Analysis of Stability and Performance of Economic Model Predictive Control with State-Independent Costs

Alireza Arastou; Erik Weyer; Ye Wang

arxiv: 2604.24054 · v1 · submitted 2026-04-27 · 🧮 math.OC · cs.SY· eess.SY

Analysis of Stability and Performance of Economic Model Predictive Control with State-Independent Costs

Alireza Arastou , Ye Wang , Erik Weyer This is my paper

Pith reviewed 2026-05-08 02:38 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY

keywords economic model predictive controldissipativityasymptotic convergencesteady-state setinput-dependent costLyapunov stabilityperiodic EMPCwater distribution networks

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The pith

Under strict dissipativity, EMPC with input-only costs drives closed-loop trajectories to the set of optimal steady states.

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

The paper examines economic model predictive control where the stage cost depends only on the control input, a setting that arises in applications such as water distribution networks and allows multiple steady states to share the same optimal input. The central result is that a strict dissipativity assumption tied to the optimal steady-state set implies asymptotic convergence of closed-loop trajectories to this set and convergence of the economic cost to its optimal value. To secure Lyapunov stability, the authors introduce a modified stage cost that keeps the optimal input unchanged while rendering a chosen equilibrium asymptotically stable, at the price of a small performance penalty. The same approach is lifted to multi-step EMPC to handle linear systems subject to periodic costs and disturbances. A water-network case study illustrates both the convergence property and the stabilizing modification.

Core claim

For EMPC schemes whose stage cost depends solely on the control input, a strict dissipativity assumption with respect to the set of optimal steady states guarantees that closed-loop trajectories converge asymptotically to this set and that the economic cost converges to the optimal steady-state cost. A modified stage cost is proposed that preserves the optimal input while guaranteeing asymptotic stability of one specific equilibrium point, accepting a slight performance loss. The framework is extended to periodic EMPC for a class of linear systems by lifting the problem to a multi-step formulation.

What carries the argument

The strict dissipativity assumption defined with respect to the set of optimal steady states, which supplies the decrease condition needed for asymptotic convergence of both trajectories and economic cost.

If this is right

Closed-loop economic cost converges to the optimal steady-state cost.
The modified stage cost yields asymptotic stability of a chosen equilibrium while keeping the same optimal input.
Lifting to multi-step EMPC extends the results to linear systems with periodic costs and disturbances.
The water-distribution-network example confirms both asymptotic convergence and the stabilizing modification in a realistic setting.

Where Pith is reading between the lines

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

The input-only cost structure may simplify controller design in systems where full state measurement is costly or unavailable.
Similar dissipativity-based arguments could apply to other networked control problems that naturally produce input-dependent objectives.
Quantifying the exact performance loss introduced by the stabilizing modification would allow explicit trade-off analysis in applications.

Load-bearing premise

The strict dissipativity assumption with respect to the set of optimal steady states must hold for the convergence claim to be valid.

What would settle it

A concrete dynamical system satisfying the strict dissipativity condition for which closed-loop trajectories under the EMPC law fail to approach the set of optimal steady states.

Figures

Figures reproduced from arXiv: 2604.24054 by Alireza Arastou, Erik Weyer, Ye Wang.

**Figure 1.** Figure 1: The proposed EMPC schemes and the obtained prop view at source ↗

**Figure 2.** Figure 2: Richmond water distribution network 9 view at source ↗

**Figure 3.** Figure 3: Demand multiplier and electricity tariff view at source ↗

**Figure 4.** Figure 4: Optimal steady state trajectory for pump view at source ↗

**Figure 5.** Figure 5: Some optimal steady-state trajectories of the water view at source ↗

**Figure 8.** Figure 8: EMPC closed-loop cost at each period From Lemma 3, the upper bound on ε can be computed. In this case study, the state constraint is given by (35b) and (36). From (36) and using the lifted state constraint in (30c), R = max x˜∈X˜ x˜ ⊤x˜ = 97.55. Choosing ε = 0.00102 guarantees that the cost difference between optimal values of modified and unmodified steady state problems is less than γ = 0.1. Denote the … view at source ↗

**Figure 11.** Figure 11: Periodic closed-loop EMPC cost using the modified view at source ↗

**Figure 10.** Figure 10: xA and uA obtained by the modified EMPC scheme steady state problem by ˜x s ε and ˜u s ε . The unique solution of the optimal steady state problem for x s ε,A and u s ε,A is shown in view at source ↗

read the original abstract

This paper studies economic model predictive Control (EMPC) schemes, where the stage cost depends only on control inputs. Such problems arise in applications like water distribution networks and differ from standard EMPC since multiple steady states can correspond to the unique optimal steady input. We show that, under a strict dissipativity assumption related to the set of optimal steady states, the closed-loop trajectories converge asymptotically to this set, ensuring convergence of the economic cost to the optimal steady state cost. To enhance Lyapunov stability, we propose a modified stage cost that preserves the optimal input while guaranteeing asymptotic stability of a specific equilibrium with a slight performance loss. The approach is further extended to EMPC of a class of linear systems with periodic costs and disturbances by lifting it to a multi-step EMPC problem for periodic operations. A case study with a water distribution network demonstrates the effectiveness of the proposed methods in achieving both asymptotic convergence and stability.

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 gives a clean but incremental stability result for EMPC with input-only costs under dissipativity, with the water network example as its main practical anchor.

read the letter

The core result is that closed-loop trajectories converge to the set of optimal steady states when the system satisfies strict dissipativity with respect to that set. Because the stage cost ignores the state, multiple equilibria can share the same optimal input, which matches the water-network setting they consider. The proof follows the usual rotated-cost and Lyapunov route once the assumption is granted. They also add a modified stage cost that keeps the optimal input but forces asymptotic stability of a single point, at the price of a small performance degradation. The periodic extension for linear systems via multi-step lifting is a straightforward adaptation that handles time-varying disturbances without much extra machinery. The water-network case study is the strongest element; it shows the closed loop behaving as predicted in simulation. The main limitation is that the dissipativity assumption is stated but not equipped with checkable conditions or a verification step for the example. The simulations confirm good behavior under the assumption, yet they do not establish whether the assumption actually holds for the network dynamics. This leaves the result formally correct but its range of applicability harder to assess. The work is aimed at researchers already working on economic MPC for resource networks or similar systems with input-only objectives. It is a modest extension of existing theory rather than a broad advance, but the example gives it enough substance to warrant referee time. I would send it to review.

Referee Report

2 major / 2 minor

Summary. The paper analyzes EMPC with state-independent stage costs (depending only on u), where multiple steady states may share the same optimal input. Under a strict dissipativity assumption w.r.t. the set of optimal steady states, it proves asymptotic convergence of closed-loop trajectories to this set (hence economic cost convergence to the optimum). It introduces a modified stage cost to achieve Lyapunov stability of a chosen equilibrium with minor performance degradation, extends the framework to linear systems with periodic costs/disturbances via multi-step lifting, and illustrates the methods on a water distribution network example.

Significance. If the dissipativity assumption is satisfied, the results offer a useful extension of EMPC theory to input-only costs, which arise in applications such as water networks. The stability modification and periodic extension are practical additions, and the case study provides concrete evidence of convergence behavior. These contributions are solid when the assumption holds, but their broader utility depends on the ease of verifying the key hypothesis.

major comments (2)

[Dissipativity assumption and main convergence result] The central convergence theorem (asymptotic convergence of trajectories to the optimal steady-state set) rests entirely on the strict dissipativity assumption w.r.t. that set. The manuscript states the assumption but supplies neither general sufficient conditions on the dynamics and input-dependent cost that guarantee dissipativity for a non-singleton set, nor any checkable test or numerical verification for the water-network example. This renders the implication from assumption to closed-loop behavior formally correct yet difficult to apply or falsify in practice.
[Case study section] In the water distribution network case study, the effectiveness demonstration assumes the dissipativity condition without providing any explicit verification, parameter values, or simulation check that the storage function and class-K function exist for the given f and l(u). This leaves the applicability of the convergence claim unconfirmed for the motivating example.

minor comments (2)

[Introduction] The introduction could more clearly distinguish the set of optimal steady states from the unique optimal input early on, to avoid potential confusion for readers unfamiliar with multi-steady-state EMPC.
[Periodic extension] In the periodic extension, the performance loss incurred by the modified cost is described qualitatively; a brief bound or comparison relative to the original economic cost would strengthen the claim of 'slight' degradation.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their constructive and detailed comments on the manuscript. We address each major comment point by point below, indicating where we will revise the paper to improve clarity and practical applicability while preserving the core theoretical contributions.

read point-by-point responses

Referee: [Dissipativity assumption and main convergence result] The central convergence theorem (asymptotic convergence of trajectories to the optimal steady-state set) rests entirely on the strict dissipativity assumption w.r.t. that set. The manuscript states the assumption but supplies neither general sufficient conditions on the dynamics and input-dependent cost that guarantee dissipativity for a non-singleton set, nor any checkable test or numerical verification for the water-network example. This renders the implication from assumption to closed-loop behavior formally correct yet difficult to apply or falsify in practice.

Authors: The convergence theorem is rigorously established under the stated strict dissipativity assumption with respect to the set of optimal steady states, which is the natural extension of standard dissipativity-based EMPC results to the case of state-independent costs. We do not supply general sufficient conditions for dissipativity on non-singleton sets, as these are typically system-dependent and their derivation would require a separate, substantial research effort beyond the scope of the present analysis. This is consistent with the broader EMPC literature, where dissipativity is posited as an assumption and verified on a case-by-case basis. To address the practical concern, we will revise the manuscript to include a discussion of numerical verification procedures and apply them explicitly to the water-network example. revision: partial
Referee: [Case study section] In the water distribution network case study, the effectiveness demonstration assumes the dissipativity condition without providing any explicit verification, parameter values, or simulation check that the storage function and class-K function exist for the given f and l(u). This leaves the applicability of the convergence claim unconfirmed for the motivating example.

Authors: We agree that the case study would benefit from explicit verification to confirm applicability. In the revised version, we will add the specific parameter values employed for the water distribution network, propose a candidate storage function, and include simulation results or direct checks demonstrating that the strict dissipativity inequality holds with a suitable class-K function for the given system dynamics and input-dependent cost. revision: yes

standing simulated objections not resolved

General sufficient conditions on the dynamics and input-dependent cost that guarantee dissipativity for a non-singleton set

Circularity Check

0 steps flagged

No circularity: results are conditional theorems under external assumptions

full rationale

The paper states its central convergence result as a theorem that holds under a strict dissipativity assumption with respect to the optimal steady-state set; this assumption is imported from the broader EMPC literature rather than being derived from or equivalent to the paper's own conclusions or fitted quantities. The subsequent construction of a modified stage cost is an explicit design choice that trades a small performance loss for Lyapunov stability of a chosen equilibrium, without any reduction of the claimed properties back to the inputs by definition. The periodic extension and water-network case study are presented as applications, not as self-referential verifications. No load-bearing step reduces by construction to a self-citation, a fitted parameter renamed as prediction, or an ansatz smuggled via prior work by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest primarily on the strict dissipativity assumption, which is a domain-specific condition that must be verified for each system rather than derived from first principles.

axioms (1)

domain assumption strict dissipativity assumption related to the set of optimal steady states
Invoked directly to establish asymptotic convergence of closed-loop trajectories to the optimal set.

pith-pipeline@v0.9.0 · 5462 in / 1173 out tokens · 73735 ms · 2026-05-08T02:38:15.340360+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 4 canonical work pages

[1]

doi: 10.1109/TAC.2010. 2101291. Matthew Ellis, Jinfeng Liu, and Panagiotis D. Christofides. Economic model predictive control: The- ory, formulations and chemical process applications

work page doi:10.1109/tac.2010 2010
[2]

doi: 10.1109/MCS.2021. 3139724. Lars Gr¨ une and Marleen Stieler. Asymptotic stabil- ity and transient optimality of economic MPC with- out terminal conditions.Journal of Process Con- trol, 24(8):1187–1196,

work page doi:10.1109/mcs.2021 2021
[3]

doi: https://doi.org/10.1016/j.jprocont.2014.05.003

ISSN 0959-1524. doi: https://doi.org/10.1016/j.jprocont.2014.05.003. Eco- nomic nonlinear model predictive control. Defeng He, Jie Luo, and Haiping Du. Dynamic negotiation-based distributed EMPC with varying consensus speeds of heterogeneous electric vehicle pla- toons.IEEE Transactions on Control Systems Tech- nology, 32(4):1495–1503,

work page doi:10.1016/j.jprocont.2014.05.003 2014
[4]

M¨ uller, David Angeli, and Frank Allg¨ ower

Matthias A. M¨ uller, David Angeli, and Frank Allg¨ ower. Transient average constraints in eco- nomic model predictive control.Automatica, 50 (11):2943–2950, 2014a. ISSN 0005-1098. doi: https://doi.org/10.1016/j.automatica.2014.10.024. URLhttps://www.sciencedirect.com/science/ article/pii/S0005109814004051. Matthias A. M¨ uller, David Angeli, Frank Allg¨ ...

work page doi:10.1016/j.automatica.2014.10.024 2014
[5]

Appendix A Lifted Dynamical System Iterating the dynamical system in (24) overTsteps yields xkT+i =AixkT + i−1X j=0 Ai−1−jB ukT+j + i−1X j=0 Ai−1−jBd dj, i= 1, . . . , T. 13 Stacking theseTstate equations yields   xkT+1 xkT+2 ... x(k+1)T   =   0 0· · ·0A 0 0· · ·0A 2 ... ... ... ... 0 0· · ·0A T   | {z } ˜A   x(k−1)T+1 x...

2012

[1] [1]

doi: 10.1109/TAC.2010. 2101291. Matthew Ellis, Jinfeng Liu, and Panagiotis D. Christofides. Economic model predictive control: The- ory, formulations and chemical process applications

work page doi:10.1109/tac.2010 2010

[2] [2]

doi: 10.1109/MCS.2021. 3139724. Lars Gr¨ une and Marleen Stieler. Asymptotic stabil- ity and transient optimality of economic MPC with- out terminal conditions.Journal of Process Con- trol, 24(8):1187–1196,

work page doi:10.1109/mcs.2021 2021

[3] [3]

doi: https://doi.org/10.1016/j.jprocont.2014.05.003

ISSN 0959-1524. doi: https://doi.org/10.1016/j.jprocont.2014.05.003. Eco- nomic nonlinear model predictive control. Defeng He, Jie Luo, and Haiping Du. Dynamic negotiation-based distributed EMPC with varying consensus speeds of heterogeneous electric vehicle pla- toons.IEEE Transactions on Control Systems Tech- nology, 32(4):1495–1503,

work page doi:10.1016/j.jprocont.2014.05.003 2014

[4] [4]

M¨ uller, David Angeli, and Frank Allg¨ ower

Matthias A. M¨ uller, David Angeli, and Frank Allg¨ ower. Transient average constraints in eco- nomic model predictive control.Automatica, 50 (11):2943–2950, 2014a. ISSN 0005-1098. doi: https://doi.org/10.1016/j.automatica.2014.10.024. URLhttps://www.sciencedirect.com/science/ article/pii/S0005109814004051. Matthias A. M¨ uller, David Angeli, Frank Allg¨ ...

work page doi:10.1016/j.automatica.2014.10.024 2014

[5] [5]

Appendix A Lifted Dynamical System Iterating the dynamical system in (24) overTsteps yields xkT+i =AixkT + i−1X j=0 Ai−1−jB ukT+j + i−1X j=0 Ai−1−jBd dj, i= 1, . . . , T. 13 Stacking theseTstate equations yields   xkT+1 xkT+2 ... x(k+1)T   =   0 0· · ·0A 0 0· · ·0A 2 ... ... ... ... 0 0· · ·0A T   | {z } ˜A   x(k−1)T+1 x...

2012