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Robust Data-Driven Predictive Control for Unknown Linear Time-Invariant Systems

Kaijian Hu, Tao Liu

TL;DR

This paper introduces a robust data-driven predictive control framework for unknown LTI systems that eliminates the need for the persistently exciting condition on pre-collected data by constructing a set of all compatible systems. At each step, it replaces the intractable min–max MPC over this set with an upper bound minimization, yielding a state-feedback gain $F_k=S_k\Gamma_k^{-1}$ obtained from LMIs. The approach is extended to handle input and output constraints, and further adapted to cases where only input-output data is available through an IO-based state construction. A key case study on an unstable batch reactor demonstrates that the proposed RDPC achieves comparable performance to DPC with much longer data, evidencing improved data efficiency and practicality for unknown, possibly unstable plants.

Abstract

This paper presents a new robust data-driven predictive control scheme for unknown linear time-invariant systems by using input-state-output or input-output data based on whether the state is measurable. To remove the need for the persistently exciting (PE) condition of a sufficiently high order on pre-collected data, a set containing all systems capable of generating such data is constructed. Then, at each time step, an upper bound of a given objective function is derived for all systems in the set, and a feedback controller is designed to minimize this bound. The optimal control gain at each time step is determined by solving a set of linear matrix inequalities. We prove that if the synthesis problem is feasible at the initial time step, it remains feasible for all future time steps. Unlike current data-driven predictive control schemes based on behavioral system theory, our approach requires less stringent conditions for the pre-collected data, facilitating easier implementation. Further, the proposed predictive control scheme features an infinite prediction horizon, potentially resulting in superior overall control performance compared to existing methods with finite prediction horizons. The effectiveness of our proposed methods is demonstrated through application to an unknown and unstable batch reactor.

Robust Data-Driven Predictive Control for Unknown Linear Time-Invariant Systems

TL;DR

This paper introduces a robust data-driven predictive control framework for unknown LTI systems that eliminates the need for the persistently exciting condition on pre-collected data by constructing a set of all compatible systems. At each step, it replaces the intractable min–max MPC over this set with an upper bound minimization, yielding a state-feedback gain obtained from LMIs. The approach is extended to handle input and output constraints, and further adapted to cases where only input-output data is available through an IO-based state construction. A key case study on an unstable batch reactor demonstrates that the proposed RDPC achieves comparable performance to DPC with much longer data, evidencing improved data efficiency and practicality for unknown, possibly unstable plants.

Abstract

This paper presents a new robust data-driven predictive control scheme for unknown linear time-invariant systems by using input-state-output or input-output data based on whether the state is measurable. To remove the need for the persistently exciting (PE) condition of a sufficiently high order on pre-collected data, a set containing all systems capable of generating such data is constructed. Then, at each time step, an upper bound of a given objective function is derived for all systems in the set, and a feedback controller is designed to minimize this bound. The optimal control gain at each time step is determined by solving a set of linear matrix inequalities. We prove that if the synthesis problem is feasible at the initial time step, it remains feasible for all future time steps. Unlike current data-driven predictive control schemes based on behavioral system theory, our approach requires less stringent conditions for the pre-collected data, facilitating easier implementation. Further, the proposed predictive control scheme features an infinite prediction horizon, potentially resulting in superior overall control performance compared to existing methods with finite prediction horizons. The effectiveness of our proposed methods is demonstrated through application to an unknown and unstable batch reactor.
Paper Structure (8 sections, 8 theorems, 84 equations, 1 figure, 1 table, 3 algorithms)

This paper contains 8 sections, 8 theorems, 84 equations, 1 figure, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Robust Data-Driven Predictive Control Problem
  4. Unconstrained Predictive Control Using Input-State-Output Data
  5. Constrained Predictive Control Using Input-State-Output Data
  6. Predictive Control Scheme Using Input-Output Data
  7. Case Study
  8. Conclusion

Key Result

Lemma 1

Let $Q\in \mathbb{R}^{n\times n}$, $P\in \mathbb{R}^{m\times m}$ be symmetric matrices and $R\in \mathbb{R}^{n\times m}$ be an $n\times m$ matrix. The following three statements are equivalent

Figures (1)

  • Figure 1: The input and output results of the closed-loop system under the RDPC, and DPC with two different lengths of data.

Theorems & Definitions (19)

  • Lemma 1: 1994Lmii
  • Lemma 2: Waarde2023
  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • Remark 4
  • Remark 5
  • Remark 6
  • ...and 9 more