First-order Space Research Articles

Finite-Time Fault-Tolerant Control via Fully Actuated System Approaches.

In this article, a finite-time fault-tolerant controller based on the fully actuated system (FAS) theory is presented to realize system stabilization and trajectory tracking. Paralleling to first-order nonlinear state space theory, the high-order FAS (HOFAS) theory contains rich controller design approaches. The existing FAS approaches can only give general global asymptotic stability results. In order to enhance the applicability of FAS approaches in fast control systems, a parameterized FAS stabilization controller based on the homogeneity principle is established for global finite-time stability. Moreover, a finite-time FAS tracking controller based on a finite-time observer is proposed for a HOFAS model with process faults. The proposed observer can yield zero-value convergence of state estimation error and fault estimation error in a finite time, and the proposed fault-tolerant controller can yield zero-value convergence of tracking error in a finite time. The main results are proved theoretically and illustrated experimentally.

Courtyards and Adjacent Spaces: Analyzing 26 Cases of Second-Order Proximity in Traditional Courtyard Houses of Yazd

Iranian traditional residential architecture is renowned for its central-courtyard houses, which are admired for their grandeur. While the courtyards and nearby spaces receive considerable artistic and historical appreciation, those situated further away often receive less attention. These areas are typically considered auxiliary and less functional for living, thereby receiving limited attention in architectural discussions. This study examines 26 traditional central-courtyard houses to investigate how spaces located farther from the courtyard (‘second-order’) compare to those directly adjacent (‘first-order’). It challenges the assumption that distance from the courtyard correlates with reduced functionality. Surprisingly, the analysis identifies similar architectural characteristics in both second-order and first-order spaces, suggesting that distant areas may serve functional roles comparable to those nearer the courtyard.

Explicit P1 Finite Element Solution of the Maxwell-Wave Equation Coupling Problem with Absorbing b. c.

In this paper, we address the approximation of the coupling problem for the wave equation and Maxwell’s equations of electromagnetism in the time domain in terms of electric field by means of a nodal linear finite element discretization in space, combined with a classical explicit finite difference scheme for time discretization. Our study applies to a particular case where the dielectric permittivity has a constant value outside a subdomain, whose closure does not intersect the boundary of the domain where the problem is defined. Inside this subdomain, Maxwell’s equations hold. Outside this subdomain, the wave equation holds, which may correspond to Maxwell’s equations with a constant permittivity under certain conditions. We consider as a model the case of first-order absorbing boundary conditions. First-order error estimates are proven in the sense of two norms involving first-order time and space derivatives under reasonable assumptions, among which lies a CFL condition for hyperbolic equations. The theoretical estimates are validated by numerical computations, which also show that the scheme is globally of the second order in the maximum norm in time and in the least-squares norm in space.

Robust adaptive dynamic surface control of high-order strict feedback systems based on fully actuated system approach

In this study, the authors propose a robust adaptive dynamic surface control method by using the fully actuated system approach for high-order strict-feedback systems (SFSs) with parameter uncertainty and disturbance. Each subsystem of the SFSs is a high-order system with a full actuation structure. In contrast to the traditional first-order state space method, the proposed control method directly treats each high-order subsystem as a whole without transforming it into a first-order system, which is a concise and efficient treatment. By introducing a series of first-order low-pass filters in each step of the design, the high-order derivatives of the virtual control law are obtained, and the complex and multiple derivation operations are transformed into simple algebraic operations. Adaptive control and robust control are combined to address parameter uncertainty and external disturbances in the system. The Lyapunov stability theory is utilised to demonstrate that all signals in the closed-loop system are uniformly ultimately bounded. The system output can effectively track the desired reference signal under specific constraints, and the tracking error can be adjusted by tweaking the parameters to converge to a sufficiently small neighbourhood around zero. Ultimately, the efficiency of the proposed control method is validated through simulations on a flexible joint manipulator system.

Improved rates for a space–time FOSLS of parabolic PDEs

We consider the first-order system space–time formulation of the heat equation introduced by Bochev and Gunzburger (in: Bochev and Gunzburger (eds) Applied mathematical sciences, vol 166, Springer, New York, 2009), and analyzed by Führer and Karkulik (Comput Math Appl 92:27–36, 2021) and Gantner and Stevenson (ESAIM Math Model Numer Anal 55(1):283–299 2021), with solution components (u1,u2)=(u,-∇xu)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(u_1,\ extbf{u}_2)=(u,-\ abla _\ extbf{x} u)$$\\end{document}. The corresponding operator is boundedly invertible between a Hilbert space U and a Cartesian product of L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2$$\\end{document}-type spaces, which facilitates easy first-order system least-squares (FOSLS) discretizations. Besides L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2$$\\end{document}-norms of ∇xu1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ abla _\ extbf{x} u_1$$\\end{document} and u2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{u}_2$$\\end{document}, the (graph) norm of U contains the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2$$\\end{document}-norm of ∂tu1+divxu2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _t u_1 +{{\\,\ extrm{div}\\,}}_\ extbf{x} \ extbf{u}_2$$\\end{document}. When applying standard finite elements w.r.t. simplicial partitions of the space–time cylinder, estimates of the approximation error w.r.t. the latter norm require higher-order smoothness of u2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{u}_2$$\\end{document}. In experiments for both uniform and adaptively refined partitions, this manifested itself in disappointingly low convergence rates for non-smooth solutions u. In this paper, we construct finite element spaces w.r.t. prismatic partitions. They come with a quasi-interpolant that satisfies a near commuting diagram in the sense that, apart from some harmless term, the aforementioned error depends exclusively on the smoothness of ∂tu1+divxu2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _t u_1 +{{\\,\ extrm{div}\\,}}_\ extbf{x} \ extbf{u}_2$$\\end{document}, i.e., of the forcing term f=(∂t-Δx)u\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f=(\\partial _t-\\Delta _x)u$$\\end{document}. Numerical results show significantly improved convergence rates.

High-order adaptive dynamic surface control for output-constrained non-linear systems based on fully actuated system approach

In this study, the authors propose a high-order adaptive dynamic surface control (HOADSC) method by using the fully actuated system-based approach for uncertain high-order strict-feedback systems (SFSs) with an output constraint. Each subsystem of the SFSs is a high-order system with a full actuation structure. In contrast to the traditional first-order state space method, the proposed control method directly treats each high-order subsystem as a whole without transforming it into a first-order system, which is a concise and efficient treatment. By introducing a series of first-order low-pass filters in each step of the design, the high-order derivatives of the virtual control law are obtained, and the complex and multiple derivation operations are transformed into simple algebraic operations. The Barrier Lyapunov function is combined to ensure that the output of the system satisfies the given constraints. And the Lyapunov theory is used to prove that all signals in the closed-loop system are ultimately uniformly bounded, and its output can accurately track the desired reference signal without violating the given constraints. The effectiveness of the proposed control method is verified by simulations of an actual manipulator system.

Algorithm 1032: Bi-cubic Splines for Polyhedral Control Nets

For control nets outlining a large class of topological polyhedra, not just tensor-product grids, bi-cubic polyhedral splines form a piecewise polynomial, first-order differentiable space that associates one function with each vertex. Akin to tensor-product splines, the resulting smooth surface approximates the polyhedron. Admissible polyhedral control nets consist of quadrilateral faces in a grid-like layout, star-configuration where n ≠ 4 quadrilateral faces join around an interior vertex, n -gon configurations, where 2n quadrilaterals surround an n -gon, polar configurations where a cone of n triangles meeting at a vertex is surrounded by a ribbon of n quadrilaterals, and three types of T-junctions where two quad-strips merge into one. The bi-cubic pieces of a polyhedral spline have matching derivatives along their break lines, possibly after a known change of variables. The pieces are represented in Bernstein-Bézier form with coefficients depending linearly on the polyhedral control net, so that evaluation, differentiation, integration, moments, and so on, are no more costly than for standard tensor-product splines. Bi-cubic polyhedral splines can be used both to model geometry and for computing functions on the geometry. Although polyhedral splines do not offer nested refinement by refinement of the control net, polyhedral splines support engineering analysis of curved smooth objects. Coarse nets typically suffice since the splines efficiently model curved features. Algorithm 1032 is a C++ library with input-output example pairs and an IGES output choice.

A first-order stabilization-free Virtual Element Method

In this paper, we introduce a new Virtual Element Method (VEM) not requiring any stabilization term based on the usual enhanced first-order VEM space. The new method relies on a modified formulation of the discrete diffusion operator that ensures stability preserving all the properties of the differential operator.

Numerical solution of two‐dimensional nonlinear Riesz space‐fractional reaction–advection–diffusion equation using fast compact implicit integration factor method

AbstractIn the present article, a finite domain is considered to find the numerical solution of a two‐dimensional nonlinear fractional‐order partial differential equation (FPDE) with Riesz space fractional derivative (RSFD). Here two types of FPDE–RSFD are considered, the first one is a two‐dimensional nonlinear Riesz space‐fractional reaction–diffusion equation (RSFRDE) and the second one is a two‐dimensional nonlinear Riesz space‐fractional reaction‐advection‐diffusion equation (RSFRADE). SFRDE is obtained by simply replacing second‐order derivative term of the standard nonlinear diffusion equation by the Riesz fractional derivative of order whereas the SFRADE is obtained by replacing the first‐order and second‐order space derivatives from the standard order advection–dispersion equation with the Riesz fractional derivatives of order . A numerical method is provided to deal with the RSFD with the weighted and shifted Grünwald–Letnikov (WSGD) approximations, for the spatial discretization. The SFRDE and SFRADE are transformed into a system of ordinary differential equations (ODEs), which have been solved using a fast compact implicit integration factor (FcIIF) with nonuniform time meshes. Finally, the demonstration of the validation and effectiveness of the numerical method is given by considering some existing models.

Design method for zoom systems based on tunable lenses

It is well known that tunable lenses, with refractive power that can be varied, e.g., by changing the curvature of a membrane, can replace the motion of lens groups in zoom systems. Similar to classical zoom systems, the performance of these systems is heavily influenced by the fundamental first-order layout. Moreover, the first-order layout sets the most important requirements for the employed tunable lenses. In this contribution, we present a method for the analysis of a large number of possible first-order solutions for typical requirements and for the selection of the most promising layouts. The first-order solution space is mapped, allowing the layouts to be automatically filtered and plotted depending on pre-defined characteristics. Ray tracing of the marginal and chief rays combined with the traditional thin lens aberration theory provide efficient estimations of the expected installation space requirements and performance for each first-order layout. Using an example, we demonstrate good agreement between these estimations and the corresponding real lens layout, optimized by commercial raytracing software. The presented design method for zoom systems based on tunable lenses is compared with similar approaches for classical zoom lenses.

Nonstationary local slope estimation via forward-backward space derivative calculation

The local slope estimated from seismic images has a variety of meaningful applications. Slope estimation based on the plane-wave destruction (PWD) method is a widely accepted technique in the seismic community. However, the PWD method suffers from sensitivity to noise in the seismic data. We have developed an improved slope estimation method based on the PWD theory that is more robust in the presence of strong random noise. The PWD operator derived in the Z-transform domain contains a phase-shift operator in space corresponding to the calculation of the first-order derivative of the wavefield in the space domain. The first-order derivative is discretized based on a forward finite difference in the traditional PWD method, which lacks the constraint from the backward direction. We have developed an improved method by discretizing the first-order space derivative based on an averaged forward-backward finite-difference calculation. The forward-backward space derivative calculation makes the space-domain first-order derivative more accurate and better at antinoise because it takes more space grids for the derivative calculation. In addition, we introduce nonstationary smoothing to regularize the slope estimation and to make it even more robust to noise. We determine the performance of the new slope estimation method with several synthetic and field data examples in different applications, including 2D/3D structural filtering, structure-oriented deblending, and horizon tracking.

Crystal flex bases and the RUM spectrum

A theory of infinite spanning sets and bases is developed for the first-order flex space of an infinite bar-joint framework, together with space group symmetric versions for a crystallographic bar-joint framework ${{\mathcal {C}}}$. The existence of a crystal flex basis for ${{\mathcal {C}}}$ is shown to be closely related to the spectral analysis of the rigid unit mode (RUM) spectrum of ${{\mathcal {C}}}$ and an associated geometric flex spectrum. Additionally, infinite spanning sets and bases are computed for a range of fundamental crystallographic bar-joint frameworks, including the honeycomb (graphene) framework, the octahedron (perovskite) framework and the 2D and 3D kagome frameworks.

Innovative mass spectrometer for high-resolution ion spectroscopy.

Conventional ion spectroscopy is inapplicable for ions produced in low concentrations or with low spectral resolutions. Hence, we constructed a high-resolution vacuum ultraviolet mass-analyzed threshold ionization (HR VUV-MATI) spectrometer composed of a four-wave frequency mixing cell capable of generating long-lasting and intense VUV laser pulses of ∼1 × 1010 photons/pulse at wavelengths of 123.6-160.0nm, a space-focused linear time-of-flight photoionization chamber with a new ion-source assembly, and a compact molecular beam chamber with a temperature-controlled pulsed nozzle for ion spectroscopy. The ion-source assembly and pulsing schemes enabled an ∼15-μs-delayed but extremely weak pulsed-field-ionization of the molecules in the zero-kinetic-energy (ZEKE) states and first-order space focusing of the generated MATI ions. These ZEKE states were effectively generated by a minute electric jitter from the high-lying Rydberg states, which were initially prepared via VUV photoexcitation. The spectral and mass resolutions (∼5cm-1 and 2400, respectively) and the signal strength were simultaneously enhanced using this spectrometer. Moreover, it could be used to measure the fine vibrational spectrum from the zero-pointlevel of the cation and the exact adiabatic ionization energy of the neutral molecule. Additionally, it could be used to measure the appearance energies of the photoproducts and elucidate the vibrational structures of the cationic isotopomers, utilizing other pulsing schemes. Furthermore, this spectrometer could be used to analyze the congested vibrational spectrum of a cation with multiple conformations. Thus, the HR VUV-MATI spectrometer-a potential alternative to photoelectron spectrometers-can be used to analyze the conformational structure-dependent reactivities.

The first-order flexibility of a crystallographic framework

Four sets of necessary and sufficient conditions are obtained for the first-order rigidity of a periodic bond-node framework C in Rd which is of crystallographic type. In particular, an extremal rank characterisation is obtained which incorporates a multi-variable matrix-valued transfer function ΨC(z) defined on the product space C⁎d=(C﹨{0})d. In general the first-order flex space is the closed linear span of polynomially weighted geometric velocity fields whose geometric multi-factors in C⁎d lie in a finite set. It is also shown that, paradoxically, a first-order rigid crystal framework may possess a nontrivial continuous motion. Examples of this phenomenon are given which are associated with aperiodic displacive phase transitions between periodic states.

Vanishing of the first-order cohomology on Olshanski spherical pairs

In this paper, we study the first-order cohomology space of countable direct limit groups related to Olshanski spherical pairs, relatively to unitary representations which do not have almost invariant vectors. In particular, we prove a variant of Delorme’s vanishing result of the first-order cohomology space for spherical representations of Olshanski spherical pairs.

Further Development of iCIPT2 for Strongly Correlated Electrons.

The efficiency of the recently proposed iCIPT2 [iterative configuration interaction (iCI) with selection and second-order perturbation theory (PT2); J. Chem. Theory Comput. 2020, 16, 2296] for strongly correlated electrons is further enhanced (by up to 20×) by using (1) a new ranking criterion for configuration selection, (2) a new particle-hole algorithm for Hamiltonian construction over randomly selected configuration state functions (CSF), and (3) a new data structure for the quick sorting of the variational and first-order interaction spaces. Meanwhile, the memory requirement is also significantly reduced. As a result, this improved implementation of iCIPT2 can handle 1 order of magnitude more CSFs than the previous version, as revealed by taking the chromium dimer and an iron-sulfur cluster, [Fe2S2(SCH3)]42-, as examples.

Parallelogram frameworks and flexible quasicrystals

The first-order flex space of the bar-joint framework $G_P$ of a parallelogram tiling $P$ is determined in terms of an explicit free basis. Applications are given to braced parallelogram frameworks and to quasicrystal frameworks associated with multigrids in the sense of de Bruijn and Beenker. In particular we characterise rigid bracing patterns, identify quasicrystal frameworks with finite dimensional flex spaces, and define a zero mode spectrum.

Generalized entropies, density of states, and non-extensivity

The concept of entropy connects the number of possible configurations with the number of variables in large stochastic systems. Independent or weakly interacting variables render the number of configurations scale exponentially with the number of variables, making the Boltzmann–Gibbs–Shannon entropy extensive. In systems with strongly interacting variables, or with variables driven by history-dependent dynamics, this is no longer true. Here we show that contrary to the generally held belief, not only strong correlations or history-dependence, but skewed-enough distribution of visiting probabilities, that is, first-order statistics, also play a role in determining the relation between configuration space size and system size, or, equivalently, the extensive form of generalized entropy. We present a macroscopic formalism describing this interplay between first-order statistics, higher-order statistics, and configuration space growth. We demonstrate that knowing any two strongly restricts the possibilities of the third. We believe that this unified macroscopic picture of emergent degrees of freedom constraining mechanisms provides a step towards finding order in the zoo of strongly interacting complex systems.

Nonlinear iteration acceleration solution for equilibrium radiation diffusion equation

This paper discusses accelerating iterative methods for solving the fully implicit (FI) scheme of equilibrium radiation diffusion problem. Together with the FI Picard factorization (PF) iteration method, three new nonlinear iterative methods, namely, the FI Picard-Newton factorization (PNF), FI Picard-Newton (PN) and derivative free Picard-Newton factorization (DFPNF) iteration methods are studied, in which the resulting linear equations can preserve the parabolic feature of the original PDE. By using the induction reasoning technique to deal with the strong nonlinearity of the problem, rigorous theoretical analysis is performed on the fundamental properties of the four iteration methods. It shows that they all have first-order time and second-order space convergence, and moreover, can preserve the positivity of solutions. It is also proved that the iterative sequences of the PF iteration method and the three Newton-type iteration methods converge to the solution of the FI scheme with a linear and a quadratic speed respectively. Numerical tests are presented to confirm the theoretical results and highlight the high performance of these Newton acceleration methods.

A second-order measure of boundary oscillations for overhang control in topology optimization

In topology optimization, oscillating boundaries may occur when geometrical overhang angle constraints are imposed for additive manufacturability. Indeed, boundary oscillations formally satisfy the constraints, but lead to designs that are ultimately unacceptable. In order to avoid such phenomenon, in this paper we formulate a measure of boundary oscillations that can be used to define filters, penalties and constraints to suppress oscillating boundaries.In particular, first we mathematically characterize the density distribution that forms a boundary oscillation. On this basis, we formulate our measure, which is a function that is positive in correspondence of “tips” of oscillating boundaries and is zero everywhere else. Such analysis is first presented in a 2D framework and later extended to 3D problems.Then, we show how this measure can be employed to formulate strategies that suppress boundary oscillations. In particular, we propose an adaptive anisotropic filter and a cost penalty that fulfill this task. Numerical experiments finally show the capabilities of our measure and of the proposed oscillation control strategies. In this context, an efficient, practical implementation in a first-order finite-element space and a 3D example are provided as well.