# Perturbation-based Regret Analysis of Predictive Control in Linear Time Varying Systems

arXiv: https://arxiv.org/abs/2106.10497 · HTML: https://mlsanity.com/paper/2106.10497/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jun 2021.

## Built on by (most cited first, 25 of 25)

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- [Online Learning for Predictive Control with Provable Regret Guarantees](https://mlsanity.com/paper/2111.15041/) (2021)
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- [Approximate Model Predictive Control for Microgrid Energy Management via Imitation Learning](https://mlsanity.com/paper/2510.20040/) (2025)
- [Learning to Control under Time-Varying Environment](https://mlsanity.com/paper/2206.02507/) (2022)
- [Sub-linear Regret in Adaptive Model Predictive Control](https://mlsanity.com/paper/2310.04842/) (2023)
- [Data-Guided Regulator for Adaptive Nonlinear Control](https://mlsanity.com/paper/2311.12230/) (2023)
- [Smoothed Online Optimization for Target Tracking: Robust and Learning-Augmented Algorithms](https://mlsanity.com/paper/2509.05930/) (2025)
- [SCaLE: Switching Cost aware Learning and Exploration](https://mlsanity.com/paper/2601.09042/) (2026)
- [Reinforcement Learning with Multi-Step Lookahead Information Via Adaptive Batching](https://mlsanity.com/paper/2601.10418/) (2026)
- [Self-Adaptive Learning and Model Predictive Control for Tracking Unknown Dynamics with No Regret](https://mlsanity.com/paper/2607.26370/) (2026)

## Abstract (arXiv)

We study predictive control in a setting where the dynamics are time-varying and linear, and the costs are time-varying and well-conditioned. At each time step, the controller receives the exact predictions of costs, dynamics, and disturbances for the future k time steps. We show that when the prediction window k is sufficiently large, predictive control is input-to-state stable and achieves a dynamic regret of O(λk T), where λ < 1 is a positive constant. This is the first dynamic regret bound on the predictive control of linear time-varying systems. Under more assumptions on the terminal costs, we also show that predictive control obtains the first competitive bound for the control of linear time-varying systems: 1 + O(λk). Our results are derived using a novel proof framework based on a perturbation bound that characterizes how a small change to the system parameters impacts the optimal trajectory.