Page - (000084) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

80 10 Nonlinearity 10.1 LinearApproximation Incellularbiology,theconcentrationofamolecule,m,oftenrespondsaninputsignal, φ, by theHill equation m= k( φ n 1+φn ) , withparametersk andn. In this example, I letk=1. Thenonlinear reactionsystem x˙1 =1−x1+u (10.1) x˙2 = x n 1 1+xn1 −γx2 describes thechange in theoutput, y= x2, asdrivenbyaHill equation response to x1 and an intrinsic decay of x2 at a rate γ . In the absence of external input, u, the internal dynamics hold the concentration of x1 at the equilibriumvalue of x∗1 =1, which in turnsets theequilibriumof theoutputat x∗2 =1/2γ .Thesystemresponds toexternalcontrolsignalsorperturbationsthroughu,whichdrivestheconcentration of x1,which in turndrives theconcentrationof x2. Wecanstudythedynamicsandcontrolofthissystembylinearizingthedynamics near a particular operatingpoint, ( xˆ1, xˆ2 ) . In this example,weuse as theoperating pointtheequilibrium, ( x∗1,x∗2 ) , intheabsenceofexternalinput,u=0.Thelinearized systemexpresses thedeviations fromtheequilibriumpoint as x˙1 =−x1+u (10.2) x˙2 =nx1/4−γx2. Figure10.1showstheresponsetoanimpulseperturbationbythenonlinearsystem and the linearapproximation. In the leftpanel,withaweak impulseandsmalldevi- ation from the equilibrium, the nonlinear and linear dynamics are nearly identical. As the impulsebecomesstronger in the rightpanels, thedeviationfromequilibrium increases and the dynamics of the linear approximation diverge from the original linear system. 10.2 Regulation We can analyze the benefits of feedback for regulating nonlinear dynamics. One approachanalyzesfeedbackforthelinearapproximationnearthetargetequilibrium. The feedback for the linear approximation should provide good regulation when applied to thenonlinear systemnear theequilibrium.