A Linear Parameter-Varying Approach to Data Predictive Control

TL;DR

This work develops direct data-driven predictive control methods for discrete-time LPV systems, replacing explicit model identification with data-driven predictors derived from the LPV Fundamental Lemma. It introduces two realizations—IO-DPC using input-output data and SS-DPC using state data—each equipped with stability and recursive feasibility guarantees via terminal ingredients computable purely from data. The authors provide procedures for computing terminal controllers and invariant sets (ellipsoidal and polyhedral MPI) without a plant model, and they address noisy data with slack variables and regularization, plus strategies for external vs internal scheduling and recursion to manage complexity. The practical relevance is demonstrated on a nonlinear unbalanced-disc example, where the LPV-DPC approach achieves competitive performance with model-based MPC and outperforms DeePC in nonlinear regimes. Overall, the paper offers a comprehensive, data-driven path to LPV-friendly predictive control with rigorous guarantees and practical implementation guidance.

Abstract

By means of the linear parameter-varying (LPV) Fundamental Lemma, we derive novel data-driven predictive control (DPC) methods for LPV systems. In particular, we present output-feedback and state-feedback-based LPV-DPC methods with terminal ingredients, which guarantee exponential stability and recursive feasibility. We provide methods for the data-based computation of these terminal ingredients. Furthermore, an in-depth analysis of the application and implementation aspects of the LPV-DPC schemes is given, including application for nonlinear systems and handling noisy data. We compare and demonstrate the performance of the proposed methods in a detailed simulation example involving a nonlinear unbalanced disc system.

Figures (5)

• Figure 1: Prediction problem for a given data-dictionary. The signals of the system are collected here in variable $w$. Figure adopted from VerhoekAbbasTothHaesaert2021.
• Figure 2: Global LPV embedding of a nonlinear system with inputs $u$ and outputs $y$ and state realization $x$. A scheduling variable $p:=\psi(x,u)$ is defined such that if the trajectory of $p$ is known, then the remaining signal relations of $y$ and $u$ are linear. To obtain an LPV representation, the connection between $\psi(x,u)$ and $p$ is severed and $p$ is assumed to be varying independently from $w$ in a bounded set $\mathbb{P} \supseteq \psi(\mathbb{X},\mathbb{U})$.
• Figure 3: Simulation results for the comparison of predictive controllers that use IO measurements on the unbalanced disc system. We compare here a NL-MPC ( ), an LPV-MPC ( ) and an LPV-DPC ( ). The left plot shows the results where the LPV predictive controllers use GS to determine the scheduling in the prediction horizon, while the right plot shows the result where the LPV predictive controllers use ISE for this purpose. The simulation results show that our LPV-DPC method results in a similar performance as nonlinear and LPV model-based approaches.
• Figure 4: Comparison of the simulation results on the unbalanced disc system with a DeePC controller ( ) and our LPV-IO-DPC controller ( ) with different reference points. The further away $\theta_\mathrm{r}$ is from the stable equilibrium $\theta_k=0$, the more the DeePC performance degrades.
• Figure 5: Comparison of the simulation results with the indirect two-step approach ( ) and the LPV-IO-DPC ( ). For the left two plots, both ${\breve{y}}$ and $y_{[k-\tau, k-1]}$ are affected by noise, while for the right two plots only ${\breve{y}}$ is noisy and $y_{[k-\tau, k-1]}$ is measured noise free.

Theorems & Definitions (14)

• Proposition 1: Simplified LPV Fundamental Lemma newpaper
• Definition 1
• Lemma 1
• Theorem 1
• proof
• Remark 2
• Remark 3
• Theorem 2: LPV-SS-DPC recursive feasibility and exponential stability
• proof
• Proposition 2: Data-driven LPV state-feedback synthesis Verhoek2022_DDLPVstatefb