[Paper Review] Data-driven Stochastic Output-Feedback Predictive Control: Recursive Feasibility through Interpolated Initial Conditions

Guanru Pan, Ruchuan Ou+1 more|arXiv (Cornell University)|Dec 15, 2022

Advanced Control Systems Optimization4 citations

TL;DR

This paper proposes a data-driven stochastic output-feedback model predictive control scheme for linear time-invariant systems with unknown dynamics and stochastic disturbances. By leveraging a stochastic variant of Willems’ fundamental lemma and interpolating between measured and predicted initial conditions, the method ensures recursive feasibility and practical stability without requiring a parametric system model, with theoretical guarantees and performance bounds provided.

ABSTRACT

The paper investigates data-driven output-feedback predictive control of linear systems subject to stochastic disturbances. The scheme relies on the recursive solution of a suitable data-driven reformulation of a stochastic Optimal Control Problem (OCP), which allows for forward prediction and optimization of statistical distributions of inputs and outputs. Our approach avoids the use of parametric system models. Instead it is based on previously recorded data using a recently proposed stochastic variant of Willems' fundamental lemma. The stochastic variant of the lemma is applicable to a large class of linear dynamics subject to stochastic disturbances of Gaussian and non-Gaussian nature. To ensure recursive feasibility, the initial condition of the OCP -- which consists of information about past inputs and outputs -- is considered as an extra decision variable of the OCP. We provide sufficient conditions for recursive feasibility and closed-loop practical stability of the proposed scheme as well as performance bounds. Finally, a numerical example illustrates the efficacy and closed-loop properties of the proposed scheme.

Motivation & Objective

• To develop a data-driven stochastic output-feedback predictive control scheme for LTI systems with unknown system matrices and stochastic disturbances.
• To ensure recursive feasibility in the presence of stochastic disturbances without relying on parametric system identification.
• To improve upon binary initial condition selection by introducing a continuous interpolation strategy between measured and predicted states.
• To provide sufficient conditions for closed-loop practical stability and performance bounds in a data-driven setting.
• To extend the stochastic fundamental lemma to enable distributional prediction of inputs and outputs from recorded data.

Proposed method

• The method uses a stochastic variant of Willems’ fundamental lemma to represent system trajectories via polynomial chaos expansions (PCE), enabling data-driven prediction of statistical distributions of inputs and outputs.
• It formulates a data-driven stochastic optimal control problem (OCP) where the initial condition—based on past inputs and outputs—is treated as an extra decision variable to ensure recursive feasibility.
• A continuous interpolation strategy is introduced to select the initial condition, blending the measured current state with the predicted state from the previous solution, improving robustness over binary selection.
• The OCP is solved recursively, with terminal ingredients (P, Γ, γ, Z_f) computed from recorded data using Hankel matrices and PCE basis functions.
• The method relies on known distributions of initial conditions and disturbances, and uses Legendre polynomials to construct a PCE basis of dimension L=39 in the numerical example.
• Stability and feasibility are guaranteed via sufficient conditions derived from the structure of the OCP and the interpolation strategy, with performance bounded by α=295.21 in the numerical example.

Experimental results

Research questions

• RQ1 Can recursive feasibility be guaranteed in data-driven stochastic output-feedback model predictive control without a parametric system model?
• RQ2 How can the initial condition in the OCP be selected to ensure recursive feasibility when system matrices are unknown and disturbances are stochastic?
• RQ3 What is the impact of continuous interpolation between measured and predicted initial conditions on closed-loop stability and performance?
• RQ4 Can the stochastic fundamental lemma be effectively leveraged to predict statistical distributions of inputs and outputs from recorded data?
• RQ5 What performance bounds can be derived for data-driven stochastic MPC under recursive feasibility?

Key findings

• The proposed scheme ensures recursive feasibility by treating the initial condition as a decision variable in the OCP, with sufficient conditions derived for feasibility and practical stability.
• The continuous interpolation strategy between measured and predicted initial conditions improves robustness compared to binary selection, enhancing recursive feasibility.
• Closed-loop practical stability is guaranteed under the proposed conditions, with the average asymptotic cost bounded above by α=295.21 in the numerical example.
• The chance constraint on output Y¹ is satisfied with probability at least 1−ε_y=0.9, as verified across 50 closed-loop realizations.
• The output Y² converges to a narrow distribution centered at 0, as shown by normalized histograms of 1000 sampled trajectories, indicating effective distributional control.
• Performance bounds are derived and validated numerically, confirming that the average cost asymptotically approaches the theoretical upper bound α=295.21.