It is well known that if the linear time invariant systemẋ = Ax + Bu, y = Cx is passive the associated incremental systemẋ = Ax + Bũ,ỹ = 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 formẋ = 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.
It is well known that if the linear time invariant system ˙ x = Ax + Bu, y = Cx is passive the associated incremental system ˙ ˜ x = A˜xA˜x + B˜uB˜u, ˜ y = C˜xC˜x, with˜(with˜ 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 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.
a b s t r a c tIt is well known that energy-balancing control is stymied by the presence of pervasive dissipation. To overcome this problem in electrical circuits, the alternative paradigm of power shaping was introduced in Ortega, Jeltsema, and Scherpen (2003) -where, as suggested by its name, stabilization is achieved shaping a function akin to the power instead of the energy function. In this paper we extend this technique to general nonlinear systems. The method relies on the solution of a PDE, which identifies the open-loop storage function. We show through some physical examples, that the power-shaping methodology yields storage functions which have units of power. To motivate the application of this control technique we illustrate the procedure with two case studies: a tunnel diode circuit and a two-tanks system.
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