Robust IDA-PBC for underactuated mechanical systems subject to matched disturbances

Abstract

The problem of robustification of interconnection and damping assignment passivity-based control for underactuated mechanical system vis-à-vis matched, constant, and unknown disturbances is addressed in the paper. This is achieved adding an outer-loop controller to the interconnection and damping assignment passivity-based control. Three designs are proposed, with the first one being a simple nonlinear PI, while the second and the third ones are nonlinear PIDs. While all controllers ensure stability of the desired equilibrium in spite of the presence of the disturbances, the inclusion of the derivative term allows us to inject further damping enlarging the class of systems for which asymptotic stability is ensured. Numerical simulations of the Acrobot system and experimental results on the disk-on-disk system illustrate the performance of the proposed controller.

Adaptive IDA‐PBC for underactuated mechanical systems with constant disturbances

“…Evaluating (7) and replacing the disturbances with their estimates, the open-loop unstable equilibrium of (33) is q * = (q * 1 , q * 2 ) with q * 1 = sin −1 (̃1∕a 3 ). 17,18 Conversely, no condition is imposed on the matched disturbancẽ2 and q * 2 can be freely chosen. It must be highlighted that this is not dependent on the specific IDA-PBC design but is common to (3) and to the integral IDA-PBC designs.…”

“…Differently from (40), the control law (42) only depends on the position q 2 ; therefore, no attempt is made to regulate q 1 . The PID IDA-PBC design 18 was employed for comparison purposes with the following parameters: k p = 0.00025; k v = 0.8; Nevertheless, there is a clear analogy in how the disturbances are accounted for in (40) and (42): in both designs, the matched component̃1 is simply added to the control input, whilẽ2 is multiplied by the same coefficient as the nonlinear term m 3 sin (q 2 ).…”

“…The solutions of the matching conditions (4), (5) are defined as follows, with k 1 , k 2 , k 3 constant parameters 18 : ( 1 + sin(q 1 ) cos(q 1 )…”

“…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.…”