Stabilisability, invariance and optimality in switched and monotone systems

Switched systems are a class of dynamic systems which consist of distinct sub-systems activated by a switching function. They are typically characterised by discontinuous dynamics at each switching instant. We propose a stabilising switching function based on comparing the Lie derivatives of candidate Lyapunov functions for each sub-system. This provides control over the active sub-system, switching frequency and trajectory of the state. The latter has an application in the design of switching functions to avoid a region of the state space. The existence of a convex combination of the sub-systems is given as a sufficient condition for the switching function to stabilise the switched system. We prove that the existence of a convex combination is equivalent to the existence of a stabilising switching function which stabilises the system independently of the initial conditions of the system. We also consider the invariant sets of switched systems which are subject to a compact and convex disturbance. We prove that invariant set of a switched system is a strict subset of the invariant set of a time-delay system that is subject to the same disturbance. The proof compares the convex hulls of the invariant sets. Given the importance of convexity to this result we derive sufficient conditions for the invariant set of a switched system to be convex. Lastly, we derive a limitation on the controllability of a positive monotone system when the output of the system is constrained below by some specified minimum. Positive monotone systems have application as models of insulin-glucose dynamics in type one diabetes. Management attempts to minimise the maximum glucose concentrations whilst avoiding the hypoglycaemic threshold. Thus the objective is to minimise the maximum of the output of the system whilst remaining above a fixed lower bound. We prove that an input to the system minimises the maximum of the output of the system if and only if the maxima and minima of the output are interlaced. Any further lowering of the maxima, by increasing the input, results in the output falling below the specified minimum. We explore the clinical implications of the control limitation and give a numerical example of the proposed switching function being used to control plasma glucose concentrations. The understanding of the control limitation informed the choice of switching surfaces to satisfy the constraints.