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Invariance is Compositional for Continuous-time Systems: From Sleekness to Lebesgue Density
Othman Cherkaoui Dekkaki, Sadek Belamfedel Alaoui, Olayo Reynaud, Mohamed Maghenem, Alessio Iovine, Adnane Saoud
TL;DR
Checking invariance for large interconnected systems is computationally difficult, and prior compositional results leave a bidirectional result unestablished. The paper introduces tangential Lebesgue-density and proves an equivalence between global product-set invariance and local subsystem invariance. A DC microgrid example demonstrates verification complexity that grows linearly with subsystem count.
Problem
For large interconnected systems, global invariance verification requires checking tangent-cone conditions over a high-dimensional safe-set boundary.
Source: §I · p. 1Method
The paper proves a bidirectional compositional invariance result and introduces tangential Lebesgue-density, a condition weaker than sleekness that supports product tangent-cone composition.
Source: §I · pp. 1–2Results
The global product set is robustly forward invariant exactly when each local set is robustly forward invariant for its subsystem under neighboring coupling inputs.
Source: §VI · p. 7Takeaways & Limitations
In the DC microgrid example, verification complexity grows linearly with the number of subsystems.
Source: §VI · p. 7
Abstract
from arXiv · showhide
This work establishes the first bidirectional compositional invariance result: robust forward invariance of a global Cartesian product set is shown to be equivalent to robust forward invariance of each local subsystem under coupling inputs from its neighbors. To facilitate this, we introduce the notion of tangential Lebesgue-density, a new condition that is weaker than classical sleekness but sufficient to ensure that the product of individual tangent cones equals the tangent cone of the product set. This equivalence reduces the curse of dimensionality by allowing the verification of a high-dimensional global system to be decomposed into a series of local sub-checks. The framework's scalability is demonstrated through a DC microgrid numerical example, confirming that the verification complexity grows only linearly with the number of subsystems.
I. INTRODUCTION
Forward invariance is well characterized for individual continuous-time systems, but checking it for large interconnected systems is computationally intractable in general. The paper establishes a bidirectional compositional result and introduces tangential Lebesgue-density to support it.
I. INTRODUCTION
- The global tangent-cone condition becomes costly at scale because it must be checked across the boundary of a state space whose dimension grows with the number of subsystems.The introduction motivates decomposing the verification into smaller subsystem-level checks.Source: §I · p. 1
- Prior compositional invariance results establish only the sufficient direction, leaving the reverse implication from global to local invariance unaddressed.The paper identifies this as the gap in existing continuous-time results.Source: §I · p. 1
- The paper establishes that global robust forward invariance of a Cartesian product set is equivalent to robust forward invariance of each local set for its subsystem.This reduces a global boundary tangent-cone check to local conditions involving each subsystem’s dynamics and local-set geometry.Source: §I · p. 1
- Tangential Lebesgue-density is strictly weaker than sleekness and is sufficient for the product of local tangent cones to be contained in the global product set’s tangent cone.It requires valid approximation times to occupy almost the entire interval near zero, enabling synchronization of local sequences.Source: §I · pp. 1–2
- The paper illustrates its framework on a DC microgrid example with 100 distributed generation units.The example is described as demonstrating scalability.Source: §I · p. 2
A. Interconnected subsystems
The framework models interconnected systems by composing subsystems whose internal inputs carry neighboring outputs. Under compatibility and regularity assumptions, the interconnected system inherits the subsystem regularity conditions.
A. Interconnected subsystems
- Each subsystem has state, external and internal inputs, dynamics, and an output map; internal inputs represent coupling from neighboring subsystems.External inputs represent exogenous disturbances, while internal inputs are determined by the interconnection.Source: A. Interconnected subsystems · p. 2
- Composition is compatible when each subsystem’s internal-input space contains the Cartesian product of its neighbors’ output spaces.The resulting interconnected system stacks subsystem states, external inputs, outputs, dynamics, and output maps.Source: A. Interconnected subsystems · p. 2
- If every subsystem satisfies Assumption 2.2, then the compatible interconnected system satisfies the same regularity assumption.The stated conditions include Lipschitz dynamics with nonempty compact values and continuous output maps.Source: A. Interconnected subsystems · pp. 2–3
B. Robust forward invariance
Robust forward invariance requires trajectories to remain in the set for all admissible initial states and inputs. The section also recalls tangent-cone concepts used in invariance analysis.
B. Robust forward invariance
- A local set is robustly forward invariant when every trajectory starting in it remains there for all times in its domain under the specified external and internal inputs.The definition quantifies over initial conditions in the set and locally measurable input signals.Source: B. Robust forward invariance · p. 3
- For a closed set, the contingent cone captures limiting feasible directions, while the Clarke tangent cone is closed, convex, and contained in the contingent cone.The section recalls these cones as central to its invariance analysis.Source: B. Robust forward invariance · p. 3
- A set is sleek at a point when its Clarke tangent cone equals its contingent cone there.Sleekness of a set requires this equality at every point in the set.Source: B. Robust forward invariance · p. 3
III. CARTESIAN PRODUCT OF TANGENT CONES
The section introduces tangential Lebesgue-density as a pointwise condition that synchronizes local tangent approximations. It supports a product-cone result while weakening the classical sleekness requirement.
III. CARTESIAN PRODUCT OF TANGENT CONES
- The section notes that a tangent direction to the product set projects to a tangent direction of each local set.The converse inclusion is the direction requiring synchronization of local approximating sequences.Source: §III · p. 3
- Tangential Lebesgue-density requires near-tangent feasible times to occupy asymptotically the full interval near zero.The condition is stronger than merely having valid times arbitrarily close to zero, but weaker than sleekness.Source: §III · pp. 3–4
- If each local set is tangentially Lebesgue-dense at its boundary points, the product of its local contingent cones is contained in the contingent cone of the product set.The proof selects common valid times across subsystems and then combines their approximating directions.Source: §III · p. 4
- Tangential Lebesgue-density lets the proof synchronize the distinct local sequences of valid times into one common sequence.This resolves the obstacle that each subsystem’s tangent direction may provide a different sequence.Source: §III · pp. 3–4
- Sleekness implies tangential Lebesgue-density, but tangential Lebesgue-density does not generally imply sleekness.The paper gives a closed-set counterexample where density holds at a boundary point despite failure of sleekness.Source: §III · pp. 4–5
IV. COMPOSITIONAL INVARIANCE
Theorem 4.1 makes robust forward invariance compositional: under its assumptions, global invariance is equivalent to local invariance of each subsystem. The result uses tangential Lebesgue-density to relate local and global tangent-cone conditions.
IV. COMPOSITIONAL INVARIANCE
- Global robust forward invariance is equivalent to robust forward invariance of each local set under the theorem’s assumptions.The equivalence also connects global and local tangent-cone conditions.Source: §IV · p. 5
- Each local set must be tangentially Lebesgue-dense at every boundary point for the stated equivalences to hold.The theorem also assumes each subsystem satisfies Assumption 2.2.Source: §IV · p. 5
- Tangential Lebesgue-density supplies the tangent-cone product property used to establish the global-to-local direction.The proof combines local conditions using this property to obtain the corresponding global tangent-cone condition.Source: §IV · pp. 5–6
- Convex sets and practical sets satisfy the density assumption through sleekness.The cited remark states that practical sets imply sleekness, and convex sets are also sleek.Source: §IV · p. 6
- Local invariance verification reduces to a tangent-cone inclusion at each local boundary point, using coupling inputs generated by neighboring sets.The check depends on the subsystem’s local dynamics and the geometry of its local safe set.Source: §IV · p. 6
V. NUMERICAL EXAMPLE
The numerical example applies the theoretical results to a DC microgrid composed of distributed generation units connected by resistive power lines.
V. NUMERICAL EXAMPLE
- The example considers a DC microgrid with N distributed generation units, each equipped with a Buck converter, an LC filter, and a local droop-based primary controller.Source: §V · p. 6
- The units are interconnected through resistive power lines.Source: §V · p. 6
- The example represents a class of large-scale networked control systems for which compositional invariance verification is practically essential.Source: §V · p. 6
A. System model
The example models each distributed generation unit as a scalar voltage system with local load disturbance and neighboring-unit voltage inputs. A droop controller regulates point-of-common-coupling voltage, and the model uses a quasi-stationary line approximation.
A. System model
- The droop controller is a decentralized static feedback law that regulates point-of-common-coupling voltage by adjusting the current reference.Source: A. System model · p. 6
- The quasi-stationary line approximation neglects line inductances because their time constants are much smaller than those of the converters.Source: A. System model · p. 6
- The closed-loop subsystem dynamics are first-order, and the example’s affine dynamics satisfy the stated Lipschitz and compact-value conditions.Source: A. System model · p. 6
- The model treats local load current as an external disturbance and neighboring units’ voltages as internal inputs.Filter capacitance and connecting-line resistance are positive model parameters.Source: A. System model · p. 6
- Every 5 s, the common load-current step alternates between approximately 1.5 A and 10.5 A, alongside per-unit offsets and 5% noise.The offsets are drawn uniformly from −1.5 A to 1.5 A.Source: A. System model · p. 6
B. Invariant sets and conditions verification
The microgrid example checks interval voltage safety sets and reports that the local tangent-cone condition holds for every unit. The theorem equates those local checks with global robust invariance.
B. Invariant sets and conditions verification
- Each DGU’s safe voltage interval is [47.2, 48.6] V, approximately ±1.5% around the 48 V nominal voltage.Source: B. Invariant sets and conditions… · p. 6
- The example reports that the tangent-cone condition is satisfied for every DGU, making the global invariance check equivalent to local robust invariance checks.Source: B. Invariant sets and conditions… · p. 6
- The local condition requires 2N scalar checks, one pair per DGU, so verification complexity is linear in N.Each check uses local parameters and direct neighbors’ safety bounds; the global tangent-cone condition ranges over an uncountably infinite boundary set in R^N.Source: B. Invariant sets and conditions… · pp. 6–7
- Earlier DC-microgrid results cited in the paper establish only that local invariance implies global invariance, not the reverse implication.Source: B. Invariant sets and conditions… · p. 7
C. Simulation results
In the N=100 DGU simulation, all voltages stay within the prescribed safe interval despite irregular load disturbances. The example also shows that verification requires only two scalar checks per DGU.
C. Simulation results
- The simulation uses 100 DGUs in a k-nearest-neighbour ring topology with k=6, with initial conditions spread uniformly across each safe interval.DGU 1 starts at the lower boundary and DGU 100 at the upper boundary.Source: C. Simulation results · p. 7
- All 100 voltage trajectories remain within [47.2, 48.6] V for t in [0, 20] s despite persistent, irregular load disturbances.The simulation starts with DGUs at the lower and upper boundaries of the safe set.Source: C. Simulation results · p. 7
- At IL_i(t) ≈ 10.5 A, the network equilibrium is near V*_i = 47.35 V; at IL_i(t) ≈ 1.5 A, it is near V*_i = 48.35 V.Voltages approach the lower boundary in the first regime and the upper boundary in the second.Source: C. Simulation results · p. 7
- For N=100, condition (d) requires only 200 scalar inequality checks; adding a DGU requires only two additional inequalities and does not require recomputing or modifying existing DGU certificates.Source: C. Simulation results · p. 7
VI. CONCLUSION
The conclusion establishes an if-and-only-if relationship between global robust invariance and local robust invariance under neighboring coupling inputs. It identifies tangential Lebesgue-density as the geometric enabler and notes scope boundaries for future extensions.
VI. CONCLUSION
- Robust forward invariance of the global Cartesian product set is equivalent to robust forward invariance of each local set under coupling inputs from neighboring invariant sets.The result makes global invariance verifiable through local subsystem conditions.Source: §VI · p. 7
- For N=100 DGUs, all trajectories remain within [47.2, 48.6] V, while condition (d) requires 200 scalar inequalities.The check count is independent of network topology.Source: Figure 1 · p. 7
- Tangential Lebesgue-density is strictly weaker than sleekness yet sufficient for the product of local tangent cones to equal the tangent cone of the product set..Source: §VI · p. 7
- The framework assumes dynamics on the whole state space R^n and treats uncontrolled systems; constrained domains and controller co-design are left as extensions.The proposed extensions include differential inclusions on constraint sets and decentralized controllers that render local sets invariant.Source: §VI · pp. 7–8