Piecewise Affine Approximation Research Articles

Robust explicit model predictive control via regular piecewise-affine approximation

This paper proposes an explicit model predictive control design approach for regulation of linear time-invariant systems subject to both state and control constraints, in the presence of additive disturbances. The proposed control law is implemented as a piecewise-affine function defined on a regular simplicial partition, and has two main positive features. First, the regularity of the simplicial partition allows one to efficiently implement the control law on digital circuits, thus achieving extremely fast computation times. Moreover, the asymptotic stability (or the convergence to a set including the origin) of the closed-loop system can be enforced a priori, rather than checked a posteriori via Lyapunov analysis.

Advanced control laws of DC–DC converters based on piecewise affine modelling. Application to a step‐down converter

This study proposes two control techniques for a buck converter operating in continuous conduction mode at a fixed switching frequency. The non-linear behaviour of this switched system is represented by a discrete piecewise affine (PWA) model. The PWA representation offers a precise approximation of the converter's dynamics in the whole operating domain and also allows the investigation of system's stability and the design of different control laws. The first control approach corresponds to a piecewise linear (PWL) state-feedback controller, designed by using a piecewise quadratic Lyapunov function and by solving a set of linear matrix inequalities. This control method guarantees the stability of the closed-loop system for a wide range of operating points. The second control strategy is a model predictive control. The constrained optimal control problem is formulated and solved using the PWA approximation as a prediction model. The explicit form of the control law is derived off-line as an affine state-feedback controller and stored in a look-up table for implementation. Both PWL and PWA controllers are validated experimentally, showing better performances in comparison with a proportional-integral or constant state-feedback controller.

Optimal trajectory planning for trains under fixed and moving signaling systems using mixed integer linear programming

The optimal trajectory planning problem for multiple trains under fixed block signaling systems and moving block signaling systems is considered. Two approaches are proposed to solve this optimal control problem for multiple trains: the greedy approach and the simultaneous approach. In each solution approach, the trajectory planning problem is transformed into a mixed integer linear programming (MILP) problem. In particular, the objective function considered is the energy consumption of trains and the nonlinear train model is approximated by a piece-wise affine model. The varying line resistance, variable speed restrictions, and maximum traction force, etc. are also included in the problem definition. In addition, the constraints caused by the leading train in a fixed or moving block signaling system are first discretized and then transformed into linear constraints using piecewise affine approximations resulting in an MILP problem. Simulation results comparing the greedy MILP approach with the simultaneous MILP approach show that the simultaneous MILP approach yields a better control performance but requires a higher computation time. Moreover, the performance of the proposed greedy and the proposed simultaneous MILP approach is also compared with that of the greedy and the simultaneous pseudospectral method, where the pseudospectral method is a state-of-the-art method for solving optimal control problems. The results show that the energy consumption and the end time violations of the greedy MILP approach are slightly larger than those of the greedy pseudospectral method, but the computation time is one to two orders of magnitude smaller. The same trend holds for the simultaneous MILP approach and the simultaneous pseudospectral method.

Vehicle Yaw Stability Control by Coordinated Active Front Steering and Differential Braking in the Tire Sideslip Angles Domain

Vehicle active safety receives ever increasing attention in the attempt to achieve zero accidents on the road. In this paper, we investigate a control architecture that has the potential of improving yaw stability control by achieving faster convergence and reduced impact on the longitudinal dynamics. We consider a system where active front steering and differential braking are available and propose a model predictive control (MPC) strategy to coordinate the actuators. We formulate the vehicle dynamics with respect to the tire slip angles and use a piecewise affine (PWA) approximation of the tire force characteristics. The resulting PWA system is used as prediction model in a hybrid MPC strategy. After assessing the benefits of the proposed approach, we synthesize the controller by using a switched MPC strategy, where the tire conditions (linear/saturated) are assumed not to change during the prediction horizon. The assessment of the controller computational load and memory requirements indicates that it is capable of real-time execution in automotive-grade electronic control units. Experimental tests in different maneuvers executed on low-friction surfaces demonstrate the high performance of the controller.

Empowering the Performance of Advanced NMPC by Multiparametric Programming—An Application to a PEM Fuel Cell System

Fuel cell (FC) systems are part of a prominent key enabling technology for achieving efficient and carbon-free electricity generation and, as such, their optimum operation is of great importance. This work presents the combination of two advanced model predictive control (MPC) methodologies to guarantee the optimal operation of a polymer electrolyte membrane (PEM) fuel cell system. More specifically, at the core of the proposed framework is a nonlinear model predictive control (NMPC) formulation that solves online a nonlinear programming (NLP) problem using a simultaneous direct transcription optimization method. The performance of the NLP solver is enhanced by a warm-start initialization and a search space reduction (SSR) technique. A piecewise affine (PWA) approximation of the variable’s feasible space is used to define the boundaries of the search space computed offline, using a multiparametric quadratic programming (mpQP) method. The proposed unified framework is developed and deployed online to an in...

Online Implementation of an Integrated Explicit and Nonlinear Model Predictive Control (exNMPC) Framework for a PEM Fuel Cell System

This work presents the use of advanced control methodologies for a Polymer Electrolyte Membrane (PEM) fuel cell system. Two Model Predictive Control (MPC) strategies, a multi-parametric and a nonlinear MPC, are combined into a framework (explicit Nonlinear MPC - exNMPC) in order to systematically exploit their synergistic benefits and enhance the behaviour of the system. The new controller solves online a dynamic simultaneous optimization problem using a warm-start procedure that considers a piecewise affine (PWA) approximation defined offline over the partition of the system's feasible space. The proposed control scheme is deployed on an industrial Supervisory Control and Data Acquisition (SCADA) system and its response is explored. Experimental studies demonstrate that the multivariable exNMPC controller exhibits a very good performance in terms of computational requirements while achieving an excellent convergence rate even at varying operating conditions.

Piecewise affine approximations of fluxes and enzyme kinetics from in vivo13C labeling experiments

SUMMARYOwing to the ever increasing amount of available information on metabolic networks and, in particular, to the increase in information content from in vivo 13C dynamic labeling experiments, this work investigates the problem of reconstructing dynamic fluxes and enzyme kinetics. The model structure is based on the use of piecewise affine approximations. The optimization procedure at the basis of the model identification is improved by separating the parameter estimation procedure into two different phases. As a first step, a dynamic flux profile in time is reconstructed using functions that are piecewise affine (in time). To achieve scalability for this step, several approaches have been developed and compared. Afterwards, the time‐dependent profiles are embedded in the concentration space and the enzyme kinetic functions for single reactions are identified independently. This is an advantage compared with standard complete kinetic network approaches, which are typically characterized by hundreds of parameters, because now only a few need to be optimized simultaneously. Additionally, different kinetic formats can be rapidly compared. The overall approach is demonstrated using an informative in silico experiment. Copyright © 2012 John Wiley & Sons, Ltd.

Optimal Piecewise Affine Approximations of Nonlinear Functions Obtained from Measurements

The paper describes a two-stage procedure for obtaining piecewise affine approximations of static nonlinearities obtained from measured data. In the first step we search for a suitable function which fits the data while minimizing the fitting error. Subsequently we show how to approximate, in an optimal fashion, the nonlinear fitting function by a piecewise affine function of pre-specified complexity. We illustrate that approximation of arbitrary nonlinear functions boils down to a series of one-dimensional approximations, rendering the procedure efficient from a computational point of view.

Data-Based Identification and Control of Nonlinear Systems via Piecewise Affine Approximation

The piecewise affine (PWA) model represents an attractive model structure for approximating nonlinear systems. In this paper, a procedure for obtaining the PWA autoregressive exogenous (ARX) (autoregressive systems with exogenous inputs) models of nonlinear systems is proposed. Two key parameters defining a PWARX model, namely, the parameters of locally affine subsystems and the partition of the regressor space, are estimated, the former through a least-squares-based identification method using multiple models, and the latter using standard procedures such as neural network classifier or support vector machine classifier. Having obtained the PWARX model of the nonlinear system, a controller is then derived to control the system for reference tracking. Both simulation and experimental studies show that the proposed algorithm can indeed provide accurate PWA approximation of nonlinear systems, and the designed controller provides good tracking performance.

Real-Time Suboptimal Model Predictive Control Using a Combination of Explicit MPC and Online Optimization

Limits on the storage space or the computation time restrict the applicability of model predictive controllers (MPC) in many real problems. Currently available methods either compute the optimal controller online or derive an explicit control law. In this paper we introduce a new approach combining the two paradigms of explicit and online MPC to overcome their individual limitations. The algorithm computes a piecewise affine approximation of the optimal solution that is used to warm-start an active set linear programming procedure. A preprocessing method is introduced that provides hard real-time execution, stability and performance guarantees for the proposed controller. By choosing a combination of the quality of the approximation and the number of online active set iterations the presented procedure offers a tradeoff between the warm-start and online computational effort. We show how the problem of identifying the optimal combination for a given set of requirements on online computation time, storage and performance can be solved. Finally, we demonstrate the potential of the proposed warm-start procedure on numerical examples.

Analysis of the PSPG method for the transient Stokes’ problem

We propose a new analysis for the PSPG method applied to the transient Stokes’ problem. Stability and convergence are obtained under different conditions on the discretization parameters depending on the approximation used in space. For the pressure we prove optimal stability and convergence only in the case of piecewise affine approximation under the standard condition on the time-step. Finally we show that the stability problems of the PSPG-method can be circumvented using an appropriate discrete reconstruction of the Laplace operator.

Realistic computable error bounds for three dimensional finite element analyses in linear elasticity

We obtain a computable estimator for the energy norm of the error in piecewise affine and piecewise quadratic finite element approximations of linear elasticity in three dimensions. We show that the estimator provides guaranteed upper bounds on the energy norm of the error as well as (up to a constant and data oscillation terms) local lower bounds.

Model-Based Predictive Traffic Control: A Piecewise-Affine Approach Based on METANET

AbstractAs an alternative to the rather intensive computations when using the nonlinear traffic flow model METANET in a model-based predictive control context, a piecewise-affine (PWA) approximation of the model is proposed. Here, several model equations amongst which the fundamental diagram are approximated by a PWA function. Some selected methods to determine this approximation are shortly discussed. In addition, we make use of model properties and physical insight to improve the PWA approximation. For the purpose of traffic control, the PWA approximation of the METANET model can be used in a model predictive control (MPC) framework. In view of the on-line optimization used by MPC and the related trade-off between accuracy and computational speed, in a case study the accuracy of various approximations is compared to the original nonlinear formulation of METANET.

Automatic Derivation of Optimal Piecewise Affine Approximations of Nonlinear Systems

AbstractThe paper proposes a method for deriving optimal piecewise affine (PWA) approximations of nonlinear systems with known analytic form. The procedure employs nonlinear optimization to derive an approximation of maximal accuracy and given complexity. We show that under mild assumptions, the task can be transformed into a series of one-dimensional approximations. An automatic procedure for generation of such approximations is discussed as well.

Explicit hybrid model predictive control: discontinuous piecewise-affine approximation and FPGA implementation

AbstractIn this paper we introduce a digital architecture implementing the explicit solution of a switched model predictive control problem. Given a mixed-logic dynamical system, we derive an explicit controller in the form of a possibly discontinuous piecewise-affine function. This function is then approximated by resorting to piecewise-affine simplicial functions, which can be implemented on a circuit by extending the representation capabilities of a previously proposed architecture to evaluate the control action. The architecture has been implemented on FPGA and validated on a benchmark example related to an air conditioning system.

Discrete extrinsic curvatures and approximation of surfaces by polar polyhedra

Duality principle for approximation of geometrical objects (also known as Eu-doxus exhaustion method) was extended and perfected by Archimedes in his famous tractate “Measurement of circle”. The main idea of the approximation method by Archimedes is to construct a sequence of pairs of inscribed and circumscribed polygons (polyhedra) which approximate curvilinear convex body. This sequence allows to approximate length of curve, as well as area and volume of the bodies and to obtain error estimates for approximation. In this work it is shown that a sequence of pairs of locally polar polyhedra allows to construct piecewise-affine approximation to spherical Gauss map, to construct convergent point-wise approximations to mean and Gauss curvature, as well as to obtain natural discretizations of bending energies. The Suggested approach can be applied to nonconvex surfaces and in the case of multiple dimensions.

Uniform tree approximation by global optimization techniques

In this paper we present adaptive algorithms for solving the uniform continuous piecewise affine approximation problem (UCPA) in the case of Lipschitz or convex functions. The algorithms are based on the tree approximation (adaptive splitting) procedure. The uniform convergence is achieved by means of global optimization techniques for obtaining tight upper bounds of the local error estimate (splitting criterion). We give numerical results in the case of the function distance to 2D and 3D geometric bodies. The resulting trees can retrieve the values of the target function in a fast way.

A fuzzy clustering based technique for piecewise affine approximation of a class of nonlinear systems

In recent years piecewise affine (PWA) modeling has developed as an attractive tool for the approximation of various complex nonlinear systems. In spite of the wide application of PWA modeling, the optimal approximation of a continuous time nonlinear system with scalar functions by the minimum number of affine systems has not been addressed properly in literature. This paper deals with a fuzzy clustering based approach for the optimal PWA approximation of a class of continuous time nonlinear systems. The technique is based on the trade-off between increasing the approximation accuracy of the various nonlinear functions and simplifying the approximation by the minimum number of subsystems. As an application, the technique is utilized to obtain a PWA approximation of the glucose regulation system. Numerical simulations depicted that, for a given number of subsystems, the derived glucose regulation model provides an optimal approximation of the original nonlinear system. The model also provided some biological insight about the interactions involved in glucose regulation.

Lebesgue piecewise affine approximation of nonlinear systems

This paper addresses a piecewise affine (PWA) approximation problem, i.e., a problem of finding a PWA system model which approximates a given nonlinear system. First, we propose a new class of PWA systems, called the Lebesgue PWA approximation systems, as a model to approximate nonlinear systems. Next, we derive an error bound of the PWA approximation model, and provide a technique for constructing the approximation model with specified accuracy. Finally, the proposed method is applied to a gene regulatory network with nonlinear dynamics, which shows that the method is a useful approximation tool.

Low Order Discontinuous Galerkin Methods for Second Order Elliptic Problems

We consider DG-methods for second order scalar elliptic problems using piecewise affine approximation in two or three space dimensions. We prove that both the symmetric and the nonsymmetric versions of the DG-method have regular system matrices without penalization of the interelement solution jumps provided boundary conditions are imposed in a certain weak manner. Optimal convergence is proved for sufficiently regular meshes and data. We then propose a DG-method using piecewise affine functions enriched with quadratic bubbles. Using this space we prove optimal convergence in the energy norm for both a symmetric and nonsymmetric DG-method without stabilization. All of these proposed methods share the feature that they conserve mass locally independent of the penalty parameter.