Initial Condition Independent Stabilisability of Switched Affine Systems

TL;DR

This work addresses Initial Condition Independent (ICI) stabilisation of switched affine systems, showing that a bounded-frequency, periodic switching function can stabilise the system for all initial conditions if and only if a stable convex combination $A\in co(\mathcal{A})$ exists. The authors connect ICI stabilisability to the average system $\dot x = Ax$ via the Baker-Campbell-Hausdorff framework and show that high-frequency switching effectively suppresses commutator terms, aligning dynamics with the averaged model. They prove equivalences among ICI stabilisability, existence of a stable convex combination, and realizability by periodic, non-vanishing switching signals, and establish determinant-based invariance under permutation of periodic activations. For affine systems, they demonstrate that ICI stabilisation requires a common equilibrium (or vanishing switching), and that practical ICI stabilisation is possible when the associated linear part is ICI stabilisable, with the average system having equilibrium $x=-A^{-1}b$. Overall, the results provide a constructive path to bounded-frequency ICI stabilisation and clarify when convergence is to a fixed point versus a limit set, guiding switching-function design in practical applications.

Abstract

We have previously demonstrated that a switched affine system is stabilisable independently of the initial condition, i.e. there exists an asymptotically stabilising switching function which is the same for all initial conditions, if and only if there exists a stable convex combination of the sub-system matrices. This result was proven by constructing a stabilising switching function of unbounded switching frequency. The current paper proves that there exists a switching function with bounded switching frequency which stabilises a switched affine system independent of its initial condition.

Figures (2)

• Figure 1: The convergence of system \ref{['eq:switchedaffine']} to the common equilibrium point $(0, -1)$ under $\sigma(\eta t)$ for $\eta = 1.1$ (orange) and $\eta = 10^{-3}$ (blue). The value of $\eta = 10^{-3}$ approximates the average system. The initial set was taken to be the unit ball centred at the origin $B(0)$. This has been scaled by $0.5$ when $\eta = 1.1$ for presentation purposes.
• Figure 2: The convergence of \ref{['eq:switchedaffine']} under $\sigma(0.5, t)$ (green) and of the average system (blue). The average system is stable converging to the equilibrium $(0,3)$ (red). For all sufficiently small $\eta > 0$ the system converges to a closed orbit (purple). The initial set was $B(0)$.

Theorems & Definitions (8)

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