Ivancevic_Applied-Diff-Geom

542 Applied Differential Geometry: A Modern Introductiontroller (4.45) are given by:(−p(x) =q(x) =L r f h(x) + β 1L r−1f)h(x) + ... + β r−1 L f h(x) + β r h(x)L g L r−1fh(x)1L g L r−1fh(x) , (4.46)which are comprised of Lie derivatives, L f h(x). Here, the relative order, r,is the smallest integer r such that L g L r−1fh(x) ≠ 0. For linear systems r isthe difference between the number of poles and zeros.To get the desired response, we choose the r parameters in the β polynomialto describe how the output will respond to the setpoint, v (pole–placement).d r ydt r + β d r−1 y1dt r−1 + ... + β dyr−1dt + β ry = v.Here is the proposed algorithm [Isidori (1989); Sastri and Isidori (1989);Wilson (2000)]):(1) Given nonlinear SISO process, ẋ = f(x, u), and output equation y =h(x), then:(2) Calculate the relative order, r.(3) Choose an rth order desired linear response using pole–placement technique(i.e., select β). For this could be used a simple rth order low–passfilter such as a Butterworth filter.(4) Construct the exact linearized nonlinear controller (4.46), using Liederivatives and perhaps a symbolic manipulator (Mathematica orMaple).(5) Close the loop and get a linear input–output black–box (see Figure4.3).(6) Verify that the result is actually linear by comparing with the desiredresponse.Relative DegreeA nonlinear SISO system,ẋ = f(x) + g(x) u,y = h(x),is said to have relative degree r at a point x o if (see [Isidori (1989); Nijmeijerand van der Schaft (1990)])