Explicit Stability Research Articles

Protein Model Quality Estimation Using Molecular Dynamics Simulation.

The estimation of protein model quality remains a challenging task and is important for protein structural model utilization. In the last decade, existing methods that rely on machine learning to deep learning have been developed and shown progressive improvement. Despite utilizing more sophisticated techniques and introducing new features, none of these methods employ explicit protein structure stability information. Hypothetically, protein model quality might be indicated by its structural stability in an in silico system disclosed by the structural difference from its initial structure. One of the possible methods to exploit such information is by implementing molecular dynamics simulations that have shown successful applications in many research fields. We present a novel approach by introducing explicit protein structure stability information using molecular dynamics simulation. Despite using only simple features, small data with no training process required, and a short molecular dynamics simulation time, our method shows comparable performance to the state-of-the-art deep learning-based method.

Modified Erlang Loss System for Cognitive Wireless Networks

This paper considers a modified Erlang loss system for cognitive wireless networks and related applications. A primary user has pre-emptive priority over secondary users, and the primary customer is lost if upon arrival all the channels are used by other primary users. Secondary users cognitively use idle channels, and they can stay (either in an infinite buffer or in an orbit) in cases where idle channels are not available upon arrival or they are interrupted by primary users. While the infinite buffer model represents the case with zero sensing time, the infinite orbit model represents the case with positive sensing time. We obtain an explicit stability condition for the cases where arrival processes of primary users and secondary users follow Poisson processes, and their service times follow two distinct arbitrary distributions. The stability condition is insensitive to the service time distributions and implies the maximal throughout of secondary users. Moreover, we extend the stability analysis to the system with outgoing calls. For a special case of exponential service time distributions, we analyze the buffered system in depth to show the effect of parameters on the delay performance and the mean number of interruptions of secondary users. Our simulations for distributions rather than exponential reveal that the mean number of terminations for secondary users is less sensitive to the service time distribution of primary users.

Solution of the generalized regularized long-wave equation with optimal spline collocation technique and implicit Crank–Nicolson as well as explicit SSP-RK43 scheme

In this work, a generalized regularized long-wave equation is solved using the optimal cubic B-spline collocation method for space discretization. Two different approaches are followed for the time-domain discretization. One is the implicit Crank–Nicolson scheme and the other is strong explicit stability preserving Runge–Kutta method of four stages and third-order. Also, the stability analysis of the techniques is carried out and it is shown that the implicit technique is unconditional stable, whereas the explicit technique is conditionally stable. These methods are applied to three problems involving a single solitary wave, the interaction of two solitary waves, and the evolution of solitons via the Maxwellian initial condition. These equations possess three invariants of motion that are mass, momentum, and energy. The value of these invariants is calculated, which is found to remain preserved for a long time. To demonstrate the robustness of both the techniques, the and error norms are calculated.

Encoding Stability into Laser Powder Bed Fusion Monitoring Using Temporal Features and Pore Density Modelling.

In laser powder bed fusion (LPBF), melt pool instability can lead to the development of pores in printed parts, reducing the part’s structural strength. While camera-based monitoring systems have been introduced to improve melt pool stability, these systems only measure melt pool stability in limited, indirect ways. We propose that melt pool stability can be improved by explicitly encoding stability into LPBF monitoring systems through the use of temporal features and pore density modelling. We introduce the temporal features, in the form of temporal variances of common LPBF monitoring features (e.g., melt pool area, intensity), to explicitly quantify printing stability. Furthermore, we introduce a neural network model trained to link these video features directly to pore densities estimated from the CT scans of previously printed parts. This model aims to reduce the number of online printer interventions to only those that are required to avoid porosity. These contributions are then implemented in a full LPBF monitoring system and tested on prints using 316L stainless steel. Results showed that our explicit stability quantification improved the correlation between our predicted pore densities and true pore densities by up to 42%.

Bounded controls for discrete-time linear systems subject to input time delay

This paper investigates the global stabilization of discrete-time linear systems with input time delay by bounded controls. Based on some special canonical forms containing time delays both in its input and state, two special discrete-time linear systems---multiple integrators and oscillators are first considered. The global stabilizing controllers are respectively established, and moreover, explicit conditions are established to guarantee the stability of the closed-loop systems. Subsequently, a concise design method is proposed for globally stabilizing general discrete-time linear system by combining the design methods for multiple integrators and oscillators. The designed controller is in the explicit form with explicit stability conditions being given, and thus is easier to use than the existing results. Finally, numerical simulations illustrate the effectiveness of the proposed approaches.

Embedded pairs for optimal explicit strong stability preserving Runge–Kutta methods

We construct a family of embedded pairs for optimal explicit strong stability preserving Runge–Kutta methods of order 2≤p≤4 to be used to obtain numerical solution of spatially discretized hyperbolic PDEs. In this construction, the goals include non-defective property, large stability region, and small error values as defined in Dekker and Verwer (1984) and Kennedy et al. (2000). The new family of embedded pairs offer the ability for strong stability preserving (SSP) methods to adapt by varying the step-size. Through several numerical experiments, we assess the overall effectiveness in terms of work versus precision while also taking into consideration accuracy and stability.

An assessment of solvers for algebraically stabilized discretizations of convection–diffusion–reaction equations

Abstract We consider flux-corrected finite element discretizations of 3D convection-dominated transport problems and assess the computational efficiency of algorithms based on such approximations. The methods under investigation include flux-corrected transport schemes and monolithic limiters. We discretize in space using a continuous Galerkin method and ℙ1 or ℚ1 finite elements. Time integration is performed using the Crank–Nicolson method or an explicit strong stability preserving Runge–Kutta method. Nonlinear systems are solved using a fixed-point iteration method, which requires solution of large linear systems at each iteration or time step. The great variety of options in the choice of discretization methods and solver components calls for a dedicated comparative study of existing approaches. To perform such a study, we define new 3D test problems for time dependent and stationary convection–diffusion–reaction equations. The results of our numerical experiments illustrate how the limiting technique, time discretization and solver impact on the overall performance.

On the stability of POD basis interpolation on Grassmann manifolds for parametric model order reduction

Proper Orthogonal Decomposition (POD) basis interpolation on Grassmann manifolds has been successfully applied to problems of parametric model order reduction (pMOR). In this work we address the necessary stability conditions for the interpolation, all defined from strong mathematical background. A first condition concerns the domain of definition of the logarithm map. Second, we show how the stability of interpolation can be lost if certain geometrical requirements are not satisfied by making a concrete elucidation of the local character of linearization. To this effect, we draw special attention to the Grassmannian exponential map and the optimal injectivity condition of this map, related to the cut–locus of Grassmann manifolds. From this, an explicit stability condition is established and can be directly used to determine the loss of injectivity in practical pMOR applications. A third stability condition is formulated when increasing the number p of POD modes, deduced from the principal angles of subspaces of different dimensions p. Definition of this condition leads to an understanding of the non-monotonic oscillatory behavior of the Reduced Order Model (ROM) error-norm with respect to the number of POD modes, and on the contrary, the well-behaved monotonic decrease of the error-norm in the two numerical examples presented herein. We have chosen to perform pMOR in hyperelastic structures using a non-intrusive approach for inserting the interpolated spatial POD ROM basis in a commercial FEM code. The accuracy is assessed by a posteriori error norms defined using the ROM FEM solution and its high-fidelity counterpart simulation. Numerical studies successfully ascertained and highlighted the implication of stability conditions which are general and can be applied to a variety of other linear or nonlinear problems involving parametrized ROMs generation based on POD basis interpolation on Grassmann manifolds.

An inversion algorithm for P-functions with applications to multi-energy CT

Multi-energy computed tomography (ME-CT) is an x-ray transmission imaging technique that uses the energy dependence of x-ray photon attenuation to determine the elemental composition of an object of interest. Mathematically, forward ME-CT measurements are modeled by a nonlinear integral transform. In this paper, local conditions for global invertibility of the ME-CT transform are studied, and explicit stability estimates quantifying the error propagation from measurements to reconstructions are provided. Motivated from the inverse problem of image reconstruction in ME-CT, an iterative inversion algorithm for the so-called P-functions is proposed. Numerical simulations for ME-CT, in two and three materials settings with an equal number of energy measurements, confirm the theoretical predictions.

Hamilton-Based Stability Criterion and Attraction Region Estimation for Grid-Tied Inverters Under Large-Signal Disturbances

As the crucial interface of renewable energy integration, the stable operation of power converters under grid faults and external disturbances has become much more important. However, synchronization instability of grid-tied inverters has been found caused by the phase-locked loop (PLL) under some critical operating conditions, such as the grid voltage sag, current surge, and transmission line faults. Due to the strong nonlinearity, transient stability analysis of inverters with PLL becomes much more complicated under large-signal perturbations and weak grid conditions. In this article, the large-signal synchronization stability of grid-tied inverters is investigated from the energy perspective. Based on the structural resemblance between the PLL and synchronous generator, the energy function is constructed to establish a port-Hamiltonian model for the grid-tied inverter. For the nonlinear inverter system, an explicit Hamilton-based stability criterion is developed for grid-tied converters according to the concept of dissipativity and LaSalle’s theorem. Moreover, the analytical stability boundary is acquired based on the proposed stability criterion and attraction region estimation, covering both controller parameters and grid parameters. To verify the effectiveness of the proposed analysis approach, extensive simulation and experimental results are presented. Compared with the conventional large-signal methods, the proposed Hamilton-based approach has explicit physical meaning and high accuracy.

A high-order partitioned fluid-structure interaction framework for vortex-induced vibration simulation

Vortex-induced vibration (VIV) is a highly nonlinear fluid-structure interaction (FSI) phenomenon, and its accurate prediction remains challenging. In this study, we developed a high-order partitioned FSI framework towards accurate simulation of the VIV phenomenon. The focus is on simulating the two-way coupled interaction between a flexibly mounted rigid body and its surrounding fluids. In the FSI framework, a high-order computational fluid dynamics method, i.e., flux reconstruction/correction procedure via reconstruction, is utilized to simulate the unsteady compressible Navier–Stokes equations in the arbitrary Lagrangian Eulerian format on unstructured moving grids. A linearized fluid-solid Riemann solver is employed to weakly couple fluid dynamics and rigid-body dynamics on the fluid-solid interface, which can effectively mitigate the so-called added-mass instability. Time marching is conducted through either the linearly implicit Rosenbrock–Wanner method or the explicit strong stability preserving Runge–Kutta method. The high-order FSI framework is verified with a challenging VIV phenomenon, i.e., VIV of a zero-mass cylinder in viscous flows, and has been used to study two VIV-related problems, namely, the nonlinear energy sink and oscillating foil based energy harvesting mechanism.

Delay-dependent input-output stability conditions for non-autonomous neutral type differential equations in a Banach space

In a Banach space we consider a class of linear non-autonomous neutral type differential equations with several delays. For the considered equations we derive explicit delay-dependent input-output stability conditions. Applications to neutral type integro-differential equations are also discussed.

Stability switches in a linear differential equation with two delays

This paper is devoted to the study of the effect of delays on the asymptotic stability of a linear differential equation with two delays \[x'(t)=-ax(t)-bx(t-\tau)-cx(t-2\tau),\quad t\geq 0,\] where \(a\), \(b\), and \(c\) are real numbers and \(\tau\gt 0\). We establish some explicit conditions for the zero solution of the equation to be asymptotically stable. As a corollary, it is shown that the zero solution becomes unstable eventually after undergoing stability switches finite times when \(\tau\) increases only if \(c-a\lt 0\) and \(\sqrt{-8c(c-a)}\lt |b| \lt a+c\). The explicit stability dependence on the changing \(\tau\) is also described.

DoD Stabilization for non-linear hyperbolic conservation laws on cut cell meshes in one dimension

In this work, we present the Domain of Dependence (DoD) stabilization for systems of hyperbolic conservation laws in one space dimension. The base scheme uses a method of lines approach consisting of a discontinuous Galerkin scheme in space and an explicit strong stability preserving Runge-Kutta scheme in time. When applied on a cut cell mesh with a time step length that is appropriate for the size of the larger background cells, one encounters stability issues. The DoD stabilization consists of penalty terms that are designed to address these problems by redistributing mass between the inflow and outflow neighbors of small cut cells in a physical way. For piecewise constant polynomials in space and explicit Euler in time, the stabilized scheme is monotone for scalar problems. For higher polynomial degrees p, our numerical experiments show convergence orders of p+1 for smooth flow and robust behavior in the presence of shocks.

Delayed Feedback Control of Hidden Chaos in the Unified Chaotic System between the Sprott C System and Yang System

In this paper, the influence of delayed feedback on the unified chaotic system from the Sprott C system and Yang system is studied. The Hopf bifurcation and dynamic behavior of the system are fully studied by using the central manifold theorem and bifurcation theory. The explicit formula, bifurcation direction, and stability of the periodic solution of bifurcation are given correspondingly. The Hopf bifurcation diagram and chaotic phenomenon are also analyzed by numerical simulation to prove the correctness of the theory. It shows that this delay control can only be applied to the hidden chaos with two stable equilibria.

A p-robust polygonal discontinuous Galerkin method with minus one stabilization

In this paper, we introduce a new stabilization for discontinuous Galerkin methods for the Poisson problem on polygonal meshes, which induces optimal convergence rates in the polynomial approximation degree [Formula: see text]. The stabilization is obtained by penalizing, in each mesh element [Formula: see text], a residual in the norm of the dual of [Formula: see text]. This negative norm is algebraically realized via the introduction of new auxiliary spaces. We carry out a [Formula: see text]-explicit stability and error analysis, proving [Formula: see text]-robustness of the overall method. The theoretical findings are demonstrated in a series of numerical experiments.

A note on the stability of pencils of plane curves

We investigate the problem of classifying pencils of plane curves of degree d up to projective equivalence. We obtain explicit stability criteria in terms of the log canonical threshold by relating the stability of a pencil to the stability of the curves lying on it.

Exponential stability of systems of vector delay differential equations with applications to second order equations

Various results and techniques, such as the Bohl-Perron theorem, a priori estimates of solutions, M-matrices and the matrix measure, are applied to obtain new explicit exponential stability conditions for the system of vector functional differential equationsx˙i(t)=Ai(t)xi(hi(t))+∑j=1n∑k=1mijBijk(t)xj(hijk(t))+∑j=1n∫gij(t)tKij(t,s)xj(s)ds,i=1,…,n. Here xi are unknown vector-functions, Ai,Bijk,Kij are matrix functions, hi,hijk,gij are delayed arguments. Using these results, we deduce explicit exponential stability tests for second order vector delay differential equations.

An explicit robust stability condition for uncertain time-varying first-order plus dead-time systems

First-order plus dead-time (FOPDT) models are broadly used in process control to represent damped dynamic processes with time delays. An explicit condition for parameter- and delay-dependent robust stability of FOPDT systems with varying uncertain parameters and delay is derived in this paper. An internal model control (IMC) approach is proposed to parameterize stabilizing controllers that satisfy the output tracking objective in time-varying FOPDT systems represented by an uncertain first-order dynamic model with a time-varying delay in the control input. The small-gain theorem is used to derive an explicit necessary and sufficient parameter-dependent robust stability condition as a function of the nominal system gain, nominal varying delay, nominal time constant, and the bounds of the parameter uncertainties. An equivalent proportional-integral-derivative (PID) controller is then extracted to facilitate the implementation of the proposed IMC-based robust control. The application of the proposed explicit robust stability condition is studied in the context of air-fuel ratio (AFR) control in lean-burn spark ignition (SI) engines with a large time-varying transport delay in the control loop due to the placement of the universal exhaust gas oxygen (UEGO) sensor downstream the catalytic converter.

Generalized splay states in phase oscillator networks.

Networks of coupled phase oscillators play an important role in the analysis of emergent collective phenomena. In this article, we introduce generalized m-splay states constituting a special subclass of phase-locked states with vanishing mth order parameter. Such states typically manifest incoherent dynamics, and they often create high-dimensional families of solutions (splay manifolds). For a general class of phase oscillator networks, we provide explicit linear stability conditions for splay states and exemplify our results with the well-known Kuramoto-Sakaguchi model. Importantly, our stability conditions are expressed in terms of just a few observables such as the order parameter or the trace of the Jacobian. As a result, these conditions are simple and applicable to networks of arbitrary size. We generalize our findings to phase oscillators with inertia and adaptively coupled phase oscillator models.