Configuration-Constrained Tube MPC

• Configuration-Constrained Tube MPC is a robust control framework that parameterizes forward-invariant tubes using a polytopic facet-vertex representation under configuration constraints.
• It guarantees state and input constraint satisfaction and recursive feasibility under both additive and multiplicative uncertainties via a convex optimization formulation.
• The approach offers flexible parameterizations—including full, homothetic, and hybrid schemes—with demonstrated runtime improvements in high-dimensional applications like quadrotor and autonomous vehicle tracking.

Configuration-Constrained Tube Model Predictive Control (CCTMPC) is an advanced robust MPC framework that parameterizes forward-invariant tubes using a polytopic facet-vertex representation subject to configuration constraints, allowing for joint online optimization of tube shape and associated vertex feedback control laws. CCTMPC state and input constraint satisfaction is guaranteed under both additive and multiplicative uncertainty, leveraging a parameterization that balances reduced conservatism with tractable convex programming (Villanueva et al., 2022, Badalamenti et al., 2024, Badalamenti et al., 20 May 2025).

1. Problem Setting and Mathematical Foundations

CCTMPC addresses uncertain discrete-time systems described by

$$x_{k+1} = A_k x_k + B_k u_k + w_k,\quad (A_k,B_k) \in \Delta := \mathrm{convh}\{(A_1,B_1),\ldots,(A_m,B_m)\}$$

with $x_k \in \mathcal{X} \subset \mathbb{R}^{n_x}$, $u_k \in \mathcal{U} \subset \mathbb{R}^{n_u}$, $w_k \in \mathcal{W} \subset \mathbb{R}^{n_x}$, and all sets convex, compact polytopes (Badalamenti et al., 20 May 2025).

Each tube cross-section is defined as

$P(y) := \{ x\in\mathbb{R}^{n_x} \mid F x \le y \}$

where $F \in \mathbb{R}^{f \times n_x}$ is a fixed facet matrix and $y \in \mathbb{R}^f$ is restricted by the configuration cone $\mathcal{E} = \{ y \mid E y \le 0 \}$ to guarantee invariant facet-vertex incidence (Villanueva et al., 2022, Badalamenti et al., 20 May 2025). The vertex map $V = \{V_j\}_{j=1}^v$, $V_j \in \mathbb{R}^{n_x \times f}$, ensures that

$x_k \in \mathcal{X} \subset \mathbb{R}^{n_x}$0

for all $x_k \in \mathcal{X} \subset \mathbb{R}^{n_x}$1.

Robust one-step reachability is encoded via the convex set

$x_k \in \mathcal{X} \subset \mathbb{R}^{n_x}$2

where $x_k \in \mathcal{X} \subset \mathbb{R}^{n_x}$3 (Badalamenti et al., 20 May 2025).

2. Full and Structured CCTMPC Formulations

The fully-parameterized CCTMPC tracking QP at time $x_k \in \mathcal{X} \subset \mathbb{R}^{n_x}$4 is given by

$x_k \in \mathcal{X} \subset \mathbb{R}^{n_x}$5

subject to

$x_k \in \mathcal{X} \subset \mathbb{R}^{n_x}$6

with reference RCI set $x_k \in \mathcal{X} \subset \mathbb{R}^{n_x}$7 computed as the unique optimizer of $x_k \in \mathcal{X} \subset \mathbb{R}^{n_x}$8 s.t. $x_k \in \mathcal{X} \subset \mathbb{R}^{n_x}$9 (Badalamenti et al., 20 May 2025). The recursive Lyapunov function

$u_k \in \mathcal{U} \subset \mathbb{R}^{n_u}$0

decreases monotonically under the dissipativity condition $u_k \in \mathcal{U} \subset \mathbb{R}^{n_u}$1, ensuring robust convergence $u_k \in \mathcal{U} \subset \mathbb{R}^{n_u}$2.

A variable-restriction framework introduces homothetic parameterization: $u_k \in \mathcal{U} \subset \mathbb{R}^{n_u}$3 producing a reduced-complexity Homothetic Tube MPC (HTMPC) QP, with significant reduction in variable and constraint counts for high-dimensional or complex tube templates. Intermediate schemes—by partially fixing vertex control laws—interpolate between full CCTMPC and HTMPC, yielding a flexible complexity-conservatism trade-off (Badalamenti et al., 20 May 2025).

3. Template Refinement and Initialization

Template complexity, characterized by facet count $u_k \in \mathcal{U} \subset \mathbb{R}^{n_u}$4 and vertex count $u_k \in \mathcal{U} \subset \mathbb{R}^{n_u}$5, determines the expressiveness and computational burden of the tube parameterization. To balance conservatism and computational tractability, an iterative template refinement algorithm is introduced:

• A convex QP determines the largest-in-the-chosen-norm RCI set $u_k \in \mathcal{U} \subset \mathbb{R}^{n_u}$6 containing specified points.
• An iterative cutting-plane procedure repeatedly tightens facial descriptions to incrementally expand the feasible invariant polytope, with monotonic improvement of tube size (Hausdorff measure) at each iteration (Proposition 4) (Badalamenti et al., 20 May 2025).
• Template initialization solves a smooth NLP over invertible transformations, vertex controls, and slack variables to ensure robust reachability for the initial template and the desired constraint sets.

These processes systematically decrease tube conservatism and are essential for scaling CCTMPC to practical, high-dimensional systems.

4. Theoretical Guarantees and Recursive Feasibility

CCTMPC guarantees, under convexity and proper terminal set design, robust recursive feasibility, robust constraint satisfaction, and Lyapunov-based asymptotic convergence. All constraints—facet-invariance, vertex mapping, robust propagation—are maintained via convex optimization (Villanueva et al., 2022, Badalamenti et al., 2024, Badalamenti et al., 20 May 2025).

For periodic or reference-tracking problems, the introduction of artificial variables for candidate periodic tubes (or RCI sets) in the optimization problem enables guaranteed recursive feasibility under arbitrary cost or reference changes, with convergence to the optimal periodic tube or invariant set under constancy (via Lyapunov function descent) (Badalamenti et al., 3 Dec 2025).

5. Computational Aspects and Trade-Offs

Computational complexity in CCTMPC grows with the number of facets $u_k \in \mathcal{U} \subset \mathbb{R}^{n_u}$7, vertices $u_k \in \mathcal{U} \subset \mathbb{R}^{n_u}$8, and uncertainty vertices $u_k \in \mathcal{U} \subset \mathbb{R}^{n_u}$9. Fully-parameterized CCTMPC incurs $w_k \in \mathcal{W} \subset \mathbb{R}^{n_x}$0 control variables, while HTMPC reduces this to $w_k \in \mathcal{W} \subset \mathbb{R}^{n_x}$1, with corresponding constraint reductions (Badalamenti et al., 20 May 2025). For complex or high-dimensional systems, homothetic or partially-parameterized schemes yield 5–15× speed-ups at the expense of moderate conservatism.

Simulation results on a triple integrator (n_x=3) and a 10-state quadrotor showcase that:

• For small $w_k \in \mathcal{W} \subset \mathbb{R}^{n_x}$2, fully-parameterized CCTMPC achieves tighter feasible domains (lower Hausdorff distances to the maximal RCI set).
• For large $w_k \in \mathcal{W} \subset \mathbb{R}^{n_x}$3, HTMPC achieves markedly reduced solve times (e.g., 1.30 ms avg for HTMPC vs. 15.06 ms for CCTMPC in the triple integrator; 22.9 ms for HTMPC vs. 334 ms for CCTMPC in the quadrotor with $w_k \in \mathcal{W} \subset \mathbb{R}^{n_x}$4) (Badalamenti et al., 20 May 2025).
• For periodic economic operation, QP approximations using Lipschitz-gradient stability preserve guarantees at orders-of-magnitude lower computation time (1.31 s vs. 26.7 s) (Badalamenti et al., 3 Dec 2025).

Guidelines: For n_x=10–20 state systems, maintain $w_k \in \mathcal{W} \subset \mathbb{R}^{n_x}$5 unless acceptable online runtimes are in the ~100 ms range; initialize with a simplex template and refine towards $w_k \in \mathcal{W} \subset \mathbb{R}^{n_x}$6; choose homothetic parameterization for $w_k \in \mathcal{W} \subset \mathbb{R}^{n_x}$7 (Badalamenti et al., 20 May 2025).

6. Comparative Analysis and Extensions

CCTMPC is systematically less conservative than rigid, homothetic, and elastic tube MPC variants, and can realize non-affine robust policies that classical disturbance-affine or fully parameterized tube MPC cannot (Villanueva et al., 2022, Badalamenti et al., 20 May 2025). In 2D, CCTMPC subsumes ETMPC; in higher-dimensions it strictly outperforms classical schemes under certain conditions.

Extensions include hierarchical CCTMPC, where tube scaling factors or zonotopic generators are made variables of upper-level (scheduling) controllers in large-scale or multi-agent systems, and periodic tracking CCTMPC for economically optimal periodic solutions (Raghuraman et al., 2022, Badalamenti et al., 3 Dec 2025). Open research avenues include precise characterization of configuration cones for general polytopes, integration with output feedback, nonlinear system extensions, and systematic template-selection algorithms.

7. Practical Implementation and Application Domains

Key application domains include autonomous vehicle tracking under uncertainty, high-dimensional aerospace systems such as quadrotors, and economically-driven periodic operation of industrial processes (Badalamenti et al., 2024, Badalamenti et al., 3 Dec 2025, Badalamenti et al., 20 May 2025). The approach integrates efficiently with modern convex QP solvers; precomputation, warm-starting, and template selection routines are recommended for real-time or high-rate sampling scenarios.

A typical workflow entails:

• Template and configuration cone selection and refinement,
• Online convex QP for tube and control optimization,
• Piecewise affine or barycentrically blended vertex control law realization,
• Recursive updating of artificial/reference-invariant tubes for reference tracking or periodicity,
• Monitoring of Lyapunov function descent for performance and stability.

This method provides a unified framework for robust, tractable control under uncertainty with clear trade-offs between controller complexity, conservatism, and runtime (Badalamenti et al., 20 May 2025, Badalamenti et al., 3 Dec 2025, Villanueva et al., 2022, Badalamenti et al., 2024).