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arxiv: 2605.23661 · v1 · pith:5ZDWVPOSnew · submitted 2026-05-22 · 📡 eess.SY · cs.SY· math.OC

Output Feedback MPC with Adaptive Tubes

Anchita Dey , Shubhendu Bhasin This is my paper

Pith reviewed 2026-05-25 03:31 UTC · model grok-4.3

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keywords adaptiveestimatesfeedbacktubecontrolframeworkgeometryinitial
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An adaptive tube-based output feedback MPC for LTI systems with parametric and additive uncertainties that updates tube geometry, constraints, and terminal sets from evolving observer estimates.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Model predictive control plans future actions while respecting limits like actuator ranges or safety bounds. When the exact system model is unknown, standard robust versions use fixed 'tubes' around predicted trajectories to guarantee safety for all possible uncertainties. This paper adds an adaptive observer that not only estimates the current state but also refines guesses about unknown parameters and the starting point. As those guesses improve, the tubes shrink and the optimization problem is retightened. The key difference from prior work is that no single linear feedback law needs to stabilize every possible parameter value at once. Instead, the tube shape itself changes with better information. The authors state that this keeps the closed-loop system recursively feasible and exponentially stable while delivering better performance as uncertainty information improves. A numerical example is mentioned to illustrate the idea.

Core claim

Recursive feasibility and robust exponential stability are established for the proposed adaptive tube MPC framework that updates constraint tightening, terminal ingredients, and tube geometry as estimates evolve.

Load-bearing premise

An adaptive observer exists that can simultaneously provide point estimates of the state, model parameters, and initial condition while jointly updating the corresponding sets containing the true parameters and initial state (abstract).

Figures

Figures reproduced from arXiv: 2605.23661 by Anchita Dey, Shubhendu Bhasin.

Figure 1
Figure 1. Figure 1: Schematic diagram for the proposed adaptive tube MPC. to the previous estimates: xˆ0t+1 ← xˆ0t , Aˆ t+1 ← Aˆ t, Bˆ t+1 ← Bˆ t, Pt+1 ← Pt, Kt+1 ← Kt, Ψt+1 ← co Ψt+1, Aˆ t Bˆ t  , Πt+1 ← co (Πt+1, {pˆt}), X0t+1 ← co X0t+1 , {xˆ0t }  . (46) The observer state is then recomputed using (12a), and the COCP components are reconstructed as described in Sec. IV, after which the COCP is solved at time t + 1. Any … view at source ↗
Figure 2
Figure 2. Figure 2: True state and state estimate trajectories, with the outer tube (in grey) guaranteed to contain the true state trajectory. 0 2 4 6 8 10 12 14 -1 0 1 2 3 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Control input. co nψ [i] 0 o i=1:3  7 . The initial set X0 is shown in grey in [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: The polytopes forming the tube cross-sectional shape at time t. 0 2 4 6 8 10 12 14 0.1 0.2 0.3 0.4 0 2 4 6 8 10 12 14 6 8 10 (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Norms of the parameter estimation error and the initial state estimation error. additionally shows, using the blue dashed line (·), the state estimate trajectory obtained when the point estimates are held fixed. As expected, this trajectory remains contained within the tube-section T xˆ ∗ 1|0 . The parameter and state estimation error norms kp˜tk2 and kx˜0t k2, and the associated variations in Kt and Pt ar… view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of RMS value of kxtk2 and the cumulative stage cost for the proposed method with and without any adaptation and [31]. vided in [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: Reduction in size of uncertainty sets: (a) Barycentric repre￾sentation of the sets Ψt containing the true parameter combination [0.05; 0.1; 0.85], using ρ (i) . (b) Sets X0t containing the initial true state x0. displayed in Figs. 6 and 7, respectively. The contraction of the uncertainty sets Ψt and X0t with time are shown in [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
read the original abstract

An output feedback model predictive control (MPC) framework with adaptive tubes is proposed for linear time-invariant systems subject to parametric and additive uncertainties. An adaptive observer provides point estimates of the system state, model parameters, and initial condition, while jointly updating the corresponding sets containing the true parameters and initial state. These estimates parameterize the constrained optimal control problem, enabling constraint tightening, terminal ingredients, and tube geometry to be updated as the estimates evolve. In contrast to standard robust tube-based MPC formulations, the proposed approach does not require a common quadratically stabilizing linear feedback gain across the parametric uncertainty set. As the available uncertainty information improves, the tube geometry evolves accordingly, resulting in an adaptive tube MPC framework with improved performance over time. Recursive feasibility and robust exponential stability are established, and a numerical example is presented.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes an output feedback MPC framework with adaptive tubes for LTI systems subject to parametric and additive uncertainties. An adaptive observer supplies point estimates of the state, parameters, and initial condition while updating the corresponding uncertainty sets; these sets are used to adaptively tighten constraints, update terminal ingredients, and evolve tube geometry online. The approach claims to avoid the need for a common quadratically stabilizing feedback gain across the uncertainty set. Recursive feasibility and robust exponential stability are asserted, and a numerical example is mentioned.

Significance. If the central claims hold, the work would provide a mechanism for performance improvement in robust tube MPC as uncertainty information is refined, reducing conservatism relative to fixed-tube designs. The relaxation of the common stabilizing gain requirement could extend applicability to broader classes of uncertain LTI systems.

major comments (2)
  1. [Abstract] Abstract: the claims of recursive feasibility and robust exponential stability are asserted without derivation steps, error bounds, or explicit conditions on the adaptive observer or set updates. This is load-bearing for the central contribution and prevents verification of the stated guarantees.
  2. [Recursive feasibility proof] Recursive feasibility argument: it is not shown how the observer-driven set updates (which may contract non-monotonically) preserve feasibility of the shifted prior optimal sequence under the new tightened constraints and updated terminal set. If the new terminal set is not invariant under the revised uncertainty, the standard tube-MPC shifting argument fails; this directly undermines the recursive-feasibility claim.
minor comments (1)
  1. [Numerical example] Numerical example: the abstract states that a numerical example is presented, yet no data, figures, or performance metrics appear in the available text, making it impossible to assess practical behavior or improvement over time.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below, clarifying the presentation of our results while indicating revisions that will strengthen the exposition.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claims of recursive feasibility and robust exponential stability are asserted without derivation steps, error bounds, or explicit conditions on the adaptive observer or set updates. This is load-bearing for the central contribution and prevents verification of the stated guarantees.

    Authors: The abstract serves as a high-level summary of the contribution. The full derivations, including error bounds on the adaptive observer, explicit conditions on the set updates, and the proofs of recursive feasibility and robust exponential stability, appear in Sections III and IV, with the main results stated as Theorems 1 and 2. To improve accessibility, we will revise the abstract to include a brief reference to these theorems and the key assumptions on the observer. revision: yes

  2. Referee: [Recursive feasibility proof] Recursive feasibility argument: it is not shown how the observer-driven set updates (which may contract non-monotonically) preserve feasibility of the shifted prior optimal sequence under the new tightened constraints and updated terminal set. If the new terminal set is not invariant under the revised uncertainty, the standard tube-MPC shifting argument fails; this directly undermines the recursive-feasibility claim.

    Authors: Theorem 1 establishes recursive feasibility by showing that the updated uncertainty sets always contain the true parameters and initial condition, and that the terminal set is recomputed to remain positively invariant under the closed-loop dynamics with the current uncertainty bounds. The proof adapts the standard shifting argument by verifying that the shifted sequence satisfies the new tightened constraints and terminal constraint because the tightening is performed with respect to the updated sets at each step. We will add a remark in Section IV explicitly highlighting the invariance property under non-monotonic updates to make this step clearer. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard tube MPC theory plus assumed adaptive observer

full rationale

The paper assumes existence of an adaptive observer that supplies point estimates and contracting sets for state, parameters, and initial condition. These sets are then used to update constraint tightening, terminal ingredients, and tube geometry in an otherwise standard output-feedback tube MPC formulation. Recursive feasibility and robust exponential stability are claimed via adaptation of the usual shifting argument from robust tube MPC literature. No load-bearing self-citation, no fitted parameter renamed as prediction, and no self-definitional reduction appear in the provided abstract or skeptic analysis; the central claims remain independent of quantities defined inside the paper itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Only the abstract is available, so the ledger is inferred from high-level claims; the framework rests on standard LTI dynamics and bounded uncertainties plus the existence of the adaptive observer.

axioms (2)
  • domain assumption The plant is linear time-invariant with bounded parametric and additive uncertainties.
    Stated in the abstract as the system class considered.
  • domain assumption An adaptive observer can jointly estimate state, parameters, and initial condition while updating the corresponding uncertainty sets.
    Central to the proposed parameterization of the MPC problem.
invented entities (1)
  • Adaptive tubes no independent evidence
    purpose: Evolving tube geometry that tightens as uncertainty estimates improve.
    Introduced to achieve less conservative performance than fixed robust tubes.

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