Algebraic Constraints Research Articles

A New Approach to Nonlinear Invariants for Hybrid Systems Based on the Citing Instances Method

In generating invariants for hybrid systems, a main source of intractability is that transition relations are first-order assertions over current-state variables and next-state variables, which doubles the number of system variables and introduces many more free variables. The more variables, the less tractability and, hence, solving the algebraic constraints on complete inductive conditions by a comprehensive Gröbner basis is very expensive. To address this issue, this paper presents a new, complete method, called the Citing Instances Method (CIM), which can eliminate the free variables and directly solve for the complete inductive conditions. An instance means the verification of a proposition after instantiating free variables to numbers. A lattice array is a key notion in this paper, which is essentially a finite set of instances. Verifying that a proposition holds over a Lattice Array suffices to prove that the proposition holds in general; this interesting feature inspires us to present CIM. On one hand, instead of computing a comprehensive Gröbner basis, CIM uses a Lattice Array to generate the constraints in parallel. On the other hand, we can make a clever use of the parallelism of CIM to start with some constraint equations which can be solved easily, in order to determine some parameters in an early state. These solved parameters benefit the solution of the rest of the constraint equations; this process is similar to the domino effect. Therefore, the constraint-solving tractability of the proposed method is strong. We show that some existing approaches are only special cases of our method. Moreover, it turns out CIM is more efficient than existing approaches under parallel circumstances. Some examples are presented to illustrate the practicality of our method.

Numerical solution of nonlinear fractal‐fractional optimal control problems by Legendre polynomials

This article is devoted to developing an accurate operational matrix (OM) method for the solution of a new category of nonlinear optimal control problems (OCPs) explained by fractal‐fractional dynamical systems. The fractal‐fractional differentiation is considered in the sense of Atangana‐Riemann‐Liouville. The method is based on the shifted Legendre polynomials (LPs). Through the way, a new OM of fractal‐fractional differentiation is extracted for the shifted LPs. The optimality conditions are converted to an algebraic system of equations by using the shifted LP expansions of the state and control variables and applying the method of constrained extrema. More precisely, these variables are approximated by the shifted LPs with undetermined coefficients. Then, these expansions are replaced in the performance index, and the Gauss‐Legendre integration method is used to approximate the performance index for extracting a nonlinear algebraic equation. In the sequel, the mentioned OM and the collocation method are used to generate some algebraic equations from the dynamical system. Finally, the method of the constrained extrema is used by coupling the algebraic constraints generated from the dynamical system and the initial condition with the algebraic equation elicited from the performance index by a set of unknown Lagrange multipliers. The accuracy of the method is studied by solving some numerical examples.

Deformed General Relativity and Quantum Black Holes Interior

Effective models of black holes interior have led to several proposals for regular black holes. In the so-called polymer models, based on effective deformations of the phase space of spherically symmetric general relativity in vacuum, one considers a deformed Hamiltonian constraint while keeping a non-deformed vectorial constraint, leading under some conditions to a notion of deformed covariance. In this article, we revisit and study further the question of covariance in these deformed gravity models. In particular, we propose a Lagrangian formulation for these deformed gravity models where polymer-like deformations are introduced at the level of the full theory prior to the symmetry reduction and prior to the Legendre transformation. This enables us to test whether the concept of deformed covariance found in spherically symmetric vacuum gravity can be extended to the full theory, and we show that, in the large class of models we are considering, the deformed covariance cannot be realized beyond spherical symmetry in the sense that the only deformed theory which leads to a closed constraints algebra is general relativity. Hence, we focus on the spherically symmetric sector, where there exist non-trivial deformed but closed constraints algebras. We investigate the possibility to deform the vectorial constraint as well and we prove that non-trivial deformations of the vectorial constraint with the condition that the constraints algebra remains closed do not exist. Then, we compute the most general deformed Hamiltonian constraint which admits a closed constraints algebra and thus leads to a well-defined effective theory associated with a notion of deformed covariance. Finally, we study static solutions of these effective theories and, remarkably, we solve explicitly and in full generality the corresponding modified Einstein equations, even for the effective theories which do not satisfy the closeness condition. In particular, we give the expressions of the components of the effective metric (for spherically symmetric black holes interior) in terms of the functions that govern the deformations of the theory.

A Reduced Model for Microbial Electrolysis Cells

Microbial electrolysis cells (MECs) are breakthrough technology of cheap hydrogen production with high efficiency. In this paper differential-algebraic equation (DAE) model of a MEC with an algebraic constraint on current was studied, simulated and validated by implementing the model on continuous-flow MECs. Then sensitivity analysis for the system was effectuated. Parameters which have the predominating influence on the current density and hydrogen production rate were defined. This sensitivity analysis was utilized in modeling and validation of the batch-cycle of MEC. After that parameters which have less influence on MEC were eliminated and simplified reduced model was obtained and validated. Finally, MEC energy productivity was maximized by optimization of operating conditions.

Maxwell-Proca theory: Definition and construction

We present a systematic construction of the most general first order Lagrangian describing an arbitrary number of interacting Maxwell and Proca fields on Minkowski spacetime. To this aim, we first formalize the notion of a Proca field, in analogy to the well known Maxwell field. Our definition allows for a non-linear realization of the Proca mass, in the form of derivative self-interactions. Consequently, we consider so-called generalized Proca/vector Galileons. We explicitly demonstrate the ghost-freedom of this complete Maxwell-Proca theory by obtaining its constraint algebra. We find that, when multiple Proca fields are present, their interactions must fulfill non-trivial differential relations in order to ensure the propagation of the correct number of degrees of freedom. These relations had so far been overlooked, which means previous multi-Proca proposals generically contain ghosts. This is a companion paper to arXiv:1905.06968 [hep-th]. It puts on a solid footing the theory there introduced.

Generalized Proca & its constraint algebra

We reconsider the construction of general derivative self-interactions for a massive Proca field. The constructed Lagrangian is such that the vector field propagates at most three degrees of freedom, thus avoiding the ghostly nature of a fourth polarisation. The construction makes use of the well-known condition for constrained systems of having a degenerate Hessian. We briefly discuss the casuistry according to the nature of the existing constraints algebra. We also explore various classes of interesting new interactions that have been recently raised in the literature. For the sixth order Lagrangian that satisfies the constraints by itself we prove its topological character, making such a term irrelevant. There is however a window of opportunity for exploring other classes of fully-nonlinear interactions that satisfy the constraint algebra by mixing terms of various order.

Reaction–permeability optimum time heating policy via process control for debinding green ceramic components

ABSTRACTThe minimisation of temperature–time schedules to thermally debind ceramic green bodies has been examined for the case when the species of binder degradation transit the green component via gas permeation in the pore space. The model is based on the spatial–temporal reaction–permeability equation in conjunction with differential equations describing the decomposition rate and temperature–time cycle, plus an algebraic constraint which limits the buildup of pressure to avoid component failure. An approach using finite elements with a process controller was compared to an approximate approach using the pseudo-steady-state modelling assumption in conjunction with the calculus of variations. The two approaches exhibit excellent agreement, which supports the accuracy of the pseudo-steady-state assumption and the usefulness of the process control strategy, both of which can save time and energy in debinding ceramic components. The process control methodology was also applied to the complex geometry of a ceramic engine valve.

Finite-Iteration Tracking of Singular Coupled Systems Based on Learning Control With Packet Losses

The finite-iteration tracking for singular coupled systems with packet-dropping learning controllers is discussed in this paper. The singular coupled systems are constructed to describe systems with some algebraic constraints, and iterative learning controllers are then designed to achieve the finite-iteration tracking of singular systems. The definition of the finite-iteration tracking is first proposed and the settling iteration is explicitly calculated. Moreover, the iterative learning controllers are designed to consider the case of packet losses. Simulation results are given to elaborate the correctness of the given theorems.

A Medical Image Segmentation Method With Anti-Noise and Bias-Field Correction

Brain magnetic resonance images (MRI) are affected by noise and bias field, which make the traditional FCM algorithm unable to segment tissue regions of MR images accurately. Based on the above problems, this paper proposes an MR image segmentation method (MPCFCM) with anti-noise and bias field correction, which implements segmentation by point-to-plane algebraic distance constraint. Different from traditional point-based clustering methods, a hyper-center of clustering (i.e., plane) model is defined, and data clustering is completed by optimizing different planes. In addition, to realize the point clustering with plane, a key problem that how to measure the distance from point to plane needs to be solved. This paper adopts the algebraic distance as a measure function, which can avoid the nonlinear problem caused by a direct calculation of the minimum distance between a point and a plane, thus simplifying the computational complexity. In the proposed algorithm, spatial distance, local variance and gray-difference of neighbors are combined to construct a new anti-noise smoothing factor for constraining the energy function so that the algorithm has better anti-noise and retains more image details. Finally, the singular value decomposition is performed on the loss energy, some information removed is re-added to the segmented image to repair it. The experimental results show that MPCFCM algorithm can better correct bias field and eliminate noise and obtain accurate image segmentation results with more details.

Modeling and discretization methods for the numerical simulation of elastic frame structures

A new model description for the numerical simulation of elastic frame structures is proposed. Instead of resolving algebraic constraints at frame nodes and incorporating them into the finite element spaces, the constraints are included explicitly in the model via new variables and enforced via Lagrange multipliers. Based on the new formulation, an inf-sup inequality for the continuous-time formulation and the finite element discretization is proved. Despite the increased number of variables in the model and the discretization, the new formulation leads to faster simulations for the stationary problem and simplifies the analysis and the numerical solution of the evolution problem describing the movement of the frame structure under external forces. The results are illustrated via numerical examples for the modeling and simulation of elastic stents.

Identification of Markov Jump Autoregressive Processes from Large Noisy Data Sets

This paper introduces a novel methodology for the identification of switching dynamics for switched autoregressive linear models. Switching behavior is assumed to follow a Markov model. The system’s outputs are contaminated by possibly large values of measurement noise. Although the procedure provided can handle other noise distributions, for simplicity, it is assumed that the distribution is Normal with unknown variance. Given noisy input-output data, we aim at identifying switched system coefficients, parameters of the noise distribution, dynamics of switching and probability transition matrix of Markovian model. System dynamics are estimated using previous results which exploit algebraic constraints that system trajectories have to satisfy. Switching dynamics are computed with solving a maximum likelihood estimation problem. The efficiency of proposed approach is shown with several academic examples. Although the noise to output ratio can be high, the method is shown to be effective in the situations where a large number of measurements is available.

Efficient numerical methods for gas network modeling and simulation

We study the modeling and simulation of gas pipeline networks, with a focus on fast numerical methods for the simulation of transient dynamics. The obtained mathematical model of the underlying network is represented by a nonlinear differential algebraic equation (DAE). With our modeling, we reduce the number of algebraic constraints, which correspond to the $(2,2)$ block in our semi-explicit DAE model, to the order of junction nodes in the network, where a junction node couples at least three pipelines. We can furthermore ensure that the $(1, 1)$ block of all system matrices including the Jacobian is block lower triangular by using a specific ordering of the pipes of the network. We then exploit this structure to propose an efficient preconditioner for the fast simulation of the network. We test our numerical methods on benchmark problems of (well-)known gas networks and the numerical results show the efficiency of our methods.

Modelling, design and control of non-isolated single-input multi-output Zeta-Buck-Boost converter

In this article, we focus on the analytical modeling of nonisolated single-input multiple-output (SIMO) dc–dc converters governed by linear differential algebraic equations (DAEs). The modeling challenge in nonisolated SIMO converters arises due to the switching among multiple DAEs governing the circuit. Averaged models of such converters, described by an ordinary differential equation, are derived using the notion of quasi-Weierstrass transformation. Using this technique, we derive the averaged model for a nonisolated Zeta–Buck–Boost (ZBB) converter. It is observed that the dynamics of the state variable corresponding to the algebraic constraint is uncontrollable and eliminating this state leads to a fourth-order completely controllable averaged model. This model is used to design a linear state-feedback controller for line regulation. In the case of bipolar SIMO converters, ZBB being an example, regulation of the two outputs can be achieved using any one of the two output integral states. Although the state-feedback controller successfully rejects the disturbance asymptotically, it is found that the deviations from the set point in the transient phase can be large. To see if smaller deviations can be achieved by appending the linear controller with a nonlinear term, we explore a nonlinear controller based on the Lyapunov redesign technique. Smaller deviations with complete disturbance rejection are possible if the disturbance is matched. When it is unmatched, small deviations are achieved at the expense of steady-state errors, whose estimate is explicitly found. The proposed model and control scheme are verified through a laboratory built hardware prototype.

Competitive exclusion in a DAE model for microbial electrolysis cells

Microbial electrolysis cells (MECs) are devices that employ electroactive bacteria to perform extracellular electron transfer, enabling hydrogen generation from biodegradable substrates. In our previous work, we developed and analyzed a differential-algebraic equation (DAE) model for MECs. The model resembles a chemostat or continuous stirred tank reactor (CSTR). It consists of ordinary differential equations for concentrations of substrate, microorganisms, and an extracellular mediator involved in electron transfer. There is also an algebraic constraint for electric current and hydrogen production. Our goal is to determine the outcome of competition between methanogenic archaea and electroactive bacteria, because only the latter contribute to electric current and the resulting hydrogen production. We investigate asymptotic stability in two industrially relevant versions of the model. An important aspect of many chemostat models is the principle of competitive exclusion. This states that only microbes which grow at the lowest substrate concentration will survive as t → ∞.We show that if methanogens can grow at the lowest substrate concentration, then the equilibrium corresponding to competitive exclusion by methanogens is globally asymptotically stable. The analogous result for electroactive bacteria is not necessarily true. In fact we show that local asymptotic stability of competitive exclusion by electroactive bacteria is not guaranteed, even in a simplified version of the model. In this case, even if electroactive bacteria can grow at the lowest substrate concentration, a few additional conditions are required to guarantee local asymptotic stability. We provide numerical simulations supporting these arguments. Our results suggest operating conditions that are most conducive to success of electroactive bacteria and the resulting current and hydrogen production in MECs. This will help identify when producing methane or electricity and hydrogen is favored.

Using a CAS/DGS to Analyze Computationally the Configuration of Planar Bar Linkage Mechanisms Based on Partial Latin Squares

Currently, the study of new isotopism invariants of partial Latin squares constitutes an open and active problem. This paper delves into this topic by analyzing computationally the configuration of planar bar linkage mechanisms for which the array formed by the lengths of their bars constitutes an empty-diagonal symmetric partial Latin square such that each one of its rows and columns has at least two non-empty cells. These assumptions enable one to define a series of algebraic and geometric constraints that can be readily implemented in any Computer Algebra or Dynamic Geometry System. In order to illustrate the different concepts and results introduced throughout the paper, it is explicitly determined and characterized the distribution of planar bar linkage mechanisms based on partial Latin squares of order up to five.

Aircraft attitude reconstruction via novel quaternion-integration procedure

Unit quaternion representation is widely used in flight simulation to overcome the limitations of the standard numerical ordinary-differential-equations (ODEs) based on three-parameters rotation variables (such as Euler angels), as they may impose kinematic singularities during aircraft's attitude reconstruction. However, these benefits do not come without a price, since the classical way of integrating rotational quaternions includes solving of differential-algebraic equations (DAEs) that requires post-integration numerical stabilization of the additional algebraic constraint enforcing the quaternion unitary norm. This can pose a problem in the case of longer flight simulations since improper numerical treatment of the quaternion-normalization constraint may induce numerical drift into the simulation results. As a remedy, the proposed novel algorithm circumvents DAE problem of quaternion integration by shifting update-integration-process from configuration manifold to the local tangential level of the incremental rotations (reducing thus integration to standard three ODEs problem at tangential Lie algebra level). This can be done due to the isomorphism of the Lie algebras of the rotational SO(3) group and the configuration manifold unit quaternion Sp(1) group. Besides avoiding DAE formulation by reducing integration process to standard three ODEs problem, the proposed algorithm also exhibits numerical advantages as it is discussed in the presented example.

Generalized ADHM equations from marginal deformations in open superstring field theory

Working within the framework of both the A∞ and the Berkovits open superstring field theory, we derive a necessary and sufficient condition for a Neveu-Schwarz marginal deformation to be exact up to third order in the deformation parameter. For a specific class of backgrounds, we find that this condition localizes on the boundary of the worldsheet moduli space, thus providing a very simple computational prescription for recovering algebraic constraints (generalized ADHM equations) which need to be satisfied by the moduli. Applying our results to the D(−1)/D3 system, we confirm up to third order that blowing up the size of the D-instanton inside the D3 brane worldvolume is an exact modulus of the full string theory. We also discuss examples of more complicated back- grounds, such as instantons on unresolved ALE spaces, as well as the spiked instantons.

Efficient solution of large-scale algebraic Riccati equations associated with index-2 DAEs via the inexact low-rank Newton-ADI method

This paper extends the algorithm of Benner et al. (2016) [10] to Riccati equations associated with Hessenberg index-2 Differential Algebratic Equation (DAE) systems. Such DAE systems arise, e.g., from semi-discretized, linearized (around steady state) Navier-Stokes equations. The solution of the associated Riccati equation is important, e.g., to compute feedback laws that stabilize the Navier-Stokes equations. Challenges in the numerical solution of the Riccati equation arise from the large-scale of the underlying systems and the algebraic constraint in the DAE system. These challenges are met by a careful extension of the inexact low-rank Newton-ADI method to the case of DAE systems. A main ingredient in the extension to the DAE case is the projection onto the manifold described by the algebraic constraints. In the algorithm, the equations are never explicitly projected, but the projection is only applied as needed. Numerical experience indicates that the algorithmic choices for the control of inexactness and line-search can help avoid subproblems with matrices that are only marginally stable. The performance of the algorithm is illustrated on a large-scale Riccati equation associated with the stabilization of Navier-Stokes flow around a cylinder.

High Order Anchoring and Reinitialization of Level Set Function for Simulating Interface Motion

A second order interface anchoring method has been developed and used with fast sweeping algorithm for reinitialization of a level set function. The algebraic anchoring formulation ensures that the location of the actual interface is preserved, leading to better mass conservation property. It also provides high order accurate algebraic constraint for solving the Eikonal equation on a finite difference grid. Geometric properties of the interface such as surface normal and curvature are subsequently computed from the reinitialized distance function. Various analytical functions for modeling distortion in level set field are considered and accuracy of reinitialization is evaluated using first and second order anchoring schemes. It is also shown that accurate computation of interface curvature requires a high order anchor in addition to a high order fast sweeping method. Mass conservation property of reinitialization is also analyzed by considering test problems from literature including the classic Rider–Kothe single vortex problem. The formulation is suitable for efficient parallelization for both distributed memory and on-node shared memory parallel systems. Scalability and performance of the reinitialization scheme on multiple architectures are demonstrated.

Model reduction techniques for port‐Hamiltonian differential‐algebraic systems

AbstractPort‐based network modeling of multi‐physics problems leads naturally to a formulation as port‐Hamiltonian differential‐algebraic systems (pHDAEs). In this way, the physical properties are directly encoded in the structure of the model. Since the state space dimension of such systems may be very large, in particular when the model is a space‐discretized partial differential‐algebraic system, in optimization and control there is a need for model reduction methods that preserve the port‐Hamiltonian (pH) structure while keeping the algebraic constraints unchanged. To combine model reduction for differential‐algebraic equations (DEAs) with port‐Hamiltonian structure preservation, we adapt power conservation based techniques from port‐Hamiltonian systems of ordinary differential equations (pHODEs) to pHDAEs. The performance of the methods is investigated for benchmark examples originating from semi‐discretized flow problems.