Source-linked AI summary
A general representation form of system data with feedback control and fundamental lemma as examples
Steven X. Ding, Linlin Li
TL;DR
The note addresses representing dynamic-system input-output data, including feedback and uncertainty. It introduces a general data representation based on complementary image and residual subspaces, and concludes that the representation exists for several additional system classes under stated conditions.
Problem
The note concerns how dynamic-system input-output data, including feedback and uncertainty dynamics, can be represented.
Source: §1 · pp. 2–3Method
The paper introduces a general data representation that decomposes input-output data into complementary image and residual subspaces.
Source: §1 · p. 3Results
The note concludes that the general data representation exists for continuous-time LTI, LTV, polynomial, and nonlinear systems under certain conditions.
Source: §2 · p. 5Takeaways & Limitations
The observer-based systems can support real-time simulation and monitoring as digital twins, and fault and attack detection as residual generators.
Source: §1 · p. 3The treatment assumes G is a proper real-rational matrix with a minimal state-space realization.
Source: §1 · p. 1
Abstract
from arXiv · showhide
This note introduces a general data representation of dynamic systems.
1 Data representation
The general data representation expresses system input-output data through image and residual components, separating the response trajectory from uncertainty dynamics. It also represents feedback configurations and supports practical simulation, monitoring, and fault or attack detection applications.
1 Data representation
- The input can be interpreted as observer-based state feedback, while residual ry represents plant uncertainties affecting y.The representation treats process input-output data as comprising a reference-signal response and uncertainty dynamics.Source: §1 · p. 2
- The observer-based systems can serve as real-time digital twins for simulation and monitoring, and as residual generators for fault and attack detection.Both systems have a single state observer at their core; online implementation computes an n-th order difference equation.Source: §1 · p. 3
- Definition 1 names representation (13) the general data representation.Source: §1 · p. 3
- The representation decomposes (u, y) into image and residual subspaces, which are complementary by the Bezout identity.Source: §1 · p. 3
- The representation holds for open- and closed-loop configurations, and a stabilising feedback controller can be described equivalently as observer-based state feedback.Source: §1 · p. 3
- The system response separates the trajectory due to the reference signal from the response to uncertainty, including faults.Source: §1 · p. 4
It can be interpreted as an approximation of the target system dynamics
The representation interprets feedback as an observer-based response to plant uncertainties and connects input-output data to a data-driven realization through Hankel matrices and a rank condition.
It can be interpreted as an approximation of the target system dynamics
- Feedback control is interpreted as an observer feeding back plant uncertainties represented by ry.The note presents this as an information aspect of feedback control and a principle of the unified framework of control and detection.Source: It can be interpreted as an… · p. 4
- The s-sample image representation can be realized from input-output data sets expressed as Hankel matrices.The input Hankel matrix is required to satisfy a rank condition, and practical applications use N >> (s + n) m + s − 1.Source: It can be interpreted as an… · pp. 4–5
- Expression (28) is the fundamental lemma, with rank condition (29) requiring persistent excitation of order (s + n) m.The lemma has been applied to data-driven MPC.Source: It can be interpreted as an… · p. 5
- The representation is also cited as useful for safe multi-agent systems and safe and secure cyber-physical control systems.These applications are described as simplifying system analysis and synthesis and supporting exploration of more complicated problem settings (Li et al., 2026b).Source: It can be interpreted as an… · p. 5
2 Conclusion
The note describes the general data representation and states that it exists for several classes of systems, with a condition attached to nonlinear systems.