A Dynamic Model-Level Method for Multimode Process Monitoring Using ω-Metric

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

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.

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.

Industrial processes generally have various operation modes, and fault detection for such processes is important. This paper proposes a method that integrates a variational Bayesian Gaussian mixture model with canonical correlation analysis (VBGMM-CCA) for efficient multimode process monitoring. The proposed VBGMM-CCA method maximizes the advantage of VBGMM in automatic mode identification and the superiority of CCA in local fault detection. First, VBGMM is applied to unlabeled historical process data to determine the number of operation modes and cluster the data in each mode. Second, local CCA models that explore input and output relationships are established. Fault detection residuals are generated in each local CCA model, and monitoring statistics are derived. Finally, a Bayesian inference probability index that integrates monitoring results from all local models is developed to increase the monitoring robustness. The effectiveness of the proposed monitoring scheme is verified through experimental studies on a numerical example and the multiphase batch-fed penicillin fermentation process. Note to Practitioners —Process monitoring is important in guaranteeing process safety and improving product quality. Large amounts of unlabeled process data with multiple operation modes generally exist in industrial processes. Labeling these data is difficult or costly. Hence, this paper presents a VBGMM-CCA method for monitoring multimode processes. The key advantage of the proposed method is that it automatically identifies the number of operation modes in historical data and clusters the data. Then, local CCA monitors are established to model the process input and output relationships. During online monitoring, the running-on operation mode is identified through a density function, and the process status is evaluated by the corresponding CCA monitor. A probabilistic monitoring index is also developed to increase the robustness of the monitoring. In comparison with the results of conventional methods, the monitoring results of the proposed approach are more reliable and informative because the process status and the type of the detected fault are presented.

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.

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

A PWA model identification method based on optimal operating region partition with the output-error minimization for nonlinear systems

Time-varying human-operator (HO) adaptation in basic manual-control tasks is barely understood, as most HO identification methods do not explicitly take into account time variations. An identification procedure based on both batch and recursive autoregressive exogenous (ARX) models is used in this paper for captur- ing HO adaptation to time-varying changes in controlled-element dynamics in compensatory tracking tasks. Conditions with constant controlled-element dynamics, matching recent experimental work, and gradual and sudden transitions in the controlled-element dynamics were compared by means of Monte Carlo simulations with different remnant noise intensities. Overall, with batch ARX model identification results for conditions with constant dynamics, it is shown that differences between remnant noise and HO dynamics, both of which are directly coupled in the ARX model structure, will bias the identified ARX model parameters. Nevertheless, recursive ARX model identification results were still found to provide a reasonably estimate of time-varying HO model parameters for the tested time-varying scenarios. A direct comparison of using a constant scalar forgetting factor and a constant forgetting matrix, containing separate forgetting factors for each ARX model parameter, showed that a forgetting matrix did not result in a significant improvement for the considered HO identification problem. A forgetting factor of = 0.99609, corresponding to a memory horizon of 256 samples at 100 Hz, is found to be optimal for all tested conditions. An evaluation of real experimental manual-control data shows that, with straightforward online implementation, the method has potential as a tool for investigating, modeling, and measuring operators’ adaptive control characteristics.

Compression Based Identification of PWA Systems

Model predictive control (MPC) is used in the CO2 removal process in the Subang field to improve its control performance. MPC is used to maintain the CO2 concentration at the sweet gas output by controlling the feed gas pressure (PIC-1101), makeup water flow rate (FIC-1102), and amine flow rate (FIC-1103). The empirical model applied to MPC to represent the process model is the auto-regressive exogenous (ARX) model. The ARX model is compared with the first order plus dead time (FOPDT) model based on the root mean square error (RMSE) between the model and the actual process, then MPC parameters are tuned which include sampling time (T), prediction horizon (P) and control horizon (M) to control for the three variables. Improved control performance is measured based on the integral square error (ISE). The results show that the ARX model is the best model for the CO2 removal process with an RMSE value of 35%-91% smaller than the FOPDT model. The optimal control parameters Prediction Horizon (P), Control Horizon (M) and Sampling Time (T) in the CO2 removal process are 75, 25 and 1 on PIC-1101, 25, 10 and 1 on FIC-1102, and 30, 25 and 1 on FIC-1103. The MPC-ARX (MPC using ARX model) can improve the control performance of 33% in the servo control and 6-56% on the regulatory control. However, not all of them showed an increase in control performance improvement from previous studies even though they had used the best model (ARX). This is due to the MPC parameter setting that is not yet appropriate, so it needs to be retuning.

Key principal components with recursive local outlier factor for multimode chemical process monitoring

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.