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<p>We compensate for the scalability issues in controller synthesis by developing a hierarchical control scheme within the framework of (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\upgamma ,\updelta \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">γ</mi> <mo>,</mo> <mi mathvariant="normal">δ</mi> </mrow> </math></EquationSource> </InlineEquation>)-similarity, which measures to what extent a potentially non-deterministic system satisfies specifications expressed as solution trajectories of a dynamical ‘specification’ system. This scheme synthesizes a controller for a non-deterministic ‘concrete’ system in three <i>hierarchical</i> steps. First, an ‘abstract’ system, which is a low-dimensional model of the concrete system, is obtained. Then, a controller is designed for the abstract system. At last, the abstract controller is refined into the concrete controller through an ‘interface’. To enable this, we introduce and characterize the notion of (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\upgamma ,\updelta \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">γ</mi> <mo>,</mo> <mi mathvariant="normal">δ</mi> </mrow> </math></EquationSource> </InlineEquation>)-abstraction that utilizes an <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathcal {L}_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">L</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> approximation metric to measure the behavioral similarity of the concrete system and its abstraction in the presence of the interface. We utilize this characterization to propose a step-by-step procedure to construct the interface. We then synthesize the abstract controller and refine it into a concrete one.</p>

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Hierarchical controller synthesis using (\(\upgamma ,\updelta \))-Similarity

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月18日

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  • Armin Pirastehzad,
  • Arjan van der Schaft,
  • Bart Besselink

摘要

We compensate for the scalability issues in controller synthesis by developing a hierarchical control scheme within the framework of ( \(\upgamma ,\updelta \) γ , δ )-similarity, which measures to what extent a potentially non-deterministic system satisfies specifications expressed as solution trajectories of a dynamical ‘specification’ system. This scheme synthesizes a controller for a non-deterministic ‘concrete’ system in three hierarchical steps. First, an ‘abstract’ system, which is a low-dimensional model of the concrete system, is obtained. Then, a controller is designed for the abstract system. At last, the abstract controller is refined into the concrete controller through an ‘interface’. To enable this, we introduce and characterize the notion of ( \(\upgamma ,\updelta \) γ , δ )-abstraction that utilizes an \(\mathcal {L}_2\) L 2 approximation metric to measure the behavioral similarity of the concrete system and its abstraction in the presence of the interface. We utilize this characterization to propose a step-by-step procedure to construct the interface. We then synthesize the abstract controller and refine it into a concrete one.