Finite-dimensional Model Research Articles

On the virtually cyclic dimension of normally poly-free groups

In this note we give an upper bound for the virtually cyclic dimension of any normally poly-free group in terms of its length. In particular, this implies that virtually even Artin groups of FC-type admit a finite-dimensional model for the classifying space with respect to the family of virtually cyclic subgroups.

Generalized entanglement capacity of de Sitter space

Near horizons, quantum fields of low spin exhibit densities of states that behave asymptotically like 1+1 dimensional conformal field theories. In effective field theory, imposing some short-distance cutoff, one can compute thermodynamic quantities associated with the horizon, and the leading cutoff sensitivity of the heat capacity is found to equal to the leading cutoff sensitivity of the entropy. One can also compute contributions to the thermodynamic quantities from the gravitational path integral. For the cosmological horizon of the static patch of de Sitter space, a natural conjecture for the relevant heat capacity is shown to equal the Bekenstein-Hawking entropy. These observations allow us to extend the well-known notion of the generalized entropy to a generalized heat capacity for the static patch of de Sitter (dS). The finiteness of the entropy and the nonvanishing of the generalized heat capacity suggest it is useful to think about dS as a state in a finite dimensional quantum gravity model that is not maximally uncertain. Published by the American Physical Society 2024

On the Picard Group of the Stable Module Category for Infinite Groups

Abstract We introduce the stable module $\infty $-category for groups of type $\Phi $ as an enhancement of the stable category defined by N. Mazza and P. Symonds. For groups of type $\Phi $ that act on a tree, we show that the stable module $\infty $-category decomposes in terms of the associated graph of groups. For groups that admit a finite-dimensional cocompact model for the classifying space for proper actions, we exhibit a decomposition in terms of the stable module $\infty $-categories of their finite subgroups. We use these decompositions to provide methods to compute the Picard group of the stable module category. In particular, we provide a description of the Picard group for countable locally finite $p$-groups.

Nonlinear solution of classical three-wave interaction via finite-dimensional quantum model

The quantum three-wave interaction, the lowest-order nonlinear interaction in plasma physics, describes energy–momentum transfer between three resonant waves in the quantum regime. We describe how it may also act as a finite-degree-of-freedom approximation to the classical three-wave interaction in certain circumstances. By promoting the field variables to operators, we quantize the classical system, show that the quantum system has more free parameters than the classical system and explain how these parameters may be selected to optimize either initial or long-term correspondence. We then numerically compare the long-time quantum–classical correspondence far from the fixed point dynamics. We discuss the Poincaré recurrence of the system and the mitigation of quantum scrambling.

Concurrent Learning Robust Adaptive Fault Tolerant Boundary Regulation of Hyperbolic Distributed Parameter Systems.

This article develops a robust adaptive boundary output regulation approach for a class of complex anticollocated hyperbolic partial differential equations subjected to multiplicative unknown faults in both the boundary sensor and actuator. The regulator design is based on the internal model principle, which amounts to stabilize a coupled cascade system, which consists of a finite-dimensional internal model driven by a hyperbolic distributed parameter system (DPS). To this end, a systematic sliding mode equipped with a backstepping approach is developed such that the robust state feedback control can be realized. Moreover, since the available information is a faulty boundary measurement at the right side point, state estimation is required. However, due to the presence of boundary unknown faults, we need to solve an issue of joint fault-state estimation. Restrictive persistent excitation conditions are usually required to guarantee the exact estimation of faults but are unrealistic in practice. To this end, a novel concurrent learning (CL) adaptive observer is proposed so that exponential convergence is obtained. It is the first time that the spirit of CL is introduced to the field of DPSs. Consequently, the observer-based adaptive boundary fault tolerant control scheme is developed, and rigorous theoretical analysis is given such that the exponential output regulation can be achieved. Finally, the effectiveness of the proposed methodology is demonstrated via comparative simulations.

Flexible variable selection in the presence of missing data.

In many applications, it is of interest to identify a parsimonious set of features, or panel, from multiple candidates that achieves a desired level of performance in predicting a response. This task is often complicated in practice by missing data arising from the sampling design or other random mechanisms. Most recent work on variable selection in missing data contexts relies in some part on a finite-dimensional statistical model, e.g., a generalized or penalized linear model. In cases where this model is misspecified, the selected variables may not all be truly scientifically relevant and can result in panels with suboptimal classification performance. To address this limitation, we propose a nonparametric variable selection algorithm combined with multiple imputation to develop flexible panels in the presence of missing-at-random data. We outline strategies based on the proposed algorithm that achieve control of commonly used error rates. Through simulations, we show that our proposal has good operating characteristics and results in panels with higher classification and variable selection performance compared to several existing penalized regression approaches in cases where a generalized linear model is misspecified. Finally, we use the proposed method to develop biomarker panels for separating pancreatic cysts with differing malignancy potential in a setting where complicated missingness in the biomarkers arose due to limited specimen volumes.

The motion of a circular foil in the field of a fixed point singularity: Integrability and asymptotic behavior

A finite-dimensional model is developed, which describes the motion of a balanced circular foil with proper circulation in the field of a fixed vortex source. The motion of the foil has been studied in two special cases: that of a fixed vortex and that of a fixed source. It is shown that in the absence of proper circulation, the fixed vortex and the fixed source have the same impact on the motion of the foil. However, adding nonzero proper circulation leads to qualitative differences in the foil's dynamics. For a fixed vortex, there exist three types of motions: the fall on a vortex in finite time, periodic and quasiperiodic motion around the vortex. The investigation of this case reduces to analysis of a Hamiltonian system with one degree of freedom. Typical phase portraits and graphs of the effective potential of the system are plotted vs the distance between the geometric center of the foil and the vortex. For a fixed source, two types of motions are possible: the fall on the source in finite time and unbounded escape from the source. For small intensities of the source, the asymptotics of escape to infinity is constructed.

Robust Cooperative Output Regulation for Networks of Hyperbolic PIDE–ODE Systems

In this paper the robust cooperative output regulation problem for multi-agent systems (MAS) with general heterodirectional hyperbolic PIDE-ODE agents is considered. This setup also covers networks of ODEs with arbitrarily long input and output delays. The output of the agents can be defined at all boundaries, in-domain and may depend on the ODE state, while disturbances act on the agents in-domain, at the boundaries, the output and the ODE. The communication network is described by a constant digraph and if its Laplacian is reducible, then heterogeneous agents are permitted also in the nominal case. The solution is based on the cooperative internal model principle, which requires to include a diffusively driven internal model in the controller. The corresponding state feedback regulator design starts with a local backstepping stabilization of the coupled hyperbolic PIDE-ODE systems. It is shown that the remaining simultaneous stabilization of the MAS can be traced back to the simultaneous stabilization of the finite-dimensional cooperative internal model. Solvability conditions in terms of the network topology and the agents transfer behavior are presented. The new design method is applied to the formation control of a platoon of uncertain heavy ropes carrying loads to verify its applicability. Simulations confirm the synchronization performance achieved by the resulting networked controller.

Passive-guaranteed modeling and simulation of a finite element nonlinear string model

This paper proposes to solve the dynamics of the Kirchhoff-Carrier nonlinear string model using the finite elements method. In order to ensure the power balance of the resulting finite dimensional model it is rewritten in the Port-Hamiltonian System (PHS) formalism. Using a discrete gradient and a quadratization of the Hamiltonian, an explicit power-preserving numerical scheme is proposed. Results of simulation are presented.

Single-Particle Model of Li-ion Battery – Model Calibration and Validation against Real Data in an Electric Vehicular Application

This paper investigates the problem of calibration and validation of a battery electrochemical model as a mandatory step towards accurate estimation of battery important variables, like state of charge (SoC) and state of health (SoH). Here, the Single Particle Model (SPM) is considered, which mathematically describes the battery internal governing phenomena by means of parabolic partial differential equations (PDEs), but whose parameters are notoriously difficult to measure or estimate. After suitable approximation of this model through a linear finite-dimensional model, a systematic procedure of SPM calibration is here proposed and validated against real data issued from battery cycling in an electric vehicular application, i.e., under standard driving cycle scenarios. In a novel approach of SoC estimation, the suitably calibrated SPM, together with measures of voltage and current, allow to analytically connect the internal spatially distributed ions’ concentrations to the equilibrium concentration, which, at its turn, is an image of battery SoC. Results suggest that SPM can reliably predict the battery internal ions’ concentrations and be further used for SoC accurate estimation.

Looking for optimal concentric ring electrodes: influence of design aspects on their performance

Concentric ring electrodes (CREs) allow improved spatial resolution, reduced crosstalk and interference, and increased bandwidth in the sensing of bioelectrical activity. A wide variety of designs have been used, but their selection is rarely well-founded. The aim of this work is to assess the implications of aspects of CRE design such as the distance between poles, their width and their maximum diameter on aspects such as the signal amplitude (and, therefore, quality), Laplacian estimation error and spatial selectivity (SS). For this purpose, a finite dimensional model of the CRE was used, and its response to the activity of an electric dipole of variable depth was simulated via finite element method modeling. Our results show that increasing the electrode size increases the error to a greater extent than the signal amplitude increases. Pole widths should be as small as possible. The middle ring of the tripolar CRE should be as far away as possible from the central disc. Tripolar CREs typically outperform bipolar CREs of the same outer diameter, significantly reducing the Laplacian estimation error and improving the SS at the cost of a small decrease in signal amplitude. Our results also show that the design of current commercial versions of CREs can be optimized. Furthermore, we propose a methodology that facilitates the selection of an adequate CRE configuration based on the specifications for CRE performance and practical aspects, such as the depth of activity sources to be recorded from and/or the maximum size of electrodes to be used. The monitoring and analysis of bioelectrical signals in a wide range of applications can benefit from the enhanced electrode design and methodology proposed in this work.

Controllability of the Main Road with an On-Ramp Section in Freeway Traffic Flow

This study examines the controllability issue pertaining to the main road with a single on-ramp segment within the context of freeway traffic flow. In this regard, a finite-dimensional nonlinear model is formulated by integrating variable speed limit (VSL) and ramp metering (RM) techniques with the controllability property of the system derived under continuous VSL and RM controllers. This allows us to simultaneously control the density of the main line and reduce the queue density of the merging sector. In the numerical experiment, numerical simulations are conducted on a sample model of a main road to investigate the null controllability of the system and validate the theoretical findings. The numerical findings suggest that it is possible to achieve a consistent queue length for the on-ramp section through the implementation of suitable VSL and RM controls. Therefore, the primary accomplishment of this study is to effectively regulate the traffic flow on the main road segment by managing the density of vehicles within a specified timeframe while also considering the queue density of the on-ramp section

Трансмерная гибридизация эволюционных алгоритмов многокритериальной оптимизации управления

The paper presents a hybrid evolutionary algorithm for multicriteria synthesis of the optimal control law for a dynamic system based on the trans-dimensional search models. The developed trans-dimensional search model belongs to the class of sequential hybridization models of the preprocessor/processor type and implies the combined use of the evolutionary algorithms of finite-dimensional and infinite-dimensional multi-objective optimization implementing the stages of global approximate and local refinement search for the optimal solutions. A finite-dimensional model of the global multi-criteria optimization is implemented using an evolutionary algorithm of the multi-criteria optimization in regard to the polyhedral dominance cone. Introduction of the uncertainty intervals of the vector indicator components weight coefficients in constructing the dominance polyhedral cone makes it possible to reduce the Pareto set uncertainty by highlighting a subset of solutions in it that are having a higher degree of balancing values for various components of the efficiency vector indicator. Evolutionary algorithm of the infinite-dimensional multi-objective optimization is a generalization of the Zoytendijk’s methods of possible directions for the class of infinite-dimensional multi-objective optimization problems and is used at the search stage. The paper provides results of a comparative analysis of various hybrid trans-dimensional models efficiency in the evolutionary search on the example of solving the problem of multicriteria synthesis of the optimal law for a bioreactor program control. Results of the computational experiments show that trans-dimensional hybridization of the evolutionary algorithms for the finite-dimensional and infinite-dimensional multicriteria control optimization provides a synergistic effect. This effect is expressed in significant increase in the accuracy of solving the problem of multicriteria control optimization in comparison with the known hybrid metaheuristic control optimization algorithms making it possible to resolve contradiction between the finite-dimensional global search model and the infinite-dimensional initial problem formulation

Multiple Active Material Lithium-Ion Batteries: Finite-Dimensional Modeling and Constrained State Estimation

Lithium-ion batteries with composite electrodes are being developed in the industry for their improved power, energy, and durability properties over classical single active material accumulators. To make the best out of multiple active material batteries, their management system needs to be fed with reliable information on the internal states, which are not available by measurements. It appears that tools developed for this purpose for single active material batteries are not suitable for multiple active material ones due to their distinctive electrical coupling rules within the electrodes. In this context, we first present a new finite-dimensional model of lithium-ion batteries for which the cathode is represented by two particles made of two different materials, and the anode is described by two graphite particles. We then use the model to design a nonlinear observer based on polytopic techniques that estimate the internal battery variables and thus the state of charge (SOC). The convergence of the observer is guaranteed under a linear matrix inequality (LMI) condition. We then exploit a technique recently proposed in the literature to modify the observer so that its state estimates remain in given plausible range at all times, while preserving its original convergence properties. This allows avoiding aberrant estimated values in the transient and may also favor the future implementation of the observer on embedded devices. Simulation results using literature data illustrate the efficiency of the observer to estimate internal battery variables.

Instantons on Sasakian 7-manifolds

AbstractWe study a natural contact instanton equation on gauge fields over 7-dimensional Sasakian manifolds, which is closely related to both the transverse Hermitian Yang–Mills (HYM) condition and the G2-instanton equation. We obtain, by Fredholm theory, a finite-dimensional local model for the moduli space of irreducible solutions. Following the approach by Baraglia and Hekmati in five dimensions [1], we derive cohomological conditions for smoothness and express its dimension in terms of the index of a transverse elliptic operator. Finally, we show that the moduli space of self-dual contact instantons is Kähler, in the Sasakian case. As an instance of concrete interest, we specialize to transversely holomorphic Sasakian bundles over contact Calabi–Yau 7-manifolds, as studied by Calvo-Andrade, Rodríguez and Sá Earp [8], and we show that in this context the notions of contact instanton, integrable G2-instanton and HYM connection coincide.

Stable data‐driven Koopman predictive control: Concentrated solar collector field case study

AbstractNon‐linearity is an inherent feature of practical systems. Although there have been significant advances in the control of nonlinear systems, the proposed methods often require considerable computational resources or rely on local linearization around equilibrium points. The Koopman operator is an infinite‐dimensional linear operator that fully captures a system's non‐linear dynamics. However, one of the major problems is identifying a Koopman finite dimensional linear model for a nonlinear system. Initiated by the Willems’ fundamental Lemma, a class of data‐driven control methods has been developed for linear systems without the need to identify the system's matrices. Motivated by these two ideas, a data‐driven Koopman‐based predictive control scheme for non‐linear systems is proposed for unknown disturbed non‐linear systems utilising a finite‐length dataset. Then, considering the uncertainty in the Koopman state variables, a robust data‐driven Koopman predictive control structure is presented. Also, the results led to the design of a data‐driven Koopman predictive control strategy with terminal components to ensure the closed‐loop stability of nonlinear systems. The proposed scheme is tested on the distributed‐parameter model of the ACUREX solar collector field (located at Almería, Spain) to regulate the field outlet temperature around a desired value. Finally, simulation results show the effectiveness of the proposed approach.

Filtering of Noisy Nonlinear Buck Converter with CPL Using Carleman Linerization

The paper includes a novel approach for filtering of nonlinear Buck converter with Constant Power Load in noisy environment based on the Carleman linearization approach. The procedure starts with the modeling of Buck converter with Constant Power Load using Hamiltonian framework resulting in the deterministic model of network i.e., a finite dimensional system of ordinary differential equations. To evaluate the effect of noise on it, a Gaussian noise is embedded in the network through switch, which results in the development of stochastic state model and noise present at measurement terminals is considered for measurement equations. Using Carleman linearization approach, the finite-dimensional nonlinear model is transmuted into a bilinear model of infinite-dimension which can be approximated for any chosen order. As a result, we arrive at the finite-dimensional bilinear model of Buck converter with augmented state vector of higher dimension than the original system. State estimation of stochastic dynamical systems in the presence of noisy measurements is an interesting problem and hence, filtering technique is adopted here. The filtering equations in Itô setting are derived for bilinear systems using probabilistic approach for state estimation. The superiority of the proposed bilinear filtering technique over the benchmark EKF is demonstrated by the numerical simulations.

A Bernstein–von-Mises theorem for the Calderón problem with piecewise constant conductivities

This note considers a finite dimensional statistical model for the Calderón problem with piecewise constant conductivities. In this setting it is shown that injectivity of the forward map and its linearisation suffice to prove the invertibility of the information operator, resulting in a Bernstein–von-Mises theorem and optimality guarantees for estimation by Bayesian posterior means.

Analytic solution of the exact Daum–Huang flow equation for particle filters

State estimation for nonlinear systems, especially in high dimensions, is a generally intractable problem, despite the ever-increasing computing power. Efficient algorithms usually apply a finite-dimensional model for approximating the probability density of the state vector or treat the estimation problem numerically. In 2007 Daum and Huang introduced a novel particle filter approach that uses a homotopy-induced particle flow for the Bayesian update step. Multiple types of particle flows were derived since with different properties. The exact flow considered in this work is a first-order linear ordinary time-varying inhomogeneous differential equation for the particle motion. An analytic solution in the interval [0,1] is derived for the scalar measurement case, which enables significantly faster computation of the Bayesian update step for particle filters.

An efficient geometric method for incompressible hydrodynamics on the sphere

We present an efficient and highly scalable geometric numerical method for two-dimensional ideal fluid dynamics on the sphere. The starting point is Zeitlin's finite-dimensional model of hydrodynamics. The efficiency stems from exploiting a tridiagonal splitting of the discrete spherical Laplacian combined with highly optimized, scalable numerical algorithms. For time-stepping, we adopt a recently developed isospectral integrator able to preserve the geometric structure of Euler's equations, in particular conservation of the Casimir functions. To overcome previous computational bottlenecks, we formulate the matrix Lie algebra basis through a sequence of tridiagonal eigenvalue problems, efficiently solved by well-established linear algebra libraries. The same tridiagonal splitting allows for computation of the stream matrix, involving the inverse Laplacian, for which we design an efficient parallel implementation on distributed memory systems. The resulting overall computational complexity is O(N3) per time-step for N2 spatial degrees of freedom. The dominating computational cost is matrix-matrix multiplication, carried out via the parallel library ScaLAPACK. Scaling tests show approximately linear scaling up to around 2500 cores for the matrix size N=4096 with a computational time per time-step of about 0.55 seconds. These results allow for long-time simulations and the gathering of statistical quantities while simultaneously conserving the Casimir functions. We illustrate the developed algorithm for Euler's equations at the resolution N=2048.