Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Conditional Generative Modeling of Stochastic LTI Systems: A Behavioral Approach

Jiayun Li, Yilin Mo

TL;DR

It is proved the convergence of the distribution of samples generated by the CGM as the size of the trajectory library increases, with an explicit characterization of the convergence rate, and integrated this generative model into predictive controllers for stochastic LTI systems.

Abstract

This paper presents a data-driven model for Linear Time-Invariant (LTI) stochastic systems by sampling from the conditional probability distribution of future outputs given past input-outputs and future inputs. It operates in a fully behavioral manner, relying solely on the current trajectory and pre-collected input-output data, without requiring explicit identification of system parameters. We refer to this model as a behavioral Conditional Generative Model (CGM). We prove the convergence of the distribution of samples generated by the CGM as the size of the trajectory library increases, with an explicit characterization of the convergence rate. Furthermore, we demonstrate that the gap between the asymptotic distribution of the proposed CGM and the true posterior distribution obtained by Kalman filter, which leverages the knowledge of all system parameters and all historical data, decreases exponentially with respect to the length of past samples. Finally, we integrate this generative model into predictive controllers for stochastic LTI systems. Numerical results verify the derived bounds and demonstrate the effectiveness of the controller equipped with the proposed behavioral CGM.

Conditional Generative Modeling of Stochastic LTI Systems: A Behavioral Approach

TL;DR

It is proved the convergence of the distribution of samples generated by the CGM as the size of the trajectory library increases, with an explicit characterization of the convergence rate, and integrated this generative model into predictive controllers for stochastic LTI systems.

Abstract

This paper presents a data-driven model for Linear Time-Invariant (LTI) stochastic systems by sampling from the conditional probability distribution of future outputs given past input-outputs and future inputs. It operates in a fully behavioral manner, relying solely on the current trajectory and pre-collected input-output data, without requiring explicit identification of system parameters. We refer to this model as a behavioral Conditional Generative Model (CGM). We prove the convergence of the distribution of samples generated by the CGM as the size of the trajectory library increases, with an explicit characterization of the convergence rate. Furthermore, we demonstrate that the gap between the asymptotic distribution of the proposed CGM and the true posterior distribution obtained by Kalman filter, which leverages the knowledge of all system parameters and all historical data, decreases exponentially with respect to the length of past samples. Finally, we integrate this generative model into predictive controllers for stochastic LTI systems. Numerical results verify the derived bounds and demonstrate the effectiveness of the controller equipped with the proposed behavioral CGM.

Paper Structure

This paper contains 22 sections, 16 theorems, 187 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Trajectory Library Construction
  4. Conditional Generative Model
  5. Performance Analysis
  6. Convergence of Conditional Distribution
  7. Optimality Gap with KF
  8. Extension to Systems with Non-Gaussian Noise
  9. Direct Data-Driven Predictive Control via Conditional Generative Modeling
  10. Predictive Controller Inspired by Scenario-based MPC
  11. Discussions: Relationship with DeePC
  12. Simulations
  13. Convergence Speed Verification
  14. Control Performance Comparison
  15. Conclusion
  16. ...and 7 more sections

Key Result

Proposition 1

Assume that the input-output data $\{\tilde{u}_t,\tilde{y}_t\}_{t=1}^K$ satisfy Assumption assump:persistently_exciting and that $T_{ini} \ge \ell$. Let $(u_{ini},y_{ini},u_f,y_f)$ be any length-$(T_{ini}+T)$ input-output trajectory that is generated by the system eq:linear_system. Then there exists Moreover, if $g$satisfies: then the associated future output is uniquely determined as $y_f=Y_fg$.

Figures (1)

  • Figure 1: Convergence speed verification of the proposed generative model as the number of trajectories $N$ (or equivalently, the length of the single trajectory) increases. The figures depict a log-log ribbon plot, with the solid line indicating the mean value and the shaded region representing the range of values.

Theorems & Definitions (34)

  • Proposition 1: Willems' lemma coulson2019datafundamental_lemma
  • Remark 1
  • Definition 1: Conditional Generative Model (CGM)
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Proposition 2
  • ...and 24 more