  Dynamical compensation (DC) provides robustness to parameter fluctuations. As
an example, DC enable control of the functional mass of endocrine or neuronal
tissue essential for controlling blood glucose by insulin through a nonlinear
feedback loop. Researchers have shown that DC is related to structural
unidentifiability and $\mathbb{P}$-invariance property, and
$\mathbb{P}$-invariance property is a sufficient and necessary condition for
the DC property. In this article, we discuss DC and $\mathbb{P}$-invariancy
from an adaptive control perspective. An adaptive controller is a self-tuning
controller used to compensate for changes in a dynamical system. To design an
adaptive controller with the DC property, it is easier to start with a
two-dimensional dynamical model. We introduce a simplified system of ordinary
differential equations (ODEs) with the DC property and extend it to a general
form. The value of the ideal adaptive control lies in developing methods to
synthesize DC to variations in multiple parameters. Then we investigate the
stability of the system with time-varying input and disturbance signals, with a
focus on the system's $\mathbb{P}$-invariance properties. This study provides
phase portraits and step-like response graphs to visualize the system's
behavior and stability properties.
