Karlene A. Hoo

AN EXOTHERMIC CONTINUOUS STIRRED TANK REACTOR IS FEEDBACK EQUIVALENT TO A LINEAR SYSTEM

This brief paper demonstrates the concept of linear feedback equivalence for an exothermic eontinu-ous stirred tank reactor with first order kinetics. Feedback control is achieved by finding a transformation for the nonlinear system which carries this system into a linear controllable system in Brunovsky canonical form. A linear state feedback controller is then designed which achieves control over a broad range of operating conditions. This example demonstrates how recent developments in nonlinear control theory can be applied to chemical systems without relying on the usual methods of local linearization.

Low-order Control-relevant Models for A Class of Distributed Parameter Systems

Abstract Accurate solutions of distributed parameter systems may be represented as the sum of an infinite series. Control design however, requires low-order models primarily due to implementation limitations. As such, developing low-order models of high fidelity is important if the objective is accurate control of the DPS. This work addresses this issue by developing a method that assures a convergent and consistent projection to a finite space. The resulting model is then subsequently used to design finite dimensional state feedback controllers. The methodology is demonstrated on two quasi-linear processes under ideal and non-ideal conditions.

Linear feedback equivalence and control of an unstable biological reactor

A novel method for the feedback control of an unstable, continuous stirred tank bioreactor is described in this paper. The models considered describe the growth of a single microorganism on a single, rate limiting substrate. A theoretical discussion demonstrates that for models of this type, a global feedback transformation exists which linearizes the state response. Such a transformation is computed using the methods of Hunt, Su and Meyer (1983). A controller is computed using linear state feedback control, and an extended Kalman filter is used to estimate unmeasured state variables. Simulation results demonstrate satisfactory control behavior over a wide region of the phase plane.

Low-order model identification for implementable control solutions of distributed parameter systems

Abstract Accurate solutions of the distributed parameter system (DPS) may be represented as the sum of an infinite series. Control design however, requires low-order models primarily due to implementation limitations. As such, developing low-order models of high fidelity is important if the objective is accurate control of the DPS. When an exact model (system of partial differential equations (PDEs)) of the system is known, this work presents a method to develop a low-order model that assures convergent and consistent projection to a finite space. The resulting low-order model can then be used to design finite dimensional controllers. When there is no available first-principle model of the system, this work introduces a novel system identification method, that combines the characteristics of singular value decomposition (SVD) and the Karhunen-Loeve (KL) expansion for DPS to arrive at a low-order model that captures the dominant characteristics of the system. Here as well, the final model form allows for the synthesis of finite order controllers. Two non-linear reactor systems that can be described by systems of PDEs are provided to demonstrate the model identification methods. Feedback controllers are then synthesized based on these models to demonstrate their accuracy for disturbance rejection.

System identification and model-based control for distributed parameter systems

A linear, low-order input/output model is identified for a nonlinear distributed parameter system using a combination of singular value decomposition and the Karhunen Loeve expansion. The model captures the dominant behavior of the system around a nominal operating point. A quadratic dynamic model-based controller (QDMC) is designed based on this low-order model. Sufficient conditions for closed-loop stability are presented and proven. The system identification method and the resulting QDMC controller are demonstrated on a nonlinear multiple-input multiple-output (MIMO) chemical reactor that produces benzene from the hydro-dealkylation of toluene.

Global linearization and control of a mixed-culture bioreactor with competition and external inhibition

Nonlinear feedback control design is demonstrated in this study for a continuous mixed-culture biological reactor model exhibiting competition and external inhibition. The model considered describes the growth of two species in a continuous stirred tank, for which one species is sensitive to an external inhibitor, and for which both species compete for the same rate-limiting substrate. It is shown that choosing the dilution rate and the inhibitor addition rate as manipulated variables admits a global linearization transformation which can be used to construct a multivariable feedback controller. The transformed state variables are found to be the cell density of the substrate insensitive species, the log ratio of cell densities, and the net difference in specific growth rates. An extended Kalman filter is used to estimate the unmeasured states when only a single measurement of the total cell mass is available. Simulated performance of the closed-loop system appears to be satisfactory for a wide region of the phase space.

A method of robust multivariate outlier replacement

Abstract Robust multivariate methods for dealing with problems caused by outliers in the data are essential especially when process data are used to validate mechanistic models, develop regression models, and in applications such as controller design and process monitoring. Gross outliers are detected easily by simple methods such as range checking, however, a multivariate outlier is very difficult to discern and techniques that rely on data to generate empirical models may produce erroneous results. In this work, a methodology to perform multivariate outlier replacement in the score space generated by principal component analysis (PCA) is proposed. The objective was to find an accurate estimate of the covariance matrix of the data so that a PCA model might be developed that could then be used for monitoring and fault detection and identification. The methodology uses the concept of winsorization to provide robust estimates of the mean (location) and S.D. (scale) iteratively, yielding a robust set of data. The paper develops the approach, discusses the concept of robust statistics and winsorization, and presents the procedures for robust multivariate outlier filtering. One simulated and two industrial examples are provided to demonstrate the approach.

Wavelet-based model reduction of distributed parameter systems

Abstract Mathematical models that describe distributed parameter systems are composed of systems of partial differential and algebraic equations. The methods that solve these systems usually yield a high-order (infinite-dimensional) solution. However, for controller synthesis and practical considerations, a low-order model is preferred. This work addresses the development of model reduction through the use of multi-resolution methods that not only yield a finite low-order model but also a representation of the systems multiscale and local behavior such that scale-specific compensation can be realized. Two systems — heat transfer along a flat metal plate, and a packed-bed reactor with axial dispersion are used to demonstrate the proposed approach.

Process Data Analysis and Interpretation

Publisher Summary This chapter proposes a perspective for integrating a wide-ranging array of technologies and managing complexity in comprehensive analysis and interpretation systems. This integrated perspective is the product of widely varying technology perspectives. To support increasingly sophisticated process management activities, the raw process data must be transformed into meaningful descriptions of process conditions. The primary purpose of pattern recognition is to determine class membership for a set of numeric input data. The performance of any given approach is ultimately driven by how well an appropriate discriminant can be defined to resolve the numeric data into a label of interest. In this context, the chapter takes a look at the alternatives to quantitative behavioral model approaches from the point of view of interpretation. These methods include a wide variety of linear and nonlinear modeling methods developed across a wide range of technical areas, including statistics, process simulation, control, and intelligent systems. The chapter discusses local and nonlocal data interpretation, Symbolic–Symbolic Interpretation, and scope of large scale operations. Three comprehensive examples illustrated in the chapter are detection of abnormal situations, data analysis of batch operation variability, and diagnosis of operating problems in a batch polymer reactor.

Stability analysis for closed-loop management of a reservoir based on identification of reduced-order nonlinear model

This objective of this study is to analyze the stability of an oil producing reservoir under closed-loop control. Given a five-spot pattern reservoir as an example, a nonlinear reduced-order model is identified and an asymptotically stabilizing controller is proposed based on the circle criterion.