Set representations are the foundation of various set-based approaches in state estimation, reachability analysis and fault diagnosis. In this paper, we investigate spectrahedral shadows, a class of nonlinear geometric objects previously studied in semidefinite programming and real algebraic geometry. We demonstrate spectrahedral shadows generalize traditional and emerging set representations like ellipsoids, zonotopes, constrained zonotopes and ellipsotopes. Analytical forms of set operations are provided including linear map, linear inverse map, Minkowski sum, intersection, Cartesian product, Minkowski-Firey Lp sum, convex hull, conic hull and polytopic map, all of which are implemented without approximation in polynomial time. In addition, we develop set validation and order reduction techniques for spectrahedral shadows, thereby establishing spectrahedral shadows as a set representation applicable to a range of set-based tasks.
Control Barrier Functions (CBFs) offer a framework for ensuring set invariance and designing constrained control laws. However, crafting a valid CBF relies on system-specific assumptions and the availability of an accurate system model, underscoring the need for systematic data-driven synthesis methods. This paper introduces a data-driven approach to synthesizing a CBF for discrete-time LTI systems using only input-output measurements. The method begins by computing the maximal control invariant set using an input-output data-driven representation, eliminating the need for precise knowledge of the system's order and explicit state estimation. The proposed CBF is then systematically derived from this set, which can accommodate multiple input-output constraints. Furthermore, the proposed CBF is leveraged to develop a minimally invasive safety filter that ensures recursive feasibility with an adaptive decay rate. To improve clarity, we assume a noise-free dataset, though the method can be extended using data-driven reachability to capture noise effects and handle uncertainty. Finally, the effectiveness of the proposed method is demonstrated on an unknown time-delay system.
Understanding the confidence with which a machine learning model classifies an input datum is an important, and perhaps under-investigated, concept. In this paper, we propose a new calibration metric, the Entropic Calibration Difference (ECD). Based on existing research in the field of state estimation, specifically target tracking (TT), we show how ECD may be applied to binary classification machine learning models. We describe the relative importance of under- and over-confidence and how they are not conflated in the TT literature. Indeed, our metric distinguishes under- from over-confidence. We consider this important given that algorithms that are under-confident are likely to be 'safer' than algorithms that are over-confident, albeit at the expense of also being over-cautious and so statistically inefficient. We demonstrate how this new metric performs on real and simulated data and compare with other metrics for machine learning model probability calibration, including the Expected Calibration Error (ECE) and its signed counterpart, the Expected Signed Calibration Error (ESCE).
This study proposes a GPA for designing optimal Quantum Sensor Circuits (QSCs) to address complex quantum physics problems. The GPA consists of two parts: the Quantum Policy Evaluation (QPE) and the Quantum Policy Improvement (QPI). The QPE performs phase estimation to generate the search space, while the QPI utilizes Grover search and amplitude amplification techniques to efficiently identify an optimal policy that generates optimal QSCs. The GPA generates QSCs by selecting sequences of gates that maximize the Quantum Fisher Information (QFI) while minimizing the number of gates. The QSCs generated by the GPA are capable of producing entangled quantum states, specifically the squeezed states. High QFI indicates increased sensitivity to parameter changes, making the circuit useful for quantum state estimation and control tasks. Evaluation of the GPA on a QSC that consists of two qubits and a sequence of R_x, R_y, and S gates demonstrates its efficiency in generating optimal QSCs with a QFI of 1. Compared to existing quantum agents, the GPA achieves higher QFI with fewer gates, demonstrating a more efficient and scalable approach to the design of QSCs. This work illustrates the potential computational power of quantum agents for solving quantum physics problems
For challenging state estimation problems arising in domains like vision and robotics, particle-based representations attractively enable temporal reasoning about multiple posterior modes. Particle smoothers offer the potential for more accurate offline data analysis by propagating information both forward and backward in time, but have classically required human-engineered dynamics and observation models. Extending recent advances in discriminative training of particle filters, we develop a framework for low-variance propagation of gradients across long time sequences when training particle smoothers. Our "two-filter'' smoother integrates particle streams that are propagated forward and backward in time, while incorporating stratification and importance weights in the resampling step to provide low-variance gradient estimates for neural network dynamics and observation models. The resulting mixture density particle smoother is substantially more accurate than state-of-the-art particle filters, as well as search-based baselines, for city-scale global vehicle localization from real-world videos and maps.
In this paper, we investigate state estimation and opacity verification problems within a decentralized observation architecture. Specifically, we consider a discrete event system whose behavior is recorded by a set of observation sites. These sites transmit the partially ordered sequences of observations that they record to a coordinator whenever a \textit{synchronization} occurs. To properly analyze the system behavior from the coordinator's viewpoint, we first introduce the notion of a Complete Synchronizing Sequence structure (CSS structure), which concisely captures the state evolution of each system state upon different information provided by the observation sites. Based on the CSS structure, we then construct corresponding current-state and initial-state estimators for offline state estimation at the coordinator. When used to verify state-isolation properties under this decentralized architecture, the use of CSS structure demonstrates a significant reduction in complexity compared with existing approaches in the literature. In particular, we discuss how to verify initial-state opacity at the coordinator, as well as a novel opacity notion, namely current-state-at-synchronization opacity.
This paper proposes a novel localization framework based on collaborative training or federated learning paradigm, for highly accurate localization of autonomous vehicles. More specifically, we build on the standard approach of KalmanNet, a recurrent neural network aiming to estimate the underlying system uncertainty of traditional Extended Kalman Filtering, and reformulate it by the adapt-then-combine concept to FedKalmanNet. The latter is trained in a distributed manner by a group of vehicles (or clients), with local training datasets consisting of vehicular location and velocity measurements, through a global server aggregation operation. The FedKalmanNet is then used by each vehicle to localize itself, by estimating the associated system uncertainty matrices (i.e, Kalman gain). Our aim is to actually demonstrate the benefits of collaborative training for state estimation in autonomous driving, over collaborative decision-making which requires rich V2X communication resources for measurement exchange and sensor fusion under real-time constraints. An extensive experimental and evaluation study conducted in CARLA autonomous driving simulator highlights the superior performance of FedKalmanNet over state-of-the-art collaborative decision-making approaches, in localizing vehicles without the need of real-time V2X communication.
State estimation remains a fundamental challenge across numerous domains, from autonomous driving, aircraft tracking to quantum system control. Although Bayesian filtering has been the cornerstone solution, its classical model-based paradigm faces two major limitations: it struggles with inaccurate state space model (SSM) and requires extensive prior knowledge of noise characteristics. We present TrackDiffuser, a generative framework addressing both challenges by reformulating Bayesian filtering as a conditional diffusion model. Our approach implicitly learns system dynamics from data to mitigate the effects of inaccurate SSM, while simultaneously circumventing the need for explicit measurement models and noise priors by establishing a direct relationship between measurements and states. Through an implicit predict-and-update mechanism, TrackDiffuser preserves the interpretability advantage of traditional model-based filtering methods. Extensive experiments demonstrate that our framework substantially outperforms both classical and contemporary hybrid methods, especially in challenging non-linear scenarios involving non-Gaussian noises. Notably, TrackDiffuser exhibits remarkable robustness to SSM inaccuracies, offering a practical solution for real-world state estimation problems where perfect models and prior knowledge are unavailable.
Decision diagrams (DDs) have emerged as an efficient tool for simulating quantum circuits due to their capacity to exploit data redundancies in quantum states and quantum operations, enabling the efficient computation of probability amplitudes. However, their application in quantum machine learning (QML) has remained unexplored. This paper introduces variational decision diagrams (VDDs), a novel graph structure that combines the structural benefits of DDs with the adaptability of variational methods for efficiently representing quantum states. We investigate the trainability of VDDs by applying them to the ground state estimation problem for transverse-field Ising and Heisenberg Hamiltonians. Analysis of gradient variance suggests that training VDDs is possible, as no signs of vanishing gradients--also known as barren plateaus--are observed. This work provides new insights into the use of decision diagrams in QML as an alternative to design and train variational ans\"atze.
This work proposes an adaptive output feedback model predictive control (MPC) framework for uncertain systems subject to external disturbances. In the absence of exact knowledge about the plant parameters and complete state measurements, the MPC optimization problem is reformulated in terms of their estimates derived from a suitably designed robust adaptive observer. The MPC routine returns a homothetic tube for the state estimate trajectory. Sets that characterize the state estimation errors are then added to the homothetic tube sections, resulting in a larger tube containing the true state trajectory. The two-tier tube architecture provides robustness to uncertainties due to imperfect parameter knowledge, external disturbances, and incomplete state information. Additionally, recursive feasibility and robust exponential stability are guaranteed and validated using a numerical example.
Optimal sensor placement is essential for minimizing costs and ensuring accurate state estimation in power systems. This paper introduces a novel method for optimal sensor placement for dynamic state estimation of power systems modeled by differential-algebraic equations. The method identifies optimal sensor locations by minimizing the steady-state covariance matrix of the Kalman filter, thus minimizing the error of joint differential and algebraic state estimation. The problem is reformulated as a mixed-integer semidefinite program and effectively solved using off-the-shelf numerical solvers. Numerical results demonstrate the merits of the proposed approach by benchmarking its performance in phasor measurement unit placement in comparison to greedy algorithms.
We propose a Neural-Enhanced Distributed Kalman Filter (NDKF) for multi-sensor state estimation in nonlinear systems. Unlike traditional Kalman filters that rely on explicit, linear models and centralized data fusion, the NDKF leverages neural networks to learn both the system dynamics and measurement functions directly from data. Each sensor node performs local prediction and update steps using these learned models and exchanges only compact summary information with its neighbors via a consensus-based fusion process, which reduces communication overhead and eliminates a single point of failure. Our theoretical convergence analysis establishes sufficient conditions under which the local linearizations of the neural models guarantee overall filter stability and provides a solid foundation for the proposed approach. Simulation studies on a 2D system with four partially observing nodes demonstrate that the NDKF significantly outperforms a distributed Extended Kalman Filter. These outcomes, as yielded by the proposed NDKF method, highlight its potential to improve the scalability, robustness, and accuracy of distributed state estimation in complex nonlinear environments.
This paper proposes a distributed prescribed-time observer for nonlinear systems representable in a block-triangular observable canonical form. Using a weighted average of neighbor estimates exchanged over a strongly connected digraph, each observer estimates the system state despite the limited observability of local sensor measurements. The proposed design guarantees that distributed state estimation errors converge to zero at a user-specified convergence time, irrespective of observers' initial conditions. To achieve this prescribed-time convergence, distributed observers implement time-varying local output injection gains that monotonically increase and approach infinity at the prescribed time. The theoretical convergence is rigorously proven and validated through numerical simulations, where some implementation issues due to increasing gains have also been clarified.
Since time immemorial an old adage has always seemed to ring true: you cannot use a high-level productive programming language like Python or R for real-time control and embedded-systems programming, you must rewrite your program in C. We present a counterexample to this mantra by demonstrating how recent compiler developments in the Julia programming language allow users of Julia and the equation-based modeling language ModelingToolkit to compile and deploy binaries for real-time model-based estimation and control. Contrary to the approach taken by a majority of modeling and simulation tools, we do not generate C code, and instead demonstrate how we may use the native Julia code-generation pipeline through LLVM to compile architecture-specific binaries from high-level code. This approach avoids many of the restrictions typically placed on high-level languages to enable C-code generation. As case studies, we include a nonlinear state estimator derived from an equation-based model which is compiled into a program that performs state estimation for deployment onto a Raspberry Pi, as well as a PID controller library implemented in Julia and compiled into a shared library callable from a C program.
Behavioural analytics provides insights into individual and crowd behaviour, enabling analysis of what previously happened and predictions for how people may be likely to act in the future. In defence and security, this analysis allows organisations to achieve tactical and strategic advantage through influence campaigns, a key counterpart to physical activities. Before action can be taken, online and real-world behaviour must be analysed to determine the level of threat. Huge data volumes mean that automated processes are required to attain an accurate understanding of risk. We describe the mathematical basis of technologies to analyse quotes in multiple languages. These include a Bayesian network to understand behavioural factors, state estimation algorithms for time series analysis, and machine learning algorithms for classification. We present results from studies of quotes in English, French, and Arabic, from anti-violence campaigners, politicians, extremists, and terrorists. The algorithms correctly identify extreme statements; and analysis at individual, group, and population levels detects both trends over time and sharp changes attributed to major geopolitical events. Group analysis shows that additional population characteristics can be determined, such as polarisation over particular issues and large-scale shifts in attitude. Finally, MP voting behaviour and statements from publicly-available records are analysed to determine the level of correlation between what people say and what they do.
Particle filtering is a Bayesian inference method and a fundamental tool in state estimation for dynamic systems, but its effectiveness is often limited by the constraints of the initial prior distribution, a phenomenon we define as the Prior Boundary Phenomenon. This challenge arises when target states lie outside the prior's support, rendering traditional particle filtering methods inadequate for accurate estimation. Although techniques like unbounded priors and larger particle sets have been proposed, they remain computationally prohibitive and lack adaptability in dynamic scenarios. To systematically overcome these limitations, we propose the Diffusion-Enhanced Particle Filtering Framework, which introduces three key innovations: adaptive diffusion through exploratory particles, entropy-driven regularisation to prevent weight collapse, and kernel-based perturbations for dynamic support expansion. These mechanisms collectively enable particle filtering to explore beyond prior boundaries, ensuring robust state estimation for out-of-boundary targets. Theoretical analysis and extensive experiments validate framework's effectiveness, indicating significant improvements in success rates and estimation accuracy across high-dimensional and non-convex scenarios.
Control loops closed over wireless links greatly benefit from accurate estimates of the communication channel condition. To this end, the finite-state Markov channel model allows for reliable channel state estimation. This paper develops a Markov jump linear system representation for wireless networked control with persistent channel state observation, stochastic message losses, and generalized packet dropout compensation. With this model, we solve the finite- and infinite-horizon linear quadratic regulation problems and introduce an easy-to-test stability condition for any given infinite-horizon control law. We also thoroughly analyze the impact of a scalar general dropout compensation factor on the stability and closed-loop performance of a rotary inverted pendulum controlled remotely through a wireless link. Finally, we validate the results numerically via extensive Monte Carlo simulations, showing the benefits of the proposed control strategy.
Data assimilation (DA) integrates observations with a dynamical model to estimate states of PDE-governed systems. Model-driven methods (e.g., Kalman, particle) presuppose full knowledge of the true dynamics, which is not always satisfied in practice, while purely data-driven solvers learn a deterministic mapping between observations and states and therefore miss the intrinsic stochasticity of real processes. Recently, score-based diffusion models learn a global diffusion prior and provide a good modeling of the stochastic dynamics, showing new potential for DA. However, their all-at-once generation rather than step-by-step transition limits their performance when dealing with highly complex stochastic processes and lacks physical interpretability. To tackle these drawbacks, we introduce FlowDAS, a generative DA framework that uses stochastic interpolants to directly learn state transition dynamics and achieve step-by-step transition to better model the real dynamics. We also improve the framework by combining the observation, better suiting the DA settings. Directly learning the underlying dynamics from collected data removes restrictive dynamical assumptions, and conditioning on observations at each interpolation step yields stable, measurement-consistent forecasts. Experiments on Lorenz-63, Navier-Stokes super-resolution/sparse-observation scenarios, and large-scale weather forecasting -- where dynamics are partly or wholly unknown -- show that FlowDAS surpasses model-driven methods, neural operators, and score-based baselines in accuracy and physical plausibility.
In this work, we investigate low-complexity remote system state estimation over wireless multiple-input-multiple-output (MIMO) channels without requiring prior knowledge of channel state information (CSI). We start by reviewing the conventional Kalman filtering-based state estimation algorithm, which typically relies on perfect CSI and incurs considerable computational complexity. To overcome the need for CSI, we introduce a novel semantic aggregation method, in which sensors transmit semantic measurement discrepancies to the remote state estimator through analog aggregation. To further reduce computational complexity, we introduce a constant-gain-based filtering algorithm that can be optimized offline using the constrained stochastic successive convex approximation (CSSCA) method. We derive a closed-form sufficient condition for the estimation stability of our proposed scheme via Lyapunov drift analysis. Numerical results showcase significant performance gains using the proposed scheme compared to several widely used methods.
Inertial odometry (IO) using only Inertial Measurement Units (IMUs) offers a lightweight and cost-effective solution for Unmanned Aerial Vehicle (UAV) applications, yet existing learning-based IO models often fail to generalize to UAVs due to the highly dynamic and non-linear-flight patterns that differ from pedestrian motion. In this work, we identify that the conventional practice of transforming raw IMU data to global coordinates undermines the observability of critical kinematic information in UAVs. By preserving the body-frame representation, our method achieves substantial performance improvements, with a 66.7% average increase in accuracy across three datasets. Furthermore, explicitly encoding attitude information into the motion network results in an additional 23.8% improvement over prior results. Combined with a data-driven IMU correction model (AirIMU) and an uncertainty-aware Extended Kalman Filter (EKF), our approach ensures robust state estimation under aggressive UAV maneuvers without relying on external sensors or control inputs. Notably, our method also demonstrates strong generalizability to unseen data not included in the training set, underscoring its potential for real-world UAV applications.