermostat that adjusts heating based on room temperature, stability is governed by the new poles of the closed-loop transfer function.
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Linear Approximation in Frequency Domain

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An Experimental Platform to Study the Closed-loop Performance of Brain-machine Interfaces

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Frequency adaptation in controlled stochastic resonance utilizing delayed feedback method: two-pole approximation for

¹Department of Applied Analysis and Complex Dynamical Systems, Graduate School of Informatics, Kyoto University, Kyoto, Japan. tutu@acs.i.kyoto-u.ac.jp

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|July 30, 2011

Summary

Stochastic resonance (SR) with time-delayed feedback control shows noise-induced switching. Optimal noise levels entrain the output signal phase to the input, enhancing SR performance.

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Area of Science:

Nonlinear Dynamics
Complex Systems Analysis

Background:

Stochastic resonance (SR) typically enhances weak signal detection in nonlinear systems.
Time-delayed feedback control is a method to influence system dynamics.

Purpose of the Study:

To investigate stochastic resonance (SR) enhanced by time-delayed feedback control.
To analyze the entrainment of output signal phase to input signal phase through controlled noise levels.

Main Methods:

Langevin equation for a bistable system without control.
Analysis of a feedback loop with a delay time of half the input signal period.
Development of a delay-coordinate series expansion method for non-Markovian systems.
Dichotomic model used for response function analysis.

Main Results:

Time-delayed feedback control induces a noise-driven oscillatory switching cycle.
Noise constructively adapts the switching cycle frequency to the input signal frequency at optimal noise levels.
Phase entrainment of the output signal to the input signal is achieved, moving from a phase-slipped state.
Power loss and response function characteristics are analyzed using a dichotomic model and Laplace transforms.

Conclusions:

The proposed delay-coordinate series expansion method is a potential tool for analyzing SR with delayed feedback.
The D-dependent behavior of response function poles characterizes power loss structure.
The study provides analytical results for correlation functions and power spectral density in this controlled SR system.