Kenneth R. Harris

On the computation of a nonlinearity measure using functional expansions

Nonlinearity measures quantify the degree of nonlinearity in dynamic systems. In this work, we present a novel approach to the computation of these measures by functional expansions. Rather than calculating an exact nonlinearity measure which is often computationally intensive, functional expansion (FEx) models are used to derive approximate lower and upper bounds on the measure. The resulting bounds are algebraic in nature and provide insight into the effect of process parameters on the nonlinearity measure. These concepts are illustrated by a benchmark CSTR problem.

Studies on the analysis of nonlinear processes via functional expansions—III: controller design

A nonlinear control design strategy is proposed based on functional expansion (FEx) models. The controller is formulated as a linear controller with a series of correction terms to account for nonlinearities in the system. Case studies include discrete control of a copolymerization reactor, ideal and one-way decoupling control of a chemical reactor, and control of a bioreactor with complex dynamics. The nonlinear control laws are shown to provide superior performance compared to their linear counterparts, while maintaining, similar flexible design features.

Studies on the analysis of nonlinear processes via functional expansions—I. Solution of nonlinear ODEs

A functional transformation approach is presented to aid in the analysis of nonlinear systems of ODEs. This approach differs from traditional linear and numerical methods in that the resulting analytic solutions reveal global information as well as the functionality of system parameters. Solutions to ODEs containing polynomial type nonlinearities are computed in a transform domain using a symbolic programming language. The fundamentals of the method are discussed, followed by two case studies demonstrating the application as well as strengths and weaknesses of the method.

Studies on the analysis of nonlinear processes via functional expansions.II. Forced dynamic responses

A functional expansion technique utilizing the Laplace-Borel transform is reviewed and applied to forced nonlinear processes with hyperbolic equilibrium points. Initial and final value theorems are presented to aid in the analysis of nonlinear systems. The expansions are shown to be equivalent to a linear transfer function with an added set of poles and zeros to capture the nonlinear dynamics of the system. The frequency response in the transform domain is derived as an analog to linear systems, with the resulting amplitude ratio and phase expressions being time dependent. The developed concepts are applied to a nonlinear CSTR example.

Control of nonlinear processes using functional expansion models

Functional expansion (FEx) models are a subclass of the general block-oriented model structure for nonlinear process systems. Controller design in this context uses the internal model control (IMC) paradigm, and one can show that the resulting controllers are easily implementable. The primary advantage arises from the fact that inverting the nonlinear dynamic operator is avoided by taking advantage of the partitioned model inverse due to the special structure of FEx models. The robust stability and performance of the closed-loop system can be analyzed by expressing the FEx model as a linear uncertain system and using the structured singular value framework. We present a case study of a polymerization reactor and, for this SISO system, analyze nominal and robust stability and performance conditions as a function of the closed-loop filter constants for a given range of the input variable.

Stability of nonlinear control systems based on low-order block-oriented models

A method to calculate a region of attraction for controllers based on low-order block-oriented models is presented. The method uses a generalization of the circle criterion, which is less conservative than other input/output stability methods since the shape of the nonlinearity is also considered. The developed method is applied to a simulation example and additional applications are discussed.

Experimental Application of Partitioned Model-Based Control to pH Neutralization†

Abstract A nonlinear control law is applied to an experimental pH neutralization process. A partitioned model is used to capture the dynamic behavior of the process, consisting of a linear ARX model and a nonlinear neural network model. When integrated into the internal model control scheme, the resulting controller is shown to have better performance compared to the linear controller for the pH neutralization process.

Control of nonlinear MIMO systems via functional expansions

Multivariable control of nonlinear processes is addressed through the use of functional expansions. Based on the internal model control scheme, the proposed controller strategy results in a flexible design method applicable to both minimum and nonminimum phase nonlinear systems. The control law can be selected as to achieve complete or partial decoupling of outputs. A simulation example is presented.

Promises and Limitations of Functional Expansions in Nonlinear Model-Based Control

The application of functional expansion (FEx) models for the analysis and control of nonlinear processes is reviewed. Nonlinear analysis tools analogous to the linear pole/zero and frequency response concepts are presented, as well as the concept of a nonlinearity measure. FEx model-based controllers are developed based on the internal model control structure. Strengths and weaknesses of the methods are discussed, and the developed concepts are applied to a simulation example.