Bilevel MPC for Linear Systems: A Tractable Reduction and Continuous Connection to Hierarchical MPC

Abstract

Model predictive control (MPC) has been widely used in many fields, often in hierarchical architectures that combine controllers and decision-making layers at different levels. However, when such architectures are cast as bilevel optimization problems, standard KKT-based reformulations often introduce nonconvex and potentially nonsmooth structures that are undesirable for real-time verifiable control. In this paper, we study a bilevel MPC architecture composed of (i) an upper layer that selects the reference sequence and (ii) a lower-level linear MPC that tracks such reference sequence. We propose a smooth single-level reduction that does not degrade performance under a verifiable block-matrix nonsingularity condition. In addition, when the problem is convex, its solution is unique and equivalent to a corresponding centralized MPC, enabling the inheritance of closed-loop properties. We further show that bilevel MPC is a natural extension of standard hierarchical MPC, and introduce an interpolation framework that continuously connects the two via move-blocking. This framework reveals optimal-value ordering among the resulting formulations and provides inexpensive a posteriori degradation certificates, thereby enabling a principled performance-computational efficiency trade-off.

Figures (6)

• Figure 3: Bilevel MPC architecture. The upper MPC selects the reference sequence $\Theta$, the lower MPC computes the input $u$ to track $\Theta$, and the plant returns the state and output $(x,y)$.
• Figure 4: Equivalence between bilevel MPC and centralized MPC: the cascade of $\mathbf{P_{1}}$ or $\mathbf{P_{2}}$ and $\mathbf{P_{L}}$ realizes the same input as $\mathbf{P_{3}}$, under Assumptions \ref{['ass:plant']}, \ref{['ass:nonsing']}, and \ref{['ass:upper']}.
• Figure 5: $\mathbf{P_{0}}$
• Figure 10: Performance degradation relative to $\mathbf{P_{2}}$ for move-blocked problems (blue circle: $\mathbf{P_{2}}$($M$), green circle: $\mathbf{P_{1}}$($M$)) and HMPC $\mathbf{P_{0}}$ (red circle), together with the upper bounds. The plot shows monotonic performance improvement as the blocking is relaxed; the upper bound (bar) captures the degradation trend well, while the tighter bound (cross) is more effective under coarse blocking; and $\mathbf{P_{2}}$($M$) with the low-rank matrix $M^\star$ (broken line) outperforms $\mathbf{P_{0}}$.
• Figure 11: Controlled trajectories with HMPC $\mathbf{P_{0}}$ (green) and BMPC $\mathbf{P_{2}}$ (blue) for quadrotor example. Although the fixed target (cross) is infeasible, both controllers converge to the optimal steady state (circle); BMPC does so while satisfying the upper-level constraint (dotted lines), whereas HMPC violates it during the transient.

Theorems & Definitions (21)

• Remark 1: Steady-state map
• Example 1: Standard block–diagonal weights
• Remark 2: Why we parameterize references via $\theta$
• Theorem 1: Feasible/solution-set inclusion
• proof
• Lemma 1: Uniqueness of solution
• proof
• Theorem 2: Equivalence to centralized MPC
• proof
• Corollary 1: Inheritance of closed-loop properties