Fundamental Lemma Research Articles

Convergence analysis of non-matching finite elements for a linear monotone additive Schwarz scheme for semi-linear elliptic problems

Abstract In this article, we are interested in the standard finite element approximation method of linear additive Schwarz iterations for a class of semi-linear elliptic problems, for two subdomains, in the context of non-matching grids. More precisely, by means of a uniform convergence result from the study by Lui and a fundamental lemma consisting of estimating, at each iteration, the gap between the continuous and the finite element Schwarz iterates, we prove that the discrete Schwarz sequences converge, in the maximum norm, to the true solution. Moreover, we also give numerical results to support the theoretical findings.

An Overview of Systems-Theoretic Guarantees in Data-Driven Model Predictive Control

The development of control methods based on data has seen a surge of interest in recent years. When applying data-driven controllers in real-world applications, providing theoretical guarantees for the closed-loop system is of crucial importance to ensure reliable operation. In this review, we provide an overview of data-driven model predictive control (MPC) methods for controlling unknown systems with guarantees on systems-theoretic properties such as stability, robustness, and constraint satisfaction. The considered approaches rely on the fundamental lemma from behavioral theory in order to predict input–output trajectories directly from data. We cover various setups, ranging from linear systems and noise-free data to more realistic formulations with noise and nonlinearities, and we provide an overview of different techniques to ensure guarantees for the closed-loop system. Moreover, we discuss avenues for future research that may further improve the theoretical understanding and practical applicability of data-driven MPC.

A robust data-driven model predictive thermal control for rack-based data center

Optimizing and controlling of air-cooled data centers cooling systems are essential for reducing energy consumption. However, traditional cooling system control strategies have been proven inadequate due to the thermodynamic processess complexity and the challenge for accuratly modeling the thermal behavior of data center (DC). Therefore, a data-driven control method whithout explicit modeling process is urgently needed. In this paper, a robust data-driven model predictive control (MPC) method for air-cooled rack-based DCs, based on Willems fundamental lemma, is proposed. The methodology is executed in three phases: Firstly, we show that the system is approximately linear time-invariant (LTI) by zonal model method and experiment. Secondly, we propose a data-driven MPC framework for air-cooled rack-based DC. Thirdly, considering the uncertainties caused by disturbances and the nonlinear dynamics, the data-driven MPC approach is tailored as a robust one. Finally, the proposed approach is verified by computational fluid dynamics simulation experiments. This approach ensures the rack-based DC cold aisle temperature to the reference temperature as well as within acceptable limits. The experiment shows that the steady-state error is guaranteed to be less than 2%, with a control accuracy increased by 2.4% compared with the regular MPC. Thus, this method provides an alternative solution for maintaining the reference temperature in air-cooled rack-based DCs by utilizing only input–output data.

Flexoelectronics of a centrosymmetric semiconductor cylindrical nanoshell

Cylindrical shell-type semiconductors are essential for sensing and energy harvesting when integrated into surfaces of certain equipment, such as spacecraft, marine devices, and portable electronics, where mechanical forces play a significant impact on charge transport. Traditionally, such functionalities are only manifested in piezoelectric or pyroelectric crystals that possess non-centrosymmetry. Here, we theoretically investigate electronic behaviors driven by strain gradient-induced flexoelectric polarization in a centrosymmetric semiconductor cylindrical nanoshell, expanding the flexoelectronics in shell structures. The governing equations and accompanying boundary conditions are formulated simultaneously using the principle of virtual work and the fundamental lemma of the calculus of variation. Electromechanical interactions through static bending and forced vibration analyses of the newly developed model are systematically investigated. Under localized force excitation, the distribution of mobile charges is manipulated by tuning loading magnitudes and areas. The effects of doping levels on electric potentials and mobile charges are explored to show the interaction mechanism between flexoelectric and semiconducting properties. Moreover, the natural frequencies and modes of all mechanical displacements, electric potentials, and carrier concentration perturbations within the shell are identified. This paper provides a new approach for designing shell-shaped sensors and energy harvesters specifically for centrosymmetric semiconductors.

Sheffer stroke operation on L-algebras via an algorithmic approach

AbstractIn this study, we introduce the Sheffer stroke L-algebra and prove some fundamental theorems, propositions and lemmas of Sheffer Stroke L-algebras. The notions of filter and ultrafilter for Sheffer stroke L-algebra are studied. We give subalgebra and normal subset definitions of a Sheffer stroke L-algebras. Moreover, a homomorphism between Sheffer stroke L-algebras is introduced and isomorphism theorems are presented. Finally, we give three new algorithms for Sheffer stroke L-algebras. Thus, it is contributed to researchers on different application areas by presenting an algorithmic approach on this subject, for the first time in the literature.

On the cubic Shimura lift to $\operatorname{PGL}(3)$: The fundamental lemma

The classical Shimura correspondence lifts automorphic representations on the double cover of \mathrm{SL}_{2} to automorphic representations on \mathrm{PGL}_{2} . Here we take key steps towards establishing a relative trace formula that would give a new global Shimura lift, from the triple cover of \mathrm{SL}_{3} to \mathrm{PGL}_{3} , and also characterize the image of the lift. The characterization would be through the non-vanishing of a certain global period involving a function in the space of the automorphic minimal representation \Theta_{\mathrm{SO}_8} for split \mathrm{SO}_{8}(\mathbb{A}) , consistent with a conjecture of Bump, Friedberg and Ginzburg (2001). In this paper, we first analyze a global distribution on \mathrm{PGL}_{3}(\mathbb{A}) involving this period and show that it is a sum of factorizable orbital integrals. The same is true for the Kuznetsov distribution attached to the triple cover of \mathrm{SL}_{3}(\mathbb{A}) . We then match the corresponding local orbital integrals for the unit elements of the spherical Hecke algebras; that is, we establish the fundamental lemma.

Arithmetic fundamental lemma for the spherical Hecke algebra

We define Hecke correspondences and Hecke operators on unitary RZ spaces and study their basic geometric properties, including a commutativity conjecture on Hecke operators. Then we formulate the arithmetic fundamental lemma conjecture for the spherical Hecke algebra. We also formulate a conjecture on the abundance of spherical Hecke functions with identically vanishing first derivative of orbital integrals. We prove these conjectures for the case U(1)×U(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{U} (1)\ imes \ extrm{U} (2)$$\\end{document}.

Data-driven uncertainty propagation for stochastic predictive control of multi-energy systems

Stochastic predictive control schemes that account for epistemic and aleatoric uncertainties, i.e. lack of model knowledge and stochastic disturbances, are of major interest for multi-energy systems. However, there exists a trade-off between model complexity, computational effort, and accuracy of uncertainty quantification. This paper attempts to assess this trade-off by comparing a recently proposed approach combining Willems’ fundamental lemma with polynomial chaos expansion to a model-based scheme that first propagates uncertainty with PCE and then considers chance constraints in the optimization. The simulation results show that the data-driven scheme yields similar performance and computational efficiency compared to the model-based scheme, with the advantage of avoiding the construction of explicit models.

Stochastic Data-Driven Predictive Control: Regularization, Estimation, and Constraint Tightening

Data-driven predictive control methods based on the Willems’ fundamental lemma have shown great success in recent years. These approaches use receding horizon predictive control with nonparametric data-driven predictors instead of model-based predictors. This study addresses three problems of applying such algorithms under unbounded stochastic uncertainties: 1) tuning-free regularizer design, 2) initial condition estimation, and 3) reliable constraint satisfaction, by using stochastic prediction error quantification. The regularizer is designed by leveraging the expected output cost. An initial condition estimator is proposed by filtering the measurements with the one-step-ahead stochastic data-driven prediction. A novel constraint-tightening method, using second-order cone constraints, is presented to ensure high-probability chance constraint satisfaction. Numerical results demonstrate that the proposed methods lead to satisfactory control performance in terms of both control cost and constraint satisfaction, with significantly improved initial condition estimation.

Frequency-Domain Data-Driven Predictive Control

In this paper, we propose a data-driven predictive control scheme based on measured frequency-domain data of the plant. This novel scheme complements the well-known data-driven predictive control (DeePC) approach based on time series data. To develop this new frequency-domain data-driven predictive control (FreePC) scheme, we introduce a novel version of Willems’ fundamental lemma based on frequency-domain data. By exploiting frequency-domain data, we allow recent direct data-driven (predictive) control methodologies to benefit from the available expertise and techniques for non-parametric frequency-domain identification in academia and industry. We prove that, under appropriate conditions, the new FreePC scheme is equivalent to the corresponding DeePC scheme. The strengths of FreePC are demonstrated in a numerical case study.

Mechanically induced electric potential and charge redistribution in laminated composite piezoelectric semiconductor circular cylindrical thin shells

We study electromechanical effects and piezotronic behaviors in a Kirchhoff–Love laminated composite piezoelectric semiconductor circular cylindrical thin shell, which is composed of a non-piezoelectric semiconductor core and two piezoelectric dielectric layers. A second-order assumption for electric potentials and carrier concentration perturbations is utilized to capture accurate descriptions of the electric field and charge distributions within the composite shell, where the zeroth-order, first-order, and second-order forms represent constant, antisymmetric, and symmetric distributions along the thickness, respectively. The field equations and boundary conditions are simultaneously derived through a principle of virtual work and the fundamental lemma of the calculus of variation. The new governing equations derived show a new coupling relation between mechanical deformation and charge behavior in the composite shell, where all mechanical deformations (extension, bending of the shell) and all distribution forms of electric potential (zeroth-, first-, second-order electric potentials) are coupled, which is different from piezoelectric semiconductor beams and plates. Then, the numerical results of electromechanical effects and piezotronic behaviors in static bending and forced vibration problems of the shell are conducted with the derived theoretical model. Numerical studies of static bending graphically show some fundamental coupling relations between mechanical forces and charge distributions in deformed piezoelectric semiconductor shells. The new results from the forced vibration analysis given by the current model show that the deflection amplitude and the electric potential distribution in the thin shell are frequency-dependent and can be adjusted by controlling the excitation frequency. In addition, the semiconducting properties (dependence of mechanical behavior on doping level) are studied in both static and dynamic problems. The mechanically controlled charge distribution suggests that shell-shaped piezoelectric semiconductor shells could serve as an effective means of sensing or energy harvesting by converting mechanical energy into electricity.

Static bending and forced vibration analyses of a piezoelectric semiconductor cylindrical shell within first-order shear deformation theory

This paper examines the mechanically induced electric potential and charge redistribution in a piezoelectric semiconductor cylindrical shell employing first-order shear deformation theory. The two-dimensional governing equations and corresponding boundary conditions are simultaneously derived through a principle of virtual work method and the fundamental lemma of the calculus of variation. For the application of electronic devices, static bending and forced vibration analyses of an open cylindrical shell under locally distributed normal forces are analytically solved with the derived theoretical model. The effects of doping level on electric potentials and mechanical displacements are presented. An interesting result shows that doping levels can alter the peak position of the zeroth-order electric potential. In addition, the first three order natural frequencies of the shell under time-harmonic load are identified, and the doping level is found to have an inhibiting effect on the first natural frequency. All the numerical results are beneficial for optimizing the design and performance of cylindrical shell-shaped sensors and energy harvesters.

Adaptive Robust Data-Driven Building Control via Bilevel Reformulation: An Experimental Result

Data-driven control approaches for the minimization of energy consumption of buildings have the potential to significantly reduce deployment costs and increase the uptake of advanced control in this sector. A number of recent approaches based on the application of Willems’ fundamental lemma for data-driven controller design from input/output measurements are very promising for deterministic linear time-invariant (LTI) systems. This article proposes a systematic way to handle unknown measurement noise and measurable process noise, and extends these data-driven control schemes to adaptive building control via a robust bilevel formulation, whose upper level ensures robustness and whose lower level guarantees prediction quality. Corresponding numerical improvements and an active excitation mechanism are proposed to enable a computationally efficient reliable operation. The efficacy of the proposed scheme is validated by a multizone building simulation and a real-world experiment on a single-zone conference building on the École Polytechnique Fédérale de Lausanne (EPFL) campus. The real-world experiment includes a 20-day nonstop test, where, without extra modeling effort, our proposed controller improves 18.4% energy efficiency against an industry-standard controller while also robustly ensuring occupant comfort.

On the Existence and Stability of Positive Solutions of Eigenvalue Problems for a Class of P-Laplacian ψ -Caputo Fractional Integro-Differential Equations

This article introduces a generalized approach for analyzing stability and establishing the existence of positive solutions in a specific type of differential equations known as p-Laplacian ψ -Caputo fractional differential equations with fractional integral boundary conditions. The study utilizes various techniques, including the analysis of Green’s function properties and the application of Guo–Krasnovelsky’s fixed point theorem on cones. By employing these methods, the research establishes novel findings concerning the existence and nonexistence of positive solutions. The investigation relies on fractional integrals, differential operators, and fundamental lemmas as fundamental tools. To assess solution stability, the Hyers–Ulam concept is employed, which extends prior research and introduces a specific definition. The article also provides numerical examples that support the obtained results, thereby demonstrating the practical applicability and accuracy of the proposed methods. Moreover, the study contributes to a deeper understanding of this subject matter and highlights real-life applications for these types of problems. Overall, this study offers a comprehensive analysis of stability and solution existence in a specific class of differential equations, with implications that extend to real-world scenarios such as engineering systems, financial modeling, population dynamics, epidemiology, and ecological studies. These types of problems arise in various fields where modeling and analyzing complex phenomena are necessary.

On a Stochastic Fundamental Lemma and Its Use for Data-Driven Optimal Control

Data-driven control based on the fundamental lemma by Willems et al. is frequently considered for deterministic LTI systems subject to measurement noise. However, besides measurement noise, stochastic disturbances might also directly affect the dynamics. In this paper, we leverage Polynomial Chaos Expansions (PCE) to extend the deterministic fundamental lemma towards stochastic systems. This extension allows to predict future statistical distributions of the inputs and outputs for stochastic LTI systems in data-driven fashion, i.e., based on the knowledge of previously recorded input-output-disturbance data and of the disturbance distribution we perform data-driven uncertainty propagation. Finally, we analyze data-driven stochastic optimal control problems and we propose a conceptual framework for data-driven stochastic predictive control. Numerical examples illustrate the efficacy of the proposed concepts.

A “fundamental lemma” for continuous-time systems, with applications to data-driven simulation

A “fundamental lemma” for continuous-time systems, with applications to data-driven simulation

Distributed data-driven predictive control for cooperatively smoothing mixed traffic flow

Cooperative control of connected and automated vehicles (CAVs) promises smoother traffic flow. In mixed traffic, where human-driven vehicles with unknown dynamics coexist, data-driven predictive control techniques allow for CAV safe and optimal control with measurable traffic data. However, the centralized control setting in most existing strategies limits their scalability for large-scale mixed traffic flow. To address this problem, this paper proposes a cooperative DeeP-LCC (Data-EnablEd Predictive Leading Cruise Control) formulation and its distributed implementation algorithm. In cooperative DeeP-LCC, the traffic system is naturally partitioned into multiple subsystems with one single CAV, which collects local trajectory data for subsystem behavior predictions based on the Willems’ fundamental lemma. Meanwhile, the cross-subsystem interaction is formulated as a coupling constraint. Then, we employ the Alternating Direction Method of Multipliers (ADMM) to design the distributed DeeP-LCC algorithm. This algorithm achieves both computation and communication efficiency, as well as trajectory data privacy, through parallel calculation. Our simulations on different traffic scales verify the real-time wave-dampening potential of distributed DeeP-LCC, which can reduce fuel consumption by over 31.84% in a large-scale traffic system of 100 vehicles with only 5%–20% CAVs.

Towards data‐driven stochastic predictive control

SummaryData‐driven predictive control based on the fundamental lemma by Willems et al. is frequently considered for deterministic LTI systems subject to measurement noise. However, little has been done on data‐driven stochastic control. In this paper, we propose a data‐driven stochastic predictive control scheme for LTI systems subject to possibly unbounded additive process disturbances. Based on a stochastic extension of the fundamental lemma and leveraging polynomial chaos expansions, we construct a data‐driven surrogate optimal control problem (OCP). Moreover, combined with an online selection strategy of the initial condition of the OCP, we provide sufficient conditions for recursive feasibility and for stability of the proposed data‐driven predictive control scheme. Finally, two numerical examples illustrate the efficacy and closed‐loop properties of the proposed scheme for process disturbances governed by different distributions.

Data-Based Optimization Control for Learning Systems

This paper is devoted to establishing a novel data-based optimization control framework of achieving the perfect tracking performances for learning systems in the absence of any model information. A modified Willems’ fundamental lemma is developed without the assumptions on the controllability and the input’s persistency of excitation, under which a data-based iterative learning control (ILC) framework is further presented. It is shown that the proposed data-based ILC can be employed to realize the perfect tracking tasks without any explicit model knowledge. A simulation example is provided to demonstrate the effectiveness of the proposed data-based ILC strategy.

Description of K‐spaces by means of J‐spaces and the reverse problem in the limiting real interpolation

AbstractWe establish conditions under which K‐spaces in the limiting real interpolation involving slowly varying functions can be described by means of J‐spaces and we also solve the reverse problem. To this end, we prove several versions of the fundamental lemma of the real interpolation theory. We apply our results to obtain density theorems for the corresponding limiting interpolation spaces.