Variable-Horizon Model Predictive Control for Switched Systems

1 Introduction

• MPC handles physical state and input constraints through finite-horizon prediction and constrained optimization, but generally requires accurate models and switching-signal information.Source: §1 · p. 1
• Unknown switching creates a circular feasibility dependency: dwell-time requirements depend on the switching-instant state, which must satisfy dwell-time-driven constraints.The pre- and post-switching models may differ substantially.Source: §1 · p. 2
• Fixed-horizon geometric approaches can impose excessively restrictive dwell-time conditions, while min-max switching MPC has high online computational demands and a restricted domain of attraction.Switch-robust invariant-set approaches have focused on constrained enforcement rather than asymptotic convergence.Source: §1 · p. 2
• The proposed variable-horizon framework steers the state into a switching feasible set, structurally decoupling dwell-time conditions from MPC constraints and aligning them with unconstrained-system requirements.Compared with Zhang et al. (2016), it reduces conservatism and provides structurally simpler conditions.Source: §1 · p. 2
• Algorithms construct the switching feasible set and characterize the domain of attraction, while the framework guarantees persistent feasibility at switching instants and closed-loop asymptotic stability.Short-horizon schemes tailor dwell time to the initial mode transition, circumvent short-horizon geometric limitations, and enlarge the domain of attraction without sacrificing relaxed dwell-time benefits for later transitions.Source: §1 · p. 2

2 Problem Formulation

• The system has stabilizable discrete-time subsystems, and its switching signal is unknown in advance.Source: §2 · p. 2
• Mode-dependent dwell time requires each interval between consecutive switches to be at least the dwell-time bound for the active subsystem.The duration of the current mode is measured from its latest switching instant.Source: §2 · p. 3
• The MPC optimization predicts subsystem dynamics subject to mode-dependent state and input constraints, a contracted switching-set constraint, and a terminal-set constraint.The objective uses state and input costs with positive-definite weighting matrices, and the terminal weight is chosen from a DARE solution.Source: §2 · p. 3
• At each switch, the horizon resets to its maximum and the control sequence steers the state into the contracted switching feasible set within that maximum horizon.As the horizon reaches its minimum, the scheme reduces to fixed-horizon MPC; the terminal constraint is T∞.Source: Figure 2 · p. 3, §2 · p. 4
• The formulation targets construction of the contracted switching feasible set, persistent feasibility, and computation of a stability-guaranteeing domain of attraction.Source: §2 · p. 4

3.1 Switching Feasible Set

• Algorithm 1 constructs the switching feasible set through iterative backward reachability, ensuring admissible predicted trajectories for states in the set.Source: §3.1 · p. 4
• If the unconstrained switched system is asymptotically stable under the mode-dependent dwell-time constraints, some λ makes both the switching terminal set and switching feasible set nonempty.Source: §3.1 · p. 4
• Co-designing the set with the variable-horizon mechanism replaces one-step invariance with multi-step reachability, allowing intermediate states outside Ξλ while respecting physical bounds.Source: §3.1 · p. 5

3.2 Persistent Feasibility

• With initial feasibility, ∆Nm ≤ τd,m, and Θ(λ, s) = λ^sΞλ∞, the MPC problem remains feasible at all future instants.Source: §3.2 · p. 5
• Using predefined dwell times to design the horizon and steer states into the switching feasible set avoids complex recursive minimum-dwell-time calculations, reducing computational complexity.The authors identify improved real-world deployability as a further advantage.Source: §3.2 · p. 5

• The domain-of-attraction computation identifies initial states for which the MPC problem is feasible, and the switching feasible set lies within each mode-specific feasible set.Source: §4 · p. 5
• If ∆Nm ≤ τd,m, the dwell times satisfy the stated conditions, and x0 lies in the computed domain Fσ(0), the closed-loop switched system is asymptotically stable.Source: §4 · p. 5
• Convergence follows for infinitely many switches from a bound shrinking with λ^2s, while finitely many switches leave the system eventually in one fixed mode, where the state converges to zero.Source: §4 · p. 6
• The proposed dwell-time condition has a simpler subsystem-wise form and less conservative bounds; its sufficiency is tied to unconstrained stability of the feedback-controlled switched system.Source: §4 · pp. 5–6
• For bounded-disturbance tube MPC, nominal-set scaling alone does not establish actual-state contraction or closed-loop stability; detailed construction and practical-stability analysis remain future work.Source: §4 · p. 6
• Shortening the horizon can degrade performance, though using the full horizon immediately after switches and a DARE-based terminal weight can mitigate truncation effects.Source: §4 · p. 6
• Increasing the maximum horizon can enlarge the feasible region but increases computational burden and required dwell time; the short-horizon scheme aims to enlarge that region without excessively long horizons.Source: §4 · p. 6

5 Short-Horizon Switched MPC

• The schemes target a feasibility squeeze: when dwell time nears the maximum horizon, the required minimum horizon may not be small enough to avoid restricting the feasible set.The proposed short-horizon schemes are introduced to mitigate this restriction.Source: §5 · p. 6

5.1 Short-Horizon Switched MPC for N max

• Intermediate predecessor sets bridge the gap when the bounded horizon cannot directly detect the terminal set at switching, guiding the state toward it.After the mode lasts sufficiently long, the scheme reduces to standard variable-horizon MPC.Source: §5.1 · p. 7
• When the maximum horizon exceeds the dwell time, the first scheme selects a shorter minimum horizon and guarantees asymptotic stability under its stated horizon, dwell-time, and initial-state conditions.The initial state must belong to F_m constructed by Algorithm 2 using the desired horizon.Source: §5.1 · p. 7

5.2 Short-Horizon Switched MPC for N max

• The second short-horizon formulation relaxes the coupling between dwell time and prediction-horizon bounds and guarantees closed-loop asymptotic stability under its stated conditions.The stability result assumes dwell time satisfying (5), suitable horizon parameters, and an initial state in the computed domain of attraction.Source: §5.2 · p. 7
• The short-horizon design systematically expands the feasible region, with its domain of attraction coinciding with the set computed by Algorithm 2 for the desired maximum horizon.For fixed minimum horizon and enlarged initial dwell time, the resulting domain of attraction coincides exactly with F_m generated using N*_m.Source: §5.2 · p. 7

Attraction domains and trajectories

• Every initial state in the computed domain is feasible under the mode-dependent dwell-time constraint, and all shown trajectories converge to the origin within 20 time steps.The switching feasible set has volume 38.5133 and is strictly contained in a set of volume 38.7460.Source: Figure 5 · p. 8, §6 · p. 8

Comparison with Ong et al. (2016)

• The proposed attraction domain strictly contains that of Ong et al. (2016), and the proposed scheme bypasses that method’s max-min optimization.The comparison reports a strictly larger volume for the proposed domain.Source: Figure 8 · p. 9, §6 · p. 9

Comparison with Zhang et al. (2016)

• For the system matrices from Zhang et al. (2016), the proposed method requires dwell times τd,1 = 2 and τd,2 = 4, which are infeasible under that method’s stricter conditions.The paper reports less restrictive dwell-time constraints than Zhang et al. (2016).Source: §6 · p. 9, Table 2 · p. 9

Comparison with Danielson et al. (2019)

• With dwell times τ1 = 4 and τ2 = 5, the proposed switching feasible set expands while the switch-RCI sets remain constrained by one-step invariance.The two constructions impose different set conditions, so neither has a general inclusion relationship with the other.Source: §6 · pp. 9–10, Figure 9 · p. 9

Computational comparison

• The proposed method has slightly higher offline computation time than Ong et al. (2016) but is faster offline than Zhang et al. (2016) and Danielson et al. (2019), with competitive online efficiency in tested configurations.For nx = 3, Danielson et al. (2019) did not converge within 300 iterations, exceeding 500 s; extending Zhang et al. (2016) to N = 7 for M = 2, nx = 2 raised its online time to 0.1709 s.Source: §6 · p. 10, Table 3 · p. 10

Short-horizon switched MPC

• The short-horizon switched MPC delivers strictly larger domains of attraction than those reported in Zhang et al. (2016), and Fig. 11 state trajectories confirm closed-loop asymptotic stability.Source: §6 · p. 10

Conclusion

• The proposed algorithm computes a switching feasible set, whose use guarantees persistent feasibility when prediction-horizon and dwell-time conditions are satisfied.The dwell-time conditions are derived from the corresponding unconstrained closed-loop subsystems and ensure stability of the switched systems under the proposed MPC.Source: §7 · pp. 10–11
• Two short-horizon switched MPC schemes relax the requirement that dwell time be less than the maximum prediction horizon and expand the domain of attraction.Source: §7 · p. 11
• The simulations with comparisons illustrate the performance of the proposed switched MPC scheme.Source: §7 · p. 11

A.1 2D Switched System with four modes

• Two additional subsystems, Modes 3 and 4, are appended to the two-dimensional system.Source: §A.1 · pp. 11–12
• Modes 3 and 4 share state constraints X3 = X4 = {x ∈ R2 : ∥x∥∞ ≤ 5} and input constraints U3 = U4 = {u ∈ R : |u| ≤ 1}.Source: §A.1 · p. 12
• For both added modes, Q3 = Q4 = I and R3 = R4 = 4; their terminal penalty matrices and feedback gains are derived via unconstrained LQR, with τd,3 = τd,4 = 2.Source: §A.1 · p. 12

A.2 3D Switched System

• The higher-dimensional test uses a three-dimensional switched system comprising two subsystems.Source: §A.2 · p. 12
• The state bounds are ∥x∥∞ ≤ 10 for subsystem 1 and ∥x∥∞ ≤ 9 for subsystem 2; the respective input bounds are 1.5 and 2.Source: §A.2 · p. 12
• The LQR weighting matrices are Q1 = 10I, Q2 = 5I, and R1 = R2 = 1, yielding the corresponding terminal penalty matrices and feedback gains.Source: §A.2 · p. 12

B Short-Horizon Switched MPC for N max

• Intermediate transitional predecessor sets bridge the restricted prediction window at switching and ensure the terminal requirement is met after sufficient mode duration.The purple regions in Fig. A.1 represent this transitional phase.Source: B Short-Horizon Switched MPC for N max · p. 12, Figure A.1 · p. 12
• The terminal constraint is a recursively constructed predecessor set that guarantees the state reaches T∞ exactly at step N*.This replaces T∞ itself as the terminal constraint in the short-horizon formulation.Source: B Short-Horizon Switched MPC for N max · p. 12
• When post-switch dwell time becomes sufficiently long, the short-horizon scheme reduces to standard variable-horizon MPC.After entering T∞, the predicted trajectory follows the local LQR law u(k) = Kσ(k)x(k).Source: B Short-Horizon Switched MPC for N max · p. 12
• Feasibility at one instant guarantees feasibility at the next, and the feasible initial states form the domain of attraction for which asymptotic stability is guaranteed.The problem is equivalent to standard switched MPC configured with maximum horizon N* and minimum horizon N_min.Source: B Short-Horizon Switched MPC for N max · p. 13

C Short-Horizon Switched MPC for N max

• With four subsystems, Zhang et al.'s τd,1 fluctuates between 2 and 7 across stages, despite unchanged first-two subsystem dynamics.In the two-subsystem case, τd,1 converges to 2 after stage 8.Source: C Short-Horizon Switched MPC for N max · p. 13
• Zhang et al.'s dwell-time variations show no clear monotonicity or physical interpretability and are attributed to combinatorial complexity in multi-step reachable-set operations.The passage distinguishes this source of conservatism from the intrinsic properties of individual subsystems.Source: C Short-Horizon Switched MPC for N max · p. 13
• τd,i = 2 for every subsystem, regardless of total mode count or prediction stage, under the proposed scheme.Table C.1 compares dwell-time constraints for Zhang et al. (2016) and the proposed scheme.Source: C Short-Horizon Switched MPC for N max · p. 13, Table C.1 · p. 13