An Input-Output Data-Driven Dissipativity Approach for Compositional Stability Certification of Interconnected LTI MIMO Systems

We propose an input-output data-driven framework for certifying the stability of interconnected multiple-input-multiple-output linear time-invariant discrete-time systems via QSR-dissipativity. That is, by using measured input-output trajectories of each subsystem, we verify dissipative properties and extract local passivity indices without requiring an explicit model identification. These passivity indices are then used to derive conditions under which the equilibrium of the interconnected system is stable. In particular, the framework identifies how the lack of passivity in some subsystems can be compensated by surpluses in others. The proposed approach enables a compositional stability analysis by combining subsystem-level conditions into a criterion valid for the overall interconnected system. We illustrate via a numerical case study, how to compute channel-wise passivity indices and infer stability guarantees directly from data with the proposed method.

💡 Research Summary

This paper presents a novel, data-driven framework for certifying the stability of interconnected linear time-invariant (LTI) multiple-input-multiple-output (MIMO) systems in a compositional manner. The core idea is to bypass the need for explicit system identification by leveraging only locally available input-output trajectory data from each subsystem.

The methodological foundation rests on two key pillars. First, the authors develop a data-based necessary and sufficient condition for verifying QSR-dissipativity of discrete-time LTI MIMO systems via a linear matrix inequality (LMI). This is achieved by constructing Hankel matrices from persistently exciting input and corresponding output data. Under assumptions of a known system lag and persistent excitation, a non-minimal state representation is derived purely from data. This representation is then refined into a minimal realization using Kalman decomposition, ultimately yielding an LMI condition (Theorem 2) that uses only data-constructed matrices, eliminating the need for knowledge of the state-space matrices (A, B, C). This formulation notably relaxes stringent assumptions like zero initial conditions required by prior methods.

Second, the paper introduces the concept of “channel-wise passivity indices.” Recognizing that different input-output channels in a MIMO system may contribute unevenly to energy exchange, the authors propose defining individual output passivity (ρ_i) and input passivity (ν_i) indices for each channel. These are embedded in diagonal matrices Q and R within the QSR-dissipativity framework. This approach reduces the conservatism inherent in using a single global passivity index for the entire system.

The main result (Theorem 3) integrates these contributions into a compositional stability certificate. It states that the origin of the overall interconnected system is stable if two conditions hold for all subsystems: (i) each subsystem, based on its local data, satisfies the data-driven LMI condition certifying it is QSR-dissipative with channel-wise indices (condition 20a), and (ii) for every pair of interconnected channels (governed by a feedback interconnection law), a specific inequality involving the four relevant indices (ρ, ν) from both subsystems is satisfied (condition 20b). This inequality formally captures how a shortage of passivity (e.g., a negative index) in one channel can be compensated by an excess of passivity (a positive index) in the interconnected channel from a neighboring subsystem.

The proposed framework enables a decentralized analysis: stability of the large-scale network can be inferred by combining these local, data-based certificates without requiring a centralized model or global data. The paper demonstrates the applicability and interpretability of the method through a numerical case study based on a DC microgrid model, showing how faults in generation units affect the channel-wise indices and the overall stability condition. This work provides a significant step towards practical, model-free stability assessment for complex networked systems.