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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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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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Controller design for an approximated nonlinear system using Wavelet Transform.

Rajeev Ranjan Pathak1, Anindita Sengupta2, Rimi Paul3

  • 1Department of Electrical & Electronics Engineering, KLE Technological University, Hubballi, India.

ISA Transactions
|August 7, 2025
PubMed
Summary
This summary is machine-generated.

This study introduces a novel Wavelet Transform (WT) method for nonlinear system control. The approach, validated on a benchmark system, offers superior performance over linearization, enhancing controller design for complex dynamics.

Keywords:
Harmony Search AlgorithmInverted PendulumLyapunov Stability analysisMinimum Description Length CriterionNonlinear systemWavelet Transform Approximated Model

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Area of Science:

  • Control Engineering
  • Signal Processing
  • Applied Mathematics

Background:

  • Nonlinear systems present significant challenges in control design.
  • Traditional linearization techniques often oversimplify system dynamics, leading to suboptimal performance.
  • Accurate system modeling is crucial for effective controller development.

Purpose of the Study:

  • To develop a method for representing nonlinear functions using Wavelet Transform (WT) coefficients for controller design.
  • To introduce an Adaptive Sliding Mode Controller (ASMC) for WT-based approximated models (WTAM).
  • To validate the proposed WT-based control strategy on a benchmark nonlinear system.

Main Methods:

  • Nonlinear function representation using Wavelet Transform (WT) coefficients.
  • Optimal Mother Wavelet (MW) selection via the Minimum Description Length (MDL) criterion.
  • Design of an Adaptive Sliding Mode Controller (ASMC) using the Harmony Search Algorithm (HSA) for tuning.
  • Lyapunov stability analysis for controller stability verification.

Main Results:

  • The WT-based approximation technique proved superior to standard linearization for a benchmark nonlinear system.
  • The designed ASMC effectively handled unbounded disturbances, parametric uncertainties, and chattering.
  • Real-time validation on a Cart Inverted Pendulum WTAM demonstrated the controller's efficiency.

Conclusions:

  • Wavelet Transform (WT) provides an effective method for approximating nonlinear functions, facilitating controller design.
  • The proposed Adaptive Sliding Mode Controller (ASMC) offers robust and stable control for nonlinear systems.
  • This WT-based approach represents a significant advancement in nonlinear control engineering.