Piecewise Affine Research Articles

Liner fleet deployment and empty container repositioning under demand uncertainty: A robust optimization approach

This paper investigates a robust optimization problem concerning the integration of fleet deployment and empty container repositioning in a shipping line network, where a fleet of vessels is dispatched to transport both laden and empty containers, aiming to fulfill a predetermined set of requests over a defined time horizon. The sizes of customer demands are uncertain and are characterized by a budgeted uncertainty set. This study aims to ascertain the vessel types assigned to each shipping route, the routing of laden containers, and the repositioning of empty containers in a manner that minimizes the total cost. Simultaneously, it ensures the feasibility of all transportation plans for any realization of demand within the uncertainty set. We introduce a path-based two-stage robust formulation for addressing the problem. In the first stage, the assignment of vessel types to each shipping route is determined, and the second stage focuses on establishing the routing of laden containers and repositioning of empty containers under a worst-case scenario. We propose the Column-and-Constraint Generation algorithm for solving the proposed robust formulation. To address large-scale size instances, we propose an acceleration technique, i.e., the piece-wise affine policy, which reduces the dimensions of the uncertainty set while maintaining a bounded compromise in solution quality. Comprehensive numerical experiments derived from real-world industries, such as the Shanghai port and CMA CGM, are conducted to validate the proposed formulation and solution methodologies.

Approximation of SBV functions with possibly infinite jump set

We prove an approximation result for functions u∈SBV(Ω;Rm) such that ∇u is p-integrable, 1≤p<∞, and g0(|[u]|) is integrable over the jump set (whose Hn−1 measure is possibly infinite), for some continuous, nondecreasing, subadditive function g0, with g0−1(0)={0}. The approximating functions uj are piecewise affine with piecewise affine jump set; the convergence is that of L1 for uj and the convergence in energy for |∇uj|p and g([uj],νuj) for suitable functions g. In particular, uj converges to u BV-strictly, area-strictly, and strongly in BV after composition with a bilipschitz map. If in addition Hn−1(Ju)<∞, we also have convergence of Hn−1(Juj) to Hn−1(Ju).

Invariant set estimation for piecewise affine dynamical systems using piecewise affine barrier function

This paper introduces an algorithm for estimating the invariant set of closed-loop controlled dynamical systems identified using single-hidden layer Rectified linear units (ReLU) neural networks or piecewise affine (PWA) functions, particularly addressing the challenge of providing safety guarantees for single-hidden layer ReLU networks commonly used in safety–critical applications. The invariant set of PWA dynamical system is estimated using single-hidden layer ReLU networks or its equivalent PWA function. This method entails formulating the barrier function as a PWA function and converting the search process into a linear optimization problem using vertices. We incorporate a domain refinement strategy to increase flexibility in case the optimization does not find a valid barrier function. Moreover, the objective of the optimization is to find a less conservative invariant set based on the current partition. Our experimental results demonstrate the effectiveness and efficiency of our approach, demonstrating its potential for ensuring the safety of PWA dynamical systems.

A Computationally Efficient Trajectory Prediction in MPC for Piecewise Affine Systems

A Computationally Efficient Trajectory Prediction in MPC for Piecewise Affine Systems

Bifurcation analysis of a two-neuron central pattern generator model for both oscillatory and convergent neuronal activities.

The neural oscillator model proposed by Matsuoka is a piecewise affine system that exhibits distinctive periodic solutions. Although such typical oscillation patterns have been widely studied, little is understood about the dynamics of convergence to certain fixed points and bifurcations between the periodic orbits and fixed points in this model. We performed fixed point analysis on a two-neuron version of the Matsuoka oscillator model, the result of which explains the mechanism of oscillation and the discontinuity-induced bifurcations such as subcritical/supercritical Hopf-like, homoclinic-like and grazing bifurcations. Furthermore, it provided theoretical predictions concerning a logarithmic oscillation-period scaling law and noise-induced oscillations observed around those bifurcations. These results are expected to underpin further investigations into oscillatory and transient neuronal activities concerning central pattern generators.

Piecewise affine power flow model for distribution network optimization using CIM aided data-driven approach

Power flow (PF) analysis serves as the basis for power system operation that is extended to the distribution levels integrated with more distributed generations (DG). Solving optimal power flow (OPF)-resembled problems for practical networks is still challenging due to unavailability of accurate PF equations adaptive to systems variations. To solve this issue, we first set up the PF simulations for complex and realistic power networks based on the common information model (CIM). This automates the recognition of network topologies and parameters in the reduced bus-branch form for system simulations and data generations. The non-convexity of OPF imposes an additional challenge, where solving the problem is still difficult for scenarios with intertemporal couplings considering energy storage. We further employ a multi-layer neural network that provides piecewise affine approximations of the PF equations with desired accuracy. The corresponding OPF-resembled problem is then reformulated as a mixed-integer optimization that facilitates fast and optimal coordination of large-scale DGs for various network operational purposes. We demonstrate the proposed framework using real CIM files from system operators, and compare against typical approaches with improved performance for voltage regulation and loss minimization.

Piecewise nonlinear approximation for non-smooth functions

Piecewise affine or linear approximation has garnered significant attention as a technique for approximating piecewise-smooth functions. In this study, we propose a novel approach: piecewise non-linear approximation based on rational approximation, aimed at approximating non-smooth functions. We introduce a method termed piecewise Padé Chebyshev (PiPC) tailored for approximating univariate piecewise smooth functions. Our investigation focuses on assessing the effectiveness of PiPC in mitigating the Gibbs phenomenon during the approximation of piecewise smooth functions. Additionally, we provide error estimates and convergence results of PiPC for non-smooth functions. Notably, our technique excels in capturing singularities, if present, within the function with minimal Gibbs oscillations, without necessitating the explicit specification of singularity locations. To the best of our knowledge, prior research has not explored the use of piecewise non-linear approximation for approximating non-smooth functions. Finally, we validate the efficacy of our methods through numerical experiments, employing PiPC to reconstruct a non-trivial non-smooth function, thus demonstrating its capability to significantly alleviate the Gibbs phenomenon.

Intelligent cooperative control for flexible landing on small celestial bodies

In view of the potential landing risks in small celestial body landing missions, the concept of flexible landing has been proposed in recent years. By employing a flexible lander with structural compliance to complex surface topography, the risks of rebound or overturning upon touchdown can be reduced. During flexible landing, the overall motion of the flexible lander is controlled by several nodes embedded in the flexible structure. Due to the flexible deformation, the motion among the nodes is intricately coupled, posing challenges to the control algorithm design. To address this problem, an intelligent cooperative control method for flexible landing is proposed in this paper. By decomposing the flexible internal force into a low-order principal term with specific physical interpretation and an unmodeled high-order residual term, the coupling relationship among the nodes is characterized. On this basis, the low-order dynamics system is reconstructed into a piecewise affine system, upon which a multi-node cooperative optimal controller is designed. A long-short-term memory recurrent neural network is further established to compensate for the unmodeled high-order term. Through the combination of the low-order principal control and the high-order intelligent compensation, a feasible approach for achieving flexible landing is obtained. Finally, the effectiveness of the proposed control method is verified via 433 Eros-based flexible landing simulations.

Sliding Homoclinic Bifurcations in a Class of Three-Dimensional Piecewise Affine Systems

This paper studies sliding homoclinic bifurcations in a class of symmetric three-zone three-dimensional piecewise affine systems. The systems have one parameter and the unperturbed systems have a pair of sliding homoclinic orbits to a saddle. Based on the analysis of the one-dimensional Poincaré maps, two types of sliding cycles are obtained from the sliding homoclinic bifurcations of the systems. In addition, two examples of sliding homoclinic orbits and sliding cycles are provided with simulations to illustrate the effectiveness of the theorems.

Identification of Hammerstein models with hysteresis nonlinearity

This paper deals with the identification of Hammerstein nonlinear models. The system nonlinearity is not necessarily static and can be of hysteresis type. The latter nonlinearity can be backlash, switch or any other memory operator having several transient cycles, but one permanent (major) cycle. The LTI (Linear Time-Invariant) block is nonparametric and is described by its impulse response. The considered system’s nonlinearity is any memory operator having one major cycle (periodic closed loop), which is bordered by any two arbitrary functions. The nonlinearity borders are not assumed to be invertible. The proposed identification algorithm can be used in static nonlinearity, e.g. polynomial nonlinearity, dead zone function or any piecewise affine functions. Unlike several previous studies, the proposed identification method is carried out using only one stage. In this approach, an input signal of piecewise constant (staircase) is used. It is shown that the estimated parameters converge to their true values.

Coexistence of singular cycles in a class of three-dimensional piecewise affine systems

Coexistence of singular cycles in a class of three-dimensional piecewise affine systems

Quantized output-feedback control of piecewise-affine systems with reachable regions of quantized measurements

This article explores the asymptotic stability and output-feedback control issues of discrete-time piecewise-affine (PWA) systems under multi-input–multi-output (MIMO) quantization. Our work features more general and practical than those previous works focusing on single-input–single-output quantization for output-feedback control of PWA systems. We choose a PWA output-feedback controller adapting to the located region of the quantized measurement. Since the measurement outputs generally contain partial state information, the determination of the located region of the quantized measurement depends on the region where the system state dwells. Thus, the controller can operate within a reduced number of regions compared to the PWA system. The leveraged Lyapunov function incorporates the region information for both the state and the quantized measurement, as well as the MIMO quantization uncertainties. We propose a method for determining the reachable regions of the quantized measurement, based on which we conduct the stability analysis and output-feedback controller implementation subsequently. In consequence, both the conservativeness and the computational burden can be alleviated compared with the previous results that incorporated all the operating regions for output-feedback controller design. Finally, our proposed control methodology is tested via a water-tank control problem to validate its competence and strengths.

End-to-end multiview fusion for building mapping from aerial images

In the domain of photogrammetry, the fusion of information from multiple views holds the potential to significantly enhance the accuracy and robustness of building mapping. While multiview observation and stereoscopic imaging form the bedrock of photogrammetric projects, current deep learning methodologies predominantly focus on orthophotos and digital surface models (DSMs), often sidelining the rich multiview information inherent in original images. Addressing this gap, we present Multiview Mapper (MVMapper), an end-to-end learning framework explicitly crafted to harness and fuse the rich semantic information from original multiview images with object-space features. MVMapper uses stereo labels for supervised building segmentation in a dual-space, encompassing both image and object domains. Additionally, it incorporates a novel piecewise affine projection method for ensuring a robust image-to-object feature transformation. Experimental results on an aerial photogrammetric dataset with a resolution of 30 cm demonstrate MVMapper’s superiority over state-of-the-art multiview data fusion methods, yielding significant improvements in segmentation and contour accuracy. Notably, the proposed piecewise affine projection method mitigates misalignment issues caused by DSM noise, enabling the effective fusion of multiview features with object-space features. Further experimentation on a separate open-source dataset demonstrates MVMapper’s substantial advantages in transferability to other contexts.

Approximate propagation of normal distributions for stochastic optimal control of nonsmooth systems

We present a method for the approximate propagation of mean and covariance of a probability distribution through ordinary differential equations (ODE) with discontinuous right-hand side. For piecewise affine systems, a normalization of the propagated probability distribution at every time step allows us to analytically compute the expectation integrals of the mean and covariance dynamics while explicitly taking into account the discontinuity. This leads to a natural smoothing of the discontinuity such that for relevant levels of uncertainty the resulting ODE can be integrated directly with standard schemes and it is neither necessary to prespecify the switching sequence nor to use a switch detection method. We then show how this result can be employed in the more general case of piecewise smooth functions based on a structure preserving linearization scheme. The resulting dynamics can be straightforwardly used within standard formulations of stochastic optimal control problems with chance constraints.

Prediction of PM2.5 with a piecewise affine model considering spatial-temporal correlation

Over the past several decades, several air pollution prevention measures have been developed in response to the growing concern over air pollution. Using models to anticipate air pollution accurately aids in the timely prevention and management of air pollution. However, the spatial-temporal air quality aspects were not properly taken into account during the prior model construction. In this study, the distance correlation coefficient (DC) between measurements made in various monitoring stations is used to identify appropriate correlated monitoring stations. To derive spatial-temporal correlations for modeling, the causality relationship between measurements made in various monitoring stations is analyzed using Transfer Entropy (TE). This work explores the process of identifying a piecewise affine (PWA) model using a larger dataset and suggests a unique hierarchical clustering-based identification technique with model structure selection. This work improves the BIRCH (Balanced Iterative Reducing and Clustering using Hierarchies) by introducing Kullback-Leibler (KL) Divergence as the dissimilarity between clusters for handling clusters with arbitrary shapes. The number of clusters is automatically determined using a cluster validity metric. The task is formulated as a sparse optimization problem, and the model structure is selected using parameter estimations. Beijing air quality data is used to demonstrate the method, and the results show that the proposed strategy may produce acceptable forecast performance.

Singular limit of periodic metric grids

We investigate the asymptotic behavior of nonlinear Schrödinger ground states on d-dimensional periodic metric grids in the limit for the length of the edges going to zero. We prove that suitable piecewise–affine extensions of such states converge strongly in H1(Rd) to the corresponding ground states on Rd. As an application of such convergence results, qualitative properties of ground states and multiplicity results for fixed mass critical points of the energy on grids are derived. Moreover, we compare optimal constants in d-dimensional Gagliardo–Nirenberg inequalities on Rd and on grids. For L2-critical and supercritical powers, we show that the value of such constants on grids is strictly related to that on Rd but, contrary to Rd, constants on grids are not attained. The proofs of these results combine purely variational arguments with new Gagliardo–Nirenberg inequalities on grids.

Stability and Tracking Recovery of Continuous-Time Markov Jump Piecewise Affine Systems Using Virtual-Sensor-Based Reconfiguration

Stability and Tracking Recovery of Continuous-Time Markov Jump Piecewise Affine Systems Using Virtual-Sensor-Based Reconfiguration

Discrete Helmholtz Decompositions of Piecewise Constant and Piecewise Affine Vector and Tensor Fields

AbstractDiscrete Helmholtz decompositions dissect piecewise polynomial vector fields on simplicial meshes into piecewise gradients and rotations of finite element functions. This paper concisely reviews established results from the literature which all restrict to the lowest-order case of piecewise constants. Its main contribution consists of the generalization of these decompositions to 3D and of novel decompositions for piecewise affine vector fields in terms of Fortin–Soulie functions. While the classical lowest-order decompositions include one conforming and one nonconforming part, the decompositions of piecewise affine vector fields require a nonconforming enrichment in both parts. The presentation covers two and three spatial dimensions as well as generalizations to deviatoric tensor fields in the context of the Stokes equations and symmetric tensor fields for the linear elasticity and fourth-order problems. While the proofs focus on contractible domains, generalizations to multiply connected domains and domains with non-connected boundary are discussed as well.

Extension of a clustering identification approach to multivariable piecewise affine systems: Application to an industrial dryer

The main objective of this paper is the extension of the clustering-based identification approach to Multi-Input Multi-Output (MIMO) PieceWise Affine systems (PWA). This approach is performed by three main steps which are data clustering, parameters matrices estimation and regions computing. Data clustering is the most important step because the performances depend on the results given by the used clustering algorithm. In the case of MIMO PWA systems, we should cluster matrices of parameters which are considered high dimensional data. However, most of the conventional clustering algorithms are not efficient since the similarity assessment which is based on the distances between objects is fruitless in high dimension space. Therefore, we propose an extension of the DBSCAN (Density Based Spatial Clustering of Applications with Noise) clustering technique to identify MIMO PWA systems. The simulation results presented in this paper illustrate the performance of the proposed method. An application to an industrial dryer of Di-Calcium Phosphate (DCP) is also presented in order to strengthen the simulation results.

Designing tractable piecewise affine policies for multi-stage adjustable robust optimization

AbstractWe study piecewise affine policies for multi-stage adjustable robust optimization (ARO) problems with non-negative right-hand side uncertainty. First, we construct new dominating uncertainty sets and show how a multi-stage ARO problem can be solved efficiently with a linear program when uncertainty is replaced by these new sets. We then demonstrate how solutions for this alternative problem can be transformed into solutions for the original problem. By carefully choosing the dominating sets, we prove strong approximation bounds for our policies and extend many previously best-known bounds for the two-staged problem variant to its multi-stage counterpart. Moreover, the new bounds are—to the best of our knowledge—the first bounds shown for the general multi-stage ARO problem considered. We extensively compare our policies to other policies from the literature and prove relative performance guarantees. In two numerical experiments, we identify beneficial and disadvantageous properties for different policies and present effective adjustments to tackle the most critical disadvantages of our policies. Overall, the experiments show that our piecewise affine policies can be computed by orders of magnitude faster than affine policies, while often yielding comparable or even better results.