Papers

Shared-control arbitration mechanisms can fail structurally when they are driven by quantities that evolve on the same automation-assisted trajectory that unsafe automation can corrupt. Under unsafe assistance, arising from faults or flawed design, the arbitration logic can adapt to the corrupted trajectory instead of resisting it. Focusing on workload regulation, this paper proposes a framework that prevents this problem by anchoring the safety constraints to an unassisted pilot trajectory. A baseline model evolving without automation assistance generates the acceptable workload envelope, while an assistance-induced deviation barrier and a closed-form mismatch bound relate actual and baseline workload evolution. The resulting control barrier function based quadratic program is always feasible and guarantees forward invariance of an augmented safe set. Simulations on an F-16 pitch-tracking problem show that the proposed method prevents workload-constraint drift under unsafe assistance.

Integrated deep reinforcement learning (DRL) and model predictive control (MPC) methods are increasingly used to control autonomous systems by combining their complementary capabilities. DRL learns control policies through interaction with the environment. MPC uses a system model to optimize control inputs while accounting for constraints. In DRL-MPC frameworks with shared control authority, both the DRL agent and the MPC controller each determine part of the control inputs. However, common learning formulations treat MPC as part of the environment and therefore do not explicitly account for MPC's contribution to control or its interaction with the DRL agent. This paper proposes a novel composite-gradient learning (CGL) method that integrates the MPC controller into the learning process by representing the DRL and MPC control inputs as a joint action and accounting for their interaction when updating the DRL agent during training. CGL is evaluated on two multi-class freeway traffic networks with different strengths of interaction between the DRL and MPC control inputs and it is compared with alternative methods that treat MPC as part of the environment or that only partially incorporate MPC into learning. The results show that CGL offers limited benefit under weak interaction, but learns higher-performing control policies than the alternative methods in a subset of training runs under strong interaction, although the average control-performance gains remain modest.

We present a framework for the modeling and analysis of geometric hybrid dynamical systems as hybrid inclusions on $C^1$-manifolds. Using tools from nonsmooth and set-valued analysis on manifolds, we derive coordinate-independent sufficient conditions for the existence of nontrivial solutions to this type of systems. We present geometric notions of uniform stability and attractivity of compact sets, and establish their equivalence to metric-based stability notions when the manifold is endowed with a Riemannian structure. We also introduce nonsmooth Lyapunov functions and a hybrid Lyapunov theorem providing sufficient conditions for uniform global asymptotic stability of compact sets. Finally, by characterizing $\omega$-limit sets of precompact solutions, we derive a hybrid invariance principle for geometric hybrid dynamical systems. The results are demonstrated through several running examples.

This paper proposes a least squares approach for the training of quadratic neural networks with regularization. The proposed methodology yields a lower bound on the solution of the training optimization problem for the case where the regularization coefficient is positive. Moreover, it yields closed-form expressions for the approximate solution and its sensitivity The lower bound is tight and the approximate solution is the optimal solution when the regularization coefficient is zero. Having a closed-form expression for the weights reduces considerably the computational time when compared with iterative numerical methods such as backpropagation that can get stuck in local minima. The proposed approach has three main contributions, namely, (i) it yields an analytical expression for the weights, (ii) an analytical expression for the sensitivity of the weights to errors in the data is also provided, (iii) it establishes a connection between the optimization to compute a lower bound and nuclear norm minimization. The proposed least squares training is successfully applied to a nonlinear system identification example where the proposed lower bound is compared with the optimal value.

Hybrid dynamical systems provide a powerful modeling framework for robotic systems, particularly in contact-rich environments. However, ensuring safety and performance in such systems remains challenging due to the intricate coupling between continuous dynamics and discrete mode transitions. In this work, we extend classical Hamilton-Jacobi (HJ) reachability analysis, a formal verification method for continuous-time nonlinear systems, to hybrid dynamical systems. Our framework characterizes safe sets for hybrid systems through a generalized value function defined over both discrete and continuous states while accounting for control constraints and model uncertainty. We additionally provide a numerical algorithm to compute this value function. Building on these safe sets, we propose two different mechanisms to integrate performance objectives. First, we introduce a hybrid least-restrictive safety filter that intervenes on both the discrete and continuous components of a nominal controller only when necessary to avoid unsafe states, thereby preserving nominal behavior whenever possible. Second, we formulate and compute hybrid backward reach-avoid tubes, enabling the simultaneous enforcement of safety and goal-reaching behavior, an extension not previously addressed within hybrid HJ reachability. This enables the synthesis of continuous and discrete control policies that guarantee both safety and task completion. We validate our framework through simulation studies and real-world experiments on a quadrupedal robot, demonstrating its effectiveness in hybrid mode planning and safety-critical applications.

In embodied cyber-physical systems, active cyberattacks pose an immediate threat not just to data, but to physical integrity and human safety. While existing security approaches excel at detection, they lack the runtime mechanisms to determine whether a disruption is tolerable or if performance degradation remains within safe operational bounds. This gap leaves autonomous systems vulnerable to graceful failure paralysis, where they cannot distinguish between a safe, degraded state and a catastrophic hazard during an ongoing attack. This paper presents RobResilience, an implementation of a formal resilience framework for embodied cyber-physical systems in a Webots simulation environment, using a PR2 robot and ROS2. The framework evaluates three predicates at runtime: tolerable disruption ($\delta$), tolerable degradation ($\gamma$), and mitigation feasibility ($\mu$), over a compromised device set derived from IDS confidence scores. When resilience is lost, the framework triggers available mitigation strategies. We evaluate our implementation through eight attack scenarios that systematically cover all possible combinations of the predicate state space, varying attack targets, degradation rates, and mitigation availability. Results confirm that the runtime behaviour of the implementation is consistent with the theoretical definitions.

Collision avoidance constraints for autonomous systems are typically formulated in position or velocity space, implicitly reacting to geometric proximity. We propose an alternative paradigm based on the adversarial time-to-collision (aTTC): the minimum time in which an adversary could achieve a collision given its dynamical constraints. By defining a control barrier function (CBF) directly in the time domain, the resulting controller is inherently anticipatory. The evading agent responds not only to whether a pursuer is on a collision course, but to how quickly it could reach one. This formulation enables velocity modulation that exploits the pursuers dynamic limits as an evasive strategy, a behavior not captured by standard distance-based CBFs. Since exact aTTC computation requires integrating the full system dynamics, we employ a lightweight neural network surrogate that admits a real-time quadratic program-based control law. We validate the approach in a 2D comparative study and a 3D multi-agent pursuit-evasion scenario, where the aTTC-based CBF outperforms a higher-order distance-based baseline by more effectively buying time against superior pursuers with a significant speed advantage.

Recently, dynamic state feedback controllers that are based on dynamic extensions have been presented for heterodirectional hyperbolic systems. In this paper, a similar concept for the control of coupled diffusion-reaction systems is suggested. The introduction of a specific controller dynamics leads to homogenized diffusion coefficients for the extended system. Then, a backstepping-based static state feedback for the dynamically extended system is designed, which, overall, results in a dynamic state feedback. Such a design allows stabilizing a more general class of parabolic systems as well as assigning arbitrary closed-loop dynamics. This can be used, e.g., to achieve a decoupled input-output behavior, which is, in general, not possible with a static state feedback. A simulation example illustrates the results.

Finite-dimensional Koopman models enable efficient linear prediction and control of nonlinear robotic systems. However, models learned purely from trajectory data may violate the energetic structure of the underlying mechanics, producing predictions that exhibit artificial energy growth and diverge under recursive propagation. This work presents a structure-preserving Koopman framework for Euler-Lagrange systems built on generalized-momentum coordinates. The momentum transformation exposes the mechanical actuation as a known, state-independent port, which is preserved explicitly in the lifted dynamics. A structure-constrained neural architecture is developed to jointly learn the lifting functions and a port-Hamiltonian Koopman generator, rendering the learned dynamics passive by construction rather than through penalty terms or post-hoc projection. A Cayley-midpoint discretization further preserves the corresponding storage-dissipation balance exactly in discrete time. These properties are established analytically by deriving the discrete storage balance and associated stability guarantees of the learned predictor. Simulation and experimental studies demonstrate improved prediction accuracy, data efficiency, and closed-loop tracking over Koopman baselines, with increasing gains for higher-dimensional systems.

This article investigates data-driven policy iteration (PI) for continuous-time linear systems without requiring an initially stabilizing policy. Standard infinite-horizon PI is not self-starting because its policy-evaluation step is well posed only when the feedback gain is stabilizing. However, verifying this property is difficult when the system matrices are unknown. To remove this requirement, we develop a finite-horizon bootstrap method. The key idea is to perform policy evaluation over a compact interval for a shifted system, where the evaluation equation is well defined for arbitrary bounded time-varying policies. We show that, for a sufficiently long horizon, the initial-time optimal gain of the shifted finite-horizon problem, when applied as a constant feedback gain, achieves a prescribed stability margin for the original system. We then derive a data-driven implementation from an off-policy identity evaluated along trajectories of the original plant. We use basis-function approximations to reconstruct the finite-horizon value matrix and policy, and we characterize the resulting error through a perturbed policy-improvement recursion. A data-driven Lyapunov certificate is further introduced to verify admissibility of the candidate gain before it is used to initialize infinite-horizon PI. Numerical studies of a batch reactor and a two-mass-spring system demonstrate the effectiveness of the proposed bootstrap method.

A walking robotic hand must use the same fingers to move its body, support its weight, and interact with the environment. We show how an anthropomorphic hand can learn these skills while retaining its finger design and position controller. Onboard power and computation make the platform self-contained. Our reinforcement learning approach accounts for the hand's unequal fingers, with training in a simulator calibrated from hardware measurements. In simulation, the hand moves faster with our reward formulation than with tuned rewards originally designed for quadrupeds. On hardware, task-specific policies enable untethered crawling, steering, and fall recovery. While supporting its own weight, the hand also executes successive keyboard commands without vision and pushes an object to targets using overhead visual feedback. These results demonstrate a compact mobile manipulator that reuses its fingers for locomotion and interaction, without a separate locomotion mechanism.

We study an optimal intervention problem for linear threshold models. This is a popular class of dynamical network systems whereby a number of agents, identified with the nodes of a graph, strategically change their binary action (0 or 1) according to a threshold rule. Specifically, an agent adopts action 1 if and only if the fraction of its neighbors in the interaction graph that do so is greater than or equal to a prescribed threshold. Assuming that a planner can modify the agents' thresholds at a cost equal to the aggregate threshold increase, we study the minimum intervention cost needed to ensure global convergence to the all-1 configuration. Our main contribution is the introduction of a new graph-theoretic quantity, called oriented path number, that is the minimum number of disjoint paths needed to cover the graph that can be oriented to form a directed acyclic graph. When thresholds are all equal to 1/2, the optimal cost is shown to coincide with the oriented path number, whereas, in the general case, it turns out to be the main ingredient of a bound on the optimal intervention cost.

Distributionally robust (DR) optimization seeks decisions that perform best under the most adverse law within a given ambiguity set, enabling the design of data-driven controllers with strong out-of-sample guarantees in the face of uncertainty. In this paper, we study the conservatism introduced by safeguarding against distributional ambiguity. Specifically, we consider the data-driven Wasserstein DR linear quadratic control problem, and we analyze the suboptimality of the corresponding solution relative to the oracle controller computed with foreknowledge of the underlying unknown uncertainty distribution. We present a sample complexity bound that characterizes the number of samples required to ensure that the true cost of the DR solution exceeds that of the oracle controller by at most a user-defined tolerance factor. Our analysis reveals that the suboptimality of the DR solution increases at most linearly with the Wasserstein radius for sufficiently small distributional ambiguity, and at most quadratically away from this local regime. Numerical simulations validate our bounds on the price of distributional robustness.

The Asset Administration Shell (AAS) is increasingly recognized as a fundamental model for the realization of and data exchange between digital twins in manufacturing. An AAS defines a hierarchical data structure to represent any type of asset throughout its entire lifecycle. In the context of AAS-based systems, comparing different AAS instances constitutes a practical challenge, as neither a widely accepted methodological framework nor a maturity model are available to systematically support such analyses. To address this gap, we propose a novel concept of AAS maturity that characterizes the extent to which established digital twin criteria are met and thus enabling comparability of AAS instances. The concepts are derived from the literature and applied through exemplification. These emerging results enable practitioners and researchers to systematically compare AAS instances and support the identification and assessment of further development steps in the digital twin engineering process.

Asset Administration Shells (AAS) provide a standardized means of representing assets and their information in manufacturing and increasingly serve as a basis for software services. However, different AAS instances vary in structure, content, and degree of completion, making it difficult to determine whether a given AAS is suitable for a specific application. This paper presents two complementary methods to support the comparison and application-oriented assessment of AAS. First, set-theoretic operations are employed to compare AAS models, enabling the identification of common, missing, and differing submodels and parameters. Second, an AAS suitability model assesses the conformity of an AAS to the requirements of a specific use case. The assessment considers structural conformity, semantic consistency, cardinality, and specification conformity and can be performed either against a reference AAS or a set of required SemanticIDs. A suitability value is derived from the identified deviations and is complemented by a detailed report of missing or non-conforming information. The proposed approach support practitioners and researchers in the comparison of evolving AAS and provide application-specific information on their suitability for manufacturing software services.

Social networks in which algorithms actively influence humans through personal recommendations are ubiquitous. While opinion dynamics is an established tool to analyze these systems, existing models typically do not capture how individual agents process personal recommendations. In this work, we introduce a model for personal recommendations that is analytically tractable and consistent with the confirmation bias phenomenon from behavioral psychology. We describe how individuals process recommendations based on prior beliefs and a sensitivity parameter using a Gaussian influence function. Using this model, we analyze the effect of different recommendation policies. For broadcast policies, where recommendations are homogeneous across the agents, a bifurcation analysis shows that the system exhibits bistability, which may result in unintended consequences. For personally targeted policies, we first derive optimal personal recommendations and then extend to robust convergence when the agents' sensitivity to personal recommendations is uncertain. The collective behavior of the proposed model and the derived policies are illustrated and evaluated through numerical simulations.

The modernization of railway systems is being driven by the need for greater efficiency, sustainability, and intelligent operation. In this review, recent advancements in dynamic scheduling, energy management, and energy storage systems (ESSs) integration within electric railway networks are analyzed. Optimization strategies for train scheduling and operation control are reviewed, with a focus on methods that reduce energy consumption and improve overall system performance. The role of energy storage technologies is analyzed in terms of energy management, peak shaving, and voltage and frequency control for practical application. The integration of artificial intelligence and advanced control strategies is also reviewed in electrified railway systems. This review provides a comprehensive overview of current research trends and outlines future directions for the development of resilient, energy efficient, and intelligent railway systems with ESSs integration.

This paper examines Smart Grid cybersecurity, emphasizing the critical distinctions between IT and OT environments. It analyzes grid architecture, substation threats, and key international standards, specifically IEC 62351, IEC 62443, and ISO 27001. Finally, it overviews latest research trends, including AI-driven threat detection.

This paper studies the cost-optimal operation of a network of charging energy hubs for electric vehicles, which provide onsite renewable energy sources and stationary battery storage and are connected with each other via DC-lines as well as with the distribution grid. Specifically, we first formulate a dynamic optimal control problem for the entire network as a convex quadratic program, whereby the charging power profiles of the individual vehicles and the energy flows between hubs and the grid are subject to optimization. Second, we propose a problem decomposition that allows for a distributed solution via ADMM algorithms that preserves global optimality guarantees and privacy of the individual stations. We showcase our framework on a case-study for the Netherlands considering a two-day ahead deterministic formulation with perfect foresight. Our results show that compared to the case where charging powers are fixed a priori, optimizing their profiles (V1G) can significantly reduce the operational costs and emissions by more than 25%. Moreover, we verify our distributed algorithm against a centralized solution, paving the way to the optimal operation of large networks and online implementations.

While indoor Internet of Things (IoT) and sensor networks increasingly rely on Wi-Fi access points (APs) to collect high-bandwidth data streams from multiple devices, conventional APs rely on static antenna deployments, whose fixed orientations are often suboptimal in dynamic propagation environments. To overcome this limitation, this paper proposes a mechanical Wi-Fi antenna control system that adaptively optimizes its 3D antenna orientation for dynamic multi-station scenarios. The proposed system autonomously actuates its physical antennas in response to perceived radio environments by combining state-specific black-box optimizers and capacity-based environment change detection. The evaluation results show that the proposed system improves channel capacity under dynamic station combinations, avoids unnecessary re-optimization under transient blockages, and triggers re-optimization after sustained environmental changes such as continuous blockage and device relocation.

The increasing electrification of railway power supply system (RPSS) intensify the operational and economic challenges at the railway power system interface. Energy storage systems (ESSs) can provide fast and flexible support to mitigate short term power spikes and to improve the energy management. However, achieving coordinated operation is challenged by the tight coupling among electrical railway operation and ESS dispatch under time-varying traction demand and network limits. This paper proposes a two-stage coordinated energy management method for electrified RPSSs that jointly optimizes railway system operation, train trajectories and ESS dispatch while explicitly accounting for traction power flow constraints. First, a day-ahead operation stage determines the train operating profiles and the ESS setting decisions to establish the baseline operating plan. Then, an intra-day rolling optimization stage based on adaptive weight economic-model predictive control (AWC-MPC) updates ESS dispatch under refreshed forecasts of traction demand and renewable output. A real Swedish railway case is utilized to minimize the energy purchase cost and the ESS cost while reducing peak grid power, demonstrating its practical applicability with 35.7% peak grid power demand and 28.8% total system cost reduction.

Reconfigurable battery systems (RBS) synthesize the converter output voltage by inserting and bypassing battery cell groups, so every switching state changes not only the source voltage but also the source impedance. This paper shows that this configuration-dependent impedance breaks a tacit assumption of optimized pulse patterns (OPP): patterns designed with the customary linear plant model overestimate their achievable current quality by up to an order of magnitude and converge to a distortion floor set by the resistance modulation. We propose impedance-aware optimized pulse patterns (IA-OPP), which embed the level-dependent source impedance in the optimization objective, combined with an event-budget formulation that treats the reconfiguration rate of the communication bus as a managed resource. An exact closed-form evaluation of the piecewise-exponential load current renders the objective smooth in the switching instants. In a simulated cascaded-bridge reference system, IA-OPP attains 0.52% current total harmonic distortion (THD) at a reconfiguration rate of 2.4 kHz, outperforming nearest-level modulation, phase-shifted carrier pulse-width modulation (PWM) at up to five times the switching rate, and a classical OPP baseline at every event budget; the advantage persists across the modulation-index range and under +/-30% parameter mismatch. For three-wire star-connected systems, masking triplen orders in the objective restores the full advantage, with the impedance-aware patterns beating their linear-model counterparts by about 20% at equal budget.

The growing integration of converter interfaced renewable energy resources (RESs) intensifies stability challenges. Energy storage system (ESS) can provide fast and flexible frequency support to mitigate frequency deviations. However, the interface converter of ESS may encounter converter-grid interaction stability issues. This paper proposes a converter-grid interaction stability guaranteed safe DRL (CIS-DRL) method for ESS integrated power systems to achieve frequency regulation. We first obtain a double DNN-based stability region to identify the guaranteed converter-grid interaction stability. Next, a novel converter-grid interaction stability Safe-TD3 (CIS-STD3) algorithm is designed that integrates a stability feasibility projection layer to map unsafe actions into stable action set before execution, enforcing converter-grid interaction stability as a hard constraint throughout learning process. The proposed approach enables ESS for grid frequency support with 100% converter-grid interaction stability without violations. Experimental results show that the proposed CIS-DRL method achieves improved frequency regulation performance while preventing unstable operating points, demonstrating its practical applicability for real time ESS frequency support.

Post-disaster phasor measurement unit (PMU) outages reduce power-system observability and degrade operator situational awareness, requiring sequential restoration under limited resources. Existing PMU restoration methods based on optimization or heuristics can generate restoration schedules, but they often provide limited support for explanation, traceability, and operator interaction. This paper proposes an agentic tool-calling framework orchestrated by a large language model (LLM) for post-disaster PMU restoration and grid observability recovery. In this framework, the LLM does not directly solve the restoration optimization problem; instead, it coordinates validated backend tools required for post-disaster restoration, including observability assessment, restoration planning, state updates, and operator verification. The framework also maintains a structured tool-call history and execution context that keep restoration decisions traceable and explainable, while enabling context-aware operator question answering during the restoration process. Simulation results on IEEE 30-bus and IEEE 57-bus systems show that the proposed framework achieves observability recovery comparable to a mixed-integer linear programming (MILP) solution, while providing tool-grounded explanations, interactive operator support, and human-overseen execution.

Power Flow (PF), Optimal Power Flow (OPF), and State Estimation (SE) are fundamental problems in power system analysis, but solving them is computationally expensive. Graph Neural Networks (GNNs) have been proposed as fast surrogates, yet existing solvers are trained for a single problem at a time, producing narrow models that must be rebuilt for each new task. We propose a more general approach: a single Heterogeneous Residual Gated Graph Convolutional Network that solves all three problems with one shared backbone. Rather than learning one mapping, the model learns a reusable representation of how the network behaves, from which PF, OPF, and SE can each be estimated. Trained jointly on the three problems across diverse topologies and loading conditions, and evaluated on the IEEE 14-bus and 118-bus systems, the shared model matches the accuracy of task-specific GNN solvers and stays robust on unseen loading levels and topologies. These results show that a single model can capture the basic operation of a power network and serve several analysis tasks at once, a first step toward a foundation model for power systems.

In ischemic stroke, changes in cerebral hemodynamics during both the ischemic and reperfusion phases strongly influence stroke outcomes. However, these hemodynamic changes remain incompletely understood. To address this challenge, we devised an imaging platform that enables time-resolved ultrasound microvascular imaging during the experimental induction of ischemia and reperfusion in a mouse model. The platform leverages our previous ultrasound imaging framework combined with continuous mechanical scanning, which acquires whole-brain blood-flow signals within 5 s. The experiments demonstrated that the proposed platform can visualize both local and whole-brain hemodynamic responses to the induction of ischemia and reperfusion, suggesting its potential for rapid and continuous whole-brain hemodynamic assessment in small-animal models.

Modern networked control systems (NCSs), cyber-physical systems (CPSs), and complex multi-agent network systems (CNSs) increasingly rely on large language models (LLMs) for high-level decision-making. However, the slow, stochastic nature of LLMs directly conflicts with the strict stability and safety guarantees required by these physical systems. This survey presents a unified analysis of how LLMs can be admitted into the control loop of NCS, CPS, and CNS without compromising closed-loop guarantees. We organize this around a core principle: the LLM operates as a slow supervisor adjusting high-level goals and constraints, while a fast, certified inner loop maintains physical stability. Under this framework, LLM integration maps directly to classical networked control challenges, where inference latency acts as delay, API failures as packet dropouts, tokenization as quantization, and hallucinations as bounded disturbances. We assess current developments across all these three domains, highlighting that rising model capabilities are frequently accompanied by a drop in formal safety assurances. Finally, we propose concrete future research directions, identifying the widespread lack of formal stability proofs as the field's central open problem.

Age of information (AoI) has become a standard design objective for timely monitoring as it measures the freshness of the latest update at a receiver. AoI alone, however, is insufficient in goal-oriented applications where decisions depend on consecutive observations. Age dispersion complements AoI by measuring the generation-time separation between consecutive updates. In this paper, we study joint freshness and dispersion control for a generate-at-will status update link modeled as a finite-state Markov wireless channel. The objective is to minimize the long-run probability that either AoI or age dispersion exceeds a prescribed threshold under an average transmission rate constraint. For channel-dependent randomized transmission policies, we derive exact matrix expressions for the stationary joint distribution of AoI and dispersion. For reversible channels, mean dispersion is lower bounded by the reciprocal of the delivery throughput, and the difference is an explicit non-negative variance term for which we establish the exact equality condition. We then formulate the adaptive joint-threshold control problem as a constrained Markov decision process and prove that it admits an exact finite-state representation whose size scales linearly with the number of channel states and the threshold levels. Simulation results show that the controller reduces joint threshold violations by 37.3% relative to adaptive AoI-only control under the same transmission budget, while increasing mean AoI by 8.3%.

Extremum seeking (ES) is a real-time optimization strategy, thus transmission delays in the feedback loop of ES have big impact on its stability. How big delay that ES control systems are able to withstand? This paper provides a potential answer to this problem. We focus on gradient-based ES for nonlinear static maps subject to known constant delays plus a small time-varying delay uncertainty. We also consider the measurement to be subject to a small disturbance. Different from a majority of existing literature addressing quadratic maps with delays by predictor feedback, this paper deals with a wider class of non-quadratic maps without any predictor or observer for delay compensation. Dither signals in modulation and demodulation are carefully designed to handle constant delays and time-varying delay uncertainties. When the nonlinear map is unknown, we offer a rigorously analytical framework of ES convergence and delay-robustness. When some a prior knowledge of nonlinear maps is available, we are able to provide a quantitative estimation on upper bounds of time delay and dither periods to keep ES systems to remain stable. A suitable choice of ES parameters guarantees practical stability for any large known constant delay.

Locational marginal emissions (LMEs) give the sensitivity of total operating carbon emissions to nodal power demand. We show that this vector with $n$ entries has a much smaller intrinsic dimension under DC optimal power flow. Within a fixed active constraint set, the LME vector lies in the span of the uniform vector and the power transfer distribution factor rows of the binding lines. Its rank $r$ is therefore at most one more than the number of binding/congested lines. This structure makes $r$ independent scalar observations necessary and sufficient for exact recovery. Across ten systems with nonzero operating emissions, from 14 to 1,354 buses, $r$ ranges from 2 to 15. For instance, on a 300 bus system, 24 dispatch simulations recover all 300 LMEs. We also derive an emissions exceedance bound under uncertain demand. The bound separates variation while the nominal active set remains unchanged, the probability of an active set change, and estimation error. Numerical results show that its usable forecast error range depends on local active set geometry.