  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 saturation is typically neglected in the design of control systems,
with compensation being made in the form of over-designing the actuator or by
post-analyzing the resulting system to ensure acceptable performance. The
bounded input control of fully actuated systems has been investigated in
multiple studies, but it is not generalized for under actuated 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 (IDA-PBC) approach. The proposed
method also offers design variables for the control law to be tuned,
considering the actuator's limit. The major 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.
