Page - (000022) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

12 2 ControlTheoryDynamics x˙1 =−a2x2+bu x˙2 = x1−a1x2+u y= x2, inwhich the rates of change in the states dependonlyon the current states and the current input. 2.2 NonlinearityandOtherProblems Classicalcontrol theoryfocusesontransfer functions.Thosefunctionsapplyonlyto linear, time-invariantdynamics.Bycontrast, state-spacemodels canbeextended to any typeofnonlinear, time-varyingprocess. Real systemsare typicallynonlinear.Nonetheless, four reasons justify the study of linear theory. First, linear analysis clarifies fundamental principles of dynamics and control. For example, feedbackoften leads to complex, nonintuitivepathwaysof causation. Linear analysis has clarified the costs and benefits of feedback in terms of trade- offs betweenperformance, stability, and robustness. Those principles carry over to nonlinear systems, although thequantitativedetailsmaydiffer. Second,many insights into nonlinear aspects of control come from linear the- ory (Isidori 1995;Khalil 2002;Astolfi et al. 2008). In addition to feedback, other principles includehowtofilter out disturbances at particular frequencies, how time delays alter dynamics and thepotential for control, how to trackexternal setpoints, and how to evaluate the costs and benefits of adding sensors tomonitor state and adjustdynamics. Third, linear theory includesmethods toanalyzedepartures frommodelassump- tions.Those linearmethodsof robustness often apply tononlinear departures from assumed linearity. One can often analyze the bounds on a system’s performance, stability, and robustness to specific typesofnonlineardynamics. Fourth, analysis of particular nonlinear systems often comes down to studying anapproximately linearizedversionof the system. If the systemstate remainsnear an equilibriumpoint, then the systemwill be approximately linear near that point. If the systemvariesmorewidely, one can sometimes consider a series of changing linearmodels that characterize the system ineach region.Alternatively, a rescaling ofanonlinear systemmay transformthedynamics intoanearly linear system. Givenaparticularnonlinearsystem,onecanalwayssimulatethedynamicsexplic- itly. Themethods one uses to understand and to control a simulated system arise mostly from the core linear theory and from theways that particular nonlinearities depart fromthat core theory.