PID Control

Deploying LLM agents in non-stationary environments requires continually revising their instructions as the task drifts, yet rewriting after every reported mistake can turn noisy feedback into persistent errors. This raises an often-overlooked scheduling question in prompt adaptation: when should feedback trigger a lasting instruction update, and at what editor-call cost? Taking inspiration from proportional-integral-derivative (PID) control, we introduce a family of lightweight gates that react to the current error, accumulate evidence over time, and roll back deterioration after an edit. We benchmark these gates against both simple baselines and advanced prompt optimizers (TextGrad and GEPA), over a diverse suite of drifting tasks, LLM parameter sizes, feedback-noise levels, and random seeds. PID-gated continual prompt adaptation yields a more favorable Pareto trade-off between task performance (high accuracy, low error) and editor-call budget than the baselines considered, pointing to control-theoretic scheduling as a simple, general tool for robust prompt adaptation under unreliable feedback.

Scaling Large Language Models (LLMs) via Mixture-of-Experts (MoE) enables massive parameter growth with nearly constant per-token computation. However, further scaling the parameter count requires increasingly sparse routing, where expert load imbalance becomes more severe. This imbalance reduces parameter utilization and training efficiency, and can undermine training stability, becoming a bottleneck to reliable scaling. In this work, we unify two representative auxiliary-loss-free methods as incomplete Proportional-Integral-Derivative (PID) controllers: DeepSeek's loss-free method acts as a fixed-step integral controller, while Kimi K3's Quantile Balancing functions as a generalized proportional controller. Building on this control perspective, we propose ID Balancing, an Integral-Derivative controller. It scales its integral term with load error and activates its derivative term only when imbalance worsens, enabling stronger corrections for large or worsening errors and smaller updates near balance. Evaluated across Top-$10$, Top-$5$, and Top-$3$ routing over $768$ experts, ID Balancing reduces worst-case backbone MaxVio and training-average backbone MinVio by over $50$% and $12$%, respectively, relative to the best baselines in the Top-$3$ setting. When the total parameter count increases from $18.9$B to $69.9$B (Top-$10$-of-$768$), ID Balancing's worst-case backbone MaxVio remains nearly unchanged and is approximately $89.6$% lower than that of the auxiliary-loss baseline. ID Balancing also maintains competitive language-modeling and downstream performance. The advantages of ID Balancing grow as sparsity increases, making it a promising solution for scaling larger, sparser MoE models.

A long-standing open problem in robot manipulator control is whether global regulation can be achieved by classical PID control. This paper provides an answer to this question for classical PID controllers with triple parameters (k_p,k_i,k_d) in R^3. We find and prove that for one-degree-of-freedom manipulators, the classical PID control guarantees global stability and asymptotic regulation under standard structural assumptions, and further derive explicit quantitative design conditions for the PID gains. However, for multi-degree-of-freedom cases, we can construct a robot manipulator satisfying the same structural assumptions for which no choice of PID gains (k_p,k_i,k_d) can achieve global asymptotic regulation. These results provide a fundamental understanding of the abovementioned open problem, revealing both the fundamental capability and intrinsic limitation of the classical PID control for robot manipulator dynamics.

Recently, residual reconstruction-based model quantization methods have achieved promising performance in low-bit post-training quantization (PTQ) by introducing cross-layer residuals to reduce error accumulated from previous layers.However, these residuals may also introduce additional bias arising from the Hessian-approximation (HA) assumption underlying reconstruction-based PTQ, leading to suboptimal quantization performance.In this work, we analyze that multiplying the residual term by a scaling coefficient provides a direct way to mitigate the HA bias associated with residual strength, while preserving accumulated-error correction. More importantly, we observe that this trade-off is module-dependent, making a single global residual strength insufficient to balance effective correction and residual-related bias across modules.Based on these observations, we propose Module-Adaptive Residual Reconstruction (MARR), which assigns a module-specific scaling coefficient to adaptively balance accumulated-error correction and residual-related HA bias for each module.To avoid expensive per-module coefficient search and obtain a stable coefficient estimate, we design a Proportional-Integral-Derivative (PID)-based adaptive update strategy that uses reconstruction error as feedback to progressively refine this coefficient. Experiments on several typical large language models (LLMs) and vision transformers (ViTs) demonstrate the effectiveness of MARR under low-bit quantization (less than or equal to 4-bit), achieving up to 20.2% performance gains on LLMs and up to 4.6% relative gains on ViTs over the residual reconstruction state-of-the-art methods.Code will be made publicly available upon acceptance.

Decentralized federated learning (FL) is a promising approach for training machine learning models on sensor networks, Internet of Things (IoT) devices, and other edge systems where no central server exists. While federated learning offers advantages such as preserving data privacy, it often suffers from non-independent and identically distributed (IID) data distributions across devices, which cause significant performance degradation. This issue is particularly severe when directly optimizing model parameters, because neural network training is inherently non-convex and standard convergence guarantees for convex optimization do not apply. Unlike existing decentralized FL methods that primarily operate in parameter space, we propose federated function-space alternating direction method of multipliers (FedF-ADMM). FedF-ADMM exploits the convexity of loss functionals within function space to derive alternating direction method of multipliers (ADMM)-based update directions, which are subsequently projected onto the parameter space via knowledge distillation. We further introduce a stabilization coefficient to enhance robustness under severe non-IID settings and analyze its behavior from a control-theoretic perspective by interpreting it as a proportional-integral (PI) term. Experiments under challenging non-IID scenarios, including settings where each device has data from only a single label, demonstrate that FedF-ADMM achieves faster and more stable convergence than existing decentralized FL methods, while attaining higher accuracy and better consensus among devices.

Behavior cloning (BC) on position-controlled robots is shaped by the PD loop that executes policy actions. We give a finite-horizon, nonasymptotic analysis of how controller gains affect BC failure. Independent sub-Gaussian action errors propagate through gain-dependent closed-loop dynamics into sub-Gaussian position errors. The resulting failure tail is controlled by controller amplification multiplied by validation loss and generalization slack, so validation loss alone can mis-rank gains. Under shape-preserving upper-bound assumptions, the analysis separates label difficulty, injection strength, and contraction, ranking compliant-overdamped gains as tightest and stiff-underdamped gains as loosest, with the mixed regimes system-dependent. In the canonical scalar second-order PD system, stationary position-error variance increases with stiffness and decreases with damping over the stable range, and exact zero-order-hold discretization inherits the ordering to leading order. This extends the error-attenuation explanation of bronars et al. (2026) to finite-horizon failure bounds.

Classical proportional--integral--derivative (PID) control remains widely used in industrial control systems, while model predictive control (MPC) is actively studied to achieve higher performance for systems with nonlinear dynamics. Model predictive path integral (MPPI) control is a sampling-based MPC method that optimizes control inputs without gradient calculations and can handle non-differentiable models and objective functions. However, conventional MPPI directly samples control-input sequences, which can produce large temporal input increments and causes the optimization dimension to grow with the prediction horizon. This study proposes MPPI--PID control, which uses MPPI to optimize PID gains online instead of directly optimizing the control-input sequences. By replacing high-dimensional input-sequence optimization with low-dimensional gain-space optimization while retaining the PID structure, the proposed formulation improves sampling efficiency and promotes smoother control inputs. Theoretical analyses are provided for a unified path-integral update, the relation between optimization dimension and effective sample size, and the temporal correlation of input perturbations induced by the PID structure. The method is evaluated on a learning-based path following of a mini forklift using a residual-learning dynamics model that combines a physical model and a neural network identified from real-machine driving data. Numerical results show that MPPI--PID improves tracking performance over fixed-gain PID, yields smaller input increments than conventional MPPI, and maintains favorable performance under reduced sampling budgets.

In this paper, a Lyapunov-based synthesis of a PI-like controller is proposed for robust trajectory tracking of an independently driven and steered four-wheel mobile robot. For the robot considered in this work, an explicit structurally verified mathematical model is used to enable systematic controller design with rigorous stability guarantees suitable for real time implementation. An augmented Lyapunov-based practical stability analysis is developed for the combined velocity-error and integral-error dynamics of the inner loop, yielding explicit bounds and sufficient conditions for practical stability and uniform ultimate boundedness of the combined velocity-error and integral-error state. The resulting control law retains a PI-like structure with model-based feedforward compensation, making it suitable for implementation on standard embedded platforms while improving robustness against configuration dependent residual dynamics and unmodelled effects. The effectiveness and robustness of the proposed design are demonstrated experimentally on a four-wheel independently steered and independently driven mobile robot platform, under both horizontal and vertical operating conditions and benchmarked against a PI controller and a sliding-mode controller.

Reasoning LLMs trained with long chain-of-thought often overthink: they spend tokens on redundant reflection and transitions that inflate cost without improving accuracy. Static activation steering (e.g.\ SEAL) suppresses such content with a fixed vector, but applies the same strength regardless of how redundant the current chunk actually is. We describe PID-steering, a training-free, decoding-time method that modulates the steering strength with a PID controller driven by a lightweight chunk-level redundancy classifier. On a subset of GSM8K with DeepSeek-R1-Distill-Qwen-1.5B, the method improves accuracy from 85.7% to 89.6% (+3.9 pp) while cutting average output length from 1026 to 790 tokens ($-$23%). We report it as a small-scale proof of concept rather than a benchmark result.