Elastic Tube-Based MPC Framework

• Elastic tube-based MPC frameworks are robust control methods that adapt tube geometry and feedback gains online to ensure recursive feasibility and asymptotic stability.
• They employ flexible parameterizations such as polytopic, zonotopic, and ellipsoidal sets to reduce conservatism and enlarge feasible regions under dynamic uncertainties.
• Advanced formulations integrate LMI constraints and Lyapunov functions to guarantee robust control performance in real-time applications like autonomous vehicles and quadrotors.

Elastic tube-based Model Predictive Control (MPC) frameworks constitute a class of robust control methodologies enabling optimization of tube geometry and feedback gains online, in response to uncertainties and changing references. Unlike rigid tube MPC—which relies on a fixed, conservatively chosen invariant set—elastic tube MPC parameterizes the tube cross-section and, often, the feedback structure as decision variables, affording reduced conservatism, enhanced tracking, and robust constraint satisfaction even when uncertainty or operating environment changes dynamically. Core concepts include polytopic, zonotopic, ellipsoidal, and configuration-constrained tube parameterizations, with recursive feasibility and robust asymptotic stability certified via Lyapunov or contractivity arguments.

1. Mathematical Formulation and Uncertainty Models

Elastic tube-based MPC frameworks are designed for discrete-time linear systems with both additive and multiplicative uncertainties: $x_{t+1} = A x_t + B u_t + w_t$ where the tuple $(A, B)$ lies in the convex hull

$$(A, B)\in\Delta = \mathrm{convh}\{(A_1, B_1),\dots,(A_p, B_p)\}$$

and the disturbance $w_t$ belongs to a compact convex set $\mathcal{W} \subset \mathbb{R}^{n_x}$. State and input constraints $x \in \mathcal{X}$, $u \in \mathcal{U}$ are convex polytopes (Badalamenti et al., 6 May 2024). This generalizes to linear parameter-varying (LPV) systems and systems with data-driven zonotopic uncertainty sets (Ghiasi et al., 24 Dec 2025).

Tube cross-sections are parameterized using polytopes, zonotopes, ellipsoids, or configuration-constrained polytopes. In configuration-constrained tube MPC, the tube at stage $t$ is given by: $$X(y_t) = \{ x \in \mathbb{R}^{n_x} \mid F x \leq y_t \}$$ where $F$ is a fixed facet-normal matrix, and $y_t$ are polytope offset parameters constrained to preserve the vertex configuration via $E y_t \leq 0$ (Badalamenti et al., 6 May 2024).

Elasticity refers to the online adjustment of parameters such as facet/scaling directions $y, \delta, \alpha$, zonotopic generators, or ellipsoid radii along the prediction horizon. This allows the tube shape to adapt to local uncertainty, state-dependent model error, and evolving control objectives.

2. Tube Parameterization and Invariance Conditions

Elastic tube MPC leverages set-theoretic constructs to guarantee that the true system trajectory remains within prescribed tubes for all admissible uncertainties. In configuration-constrained parametrizations, the tube is defined by fixed facet normals and optimized offsets. One-step invariance is enforced by requiring that, for each triple $(y, u, y^+)$,

$F(A V_j y + B U_j u) + d \leq y^+$

for all vertices $j$, admissible $(A,B)$, and worst-case disturbance directions $d$ (Badalamenti et al., 6 May 2024). The constraint set is denoted $\mathcal{S}$, and recursive forward invariance (RFIT) is certified if the sequence $(y_t, u_t, y_{t+1})$ remains in $\mathcal{S}$.

For zonotopic or ellipsoidal tubes, cross-sections are written as scaled sets of the form: $$Z_k = \langle c_k, G \Delta_k \rangle = \{ c_k + G \Delta_k \xi : \|\xi\|_\infty \leq 1 \}$$ where $\Delta_k = \text{diag}(\delta_k)$, with scaling vector $\delta_k$ optimized at each time step (Diaconescu et al., 24 Sep 2025, Ghiasi et al., 24 Dec 2025). Ellipsoidal tubes employ: $$\mathcal{X}_{l|k} = z_{l|k} \oplus \alpha_{l|k} \bar{\mathcal{X}}$$ where $\bar{\mathcal{X}} = \{ x \mid x^\top P x \leq 1 \}$ is a fixed shape and $\alpha_{l|k}$ is a scaling radius (Parsi et al., 2022).

Contractivity and invariance are ensured via linear or semidefinite constraints that propagate inclusion of successor tube sections under system dynamics and additive uncertainties. These take the form of robust set-inclusion conditions or LMIs.

3. MPC Optimization Problem and Algorithmic Structure

The elastic tube-MPC optimization problem, as formulated in (Badalamenti et al., 6 May 2024), utilizes the following decision variables:

• Tube parameters over the horizon ($\{y_k\}$, $\{\delta_k\}$, or $\{\alpha_k\}$)
• Vertex inputs or feedforward actions ($\{u_k\}$, $\{v_k\}$)
• Terminal invariant set parameters

Constraints include:

• Initial-state consistency: $F x_t \leq y_0$
• Recursive tube inclusion: $(y_k, u_k, y_{k+1}) \in \mathcal{S}$
• Terminal contractivity: $$(y_N, u_N, \gamma y_N + (1-\gamma) y_s) \in \mathcal{S}$$ for $\gamma \in [0,1)$

The objective comprises stage tracking cost, terminal set cost, and steady-state/reference cost. Formally,

$$s(y_k - y_s, u_k - u_s) = \|(y_k - y_s, u_k - u_s)\|_Q^2$$

$$M(y_N, y_s, u_s) = \min_{u^+} \{ \|(y_N-y_s, u^+ - u_s)\|_P^2 \mid (y_N, u^+, \gamma y_N + (1-\gamma) y_s)\in\mathcal{S} \}$$

The receding-horizon MPC algorithm entails:

1. Measurement of the current state and reference
2. Solution of the convex QP (or SDP/LMI for ellipsoidal/zontopic cases) yielding tube parameters and control sequences
3. Extraction of the first control action via convex-combination of vertex inputs, using barycentric coordinates for the current state within the computed tube (Badalamenti et al., 6 May 2024)
4. Application and propagation of the system, with instant adaptation to reference or uncertainty changes—no offline recomputation required

Numerical studies demonstrate real-time tractability with variable counts scaling linearly in horizon length and tube complexity (e.g., $(N+1)m + (N+1)v n_u + m$ for configuration-constrained tubes).

4. Recursive Feasibility, Robustness, and Stability

Robust recursive feasibility and asymptotic stability of elastic tube-MPC are established via Lyapunov- and contraction-based arguments. The contractive terminal constraint $\gamma$ ensures that the cost

$M(y, y_s, u_s)$

acts as a strict Lyapunov function, with descent established via in-horizon cost-to-travel minimization

$$\min_{y^+}\{ M(y^+, y_s, u_s) + V(y, y^+, y_s, u_s) \} \leq M(y, y_s, u_s)$$

where $V(y, y^+, y_s, u_s)$ is the one-step cost-to-go.

Corollary statements ensure that feasibility at the initial time propagates under arbitrary changes in references, and that convergence to the optimal robust-invariant steady state occurs under constant reference scenarios (Badalamenti et al., 6 May 2024). Similar recursive feasibility, robust constraint satisfaction, and attractivity results are established for zonotopic, ellipsoidal, and hierarchical adjustable-tube frameworks (Ghiasi et al., 24 Dec 2025, Diaconescu et al., 24 Sep 2025, Raghuraman et al., 2022).

Lyapunov functions for elastic tubes use either strict convex quadratic forms over parameter differences, or facet-wise piecewise-linear maximizers over polyhedral sections, certifying exponential decay under contractive tube updates.

5. Complexity, Conservatism, and Parameterization Trade-Offs

Elastic tube-MPC frameworks admit multiple parameterization strategies along the prediction horizon, trading off computational complexity against domain of attraction (DOA):

• Scenario tubes: Maximum DOA via per-vertex parameterization, but exponential complexity in prediction steps (Hanema et al., 2019)
• Homothetic tubes: Minimal complexity (linear in horizon length), smaller DOA due to fixed-shape cross-sections
• Mixed scenario–homothetic tubes: Tunable complexity/DOA ratio by switching template after initial steps; proven recursive feasibility and closed-loop stability

For zonotopic tubes, the scaled-inclusion test enables linear constraints in the tube scaling variables, avoiding the combinatorial explosion of classical polyhedral set containment (which grows as $O(\binom{D}{n-1})$) (Diaconescu et al., 24 Sep 2025, Ghiasi et al., 24 Dec 2025). The “Φ” variant achieves highest flexibility at higher computational cost, while precomputed “Γ” or “Φ₀” templates reduce variable count and solve time at a marginal DOA penalty.

Ellipsoidal and LMI-based elastic tubes retain tractable complexity scaling with $O(n_x^2 N)$ in the number of variables, facilitating real-time application for medium and large-scale systems (Parsi et al., 2022).

6. Practical Applications and Numerical Examples

Elastic tube-MPC has been demonstrated in autonomous vehicle lane-change, unstable linear systems, quadrotor tracking, mass–spring–damper chains, pendulum and obstacle avoidance testbeds (Badalamenti et al., 6 May 2024, Ghiasi et al., 24 Dec 2025, Parsi et al., 2022, Morozov et al., 2020). Key empirical findings include:

• The configuration-constrained scheme achieves feasible regions closest to the true maximal robust control invariant set, with Hausdorff distance reductions from 1.10 (rigid tube MPC) to 0.0179 (configuration-constrained tube-MPC) (Badalamenti et al., 6 May 2024)
• Zonotopic elasticity enables larger feasibility margins and disturbance resilience even in high-dimensional scenarios where polyhedral schemes are infeasible
• Adaptive tube-gain co-design yields λ-contractive tubes with robust positive invariance, improved feasibility, and exponential stability of the closed-loop error dynamics (Ghiasi et al., 24 Dec 2025)
• Hierarchical adjustable-tube MPC reduces centralized computational overhead by an order of magnitude in aircraft power systems, with online zonotopic RPI/set tightening computed in milliseconds (Raghuraman et al., 2022)
• Ellipsoidal tube MPC scales to 50-state problems in < 100 ms, with cost and domain-of-attraction benefits over polytopic tubes (Parsi et al., 2022)

7. Conclusions and Research Directions

Elastic tube-based MPC frameworks, particularly the configuration-constrained polytopic tube approach, advance robust control for systems with structured uncertainty, enabling online adaptation of both tube geometry and feedback gains. These frameworks reduce conservatism, enlarge feasible regions, and maintain tractable convex optimization programs. Ongoing research encompasses:

• Data-driven refinement of disturbance/model sets via online identification and prior-consistency enforcement (Ghiasi et al., 24 Dec 2025)
• Trade-off analysis between tube parameterization richness and computational resources (Diaconescu et al., 24 Sep 2025, Hanema et al., 2019)
• Extension to nonlinear, distributed, and hybrid systems via system-level parameterization, FIR constraints, and scalable convex relaxations (Sieber et al., 2021, Sieber et al., 2021, Raghuraman et al., 2022)
• Integration with scalable QP/SDP solvers for deployment in safety-critical automotive and aerospace applications

The elasticity in tube-based MPC represents a critical enabling factor for high-performance, robust tracking under uncertainty, with rigorous guarantees—recursively feasible control laws, robust positive invariance, and asymptotic stability—realized through principled convex optimization (Badalamenti et al., 6 May 2024, Ghiasi et al., 24 Dec 2025, Parsi et al., 2022, Hanema et al., 2019).