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arxiv: 2605.04656 · v1 · submitted 2026-05-06 · 📡 eess.SY · cs.SY

Adaptive MPC for Constrained Trajectory Tracking of Uncertain LTI System with Input-Rate Limits

Bishal Dey , Abhishek Dhar , Sumit kr. Pandey , Anindita Sengupta This is my paper

Pith reviewed 2026-05-08 17:22 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords adaptive MPCtrajectory trackingparametric uncertaintyinput-rate constraintsrecursive feasibilityLyapunov stabilityLTI systemsconstrained control
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The pith

An adaptive MPC framework guarantees recursive feasibility and stability for trajectory tracking of uncertain LTI systems with input-rate limits.

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

The paper addresses the challenge of tracking trajectories in discrete-time linear time-invariant systems that have unknown but bounded parameters and must respect hard limits on states, inputs, and how fast inputs can change. It shows that by feeding online estimates of the parameters into a specially reformulated model predictive controller and pairing it with an adaptive learning scheme, the optimization problem stays feasible from one step to the next even though the allowed control values change over time because of the rate limits. This setup also lets the authors prove that the tracking error goes to zero and the states stay bounded using a Lyapunov function. Readers should care because many practical systems, from autonomous vehicles to industrial robots, face exactly these uncertainties and actuator speed limits, yet most prior control designs either assume perfect knowledge or drop some of the constraints.

Core claim

This paper claims that an adaptive model predictive control framework, which systematically incorporates estimated system parameters and a suitably designed adaptive learning process, overcomes the challenges of achieving zero tracking error under unknown parameters and the temporal coupling from input-rate constraints. The reformulated MPC ensures recursive feasibility of the optimization routine despite the time-varying admissible control set, and Lyapunov-based analysis guarantees closed-loop stability with convergence of the tracking error and boundedness of system states.

What carries the argument

The reformulated MPC optimization routine that uses estimated parameters together with an adaptive learning process to handle the time-varying admissible control set induced by input-rate constraints.

Load-bearing premise

The discrete-time LTI system has bounded parametric uncertainty and a suitably designed adaptive learning process can restore recursive feasibility for the time-varying admissible control set from input-rate constraints.

What would settle it

A case where the adaptive estimates cause the MPC optimization to become infeasible or the tracking error to grow unbounded despite the uncertainty bounds being respected would disprove the feasibility and stability results.

Figures

Figures reproduced from arXiv: 2605.04656 by Abhishek Dhar, Anindita Sengupta, Bishal Dey, Sumit kr. Pandey.

Figure 1
Figure 1. Figure 1: System States 0 5 10 15 20 25 30 35 40 45 Time (s) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 s e k view at source ↗
Figure 2
Figure 2. Figure 2: Error States 0 5 10 15 20 25 30 35 40 45 Time (s) -3 -2 -1 0 1 2 3 u k view at source ↗
Figure 3
Figure 3. Figure 3: Control Input 0 5 10 15 20 25 30 35 40 45 Time (s) -3 -2 -1 0 1 2 3 u k view at source ↗
Figure 4
Figure 4. Figure 4: Rate of Change of Control Input [−0.4, 0.15; 0.35, −0.5] and Bˆ 0 = [1, 0.1; 0.1, 0.8]. The prediction horizon length N is set to 5. To assess the performance of the proposed control framework under various conditions, simulations are performed with 10 different initial parameter estimates and 10 different initial states distributed throughout the prescribed constraint region. 0 5 10 15 20 25 30 35 40 45 T… view at source ↗
read the original abstract

This paper addresses the trajectory-tracking problem for discrete-time linear time-invariant systems with bounded parametric uncertainty, subject to hard constraints on system states, control inputs, and input rates. Unlike existing methods, which often consider only partial uncertainty, omit input-rate or state constraints, or focus on regulation problems, this work provides a systematic adaptive model predictive control (MPC) solution for constrained trajectory tracking under full parametric uncertainty. Determining the control input required to achieve zero tracking error under unknown parameters is challenging. Simultaneously, trajectory tracking under uncertainty with input-rate constraints induces temporal coupling in the control sequence, resulting in a time-varying admissible control set and rendering standard recursive feasibility arguments inapplicable. These challenges are overcome by systematically utilizing the estimated system parameters, coupled with a suitably designed adaptive learning process within a reformulated MPC framework. The recursive feasibility of the proposed MPC optimization routine is then rigorously established despite the time-varying admissible control set induced by input-rate constraints. Closed-loop stability is guaranteed via Lyapunov-based analysis, ensuring convergence of the tracking error and boundedness of system states. Simulation results validate the effectiveness of the pr

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 paper proposes an adaptive MPC framework for discrete-time LTI systems with bounded parametric uncertainty, addressing constrained trajectory tracking under hard state, input, and input-rate constraints. It reformulates the MPC optimization using online parameter estimates and a tailored adaptive learning process to handle the time-varying admissible control set induced by rate limits, claims rigorous recursive feasibility despite this coupling, and establishes closed-loop stability via Lyapunov analysis, with simulations validating tracking error convergence and state boundedness.

Significance. If the feasibility and stability results hold as claimed, the work would meaningfully extend adaptive MPC literature by systematically addressing full parametric uncertainty in tracking (rather than regulation) problems that include input-rate constraints, which create temporal dependencies absent from many standard formulations. The combination of online adaptation with a reformulated optimization to restore feasibility could enable safer application in domains like robotics or vehicle control where rate limits and uncertainty coexist.

minor comments (3)
  1. The abstract is truncated mid-sentence ('the pr'); ensure the concluding sentence on simulation validation is completed in the final manuscript.
  2. In the simulation section, the specific system matrices, uncertainty bounds, and comparison baselines (e.g., non-adaptive MPC or other adaptive schemes) are not detailed enough to allow reproduction or assessment of the claimed performance gains.
  3. Notation for the time-varying admissible set and the adaptive update law should be introduced with explicit definitions early in the problem formulation to improve readability of the subsequent feasibility argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation for minor revision. The referee's assessment correctly identifies the key contributions regarding adaptive MPC for constrained trajectory tracking under full parametric uncertainty and input-rate limits. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's derivation chain relies on standard adaptive MPC reformulation to handle time-varying admissible sets from input-rate constraints, followed by recursive feasibility arguments and Lyapunov-based closed-loop stability proofs. These steps are presented as rigorous extensions of existing methods without reducing to self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations. The abstract and description indicate that feasibility and stability follow from the constructed adaptive learning process and MPC optimization, which remain independent of the target results. No quoted equations or steps exhibit the enumerated circular patterns, making the central claims self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard domain assumptions for adaptive MPC; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (2)
  • domain assumption The plant is a discrete-time LTI system with bounded parametric uncertainty
    Explicitly stated as the problem setting in the abstract.
  • domain assumption An adaptive learning process can be designed to restore recursive feasibility for the time-varying admissible control set
    Invoked to overcome the challenge of input-rate constraints.

pith-pipeline@v0.9.0 · 5505 in / 1347 out tokens · 26674 ms · 2026-05-08T17:22:07.003925+00:00 · methodology

discussion (0)

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Reference graph

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