From Data to Predictive Control: A Framework for Stochastic Linear Systems with Output Measurements

TL;DR

The paper addresses robust predictive control for stochastic discrete-time LTI systems with partial measurements by bridging data-driven parameter estimation and model-based control. It introduces D2PC, which combines EM-based parameter identification under structural constraints, asymptotically correct uncertainty quantification forming Θ_δ, robust dynamic output-feedback synthesis via an LFR and S-procedure, and a predictive control scheme using ellipsoidal homothetic tubes and chance constraints. The approach guarantees recursive feasibility and probabilistic constraint satisfaction while providing an asymptotic bound on the closed-loop cost, demonstrated on a 10-dimensional spring-mass-damper example. This framework enables principled integration of data-driven estimation with robust predictive control, offering scalable guarantees for systems with unbounded stochastic noise and partial state measurements.

Abstract

We introduce data to predictive control, D2PC, a framework to facilitate the design of robust and predictive controllers from data. The proposed framework is designed for discrete-time stochastic linear systems with output measurements and provides a principled design of a predictive controller based on data. The framework starts with a parameter identification method based on the Expectation-Maximization algorithm, which incorporates pre-defined structural constraints. Additionally, we provide an asymptotically correct method to quantify uncertainty in parameter estimates. Next, we develop a strategy to synthesize robust dynamic output-feedback controllers tailored to the derived uncertainty characterization. Finally, we introduce a predictive control scheme that guarantees recursive feasibility and satisfaction of chance constraints. This framework marks a significant advancement in integrating data into robust and predictive control schemes. We demonstrate the efficacy of D2PC through a numerical example involving a 10-dimensional spring-mass-damper system.

Figures (9)

• Figure 1: Illustration of the proposed D2PC framework.
• Figure 2: Illustration of the linear fractional representation for the closed loop system \ref{['eq:closed_loop_lfr']}.
• Figure 3: Illustration for the spring-mass-damper system.
• Figure 4: Prediction error (normalized w.r.t true system) across identified models and the true system, error bars denote $\pm 3$ standard deviations. LS denotes models estimated using the least squares method, with the corresponding order.
• Figure 5: Guaranteed closed-loop $\mathcal{H}_2$-norm (normalized w.r.t. nominal) versus considered probability level $\delta$. Proposed method with full-block S-procedure (Lemma \ref{['lem:multiplier']}) is solid, over-approximation (Prop. \ref{['prop:alternative_unc_set']}) is dashed, and performance of nominal LQG with estimated system model is dotted.