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Second Order systems II

Second Order systems II

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.
If ζ...

Stability

Stability

The time response of a linear time-invariant (LTI) system can be divided into transient and steady-state responses. The transient response represents the system's initial reaction to a change in input and diminishes to zero over time. In contrast, the steady-state response is the behavior that persists after the transient effects have faded.
The stability of an LTI system is determined by the roots of its characteristic equation, known as poles. A system is stable if it produces a bounded...

BIBO stability of continuous and discrete -time systems

BIBO stability of continuous and discrete -time systems

System stability is a fundamental concept in signal processing, often assessed using convolution. For a system to be considered bounded-input bounded-output (BIBO) stable, any bounded input signal must produce a bounded output signal. A bounded input signal is one where the modulus does not exceed a certain constant at any point in time.
To determine the BIBO stability, the convolution integral is utilized when a bounded continuous-time input is applied to a Linear Time-Invariant (LTI) system.

Transient and Steady-state Response

Transient and Steady-state Response

In control systems, test signals are essential for evaluating performance under various conditions. The ramp function is effective for systems undergoing gradual changes, while the step function is suitable for assessing systems facing sudden disturbances. For systems subjected to shock inputs, the impulse function is the most appropriate test signal.
These test signals are integral in designing control systems to exhibit two key performance aspects: transient response and steady-state response.

Second Order systems I

Second Order systems I

A servo system exemplifies a second-order system, featuring a proportional controller and load elements that ensure the output position aligns with the input position. The relationship between these components is described by a second-order differential equation. Applying the Laplace transform under zero initial conditions yields the transfer function, showing how inputs are converted to outputs in the system.
By reinterpreting the system, one can derive the closed-loop transfer function, which...

State Space Representation

State Space Representation

The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
Consider an RLC circuit, a...

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Related Experiment Video

Updated: May 7, 2026

Fabrication and Testing of Microfluidic Optomechanical Oscillators

Fabrication and Testing of Microfluidic Optomechanical Oscillators

Published on: May 29, 2014

The unsaturated bistable stochastic resonance system.

Wenli Zhao¹, Juan Wang, Linze Wang

¹School of Mechanical Engineering, Hangzhou Dianzi University, Hangzhou 310018, China.

Chaos (Woodbury, N.Y.)

|October 5, 2013

Summary

This summary is machine-generated.

This study introduces an improved piecewise bistable system for enhanced stochastic resonance (SR). The new system significantly boosts signal-to-noise ratio (SNR), outperforming traditional saturation bistable systems in weak signal extraction.

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Last Updated: May 7, 2026

Fabrication and Testing of Microfluidic Optomechanical Oscillators

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All-electronic Nanosecond-resolved Scanning Tunneling Microscopy: Facilitating the Investigation of Single Dopant Charge Dynamics

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Published on: January 19, 2018

Area of Science:

Nonlinear Dynamics
Stochastic Resonance
Signal Processing

Background:

Classical continuous bistable systems exhibit output saturation, limiting their performance in stochastic resonance (SR).
Understanding the impact of saturation on SR is crucial for optimizing signal detection in noisy environments.

Purpose of the Study:

To investigate the effect of output saturation on the stochastic resonance (SR) phenomenon in classical bistable systems.
To propose and analyze a novel piecewise bistable SR system (unsaturated bistable system) for improved performance.
To compare the signal-to-noise ratio (SNR) and weak signal extraction capabilities of the proposed system against the saturation bistable system.

Main Methods:

Theoretical analysis using adiabatic approximation theory to derive the signal-to-noise ratio (SNR) expression.
Numerical simulations to evaluate system performance in extracting weak signals from strong background noise.

Main Results:

The proposed unsaturated bistable system demonstrates a significantly improved SNR compared to the saturation bistable system.
Numerical simulations confirm the superior performance of the unsaturated bistable system in extracting weak signals from strong noise.
The piecewise bistable SR system effectively overcomes the limitations imposed by output saturation.

Conclusions:

The unsaturated bistable system offers a substantial enhancement in stochastic resonance performance.
This novel system provides a more effective approach for weak signal detection in the presence of significant background noise.
The findings have implications for improving signal processing techniques in various scientific and engineering fields.