An improved Wien bridge oscillator (original) (raw)

Consider the smooth (C") nonlinear dynamical system dx/dr = f (.x) + g(x) U with f and g locally defined smooth \rector fields and /I a C" function. Equation (A.l) is referred as the triple (f , g , h). The set hK'(0) = { x : y = h (x) = O)} is assumed to be a lo cull^ regukur ititegrohle niunrfold (Boothby [ll]). L,v denotes the Lie dcrivative (directional derivative [ll]) of the smooth function 11 in the direction of the vector field +. Definrrioti A.1 /7]: A variable structure feedback control law: .p = h (x) (A.1) U + (x) , U-(.x) , for y > 0 for y < 0 is said to locally create a sliding regirtit. on hK'(0) if and only if lim L,+gu+ h < O and lim L,+gu-/7 > 0 (A.3) y-+ o ,- .-0 where u ' (x) and U-(x) are given smooth scalar feedback control functions. Without loss of generality we assume that, locally, u ' (x) > u-(x) .