Bounded inputs total energy shaping for a class of underactuated mechanical systems

Abstract

Designing control systems with bounded input is a practical consideration since realizable physical systems are limited by the saturation of actuators. The actuators' saturation degrades the performance of the control system, and in extreme cases, the stability of the closed‐loop system may be lost. However, actuator's saturation is typically neglected in the design of control systems, with compensation being made in the form of overdesigning the actuator or by postanalyzing the resulting system to ensure acceptable performance. The bounded input control of fully actuated mechanical systems has been investigated in multiple studies, but it is not generalized for underactuated mechanical systems. This article proposes a systematic framework for finding the upper bound of control effort in underactuated systems, based on interconnection and the damping assignment passivity based control approach. The proposed method also offers design variables for the control law to be tuned, considering the actuator's limit. The primary difficulty in finding the control input upper bounds is the velocity‐dependent kinetic energy‐related terms. Thus, the upper bound of velocity is computed using a suitable Lyapunov candidate as a function of closed‐loop system parameters. The validity and application of the proposed method are investigated in detail through two benchmark systems simulations.

Integral passivity‐based control of underactuated mechanical systems with actuator dynamics and constant disturbances

“…where S 13 and S 23 are given in (15), 𝜍 is given in (14), and the integral states 𝜁 0 and 𝜁 1 are defined by the update laws…”

“…12,13 Further works have investigated the effect of actuator saturation. 14 In other cases, the actuator dynamics has been neglected for simplicity, considering that many actuators have higher bandwidth compared to the corresponding mechanical subsystems. 15 In general, accounting for the actuator dynamics results in a system that is not input affine and where the disturbances affecting the mechanical subsystem may not be directly canceled by the control input.…”

“…More recently, an IDA‐PBC design has been proposed for underactuated mechanical systems with fluidic actuation 12,13 . Further works have investigated the effect of actuator saturation 14 . In other cases, the actuator dynamics has been neglected for simplicity, considering that many actuators have higher bandwidth compared to the corresponding mechanical subsystems 15 .…”

Integral passivity‐based control of underactuated mechanical systems with state‐dependent matched disturbances

“…Proof. Consider the storage function W d defined in (11), which is positive definite by design. Computing the time derivative of W d along the trajectories of the closed-loop system (10) while simplifying skew symmetric terms in S 22 and S 33 yields…”

Energy shaping control of underactuated mechanical systems with fluidic actuation

“…Recently, more sophisticated designs were proposed to compensate the effects of matched disturbances 10 and of unmatched disturbances under some structural assumptions 11 . In addition, the effect of input saturation was investigated in Reference 12.…”