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arxiv: 1907.10169 · v1 · pith:KSLW2ZOLnew · submitted 2019-07-23 · 🧮 math.OC

Distributed Model Predictive Control Under Inexact Primal-Dual Gradient Optimization Based on Contraction Analysis

Yanxu Su , Yang Shi , Changyin Sun This is my paper

Pith reviewed 2026-05-24 16:56 UTC · model grok-4.3

classification 🧮 math.OC
keywords distributed model predictive controlprimal-dual gradientcontraction analysisinexact optimizationcoupled constraintsrecursive feasibilitystability analysis
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The pith

Distributed MPC for linear systems with coupled constraints remains recursively feasible and stable under inexact primal-dual gradient solutions.

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

The paper develops a distributed model predictive control approach for discrete-time linear systems subject to globally coupled constraints. The dual problem is solved distributively via primal-dual gradient optimization with Laplacian consensus, and constraint tightening permits early termination of the iterations. Contraction theory is used to analyze convergence of the discrete-time primal-dual dynamics to a nonlinear objective. This setup allows proofs of recursive feasibility and closed-loop stability even when the optimization is solved inexactly.

Core claim

Under assumptions on the system and contraction conditions, the DMPC strategy based on inexact primal-dual gradient optimization with distributed consensus achieves recursive feasibility and stability of the closed-loop system.

What carries the argument

Primal-dual gradient optimization using Laplacian consensus, combined with constraint tightening and analyzed by contraction theory for discrete-time dynamics.

If this is right

  • The optimization can terminate prematurely while maintaining convergence guarantees.
  • Recursive feasibility holds for the DMPC problem despite inexact solutions.
  • Closed-loop stability is guaranteed under the inexact optimization.
  • Distributed implementation is enabled through consensus on the dual variables.

Where Pith is reading between the lines

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

  • This method could apply to systems where exact optimization is computationally prohibitive.
  • Similar contraction-based analysis might be used for other distributed optimization problems in control.
  • The approach suggests potential for reducing computation in large networks of subsystems.

Load-bearing premise

The discrete-time linear system properties and the conditions for contraction theory to prove convergence of the primal-dual gradient dynamics are satisfied.

What would settle it

An example where the closed-loop system becomes unstable or the optimization loses feasibility when using the inexact primal-dual solutions under the paper's stated assumptions.

Figures

Figures reproduced from arXiv: 1907.10169 by Changyin Sun, Yang Shi, Yanxu Su.

Figure 3
Figure 3. Figure 3: States of each subsystem. Define the objective function for the DMPC optimization problem as 0 5 10 15 20 25 30 t 0 0.2 0.4 0.6 0.8 u(t) = 0.1 = 0.3 = 0.5 = 0.7 = 0.9 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 1
Figure 1. Figure 1: Double-water tanks systems. 0 5 10 15 20 25 30 t -0.5 0 0.5 1 u(t) Subsystem 1 Subsystem 2 Subsystem 3 Subsystem 4 Total [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison on ǫ. Ji (xi , ui) = X N s=0  kxi (s|t)k 2 Qi + kui (s|t)k 2 Ri  +kxi (N|t)k 2 Pi (63) where N = 8, Qi = 10I, Ri = 1, Pi =  31.7459 9.8300 9.8300 56.3415 (64) [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

This paper develops a distributed model predictive control (DMPC) strategy for a class of discrete-time linear systems with consideration of globally coupled constraints. The DMPC under study is based on the dual problem concerning all subsystems, which is solved by means of the primal-dual gradient optimization in a distributed manner using Laplacian consensus. To reduce the computational burden, the constraint tightening method is utilized to provide a capability of premature termination with guaranteeing the convergence of the DMPC optimization. The contraction theory is first adopted in the convergence analysis of the primal-dual gradient optimization under discrete-time updating dynamics towards a nonlinear objective function. Under some reasonable assumptions, the recursive feasibility and stability of the closed-loop system can be established under the inexact solution. A numerical simulation is given to verify the performance of the proposed strategy.

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 / 2 minor

Summary. The paper develops a distributed MPC strategy for discrete-time linear systems subject to globally coupled constraints. The dual problem is solved in a distributed fashion via primal-dual gradient iterations that incorporate Laplacian consensus; constraint tightening is introduced to permit early termination of the optimizer. Contraction theory is applied to establish convergence of the resulting discrete-time nonlinear dynamics, from which recursive feasibility and closed-loop stability are claimed to follow under inexact solutions, with a numerical example provided for illustration.

Significance. If the contraction metric and associated bounds can be rigorously verified for the specific dual function induced by the coupled constraints, the approach would supply a principled route to lowering per-iteration computation in DMPC while retaining stability guarantees, extending standard primal-dual and consensus techniques to the inexact setting.

major comments (2)
  1. [Convergence analysis section] Convergence analysis section: the manuscript invokes contraction theory for the discrete-time primal-dual gradient map augmented by Laplacian consensus, yet provides neither an explicit construction of a contraction metric nor a verification that the required step-size and Lipschitz conditions hold for the dual function arising from the globally coupled linear constraints; because this property is load-bearing for the subsequent recursive-feasibility and Lyapunov-stability arguments, the link between inexact optimization and closed-loop guarantees remains unestablished.
  2. [Recursive feasibility and stability section] Recursive feasibility and stability section: the error bounds relating the inexact primal-dual iterates to the exact optimizer (and thence to the tightened constraint sets) are not derived; without quantitative propagation of the contraction residual into the MPC feasibility and Lyapunov decrease conditions, the central claim that stability holds under premature termination cannot be assessed.
minor comments (2)
  1. [Abstract] The abstract states that results hold 'under some reasonable assumptions' without enumerating them; listing the key assumptions (e.g., on system matrices, graph connectivity, and step-size selection) would improve readability.
  2. [Notation and problem formulation] Notation for the dual variables and the consensus-augmented update rule should be introduced with an explicit equation reference when first used, to avoid ambiguity when the contraction metric is later discussed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and will revise the manuscript to strengthen the rigor of the analysis as suggested.

read point-by-point responses

  1. Referee: [Convergence analysis section] Convergence analysis section: the manuscript invokes contraction theory for the discrete-time primal-dual gradient map augmented by Laplacian consensus, yet provides neither an explicit construction of a contraction metric nor a verification that the required step-size and Lipschitz conditions hold for the dual function arising from the globally coupled linear constraints; because this property is load-bearing for the subsequent recursive-feasibility and Lyapunov-stability arguments, the link between inexact optimization and closed-loop guarantees remains unestablished.

    Authors: We agree that the current version does not provide an explicit construction of the contraction metric or a direct verification of the step-size and Lipschitz conditions for the specific dual function. In the revised manuscript we will add an explicit construction of a contraction metric tailored to the discrete-time primal-dual gradient map with Laplacian consensus, together with a verification that the required conditions hold for the dual function induced by the globally coupled linear constraints under the paper's stated assumptions. revision: yes

  2. Referee: [Recursive feasibility and stability section] Recursive feasibility and stability section: the error bounds relating the inexact primal-dual iterates to the exact optimizer (and thence to the tightened constraint sets) are not derived; without quantitative propagation of the contraction residual into the MPC feasibility and Lyapunov decrease conditions, the central claim that stability holds under premature termination cannot be assessed.

    Authors: We acknowledge that explicit quantitative error bounds and their propagation into the feasibility and stability conditions are not derived in the present manuscript. The revised version will include derivations of the error bounds between the inexact primal-dual iterates and the exact optimizer, followed by a quantitative propagation of the contraction residual into the tightened constraint sets, recursive feasibility, and Lyapunov decrease conditions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard contraction theory and external assumptions.

full rationale

The paper adopts contraction theory to analyze convergence of the discrete-time primal-dual gradient dynamics and uses it to establish recursive feasibility and stability under inexact termination. This is presented as an application of an existing mathematical framework to the DMPC setting with Laplacian consensus, without any quoted steps that reduce predictions to fitted inputs by construction, self-definitional mappings, or load-bearing self-citations whose validity is internal to the paper. The central claims rest on stated assumptions about the linear system and contraction conditions, which are treated as given rather than derived from the paper's own outputs. No renaming of known results or smuggling of ansatzes via self-citation is exhibited in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on domain assumptions about linear systems and the applicability of contraction theory to the optimization dynamics; no free parameters or invented entities are identifiable from the abstract.

axioms (2)
  • domain assumption Systems are discrete-time linear with globally coupled constraints.
    Explicitly stated as the class of systems under study.
  • domain assumption Contraction theory applies to the discrete-time primal-dual gradient dynamics for the nonlinear objective.
    Invoked as the basis for the convergence analysis and stability proofs.

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