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State Space Representation01:27

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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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State-space representation is a powerful tool for simulating physical systems on digital computers, necessitating the conversion of the transfer function into state-space form. Consider an nth-order linear differential equation with constant coefficients, like those encountered in an RLC circuit. The state variables are selected as the output and its n−1 derivatives. Differentiating these variables and substituting them back into the original equation produces the state equations.
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The conversion of state-space representation to a transfer function is a fundamental process in system analysis. It provides a method for transitioning from a time-domain description to a frequency-domain representation, which is crucial for simplifying the analysis and design of control systems.
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Overlapping state-input-output decomposition of complex systems with guaranteed structural observability and

Sahar Maleki1, Mehdi Rahmani1, Hassan Zarabadipour1

  • 1Department of Electrical Engineering, Imam-Khomeini International University, Qazvin, Iran.

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|April 16, 2026
PubMed
Summary

This study introduces an overlapping decomposition method for complex dynamic systems, ensuring subsystem observability and controllability. The approach systematically detects disjoint and overlapping structures, simplifying analysis and control.

Keywords:
Complex dynamic systemsGraph theoryOptimizationOverlapping -decompositionStructural observability and controllability

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Area of Science:

  • Control Systems Engineering
  • Systems Theory
  • Optimization

Background:

  • Complex dynamic systems present significant analytical and control challenges.
  • Decomposition into lower-order subsystems is a key strategy to manage complexity.
  • Existing methods may not effectively handle overlapping subsystem structures.

Purpose of the Study:

  • To propose an overlapping ϵ-decomposition method for complex dynamic systems.
  • To guarantee structural observability and controllability of the decomposed subsystems.
  • To systematically detect both disjoint and overlapping structures within dynamic systems.

Main Methods:

  • Formulation of a single optimization problem for system decomposition.
  • Development of an approach based on optimization and graph theory.
  • Extension of decomposition to ensure structural observability and controllability.

Main Results:

  • A novel overlapping ϵ-decomposition technique is presented.
  • The method effectively decomposes systems into weakly coupled subsystems with potential overlaps.
  • Simulations demonstrate the approach's effectiveness on ship boiler and power system case studies.

Conclusions:

  • The proposed optimization-based decomposition systematically identifies system structures.
  • The method ensures essential properties like structural observability and controllability.
  • This approach offers a robust tool for analyzing and controlling complex dynamic systems.