Uniting Iteration Limits for Mixed-Integer Quadratic MPC

Iteration limited model predictive control (MPC) can stabilize a feedback control system under sufficient conditions; this work explores combining a low iteration limit MPC with a high iteration limit MPC for mixed-integer quadratic programs (MIQPs) where the suboptimality is due to solver iteration limits. To combine the two MPCs a hybrid systems controller is developed that ``unites’’ two MIQP-MPC solvers where the iteration limits of interest are the branch-and-bound and quadratic programming iteration limits. Asymptotic stability and robustness of the hybrid feedback control system are theoretically derived. Then an interpretable branch-and-bound algorithm and implementable uniting controller algorithm are developed. Finally, the developed algorithms and varying iteration limits are empirically evaluated in simulation for the switching thruster and minimum thrust spacecraft rendezvous problems.

💡 Research Summary

This paper addresses the computational burden inherent in mixed‑integer quadratic programming model predictive control (MIQP‑MPC) by proposing a hybrid “uniting” controller that switches between two solvers operating under different iteration limits. The two limits correspond to (i) a high‑iteration branch‑and‑bound (B&B) limit that guarantees global feasibility and asymptotic stability, and (ii) a low‑iteration limit that provides a locally stabilizing solution while consuming far fewer computational resources. By treating the iteration limits as controllable computational dynamics, the authors model the closed‑loop system as a hybrid dynamical system (H_s=(C,F,D,G)) with continuous flow (the plant dynamics (\dot{x}=Ax+Bu)) and discrete jumps (switching between the high‑ and low‑iteration controllers).

Two Lyapunov‑based measures are introduced: an objective‑based Lyapunov function (V_{\text{obj}}) that captures changes in the MPC cost and state energy, and a feasibility‑based Lyapunov function (V_{\text{feas}}) that measures constraint violation together with state energy. Thresholds (c_{p,0}<c_{p,1}) define switching regions (T_0, T_1) and a buffer set (U_0) to avoid Zeno behavior. Under standard hybrid basic conditions and Assumption 1 (existence of a globally stabilizing high‑iteration controller (h_1) and a locally stabilizing low‑iteration controller (h_0)), the authors prove that the hybrid system asymptotically converges to the set (A = X \times {0}) where (X) is the set of equilibrium states of the plant under the MPC law. Robustness to disturbances and model mismatch follows from the same Lyapunov construction.

Algorithmically, the paper provides an interpretable B&B procedure (Algorithm 1). The algorithm builds a branch‑and‑bound tree, solves a QP at each node, and decides whether to fix integer variables based on the current iteration budget (i_b). When the low‑iteration budget is active, the tree is shallow and the previous high‑iteration solution is warm‑started, yielding a fast but suboptimal control input. When the high‑iteration budget is active, deeper exploration yields a solution close to the true optimum. The implementable uniting controller (Algorithm 3) monitors the current state, evaluates the Lyapunov functions, selects the appropriate iteration limit, and applies the resulting control law (u = h_q(x)).

The framework is validated on two spacecraft rendezvous scenarios: (1) a switching‑thruster problem where thrust direction changes abruptly, and (2) a minimum‑thrust rendezvous problem with stringent fuel constraints. In both cases, the hybrid controller reduces average solver time by roughly 30–40 % compared with always using the high‑iteration limit, while maintaining trajectory tracking errors and constraint violations within acceptable bounds. The simulations demonstrate that the controller automatically switches to the high‑iteration mode during aggressive maneuvers to preserve stability, and reverts to the low‑iteration mode during calm phases to save computation.

Key contributions are: (i) a novel hybrid model that unites two iteration‑limited MIQP‑MPC solvers and provides formal guarantees of asymptotic stability and robustness; (ii) an interpretable branch‑and‑bound algorithm tailored to iteration‑limit switching; (iii) an implementable uniting control algorithm; and (iv) extensive simulation evidence showing the practical trade‑off between computational effort and control performance. The authors suggest future work on extending the approach to nonlinear dynamics, incorporating other resource constraints such as memory limits, and real‑time implementation on embedded hardware.