- The paper establishes that contractivity in affine systems is equivalent to offset and 0-stability through a quadratic difference form approach.
- It reformulates stability criteria using Schur conditions and LMI formulations, enabling computationally tractable verification.
- The paper demonstrates that linear controllers suffice for contraction, while affine controllers are needed only for fixed-point assignment.
Stability, Contraction, and Controller Synthesis for Affine Systems
Introduction
This paper presents a comprehensive, representation-free analysis of stability, contraction, and control for discrete-time affine systems within the behavioral framework. Affine systems are positioned as an intermediate class between linear and fully nonlinear systems, motivated both by their prevalence in physics-derived models and their utility in data-driven predictive control. The behavioral approach, which focuses on sets of system trajectories rather than specific representations (e.g., state-space or input-output models), underpins the derivations and results.
Recent work has identified nontrivial distinctions between affine and linear behavioral systems, particularly in data-driven control settings. The paper systematically addresses open questions regarding stability (particularly contraction as an affine analogue to linear stability), implementability of reference behaviors, and the sufficiency of linear versus affine controllers for achieving contractive closed-loop behavior.
Stability and Contraction for Affine Behaviors
Affine systems, unlike linear ones, generally do not possess the origin as an equilibrium and may indeed lack equilibria in the usual sense. This motivates the adoption of contraction, rather than classical Lyapunov stability, as the core concept: an affine system is contractive if every pair of trajectories asymptotically coalesce.
The authors establish that for any affine behavior B, contractivity, offset stability (existence of convergence to a unique constant trajectory), and $0$-stability of the associated difference behavior $\dif(B)$ are equivalent. The difference behavior $\dif(B)$, always linear, encapsulates the deviations between pairs of trajectories. This equivalence allows established linear theory to be leveraged for affine systems.
They then generalize the use of quadratic Lyapunov functions in state-space models to quadratic difference forms (QDFs), showing that contractivity of an affine behavior is equivalent to the existence of a QDF on $\dif(B)$ satisfying suitable positivity and decrease conditions. Contrary to norm-based contraction metrics, the QDF-based approach is also applicable when no compatible vector norm exists.
System Representations and Numeric Characterizations
The results are formulated without reliance on specific representations, but the authors provide translation of the main contractivity criteria into offset kernel and state-space forms:
- An affine behavior B is contractive if and only if its offset kernel polynomial matrix R is Schur (all roots in the open unit disk), or equivalently, its state matrix A (for suitable realization) is Schur.
- The existence of a QDF as a contraction metric, required only for the difference behavior, leads to linear matrix inequality (LMI) conditions, making verification computationally tractable in practical scenarios.
Controller Synthesis and Implementability
Given a partitioned affine system with controllable variables, the paper investigates the realization of reference behaviors (targets for control) under interconnection with linear or affine controllers, again in a representation-free manner. The principal implementability criterion provided is:
A reference behavior R is implementable by an affine controller if and only if R is contained in the projection of the system onto the to-be-controlled variables, and, for some trajectory, all perturbations induced by the difference behavior subject to fixed control are also contained in $0$0.
This is equivalent to the established behavioral implementability conditions but generalized to the affine setting with finite cardinality and convexity, hence suited for algebraic computational procedures. Importantly, the authors highlight that under data-driven system identification (for instance, via trajectory columns), these conditions remain testable via linear algebra.
Existence of Contractive References and Regular Interconnection
Attention shifts to the question of robust feedback design: given an affine system, when does there exist a contractive, regularly implementable reference? Here, regular interconnection ensures controllers do not over-constrain the system beyond its input degrees of freedom. The analysis relies on input/output cardinality derived from the behavioral structure.
Detectability and offset stabilizability (read via the difference behavior) are shown to be necessary and sufficient for existence of such a reference. The characterization is rigorous: detectability is equivalent to contractivity of behaviors with fixed control, and offset stabilizability requires the difference behavior to admit $0$1-stabilizing feedback. The authors give translation of these concepts into both kernel and state-space representations.
Sufficiency of Linear Controllers
A central contribution, with implications for control synthesis, is the demonstration that linear controllers always suffice for contracting the closed-loop behavior of affine systems, provided detectability and offset stabilizability conditions are met. Affine controllers, while not needed for contraction, are shown to be precisely those required for placement of the closed-loop fixed point (i.e., for achieving desired equilibrium behaviors).
Implications and Future Perspectives
The results cement the behavioral framework as a natural, representation-free setting for studying stability and control of affine systems, harmonizing data-driven, input/output, and physical modeling viewpoints. The identification of the difference behavior as the structural bridge to linear theory greatly expands the practical reach of data-driven control algorithms, extending known theoretical guarantees to the broad class of affine models.
The explicit sufficiency of linear controllers for contraction, and the necessity of affine components only for equilibrium assignment, establishes crisp separation between stabilization and set-point placement challenges. The detailed algebraic and LMI-based results pave the way for future research on numerically robust data-driven controller design, the extension to constrained and hybrid systems, and further application to nonlinearization techniques.
Conclusion
The paper presents rigorous, representation-free necessary and sufficient conditions for offset stability, contraction, and implementability in affine discrete-time behaviors, connecting these to quadratic difference forms and yielding computationally tractable tests. The results clarify the roles of linear and affine controllers for achieving closed-loop objectives, and establish a pathway for systematic, data-driven stability analysis and synthesis in affine system settings. This framework invites further inquiry into scalability, robustness to modeling error, and transferability to nonlinear control paradigms.
