Discounted MPC and infinite-horizon optimal control under plant-model mismatch: Stability and suboptimality

Dragan Ne\v{s}i\'c; Karl Worthmann; Mathieu Granzotto; Robert H. Moldenhauer; Romain Postoyan

arxiv: 2604.08521 · v1 · submitted 2026-04-09 · 🧮 math.OC · cs.SY· eess.SY

Discounted MPC and infinite-horizon optimal control under plant-model mismatch: Stability and suboptimality

Robert H. Moldenhauer , Karl Worthmann , Romain Postoyan , Dragan Ne\v{s}i\'c , Mathieu Granzotto This is my paper

Pith reviewed 2026-05-10 16:50 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY

keywords model predictive controlplant-model mismatchexponential stabilitysuboptimality boundinfinite-horizon controlsurrogate modeldiscounted costs

0 comments

The pith

Model predictive control using a surrogate model guarantees exponential stability and suboptimality under proportional plant-model mismatch, uniformly in the horizon length.

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

The paper establishes that both model predictive control and infinite-horizon optimal control remain exponentially stable when computed with a surrogate model that differs from the true plant. It assumes mismatch bounds proportional to states and controls to keep the origin an equilibrium point. Under model continuity and cost-controllability, stability holds, along with a bound showing the closed-loop cost is close to the surrogate's optimal cost. These results apply uniformly regardless of horizon length, highlighting tradeoffs with discounting and mismatch size.

Core claim

The central claim is that under continuity of the surrogate model and a cost-controllability assumption, plant-model mismatch bounded proportionally to states and controls ensures exponential stability of the closed loop for MPC and infinite-horizon problems, while also providing a suboptimality bound that recovers the surrogate optimal cost, with robustness independent of the horizon length.

What carries the argument

The proportional mismatch bounds that preserve the equilibrium at the origin, analyzed within a unified quadratic-cost framework for finite and infinite horizons.

If this is right

Stability and suboptimality guarantees are independent of the prediction horizon length.
Discounting allows handling larger mismatches while maintaining stability.
The achieved cost approaches the surrogate optimum as mismatch vanishes.
Both discounted and undiscounted cases are covered by the same analysis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This implies that simplified models can be used reliably in control design if their error scales with signal amplitude.
The uniformity in horizon suggests benefits for long-horizon planning without increased robustness requirements.
Extensions might consider time-varying mismatch bounds or nonlinear costs under similar assumptions.

Load-bearing premise

Plant-model mismatch is bounded proportionally to the norms of the state and control, ensuring the origin remains an equilibrium for the real plant.

What would settle it

Observing that the closed-loop system diverges or converges sub-exponentially for some initial condition despite the mismatch satisfying the proportional bound would falsify the stability result.

Figures

Figures reproduced from arXiv: 2604.08521 by Dragan Ne\v{s}i\'c, Karl Worthmann, Mathieu Granzotto, Robert H. Moldenhauer, Romain Postoyan.

**Figure 1.** Figure 1: Bounds κ1,N (s) of Proposition 2 that apply only to horizon length N (solid in color), bounds κ1,N (s)+FN (1+s) 2 that apply to all horizon lengths (dashed in color), and bound κ1 of Proposition 3 constructed as lower envelope of dashed lines (solid black), for γ = 1, L = 1.1, B = 10. hold for OCP (4). Finally, let the map g of system (1) satisfy |f − g|S < ∞ on some set S ⊆ R n. Then, the inequalities γV … view at source ↗

**Figure 2.** Figure 2: Suboptimality index α f,g γ,N without plant-model mismatch (blue), calculated with κ1,N instead of κ1 (red), calculated with κe1 as in (15) (yellow), and suboptimality index given in [25] (purple). In the top panel (|f − g|S = 5 · 10−5 ) the purple curve is negative, whereas in the bottom panel (|f − g|S = 10−12) the first three curves are indistinguishable. REFERENCES [1] G. Grimm, M. Messina, S. Tuna, an… view at source ↗

read the original abstract

We study closed-loop stability and suboptimality for MPC and infinite-horizon optimal control solved using a surrogate model that differs from the real plant. We employ a unified framework based on quadratic costs to analyze both finite- and infinite-horizon problems, encompassing discounted and undiscounted scenarios alike. Plant-model mismatch bounds proportional to states and controls are assumed, under which the origin remains an equilibrium. Under continuity of the model and cost-controllability, exponential stability of the closed loop can be guaranteed. Furthermore, we give a suboptimality bound for the closed-loop cost recovering the optimal cost of the surrogate. The results reveal a tradeoff between horizon length, discounting and plant-model mismatch. The robustness guarantees are uniform over the horizon length, meaning that larger horizons do not require successively smaller plant-model mismatch.

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.

Desk Editor's Note private letter to a colleague

This paper unifies stability and suboptimality bounds for MPC under linear plant-model mismatch, with the key feature that the guarantees stay uniform as the horizon grows and cover both discounted and undiscounted cases.

read the letter

The central result is that, under standard continuity and cost-controllability assumptions plus mismatch bounds linear in state and input, the closed-loop system is exponentially stable and the realized cost stays within a fixed factor of the surrogate optimum. The uniformity over horizon length is the part that stands out; most earlier mismatch analyses tighten the allowed error as the horizon increases, which this avoids. The framework handles finite-horizon, infinite-horizon, and discounted problems in one set of arguments built on Lyapunov functions and controllability, so the tradeoff between horizon, discount factor, and mismatch size appears explicitly. That is useful for anyone who wants to pick a longer horizon without having to re-derive a smaller mismatch tolerance. The suboptimality bound that recovers the surrogate optimum is also stated cleanly. The derivations rest on the usual quadratic-cost Lyapunov arguments rather than anything circular or fitted, which keeps the logic straightforward. The assumptions are the usual ones in this literature, and the origin remains an equilibrium under the proportional mismatch, so nothing exotic is hidden there. The main limitation is that everything is stated for quadratic costs; extending the same uniformity to general nonlinear stage costs would require additional work, though that is outside the paper's stated scope. The bounds themselves are likely conservative because they use worst-case linear mismatch, but the paper does not claim tightness. Overall the claims line up with the assumptions and the uniformity result addresses a known practical gap. This is for people working on robust MPC theory or on implementation where model error is inevitable. A reader who already knows the standard Lyapunov-MPC toolkit will see the incremental advance immediately. It is worth sending to a serious referee; the proofs are checkable from the given assumptions and the uniformity claim is the sort of concrete improvement that journals in this area look for.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a unified framework for analyzing closed-loop stability and suboptimality of both finite-horizon discounted MPC and infinite-horizon optimal control when the controller is designed using a surrogate model that differs from the true plant. It assumes quadratic stage costs and plant-model mismatch bounds that are linear in the state and input (ensuring the origin remains an equilibrium for both systems). Under the standing assumptions of model continuity and cost-controllability, the paper proves exponential stability of the closed loop and derives an explicit suboptimality bound showing that the closed-loop cost recovers the optimal cost of the surrogate. The results identify trade-offs among prediction horizon, discount factor, and mismatch size, with the distinctive claim that all robustness guarantees are uniform with respect to the horizon length.

Significance. If the derivations hold, the contribution is meaningful for robust MPC theory. The uniformity of the stability and suboptimality bounds with respect to horizon length is a practically relevant feature that removes the usual requirement of successively tighter mismatch tolerances for longer horizons. The unified treatment of discounted/undiscounted and finite/infinite-horizon cases under a single set of quadratic-cost assumptions broadens the applicability of the results to a range of uncertain systems.

minor comments (3)

[Abstract] The abstract and introduction would benefit from a brief statement of the precise mismatch bound (e.g., the constants multiplying ||x|| and ||u||) so that readers can immediately assess the strength of the uniformity claim.
[Introduction] Notation distinguishing the true plant trajectory from the surrogate-model trajectory should be introduced consistently at the first use (currently the distinction appears only after several paragraphs).
[Abstract] The statement that the robustness guarantees are 'uniform over the horizon length' would be clearer if accompanied by an explicit remark on whether the constants in the exponential decay rate and suboptimality bound are independent of N.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive review, accurate summary of our contributions, and recommendation for minor revision. The emphasis on the uniformity of the robustness guarantees with respect to horizon length is particularly appreciated, as this is a central feature of the framework.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's derivation chain rests on standard assumptions (continuity of the model, cost-controllability, and plant-model mismatch bounds linear in states/controls that preserve the origin as equilibrium) together with Lyapunov-based stability arguments and suboptimality estimates. These are externally verifiable conditions from the MPC literature and do not reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations whose validity depends on the present work. The claimed uniformity of robustness bounds over horizon length follows directly from the stated hypotheses without circular renaming or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on standard domain assumptions from optimal control rather than new free parameters or invented entities.

axioms (3)

domain assumption Continuity of the model
Invoked to guarantee existence of solutions and stability properties.
domain assumption Cost-controllability
Standard assumption used to obtain suboptimality and stability bounds in MPC.
domain assumption Plant-model mismatch bounded proportionally to states and controls
Core modeling assumption that keeps the origin an equilibrium.

pith-pipeline@v0.9.0 · 5462 in / 1337 out tokens · 31722 ms · 2026-05-10T16:50:38.082511+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages · 1 internal anchor

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J. K ¨ohler, R. Soloperto, M. A. M ¨uller, and F. Allg ¨ower, “A compu- tationally efficient robust model predictive control framework for un- certain nonlinear systems,”IEEE Transactions on Automatic Control, vol. 66, no. 2, pp. 794–801, 2020

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Stability of data-driven Koopman MPC with terminal conditions

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work page internal anchor Pith review Pith/arXiv arXiv 2026

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S. J. Kuntz and J. B. Rawlings, “Beyond inherent robustness: strong stability of MPC despite plant-model mismatch,”IEEE Transactions on Automatic Control, vol. 71, no. 2, pp. 780–792, 2026

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Shorter horizons for model predictive control,

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Analysis of unconstrained nonlinear MPC schemes with time varying control horizon,

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M. Granzotto, R. Postoyan, L. Bus ¸oniu, D. Ne ˇsi´c, and J. Daafouz, “Finite-horizon discounted optimal control: Stability and perfor- mance,”IEEE Trans. on Aut. Cont., vol. 66, no. 2, pp. 550–565, 2021

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Conditions for MPC based stabi- lization of sampled-data nonlinear systems via discrete-time approx- imations,

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