Learning-Based Homothetic Tube MPC

• The paper introduces a learning-based homothetic tube MPC that dynamically refines uncertainty sets using online measurements, significantly reducing conservatism.
• It combines classical tube MPC with adaptive learning techniques to provide robust constraint satisfaction and closed-loop stability under parametric uncertainties and disturbances.
• Empirical evaluations show improved feasibility, reduced conservatism, and real-time computation with rigorous probabilistic and stability guarantees.

Learning-based homothetic tube Model Predictive Control (MPC) refers to a paradigm for robust output-feedback control of systems with parametric uncertainty and unknown-but-bounded disturbances, where uncertainty sets are dynamically refined online using learning or set-membership estimation. This approach combines the classical homothetic tube MPC structure with adaptive set updates, enabling the controller to reduce conservatism and maintain rigorous guarantees on constraint satisfaction and closed-loop stability in the face of evolving model knowledge.

1. Fundamental Principles

Homothetic tube MPC constructs a sequence of polytopic tubes around nominal state (or state estimate) trajectories. Each tube section is a scaled (homothetic) copy of a fixed base polytope. Guaranteeing that the true plant state remains within these tubes at each prediction step allows robust handling of disturbance and modeling uncertainty. In learning-based variants, the representation or size of uncertainty sets—in particular, disturbance sets and parameter sets—is adapted online using measurements, learning rules, or set-membership identification, leading to adaptive, less conservative tube geometry as the quality of model knowledge improves. This coupling is key for optimizing performance without sacrificing recursive feasibility or constraint robustness (Gao et al., 6 May 2025, Dey et al., 6 Feb 2025, Dey et al., 16 Mar 2026, Parsi et al., 2022).

2. System and Uncertainty Modeling

The systems of interest are discrete-time (possibly continuous-time) linear or nonlinear plants with affine parametric dependence and additive disturbances: $$x_{k+1} = A(\psi)x_k + B(\psi)u_k + d_k, \quad y_k = Cx_k$$ where $\psi$ is an unknown parameter vector in a convex compact set $\Psi$. The disturbance $d_k$ or $w_k$ lies in an unknown, compact set $W^*\subseteq W$, with $W$ a conservative initial guess. State and input constraints are imposed via convex polytopes, typically $x_k\in \mathbb{X}$ and $u_k\in\mathbb{U}$, all containing the origin (Gao et al., 6 May 2025, Dey et al., 6 Feb 2025).

Uncertainty sets for parameters, disturbance, and estimation error are recursively constructed using set-membership principles, for example by intersecting prior feasible sets with hyperplane-defined “non-falsified” regions derived from new measurements (Dey et al., 16 Mar 2026, Parsi et al., 2022, Gao et al., 6 May 2025). In output-feedback cases, a robust adaptive observer provides online state and parameter estimates, propagating error set dynamics and recursively updating polytopic error bounds (Dey et al., 6 Feb 2025).

3. Homothetic Tube Construction and Propagation

The predicted state (or state estimate) at time $i$ and current time $\psi$0 is decomposed as $\psi$1, where $\psi$2 tracks the nominal trajectory and $\psi$3 lies within a homothetic section $\psi$4 for base polytope $\psi$5 and scaling $\psi$6 (Gao et al., 6 May 2025). Error set propagation interprets the closed-loop error dynamics with feedback gain $\psi$7 (ensuring $\psi$8 stability): $\psi$9 To guarantee that $\Psi$0 holds forward in time, robust propagation inequalities are imposed: $\Psi$1 where $\Psi$2 and $\Psi$3 denote maximal images under $\Psi$4 and $\Psi$5 over $\Psi$6 and $\Psi$7, respectively.

In output-feedback or parameter-adaptive cases, a “two-tier” tube structure is used, where an inner tube (around the current state estimate trajectory) addresses nominal error and an outer tube is defined by Minkowski addition with estimation error sets, capturing the total uncertainty envelope for the true state (Dey et al., 6 Feb 2025).

4. Online Learning and Set Update Algorithms

Three main learning mechanisms for updating uncertainty sets are central in this context:

• Disturbance set approximation: LP-based scenario optimization is used to fit a disturbance set from observed samples, with updates adding new samples and preserving outer approximations. The resulting sequence $\Psi$8 is non-decreasing and converges to minimal-volume estimates with rigorous statistical gap bounds (Gao et al., 6 May 2025).
• Parameter set refinement: Set-membership identification exploits system measurements and knowledge of bounded disturbances to intersect prior parameter uncertainty (e.g., a polytope) with regions non-falsified by recent data, yielding $\Psi$9 and thereby reducing allowable model mismatch (Dey et al., 16 Mar 2026, Parsi et al., 2022). Associated parameter estimates are updated via projected normalized-gradient or filtered-regression laws, always projected back into the feasible set.
• Estimation error tracking: In output-feedback architectures, error set recursion and robust adaptive observers update (potentially time-varying) polytopic bounds for state and parameter estimation errors, feeding into tube constraint tightening (Dey et al., 6 Feb 2025).

These mechanisms enable the MPC to dynamically adjust tube cross-sections, tighten constraints, and reduce conservatism as models adapt.

5. MPC Optimization Problem Formulation

Homothetic tube MPCs define a convex optimization at each sample, with variables comprising nominal trajectories ($d_k$0 or $d_k$1), tube scaling factors ($d_k$2), and in some cases control vertices or additional slack/dual variables (for exact set-membership embedding (Parsi et al., 2022)). Standard objectives penalize nominal state and input costs as well as tube width (to favor reduced conservatism):

$d_k$3

Constraints include:

• Initial membership: $d_k$4
• Tube propagation: As above, to ensure $d_k$5 for all $d_k$6
• State/input constraint tightening: e.g., $d_k$7
• Terminal constraints: Nominal terminal state and tube section within a robust invariant set at horizon end
• Additional constraints, when relevant, on state/parameter estimation error sets and projection of parameter updates

This problem is a QP/LP (with complexity $d_k$8 for typical instances (Gao et al., 6 May 2025)). All constraints can be precomputed for tractability, and only a small number of online LPs are required per step for set update and disturbance bound calculation.

6. Recursive Feasibility, Robustness, and Stability Guarantees

A key property of learning-based homothetic tube MPC is the guarantee of recursive feasibility and robust constraint satisfaction throughout online operation. The central theorems (e.g., (Dey et al., 6 Feb 2025, Dey et al., 16 Mar 2026, Gao et al., 6 May 2025, Parsi et al., 2022)) demonstrate that, provided the optimization is feasible at initialization, the closed-loop solution remains feasible for all future times with quantifiable probabilistic or robust guarantees. This is enabled by contractive tube design, Lyapunov-based terminal conditions, and rigorous statistical bounds on set learning (scenario results).

Most formulations ensure robust exponential stability of the closed-loop system, with the true plant state converging to a small invariant neighborhood around the origin (size determined by disturbance bound and residual uncertainty). The two-tier tube construction guarantees robustness to estimation and model errors in the presence of incomplete state information (Dey et al., 6 Feb 2025).

Statistical results ensure that if the initial learned disturbance set covers the true disturbance set except for an $d_k$9-fraction (with confidence $w_k$0), then feasibility is preserved with the same bound for all $w_k$1 by monotonicity of the learning procedure (Gao et al., 6 May 2025).

7. Applications and Empirical Performance

Learning-based homothetic tube MPC has been evaluated on a variety of benchmarks, including high-dimensional state spaces, output-feedback plants with incomplete measurements, platooning scenarios, and planar quadrotor navigation under parametric and disturbance uncertainty (Gao et al., 6 May 2025, Dey et al., 6 Feb 2025, Sasfi et al., 2022). Empirical evidence demonstrates the following:

• Reduced conservatism: Learning-based tube adaptation yields feasible sets up to twice the size of those for conventional robust MPC and at least 20% larger than previous learning-based rigid-tube schemes, as demonstrated in vehicle platooning (Gao et al., 6 May 2025).
• Improved constraint satisfaction: In cases where conventional robust or rigid-tube MPC fails to find feasible solutions, the learning-based homothetic tube architecture succeeds due to sharper uncertainty set approximations.
• Computational tractability: Typical online computation times for homothetic tube learning-based MPC are under 0.25 s for moderate-scale linear systems and compatible with real-time operation (Gao et al., 6 May 2025).
• Rigorous probabilistic guarantees: Empirical feasibility matches predicted statistical bounds, achieving $w_k$2 feasibility over 100 repeated runs.

A plausible implication is that learning-based homothetic tubes, when coupled with appropriate observer design and set-learning frameworks, enable practical deployment of robust output-feedback MPC with tight feasibility and safety margins in uncertain environments.

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References: (Gao et al., 6 May 2025, Dey et al., 6 Feb 2025, Dey et al., 16 Mar 2026, Parsi et al., 2022, Gros et al., 2020, Sasfi et al., 2022)