Integral Control of Port-Hamiltonian Systems: Nonpassive Outputs Without Coordinate Transformation

Abstract

Abstract-In this paper we present a method for the addition of integral action to non-passive outputs of a class of portHamiltonian systems. The proposed integral controller is a dynamic extension, constructed from the open loop system, such that the closed loop preserves the port-Hamiltonian form. It is shown that the controller is able to reject the effects of both matched and unmatched disturbances, preserving the regulation of the non-passive outputs. Previous solutions to this problem have relied on a change of coordinates whereas the presented solution is developed using the original state vector and, therefore, retains its physical interpretation. In addition, the resulting closed loop dynamics have a natural interpretation as a Control by Interconnection scheme.

Adaptive IDA‐PBC for underactuated mechanical systems with constant disturbances

“…Consequently, the equilibrium (q, p) = (q * , 0) is asymptotically stable if z = 0. According to Proposition 1, z is vanishing exponentially, and hence, (q * , 0) is also a locally stable equilibrium for the complete system (19)- (20). In order to prove global asymptotic stability, it is necessary to establish boundedness of the trajectories q(t), p(t).…”

“…According to Proposition 1, z is vanishing exponentially, and hence, (q * , 0) is also a locally stable equilibrium for the complete system (19)- (20). According to Proposition 1, z is vanishing exponentially, and hence, (q * , 0) is also a locally stable equilibrium for the complete system (19)- (20).…”

“…Computing the time derivative of W ′ along the trajectories of (19)- (20) giveṡ Computing the time derivative of W ′ along the trajectories of (19)- (20) giveṡ…”

“…7,8 In parallel, the effect of continuous and smooth physical dissipation within IDA-PBC was studied in the works Delgado and Kotyczka 9 and Gómez-Estern and Van der Schaft, 10 while Dahl friction on actuated joints was considered in the work of Sandoval et al 11 Recently, an IDA-PBC design with adaptive friction compensation was proposed in the work of Franco. More sophisticated integral IDA-PBC designs for underactuated mechanical systems with matched disturbances, constant or bounded, were proposed in the works of Donaire et al 18 and Haddad et al 19 The extension of integral IDA-PBC to the case of unmatched disturbances for mechanical systems with constant inertia matrix was investigated in the work of Ferguson et al 20 In this respect, it must be highlighted that unmatched disturbances represent a particularly challenging condition even for linear systems, since control input and disturbances are not linearly dependent. Furthermore, an H-infinity control was proposed for the inertia-wheel pendulum with bounded disturbances in the work of Binh et al 14 and an adaptive neural network control was employed to compensate friction forces for the Furuta pendulum in the work of Moreno-Valenzuela et al 15 In summary, most research on nonlinear underactuated mechanical systems with disturbances has been focusing on friction, while the constant disturbance rejection problem has remained relatively unexplored.…”

“…Initial results in this FRANCO respect were presented in the works of Teo et al 16 and Ryalat et al, 17 where integral control was overlaid to IDA-PBC in order to compensate constant matched disturbances (ie, only affecting the actuated joints). More sophisticated integral IDA-PBC designs for underactuated mechanical systems with matched disturbances, constant or bounded, were proposed in the works of Donaire et al 18 and Haddad et al 19 The extension of integral IDA-PBC to the case of unmatched disturbances for mechanical systems with constant inertia matrix was investigated in the work of Ferguson et al 20 In this respect, it must be highlighted that unmatched disturbances represent a particularly challenging condition even for linear systems, since control input and disturbances are not linearly dependent. 21,22 Additionally, differently from friction, constant disturbances do not vanish at equilibrium.…”