Passivity of Nonlinear Incremental Systems: Application to PI Stabilization of Nonlinear RLC Circuits

It is well known that if the linear time invariant system x = Ax + Bu, y = Cx is passive the associated incremental system x~ = Ax˜ + Bũ, y = Cx˜, with (~ = (.) - (.)*, u* , y* the constant input and output associated to an equilibrium state x*, is also passive. In this paper, we identify a class of nonlinear passive systems of the form x = f(x) + gu, y = h(x) whose incremental model is also passive. Using this result we then prove that general nonlinear RLC circuits with convex and proper electric and magnetic energy functions and passive resistors with monotonic characteristic functions are globally stabilizable with linear PI control.