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Causality-Informed Data-Driven Predictive Control

Malika Sader, Yibo Wang, Dexian Huang, Chao Shang, Biao Huang

TL;DR

The paper tackles the degradation of data-driven predictive control (DDPC) performance under uncertainty by diagnosing lack of causality as a key source of high prediction variance. It develops a causality-informed DDPC framework using $LQ$ factorization, introducing a causal $oldsymbol{ extgamma}$-DDPC (C-$oldsymbol{ extgamma}$-DDPC) and its regularized variant (RC-$oldsymbol{ extgamma}$-DDPC) that enforce a strictly causal multi-step predictor with minimal additional complexity. By connecting causality to an explicit $LQ$-based representation, the authors derive a simple yet powerful reformulation that reduces non-causal residuals and balances control cost against implicit predictor identification. Numerical studies on stochastic LTI and nonlinear systems, plus a simulated industrial heating furnace, show that enforcing causality improves prediction accuracy and control performance, particularly under noise, model mismatch, or nonlinearity, with competitive computation times relative to noncausal approaches. The work offers a practical, data-driven pathway to more reliable predictive control in industrial applications where uncertainty and nonlinearities are prevalent.

Abstract

As a useful and efficient alternative to generic model-based control scheme, data-driven predictive control is subject to bias-variance trade-off and is known to not perform desirably in face of uncertainty. Through the connection between direct data-driven control and subspace predictive control, we gain insight into the reason being the lack of causality as a main cause for high variance of implicit prediction. In this article, we seek to address this deficiency by devising a novel causality-informed formulation of direct data-driven control. Built upon LQ factorization, an equivalent two-stage reformulation of regularized data-driven control is first derived, which bears clearer interpretability and a lower complexity than generic forms. This paves the way for deriving a two-stage causality-informed formulation of data-driven predictive control, as well as a regularized form that balances between control cost minimization and implicit identification of multi-step predictor. Since it only calls for block-triangularization of a submatrix in LQ factorization, the new causality-informed formulation comes at no excess cost as compared to generic ones. Its efficacy is investigated based on numerical examples and application to model-free control of a simulated industrial heating furnace. Empirical results corroborate that the proposed method yields obvious performance improvement over existing formulations in handling stochastic noise and process nonlinearity.

Causality-Informed Data-Driven Predictive Control

TL;DR

The paper tackles the degradation of data-driven predictive control (DDPC) performance under uncertainty by diagnosing lack of causality as a key source of high prediction variance. It develops a causality-informed DDPC framework using factorization, introducing a causal -DDPC (C--DDPC) and its regularized variant (RC--DDPC) that enforce a strictly causal multi-step predictor with minimal additional complexity. By connecting causality to an explicit -based representation, the authors derive a simple yet powerful reformulation that reduces non-causal residuals and balances control cost against implicit predictor identification. Numerical studies on stochastic LTI and nonlinear systems, plus a simulated industrial heating furnace, show that enforcing causality improves prediction accuracy and control performance, particularly under noise, model mismatch, or nonlinearity, with competitive computation times relative to noncausal approaches. The work offers a practical, data-driven pathway to more reliable predictive control in industrial applications where uncertainty and nonlinearities are prevalent.

Abstract

As a useful and efficient alternative to generic model-based control scheme, data-driven predictive control is subject to bias-variance trade-off and is known to not perform desirably in face of uncertainty. Through the connection between direct data-driven control and subspace predictive control, we gain insight into the reason being the lack of causality as a main cause for high variance of implicit prediction. In this article, we seek to address this deficiency by devising a novel causality-informed formulation of direct data-driven control. Built upon LQ factorization, an equivalent two-stage reformulation of regularized data-driven control is first derived, which bears clearer interpretability and a lower complexity than generic forms. This paves the way for deriving a two-stage causality-informed formulation of data-driven predictive control, as well as a regularized form that balances between control cost minimization and implicit identification of multi-step predictor. Since it only calls for block-triangularization of a submatrix in LQ factorization, the new causality-informed formulation comes at no excess cost as compared to generic ones. Its efficacy is investigated based on numerical examples and application to model-free control of a simulated industrial heating furnace. Empirical results corroborate that the proposed method yields obvious performance improvement over existing formulations in handling stochastic noise and process nonlinearity.
Paper Structure (15 sections, 5 theorems, 25 equations, 5 figures, 3 tables, 1 algorithm)

This paper contains 15 sections, 5 theorems, 25 equations, 5 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basics of DDPC
  4. $\gamma$-DDPC via LQ factorization
  5. Causality-Informed Data-Driven Predictive Control Formulation
  6. Causal form of SPC
  7. Causal $\gamma$-DDPC
  8. Regularized causal $\gamma$-DDPC
  9. Numerical Case Studies
  10. DDPC of stochastic LTI systems
  11. DDPC of nonlinear systems
  12. Predictive Control of A Simulated Industrial Heating Furnace
  13. Data collection under open-loop conditions
  14. Data collection under closed-loop conditions
  15. Conclusion

Key Result

Theorem 1

(markovsky2008data) For system equation: LTI system with $w(t)=0$ and $v(t)=0$, $U_d = \mathcal{H}_L({u}_{d,[1,N]})$ and $Y_d = \mathcal{H}_L({y}_{d,[1,N]})$ are block Hankel matrices of inputs and outputs. Consider the past input-output data $u_{\rm p}= u_{[t-L_{\rm p}:t-1]}$ and $y_{\rm p}= y_{[t- with $Z_p = [U_p^\top~~ Y_p^\top]^\top$ and $z_p = [u_p^\top~~ y_p^\top]^\top$, where $U_{\rm p} =

Figures (5)

  • Figure 1: Control performance of different approaches of LTI system with open-loop data collection in $100$ Monte Carlo simulations.
  • Figure 2: Control performance of different approaches of nonlinear system with open-loop data collection ($N_d=200$).
  • Figure 3: Structural diagram of industrial tubular furnace
  • Figure 4: Outputs of the tubular furnace system controlled by different controllers using open-loop data, including $\gamma$-DDPC, S-DDPC, the proposed C-$\gamma$-DDPC and RC-$\gamma$-DDPC ($\lambda=5\times10^{-3}$, $\mu=0.1$).
  • Figure 5: Outputs of the tubular furnace system controlled by different controllers using closed-loop data, including $\gamma$-DDPC, S-DDPC, the proposed C-$\gamma$-DDPC and RC-$\gamma$-DDPC.

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2: Equivalence to SPC breschi2023data
  • Lemma 1
  • Lemma 2
  • proof
  • Remark 1
  • Theorem 3: Equivalence between C-$\gamma$-DDPC and C-SPC
  • proof
  • Remark 2
  • Remark 3: Comparison with segmented DDPC (S-DDPC) o2022data