Robust tube-based MPC for switched systems with bounded disturbances

Develop a robust tube-based model predictive control formulation for switched systems subject to bounded additive disturbances by constructing a robust nominal switching target set that jointly addresses constraint tightening, inter-step reachability or invariance, tube re-centering under admissible mode transitions, and an appropriate practical-stability condition, and establish the resulting practical-stability guarantees.

Background

The paper establishes a variable-horizon switched MPC framework for constrained switched linear systems with unknown switching signals and mode-dependent dwell-time constraints. The authors note that this framework could be extended to bounded additive disturbances using a tube-based MPC construction: local feedback gains and robust positively invariant error sets would generate tightened nominal state and input constraints, after which the predecessor-set algorithms could be applied to the nominal system.

The unresolved issue is the construction and analysis of the robust nominal switching target set. Because the mode-dependent error tube does not contract with the scalar factor used to scale the nominal switching feasible set, nominal-set scaling alone cannot guarantee contraction of the actual-state set or closed-loop stability. The future work therefore requires a joint treatment of constraint tightening, inter-step reachability or invariance, tube re-centering across admissible mode transitions, and practical stability.

References

The proposed framework can be extended to a robust tube-based MPC formulation for switched systems subject to bounded additive disturbances. Following , for each mode $m$, a local gain $K_ m$ and a robust positively invariant set $\mathbb Z_m$ can be constructed for the error dynamics, yielding the tightened nominal constraints of $\hat{\mathbb{X}_m = \mathbb{X}_m \ominus \mathbb{Z}_m$ and $\hat{\mathbb{U}_m = \mathbb{U}_m \ominus K_m \mathbb{Z}_m$. The predecessor-set computations by Algorithms 1 and 2 are then performed for the nominal system using these tightened constraint sets to construct the corresponding nominal terminal and switching feasible sets. Let $\hat\Theta_m(\lambda,s)$ denote a robust nominal switching target set to be designed. Its construction should jointly address constraint tightening, inter-step reachability or invariance, tube re-centering under admissible mode transitions, and an appropriate practical stability condition. Since the mode-dependent tube does not contract with $\lambda$, nominal-set scaling alone could not establish contraction of the actual-state set or closed-loop stability. The detailed set construction and practical-stability analysis are left for future investigation.

— Variable-Horizon Model Predictive Control for Switched Systems  (2609.29100 - Zhao et al., 24 Sep 2026) in Remark following Theorem 2, Section 4 (Stability)