2D piecewise affine models approximate real continuous dynamics up to invariant sets

Mathematical modeling of biological regulatory networks provides valuable insights into the structural and dynamical properties of the underlying systems. While dynamic models based on differential equations provide quantitative information on the biological systems, qualitative models that rely on the logical interactions among the components provide coarse-grained descriptions useful for systems whose mechanistic underpinnings remain incompletely understood. The middle ground class of piecewise affine differential equation models was proven informative for systems with partial knowledge of kinetic parameters. In this work we provide a comparison of the dynamic characteristics of these three approaches applied on several biological regulatory network motifs. Specifically, we compare the attractors and state transitions in asynchronous Boolean, piecewise affine and Hill-type continuous models. Our study shows that while the fixed points of asynchronous Boolean models are observed in continuous Hill-type and piecewise affine models, these models may exhibit different attractors under certain conditions. Overall, qualitative models are suitable for systems with limited knowledge of quantitative information. On the other hand, when practical, using quantitative models can provide detailed information about additional real-valued attractors not present in the qualitative models.

This work presents an algorithm that, given a piecewise affine (PWA) model, derives an equivalent PWA model that is minimal in the number of regions. The algorithm is based on the cells of the hyperplane arrangement that are already given when the PWA model is the result of the mode enumeration algorithm. In particular, the algorithm executes a branch and bound search on the markings of the cells of the hyperplane arrangement assuring optimality. As we refrain from solving additional LPs, the algorithm is not only optimal but also computationally attractive. The applicability of the algorithm can be extended to derive minimal PWA representations of general PWA models by first computing the hyperplane arrangement. An example illustrates the algorithm and shows its computational effectiveness.

Piecewise affine modeling and hybrid optimal control of intelligent vehicle longitudinal dynamics for velocity regulation

A PWA model identification method for nonlinear systems using hierarchical clustering based on the gap metric

SummaryThe multiple model approach provides a powerful tool for identification and control of nonlinear systems. Among different multiple model structures, the piecewise affine (PWA) models have drawn most of the attention in the past two decades. However, there are two major issues for the PWA model‐based identification and control: the curse of dimensionality and the computational complexity. To resolve these two issues, we propose a novel multiple model approach in this paper. Different from PWA models in which all dimensions of the regressor space are engaged in the partitioning, the key idea of the proposed multiple model architecture is to partition only the range of the control input u(k) at time k (the instant of interest in the control problem) into several intervals and identify a local model that is linear in u(k) within each interval. On the basis of Taylor's theorem, a theoretical upper bound for the approximation error of the model structure can also be obtained. With the proposed multiple model architecture, a switching control algorithm is derived to control nonlinear systems on the basis of the weighted one‐step‐ahead predictive control method and constrained optimization techniques. In addition, the upper bound for the tracking error using this switching control strategy is also analyzed rigorously under certain assumptions. Finally, both simulation studies and experimental results demonstrate the effectiveness of the proposed multiple model architecture and switching control algorithm. Copyright © 2017 John Wiley & Sons, Ltd.

A multimode process possesses dynamic behaviors, especially in between-mode transitions. Motivated by that dynamic models are well-suited for interpretation of process dynamics, we initially propose a dynamic model-level method for multimode process monitoring to detect abnormal operations quickly and accurately. It mainly includes three parts. First, a piecewise affine (PWA) model is identified offline to represent different dynamic patterns of a multimode process, which can avoid model redundancy compared with existing achievements. Owing to the utilization of the PWA model, the proposed method is still valid when unknown transitions occur. Second, in online monitoring, local autoregressive exogenous (ARX) models, obtained by just-in-time learning (JITL), can sufficiently extract local dynamic patterns so as to facilitate evaluating dynamic changes in the process. Finally, a novel measurement, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\omega $ </tex-math></inline-formula> -metric, is devised to analyze the dynamic variations in the process by directly measuring the similarity between local models and the PWA model, which is more beneficial for detecting abnormal operations than data distances. The superiority of the proposed method is validated by a continuous stirred tank reactor (CSTR) and a real air separation process compared with the existing multimode process monitoring approaches.

In the proposed study, a Hybrid Model Predictive Controller is introduced for cruise control of an automobile model. The presented model consists of the engine, the gearbox, and the transmission dynamics, where the aerodynamics force and elastic friction between the tires and road are taken into account. Through Piecewise Linearization of nonlinearities in the system; (torque)-(throttle)-(angular velocity) of engine and (aerodynamic drag force)-(automobile velocity), a comprehensive piecewise linear model for the system is obtained. Then combined with the switch and shift between engaged gears in gearbox, the Piecewise Affine (PWA) model for the vehicle dynamics is acquired. As far as the control design is concerned, the cruise control problem for tracking a desired speed fashion is addressed by a MPC-based controller design. The proposed control approach is based on the online model predictive control, applied on the obtained PWA dynamics. The highlighted novelties of the presented research work are summarized as: first a more complete model is examined due to the consideration of a realistic model for engine. This improvement makes the polyhedron regions of the PWA system dependent to both state variable (i.e., velocity) and input signals (i.e., throttle and engaged gear) which brings the complexity to the design of control procedure. Second, due to the switch in the dynamics and dependence of our PWA model to discrete input (gear shift), the desperate need to solve the optimization problem through mixed integer programming, which needs high computation effort specially for our system, seems inevitable. We triumph over this challenge through introducing “possible gear shift scenario” sets. Hence, by constraining the optimization problem to the introduced logical sets, the problem still remains convex optimization type and the computation volume is reduced. In addition, we hired branch and bound method which allowed us to have large problems to be solved in a tractable amount of time and computation resources. At last, some simulations are presented to exhibit the performance of the proposed method.

Compression Based Identification of PWA Systems

A survey on switched and piecewise affine system identification

In this paper we propose a bilevel programming formulation of the piecewise affine regression problem, where the upper level fixes the partition of the regressor domain and classifies the data points, and the lower level computes parameter estimates of the affine models in each region of the partition. The proposed formulation accommodates a general class of regression problems, where parameter estimates in each region of the partition are based on a prediction error criterion, while the overall piecewise affine model is selected according to a possibly different criterion. Due to the use of binary variables for the classification task, the proposed approach is typically viable only for small data sets and number of submodels. Still, it can be used to formulate and solve several problems of practical interest, where it is important to carry out simultaneously the three tasks of data classification, parameter estimation and estimation of the partition of the regressor domain. This is demonstrated through an application of the proposed approach to the pick-and-place machine data set.

In this chapter, a new identification method for nonparametric piecewise affine (PWA) models is introduced. The method is based on the nonparametric data-based representation of PWA maps and the data compression with the l1 optimization technique, which enable the method to deal with large data sets. This method can be applied to a wide range of modeling problems, and an example with a DC motor system is shown to verify the usability of the method. Also, a method for choosing appropriate compression ratio is discussed.

Set Membership Identification of Piecewise Affine Models

This paper proposes a distributed model predictive control (DMPC) method with guaranteed stability for system with control coupled information and coupled constraints among the subsystems, which are described by piecewise affine (PWA) model. According to the condition of control input coefficient matrix of the PWA model, two frameworks are established. The DMPC optimal problem of each subsystem is converted into mixed integer quadratic programming (MIQP) to solve and obtain the control actions. To ensure the stability, equality constraint is applied.

This paper addresses the conversion of discrete-time piecewise affine (PWA) state space models into input-output form. Necessary and sufficient conditions for the existence of equivalent input-output representations of a given PWA state space model are derived. Connections to the observability properties of PWA models are investigated. Under a technical assumption, it is shown that every finite-time observable PWA model admits an equivalent input-output representation. When an equivalent input-output model exists, a constructive procedure is presented to derive its equations. Several examples illustrate the proposed results.

Multimodeling vs PieceWise Affine modeling for the identification of open channel systems