Sampled-data Systems: Stability, Contractivity and Single-iteration Suboptimal MPC

This paper analyzes the stability of interconnected continuous-time (CT) and discrete-time (DT) systems coupled through sampling and zero-order hold mechanisms. The DT system updates its output at regular intervals $T>0$ by applying an $n$-fold composition of a given map. This setup is motivated by online and sampled-data implementations of optimization-based controllers - particularly model predictive control (MPC) - where the DT system models $n$ iterations of an algorithm approximating the solution of an optimization problem.
We introduce the concept of a reduced model, defined as the limiting behavior of the sampled-data system as $T \to 0^+$ and $n \to +\infty$. Our main theoretical contribution establishes that when the reduced model is contractive, there exists a threshold duration $T(n)$ for each iteration count $n$ such that the CT-DT interconnection achieves exponential stability for all sampling periods $T < T(n)$. Finally, under the stronger condition that both the CT and DT systems are contractive, we show exponential stability of their interconnection using a small-gain argument. Our theoretical results provide new insights into suboptimal MPC stability, showing that convergence guarantees hold even when using a single iteration of the optimization algorithm - a practically significant finding for real-time control applications.