Commutator-Driven Stability Bounds for Periodic Switching
Provides a rigorous, computable stability certificate for periodically switched linear systems, addressing a fundamental gap in averaging theory for control engineers.
The paper derives an explicit operator-norm bound for the mismatch between periodically switched and averaged propagators, showing that stability of the averaged model guarantees exponential stability of the switched system when the switching period is below a computable threshold. The bound depends on pairwise commutator norms and yields a semidefinite program for duty-dependent Lyapunov metrics.
Averaged models are widely used to analyze periodically switched linear systems, yet the stability of the averaged flow does not automatically guarantee the stability of the true switched dynamics. The discrepancy arises from noncommutativity among the subsystem generators, so stability certificates benefit from bounds that expose this dependence in a form compatible with Lyapunov contraction metrics. We derive an explicit operator-norm bound for the one-period mismatch between the switched and averaged propagators, in which the leading-order error depends explicitly on the pairwise commutator norms of the scaled mode generators, with a closed-form prefactor depending only on the generator norms. This bound yields a computable threshold for the switching period below which the switched system inherits exponential stability from its averaged model, uniformly certified over admissible duty fractions. The analysis extends to an arbitrary number of switching modes via telescoping induction, and a semidefinite program provides sampled duty-dependent Lyapunov metrics for implementing the certificate.