  Recently, a number of counter examples have surfaced where Linear
Parameter-Varying (LPV) control synthesis applied to achieve asymptotic output
tracking and disturbance rejection for a nonlinear system, fails to achieve the
desired asymptotic tracking and rejection behavior even when the scheduling
variations remain in the bounded region considered during design. It has been
observed that the controlled system may exhibit an oscillatory motion around
the equilibrium point in the presence of a bounded constant input disturbance
even if integral action is present. This work aims at investigating how and why
the baseline Lyapunov stability notion, currently widely used in the LPV
framework, fails to guarantee the desired system behavior. Specifically, it is
shown why the quadratic Lyapunov concept is insufficient to always guarantee
asymptotic stability under reference tracking and disturbance rejection
scenarios, and why an equilibrium independent stability notion is required for
LPV stability analysis and synthesis of controllers. The introduced concepts
and the apparent pitfalls are demonstrated via a simulation example.
