Showing papers on "Piecewise published in 2020"

Non-Hermitian Bulk-Boundary Correspondence and Auxiliary Generalized Brillouin Zone Theory

Abstract: We provide a systematic and self-consistent method to calculate the generalized Brillouin zone (GBZ) analytically in one-dimensional non-Hermitian systems, which helps us to understand the non-Hermitian bulk-boundary correspondence. In general, a $n$-band non-Hermitian Hamiltonian is constituted by $n$ distinct sub-GBZs, each of which is a piecewise analytic closed loop. Based on the concept of resultant, we can show that all the analytic properties of the GBZ can be characterized by an algebraic equation, the solution of which in the complex plane is dubbed as auxiliary GBZ (aGBZ). We also provide a systematic method to obtain the GBZ from aGBZ. Two physical applications are also discussed. Our method provides an analytic approach to the spectral problem of open boundary non-Hermitian systems in the thermodynamic limit.

The Influence of the Activation Function in a Convolution Neural Network Model of Facial Expression Recognition

Abstract: The convolutional neural network (CNN) has been widely used in image recognition field due to its good performance. This paper proposes a facial expression recognition method based on the CNN model. Regarding the complexity of the hierarchic structure of the CNN model, the activation function is its core, because the nonlinear ability of the activation function really makes the deep neural network have authentic artificial intelligence. Among common activation functions, the ReLu function is one of the best of them, but it also has some shortcomings. Since the derivative of the ReLu function is always zero when the input value is negative, it is likely to appear as the phenomenon of neuronal necrosis. In order to solve the above problem, the influence of the activation function in the CNN model is studied in this paper. According to the design principle of the activation function in CNN model, a new piecewise activation function is proposed. Five common activation functions (i.e., sigmoid, tanh, ReLu, leaky ReLus and softplus–ReLu, plus the new activation function) have been analysed and compared in facial expression recognition tasks based on the Keras framework. The Experimental results on two public facial expression databases (i.e., JAFFE and FER2013) show that the convolutional neural network based on the improved activation function has a better performance than most-of-the-art activation functions.

Multiple Lyapunov Functions Analysis Approach for Discrete-Time-Switched Piecewise-Affine Systems Under Dwell-Time Constraints

Abstract: In this technical note, the asymptotic stability analysis and state-feedback control design are investigated for a class of discrete-time switched nonlinear systems via the smooth approximation technique. The modal dwell-time switching property is considered to constrain switchings between nonlinear subsystems. A kind of autonomous switchings, i.e., state-partition-dependent switching, is introduced within each approximated piecewise-affine (PWA) subsystem. Combining with the state partition information, the novel asymptotic stability conditions with less conservatism are derived for the induced switched PWA systems with dual switching mechanism based on the approach of multiple Lyapunov functions in piecewise quadratic form. Then, the design of PWA state-feedback controllers is implemented. The effectiveness of the obtained theoretical results is demonstrated by a numerical example.

On an Improved State Parametrization for Soft Robots With Piecewise Constant Curvature and Its Use in Model Based Control

Abstract: Piecewise constant curvature models have proven to be an useful tool for describing kinematics and dynamics of soft robots. However, in their three dimensional formulation they suffer from many issues limiting their range of applicability - as discontinuities and singularities - mainly concerning the straight configuration of the robot. In this work we analyze these flaws, and we show that they are not due to the piecewise constant curvature assumption itself, but that instead they are a byproduct of the commonly employed direction/angle of bending parametrization of the state. We therefore consider an alternative state representation which solves all the discussed issues, and we derive a model based controller based on it. Examples in simulation are provided to support and describe the theoretical results. When using the novel parametrization, the system is able to perform more complex tasks, with a strongly reduced computational burden, and without incurring in spikes and discontinuous behaviors.

Fast Differentiable Sorting and Ranking

Abstract: The sorting operation is one of the most commonly used building blocks in computer programming. In machine learning, it is often used for robust statistics. However, seen as a function, it is piecewise linear and as a result includes many kinks where it is non-differentiable. More problematic is the related ranking operator, often used for order statistics and ranking metrics. It is a piecewise constant function, meaning that its derivatives are null or undefined. While numerous works have proposed differentiable proxies to sorting and ranking, they do not achieve the $O(n \log n)$ time complexity one would expect from sorting and ranking operations. In this paper, we propose the first differentiable sorting and ranking operators with $O(n \log n)$ time and $O(n)$ space complexity. Our proposal in addition enjoys exact computation and differentiation. We achieve this feat by constructing differentiable operators as projections onto the permutahedron, the convex hull of permutations, and using a reduction to isotonic optimization. Empirically, we confirm that our approach is an order of magnitude faster than existing approaches and showcase two novel applications: differentiable Spearman's rank correlation coefficient and least trimmed squares.

Multi-Stage Stochastic Programming to Joint Economic Dispatch for Energy and Reserve With Uncertain Renewable Energy

Abstract: To address the uncertain renewable energy in the day-ahead optimal dispatch of energy and reserve, a multi-stage stochastic programming model is established in this paper to minimize the expected total costs. The uncertainties over the multiple stages are characterized by a scenario tree and the optimal dispatch scheme is cast as a decision tree which guarantees the flexibility to decide the reasonable outputs of generation and the adequate reserves accounting for different realizations of renewable energy. Most importantly, to deal with the “Curse of Dimensionality” of stochastic programming, stochastic dual dynamic programming (SDDP) is employed, which decomposes the original problem into several sub-problems according to the stages. Specifically, the SDDP algorithm performs forward pass and backward pass repeatedly until the convergence criterion is satisfied. At each iteration, the original problem is approximated by creating a linear piecewise function. Besides, an improved convergence criterion is adopted to narrow the optimization gaps. The results on the IEEE 118-bus system and real-life provincial power grid show the effectiveness of the proposed model and method.

Asymptotically Optimal Planning under Piecewise-Analytic Constraints

Abstract: We present the first asymptotically optimal algorithm for motion planning problems with piecewise-analytic differential constraints, like manipulation or rearrangement planning. This class of problems is characterized by the presence of differential constraints that are local in nature: a robot can only move an object once the object has been grasped. These constraints are not analytic and thus cannot be addressed by standard differentially constrained planning algorithms. We demonstrate that, given the ability to sample from the locally reachable subset of the configuration space with positive probability, we can construct random geometric graphs that contain optimal plans with probability one in the limit of infinite samples. This approach does not require a hand-coded symbolic abstraction. We demonstrate our approach in simulation on a simple manipulation planning problem, and show it generates lower-cost plans than a sequential task and motion planner.

Piecewise optimal fractional reproducing kernel solution and convergence analysis for the atangana–baleanu–caputo model of the lienard’s equation

Abstract: In this paper, an attractive reliable analytical technique is implemented for constructing numerical solutions for the fractional Lienard’s model enclosed with suitable nonhomogeneous initial condi...

Deep Physical Informed Neural Networks for Metamaterial Design

Abstract: In this paper, we propose a physical informed neural network approach for designing the electromagnetic metamaterial. The approach can be used to deal with various practical problems such as cloaking, rotators, concentrators, etc. The advantage of this approach is the flexibility that we can deal with not only the continuous parameters but also the piecewise constants. As our best knowledge, there is no other faster and much efficient method to deal with these problems. As a byproduct, we propose a method to solve high frequency Helmholtz equation, which is widely used in physics and engineering. Some benchmark problems have been solved in numerical tests to verify our method.

Consensus in High-Power Multiagent Systems With Mixed Unknown Control Directions via Hybrid Nussbaum-Based Control.

Abstract: This work investigates the consensus tracking problem for high-power nonlinear multiagent systems with partially unknown control directions. The main challenge of considering such dynamics lies in the fact that their linearized dynamics contain uncontrollable modes, making the standard backstepping technique fail; also, the presence of mixed unknown control directions (some being known and some being unknown) requires a piecewise Nussbaum function that exploits the a priori knowledge of the known control directions. The piecewise Nussbaum function technique leaves some open problems, such as Can the technique handle multiagent dynamics beyond the standard backstepping procedure? and Can the technique handle more than one control direction for each agent? In this work, we propose a hybrid Nussbaum technique that can handle uncertain agents with high-power dynamics where the backstepping procedure fails, with nonsmooth behaviors (switching and quantization), and with multiple unknown control directions for each agent.

Spline collocation methods for systems of fuzzy fractional differential equations

Abstract: In this paper, systems of fuzzy fractional differential equations with a lateral type of the Hukuhara derivative and the generalized Hukuhara derivative are numerically studied. Collocation method on discontinuous piecewise polynomial spaces is proposed. Convergence of the proposed method is analyzed. The superconvergent results on the graded mesh are studied. Examples are provided to support theoretical results. Finally, the effect of uncertainty in a diabetes model and its resulting complications is investigated as a practical application.

A New 4D Four-Wing Memristive Hyperchaotic System: Dynamical Analysis, Electronic Circuit Design, Shape Synchronization and Secure Communication

Abstract: In this paper, a simple four-wing chaotic attractor is first proposed by replacing the constant parameters of the Chen system with a periodic piecewise function. Then, a new 4D four-wing memristive...

Barrier Lyapunov function based adaptive finite-time control for hypersonic flight vehicles with state constraints.

Abstract: This paper investigates the finite-time tracking control problem of the hypersonic flight vehicle (HFV) with state constraints. Firstly, a control-oriented model is introduced to enable the application of adaptive backstepping scheme. To meet strict requirements in terms of working conditions of HFV, barrier Lyapunov function is adopted to constrain the tracking errors, while piecewise saturation function is constructed to restrict the virtual signals. To guarantee the finite-time convergent property of HFV dynamics, an adaptive scheme in accordance with finite-time stability theory is designed. Meanwhile, a sliding mode differentiator is employed to estimate the derivatives of the virtual control laws. Novel auxiliary systems are then designed to consider the side effects of the possible saturation and to maintain the finite-time convergent property. In the final stage, the effectiveness and performance of the proposed method is demonstrated by numerical simulations.

Fuzzy-Model-Based Output Feedback Sliding-Mode Control for Discrete-Time Uncertain Nonlinear Systems

Abstract: This paper addresses the output feedback sliding-mode control (SMC) problem for discrete-time uncertain nonlinear systems through Takagi–Sugeno fuzzy dynamic models. Combining with the sliding surface, a descriptor system is constructed to characterize the sliding motion dynamics. Sufficient conditions for asymptotic stability analysis of the sliding motion are attained by the piecewise quadratic Lyapunov functions within a convex optimization setup. Two SMC design approaches are proposed to ensure the finite-time convergence of the sliding surface. Two simulation examples are presented to show the effectiveness of the proposed approaches.

Compound Regularization of Full-Waveform Inversion for Imaging Piecewise Media

Abstract: Full-waveform inversion (FWI) is an iterative nonlinear waveform matching procedure, which seeks to reconstruct unknown model parameters from partial waveform measurements. The nonlinear and ill-posed nature of FWI requires sophisticated regularization techniques to solve it. In most applications, the model parameters may be described by physical properties (e.g., wave speeds, density, attenuation, and anisotropy) that are piecewise functions of space. Compound regularizations are, thus, beneficial to capture these different functions by FWI. We consider different implementations of compound regularizations in the wavefield reconstruction inversion (WRI) method, a formulation of FWI that extends its search space and mitigates the so-called cycle skipping pathology. Our hybrid regularizations rely on the Tikhonov and total variation (TV) functionals, from which we build two classes of hybrid regularizers: the first class is simply obtained by a convex combination (CC) of the two functionals, while the second relies on their infimal convolution (IC). In the former class, the model parameters are required to simultaneously satisfy different priors, while in the latter, the model is broken into its basic components, each satisfying a distinct prior (e.g., smooth, piecewise constant, and piecewise linear). We implement these compound regularizations in WRI using the alternating direction method of multipliers (ADMM). Then, we assess our regularized WRI for seismic imaging applications. Using a wide range of subsurface models, we conclude that the compound regularizer based on IC leads to the lowest error in the parameter reconstruction compared to that obtained with the CC counterpart and the Tikhonov and TV regularizers when used independently.

Recovering piecewise constant refractive indices by a single far-field pattern

Abstract: We are concerned with the inverse scattering problem of recovering an inhomogeneous medium by the associated acoustic wave measurement. We prove that under certain assumptions, a single far-field pattern determines the values of a perturbation to the refractive index on the corners of its support. These assumptions are satisfied for example in the low acoustic frequency regime. As a consequence if the perturbation is piecewise constant with either a polyhedral nest geometry or a known polyhedral cell geometry, such as a pixel or voxel array, we establish the injectivity of the perturbation to far-field map given a fixed incident wave. This is the first unique determinancy result of its type in the literature, and all of the existing results essentially make use of infinitely many measurements.

Coordination Control for Uncertain Networked Systems Using Interval Observers

Abstract: In this article, we take the coordination control problem of linear time-invariant networked systems with uncertain additive disturbance and uncertain initial states into consideration. A distributed interval observer is first constructed for uncertain networked systems in which the control algorithm of each agent involves only the upper bound information and the lower bound information of the interval observer associated with itself and its neighbors, respectively. With the help of the cooperativity theory, it is proved that the interval observer can estimate the piecewise state for each agent and the interval-observer-based control algorithm can drive the uncertain system to achieve coordination behavior. Then, time-varying coordinate transformation is introduced to construct a novel interval observer which can eliminate the cooperativity premise on the system matrices and bound the states of all agents in real time. It is shown that the novel interval-observer-based control algorithm can guide the uncertain system to reach coordinated behavior. Finally, the numerical simulations are provided to verify the theoretical results.

Superconvergence of $$C^0$$-$$Q^k$$ Finite Element Method for Elliptic Equations with Approximated Coefficients

Abstract: We prove that the superconvergence of $$C^0$$-$$Q^k$$ finite element method at the Gauss–Lobatto quadrature points still holds if variable coefficients in an elliptic problem are replaced by their piecewise $$Q^k$$ Lagrange interpolants at the Gauss–Lobatto points in each rectangular cell. In particular, a fourth order finite difference type scheme can be constructed using $$C^0$$-$$Q^2$$ finite element method with $$Q^2$$ approximated coefficients.

A Novel Piecewise Affine Filtering Design for T–S Fuzzy Affine Systems Using Past Output Measurements

Abstract: This paper tackles the problem of piecewise affine memory filtering design for the discrete-time norm-bounded uncertain Takagi–Sugeno fuzzy affine systems. The objective is to design an admissible filter using past output measurements of the system, guaranteeing the asymptotic stability of the filtering error system with a given ${\mathscr H}_{\infty }$ performance index. Based on the piecewise fuzzy Lyapunov functions and the projection lemma, a new sufficient condition for $\mathscr H_{\infty }$ filtering performance analysis is first derived, and then the filter synthesis is carried out. It is shown that the filter gains can be obtained by solving a set of linear matrix inequalities. In addition, it is also shown that the filtering performance can be improved with the increasing number of past output measurements used in the filtering design. Finally, two examples are presented to show the advantages and effectiveness of the proposed approach.

Polynomial Augmented Extended Kalman Filter to Estimate the State of Charge of Lithium-Ion Batteries

Abstract: In battery-electric vehicles, an accurate knowledge of the current state of charge (SOC) of the battery is crucial for safe and efficient operation. Offset-free SOC estimation for Lithium-ion batteries (LIB) during operation still is a challenging problem, due to the uncertain output nonlinearity present in battery models. In battery management systems employed in such vehicles, the age- and cycle-dependent relationship between the open circuit voltage (OCV) and the state of charge of each cell (SOC) can be kept track of using lookup tables. Between the scattered data points, linear interpolation is often used. Another common approach is to switch between models, each with a different piecewise polynomial fit of low order. In this contribution, a single polynomial fit of a higher order with state-dependent coefficients for the whole SOC-OCV range is proposed, so as to avoid switching and facilitate usage of Taylor-based linearization algorithms like in the extended Kalman filter. While the polynomial order is high, the number of parameters is only three; the parameters are estimated and updated by the Kalman filter itself. This augmentation of the observer's state vector with said polynomial parameters is inspired by the idea of an increased “stochastization,” introducing redundancy that aids to achieve a more accurate SOC estimation. In this sense, the algorithm can be described as a model-adaptive extended Kalman estimator. Two variants of a polynomial EKF are investigated: polynomial EKF with output and with state nonlinearity models. Their performances are compared in real experiments using a dedicated test bench based on a Samsung ICR18650-26F cell, demonstrating the effectiveness of the algorithms. The variant that includes the terminal voltage in the state vector, i.e. the state nonlinearity variant of the EKF, shows better result, also compared to the existing literature.

Non-convex Total Variation Regularization for Convex Denoising of Signals

Abstract: Total variation (TV) signal denoising is a popular nonlinear filtering method to estimate piecewise constant signals corrupted by additive white Gaussian noise. Following a ‘convex non-convex’ strategy, recent papers have introduced non-convex regularizers for signal denoising that preserve the convexity of the cost function to be minimized. In this paper, we propose a non-convex TV regularizer, defined using concepts from convex analysis, that unifies, generalizes, and improves upon these regularizers. In particular, we use the generalized Moreau envelope which, unlike the usual Moreau envelope, incorporates a matrix parameter. We describe a novel approach to set the matrix parameter which is essential for realizing the improvement we demonstrate. Additionally, we describe a new set of algorithms for non-convex TV denoising that elucidate the relationship among them and which build upon fast exact algorithms for classical TV denoising.

Piecewise linearity, freedom from self-interaction, and a Coulomb asymptotic potential: three related yet inequivalent properties of the exact density functional

Abstract: The exact energy functional of density functional theory (DFT) is well known to obey various constraints. Three conditions that must be obeyed by the exact energy functional, but may or may not be obeyed by approximate ones, are often pointed out as important in general and for accurate computation of spectroscopic observables in particular. These are: (1) piecewise linearity as a function of the fractional particle number, (2) freedom from one-electron self-interaction, and (3) for a finite system, the functional derivative with respect to the density results in an asymptotic -1/r potential (in Hartree atomic units), where r is the distance from the system center. In this overview, we explain what these conditions are, what they address, and why each one is of importance for spectroscopy. We then show, using specific examples from the literature, that these three properties are related, but are not equivalent and need to be assessed individually.

Topology optimization with local stress constraints: a stress aggregation-free approach

Abstract: This paper presents a consistent topology optimization formulation for mass minimization with local stress constraints by means of the augmented Lagrangian method. To solve problems with a large number of constraints in an effective way, we modify both the penalty and objective function terms of the augmented Lagrangian function. The modification of the penalty term leads to consistent solutions under mesh refinement and that of the objective function term drives the mass minimization towards black and white solutions. In addition, we introduce a piecewise vanishing constraint, which leads to results that outperform those obtained using relaxed stress constraints. Although maintaining the local nature of stress requires a large number of stress constraints, the formulation presented here requires only one adjoint vector, which results in an efficient sensitivity evaluation. Several 2D and 3D topology optimization problems, each with a large number of local stress constraints, are provided.

Parametrized equation of state for neutron star matter with continuous sound speed

Abstract: We present a generalized piecewise polytropic parametrization for the neutron-star equation of state using an ansatz that imposes continuity in not only pressure and energy density, but also in the speed of sound. The universe of candidate equations of state is shown to admit preferred dividing densities, determined by minimizing an error norm consisting of integral astrophysical observables. Generalized piecewise polytropes accurately reproduce astrophysical observables, such as mass, radius, tidal deformability and mode frequencies, as well as thermodynamic quantities, such as the adiabatic index. This makes the new equations of state useful for parameter estimation from gravitational waveforms. Since they are differentiable, generalized piecewise polytropes can improve pointwise convergence in numerical relativity simulations of neutron stars. Existing implementations of piecewise polytropes can easily accommodate this generalization with the same number of free parameters. Optionally, generalized piecewise polytropes can also accommodate adjustable jumps in sound speed, which allows them to capture phase transitions in neutron star matter.