Configuration-Constrained Tube MPC for Periodic Operation

TL;DR

The paper develops a robust, single-layer CCTMPC framework for periodic operation under polytopic linear differential inclusions, using configuration-constrained polytopic tubes to guarantee forward invariance and economic optimization over a periodic trajectory. It proves recursive feasibility and convergence to the optimal periodic invariant tube when the economic criterion is constant, and introduces a quadratic over-approximation of the periodic cost to obtain a computationally efficient QP formulation without sacrificing stability. The method is validated on a low-dimensional illustrative example and a higher-dimensional ball-plate system, showing effective tracking of the optimal PFIT and significant speedups with the QP-based approach. Practical considerations such as warm-starting and offline PFIT precomputation are discussed, highlighting pathways to further reduce online computational burden in real-time implementations.

Abstract

Periodic operation often emerges as the economically optimal mode in industrial processes, particularly under varying economic or environmental conditions. This paper proposes a robust model predictive control (MPC) framework for uncertain systems modeled as polytopic linear differential inclusions (LDIs), where the dynamics evolve as convex combinations of finitely many affine control systems with additive disturbances. The robust control problem is reformulated as a convex optimization program by optimizing over configuration-constrained polytopic tubes and tracks a periodic trajectory that is optimal for a given economic criterion. Artificial variables embedded in the formulation ensure recursive feasibility and robust constraint satisfaction when the economic criterion is updated online, while guaranteeing convergence to the corresponding optimal periodic tube when the criterion remains constant. To improve computational efficiency, we introduce a quadratic over-approximation of the periodic cost under a Lipschitz continuity assumption, yielding a Quadratic Program (QP) formulation that preserves the above theoretical guarantees. The effectiveness and scalability of the approach are demonstrated on a benchmark example and a ball-plate system with eight states.

Figures (3)

• Figure 1: Closed-loop evolution of System \ref{['eq:2DIntegrator']}. Dotted lines indicate state trajectories, with red dots marking states at $t=0$ and at the reference change. For $t\in[0,12]$, $x^{ref}_{[t+j]} = (3\cos(\theta_{[t+j]}),\frac{1}{2}\sin(\theta_{[t+j]}))+(\frac{3}{2},0)$, with $\theta_{[t+j]}=-(\frac{[t+j]}{T}2\pi)-\frac{\pi}{2}$. When $t\ge13$, $x^{ref}_{[t+j]}=(-10,0)$. Sets $X(y_0^*)$ and $X(z_0^*)$ are shown for $t\leq5$ and $t\in[13,21]$.
• Figure 2: Closed-loop evolution of system \ref{['eq:BallPlateSysEq']} under Problem \ref{['eq:periodic_MPC_implementation_1']} (yellow trajectory) and its QP approximation in Problem \ref{['eq:periodic_MPC_implementation_2']} (black trajectory), projected onto the $(x_b^1,x_b^2)$-plane. The white region denotes the feasible state set $\mathcal{X}$, and the dashed red ellipse represents the periodic output reference $r_{[t+j]} = (0.25\cos\theta_{[t+j]},\,0.2\sin\theta_{[t+j]})$, with $\theta_{[t+j]} =\pi -\frac{[t+j]}{T}2\pi$. . The system is initialized at the origin.
• Figure 3: Lyapunov functions for system \ref{['eq:2DIntegrator']} (left) and \ref{['eq:BallPlateSysEq']} (right). The vertical dashed line indicates a change in the reference.

Theorems & Definitions (11)

• Definition 1
• Definition 2
• Proposition 1
• Proposition 2
• proof
• Theorem 1
• proof
• Corollary 1
• proof
• Theorem 2