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Reachability of dimension-bounded linear systems.

Yiliang Li1, Haitao Li2, Jun-E Feng1

  • 1School of Mathematics, Shandong University, Jinan 250100, China.

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Summary
This summary is machine-generated.

This study investigates the reachability of dimension-bounded linear systems. We present methods to determine reachability and show that reachable subsets form linear spaces, crucial for control theory analysis.

Keywords:
annihilator polynomialdimension-bounded linear systemreachable subsetstate dimension

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Area of Science:

  • Control Theory
  • Linear Systems Analysis
  • State-Space Methods

Background:

  • Dimension-bounded linear systems exhibit time-varying state dimensions, posing challenges for traditional reachability analysis.
  • Understanding the evolution of reachable states is critical for designing effective control strategies.

Purpose of the Study:

  • To investigate the reachability properties of dimension-bounded linear systems.
  • To develop methods for assessing reachability and characterizing reachable subsets.
  • To explore the relationship between invariant spaces and reachable subsets over time.

Main Methods:

  • Derivation of state dimension expressions for time-varying systems.
  • Development of a novel method for judging the reachability of vector spaces.
  • Proof that the t-step reachable subset constitutes a linear space.
  • Utilization of rank conditions for verifying t-step state reachability.
  • Application of annihilator polynomials to analyze invariant spaces and reachable subsets.

Main Results:

  • A method for judging the reachability of a given vector space $ \mathcal{V}_{r} $ is proposed.
  • The t-step reachable subset is proven to be a linear space, with a computational method provided.
  • t-step reachability of a given state is verified using a rank condition.
  • The relationship between invariant spaces and reachable subsets after an invariant time point $ t^{\ast} $ is illustrated using annihilator polynomials.
  • Inclusion relations between reachable subsets at different time points are demonstrated.

Conclusions:

  • The study provides a comprehensive framework for analyzing the reachability of dimension-bounded linear systems.
  • The findings contribute to a deeper understanding of state-space evolution and control system design.
  • The developed methods and theoretical insights are valuable for both theoretical research and practical applications in control engineering.