Page - (000087) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

10.4 FeedbackLinearization 83 10.4 FeedbackLinearization Consider the simplenonlinear systemwithan input x˙= x2+u, in which the output is equal to the single state, y= x (Khalil 2002, p. 473). The equilibrium x∗ =0 isunstablebecauseanyperturbation fromtheequilibriumleads touncontrolledgrowth. The error deviation from equilibrium is x. Classical negative linear feedback would apply the control inputu=−kx, inwhich the feedback isweighted by the gain,k.Theclosed-loopsystembecomes x˙=−kx+x2. This systemhas a locally stable equilibriumat zero and anunstable equilibriumat k. For aperturbation that leaves x< k, the systemreturns to its stable equilibrium. For a perturbation that pushes x beyond k, the systemgrowswithout bound.Thus, linear feedbackprovides localstability.Thestronger thefeedback,with largerk, the broader the local regionof stability. Inthiscase,linearfeedbacktransformsanunstableopen-loopsystemintoalocally stable closed-loop system.However, the closed-loop systemremainsnonlinear and prone to instability. Ifwechoose feedback tocancel thenonlinearity,u=−kx−x2, thenweobtain the linearly stableclosed-loopsystem, x˙=−kx. Oncewe have a linear closed-loop system,we can treat that system as a linear open-loop subsystem, anduse linear techniques todesign controllers and feedback toachieveperformancegoals. Forexample,wecouldconsider thefeedbacklinearizeddynamicsas x˙=−kx+ v, inwhich v is an input into this new linearized subsystem.Wecould thendesign feedbackcontrol through the input v to achievevariousperformancegoals, suchas improved regulation todisturbanceor improved trackingofan input signal. Anonlinear systemcanbe linearizedby feedback if the states canbewritten in the form x˙= f(x)+g(x)u. (10.3) Suchsystemsarecalled input linear,because thedynamicsare linear in the input,u. These systems are also calledaffine in input, because a transformationof the form a+bu is an affine transformationof the input,u.Here, f and gmaybenonlinear functionsof x, butdonotdependonu. In the example x˙= x2+u, we easily found the required feedback to can- cel the nonlinearity. For more complex nonlinearities, geometric techniques have beendeveloped tofind the linearizing feedback (Slotine andLi1991; Isidori 1995;