Flexible-step MPC for Unknown Linear Time-Invariant Systems
Markus Pietschner, Christian Ebenbauer, Bahman Gharesifard, Raik Suttner
TL;DR
The paper tackles stabilization of unknown discrete-time LTI systems using MPC without prior data. It introduces a flexible-step MPC framework that applies a variable number of optimized inputs per iteration and enforces stability with a generalized discrete-time control Lyapunov function (GDCLF) inside the OCP, effectively decoupling optimization from stabilization. For unknown systems, online data-driven estimates of $(A,B)$ are maintained and updated through a data list and a PE-based exploration mechanism, with convergence guarantees showing that, under feasibility, the state converges to the origin. Simulations on a hard-to-learn system demonstrate fast exponential convergence and robustness to measurement noise, achieving stabilization with minimal exploratory disturbance. The approach offers a practical, data-driven path to stabilize unknown LTI systems and suggests avenues for extensions to nonlinear or structured-model settings.
Abstract
We propose a novel flexible-step model predictive control algorithm for unknown linear time-invariant discrete-time systems. The goal is to asymptotically stabilize the system without relying on a pre-collected dataset that describes its behavior in advance. In particular, we aim to avoid a potentially harmful initial open-loop exploration phase for identification, since full identification is often not necessary for stabilization. Instead, the proposed control scheme explores and learns the unknown system online through measurements of inputs and states. The measurement results are used to update the prediction model in the finite-horizon optimal control problem. If the current prediction model yields an infeasible optimal control problem, then persistently exciting inputs are applied until feasibility is reestablished. The proposed flexible-step approach allows for a flexible number of implemented optimal input values in each iteration, which is beneficial for simultaneous exploration and exploitation. A generalized control Lyapunov function is included into the constraints of the optimal control problem to enforce stability. This way, the problem of optimization is decoupled from the problem of stabilization. For an asymptotically stabilizable unknown control system, we prove that the proposed flexible-step algorithm can lead to global convergence of the system state to the origin.