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Set-Theoretic Direct Data-driven Predictive Control

Mohammad Bajelani, Walter Lucia, Klaske van Heusden

TL;DR

By considering the class of constrained LTI systems with unknown time delays, a set-theoretic direct data-driven predictive controller is proposed to provide closed-loop guarantees and guarantees finite-time convergence and recursive feasibility, independent of objective function tuning.

Abstract

Designing the terminal ingredients of direct data-driven predictive control presents challenges due to its reliance on an implicit, non-minimal input-output data-driven representation. By considering the class of constrained LTI systems with unknown time delays, we propose a set-theoretic direct data-driven predictive controller that does not require a terminal cost to provide closed-loop guarantees. In particular, first, starting from input/output data series, we propose a sample-based method to build N-step input output backward reachable sets. Then, we leverage the constructed family of backward reachable sets to derive a data-driven control law. The proposed method guarantees finite-time convergence and recursive feasibility, independent of objective function tuning. It requires neither explicit state estimation nor an explicit prediction model, relying solely on input-output measurements; therefore, unmodeled dynamics can be avoided. Finally, a numerical example highlights the effectiveness of the proposed method in stabilizing the system, whereas direct data-driven predictive control without terminal ingredients fails under the same conditions.

Set-Theoretic Direct Data-driven Predictive Control

TL;DR

By considering the class of constrained LTI systems with unknown time delays, a set-theoretic direct data-driven predictive controller is proposed to provide closed-loop guarantees and guarantees finite-time convergence and recursive feasibility, independent of objective function tuning.

Abstract

Designing the terminal ingredients of direct data-driven predictive control presents challenges due to its reliance on an implicit, non-minimal input-output data-driven representation. By considering the class of constrained LTI systems with unknown time delays, we propose a set-theoretic direct data-driven predictive controller that does not require a terminal cost to provide closed-loop guarantees. In particular, first, starting from input/output data series, we propose a sample-based method to build N-step input output backward reachable sets. Then, we leverage the constructed family of backward reachable sets to derive a data-driven control law. The proposed method guarantees finite-time convergence and recursive feasibility, independent of objective function tuning. It requires neither explicit state estimation nor an explicit prediction model, relying solely on input-output measurements; therefore, unmodeled dynamics can be avoided. Finally, a numerical example highlights the effectiveness of the proposed method in stabilizing the system, whereas direct data-driven predictive control without terminal ingredients fails under the same conditions.

Paper Structure

This paper contains 10 sections, 2 theorems, 22 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Statement
  3. Constrained LTI systems
  4. Input-output framework
  5. Set-Theoretic MPC
  6. Problem of interest
  7. Sample-based N-step Input-output Backward Reachable Sets
  8. Set-Theoretic Data-Driven Predictive Control
  9. Numerical Results and Discussion
  10. Conclusion

Key Result

Theorem 1

Let $u^d$ be persistently exciting of order $L+n$, and ${\{u^d_t},{y^d_t\}}_{t=0}^{N_{0}-1}$ a trajectory of system $G$. Then, ${\{\bar{u}},{\bar{y}}\}$ is a trajectory of system $G$ if and only if there exists $\alpha \in \mathbb{R}^{N_{0}-L+1}$ such that

Figures (6)

  • Figure 1: Flowchart of the overall process: (A) a single experiment generating an input-output dataset; (B) the sample-based method for computing N-IOBRS; and (C) the ST-DDPC developed from the sample-based N-IOBRS.
  • Figure 2: Visualization of an extended trajectory for an input-output trajectory, assuming $T_{\text{ini}}=2$, integrating both past data (historical measurements) and future data (predicted trajectory).
  • Figure 3: A visualization example of sample-based N-step input-output reachable sets for two nested sets: $\hat{\Xi}^{l-1}$ is the target set, $\hat{\Xi}^l$ is the corresponding N-IOBRS, $\xi_{[i,j]}^l$ is the $j^{th}$ element of $i^{th}$ sampled extended trajectory $\bar{\xi}_{i}^l$.
  • Figure 4: Projection of N-step Input-Output Backward Reachable Sets (N-IOBRS) onto the output space, $Proj_{[y_{-1},y_{-2}]}(\Xi^{1}), \ldots, Proj_{[y_{-1},y_{-2}]}(\Xi^{5})$, alongside the realized extended trajectories of DDPC and ST-DDPC.
  • Figure 5: Input-output trajectory of the proposed method, ST-DDPC, along with the corresponding realized set index.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Definition 1: System's Lag 9654975
  • Definition 2: LTI System's Trajectory 9654975
  • Definition 3: Convex Hull
  • Definition 4: Polytope
  • Definition 5: Persistently Excitation
  • Theorem 1: Fundamental Lemma berberich2020robust
  • Definition 6: Extended Stateberberich2021design
  • Definition 7: Extended Trajectory
  • Definition 8: Input-output Control Invariant Set
  • Definition 9: Input-output Equilibrium Point
  • ...and 5 more