Ideal adaptive control in biological systems: an analysis of \(\mathbb {P}\)-invariance and dynamical compensation properties

Dynamical compensation (DC) provides robustness to parameter fluctuations. As an example, DC enables 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 the structural unidentifiability and the \(\mathbb {P}\) $\mathbb{P}$ -invariance property. The \(\mathbb {P}\) $\mathbb{P}$ -invariance property is a sufficient and necessary condition for the DC property. DC has been seen in systems with at least three dimensions. In this article, we discuss DC and \(\mathbb {P}\) $\mathbb{P}$ -invariance from an adaptive control perspective. An adaptive controller automatically adjusts its parameters to optimise performance, maintain stability, and deal with uncertainties in a system.

We initiate our analysis by introducing a simplified two-dimensional dynamical model with DC, fostering experimentation and understanding of the system’s behavior. We explore the system’s behavior with time-varying input and disturbance signals, with a focus on illustrating the system’s \(\mathbb {P}\) $\mathbb{P}$ -invariance properties in phase portraits and step-like response graphs.

We show that DC can be seen as a case of ideal adaptive control since the system is invariant to the compensated parameter.