Algebraic Constraints Research Articles

Spinorial discrete symmetries and adjoint structures

Extending the investigations about the theory of duals, we analyze duals built up with the aid of discrete symmetry operators. We scrutinize algebraic and physical constraints (encompassing them in a theoretical scope) in order to verify which combination of discrete symmetries may compose a physical and mathematical well-posed spinorial dual. In this scenario, we relate the Lounesto classification with several other spinor classification possibilities, attempting to connect classes and physical constraints. Several possibilities are investigated.

Event-triggered control for switched singular systems with asynchronous switching and state jumps

This work discusses the event-triggered control (ETC) issue for switched singular systems (SSS) with asynchronous switching and state jumps. Firstly, a modified consistency projector is presented to describe the transient state jumps at switching moments, which shows that the state jumps are caused by switching law and algebraic constraint. Then, a discontinuous multiple Lyapunov function is presented under the event-triggered state feedback control, and several sufficient conditions are derived to ensure that the closed-loop systems are regular, impluse-free and globally uniformly exponentially stable (GUES). Then, a Luenberger-like observer is constructed to estimate continuous and jumping states, some new sufficient conditions in terms of linear matrix inequalities (LMIs) are obtained by using matrix decoupling techniques under the observer-based ETC. Furthermore, a lower bound for the event-triggered time interval to exclude the Zeno phenomenon is given. Finally, two examples and an application are presented to demonstrate the validity of the results.

Convex Optimization for Rendezvous and Proximity Operation via Birkhoff Pseudospectral Method

Rapid and accurate rendezvous and proximity operations for spacecraft are crucial to the success of most space missions. In this paper, a sequential convex programming method, combined with the first-order and second-order Birkhoff pseudospectral methods, is proposed for the autonomous rendezvous and proximity operations of spacecraft. The original nonlinear and nonconvex close-range rendezvous problem with thrust constraints and no-fly zone constraints is converted into its convex version by using the sequential convexification techniques; then, the Birkhoff pseudospectral method is used to transcribe the dynamic constraints into a series of linear algebraic equality constraints, in other words, a convex second-order conic programming problem with a relatively small condition number. Thus, the resulting problem can be accurately and efficiently solved by a convex solver. The simulation results indicate that the proposed methods, especially the second-order Birkhoff pseudospectral method, have obvious advantages over other methods in computational efficiency and sensitivity.

Имитационная модель силового трехфазного трансформатора бронестержневой конструкции с витым магнитопроводом

Background. One of the areas of industry digitalization is associated with the concept of digital twins, which allow simulating the operation of real devices in various modes. At present, simulation models of a three-phase rod-type transformer have been developed in the MatLab Simulink SimPowerSystem environment. The models can serve as the basis to design a digital twin of a real device. The problem is to find the most universal method for mathematical description of a transformer of unspecified design to develop its simulation model. The purpose of this study is to develop a simulation model of a three-phase shell-type transformer core with a twisted magnetic circuit and to study this model when the transformer is operating in steady and transient modes. Materials and methods. The authors have applied the methods to model electrical circuits based on the theory of ordinary differential equations, a simulation method using the MatLab Simulink SimPowerSystem software package. Results. The structural futures of the transformers with shell-core design and the main assumptions when developing the simulation model have been assessed. A system of nonlinear equations has been developed to find the flows and MMF generated by the windings of the device. A simulation model of a three-phase shell-core transformer with a twisted magnetic core has been developed. It is based on the obtained equations using the Algebraic Constraint block with the Solver function in MatLab Simulink SimPowerSystem. The results of simulation of dynamic processes in the transformer in various modes are presented. Conclusions. The results of the study can be used to design power transformers in engineering companies and under production conditions. It is also possible to use the developed models in case of power transformer operation to analyze the static and dynamic modes of operation of sections of electrical networks.

New approach for the design and optimization of a quarter-car suspension system

ABSTRACT In this research study, a new approach was applied for the design of a quarter-car suspension. The objective was to determine the optimal design parameters of the suspension that satisfy the required static and dynamic constraints. The design variables were defined using intervals of values and all the imposed constraints were integrated in an NCSP solving tool. The dynamic performance of the suspension system was integrated as algebraic constraints and ordinary differential equations by intervals. Linking dynamic and static sizing steps in the same phase eliminated over-sizing issue of the system and helped save the computation time. In order to reduce the search area of the design parameters, the ModelCenter coupled with Matlab tool were used to perform parametric studies and get the optimal dimensions.

A general numerical scheme for the optimal control of fractional Birkhoffian systems

This paper gives a general numerical scheme for the optimal control problem of fractional Birkhoffian systems. The fractional forced Birkhoff equations within Riemann–Liouville fractional derivatives are derived from the fractional Pfaff–Birkhoff–d’Alembert principle which includes the control as an external force term. Following the strategy of variational integrators, the fractional Pfaff–Birkhoff–d’Alembert principle is directly discretized to develop the equivalent discrete fractional forced Birkhoff equations that served as the equality constraints of the optimization problem. Together with the initial and final state constraints on the configuration space, the original optimal control problem is converted into a nonlinear optimization problem subjected to a system of algebraic constraints, which can be solved by existing algorithms. An illustrative example is given to show the efficiency and simplicity of the proposed method.

Optimal discrete-time unbiased filtering for systems with unknown inputs

This paper concerns the H ∞ functional filtering design problem for discrete-time systems with unknown inputs. Our interest is focused on the unbiased filtering satisfying the H ∞ performance. First the unbiasedness conditions of the estimation error are presented. The dynamic of this error is dependent on the present and future values of the noise. Different cases are presented, the first one is based on the introduction of an algebraic constraint leading to the dynamic error dependent only on the present noise, the second case is more general and can be treated by modifying the performance index or by using the descriptor systems formulation. The conditions of the existence of an unbiased filter are presented and the designs of the different filters, based on the bounded real lemma, are given in the form of the linear matrix inequalities (LMI). The presented approach unifies the design of functional, reduced-order and full-order filters for discrete-time systems. Three numerical examples are presented to illustrate the approach described in the paper and shows the performance of the proposed functional filter.

Feedback control approaches for restoration of power grids from blackouts

The automated restoration of power systems with variable energy resources is a timely problem to tackle. Automated restoration advice can support operators in deciding on strategic actions to restore power grids from a blackout with a mix of conventional and renewable generation resources. To this end, this paper frames the restoration process of power grids with solar resources as a nonlinear dynamic model with algebraic constraints in discrete time which is steered by feedback control loops. We discuss two feedback-control strategies based on greedy and reinforcement learning algorithms, and contrast their performance with restoration plans generated by a mixed-integer linear program. We found that the reinforcement learning algorithm infers restoration actions faster than the greedy one. However, the tuning process of the reinforcement learning parameters is slower than for the greedy one.

Asymptotic stability and synchronization of fractional delayed memristive neural networks with algebraic constraints

The asymptotic stability and synchronization of fractional delayed memristive neural networks with algebraic constraints in Riemann–Liouville sense will be investigated in this article. First, algebraic constraints are introduced for the first time into the existing fractional delayed memristive neural networks, and a new fractional singular delayed memristive neural networks (FSDMNNs) model is presented. Then, within the framework of Filippov’s solution, a less conservative result for the asymptotic stability of FSDMNNs is obtained by Lyapunov–Krasovskii functional. Subsequently, the appropriate feedback scheme and adaptive scheme are designed to synchronize FSDMNNs and two sufficient conditions are acquired. In addition, the results not only address the influence of delays and algebraic constraints, but can also easily detect and synchronize the actual memristive neural networks. Finally, numerical simulations frankly confirm the correctness and validity of the derived results.

Real-time path planning for autonomous vehicle based on teaching–learning-based optimization

This paper presents an online path planning approach for an autonomous tracked vehicle in a cluttered environment based on teaching–learning-based optimization (TLBO), considering the path smoothness, and the potential collision with the surrounding obstacles. In order to plan an efficient path that allows the vehicle to be autonomously navigated in cluttered environments, the path planning problem is solved as a multi-objective optimization problem. First, the vehicle perception is fully achieved by means of inertial measurement unit (IMU), wheels odometry, and light detection and ranging (LiDAR). In order to compensate the sensors drift to achieve more reliable data and improve the localization estimation and corrections, data fusion between the outputs of wheels odometry, LiDAR, and IMU is made through extended Kalman filter (EKF). Then, TLBO is proposed and applied to determine the optimum online path, where the objectives are to find the shortest path to reach the target destination, and to maximize the path smoothness, while avoiding the surrounding obstacles, and taking into account the vehicle dynamic and algebraic constraints. To check the performance of the proposed TLBO algorithm, it is compared in simulation to genetic algorithm (GA), particle swarm optimization (PSO), and a hybrid GA–PSO algorithm. Finally, real-time experiments based on robot operating system (ROS) implementation are conducted to validate the effectiveness of the proposed path planning algorithm.

Evolution equations dynamical system of the Lemaître–Tolman–Bondi metric containing coupled dark energy

In this paper, we consider inhomogeneous spherically symmetric models based on the Lemaître–Tolman–Bondi (LTB) metric, assuming as its source an interactive mixture of ordinary baryonic matter, cold dark matter and dark energy with a coupling term proportional to the addition of energy densities of both dark fluids. We reduce Einstein’s field equations to a first-order seven-dimensional autonomous dynamical system of evolution equations and algebraic constraints. We study in detail the evolution of the energy density and spatial curvature profiles along the phase–space by means of two subspace projections: a three-dimensional projection associated with the solutions of the Friedman–Lemaître–Robertson–Walker metric (invariant subspace) and a four-dimensional projection describing the evolution of the inhomogeneous fluctuations. We also classify and study the critical points of the system in comparison with previous work on similar sources, as well as solving numerically the equations for initial energy density and curvature profiles that lead to a spherical bounce whose collapsing time we estimate appropriately.

Gauge symmetry of unimodular gravity in Hamiltonian formalism

We work out the description of the gauge symmetry of unimodular gravity in the constrained Hamiltonian formalism. In particular, we demonstrate how the transversality conditions restricting the diffeomorphism parameters emerge from the algebra of the Hamiltonian constraints. The alternative form is long known as parametrizing the volume preserving diffeomorphisms by unrestricted two-forms instead of the transverse vector fields. This gauge symmetry is reducible. We work out the Hamiltonian description of this form of unimodular gravity (UG) gauge symmetry. Becchi-Rouet-Stora-Tyutin--Batalin-Fradkin-Vilkovisky (BFV-BRST) Hamiltonian formalism is constructed for both forms of the UG gauge symmetry. These two BRST complexes have a subtle inequivalence: Their BRST cohomology groups are not isomorphic. In particular, for the first complex, which is related to the restricted gauge parameters, the cosmological constant does not correspond to any nontrivial BRST cocycle, while for the alternative complex it does. In the wording of physics, this means $\mathrm{\ensuremath{\Lambda}}$ is a fixed parameter defined by the field asymptotics rather than the physical observable from the standpoint of the first complex. The second formalism views $\mathrm{\ensuremath{\Lambda}}$ as the observable with unrestricted initial data.

Thermal-Enhanced Adaptive Interval Estimation in Battery Packs With Heterogeneous Cells

The internal states of lithium-ion batteries need to be carefully monitored during operation to manage energy and safety. In this article, we propose a thermal-enhanced adaptive interval observer for state-of-charge (SOC) and temperature estimation for a battery pack. For a large battery pack with hundreds or thousands of heterogeneous cells, each individual cell characteristic is different from others. Practically, applying estimation algorithms on each and every cell would be mathematically and computationally intractable since battery packs are often characterized by combinations of differential equations (state dynamics) and algebraic constraints (Kirchhoff’s laws). These issues are tackled using an interval observer based on monotone/cooperative system theory, whose novelty lies in considering cell heterogeneity and state-dependent parameters as unknown, but bounded uncertainties. The resulting interval observer maps the bounded uncertainties to a feasible set of SOC and temperature estimation for all cells in the pack at each time instant. This work also addresses the significant conservatism under extreme conditions with large currents via a thermal-enhanced adaptive scheme. The proposed interval estimation is scalable and computationally tractable since it is independent of the number of cells in a pack, as numerically demonstrated in a comparison with respect to a state-of-the-art single-cell state observer. The stability and inclusion of the adaptive interval observer are proven and validated through simulations.

Integration of design and NMPC-based control for chemical processes under uncertainty: An MPCC-based framework

The use of nonlinear model predictive control (NMPC) for the integration of design and control remains as an open area of research. When NMPC is incorporated into the framework for design and control, this results into a bilevel problem formulation. The search of a solution for a bilevel problem is often carried out with the implementation of systematic iterative methodologies. In this work, we implement the classical KKT transformation strategy to transform the original bilevel problem for simultaneous design and NMPC-based control into a single-level dynamic optimization problem. This single-level formulation is referred to as a mathematical program with complementarity constraints (MPCC). Violations to constraint qualifications (CQs) are avoided through the implementation of several reformulation techniques. This facilitates the convergence of the MPCC for optimal design and control. The use of an MPCC-based formulation allows to fully incorporate the NMPC’s necessary conditions for optimality into the design problem as a set of algebraic constraints. Accordingly, the MPCC is solved as a conventional NLP, i.e., we avoid the use of decomposition or simplification methodologies for the solution of the bilevel problem. This guarantees optimality in the solution for integrated design and NMPC-based control. This strategy was implemented in three case studies involving small, medium-scale, and highly nonlinear systems to demonstrate the application of MPCCs for design and NMPC-based control under process disturbances and uncertainty.

Likelihood Maximization and Moment Matching in Low SNR Gaussian Mixture Models

AbstractWe derive an asymptotic expansion for the log‐likelihood of Gaussian mixture models (GMMs) with equal covariance matrices in the low signal‐to‐noise regime. The expansion reveals an intimate connection between two types of algorithms for parameter estimation: the method of moments and likelihood optimizing algorithms such as Expectation‐Maximization (EM). We show that likelihood optimization in the low SNR regime reduces to a sequence of least squares optimization problems that match the moments of the estimate to the ground truth moments one by one. This connection is a stepping stone towards the analysis of EM and maximum likelihood estimation in a wide range of models. A motivating application for the study of low SNR mixture models is cryo‐electron microscopy data, which can be modeled as a GMM with algebraic constraints imposed on the mixture centers. We discuss the application of our expansion to algebraically constrained GMMs, among other example models of interest. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Jordan and Einstein frames from the perspective of ω=−3/2 Hamiltonian Brans-Dicke theory

We carefully perform a Hamiltonian Dirac's constraint analysis of the $\ensuremath{\omega}=\ensuremath{-}\frac{3}{2}$ Brans-Dicke theory with the Gibbons-Hawking-York boundary term. The Poisson brackets are computed via functional derivatives. After a brief summary of the results for the $\ensuremath{\omega}\ensuremath{\ne}\ensuremath{-}\frac{3}{2}$ case [G. Gionti S. J., Canonical analysis of Brans-Dicke theory addresses Hamiltonian inequivalence between the Jordan and Einstein frames, Phys. Rev. D 103, 024022 (2021)] we derive all Hamiltonian Dirac's constraints and constraint algebra in both the Jordan and the Einstein frames. Confronting and contrasting Dirac's constraint algebra in both frames, it is shown that they are not equivalent. This highlights that the transformations from the Jordan to the Einstein frames are not Hamiltonian canonical transformations.

An effective model for the quantum Schwarzschild black hole

We present an effective theory to describe the quantization of spherically symmetric vacuum motivated by loop quantum gravity. We include anomaly-free holonomy corrections through a canonical transformation and a linear combination of constraints of general relativity, such that the modified constraint algebra closes. The system is then provided with a fully covariant and unambiguous geometric description, independent of the gauge choice on the phase space. The resulting spacetime corresponds to a singularity-free (black-hole/white-hole) interior and two asymptotically flat exterior regions of equal mass. The interior region contains a minimal smooth spacelike surface that replaces the Schwarzschild singularity. We find the global causal structure and the maximal analytical extension. Both Minkowski and Schwarzschild spacetimes are directly recovered as particular limits of the model.

The hidden simplicity of metabolic networks is revealed by multireaction dependencies.

Understanding the complexity of metabolic networks has implications for manipulation of their functions. The complexity of metabolic networks can be characterized by identifying multireaction dependencies that are challenging to determine due to the sheer number of combinations to consider. Here, we propose the concept of concordant complexes that captures multireaction dependencies and can be efficiently determined from the algebraic structure and operational constraints of metabolic networks. The concordant complexes imply the existence of concordance modules based on which the apparent complexity of 12 metabolic networks of organisms from all kingdoms of life can be reduced by at least 78%. A comparative analysis against an ensemble of randomized metabolic networks shows that the metabolic network of Escherichia coli contains fewer concordance modules and is, therefore, more tightly coordinated than expected by chance. Together, our findings demonstrate that metabolic networks are considerably simpler than what can be perceived from their structure alone.

Isotope-assisted metabolic flux analysis as an equality-constrained nonlinear program for improved scalability and robustness.

Stable isotope-assisted metabolic flux analysis (MFA) is a powerful method to estimate carbon flow and partitioning in metabolic networks. At its core, MFA is a parameter estimation problem wherein the fluxes and metabolite pool sizes are model parameters that are estimated, via optimization, to account for measurements of steady-state or isotopically-nonstationary isotope labeling patterns. As MFA problems advance in scale, they require efficient computational methods for fast and robust convergence. The structure of the MFA problem enables it to be cast as an equality-constrained nonlinear program (NLP), where the equality constraints are constructed from the MFA model equations, and the objective function is defined as the sum of squared residuals (SSR) between the model predictions and a set of labeling measurements. This NLP can be solved by using an algebraic modeling language (AML) that offers state-of-the-art optimization solvers for robust parameter estimation and superior scalability to large networks. When implemented in this manner, the optimization is performed with no distinction between state variables and model parameters. During each iteration of such an optimization, the system state is updated instead of being calculated explicitly from scratch, and this occurs concurrently with improvement in the model parameter estimates. This optimization approach starkly contrasts with traditional “shooting” methods where the state variables and model parameters are kept distinct and the system state is computed afresh during each iteration of a stepwise optimization. Our NLP formulation uses the MFA modeling framework of Wiechert et al. [1], which is amenable to incorporation of the model equations into an NLP. The NLP constraints consist of balances on either elementary metabolite units (EMUs) or cumomers. In this formulation, both the steady-state and isotopically-nonstationary MFA (inst-MFA) problems may be solved as an NLP. For the inst-MFA case, the ordinary differential equation (ODE) system describing the labeling dynamics is transcribed into a system of algebraic constraints for the NLP using collocation. This large-scale NLP may be solved efficiently using an NLP solver implemented on an AML. In our implementation, we used the reduced gradient solver CONOPT, implemented in the General Algebraic Modeling System (GAMS). The NLP framework is particularly advantageous for inst-MFA, scaling well to large networks with many free parameters, and having more robust convergence properties compared to the shooting methods that compute the system state and sensitivities at each iteration. Additionally, this NLP approach supports the use of tandem-MS data for both steady-state and inst-MFA when the cumomer framework is used. We assembled a software, eiFlux, written in Python and GAMS that uses the NLP approach and supports both steady-state and inst-MFA. We demonstrate the effectiveness of the NLP formulation on several examples, including a genome-scale inst-MFA model, to highlight the scalability and robustness of this approach. In addition to typical inst-MFA applications, we expect that this framework and our associated software, eiFlux, will be particularly useful for applying inst-MFA to complex MFA models, such as those developed for eukaryotes (e.g. algae) and co-cultures with multiple cell types.

Fast Solution of Fully Implicit Runge--Kutta and Discontinuous Galerkin in Time for Numerical PDEs, Part II: Nonlinearities and DAEs

Fully implicit Runge--Kutta (IRK) methods have many desirable accuracy and stability properties as time integration schemes, but high-order IRK methods are not commonly used in practice with large-scale numerical PDEs because of the difficulty of solving the stage equations. This paper introduces a theoretical and algorithmic framework for solving the nonlinear equations that arise from IRK methods (and discontinuous Galerkin discretizations in time) applied to nonlinear numerical PDEs, including PDEs with algebraic constraints. Several new linearizations of the nonlinear IRK equations are developed, offering faster and more robust convergence than the often-considered simplified Newton, as well as an effective preconditioner for the true Jacobian if exact Newton iterations are desired. Inverting these linearizations requires solving a set of block 2 x 2 systems. Under quite general assumptions, it is proven that the preconditioned 2 x 2 operator's condition number is bounded by a small constant close to one, independent of the spatial discretization, spatial mesh, and time step, and with only weak dependence on the number of stages or integration accuracy. Moreover, the new method is built using the same preconditioners needed for backward Euler-type time stepping schemes, so can be readily added to existing codes. The new methods are applied to several challenging fluid flow problems, including the compressible Euler and Navier--Stokes equations, and the vorticity-streamfunction formulation of the incompressible Euler and Navier--Stokes equations. Up to 10th-order accuracy is demonstrated using Gauss IRK, while in all cases fourth-order Gauss IRK requires roughly half the number of preconditioner applications as required by standard Singly diagonally implicit Runge--Kutta methods.