Page - (000083) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

Chapter10 Nonlinearity Realsystemsarenonlinear.Beforediscussingnonlineartheory,Ireviewthreereasons why thecore theoryofcontrol focuseson linear analysis. First, feedback compensates formodel uncertainty. Supposeweanalyze a feed- back systembasedona linearmodel of dynamics, and the truemodel is nonlinear. If the linearmodel captures essential aspects of the dynamics sufficiently, the true nonlinearfeedbacksystemwilloftenhavethesamequalitativeresponsecharacteris- ticsas themodeled linearsystem.AsVinnicombe(2001,p.xvii)emphasized:“One of the key aims of using feedback is tominimize the effects of lack of knowledge about a systemwhich is tobecontrolled.” Second, the fundamental principles of control systems apply to both linear and nonlinear dynamics. The comprehensive theory for linear systemsprovides insight intononlinearsystems.Forexample,strongfeedbacksignalsoftenhelptominimize error but can create instabilities. Controllers can be added at different points in a system to filter signals or modulate inputs that drive processes. Primary goals of analysis and design often emphasize stability, disturbance rejection, regulation or tracking.Certaintradeoffsinevitablyarise.Integralcontrolsmoothsresponsetoward a long-term target. Derivative control improves the speed of response by using a simpleprediction. Third, themain tools for the analysis and design of nonlinear systems typically extend the tools developed for linear systems. For example, nonlinear systems are approximately linear near a particular operating point. One can study the linear approximation around that point, and then switch to an alternative linear approxi- mation as the systemmoves to another operating domain. By piecing together the different linear approximations in neighboring domains, one develops a sequence of linear systems that together capturemuchof thenonlinear characteristics.Other toolsofnonlinearanalysis typically leveragethedeepinsightsandmethodsof linear analysis (SlotineandLi1991; Isidori1995;Khalil 2002). This chapterpresents a fewbrief illustrationsofnonlinear control systems. ©TheAuthor(s)2018 S.A.Frank,ControlTheoryTutorial, SpringerBriefs inAppliedSciences andTechnology,https://doi.org/10.1007/978-3-319-91707-8_10 79