Piecewise Affine Approximation Research Articles

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.

Convexified Open-Loop Stochastic Optimal Control for Linear Systems With Log-Concave Disturbances

We consider open-loop solutions to the stochastic optimal control of a linear dynamical system with an additive non-Gaussian, log-concave disturbance. We propose a novel, sampling-free approach, based on characteristic functions and convex optimization, to cast the stochastic optimal control problem as a difference-of-convex program. Our method invokes higher moments, resulting in less conservatism compared to moment-based approaches. We employ piecewise affine approximations and the convex-concave procedure for efficient solution via standard conic solvers. We demonstrate that the proposed solution is competitive with sampling and moment based approaches, without compromising probabilistic constraints.

Error Bounds for Compositions of Piecewise affine Approximations

Nonlinear expressions in dynamic models are often approximated by piecewise affine (PWA) functions to simplify analysis or reduce computational costs. Rather than directly approximating complicated multivariate functions in a high-dimensional space, these can instead be decomposed into a collection of simpler functions, which are then approximated individually and recomposed. This paper provides efficient methods to generate PWA approximations of nonlinear functions via functional decomposition. The key contributions focus on placing breakpoints for PWA approximations to satisfy a desired error tolerance and on bounding the error of PWA compositions as a function of the error tolerance for each component. The proposed methods are used to systematically construct a PWA approximation for a complicated function, either to within a desired error tolerance or to a given level of complexity in the approximation.

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.

Sensitivity Analysis for Piecewise-Affine Approximations of Nonlinear Programs With Polytopic Constraints

Sensitivity Analysis for Piecewise-Affine Approximations of Nonlinear Programs With Polytopic Constraints

Lattice piecewise affine approximation of explicit nonlinear model predictive control with application to trajectory tracking of mobile robot

AbstractTo promote the widespread use of mobile robots in diverse fields, the performance of trajectory tracking must be ensured. To address the constraints and nonlinear features associated with mobile robot systems, nonlinear model predictive control (MPC) is applied to realize trajectory tracking of mobile robots. Specifically, to alleviate the online computational complexity of nonlinear MPC, this paper devises a lattice piecewise affine (PWA) approximation method that can approximate both the nonlinear system and control law of explicit nonlinear MPC. The kinematic model of the mobile robot is successively linearized along the trajectory to obtain a linear time‐varying description of the system, which is then expressed using a lattice PWA model. Subsequently, the nonlinear MPC problem can be transformed into a series of linear MPC problems. Furthermore, to reduce the complexity of online calculation of multiple linear MPC problems, the optimal solution of the linear MPCs is approximated by using the lattice PWA models. That is, for different sampling states, the optimal control inputs are obtained offline, and lattice PWA approximations are constructed for the state control pairs. Simulations are performed to evaluate the performance of the method in comparison with the state‐of‐the‐art methods, and the results show that the method has a higher online computing speed and can decrease the offline computing time without significantly increasing the tracking error.

An adaptive time stepping scheme for rate-independent systems with nonconvex energy

Abstract We investigate a local incremental stationary scheme for the numerical solution of rate-independent systems. Such systems are characterized by a (possibly) nonconvex energy and a dissipation potential, which is positively homogeneous of degree one. Due to the nonconvexity of the energy, the system does in general not admit a time-continuous solution. In order to resolve these potential discontinuities, the algorithm produces a sequence of state variables and physical time points as functions of a curve parameter. The main novelty of our approach in comparison to existing methods is an adaptive choice of the step size for the update of the curve parameter, depending on a prescribed tolerance for the residua in the energy-dissipation balance, and in a complementarity relation concerning the so-called local stability condition. It is proven that, for tolerance tending to zero, the piecewise affine approximations generated by the algorithm converge (weakly) to a so-called parametrized balanced viscosity solution. Numerical experiments illustrate the theoretical findings, and show that an adaptive choice of the step size indeed pays off as they lead to a significant increase of the step size during sticking and in viscous jumps.

Parametric Piecewise-affine Approximation of Nonlinear Systems: A Cut-Based Approach

Piecewise-affine (PWA) approximations are widely used among hybrid modeling frameworks as a way to increase computational efficiency in nonlinear control and optimization problems. A variety of approaches to construct PWA approximations have been proposed, most of which are tailored to specific application areas by using some prior knowledge of the system in their assumptions and/or steps. In this paper, a parametric method is proposed to identify PWA approximations of nonlinear systems, without any prior knowledge of their dynamics or application requirements. The algorithm defines the regions parametrically using hyperplanes to cut the domain, and increases the number of regions iteratively until a user-defined error tolerance criterion is met. General remarks are given on the algorithm's implementation and a case study is provided to illustrate its application to vehicle dynamics.

Temporal Logic Control of Nonlinear Stochastic Systems Using a Piecewise-Affine Abstraction

Automatically synthesizing controllers for continuous-state nonlinear stochastic systems, while giving guarantees on the probability of satisfying (infinite-horizon) temporal logic specifications crucially depends on abstractions with a quantified accuracy. For this similarity quantification, approximate stochastic simulation relations are often used. To handle the nonlinearity of the system effectively, we use finite-state abstractions based on piecewise-affine approximations together with tailored simulation relations that leverage the local affine structure. In the end, we synthesize a robust controller for a nonlinear stochastic Van der Pol oscillator.

Robust multiplexed piecewise affine MPC-based decentralized control of multi-satellite formations using aerodynamic forces

Aerodynamic forces are well suited to be exploited for Low Earth Orbit satellite formation control. However, some critical gaps still exist in control schemes suitable for multi-satellite formation applications, especially when facing atmospheric environmental uncertainties. This paper proposes a scalable and flexible decentralized control scheme for the multi-satellite formation using only aerodynamic forces. This scheme adopts a strategy of updating the control inputs alternatively and is applicable to various formation configurations with different architectures without increasing the computational complexity and communication pressure of each satellite. Aerodynamic forces are changed by reorienting the satellite, and the linear formation control model with pointing angles as inputs is obtained by the PWA (piecewise affine) approximation method. A decentralized control scheme is proposed in which each satellite only has to communicate with its neighboring satellites and update its own inputs. This control scheme allows various generalizations of formation architectures, such as one that includes local chief satellites. To deal with model uncertainties, an improved MPC algorithm named Robust PWA-MMPC is developed using the constraint tightening approach, and then its feasibility and stability issues are analyzed. Hard-in-the-loop simulations are carried out for formation maintenance and reconfiguration, in which aerodynamic force uncertainties are considered.

Multiplexed MPC attitude control of a moving mass satellite using dual-rate piecewise affine model

This paper proposes a Multiplexed MPC algorithm for the dual-rate piecewise affine (PWA) model to control the attitude of a low-Earth-orbit (LEO) moving mass satellite. Although still facing an inherent inertial disturbance problem, the moving mass technology is a promising way to counter the strong aerodynamic torques in LEO. To make the satellite attitude fully actuated, a magnetorquer is used to cooperate with the moving mass system, resulting in a two-input control system. Through a strategy that updates inputs asynchronously, the proposed algorithm effectively balances the inertial disturbances caused by mass motions, the system's dynamic performance, and the controller's computational complexity. Error attitude kinematic and dynamic equations are derived, to convert the tracking problem into an equivalent regulation problem. The PWA approximation is used to transform the nonlinear attitude control equations described by error modified Rodrigues parameter to a linear state-space model for MPC design. Then, a new form of Multiplexed MPC, namely dual-rate PWA-MMPC, is developed for this dual-rate PWA model, which updates two control inputs at different rates. Despite only finding sub-optimal solutions to the original problem by stability analysis, this algorithm outperforms other control strategies through its ability to coordinate multiple inputs, deal with constraints, and reduce computational complexity. Hardware-in-the-loop simulations are conducted to demonstrate the effectiveness of the proposed methodology for the attitude control.

A stochastic optimization framework for integrated scheduling and control under demand uncertainty

Increased globalization and energy market deregulation are requiring process industries to respond more rapidly to fluctuations in demand levels, and utility and raw material prices, in order to remain competitive. In this study, a two-stage stochastic approach is proposed to account for demand uncertainty in a closed-loop dynamic real-time optimization (CL-DRTO) formulation that includes scheduling decisions. The CL-DRTO problem utilizes a prediction of the closed-loop response of the plant under the action of constrained MPC. The CL-DRTO system is executed in a rolling horizon fashion to compute economically optimal operation that is communicated to the plant through set-point trajectories assigned to the plant MPC. Nonlinear plant models are approximated using linear and piecewise affine (PWA) approximations, allowing the integrated CL-DRTO problem to be formulated as a mixed-integer linear program (MILP). Case studies demonstrate significantly higher expected profit using the proposed formulation than with a deterministic formulation utilizing the expected demand.

Coordinated yaw stability control for extreme path tracking of 4WIDEVs based on predictive control

To improve safety and stability of four-wheel independent drive electric vehicles (4WIDEVs) in extreme path tracking maneuvers, a hierarchical yaw stability control scheme based on predictive control is proposed in this work. The upper-level stability control strategy coordinates active front steering (AFS) and direct yaw moment control (DYC), yielding desired front steering angle and longitudinal force of each tire. The control task is formulated into a finite horizon optimal control problem, which is solved by hybrid model predictive control (hMPC) algorithm. The lower-level slip ratio control strategy calculates and tracks the optimal longitudinal slip ratio of each tire for implementation of the longitudinal tire force distributions in a wide range of road friction coefficient. Based on a qualified piecewise affine (PWA) approximation of lateral tire force models, the stable regions of vehicle states under multiple varying inputs are obtained by analyzing trajectories of fixed points. Furthermore, the control modes of the upper-level control strategy with different control inputs and state constraints are determined by judging bifurcation point of system. Finally, the proposed control approach is verified and compared through the dSPACE-based hardware-in-the-loop (HIL) experiments. The results indicate that the proposed yaw stability control strategy can improve yaw stability performance and reduce the tracking error significantly in extreme situations.

Formula omitted] control for continuous-time Markov jump nonlinear systems with piecewise-affine approximation

This paper investigates the problem of stability, H∞ performance analysis, and H∞ control for continuous-time Markov jump nonlinear systems, where the nonlinear subsystems are approximated by the piecewise-affine technique. The proposed Markov jump piecewise-affine systems contain different modes and regions, both of which are determined by Markov chains and piecewise-affine partitions, respectively. A new admissible adjacent region switching paths (AARSPs) algorithm is proposed for the first time in the continuous-time domain to decrease the conservatism of the complete adjacent region switching paths (CARSPs) algorithm. This new algorithm optimizes the path selection conditions of the next instantaneous time region switching in the CARSPs algorithm, and effectively reduces the computational complexity and the conservatism of the CARSPs algorithm. Furthermore, a state-feedback piecewise-linear controller is designed by means of the ellipsoidal outer approximation estimation method, such that the corresponding closed-loop system is stochastically stable and has a guaranteed H∞ performance index. Finally, the effectiveness and practicability of both the AARSPs algorithm and the piecewise-linear control strategy are fully demonstrated via two illustrative examples including a class of tunnel diode circuit systems.

On passivity, feedback passivity, and feedback passivity over erasure network: a piecewise affine approximation approach

In this paper, we deal with the problem of local passivity and local feedback passivation of smooth discrete-time nonlinear systems by considering their piecewise affine approximations. Sufficient conditions are derived for local passivity and local feedback passivity. These results are then extended to systems that operate over Gilbert-Elliott type communication channels. One of the main features of the approach presented in this paper is that it allows us to design the controller by solving a few linear matrix inequalities (LMIs) that are subject to certain constraints. As a special case, results for feedback passivity of piecewise affine systems over a lossy channel are also derived.

Implicit-explicit multistep formulations for finite element discretisations using continuous interior penalty

We consider a finite element method with symmetric stabilisation for the discretisation of the transient convection–diffusion equation. For the time-discretisation we consider either the second order backwards differentiation formula or the Crank–Nicolson method. Both the convection term and the associated stabilisation are treated explicitly using an extrapolated approximate solution. We prove stability of the method and the τ2 + hp+½ error estimates for the L2-norm under either the standard hyperbolic CFL condition, when piecewise affine (p = 1) approximation is used, or in the case of finite element approximation of order p ≥ 1, a stronger, so-called 4/3-CFL, i.e. τ ≤ Ch4/3. The theory is illustrated with some numerical examples.

Multilevel Selective Harmonic Modulation by Duality*

We address the Selective Harmonic Modulation (SHM) problem in power electronic engineering, consisting in designing a multilevel staircase control signal with some prescribed frequencies to improve the performances of a converter. In this work, SHM is addressed through an optimal control methodology based on duality, in which the admissible controls are piece-wise constant functions, taking values only in a given finite set. To fulfill this constraint, the cornerstone of our approach is the introduction of a special penalization in the cost functional, in the form of a piece-wise affine approximation of a parabola. In this manner, we build optimal multilevel controls having the desired staircase structure.

Compressive MRI quantification using convex spatiotemporal priors and deep encoder-decoder networks.

We propose a dictionary-matching-free pipeline for multi-parametric quantitative MRI image computing. Our approach has two stages based on compressed sensing reconstruction and deep learned quantitative inference. The reconstruction phase is convex and incorporates efficient spatiotemporal regularisations within an accelerated iterative shrinkage algorithm. This minimises the under-sampling (aliasing) artefacts from aggressively short scan times. The learned quantitative inference phase is purely trained on physical simulations (Bloch equations) that are flexible for producing rich training samples. We propose a deep and compact encoder-decoder network with residual blocks in order to embed Bloch manifold projections through multi-scale piecewise affine approximations, and to replace the non-scalable dictionary-matching baseline. Tested on a number of datasets we demonstrate effectiveness of the proposed scheme for recovering accurate and consistent quantitative information from novel and aggressively subsampled 2D/3D quantitative MRI acquisition protocols.

Topology-induced dynamics in a network of synthetic oscillators with piecewise affine approximation.

In synthetic biology approaches, minimal systems are used to reproduce complex molecular mechanisms that appear in the core functioning of multi-cellular organisms. In this paper, we study a piecewise affine model of a synthetic two-gene oscillator and prove existence and stability of a periodic solution for all parameters in a given region. Motivated by the synchronization of circadian clocks in a cluster of cells, we next consider a network of N identical oscillators under diffusive coupling to investigate the effect of the topology of interactions in the network's dynamics. Our results show that both all-to-all and one-to-all coupling topologies may introduce new stable steady states in addition to the expected periodic orbit. Both topologies admit an upper bound on the coupling parameter that prevents the generation of new steady states. However, this upper bound is independent of the number of oscillators in the network and less conservative for the one-to-all topology.