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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.
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Uncertainty: Overview

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In analytical chemistry, we often perform repetitive measurements to detect and minimize inaccuracies caused by both determinate and indeterminate errors. Despite the cares we take, the presence of random errors means that repeated measurements almost never have exactly the same magnitude. The collective difference between these measurements - observed values - and the estimated or expected value is called uncertainty. Uncertainty is conventionally written after the estimated or expected value.

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The atomic mass of an element varies due to the relative ratio of its isotopes. A sample's relative proportion of oxygen isotopes influences its average atomic mass. For instance, if we were to measure the atomic mass of oxygen from a sample, the mass would be a weighted average of the isotopic masses of oxygen in that sample. Since a single sample is not likely to perfectly reflect the true atomic mass of oxygen for all the molecules of oxygen on Earth, the mass we obtain from this...

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An experiment often consists of more than a single step. In this case, measurements at each step give rise to uncertainty. Because the measurements occur in successive steps, the uncertainty in one step necessarily contributes to that in the subsequent step. As we perform statistical analysis on these types of experiments, we must learn to account for the propagation of uncertainty from one step to the next. The propagation of uncertainty depends on the type of arithmetic operation performed on...

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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A flexible bounded stochastic framework for uncertainty and reliability in physical systems.

Tmader Alballa¹, Abdus Saboor², Muhammad Danish³

¹Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh, 11671, Saudi Arabia.

Scientific Reports

|April 30, 2026

Summary

This summary is machine-generated.

A new flexible statistical framework models bounded random variables using a beta transformation of the Kumaraswamy distribution. This provides a robust tool for uncertainty quantification and reliability analysis in engineering and physical systems.

Keywords:

Bounded-response modeling Engineering measurements Environmental data Maximum likelihood estimation Statistical model Unit-interval data

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Area of Science:

Statistics
Probability Theory
Reliability Engineering

Background:

Bounded random variables are common in physical and engineering systems, representing constrained measurements.
Existing models may lack flexibility in capturing diverse boundary behaviors and hazard rate structures.

Purpose of the Study:

Introduce a novel four-parameter bounded stochastic framework.
Develop rigorous theoretical properties and analytical tools for the new model.
Demonstrate practical relevance through an application to engineered system data.

Main Methods:

Beta transformation of the Kumaraswamy (Kw) baseline distribution.
Derivation of theoretical properties: identifiability, limiting behavior, PDF and HR shapes.
Calculation of moments, probability-weighted moments, and information-theoretic measures (Shannon, Rényi, Tsallis entropies, Kullback-Leibler divergence).
Development of likelihood-based inference, including score functions, Fisher information, and maximum likelihood estimators (MLE) properties.

Main Results:

A flexible four-parameter family of distributions for bounded variables is established.
Closed-form expressions for moments and probability-weighted moments are derived.
Comprehensive information-theoretic characterizations are provided.
Asymptotic properties of MLEs are established, facilitating inference.

Conclusions:

The proposed framework offers a mathematically rigorous and interpretable approach for analyzing bounded data.
It enhances uncertainty quantification and reliability analysis in various scientific and engineering domains.
The model's flexibility allows for diverse boundary behaviors and hazard rate structures.