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.

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.

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.

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.

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.

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.

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...

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.

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...

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.

Abstract: This article investigates the H∞ stochastic tracking control problem for uncertain fuzzy Markovian hybrid switching systems by using a fuzzy switching dynamic adaptive control approach. The long and the short is to construct multiple piecewise stochastic Lyapunov functions which provide an effective tool for designing hybrid switching law and fuzzy switching dynamic adaptive law. A hybrid switching law, including both stochastic switching and deterministic switching, is designed to represent more general switching scenarios, which can improve the H∞ adaptive tracking performance through offering a running time before stochastic switching for the adaptive control strategy to work well. A fuzzy switching dynamic adaptive control technique is developed such that all signals of the tracking error equation are bounded, and the system state trajectory tracks the reference model state trajectory under a disturbance attenuation level as closely as possible. Finally, an application study verifies the effectiveness of the acquired methods.

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.

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.

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.

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.

Abstract: This article aims to systematically address the leader-following consensus issue for continuous-time multiagent systems with semi-Markov jump parameters. A more universal random process, semi-Markov process, is utilized to model the variations of parameters which may result from the complexity of environment. The innovative hybrid event-triggered strategy associated with an improved threshold function is employed for economizing the network bandwidth resources. It presents an exponential decay term in the threshold function for decreasing the transmission frequency of superfluous data packets, ulteriorly reducing triggering times. The dominating core concentrates on the design of a controller interrelated with the hybrid event-triggered condition, which can be implemented to ensure the leader-following consensus of the considered systems. On the strength of time-dependent and piecewise Lyapunov–Krasovskii functional, several criteria, which are capable of ensuring the leader-following consensus of the multiagent systems, are derived. Ultimately, two suitable numerical examples and a meaningful practical one are utilized to demonstrate the validity and potential of the derived results.

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.

Abstract: Assuming stationarity is unrealistic in many time series applications. A more realistic alternative is to assume piecewise stationarity, where the model can change at potentially many change points...

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.

Abstract: Referring to continuous-time chaotic dynamical systems, this paper investigates the inverse full state hybrid function projective synchronization (IFSHFPS) of non-identical systems characterized by different dimensions. By taking a master system of dimension n and a slave system of dimension m, the method enables each master system state to be synchronized with a linear combination of slave system states, where the scaling factor of the linear combination can be any arbitrary differentiable function. The approach, based on the Lyapunov stability theory and stability of linear continuous-time systems, presents some useful features: (i) it enables non-identical chaotic systems with different dimension $$nm$$ to be synchronized; (ii) it can be applied to a wide class of chaotic (hyperchaotic) systems for any differentiable scaling function; (iii) it is rigorous, being based on two theorems, one for the case $$nm$$ . Two different numerical examples are reported. The examples clearly highlight the capability of the conceived approach in effectively achieving synchronized dynamics for any differentiable scaling function.

Abstract: We present large sample results for partitioning-based least squares nonparametric regression, a popular method for approximating conditional expectation functions in statistics, econometrics and machine learning. First, we obtain a general characterization of their leading asymptotic bias. Second, we establish integrated mean squared error approximations for the point estimator and propose feasible tuning parameter selection. Third, we develop pointwise inference methods based on undersmoothing and robust bias correction. Fourth, employing different coupling approaches, we develop uniform distributional approximations for the undersmoothed and robust bias-corrected $t$-statistic processes and construct valid confidence bands. In the univariate case, our uniform distributional approximations require seemingly minimal rate restrictions and improve on approximation rates known in the literature. Finally, we apply our general results to three partitioning-based estimators: splines, wavelets and piecewise polynomials. The Supplemental Appendix includes several other general and example-specific technical and methodological results. A companion $\mathsf{R}$ package is provided.

Abstract: We introduce an incremental learning method for the optimal construction of rule-based granular systems from numerical data streams. The method is developed within a multiobjective optimization framework considering the specificity of information, model compactness, and variability and granular coverage of the data. We use $\alpha$ -level sets over Gaussian membership functions to set model granularity and operate with hyperrectangular forms of granules in nonstationary environments. The resulting rule-based systems are formed in a formal and systematic fashion. They can be useful in time series modeling, dynamic system identification, predictive analytics, and adaptive control. Precise estimates and enclosures are given by linear piecewise and inclusion functions related to optimal granular mappings.

Abstract: We study a family of $H^m$-conforming piecewise polynomials based on artificial neural network, named as the finite neuron method (FNM), for numerical solution of $2m$-th order partial differential equations in $\mathbb{R}^d$ for any $m,d \geq 1$ and then provide convergence analysis for this method. Given a general domain $\Omega\subset\mathbb R^d$ and a partition $\mathcal T_h$ of $\Omega$, it is still an open problem in general how to construct conforming finite element subspace of $H^m(\Omega)$ that have adequate approximation properties. By using techniques from artificial neural networks, we construct a family of $H^m$-conforming set of functions consisting of piecewise polynomials of degree $k$ for any $k\ge m$ and we further obtain the error estimate when they are applied to solve elliptic boundary value problem of any order in any dimension. For example, the following error estimates between the exact solution $u$ and finite neuron approximation $u_N$ are obtained. $$ \|u-u_N\|_{H^m(\Omega)}=\mathcal O(N^{-{1\over 2}-{1\over d}}). $$ Discussions will also be given on the difference and relationship between the finite neuron method and finite element methods (FEM). For example, for finite neuron method, the underlying finite element grids are not given a priori and the discrete solution can only be obtained by solving a non-linear and non-convex optimization problem. Despite of many desirable theoretical properties of the finite neuron method analyzed in the paper, its practical value is a subject of further investigation since the aforementioned underlying non-linear and non-convex optimization problem can be expensive and challenging to solve. For completeness and also convenience to readers, some basic known results and their proofs are also included in this manuscript.

Abstract: Two interpolation methods are presented, both of which use multiple Piecewise Cubic Hermite Interpolating Polynomials (PCHIPs). The first method is based on performing 16 PCHIPs on 8 rotate...