On the design of terminal ingredients for data-driven MPC

Abstract

We present a model predictive control (MPC) scheme to control unknown linear time-invariant systems using only measured input-output data and no model knowledge. The scheme includes a terminal cost and a terminal set constraint on an extended state containing past input-output values. We provide an explicit design procedure for the corresponding terminal ingredients that only uses measured input-output data. Further, we prove that the MPC scheme based on these terminal ingredients exponentially stabilizes the desired setpoint in closed loop. Finally, we illustrate the advantages over existing methods with a numerical example.

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

“…The proof is similar to stability arguments in model-based MPC [21] with the additional difficulty that the cost of (3) depends on the output and is thus only positive semi-definite in the internal state. The assumption that the cost of ( 3) is quadratically upper bounded is not restrictive and it holds, e.g., for compact constraints if (𝑢 𝑠 , 𝑦 𝑠 ) ∈ int(U × Y) (see [7,8] for details).…”

“…Since Theorem 1 provides an equivalent parametrization of system trajectories, its applicability is not limited to MPC schemes with terminal equality constraints as above. In particular, it can be used to design more sophisticated MPC schemes with general terminal ingredients, i.e., a terminal cost and a terminal region constraint, see [8] for details. Similar to terminal ingredients in model-based MPC, this has the advantage of increasing the region of attraction and improving robustness in closed loop.…”

“…Due to this combination and the controllability argument used to prove stability in [7], the theoretical guarantees are only valid locally for a one-step MPC scheme [7,Remark 4]. When removing the terminal equality constraints (4d) as in [9] or replacing them by general terminal ingredients [8], then comparable closed-loop guarantees can also be given for a one-step scheme. Theorem 3]) Suppose 𝐿 ≥ 2𝑛, 𝑢 𝑠 = 0 ∈ int(U), and 𝑢 𝑑 is persistently exciting of order 𝐿 + 2𝑛.…”

“…Fig. 2 shows the closed-loop cost depending on the parameter 𝜆 𝛼 with all other parameters as in (8). Although the cost strongly depends on 𝜆 𝛼 , it can be seen that a wide range of values 𝜆 𝛼 ∈ [2 • 10 −5 , 0.01] leads to a good performance, i.e., 𝐽 ≤ 1.5 • 10 5 .…”

“…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.…”