Abstract
We introduce (γ,δ)-similarity, a notion of system comparison that measures to what extent two stable linear dynamical systems behave similarly in an input-output sense. This behavioral similarity is characterized by measuring the sensitivity of the difference between the two output trajectories in terms of the external inputs to the two potentially non-deterministic systems. As such, (γ,δ)-similarity is a notion that characterizes <italic>approximation</italic> of input-output behavior, whereas existing notions of simulation target equivalence. Next, as this approximation is specified in terms of the <inline-formula><tex-math notation="LaTeX">$\mathcal {L}_{2}$</tex-math></inline-formula> signal norm, (γ,δ)-similarity allows for integration with existing methods for analysis and synthesis of control systems, in particular, robust control techniques. We characterize the notion of (γ,δ)-similarity as a linear matrix inequality feasibility problem and derive its interpretation in terms of transfer matrices. Our study on the compositional properties of (γ,δ)-similarity shows that the notion is preserved through series and feedback interconnections. This highlights its potential application in compositional reasoning, namely abstraction and modular synthesis of large-scale interconnected dynamical systems. We further illustrate our results in an electrical network example.
Original language | English |
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Pages (from-to) | 8617-8632 |
Number of pages | 16 |
Journal | IEEE Transactions on Automatic Control |
Volume | 69 |
Issue number | 12 |
Early online date | 18-Jun-2024 |
DOIs | |
Publication status | Published - Dec-2024 |
Keywords
- Abstraction
- approximation
- Cognition
- compositional reasoning
- Control theory
- Dynamical systems
- Linear systems
- non-deterministic systems
- Sensitivity
- simulation relation
- Trajectory
- Vectors