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

Abstract

We provide a comprehensive review and practical implementation of a recently developed model predictive control (MPC) framework for controlling unknown systems using only measured data and no explicit model knowledge. Our approach relies on an implicit system parametrization from behavioral systems theory based on one measured input-output trajectory. The presented MPC schemes guarantee closed-loop stability for unknown linear time-invariant (LTI) systems, even if the data are affected by noise. Further, we extend this MPC framework to control unknown nonlinear systems by continuously updating the data-driven system representation using new measurements. The simple and intuitive applicability of our approach is demonstrated with a nonlinear four-tank system in simulation and in an experiment.

Linear tracking MPC for nonlinear systems Part II: The data-driven case

“…The result shows that the nominal MPC scheme based on repeatedly solving Problem ( 16) with σ = 0 in an nstep fashion (compare Algorithm 1) practically exponentially stabilizes the optimal reachable equilibrium ξ sr in closed loop in the presence of input disturbances. To be precise, the decay bound (39) in combination with the lower and upper bounds in (38) implies that the closed loop converges to a neighborhood of ξ sr , the size of which shrinks if the disturbance bound (denoted by ε with a slight abuse of notation) is small. The guaranteed region of attraction is then given by V (ξ 0 ) ≤ V ROA and, in particular, for a larger size V ROA the maximal disturbance bound ε ensuring the closed-loop properties decreases.…”

“…For simplicity, we assume w.l.o.g. that N n ∈ I ≥0 , i.e., N is divisible by n. Note that ξN lies in the set {ξ | V (ξ , D N ) ≤ V ROA }, which is compact due to the lower bound (38) and Assumption 5, i.e., compactness of the (linearized) steadystate manifold [20, Assumption 5]. Similar to the proof of [20, Proposition 2], Lipschitz continuity of the dynamics (41) and compactness of U imply that the union of the N -step reachable sets of the linearized and the nonlinear dynamics (compare [51]) starting in {ξ | V (ξ , D N ) ≤ V ROA }, which we denote by X, is compact.…”

“…for some β V,3 ∈ K ∞ and any t = N + ni, i ∈ I ≥0 . Exploiting the lower and upper bounds in (38), we obtain…”

“…The proof is divided into four parts. We first show the lower and upper bounds (38) In combination with (1), this implies…”

“…This is possible since we implicitly encourage the closed-loop trajectory to remain in vicinity of the steady-state manifold, where our data-driven prediction model is a good approximation of the underlying nonlinear dynamics. Initial ideas in this direction have been discussed in [38], however, without any theoretical analysis. Finally, the presented results are also related to offset-free MPC [39], which deals with setpoint tracking based on an uncertain model, whereas our approach achieves asymptotic convergence to the setpoint using only input-output data.…”