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BIBO stability of continuous and discrete -time systems01:24

BIBO stability of continuous and discrete -time systems

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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.
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Cruise control systems in cars are designed as multi-input systems to maintain a driver's desired speed while compensating for external disturbances such as changes in terrain. The block diagram for a cruise control system typically includes two main inputs: the desired speed set by the driver and any external disturbances, such as the incline of the road. By adjusting the engine throttle, the system maintains the vehicle's speed as close to the desired value as possible.
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Multimachine stability analysis is crucial for understanding the dynamics and stability of power systems with multiple synchronous machines. The objective is to solve the swing equations for a network of M machines connected to an N-bus power system.
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Linear Approximation in Time Domain01:21

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Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
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Linear time-invariant Systems01:23

Linear time-invariant Systems

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A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
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Mechanistic models are utilized in individual analysis using single-source data, but imperfections arise due to data collection errors, preventing perfect prediction of observed data. The mathematical equation involves known values (Xi), observed concentrations (Ci), measurement errors (εi), model parameters (ϕj), and the related function (ƒi) for i number of values. Different least-squares metrics quantify differences between predicted and observed values. The ordinary least...
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Data-driven prediction of multistable systems from sparse measurements.

Bryan Chu1, Mohammad Farazmand1

  • 1Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205, USA.

Chaos (Woodbury, N.Y.)
|July 9, 2021
PubMed
Summary
This summary is machine-generated.

We developed a data-driven method using sparsity-promoting metric-learning (SPML) to predict the final states of complex systems from limited data. This approach accurately forecasts system behavior using sparse measurements, crucial for pattern formation and biological models.

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Area of Science:

  • Complex Systems
  • Computational Mathematics
  • Applied Physics

Background:

  • Multistable systems exhibit multiple stable states, making their long-term behavior prediction challenging.
  • Sparse spatial measurements are often the only feasible data acquisition method for complex systems.
  • Predicting asymptotic states is vital for understanding pattern formation and biological dynamics.

Purpose of the Study:

  • To develop a data-driven, semi-supervised classification method for predicting the asymptotic states of multistable systems using sparse measurements.
  • To introduce a novel sparsity-promoting metric-learning (SPML) optimization for quantifying proximity to precomputed states.
  • To ensure the learned metric is compatible with existing data and computable from sparse observations.

Main Methods:

  • A semi-supervised classification approach was employed.
  • Sparsity-Promoting Metric-Learning (SPML) optimization was introduced to learn a metric from precomputed data.
  • The method was validated on a reaction-diffusion equation and a FitzHugh-Nagumo model.

Main Results:

  • The SPML optimization was proven to be convex and non-degenerate.
  • For a reaction-diffusion equation, SPML achieved 95% accuracy in predicting asymptotic behavior from two-point measurements.
  • For the FitzHugh-Nagumo model, SPML achieved 90% accuracy from one-point measurements, also identifying optimal measurement locations.

Conclusions:

  • The developed data-driven method effectively predicts asymptotic states of multistable systems using sparse measurements.
  • SPML provides an accurate and efficient tool for analyzing complex systems in pattern formation and computational neuroscience.
  • The learned metric guides efficient data acquisition for accurate state prediction.