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Minimal data rate stabilization of nonlinear systems over networks with large delays
Claudio De Persis
TL;DR
The paper addresses stabilization of nonlinear feedforward systems when finite-rate feedback is quantized, impulsive, and delayed by an arbitrarily large amount. It designs a hybrid encoder–decoder–controller using coordinate transformations and nested saturation, and proves stabilization at an average data rate arbitrarily close to the infimal one.
Problem
The paper asks whether nonlinear feedforward systems can be stabilized with minimal-rate feedback when communication introduces quantization, impulses, and arbitrarily large delay.
Method
The paper designs a hybrid delayed encoder, decoder, and nested saturated controller using recursive feedforward construction and coordinate transformations.
Results
The system is semi-globally asymptotically and locally exponentially stabilizable with an average data rate arbitrarily close to the infimal one.
Takeaways & Limitations
Minimal data rate stabilization extends to nonlinear feedforward systems despite quantization noise, impulsive transmissions, and arbitrarily large communication delay.
Takeaways & Limitations
The stability formulation initially omits a Lyapunov simple-stability requirement, although the origin is also shown to be stable.
Abstract
from arXiv · showhide
Control systems over networks with a finite data rate can be conveniently modeled as hybrid (impulsive) systems. For the class of nonlinear systems in feedfoward form, we design a hybrid controller which guarantees stability, in spite of the measurement noise due to the quantization, and of an arbitrarily large delay which affects the communication channel. The rate at which feedback packets are transmitted from the sensors to the actuators is shown to be arbitrarily close to the infimal one.
1 Introduction
The paper studies stabilization under finite-bandwidth feedback, extending well-understood linear data-rate results to nonlinear feedforward systems with communication delays, quantization, and impulsive transmissions.
- 1 Introduction: Linear systems admit a minimal data rate above which stabilization is always possible.This rate is described as proportional to the inverse product of the unstable eigenvalues of the system matrix.
- 1 Introduction: Small feedback packets can increase the number of tasks carried out simultaneously and support explicit procedures.
- 1 Introduction: Feedforward nonlinear systems can be stabilized despite actuator saturation.
- 1 Introduction: The contribution extends minimal-data-rate stabilization to arbitrarily delayed channels with quantization error and impulsive packet transmission.Existing delay-free or continuous-time delay methods do not directly handle all three effects together.
2 Preliminaries
The paper models delayed network feedback using sampled packets, encoders, decoders, and nested saturated control for nonlinear feedforward systems. The construction synchronizes delayed state reconstructions while accounting for quantization and impulsive updates.
- 2 Preliminaries: The plant is a nonlinear feedforward system with scalar states, feedforward state vectors, C2 nonlinearities, and bounded initial conditions.A known componentwise bound on the initial state is assumed.
- 2 Preliminaries: Feedback packets are transmitted at increasing sampling times, contain N(tk) bits, and arrive after a known delay θ.The transmission schedule is bounded by known positive constants Tm and TM.
- 2 Preliminaries: The average data rate counts the total transmitted bits over the elapsed sampling interval.The encoder samples the state at each transmission time and converts it into symbols for the channel.
- 2.1 Encoder in the delay-free case: In the delay-free encoder, ξ tracks the state while ℓ stores quantization-region edge lengths, and Λ is Schur stable.At transmission times, the quantization cuboid is split into 2^n subregions and the selected region is communicated using symbols in {0, ±1}.
- 2.1 Encoder in the delay-free case: The decoder reconstructs the state estimate and applies a nested saturated feedback controller when encoder and decoder initialize identically.Under this agreement, ξ(t)=ψ(t) and ℓ(t)=ν(t) for all t.
- 2.2 Encoders for delayed channels: For delayed channels, coordinate transformations modify encoder and decoder maps, while delayed auxiliary states reproduce the decoder trajectory.The proposed dynamics use delayed samples and resets at both transmission and reception times.
- 2.2 Encoders for delayed channels: The decoder initially applies zero control because it has not yet received the first state sample, then reconstructs ξ(t−θ) from received packets.The control law uses the reconstructed delayed state through ψ(t).
- 2.2 Encoders for delayed channels: The delayed encoder reconstructs the control-related trajectory using delayed state and encoder variables, with initial histories specified before the first reception.
3 Main result
The paper formulates stabilization of nonlinear feedforward systems with quantized measurements and arbitrarily large communication delays, targeting rates arbitrarily close to the infimal rate. Its theorem establishes semi-global asymptotic and local exponential stabilizability under the stated controller and parameter conditions.
- Problem: The problem is to stabilize the closed-loop system for any delay θ while transmitting at an average rate arbitrarily close to the infimal one.The measured state is sampled, encoded into N(t_k) bits, and sent from sensors to the controller.
- Definition: The definition requires eventual confinement to a compact set, exponential convergence after time T, and average rate R_av < R̂.These conditions define semi-global asymptotic and local exponential stabilizability using an average data rate arbitrarily close to the infimal one.
- Caveat: The definition omits a Lyapunov simple-stability requirement, although the paper states that stability of the origin can nevertheless be proved.For each ε > 0, sufficiently small initial state norm implies |x(t)| < ε for all t ≥ θ0.
- Novelty: The contribution extends prior nonlinear feedforward stabilization results to impulses, quantization noise, and delayed control action, while neglecting parametric uncertainty.Earlier results separately treated some of these features but not their combination.
- Theorem: The main result states that, under the theorem’s parameter conditions, the system is semi-globally asymptotically and locally exponentially stabilizable with an average data rate arbitrarily close to the infimal one.The theorem selects controller and encoder parameters to accommodate the delay and quantization setting.
- Interpretation: The paper describes the result as a nonlinear generalization of the data rate theorem for linear systems.The linearization at the origin is identified as a chain of integrators.
4 Change of coordinates
The analysis transforms the process–decoder system into recursive coordinates and a rescaled time variable, exposing subsystem dynamics, encoder errors, and impulsive updates. This form supports the recursive design of the encoder, decoder, and nested saturated controller under a boundedness condition.
- Coordinate transformation: The process and decoder are transformed into a special form using coordinate transformations developed in earlier work.The transformed representation is used before analyzing the recursive controller and encoder–decoder design.
- Recursive design: The design proceeds recursively across i = 1, 2, …, n, focusing at each step on the last n−i+1 system equations and corresponding encoder, decoder, and controller components.The procedure then advances to the subsystem with one fewer equation.
- Rescaling: The time scale is changed according to τ := θ/κ, r_k := t_k/κ, and ρ_k := θ_k/κ.The input and state coordinates are also changed to define Z_i, E_i, and P_i.
- Transformed dynamics: In the transformed dynamics, Z_i follows the process subsystem, E_i captures decoder-state error relative to delayed process coordinates, and P_i is constant between impulses.At impulse times, E_i is reset using P_i and P_i is updated by Λ_i.
- Condition: The transformed subsystem equations are used under the condition |Z_i+1|_∞ ≤ (Mκ)/(L(n + 1)!).The coordinate construction also introduces nonsingular positive matrices Φ_i.
- Controller: The controller in the new coordinates is a nested saturated feedback law combining current error coordinates with delayed transformed states.Its recursive expression uses the gains σ_i and intermediate terms λ̂_i−1.
5 Analysis
The analysis establishes stability through an inductive, cascade-based treatment of quantization errors, boundedness, saturation, impulses, and delays. Exponential stability of the error subsystem supports exponential stability of the full closed loop.
- Error subsystem: The quantization-error subsystem admits an exponential Lyapunov function, despite impulsive updates and delayed feedback.The error dynamics are modeled as an impulsive system whose jump matrix is Schur stable.
- Inductive analysis: The proof proceeds inductively from subsystem i=n to i=1, preserving boundedness and quantization-error bounds at each step.The induction transfers the hypothesis from i to i−1 and eventually covers the complete system.
- Inductive analysis: After finite time, all saturation functions enter and remain in their linear operation regions.The repeated application of the boundedness lemma yields this regime before the final exponential-stability argument.
- Cascade stability: Quantization noise drives the z-subsystem, forming a cascade whose exponential stability follows from a Lyapunov-Krasovskii functional.This extends the corresponding no-noise, no-impulse stability analysis to the delayed quantized setting.
- Final stability result: The closed-loop system is semi-globally asymptotically and locally exponentially stabilized by the proposed controller.The result is obtained after returning from the transformed coordinates to the original system coordinates.
- Rate minimality: The average rate can remain below any prescribed positive threshold because stability holds for arbitrarily large transmission periods Tm.The proof uses Rav = 2^n/Tm and shows stability is unaffected by increasing Tm.
6 Conclusion
The paper shows that minimal-data-rate stabilization remains possible for feedforward nonlinear systems with arbitrarily large feedback delays. In suitable coordinates, the closed loop becomes a cascade of impulsive nonlinear systems with delayed feedback.
- Conclusion: Minimal-data-rate stabilization is possible despite an arbitrarily large transmission delay in the communication channel.The result concerns nonlinear systems in feedforward form modeled through impulsive control systems.
APPENDIX
The appendix verifies the recursive error bounds and stability properties used in the main analysis. It shows that the error dynamics evolve through continuous intervals and quantized impulsive updates governed by Schur-stable matrices.
- Error dynamics: The appendix derives the error dynamics for each subsystem from the transformed system equations and encoder-decoder definitions.The resulting dynamics include delayed decoder states and nonlinear terms evaluated at error-plus-delayed-state variables.
- Error bounds: For the final subsystem, choosing Λn = 1/2 preserves the bound |en(r)| ≤ pn(r)/2 across successive transmission intervals.The argument iterates the bound from one interval to the next.
- Error dynamics: Between impulses, the error component ei follows delayed nonlinear dynamics, while transmission instants introduce discrete updates.The appendix explicitly separates the cases r ≠ ρk and r = ρk.
- Recursive stability: The matrix Λi is Schur stable whenever Λi+1 is Schur stable, enabling the recursive stability argument.This property is used to propagate stability upward through the subsystem indices.