Robust Constraint Satisfaction in Data-Driven MPC

Abstract

We propose a purely data-driven model predictive control (MPC) scheme to control unknown linear time-invariant systems with guarantees on stability and constraint satisfaction in the presence of noisy data. The scheme predicts future trajectories based on data-dependent Hankel matrices, which span the full system behavior if the input is persistently exciting. This paper extends previous work on data-driven MPC by including a suitable constraint tightening which ensures that the closed-loop output trajectory satisfies desired pointwise-in-time constraints. Furthermore, we provide estimation procedures to compute system constants related to controllability and observability, which are required to implement the constraint tightening. The practicality of the proposed approach is illustrated via a numerical example.

Controller design for robust invariance from noisy data

“…With d = 0 ∈ D δ , this concludes the existence of d ∈ D δ and V ∈ V T , so the assumption of Fact 1 is verified. Then, we apply Fact 1 with clear correspondences between the quantities in (27) and those in (3), and obtain that ( 16) holds if and only if for all x ∈ S, there exists E ∈ R ns×(n d +T n d ) such that (23) holds.…”

“…While early use of this insight in the context of data-driven control has appeared, e.g., in [31], [33], recent contributions have emphasized its role in the constrained optimal control of unknown stochastic linear systems [13], [14] and in the derivation of explicit formulas for data-driven control with optimal and robust performance [19]. Since then, many very recent works in data-driven control have appeared, among which [4], [35], [5], [32], [36], [3], [23], [6]. In the context of stabilization or H 2 , H ∞ control (instead of robust invariance), data-driven approaches with noisy data were considered in [19], [36], [4].…”

“…Although the quadratic Lyapunov functions considered therein for stabilization may be relied upon to build robustly invariant sublevel sets, we consider here the different setting of polyhedral constraints. The consideration of state constraints relates this work to data-driven predictive control such as [34], [13], [3]. Unlike these works, our focus is to provide, based on noisy data, necessary and sufficient conditions for when a given polyhedral set can be made robustly invariant.…”

Constrained nonlinear output regulation using model predictive control -- extended version

“…However, offline verification of the corresponding detectability and stabilizability conditions (Ass. [3][4] with such an implicit data based model is still an open research topic, compare, e.g., [40] where controllability/observability properties are verified using data.…”

On the design of terminal ingredients for data-driven MPC

“…While Coulson et al (2020) derive open-loop robustness guarantees of the scheme, Berberich et al (2021) prove desirable closed-loop stability and robustness properties, even if the measured data are affected by noise. Further theoretical results on robust constraint satisfaction based on noisy data and on a tracking MPC formulation are derived by Berberich et al (2020b) and Berberich et al (2020a), respectively. Notably, the existing works with closed-loop guarantees by Berberich et al (2021Berberich et al ( , 2020a require terminal equality constraints, which may result in a small region of attraction and a small robustness margin.…”