PaperlayerView exact revision on arXiv

Source-linked AI summary

Context-Enriched Performance Boosting via Operator Decomposition

Leonardo Massai, Sebastiano Messina, Nicolas Kirsch, Giancarlo Ferrari-Trecate

arXiv:2609.05158v1eess.SY

TL;DR

Context-aware control needs to use exogenous signals without sacrificing PB’s closed-loop Lp-stability guarantees. The paper introduces a structured factorization that separates stable disturbance processing from bounded contextual mixing, and evaluates it in moving-gate navigation. The factorized controller achieves the best reported overall performance, including an 83.45% success rate and 16.53% crash rate.

  • Problem

    PB controllers can be difficult to learn from reconstructed disturbances alone, while contextual signals are generally not ℓp sequences and cannot be directly fed into Lp-bounded operators without invalidating guarantees.

  • Method

    The paper factorizes context-enriched PB operators into an Lp-stable disturbance-driven module and a uniformly bounded matrix-valued mixer depending on disturbances and context.

  • Results

    83.45% success rate, 16.53% crash rate, 0.1168 average crossing error, and 12.230 average cost are achieved by C. Factorization in the moving-gate experiment.

  • Takeaways & Limitations

    The factorization preserves nominal PB stability guarantees while enabling general contextual information and multi-channel mixing, outperforming context-agnostic PB, MAD, and rPB in the reported navigation experiment.

Abstract

from arXiv · show

Performance Boosting (PB) is a control framework that, for a pre-stabilized system subject to $\mathcal L_p$ process disturbances, parametrizes the controllers that preserve closed-loop $\mathcal L_p$-stability through a causal $\mathcal L_p$-stable operator mapping reconstructed disturbances to corrective control actions. Although this permits optimization over expressive stability-preserving controllers, learning a desired policy from disturbance information alone can be difficult. We introduce a structured factorization for context-enriched, multi-input PB operators. The proposed architecture combines an $\mathcal L_p$-stable dynamical module that processes reconstructed disturbances with a uniformly bounded matrix-valued mixer depending on disturbances and contextual signals. Under the standard PB assumptions, this factorization preserves closed-loop $\mathcal L_p$-stability by construction. Moreover, on a weighted-envelope disturbance domain, we prove that the factorization is necessary and sufficient for causal operators satisfying a context-uniform envelope-preservation property. A numerical moving-gate navigation experiment demonstrates the advantages of the proposed architecture over context-agnostic PB, MAD, and reference-aware PB baselines.

I. INTRODUCTION

Context-aware control is motivated by objectives that depend on exogenous signals, while existing approaches often lack a generalized mechanism combining rich context with strict closed-loop Lp-stability guarantees.

  • Modern control objectives increasingly depend on exogenous signals such as energy prices and obstacle positions.
  • Traditional controllers are largely context-agnostic, limiting environmental awareness and anticipatory adaptation.
  • MPC and learning-based approaches incorporate contextual information but face challenges in maintaining constructive closed-loop guarantees for time-varying signals.
  • MAD and rPB incorporate selected exogenous inputs but do not provide a generalized multi-input, matrix-valued factorization for arbitrary contextual signals.
  • The paper extends PB to general contextual signals through a structured operator decomposition while retaining its strict Lp-stability guarantees.

II. NOTATION AND BASIC DEFINITIONS

This section establishes notation for vector, matrix, and sequence norms, causal and Lp-stable operators, and the timewise matrix–vector product used later.

  • Vectors, matrices, and sequences are distinguished by lowercase letters, uppercase letters, and bold letters, respectively.
  • The same induced norm notation is used for vectors and matrices, yielding ∥Av∥≤∥A∥∥v∥.
  • The spaces ℓp and ℓ∞ collect matrix-valued sequences according to summability and uniform boundedness norms.
  • A causal operator maps each output at time t using only inputs through time t.
  • An Lp-stable operator is causal and maps ℓp inputs to ℓp outputs.
  • The timewise matrix–vector product applies each matrix At to the vector vt at the same time index.

III. PERFORMANCE BOOSTING FRAMEWORK

PB improves a pre-stabilized system by reconstructing process disturbances and mapping them through a causal Lp-stable operator, but disturbance-only inputs can make desired policies difficult to learn.

  • The system state evolves from past states and controls plus an unknown process disturbance, with disturbances assumed to belong to ℓp.
  • PB uses an internal plant model to reconstruct the disturbance affecting the plant and computes control through u=M(bw).
  • When the pre-stabilized system F and model assumptions hold, searching over causal Lp-stable operators preserves closed-loop stability by construction.
  • The standard PB operator observes only reconstructed disturbances, so imitating state feedback may require internally reconstructing the state by resimulating plant dynamics.
  • Contextual signals are generally not ℓp sequences, so directly feeding them into an Lp-bounded operator would invalidate the theoretical guarantees.

IV. MAIN RESULTS

The proposed factorization separates stable disturbance processing from bounded context-dependent mixing, generalizing earlier context-aware PB variants while remaining suitable for practical parametrization.

  • The factorization combines a bounded matrix-valued component depending on disturbance and context with an ℓp component depending only on disturbance.
  • The mixer–feature structure generalizes MAD and rPB to arbitrary contextual signals, multidimensional disturbance features, and matrix-valued mixing across control channels.
  • MAD is recovered with a scalar disturbance magnitude and bounded state-dependent direction, while rPB is recovered with a diagonal reference-dependent mixer.
  • The latent dimension s is a practical architectural hyperparameter controlling the disturbance-feature dimension rather than a representability requirement.
  • Stable RENs or SSMs can parametrize the disturbance module, while bounded-output neural networks can enforce the mixer constraint.

A. Operator decomposition and Lp-stability

The proposed operator factorization combines a stable disturbance-processing module with a bounded mixer, preserving causality and closed-loop Lp-stability by construction.

  • The factorization is sufficient for Lp-stability because a bounded mixer applied timewise to an ℓp feature sequence remains in ℓp.The proof also uses causality of both factors and the timewise product definition.
  • The resulting operator M is causal when both Mp and M∞ are causal.
  • The stability bound is ∥M(bw, z)∥p ≤ ∥M∞(bw, z)∥∞∥Mp(bw)∥p.

B. Non-conservative factorization in specific noise regime

On a weighted-envelope disturbance domain, the bounded-mixer factorization is exact for causal operators with context-uniform envelope preservation, while outside that regime necessity can fail.

  • Disturbance regime: Weighted envelope spaces restrict disturbances to sequences uniformly dominated by a time-varying template whose inverse is ℓp-summable.These spaces include finite-support transients and some polynomially decaying disturbances.
  • Envelope preservation: Uniformity over contexts is essential because it excludes responses that can be delayed arbitrarily by the context.This regularity is what enables a bounded-mixer representation in the studied regime.
  • Non-conservative factorization: The factorization is necessary and sufficient for causal operators satisfying context-uniform envelope preservation on the considered weighted-envelope domain.The result requires a context-uniform bound on the mixer for every admissible disturbance.
  • Failure outside the regime: Without envelopes, the first-trigger echo is an admissible ℓp operator but cannot admit a context-uniform bounded-mixer factorization.The trigger can occur at any time, forcing the disturbance feature to remain bounded away from zero indefinitely.
  • Interpretation: The existence proof uses a mathematical witness for Mp, whereas practical implementations design Mp and M∞ directly.The witness is not intended as an implementation prescription.
  • Scope: The converse representability result is not guaranteed outside the weighted-envelope domain, although the sufficient stability guarantee remains valid.The theorem concerns operator classes and does not establish completeness of a particular finite-dimensional parametrization.
  • Model mismatch: Under model mismatch, closed-loop Lp-stability can still be preserved by a small-gain argument when the boosting gain is sufficiently small.The uniformly bounded mixer supports specifying a desired overall gain for robustification.
  • Realizable processors: Stable REN and SSM disturbance processors satisfy the exponential-forgetting condition that preserves polynomial weighted envelopes.Finite cascades preserve the required decay estimate, so the commonly used processors remain within the theorem’s regime.

V. NUMERICAL EXPERIMENT: CONTINUOUS MOVING-GATE NAVIGATION

The numerical study evaluates context-aware PB in a moving-gate navigation task where a planar robot must reach the origin while using causal gate observations.

  • The experiment compares context-aware PB with context-agnostic PB, MAD, and rPB architectures.
  • The task requires a planar robot to reach the origin while crossing a wall through a continuously moving gate.
  • The controller uses causal gate observations and recent history to decide when and where to cross.
  • The corrective operator retains the structural Lp constraint imposed by the proposed factorization.

A. The setup

The experiment studies a pre-stabilized planar robot that must reach the origin while avoiding a continuously moving gate and process disturbances. Its PB correction is optimized using reconstructed disturbances and a nonlinear task objective combining target, gate-tracking, collision-avoidance, and control-energy terms.

  • System and disturbances: The robot is modeled as a pre-stabilized double integrator with dissipative quadratic drag and additive process disturbances.The nominal system is stabilized around the origin before PB correction is applied.
  • System and disturbances: Disturbances combine small Gaussian background noise with transient velocity gusts whose amplitudes and durations are randomly sampled.The disturbance sequence is extended by zero after horizon T = 160.
  • PB setup: PB reconstructs the disturbance online from one-step nominal prediction errors, and under the stated dynamics this reconstruction is exact.The reconstructed disturbance satisfies b_w,t = w_t at every step.
  • Moving-gate environment: The moving gate follows a clipped Ornstein–Uhlenbeck process, producing smooth, persistent motion throughout each episode.The reported parameters are τg = 60 and rg = 0.50, with the gate constrained to [−0.95, 0.95].
  • Optimization objective: The corrective input steers the robot to the origin while avoiding corridor and moving-wall collisions through a nonlinear objective.The objective includes terminal and stage target cost, moving-gate tracking and collision avoidance, control energy, and corridor collision avoidance.

B. Context, PB controller design and training procedure

The context-aware controller augments reconstructed-disturbance processing with a bounded causal context signal describing gate motion and robot geometry. The experiment trains and evaluates this factorized design against context-agnostic PB, MAD, and rPB using navigation and control-effort metrics.

  • Context representation: The context signal is a bounded design input summarizing gate position, motion, recent average position, robot–gate geometry, goal displacement, and velocity.Its components include causal gate velocity, an exponential moving average, normalized position features, and current velocity.
  • Controller design: The proposed factorization uses a stable disturbance-processing SSM and a pointwise bounded mixer that depends on reconstructed disturbances and context.The implementation uses s = 16, eight SSM layers, and a spectrally normalized MLP mixer reshaped into a 2×16 matrix.
  • Baselines: The context-agnostic PB baseline keeps the factorized architecture but fixes z_t = 0, preventing conditioning on the realized gate trajectory.MAD and rPB are compared as parameter-matched special cases that receive the complete context signal.
  • Training procedure: Training uses fresh batches of 4096 trajectories of length T = 160 with independently sampled initial conditions, gate trajectories, and disturbance sequences.Each base scenario is paired with a gate-reflected companion using the same initial condition and disturbance sequence.
  • Evaluation: Evaluation reports success, crash, goal, average crossing error, average control energy, and average task cost.Success requires both collision-free gate crossing and termination within ε_goal = 0.18 of the origin.

C. Results

The continuous moving-gate experiments compare controller architectures and progressively richer context features. Context-aware factorization improves collision avoidance and overall performance, with richer gate information yielding further gains.

  • Architecture comparison: The context-agnostic PB controller succeeds in only 28.20% of test episodes and cannot condition corrections on realized gate motion.Goal success is approximately 100% for every architecture after crossing, indicating that collision with the moving wall is the main failure source.
  • Architecture comparison: C. Factorization achieves the best overall result, with an 83.45% success rate, a 16.53% crash rate, smallest average crossing error of 0.1168, and lowest average cost of 12.230.Its average control energy is 1.493, close to MAD’s 1.482.
  • Experimental setup: Across 4096 test episodes, Panel (a) compares C. Factorization with context-agnostic PB, MAD, and rPB, while Panel (b) varies supplied context features.Success requires collision-free gate passage and terminal goal attainment.
  • Context ablation: With minimal gate-aware context z(1)_t, success rises to 79.39% from 28.93% without gate information, while collision rate falls to 20.58%.The gate-crossing error decreases from 0.4096 to 0.1265, and total cost falls from 51.036 to 14.257.
  • Context ablation: Success increases from 79.39% with z(1)_t to 80.81% with z(2)_t and 83.45% with complete context z(3)_t.The complete context gives a 16.53% crash rate, crossing error 0.1168, and average cost 12.230.
  • Context ablation: Additional full-context features, particularly causally observed gate velocity and relative target position, improve adaptation with control energy increasing modestly from 1.449 to 1.493.A challenging realization shows C. Factorization adapting to rapid late gate motion while the other controllers collide with the wall.

VI. CONCLUSIONS

The paper introduces a context-enriched PB factorization that combines stable disturbance processing with bounded contextual mixing. It establishes stability and factorization results, then demonstrates improved moving-gate control relative to context-agnostic and contextual baselines.

  • Contribution: The proposed factorization combines an Lp-stable disturbance-driven dynamical module with a uniformly bounded context-dependent mixer for multi-input PB controllers.The architecture is intended to inject general contextual information while retaining nominal PB stability guarantees.
  • Theory: The factorization is sufficient for Lp-stability and exact on a weighted-envelope disturbance domain for causal operators satisfying context-uniform envelope preservation.A broad family of fading-memory operators satisfies this regularity condition.
  • Empirical conclusion: In the continuous moving-gate experiment, contextual information substantially improves collision avoidance and overall control performance relative to context-agnostic PB, MAD, and rPB.Figure 2 illustrates safe crossing by C. Factorization under rapid late gate motion while the other controllers collide.
Loading 2609.05158v1…