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˜+Bu˜,y˜=Cx˜, with (·)˜=(·)-(·)⋆, u⋆,y⋆u⋆,y⋆ the constant input and output associated to an equilibrium state x⋆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 a large class of nonlinear RLC circuits with strictly convex electric and magnetic energy functions and passive resistors with monotonic characteristic functions are globally stabilizable with linear PI control.