Piecewise Affine Function Research Articles

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.

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.

Template-based piecewise affine regression

Abstract We study the problem of fitting a piecewise affine (PWA) function to input–output data. Our algorithm divides the input domain into finitely many regions whose shapes are specified by a user-provided template and such that the input–output data in each region are fit by an affine function within a user-provided error tolerance. We first prove that this problem is NP-hard. Then, we present a top-down algorithmic approach for solving the problem. The algorithm considers subsets of the data points in a systematic manner, trying to fit an affine function for each subset using linear regression. If regression fails on a subset, the algorithm extracts a minimal set of points from the subset (an unsatisfiable core) that is responsible for the failure. The identified core is then used to split the current subset into smaller ones. By combining this top-down scheme with a set-covering algorithm, we derive an overall approach that provides optimal PWA models for a given error tolerance, where optimality refers to minimizing the number of pieces of the PWA model. We demonstrate our approach on three numerical examples that include PWA approximations of a widely used nonlinear insulin–glucose regulation model and a double inverted pendulum with soft contacts.

A two-point heuristic to calculate the stepsize in subgradient method with application to a network design problem

We introduce a heuristic rule for calculating the stepsize in the subgradient method for unconstrained convex nonsmooth optimization which, unlike the classic approach, is based on retaining some information from previous iteration. The rule is inspired by the well known two-point stepsize by Barzilai and Borwein (BB) [6] for smooth optimization and it coincides with (BB) in case the function to be minimised is convex quadratic.Under the use of appropriate safeguards we demonstrate that the method terminates at a point that satisfies an approximate optimality condition.The proposed approach is tested in the framework of Lagrangian relaxation for integer linear programming where the Lagrangian dual requires maximization of a concave and nonsmooth (piecewise affine) function. In particular we focus on the relaxation of the Minimum Spanning Tree problem with Conflicting Edge Pairs (MSTC). Comparison with classic subgradient method is presented. The results on some widely used academic test problems are provided too.

Mixed Integer Linear Programming Based Speed Profile Optimization for Heavy-Haul Trains

Automatic heavy-haul train (HHT) operation technology has recently received considerable attention in the field of rail transportation. In this paper, a discrete-time-based mathematical formulation is proposed to address the speed profile optimization problem in order to ensure the safe, efficient, and economical operation of heavy-haul trains (HHTs). Due to the presence of long and steep downgrades (LSDs) on some heavy-haul lines, the brake forces of the HHT are typically jointly determined by air braking and electric braking. The time characteristics of the air braking, such as the command delay and the change process caused by the air pressure, are taken into account, and then formulas are presented to calculate the air brake force. In addition, the influence of the neutral section on the control of the electric braking is considered via space-based constraints. The resulting problem is a nonlinear optimal control problem. To achieve linearization, auxiliary 0-1 binary variables and the big-M approach are introduced to transform the nonlinear constraints regarding slope, curve, neutral section, air brake force, and air-filled time into linear constraints. Moreover, piecewise affine (PWA) functions are used to approximate the basic resistance of the HHT. Finally, a mixed integer linear programming (MILP) model is developed, which can be solved by CPLEX. The experiments are carried out using data from a heavy-haul railway line in China, and the results show that the proposed approach is effective and flexible.

Piecewise Affine Maximum Torque per Ampere for the Wound Rotor Synchronous Machine

Power-efficient torque control of the Wound Rotor Synchronous Machine (WRSM) below base speed requires a minimization of electrical losses, namely copper losses. In this paper, the Maximum Torque per Ampere (MTPA) optimization problem is presented and solved using a convex pareto frontier of simulated or measured data points. Three filtered solution sets are mapped to current space using piecewise affine functions, which approximate the current using a piecewise linear function for a given torque. This set of piecewise linear functions enables a machine controller to implement MTPA online or as an offline lookup table. Simulated and experimental results are presented for a 65 kW, FEA-sampled WRSM. Compared to linear MTPA, the PWA MTPA functions are shown to reduce torque error by > 25%, reduce average copper loss by > 20%.

Nonlocal Error Bounds for Piecewise Affine Functions

Nonlocal Error Bounds for Piecewise Affine Functions

Bundle Enrichment Method for Nonsmooth Difference of Convex Programming Problems

The Bundle Enrichment Method (BEM-DC) is introduced for solving nonsmooth difference of convex (DC) programming problems. The novelty of the method consists of the dynamic management of the bundle. More specifically, a DC model, being the difference of two convex piecewise affine functions, is formulated. The (global) minimization of the model is tackled by solving a set of convex problems whose cardinality depends on the number of linearizations adopted to approximate the second DC component function. The new bundle management policy distributes the information coming from previous iterations to separately model the DC components of the objective function. Such a distribution is driven by the sign of linearization errors. If the displacement suggested by the model minimization provides no sufficient decrease of the objective function, then the temporary enrichment of the cutting plane approximation of just the first DC component function takes place until either the termination of the algorithm is certified or a sufficient decrease is achieved. The convergence of the BEM-DC method is studied, and computational results on a set of academic test problems with nonsmooth DC objective functions are provided.

Extending the network calculus algorithmic toolbox for ultimately pseudo-periodic functions: pseudo-inverse and composition

Network Calculus (NC) is an algebraic theory that represents traffic and service guarantees as curves in a Cartesian plane, in order to compute performance guarantees for flows traversing a network. NC uses transformation operations, e.g., min-plus convolution of two curves, to model how the traffic profile changes with the traversal of network nodes. Such operations, while mathematically well-defined, can quickly become unmanageable to compute using simple pen and paper for any non-trivial case, hence the need for algorithmic descriptions. Previous work identified the class of piecewise affine functions which are ultimately pseudo-periodic (UPP) as being closed under the main NC operations and able to be described finitely. Algorithms that embody NC operations taking as operands UPP curves have been defined and proved correct, thus enabling software implementations of these operations. However, recent advancements in NC make use of operations, namely the lower pseudo-inverse, upper pseudo-inverse, and composition, that are well-defined from an algebraic standpoint, but whose algorithmic aspects have not been addressed yet. In this paper, we introduce algorithms for the above operations when operands are UPP curves, thus extending the available algorithmic toolbox for NC. We discuss the algorithmic properties of these operations, providing formal proofs of correctness.

Piecewise Affine Modeling of Wound-Rotor Synchronous Machines for Real-Time Motor Control

High-performance control of Wound Rotor Synchronous Machines (WRSM) motor drives requires a nonlinear magnetic model that captures magnetic saturation and cross saturation. This research proposes a generalized Piece-wise Affine (PWA) magnetic and state-space model that divides the current and flux space into simplices. PWA functions are obtained from experimental or Finite Element Analysis (FEA) data leveraging the Delaunay triangulation. Error margins and memory requirements can be optimized for given applications by varying the number of PWA functions. The PWA domains can be irregular and an algorithm to optimize accuracy for a given number of PWA domains is proposed. The PWA performance is compared with a high-fidelity magnetic model. Irregular PWA functions have lower error than regularly gridded tables at reduced size: average error is improved from 15% to 1%; peak error is improved from 20% to 9%. Finally, the PWA state-space model is validated on an experimental testbench with a real-time Micro-Controller Unit (MCU). The state space equations are verified using steady-state and dynamic tests.

Towards rigorous understanding of neural networks via semantics-preserving transformations

AbstractIn this paper, we present an algebraic approach to the precise and global verification and explanation of Rectifier Neural Networks, a subclass of Piece-wise Linear Neural Networks (PLNNs), i.e., networks that semantically represent piece-wise affine functions. Key to our approach is the symbolic execution of these networks that allows the construction of semantically equivalent Typed Affine Decision Structures (TADS). Due to their deterministic and sequential nature, TADS can, similarly to decision trees, be considered as white-box models and therefore as precise solutions to the model and outcome explanation problem. TADS are linear algebras, which allows one to elegantly compare Rectifier Networks for equivalence or similarity, both with precise diagnostic information in case of failure, and to characterize their classification potential by precisely characterizing the set of inputs that are specifically classified, or the set of inputs where two network-based classifiers differ. All phenomena are illustrated along a detailed discussion of a minimal, illustrative example: the continuous XOR function.

A stochastic adaptive dynamics model for bacterial populations with mutation, dormancy and transfer

This paper introduces a stochastic adaptive dynamics model for the interplay of several crucial traits and mechanisms in bacterial evolution, namely dormancy, horizontal gene transfer (HGT), mutation and competition. In particular, it combines the recent model of Champagnat, Meleard and Tran (2021) involving HGT with the model for competition-induced dormancy of Blath and Tobias (2020). Our main result is a convergence theorem which describes the evolution of the different traits in the population on a `doubly logarithmic scale' as piece-wise affine functions. Interestingly, even for a relatively small trait space, the limiting process exhibits a non-monotone dependence of the success of the dormancy trait on the dormancy initiation probability. Further, the model establishes a new `approximate coexistence regime' for multiple traits that has not been observed in previous literature.

Signal Approximations Based on Nonlinear and Optimal Piecewise Affine Functions

Signal Approximations Based on Nonlinear and Optimal Piecewise Affine Functions

A decentralized feedback approach for flow control in highway traffic networks

Finite-time optimal feedback control for highway traffic networks under information constraints is studied. The primary objective is to design a fully decentralized state-feedback controller with ramp metering and variable speed limit control actions, with no information exchange between local controllers, that ensures feasibility and improves a global performance index. The design of a decentralized feedback controller with a one-hop information structure is examined, wherein the optimum outflow rate from each segment of the network depends only on the state of that segment and the state of all other segments directly connected to the downstream junction of that segment. It is demonstrated that when cost/constraints can be modeled or approximated by piecewise-affine functions, the control law has a closed-form state-feedback realization. The decentralization is based on the truncation of non-local variables in local optimization problems. The resulting decentralized control scheme has a simple closed-form representation and is scalable to arbitrary large networks; moreover, we demonstrate that, under certain conditions, with respect to certain meaningful performance indexes, the performance loss due to decentralization is zero; namely, the centralized optimal controller has a decentralized realization with a one-hop information structure.

Trajectory Optimization for High-Speed Trains via a Mixed Integer Linear Programming Approach

This paper proposes a trajectory optimization approach for high-speed trains to reduce traction energy consumption and increase riding comfort. Besides, the proposed approach can also achieve energy-saving effects by optimizing the operation time between stations. First, an optimization model is developed by defining the objective function as a trade-off function of the traction energy consumption and riding comfort. In addition to constraints in the classic optimal train control model, three new factors–the discrete throttle settings, neutral zones, and sectionalized tunnel resistance–are considered. Then, the model is discretized and turned into a multi-step decision optimization problem. All the nonlinear constraints are approximated using piecewise affine (PWA) functions, and the trajectory optimization problem is turned into a mixed integer linear programming (MILP) problem which can be solved by existing solvers CPLEX and YALMIP. Finally, some case studies with real-world data sets are conducted to present the effectiveness of the proposed approach. The simulation results are compared with the practical running data of trains, which shows that the proposed model and the optimization approach save energy and improve the riding comfort.

Finding the set of global minimizers of a piecewise affine function

Coexhausters are families of convex compact sets that allow one to represent the approximation of the increment of a function at a given point in the form of minmax or maxmin of affine functions. We demonstrate that this representation can be used to define a piecewise affine function and therefore coexhausters are a natural technique for studying the problem of finding a global minimum of piecewise affine functions. All the conditions and methods in the current study were obtained by means of coexhausters theory. Firstly, we apply coexhauster based conditions to state and prove necessary and sufficient conditions for a piecewise affine function to be bounded from below. Secondly, we use coexhausters to construct a simple method which allows one to get the minimum value of the studied function and the corresponding set of all its global minimizers. Illustrative numerical examples are provided throughout the paper.

Data-driven adjustable robust solution to voltage-regulation problem in PV-rich distribution systems

Adjustable robust counterpart (ARC) methodology has been recently utilised in distribution system operation to allow fast-acting devices, such as PV inverters, to take live recourse actions in response to operating point deviations. Available ARC models are used to obtain the controller parameters based on the worst-case realisations. However, we show that such a strategy might lead to over-conservative results when uncertainty realises away from the worst-case. To close this gap, this paper extends ARC to allow exploiting probabilistic information of uncertainty within the optimisation problem. This results in controller parameters that not only secure the worst-case but also work better in expectation. We formulate our approach as a linear program that minimises the real power curtailment (RPC) and reactive power usage of PV inverters while maintaining network voltages in an acceptable range. In contrast to the conventional ARC, our approach considers a segmented uncertainty set, and assigns a probability to each segment based on historical data. It then optimises to find inverter controller parameters in the form of piecewise affine functions, with each piece associated with one segment in the segmented uncertainty set. In live operation, the controllers use the real-time local measurements and assigned piecewise affine functions to generate real and reactive power set-points for the inverters. The performance of the proposed approach is validated using Monte-Carlo simulations on IEEE 37-bus and 906-bus networks. The simulations show that the proposed approach improves the performance of ARC by 30% to 60% over a variety of experiments.

QPALM: a proximal augmented lagrangian method for nonconvex quadratic programs

We propose QPALM, a nonconvex quadratic programming (QP) solver based on the proximal augmented Lagrangian method. This method solves a sequence of inner subproblems which can be enforced to be strongly convex and which therefore admit a unique solution. The resulting steps are shown to be equivalent to inexact proximal point iterations on the extended-real-valued cost function, which allows for a fairly simple analysis where convergence to a stationary point at an \(R\)-linear rate is shown. The QPALM algorithm solves the subproblems iteratively using semismooth Newton directions and an exact linesearch. The former can be computed efficiently in most iterations by making use of suitable factorization update routines, while the latter requires the zero of a monotone, one-dimensional, piecewise affine function. QPALM is implemented in open-source C code, with tailored linear algebra routines for the factorization in a self-written package LADEL. The resulting implementation is shown to be extremely robust in numerical simulations, solving all of the Maros-Meszaros problems and finding a stationary point for most of the nonconvex QPs in the Cutest test set. Furthermore, it is shown to be competitive against state-of-the-art convex QP solvers in typical QPs arising from application domains such as portfolio optimization and model predictive control. As such, QPALM strikes a unique balance between solving both easy and hard problems efficiently.

Coordinate sizing of energy storage and transmission line for a remote renewable power plant

To protect the environment and reduce carbon emissions, renewable power generation has been growing rapidly during the past decade. Renewable energy resources are sometimes far away from the main grid, leading to expensive grid-connection transmission lines. Deploying on-site energy storage can smooth the output power and help to reduce the renewable power spillage and the requirement of transmission line capacity. This paper presents a method to coordinately size on-site energy storage and grid-connection transmission line for a remote renewable power plant, minimising the total investment cost subject to the constraint of renewable curtailment risk. Through an optimal operation model, the renewable curtailment is proven to be a piecewise affine function of capacity parameters and renewable power generation, and a linear programming-based algorithm is proposed to generate an approximate expression. A distributionally robust optimisation model is proposed to determine the sizes. The renewable generation uncertainty is modelled by a Wasserstein-metric-based ambiguity set containing probability distributions around the empirical distribution constructed from the historical data. The utilisation rate is ensured in the worst-case distribution. The sizing problem is transformed into a tractable linear program. The case study demonstrates the effectiveness of the proposed method.

Fast computation of binary search tree for PWA functions representation using intersection classification

Explicit Model Predictive Control (eMPC) aims to overcome the runtime computational effort from Model Predictive Control (MPC) by computing offline the overall optimization procedure. The result from eMPC is a Piecewise Affine (PWA) function that defines a relationship between the system state and the optimal control action. To improve the PWA evaluation during runtime, Binary Search Trees (BST) are employed to represent PWA functions, which may be time prohibitive considering the total time to obtain the tree. This note presents a new classification strategy of polyhedral regions with respect to hyperplanes to build BST for PWA functions. The proposed solution is based on the verification of intersection between regions and hyperplanes, resulting in a decrease by almost half in the number of optimization problems, and consequently, in the total time to build the BST, which is the main time-consuming task. The computational time improvement is verified on numerical examples of different sizes, where the same BST is obtained for the original and the proposed classification methods.