Robust stability analysis of a simple data-driven model predictive control approach

Abstract

In this paper, we provide a theoretical analysis of closed-loop properties of a simple data-driven model predictive control (MPC) scheme. The formulation does not involve any terminal ingredients, thus allowing for a simple implementation without (potential) feasibility issues. The proposed approach relies on an implicit description of linear time-invariant systems based on behavioral systems theory, which only requires one input-output trajectory of an unknown system. For the nominal case with noise-free data, we prove that the data-driven MPC scheme ensures exponential stability for the closed loop if the prediction horizon is sufficiently long. Moreover, we analyze the robust data-driven MPC scheme for noisy output measurements for which we prove closed-loop practical exponential stability. The advantages of the presented approach are illustrated with a numerical example.

Data-Driven Resilient Predictive Control under Denial-of-Service

“…Moreover, the data-driven MPC problem in [32] is non-convex, which is NP-hard in general and computationally challenging. Inspired by [45], the cost function J * L in the present work is modified by adding noise bound as the coefficients of h(t) 2 and g(t) 2 . In this manner, the non-convex constraint h i (t) ∞ ≤ n(1 + g(t) 1 ) (where • 1 , and • ∞ are the 1 -, and ∞ -norm of vectors, respectively) employed in [32] to guarantee closed-loop stability for their data-driven MPC, is not needed.…”

“…Since J * L (z sr ) > 0, it follows from (44) that V sr ≥ γW (z sr ) ≥ γλ P z sr 2 . It remains to establish the upper bound in (45). To this end, let us start by constructing a candidate solution (ḡ(s r ), h(s r ), ū(s r ), ȳ(s r )) of Problem (20), in which, the input sequence is expected to bring the state x (and its corresponding output y) of system (3) to a ball around the origin whose size depends on the magnitude of noise v in L steps.…”

“…In Theorem 20) can be shown similarly to that of Theorem 1, and the closed-loop system achieves global ISS provided that condition (1/ν f ) + (1/ν d ) < 1 − ((n x − 1)/ν f ) on DoS attacks is met. In addition, a data-driven MPC scheme without the terminal constraint (20c) was recently proposed in [45], under which global ISS can be ensured if the prediction horizon (i.e., L) is sufficiently large. Condition on L implies that this scheme also requires sufficiently many pre-collected trajectories (i.e., large N ).…”

“…In this manner, system (3) can achieve maximum resilience against DoS attacks at the price of increased computational complexity. Stability analysis for these two schemes can be performed by leveraging the results in [32] and [45]. Remark 9 (DoS attacks also in controller-to-plant channel).…”

Data-driven model predictive control: closed-loop guarantees and experimental results

“…In this paper, we provide an overview of recent advances in data-driven MPC based on [22]. We focus on MPC schemes with guaranteed closed-loop stability and robustness properties in case of LTI systems [7,5,6,8,9]. Additionally, we demonstrate how such MPC schemes can be modified to control unknown nonlinear systems using only measured data.…”