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Asynchronous Model Predictive Control Under Model Mismatch: Stability and Performance Guarantees

Changrui Liu, Anil Alan, Shengling Shi, Bart De Schutter

arXiv:2609.08836v1eess.SYcs.SY

TL;DR

Model mismatch is unavoidable, and existing CE-MPC analyses offer limited guarantees for asynchronous feedback. The paper develops a perturbation-based analysis that gives stability and performance guarantees relating horizon, feedback frequency, and mismatch, subject to an equilibrium-preserving mismatch assumption.

  • Problem

    A unified stability and suboptimality analysis for asynchronous CE-MPC under model mismatch, with explicit dependence on inter-execution steps, remains absent.

    Source: §I · p. 2

  • Method

    The paper combines nominal asynchronous MPC analysis with value-function perturbation analysis to derive CE-MPC guarantees under model mismatch.

    Source: §I · p. 2, C. Analysis Pipeline · p. 6

  • Results

    The analysis provides stability and worst-case infinite-horizon performance guarantees that jointly characterize horizon, feedback frequency, and model mismatch.

    Source: §I · p. 2, §VII · pp. 7–8

  • Takeaways & Limitations

    The bounds show that shorter inter-execution steps and longer prediction horizons reduce the suboptimality gap relative to IHOPC.

    Source: §I · p. 2

    The guarantees assume model mismatch does not change the true model's equilibrium.

    Source: §I · p. 2

Abstract

from arXiv · show

Certainty-equivalence model predictive control (CE-MPC) is widely used for its simplicity and efficiency, but theoretical guarantees under asynchronous feedback remain limited. This paper establishes stability and performance guarantees for asynchronous CE-MPC of input-constrained nonlinear systems. We first derive a nominal stability condition and competitive-ratio bound that explicitly account for inter-execution intervals without prescribing a feedback mechanism. A value-function perturbation analysis for quadratic stage costs then accommodates additive, potentially non-smooth model mismatch without constraint qualification conditions. Combining these results yields stability criteria and competitive-ratio bounds for CE-MPC under general asynchronous feedback, including event/self-triggered and multi-step MPC. The guarantees explicitly relate prediction horizon, inter-execution time, and uncertainty magnitude, quantifying performance degradation relative to an ideal infinite-horizon controller. These results clarify tradeoffs between feedback frequency, model accuracy, and horizon length, guiding asynchronous MPC design using approximate or learned models.

I. INTRODUCTION

The paper studies how model mismatch and asynchronous feedback jointly affect CE-MPC guarantees, and develops an analysis for input-constrained nonlinear systems. Its framework relates stability and infinite-horizon performance to the prediction horizon, inter-execution steps, and mismatch.

I. INTRODUCTION

  • A unified analysis of asynchronous CE-MPC under mismatch is absent, while prior CE-MPC suboptimality analyses mostly assume time-triggered feedback.Trigger-specific bounds can obscure how inter-execution steps directly affect performance.Source: §I · pp. 1–2
  • The analysis quantifies nominal stability and performance, then bounds how asynchronous execution and model mismatch affect CE-MPC stability and suboptimality.The bounds depend on prediction horizon and maximum inter-execution step; the analysis assumes mismatch does not change the true model's equilibrium.Source: §I · p. 2

A. Notation

  • The notation defines I_a:b as the natural numbers in the interval [a,b] and uses the 2-norm by default.The paper also uses positive-definite functions and the standard comparison-function classes K and KL.Source: A. Notation · p. 2

B. System Description & Optimal Control

  • The system model separates nominal dynamics from unknown time-varying mismatch, with a shared equilibrium and bounded mismatch assumed.The mismatch need not be smooth, and the control objective is asymptotic regulation to the origin with quadratic stage cost.Source: B. System Description & Optimal Control · pp. 2–3

C. Asynchronous Model Predictive Control

Asynchronous CE-MPC solves its finite-horizon optimization only at selected feedback times, using the nominal model. The maximum interval between executions is bounded by the prediction horizon, and special timing choices recover multi-step or time-triggered CE-MPC.

C. Asynchronous Model Predictive Control

  • At each triggering time, CE-MPC solves a finite-horizon problem using the nominal model, prediction horizon N, and quadratic terminal cost.The terminal cost is required to satisfy ℓ_T(x) ≥ ℓ⋆(x).Source: C. Asynchronous Model Predictive Control · p. 3
  • The feedback sequence starts at t₁ = 0, and each inter-execution step is at most M, with M ≤ N − 1.The control loop closes only at the selected feedback times.Source: C. Asynchronous Model Predictive Control · p. 3
  • When every inter-execution step equals M, the scheme is multi-step CE-MPC; setting M = 1 yields time-triggered CE-MPC.Between feedback times, the controller applies the planned inputs.Source: C. Asynchronous Model Predictive Control · p. 3

III. PROBLEM FORMULATION

The formulation defines the finite-horizon CE-MPC value function and asynchronous closed-loop performance, then sets stability and ideal-performance comparison as its objectives.

III. PROBLEM FORMULATION

  • The optimal value of the horizon-N MPC problem defines the CE-MPC value function V_N(·; f).The analysis pairs this value function with nominal cost controllability and a relaxed control Lyapunov property for the terminal cost.Source: §III · p. 3
  • Nominal cost controllability bounds the MPC value function relative to stage cost, while the terminal cost is assumed to satisfy a relaxed control Lyapunov property.These conditions depend only on nominal dynamics; their parameters can be approximated numerically offline.Source: §III · p. 4
  • The asynchronous controller applies its MPC-selected inputs over inter-execution ranges, with consecutive ranges joined by a boundary condition.The resulting infinite-horizon performance is defined for the specified feedback-time sequence and initial state.Source: §III · p. 4
  • The analysis seeks conditions for asymptotic stabilization of the true system and an explicit bound on infinite-horizon performance relative to the ideal infinite-horizon optimum.The bound quantifies the suboptimality associated with finite horizon, plant-model mismatch, and asynchronous feedback.Source: §III · p. 4

IV. ANALYSIS OF NOMINAL ASYNCHRONOUS MODEL PREDICTIVE CONTROL

The paper first characterizes nominal asynchronous MPC, then extends the analysis to model mismatch using value-function perturbation bounds. The results cover arbitrary asynchronous feedback and allow time-varying, non-smooth mismatch.

IV. ANALYSIS OF NOMINAL ASYNCHRONOUS MODEL PREDICTIVE CONTROL

  • The nominal analysis extends multi-step MPC results to arbitrary asynchronous feedback and makes stability conditions and performance bounds explicit in the maximum open-loop interval M.It also provides a foundation for the subsequent mismatch analysis.Source: §IV · p. 4
  • When δ_N,M < 1, nominal asynchronous MPC is asymptotically stable and its infinite-horizon performance has a competitive-ratio bound; for fixed M, increasing N drives δ_N,M to 0 and the bound to 1.For finite N, δ_N,M increases with M, so performance may degrade as the maximum inter-execution step expands.Source: §IV · pp. 4–5

V. STABILITY & PERFORMANCE ANALYSIS UNDER PLANT-MODEL MISMATCH

  • The mismatch results accommodate arbitrary asynchronous feedback and time-varying, non-smooth mismatch without restrictive constraint qualifications.The approach differs from prior time-triggered analysis based on sensitivity under constraint qualifications.Source: §V · p. 5, A. Perturbation Analysis · p. 5

A. Perturbation Analysis

  • Under model mismatch, the perturbation analysis bounds state-driven changes in the CE-MPC value function, the functional discrepancy between CE-MPC and true-MPC value functions, and accumulated stage-cost error excluding terminal cost.These bounds provide the basis for the subsequent closed-loop stability and suboptimality analysis.Source: A. Perturbation Analysis · pp. 5–6

B. Competitive-Ratio Performance Bound

The theorem gives stability and a competitive-ratio performance bound when horizon, maximum inter-execution step, and mismatch jointly satisfy a stability condition. Its bounds characterize mismatch-related degradation and the effects of feedback frequency and horizon.

B. Competitive-Ratio Performance Bound

  • When ρ_N,M(ε_f) < 1, asynchronous CE-MPC is asymptotically stable at the origin and its infinite-horizon performance satisfies a competitive-ratio bound.The condition jointly depends on prediction horizon N, maximum inter-execution step M, and mismatch level ε_f.Source: B. Competitive-Ratio Performance Bound · p. 6
  • Greater model mismatch reduces the stability margin and increases the worst-case performance bound, with these effects especially pronounced for larger M.Maintaining a target performance level under mismatch therefore requires more frequent feedback than in the nominal case.Source: B. Competitive-Ratio Performance Bound · p. 6
  • Under mismatch, the competitive ratio need not vary monotonically with horizon N, and minimizing it can provide a coarse warm start for horizon tuning.The stability condition remains useful for deriving permissible parameter ranges; under perfect modeling and fixed M, the bound approaches 1 as N grows.Source: B. Competitive-Ratio Performance Bound · p. 6

C. Analysis Pipeline

The analysis pipeline combines nominal asynchronous MPC guarantees with value-function perturbation bounds to establish robust stability and performance for asynchronous CE-MPC. The resulting competitive ratio can be computed from the mismatch level and nominal model, while the stability condition is sufficient and the performance bound conservative.

C. Analysis Pipeline

  • The pipeline derives state- and model-driven value-function perturbation bounds, combines them with nominal asynchronous MPC guarantees, and establishes robust stability and performance in Theorem 2.It is tailored to general terminal costs characterized by rCLFs.Source: C. Analysis Pipeline · p. 6
  • The stability condition is sufficient, and the performance bound is inherently conservative.The analysis can be tightened under stronger system assumptions or more structured mismatch by sharpening its stability index and bounding functions.Source: C. Analysis Pipeline · pp. 6–7
  • The competitive ratio depends only on the mismatch level and nominal model, not the inaccessible true model.Source: C. Analysis Pipeline · p. 7
  • The pipeline also applies when the nominal model varies over time while the true model is static, as in learning-based MPC with online estimates fluctuating near the true model.Source: C. Analysis Pipeline · p. 7

VI. NUMERICAL EXAMPLE

The numerical example finds that the theoretical competitive-ratio bound holds across the tested asynchronous execution and mismatch settings, while tracking performance deterioration as either increases. The conclusion states that the framework characterizes how horizon, feedback frequency, and mismatch jointly affect stability and worst-case infinite-horizon performance.

VI. NUMERICAL EXAMPLE

  • The simulations vary M at fixed εf = 0.01 and vary εf at fixed M = 5, evaluating random and fixed inter-execution intervals.The setup uses a horizon N = 25 and restricts M ≤ 8 to satisfy the stated stability condition for εf ≤ 0.01.Source: §VI · p. 7
  • The competitive-ratio bound consistently holds in the simulations and captures observed performance deterioration as maximum inter-execution step M and mismatch level εf increase.The bound is conservative but parameter-dependent, rather than an empirical fit.Source: §VI · p. 7, Figure 2 · p. 7, Figure 3 · p. 7

VII. CONCLUSIONS AND FUTURE WORK

  • The framework characterizes how prediction horizon, feedback frequency, and model mismatch jointly influence stability and worst-case infinite-horizon performance.Source: §VII · pp. 7–8
  • Future research directions include tighter perturbation bounds and extending the framework to multi-agent systems with distributed MPC.Source: §VII · p. 8

APPENDIX I DETAILS OF SECTION IV

The appendix defines nominal closed-loop notation and auxiliary results, then uses relaxed dynamic programming arguments to establish asymptotic stability and a performance bound for asynchronous MPC without model mismatch.

APPENDIX I DETAILS OF SECTION IV

  • The nominal k-step closed-loop state is recursively defined under a state-feedback policy, with ℓπ(x) denoting the stage cost under that policy.Source: Appendix I · p. 8

A. Auxiliary Lemmas

B. Proof of Theorem 1

  • The proof establishes the performance bound by combining the preceding inequalities and taking the infinite-execution limit.Source: B. Proof of Theorem 1 · p. 9
  • For nominal asynchronous MPC, the condition δN(M) < 1 implies asymptotic stability of the origin.The proof uses relaxed dynamic programming arguments and positive definiteness of the stage cost.Source: B. Proof of Theorem 1 · p. 9

A. Auxiliary Lemmas

The auxiliary results bound state perturbations and relate value-function estimates to quadratic stage costs, supporting later perturbation analysis.

A. Auxiliary Lemmas

  • The state-perturbation bounds are established by induction and have symmetric upper-bound structures for the two compared trajectories.Source: A. Auxiliary Lemmas · p. 9
  • Lemma 2 follows from strong convexity of quadratic stage costs ℓx(x) = x⊤Qx with Q ≻ 0.Source: A. Auxiliary Lemmas · p. 9
  • Lemma 6 supplies an inequality for vectors x and y under a positive definite matrix Q, derived using Cauchy–Schwarz.Source: A. Auxiliary Lemmas · p. 9

B. Supporting Propositions

The supporting propositions bound value-function perturbations from state and model mismatch, providing ingredients for the asynchronous CE-MPC guarantees. For Lipschitz-continuous models, one bound starts from the trajectory deviation caused by different initial states under identical inputs.

B. Supporting Propositions

  • The Lipschitz-model result quantifies trajectory deviation under identical input sequences as a function of the initial-state difference.The cited result uses the model's Lipschitz constant and the step index.Source: B. Supporting Propositions · p. 10
  • Different initial states under identical inputs yield a bounded state-trajectory deviation for Lipschitz-continuous models, supporting a bound on general value-function perturbation.The state-driven perturbation bound is identified as (15).Source: B. Supporting Propositions · p. 10
  • The propositions establish separate state-driven and model-driven perturbation bounds, identified as (16) and (17).The proof passages identify the state-driven bound as (16) and the model-driven bound as (17).Source: B. Supporting Propositions · pp. 9, 11

C. Proof of Theorem 2

The proof combines nominal dissipation with perturbation bounds to establish stability and a performance bound for asynchronous CE-MPC. Asymptotic stability holds when the stated stability index is below one.

C. Proof of Theorem 2

  • The proof combines the nominal dissipation inequality with perturbation bounds to analyze the value-function difference across consecutive active-feedback times.That difference is the forward difference of the CE-MPC value function.Source: C. Proof of Theorem 2 · p. 11
  • Asymptotic stability of the origin under asynchronous CE-MPC with model mismatch is ensured provided that ρ_N,M(ε_f) < 1.The proof invokes standard MPC stability arguments via relaxed dynamic programming.Source: C. Proof of Theorem 2 · p. 11
  • Theorem 2's performance bound is established by combining the accumulated-cost inequality with the subsequent proof steps.The displayed inequality bounds accumulated stage cost by the CE-MPC value and then by a factor-scaled comparison value.Source: C. Proof of Theorem 2 · p. 12
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