⚡ electrical engineering

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

This paper establishes the exponential stability of interconnected continuous-time and discrete-time systems, particularly in the context of single-iteration suboptimal Model Predictive Control, by proving that stability is guaranteed when the underlying reduced model is contractive and the sampling period is sufficiently small.

Original authors: Yiting Chen, Francesco Bullo, Emiliano Dall'Anese

Published 2026-07-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yiting Chen, Francesco Bullo, Emiliano Dall'Anese

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to steer a very fast, complex car (the Continuous-Time System) using a GPS navigation app (the Discrete-Time System).

In a perfect world, the GPS would calculate the perfect steering angle instantly and update it continuously as you drive. But in the real world, computers aren't magic. The GPS has to:

Sample your current location (take a snapshot).
Compute the best route. This isn't done instantly; it runs a loop of calculations (iterations) to get closer to the perfect answer.
Hold that answer steady until the next snapshot is taken.

This paper is about making sure the car doesn't crash when the GPS is slightly delayed and only runs a few calculation loops before giving you a "good enough" answer.

Here is the breakdown of their findings using simple analogies:

1. The Two Systems Working Together

Think of the car as a Continuous System (it moves smoothly all the time) and the GPS as a Discrete System (it only speaks up at specific moments, like every 0.1 seconds).

The Problem: If the GPS waits too long between updates (the "sampling period" is too big), or if it stops calculating too early (too few "iterations"), the car might start swerving wildly and crash.
The Goal: The authors wanted to find the "safe zone." How fast must the GPS update? How many calculation loops must it run to keep the car stable?

2. The "Reduced Model" (The Ideal GPS)

The authors invented a concept called the Reduced Model. Imagine a magical version of your GPS that:

Updates infinitely fast (zero delay).
Calculates the perfect route instantly (infinite calculation loops).

If this magical, perfect GPS would keep the car stable, the authors asked: "Can we still keep the car stable with our real, slower, 'good enough' GPS?"

3. The Big Discovery: The "One-Step" Miracle

This is the paper's most surprising finding. Usually, engineers think you need to run a complex algorithm many times to get a stable result. You might think, "I need to let the GPS calculate for 10 loops before I trust it."

The paper proves you don't.

They showed that as long as the "Perfect GPS" (the Reduced Model) is stable, you can use a real GPS that only runs just ONE calculation loop before giving you an answer, and the car will still be safe—provided you update the GPS frequently enough.

Analogy: Imagine trying to balance a broom on your hand. Usually, you think you need to make tiny, perfect adjustments constantly. This paper says: "Actually, you can make a big, slightly imperfect adjustment, as long as you do it very, very quickly."

4. The Two Rules for Stability

The paper gives two main ways to guarantee the car won't crash:

Rule A (The Strict Coach): If the car itself is naturally very stable (like a self-balancing scooter) AND the GPS is good at refining its answers, then the system is safe no matter how slow the GPS is or how few loops it runs. This is a very strong condition, but it's hard to guarantee in real life.
Rule B (The Fast Runner - The Main Result): If the "Perfect GPS" (the Reduced Model) is stable, then the real system is safe IF the time between updates is short enough.
- The paper provides a specific formula for this "short enough" time.
- Key Insight: The more calculation loops you run (the more you let the GPS think), the slower you can update it. But if you only run one loop, you just need to update it very fast.

5. Why This Matters for "Model Predictive Control" (MPC)

The paper specifically mentions Model Predictive Control (MPC). Think of MPC as a super-smart driver that looks 5 seconds into the future to decide what to do right now.

The Old Problem: Solving the "look 5 seconds ahead" math is hard. Doing it perfectly takes too long for a real car. So, engineers often had to compromise: either run the math slowly (which is dangerous) or stop the math early (which might be unstable).
The New Insight: This paper proves that you can stop the math immediately after the first step (a "single iteration") and still be safe, as long as you refresh that decision very frequently. This is a huge deal for real-time control because it means you can use much simpler, faster computers to control complex systems without them crashing.

Summary

The paper says: You don't need a perfect, slow computer to control a fast system. You just need a "good enough" computer that updates its "good enough" answers very, very quickly. Even if the computer only does one tiny calculation before speaking, the system remains stable as long as the "perfect version" of that system would have been stable.

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

Problem Statement

This paper addresses the stability analysis of interconnected continuous-time (CT) and discrete-time (DT) systems coupled via sampling and zero-order hold (ZOH) mechanisms. The specific architecture involves a CT plant $\dot{x}(t) = f(x(t), z(t))$ and a DT subsystem that updates its output $z$ at regular intervals $T$ . Crucially, the DT subsystem does not compute the optimal control law $z^*(x)$ in closed form; instead, it performs $n$ iterations of an algorithm defined by a map $G(x, z)$ to approximate the solution.

This setup models practical implementations of optimization-based controllers, such as Model Predictive Control (MPC), where computational constraints prevent solving the optimization problem to full convergence ( $n \to \infty$ ) and where control inputs are updated at discrete sampling times ( $T > 0$ ). The core challenge is to determine conditions under which this sampled-data interconnection remains exponentially stable, particularly when the number of iterations $n$ is small (e.g., $n=1$ ) and the sampling period $T$ is finite.

Methodology

The authors employ contraction theory and Lipschitz analysis to study the stability of the interconnected system. The methodology proceeds through the following steps:

Reduced Model (RM) Definition: The paper formalizes the concept of a "reduced model," defined as the limiting behavior of the system as the sampling period $T \to 0^+$ and the number of iterations $n \to +\infty$ . In this limit, the DT subsystem instantly computes the fixed point $z^*(x)$ such that $z^*(x) = G(x, z^*(x))$ , resulting in the closed-loop CT system $\dot{x} = f(x, z^*(x))$ .
Time-Scale Separation: The parameters $T$ (sampling interval) and $n$ (iteration count) are treated as inducing a time-scale separation between the "slow" plant dynamics and the "fast" computational dynamics of the controller.
Stability Analysis Framework:
- Small-Gain Approach: The authors first derive stability conditions based on a small-gain argument, requiring both the CT plant and the DT map to be contractive. This leads to a condition independent of $T$ and $n$ .
- Reduced Model Approach: The primary contribution involves analyzing the case where only the Reduced Model is contractive. The authors establish that if the RM is contractive, there exists a threshold sampling period $T(n)$ dependent on the iteration count $n$ and the Lipschitz constants of the system maps.
Mathematical Tools: The analysis utilizes:
- One-sided Lipschitz constants (log-norms) to characterize contraction rates.
- Composite norms to bound the distance between trajectories of the interconnected system.
- Matrix spectral radius analysis to derive explicit bounds for $T(n)$ ensuring the discrete-time update map is Schur stable.

Key Contributions and Results

1. Formalization of the Reduced Model

The paper defines the Reduced Model (RM) as the system $\dot{x} = f(x, z^*(x))$ , where $z^*(x)$ is the unique fixed point of the map $G(x, \cdot)$ . The authors demonstrate that while the RM may be contractive, the actual sampled-data system can become unstable if the sampling period $T$ is too large, even if $n$ is large. This highlights that stability of the RM does not automatically guarantee stability of the sampled implementation without constraints on $T$ .

2. Stability via Small-Gain Conditions (Theorem 3.4)

Under the strong assumption that both the CT plant dynamics ( $x \mapsto f(x, z)$ ) and the DT map ( $z \mapsto G(x, z)$ ) are contractive, the authors prove that the interconnected system is globally exponentially stable (GES) and discrete-time contractive for any $n \in \mathbb{Z}_{>0}$ and any $T > 0$ , provided a specific small-gain condition on the interconnection gains is met. This condition is stricter than necessary for the main result but offers robustness independent of sampling rates.

3. Stability via Reduced Model Contractivity (Theorem 3.6)

The paper's main theoretical contribution is establishing that if the Reduced Model is contractive (i.e., $z^*(x)$ stabilizes the plant) and the DT map is contractive, then the interconnected system is GES provided the sampling period $T$ is sufficiently small.

Threshold $T(n)$ : The authors derive an explicit, albeit conservative, expression for a threshold duration $T(n)$ .
Dependence on $n$ : The threshold $T(n)$ is strictly increasing with respect to $n$ . This implies that performing more iterations allows for a larger sampling period while maintaining stability.
Single-Iteration Guarantee: Crucially, for any $n \ge 1$ , there exists a $T(n) > 0$ such that stability is guaranteed. This means stability can be achieved even with a single iteration ( $n=1$ ) of the optimization algorithm, provided the sampling is fast enough.

4. Application to Model Predictive Control (MPC)

The theoretical results are applied to suboptimal MPC.

Contrast with Existing Literature: Existing stability analyses for online or time-distributed MPC often require $n > 1$ or assume the optimization converges. This paper shows that for linear time-invariant (LTI) systems, a single iteration ( $n=1$ ) is sufficient to ensure stability if $T$ is small enough.
Numerical Validation: The authors present numerical experiments on a double integrator.
- In an unconstrained case, the system remains stable with $n=1$ and small $T$ .
- In a constrained case (using soft constraints), the Reduced Model is only locally contractive. The authors verify that the system remains stable within a specific region of attraction when $n=1$ and $T$ is sufficiently small, validating the local stability corollary.

Significance and Claims

The paper claims to provide new insights into the stability of suboptimal MPC and optimization-based control in real-time settings. Its primary significance lies in:

Rigorous Single-Iteration Stability: To the best of the authors' knowledge, this is the first work to rigorously establish that a single iteration ( $n=1$ ) of an optimization solver is sufficient to guarantee the exponential stability of the closed-loop system, provided the sampling period is sufficiently small. This challenges the conventional requirement for multiple iterations or full convergence in real-time MPC.
Quantitative Sampling Bounds: The paper provides explicit formulas for the maximum allowable sampling period $T(n)$ as a function of the number of iterations and system Lipschitz constants. This offers a concrete design guideline for engineers implementing sampled-data optimization controllers.
Extension of Contraction Theory: The work extends classical small-gain contraction theorems (typically applied to CT-CT interconnections) to CT-DT architectures, bridging the gap between continuous-time control theory and discrete-time computational constraints.

The authors maintain a modest tone, noting that the derived bounds for $T(n)$ may be conservative and that future work will explore local contractivity for systems with hard constraints and the design of observers for such interconnected systems.