Design Principles for Sequence Models via Coefficient Dynamics

Deep sequence models, ranging from Transformers and State Space Models (SSMs) to more recent approaches such as gated linear RNNs, fundamentally compute outputs as linear combinations of past value vectors. To draw insights and systematically compare such architectures, we develop a unified framework that makes this output operation explicit, by casting the linear combination coefficients as the outputs of autonomous linear dynamical systems driven by impulse inputs. This viewpoint, in spirit substantially different from approaches focusing on connecting linear RNNs with linear attention, reveals a common mathematical theme across diverse architectures and crucially captures softmax attention, on top of RNNs, SSMs, and related models. In contrast to new model proposals that are commonly evaluated on benchmarks, we derive design principles linking architectural choices to model properties. Thereby identifying tradeoffs between expressivity and efficient implementation, geometric constraints on input selectivity, and stability conditions for numerically stable training and information retention. By connecting several insights and observations from recent literature, the framework both explains empirical successes of recent designs and provides guiding principles for systematically designing new sequence model architectures.

Task-Level Insights from Eigenvalues across Sequence Models

Although softmax attention drives state-of-the-art performance for sequence models, its quadratic complexity limits scalability, motivating linear alternatives such as state space models (SSMs). While these alternatives improve efficiency, their fundamental differences in information processing remain poorly understood. In this work, we leverage the recently proposed dynamical systems framework to represent softmax, norm and linear attention as dynamical systems, enabling a structured comparison with SSMs by analyzing their respective eigenvalue spectra. Since eigenvalues capture essential aspects of dynamical system behavior, we conduct an extensive empirical analysis across diverse sequence models and benchmarks. We first show that eigenvalues influence essential aspects of memory and long-range dependency modeling, revealing spectral signatures that align with task requirements. Building on these insights, we then investigate how architectural modifications in sequence models impact both eigenvalue spectra and task performance. This correspondence further strengthens the position of eigenvalue analysis as a principled metric for interpreting, understanding, and ultimately improving the capabilities of sequence models.

Model Predictive Control with Reference Learning for Soft Robotic Intracranial Pressure Waveform Modulation

This paper introduces a learning-based control framework for a soft robotic actuator system designed to modulate intracranial pressure (ICP) waveforms, which is essential for studying cerebrospinal fluid dynamics and pathological processes underlying neurological disorders. A two-layer framework is proposed to safely achieve a desired ICP waveform modulation. First, a model predictive controller (MPC) with a disturbance observer is used for offset-free tracking of the system's motor position reference trajectory under safety constraints. Second, to address the unknown nonlinear dependence of ICP on the motor position, we employ a Bayesian optimization (BO) algorithm used for online learning of a motor position reference trajectory that yields the desired ICP modulation. The framework is experimentally validated using a test bench with a brain phantom that replicates realistic ICP dynamics in vitro. Compared to a previously employed proportional-integral-derivative controller, the MPC reduces mean and maximum motor position reference tracking errors by 83 % and 73 %, respectively. In less than 20 iterations, the BO algorithm learns a motor position reference trajectory that yields an ICP waveform with the desired mean and amplitude.

ZipMPC: Compressed Context-Dependent MPC Cost via Imitation Learning

The computational burden of model predictive control (MPC) limits its application on real-time systems, such as robots, and often requires the use of short prediction horizons. This not only affects the control performance, but also increases the difficulty of designing MPC cost functions that reflect the desired long-term objective. This paper proposes ZipMPC, a method that imitates a long-horizon MPC behaviour by learning a compressed and context-dependent cost function for a short-horizon MPC. It improves performance over alternative methods, such as approximate explicit MPC and automatic cost parameter tuning, in particular in terms of i) optimizing the long term objective; ii) maintaining computational costs comparable to a short-horizon MPC; iii) ensuring constraint satisfaction; and iv) generalizing control behaviour to environments not observed during training. For this purpose, ZipMPC leverages the concept of differentiable MPC with neural networks to propagate gradients of the imitation loss through the MPC optimization. We validate our proposed method in simulation and real-world experiments on autonomous racing. ZipMPC consistently completes laps faster than selected baselines, achieving lap times close to the long-horizon MPC baseline. In challenging scenarios where the short-horizon MPC baseline fails to complete a lap, ZipMPC is able to do so. In particular, these performance gains are also observed on tracks unseen during training.

Time-Varying Coverage Control: A Distributed Tracker-Planner MPC Framework

Time-varying coverage control addresses the challenge of coordinating multiple agents covering an environment where regions of interest change over time. This problem has broad applications, including the deployment of autonomous taxis and coordination in search and rescue operations. The achievement of effective coverage is complicated by the presence of time-varying density functions, nonlinear agent dynamics, and stringent system and safety constraints. In this paper, we present a distributed multi-agent control framework for time-varying coverage under nonlinear constrained dynamics. Our approach integrates a reference trajectory planner and a tracking model predictive control (MPC) scheme, which operate at different frequencies within a multi-rate framework. For periodic density functions, we demonstrate closed-loop convergence to an optimal configuration of trajectories and provide formal guarantees regarding constraint satisfaction, collision avoidance, and recursive feasibility. Additionally, we propose an efficient algorithm capable of handling nonperiodic density functions, making the approach suitable for practical applications. Finally, we validate our method through hardware experiments using a fleet of four miniature race cars.

A robust and adaptive MPC formulation for Gaussian process models

In this paper, we present a robust and adaptive model predictive control (MPC) framework for uncertain nonlinear systems affected by bounded disturbances and unmodeled nonlinearities. We use Gaussian Processes (GPs) to learn the uncertain dynamics based on noisy measurements, including those collected during system operation. As a key contribution, we derive robust predictions for GP models using contraction metrics, which are incorporated in the MPC formulation. The proposed design guarantees recursive feasibility, robust constraint satisfaction and convergence to a reference state, with high probability. We provide a numerical example of a planar quadrotor subject to difficult-to-model ground effects, which highlights significant improvements achieved through the proposed robust prediction method and through online learning.

Optimal kernel regression bounds under energy-bounded noise

Non-conservative uncertainty bounds are key for both assessing an estimation algorithm's accuracy and in view of downstream tasks, such as its deployment in safety-critical contexts. In this paper, we derive a tight, non-asymptotic uncertainty bound for kernel-based estimation, which can also handle correlated noise sequences. Its computation relies on a mild norm-boundedness assumption on the unknown function and the noise, returning the worst-case function realization within the hypothesis class at an arbitrary query input location. The value of this function is shown to be given in terms of the posterior mean and covariance of a Gaussian process for an optimal choice of the measurement noise covariance. By rigorously analyzing the proposed approach and comparing it with other results in the literature, we show its effectiveness in returning tight and easy-to-compute bounds for kernel-based estimates.

Finite-Sample-Based Reachability for Safe Control with Gaussian Process Dynamics

Gaussian Process (GP) regression is shown to be effective for learning unknown dynamics, enabling efficient and safety-aware control strategies across diverse applications. However, existing GP-based model predictive control (GP-MPC) methods either rely on approximations, thus lacking guarantees, or are overly conservative, which limits their practical utility. To close this gap, we present a sampling-based framework that efficiently propagates the model's epistemic uncertainty while avoiding conservatism. We establish a novel sample complexity result that enables the construction of a reachable set using a finite number of dynamics functions sampled from the GP posterior. Building on this, we design a sampling-based GP-MPC scheme that is recursively feasible and guarantees closed-loop safety and stability with high probability. Finally, we showcase the effectiveness of our method on two numerical examples, highlighting accurate reachable set over-approximation and safe closed-loop performance.

Performance-driven Constrained Optimal Auto-Tuner for MPC

A key challenge in tuning Model Predictive Control (MPC) cost function parameters is to ensure that the system performance stays consistently above a certain threshold. To address this challenge, we propose a novel method, COAT-MPC, Constrained Optimal Auto-Tuner for MPC. With every tuning iteration, COAT-MPC gathers performance data and learns by updating its posterior belief. It explores the tuning parameters' domain towards optimistic parameters in a goal-directed fashion, which is key to its sample efficiency. We theoretically analyze COAT-MPC, showing that it satisfies performance constraints with arbitrarily high probability at all times and provably converges to the optimum performance within finite time. Through comprehensive simulations and comparative analyses with a hardware platform, we demonstrate the effectiveness of COAT-MPC in comparison to classical Bayesian Optimization (BO) and other state-of-the-art methods. When applied to autonomous racing, our approach outperforms baselines in terms of constraint violations and cumulative regret over time.

Moving Horizon Estimation for Simultaneous Localization and Mapping with Robust Estimation Error Bounds

This paper presents a robust moving horizon estimation (MHE) approach with provable estimation error bounds for solving the simultaneous localization and mapping (SLAM) problem. We derive sufficient conditions to guarantee robust stability in ego-state estimates and bounded errors in landmark position estimates, even under limited landmark visibility which directly affects overall system detectability. This is achieved by decoupling the MHE updates for the ego-state and landmark positions, enabling individual landmark updates only when the required detectability conditions are met. The decoupled MHE structure also allows for parallelization of landmark updates, improving computational efficiency. We discuss the key assumptions, including ego-state detectability and Lipschitz continuity of the landmark measurement model, with respect to typical SLAM sensor configurations, and introduce a streamlined method for the range measurement model. Simulation results validate the considered method, highlighting its efficacy and robustness to noise.