pith. sign in

arxiv: 2604.12840 · v1 · submitted 2026-04-14 · 🧮 math.OC · cs.SY· eess.SY

On stability and non-averaged performance of economic MPC with terminal conditions for optimal periodic operation

Jonas Mair , Lukas Schwenkel , Matthias A. M\"uller , Frank Allg\"ower This is my paper

Pith reviewed 2026-05-10 15:21 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords economic MPCperiodic operationasymptotic stabilityterminal conditionsCesaro summationnon-averaged performanceoptimal periodic operation
0
0 comments X

The pith

Economic MPC with terminal conditions guarantees asymptotic stability and non-averaged performance for systems with optimal periodic operation.

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

The paper shows that an economic model predictive controller equipped with terminal conditions can drive a system to its best periodic behavior rather than a fixed steady state. This stability result holds for a broader class of systems than earlier approaches allowed. By measuring the closed-loop cost with a Cesaro summation, which takes the limit of average costs over expanding intervals, the authors obtain a performance bound that does not depend on ordinary time-averaging. Readers should care because many processes in energy, chemistry, and manufacturing achieve lower costs by cycling between states, and this work supplies the stability and performance theory needed to use MPC in those cases.

Core claim

We derive asymptotic stability guarantees of an economic model predictive control scheme with terminal conditions for systems with optimal periodic operation for a more general setup than existing methods can handle. Moreover, we establish a non-averaged closed-loop performance bound by defining the closed-loop cost via a Cesaro summation instead of ordinary summation. Such a non-averaged performance bound provides new insights for systems with periodic optimal operation.

What carries the argument

Terminal conditions added to the economic MPC finite-horizon problem that enable a Lyapunov argument for convergence to a periodic orbit, paired with the Cesaro-mean definition of the infinite-horizon closed-loop cost.

Load-bearing premise

The system admits an optimal periodic operation and the terminal conditions satisfy the necessary assumptions to enable the stability proof in the claimed more general setup.

What would settle it

A concrete dynamical system that possesses an optimal periodic orbit, for which the stated terminal conditions are imposed in the MPC problem, yet the closed-loop trajectories fail to converge to that orbit or the Cesaro-based cost bound is violated.

Figures

Figures reproduced from arXiv: 2604.12840 by Frank Allg\"ower, Jonas Mair, Lukas Schwenkel, Matthias A. M\"uller.

Figure 1
Figure 1. Figure 1: Illustration of the non-minimal optimal periodic orbit [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Closed-loop state trajectory xµ5 (k, 0.3) over the time index k. However, if the optimal periodic orbit is in fact minimal, then Assumption 5 is implied by Assumptions 1, 3 and 4 and therefore it is in this case not restrictive. Proposition 12 (Relaxation for minimal orbits): Let As￾sumptions 1, 3 and 4 hold. Then, Assumption 5 holds if the optimal periodic orbit Π⋆ is minimal. Proof: Let V˜min := minx∈Xf … view at source ↗
read the original abstract

Operation at steady state is often not optimal when optimizing over an economic cost objective. In many cases, periodic operation yields better performance. Therefore, we derive asymptotic stability guarantees of an economic model predictive control scheme with terminal conditions for systems with optimal periodic operation for a more general setup than existing methods can handle. Moreover, we establish a non-averaged closed-loop performance bound by defining the closed-loop cost via a Ces\`aro summation instead of ordinary summation. Such a non-averaged performance bound provides new insights for systems with periodic optimal operation.

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

0 major / 3 minor

Summary. The manuscript develops an economic MPC scheme with terminal conditions for nonlinear systems whose optimal operation is periodic rather than steady-state. It proves asymptotic stability of the closed-loop system under a generalized set of assumptions on the periodic orbit and terminal ingredients that extend beyond prior results, and derives a non-averaged performance bound by replacing the standard sum with a Cesàro mean of the stage cost along closed-loop trajectories.

Significance. If the central derivations hold, the contribution is significant for economic MPC theory: it enlarges the class of systems for which terminal-condition-based stability can be certified when periodicity is optimal, and the Cesàro-based bound supplies a tighter, non-averaged characterization of transient and periodic performance that averaged-cost analyses cannot capture. These results are directly relevant to process-control and energy-management applications where periodic operation improves economic performance.

minor comments (3)
  1. [§2] §2 (Preliminaries): the definition of the Cesàro mean and its relation to the standard infinite-horizon cost should be stated explicitly with the precise limit expression before it is used in the performance theorem.
  2. [§4] §4 (Stability proof): the argument that the terminal cost and terminal set satisfy the required decrease condition for the generalized periodic case is only sketched; a self-contained lemma stating the exact inequality (analogous to Eq. (12) in earlier works) would improve readability.
  3. [Numerical example] Figure 2 and the accompanying simulation: the caption should clarify whether the plotted trajectories are for the nominal or perturbed system and whether the cost is the Cesàro or ordinary sum.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript and the recommendation for minor revision. The assessment of the significance of the stability guarantees and Cesàro-based non-averaged performance bounds for economic MPC under periodic optimal operation is appreciated.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives asymptotic stability guarantees and non-averaged performance bounds for economic MPC under optimal periodic operation using terminal conditions. These are standard Lyapunov-based stability proofs in control theory that rely on structural assumptions (existence of optimal periodic orbit, suitable terminal ingredients) rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation chain. The Cesàro summation for performance is a definitional choice for the bound, not a reduction of the result to its inputs. No equation or claim reduces by construction to prior outputs within the paper; the derivation chain is self-contained against external benchmarks in MPC literature.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard mathematical assumptions in control theory and the domain assumption of optimal periodic operation. No free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption Standard assumptions on the system dynamics and cost functions for MPC stability analysis.
    Typical for economic MPC papers; invoked implicitly in deriving stability guarantees.
  • domain assumption Existence of an optimal periodic orbit for the system.
    Central to the setup for periodic operation.

pith-pipeline@v0.9.0 · 5396 in / 1450 out tokens · 56097 ms · 2026-05-10T15:21:24.959508+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

  1. [1]

    A tutorial review of economic model predictive control methods,

    M. Ellis, H. Durand, and P. D. Christofides, “A tutorial review of economic model predictive control methods,”Journal of Process Control, vol. 24, no. 8, pp. 1156–1178, 2014

    work page 2014

  2. [2]

    Economic Nonlinear Model Predictive Control,

    T. Faulwasser, L. Gr ¨une, and M. A. M ¨uller, “Economic Nonlinear Model Predictive Control,”Found. Trends in Syst. Control, vol. 5, no. 1, pp. 224–409, 2018

    work page 2018

  3. [3]

    Economic optimization using model predictive control with a terminal cost,

    R. Amrit, J. B. Rawlings, and D. Angeli, “Economic optimization using model predictive control with a terminal cost,”Annual Reviews in Control, vol. 35, no. 2, pp. 178–186, Dec. 2011

    work page 2011

  4. [4]

    Economic receding horizon control without terminal con- straints,

    L. Gr ¨une, “Economic receding horizon control without terminal con- straints,”Automatica, vol. 49, no. 3, pp. 725–734, 2013

    work page 2013

  5. [5]

    On average performance and stability of economic model predictive control,

    D. Angeli, R. Amrit, and J. B. Rawlings, “On average performance and stability of economic model predictive control,”IEEE Transactions on Automatic Control, vol. 57, no. 7, pp. 1615–1626, Jul. 2012

    work page 2012

  6. [6]

    Asymptotic stability and transient optimality of economic mpc without terminal conditions,

    L. Gr ¨une and M. Stieler, “Asymptotic stability and transient optimality of economic mpc without terminal conditions,”Journal of Process Control, vol. 24, no. 8, pp. 1187–1196, Aug. 2014

    work page 2014

  7. [7]

    On non-averaged performance of economic MPC with terminal conditions,

    L. Gr ¨une and A. Panin, “On non-averaged performance of economic MPC with terminal conditions,” in54th IEEE Conf. on Decision and Control (CDC), 2015, pp. 4332–4337

    work page 2015

  8. [8]

    On the role of dissipativity in economic model predictive control,

    M. A. M ¨uller, L. Gr¨une, and F. Allg¨ower, “On the role of dissipativity in economic model predictive control,”Proc. 5th IFAC Conf. Nonlinear Model Predictive Control (NMPC), pp. 110–116, 2015

    work page 2015

  9. [9]

    Economic model predictive control with- out terminal constraints for optimal periodic behavior,

    M. A. M ¨uller and L. Gr¨une, “Economic model predictive control with- out terminal constraints for optimal periodic behavior,”Automatica, vol. 70, pp. 128–139, Aug. 2016

    work page 2016

  10. [10]

    On discount functions for economic model predictive control without terminal con- ditions,

    L. Schwenkel, D. Briem, M. A. M ¨uller, and F. Allg¨ower, “On discount functions for economic model predictive control without terminal con- ditions,” inSymposium on Systems Theory in Data and Optimization. Springer, 2024, pp. 231–246

    work page 2024

  11. [11]

    Linearly discounted economic MPC without terminal conditions for periodic optimal operation,

    L. Schwenkel, A. Hadorn, M. A. M ¨uller, and F. Allg ¨ower, “Linearly discounted economic MPC without terminal conditions for periodic optimal operation,”Automatica, vol. 159, p. 111393, 2024

    work page 2024

  12. [12]

    Periodic optimal control, dissi- pativity and MPC,

    M. Zanon, L. Gr ¨une, and M. Diehl, “Periodic optimal control, dissi- pativity and MPC,”IEEE Transactions on Automatic Control, vol. 62, no. 6, pp. 2943–2949, 2016

    work page 2016

  13. [13]

    Analysis of economic model predictive control with terminal penalty functions on generalized optimal regimes of operation,

    Z. Dong and D. Angeli, “Analysis of economic model predictive control with terminal penalty functions on generalized optimal regimes of operation,”International Journal of Robust and Nonlinear Control, vol. 28, no. 16, pp. 4790–4815, 2018

    work page 2018

  14. [14]

    The ces `aro value iteration,

    J. Mair, L. Schwenkel, M. A. M ¨uller, and F. Allg ¨ower, “The ces `aro value iteration,”IEEE Control Systems Letters, 2025

    work page 2025

  15. [15]

    On Periodic Dissipativity Notions in Economic Model Predictive Control,

    J. K ¨ohler, M. A. M ¨uller, and F. Allg ¨ower, “On Periodic Dissipativity Notions in Economic Model Predictive Control,”IEEE Control Sys- tems Letters, vol. 2, no. 3, pp. 501–506, Jul. 2018

    work page 2018

  16. [16]

    H. K. Khalil and J. W. Grizzle,Nonlinear systems. Prentice hall Upper Saddle River, NJ, 2002, vol. 3

    work page 2002

  17. [17]

    J. B. Rawlings and D. Q. Mayne,Model Predictive Control: Theory and Design. Madison, WI, USA: Nob Hill Publishing, 2009

    work page 2009

  18. [18]

    G. H. Hardy,Divergent Series. Oxford, U.K.: Oxford Univ. Press, 1949. APPENDIX Lemma 21 (Monotonic increase of ˜J ces N ):Let Assumption 3 be satisfied and letx∈X ∞ andu∈U ∞(x). Then, ˜J ces N+1 (x, u)≥ ˜J ces N (x, u)and ˜J ces ∞ (x, u)≥ ˜J ces N (x, u) for allN∈N. Proof:Since(1− k N+1 )≥(1− k N )for allk∈I [0,N] and ˜ℓ(x, u)≥0, for anyxandu, we infer ˜J...

    work page 1949