Learning-based Homothetic Tube MPC

TL;DR

This work addresses robust MPC for discrete-time linear systems with an unknown true disturbance set $\mathbb{W}_{\rm true}$ by learning a parameterised disturbance set $\mathcal{W}(v,\bm{\theta},\rho)$ that scales $\mathbb{W}$ in different directions. The learned set $\hat{\mathbb{W}}_k$ is updated online via LP-based optimization grounded in scenario approach theory, and is integrated into a learning-based homothetic tube MPC that enforces finite-horizon constraints through tube scaling factors $\alpha_{i|k}$. The authors establish probabilistic guarantees on the approximation quality of $\hat{\mathbb{W}}_k$ and prove probabilistic recursive feasibility for the MPC under the learned disturbance set, with a fixed horizon parameter $\nu$ to maintain computational tractability. Numerical examples, including a platooning scenario, illustrate that the proposed method yields larger feasible regions and improved feasibility compared with state-of-the-art MPC schemes, while maintaining reasonable computation times. Overall, the approach provides a data-driven, computationally efficient path to reduce conservativeness in tube MPC by online uncertainty quantification and learning.

Abstract

In this paper, we study homothetic tube model predictive control (MPC) of discrete-time linear systems subject to bounded additive disturbance and mixed constraints on the state and input. Different from most existing work on robust MPC, we assume that the true disturbance set is unknown but a conservative surrogate is available a priori. Leveraging the real-time data, we develop an online learning algorithm to approximate the true disturbance set. This approximation and the corresponding constraints in the MPC optimisation are updated online using computationally convenient linear programs. We provide statistical gaps between the true and learned disturbance sets, based on which, probabilistic recursive feasibility of homothetic tube MPC problems is discussed. Numerical simulations are provided to demonstrate the efficacy of our proposed algorithm and compare with state-of-the-art MPC algorithms.

Figures (4)

• Figure 1: Learning-based tube MPC framework, where $x_k$, $u_k$, $w_k$ are state, control input, and disturbance respectively; $\hat{\mathbb{W}}_k$ is the learned disturbance set; and $\hat{w}_{\max,k}$ is the worst-case estimation.
• Figure 2: The disturbance set and feasible region. (a) Learned disturbance set by homothetic approach and rigid approach of gao2023learning with $|\mathcal{I}_0|=30$, the worst-case disturbance set, and the true disturbance set; (b) The initial feasible region of the learning-based homothetic MPC, the learning-based rigid MPC, and the conventional homothetic MPC with $|\mathcal{I}_0|=30$.
• Figure 3: Trajectories and control inputs of the relative model. (a) State trajectories for three methods with the initial state as state 1 in Fig. \ref{['Fig: disturbance set and feasible region']}(b); (b) State trajectory for three methods with the initial state as state 2 in Fig. \ref{['Fig: disturbance set and feasible region']}(b).
• Figure 4: (a) Control inputs of the follower agent by three methods corresponding to the state trajectories in Fig. \ref{['Fig:path']}(a); (b) Control inputs of the follower agent by three methods corresponding to the state trajectories in Fig. \ref{['Fig:path']}(b).

Theorems & Definitions (11)

• Lemma 1: kouvaritakis2016model
• Lemma 2: kouvaritakis2016model
• Lemma 3
• proof
• Proposition 1
• proof
• Theorem 1: 5531078
• Theorem 2
• Lemma 4
• proof