Comparison of Nondeterministic Stable Linear Systems by ($\gamma,\delta$)-Similarity

We introduce ($\gamma,\delta$)-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 nondeterministic systems. As such, ($\gamma,\delta$)-similarity is a notion that characterizes approximation of input–output behavior, whereas existing notions of simulation target equivalence. Next, as this approximation is specified in terms of the $\mathcal {L}_{2}$ signal norm, ($\gamma,\delta$)-similarity allows for integration with existing methods for analysis and synthesis of control systems, in particular, robust control techniques. We characterize the notion of ($\gamma,\delta$)-similarity as a linear matrix inequality feasibility problem and derive its interpretation in terms of transfer matrices. Our study on the compositional properties of ($\gamma,\delta$)-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.