Tobias Schweickhardt

On System Gains, Nonlinearity Measures, and Linear Models for Nonlinear Systems

This article is concerned with the assessment and derivation of (best) linear models for nonlinear systems. We introduce a general framework reminiscent of uncertainty descriptions in linear robust control theory in order to define a family of six model quality indices for linear models of nonlinear systems. Each quality index corresponds to the gain of an error system, where the nonlinear system can be represented as the interconnection of a nominal linear model and this error system. For an analysis in a certain region of operation, these gains are considered over a specified signal set. A best linear model is a linear model that minimizes the quality index and the minimal quality index serves at the same time as a measure of nonlinearity of the nonlinear system. We show that the model quality indices and nonlinearity measures can be defined under very weak assumptions, and we give conditions that guarantee the existence of optimal linear models. We discuss the structure of the problem of computing the nonlinearity measures. We show that the local linearization of a state space system is a best linear model for the limiting case of a vanishing operating range for one of the measures. We show the relation between the steady-state behavior of a system and its gain and nonlinearity measures. Furthermore we give linear models and upper bounds for nonlinearity measures for systems composed of a linear dynamic and a nonlinear static part. In the case of scalar (SISO) systems, these bounds are given by the sector bounds of the nonlinearity. Finally, it is shown that a lower bound on the nonlinearity measures can be derived using harmonic analysis. Several small examples serve to illustrate the results and the underlying ideas.

Linear modeling error and steady-state behaviour of nonlinear dynamical systems

We consider different measures of nonlinearity that quantify the modeling error of optimal linear models for nonlinear systems. We show that linear dynamic approximations have no advantage over linear static approximations for static nonlinear functions in any of the considered measures. Simplified formulae for scalar nonlinearities are derived using the notion of a sector nonlinearity. We show that the steady state behaviour of a nonlinear system gives rise to a lower bound on the nonlinearity measures. Furthermore, some results for composite nonlinear systems are given.

GOOD OR BAD – WHEN IS PLANT NONLINEARITY AN OBSTACLE FOR CONTROL?

Abstract Virtually every real process is a nonlinear system. Nevertheless, linear system analysis and linear controller design methods have proven to be adequate in many applications. On the other hand, there are nonlinear processes that require or benefit from nonlinear control. Therefore, recognizing a system as being nonlinear does not suffice, but the extent and severity of a systems inherent nonlinearity is the crucial characteristic in order to decide whether linear system analysis and controller synthesis methods are adequate. The introduction of nonlinearity measures is an attempt to systematically approach this problem. In this contribution, we review existing approaches to nonlinearity assessment, we state the most important results and we give a glance ahead to what might be expected from this field in the future.

Integration of e-learning modules in automatic control education

Abstract Despite the many advantages that e-learning has to offer in education and especially in the complex and theoretical field of control education, it appears that the hype of e-learning has somewhat decreased. One of the reasons is the huge effort that is often necessary to setup and maintain e-learning services. An approach to overcome this dilemma is the concept of blended learning. This paper shows some examples of how e-learning applications can be integrated in a traditional control course and how the best of both worlds can be combined. Another promising approach that we discuss in this paper is the integration of computer-games in education to intensify the motivation of the students.

Plant and Control-Relevant Nonlinearity Analysis of a CSTR: A Case Study

Abstract Control-relevant nonlinearity analysis has gained importance with the need to control nonlinear processes more tightly. To this effect, the recently proposed closed-loop optimal control law (OCL) nonlinearity measure quantifies the degree of nonlinearity of the optimal state feedback controller for a given control problem. The current paper attempts to gain more insight into the characteristics and behavior of the formulation through a case study using a CSTR model. The multivariate character and the correlation of plant and control-relevant nonlinearity are analyzed with particular focus.

An approach to linear control of nonlinear processes

Nonlinear controller design is a demanding task. Therefore, linear methods are favored over nonlinear ones in practice although it is not always clear a priori whether the former are appropriate in face of a nonlinear process. Nonlinearity measures are a means to quantify the nonlinearity inherent to a process. However, as yet there were no rule on how to design a controller based on the results of nonlineariiy assessment. In this paper, an approach is presented that integrates nonlinearity assessment and the design of linear controllers for nonlinear systems by using methods from linear robust control.

Integration und interaction: möglichkeiten des einsatzes von notebooks und internet in der regelungstechnischen lehre (Integration and interaction: opportunities of using notebooks and internet for teaching in control engineering)

Abstract In Schulen werden bereits seit einigen Jahren Notebooks für den Einsatz in der Lehre erprobt und auch schon erfolgreich eingesetzt. Neu hingegen sind die aktuellen Bestrebungen, Laptops auch in die Hochschul-Ausbildung einzubinden. Der erste Teil dieses Beitrags gibt einen Überblick über das Konzept des BMBF-Projekts Notebook University. Im zweiten Teil wird der konkrete Einsatz von Notebooks für die regelungstechische Ausbildung anhand des Projekts eLearnIng. beschrieben.