⚡ electrical engineering

System Level Synthesis for Affine Control Policies: Model Based and Data-Driven Settings

This paper introduces a novel closed-loop parameterization that extends System Level Synthesis (SLS) to time-varying affine control policies in both model-based and data-driven settings, enabling equivalent formulations of general Model Predictive Control (MPC) problems that achieve performance comparable to traditional approaches.

Original authors: Lukas Schüepp, Giulia De Pasquale, Florian Dörfler, Carmen Amo Alonso

Published 2026-01-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Lukas Schüepp, Giulia De Pasquale, Florian Dörfler, Carmen Amo Alonso

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

The Big Picture: Teaching a Robot to Drive Without a Map

Imagine you are trying to teach a robot to drive a car from point A to point B. The car has a weird quirk: it always drifts slightly to the right because of a heavy weight in the trunk (this is the "affine" part of the system).

For a long time, engineers had two main ways to solve this:

The Map Method (Model-Based): You build a perfect mathematical map of the car's physics. You calculate exactly how the steering wheel needs to turn to counteract that heavy weight.
The Memory Method (Data-Driven): You don't know the physics, but you have a video recording of the car driving perfectly in the past. You tell the robot, "Just do what worked in the video."

The Problem:
Most existing "smart" control methods (like the one called System Level Synthesis or SLS) are great at handling big, complex systems, but they are very rigid. They only work if the car behaves in a perfectly straight line (linear). If the car has that heavy weight causing a constant drift, or if the road conditions change, these old methods struggle. They try to force a straight-line solution onto a curved problem, which isn't efficient.

The New Idea: The "Drift-Aware" GPS

This paper introduces a new way to control systems called Affine System Level Synthesis.

Think of the old method as a GPS that only knows how to drive on a perfectly flat, straight highway. It gets confused if the road tilts or if the car has a heavy load.

The authors' new method is like a GPS that understands "drift." It realizes that sometimes you need to steer slightly left just to stay straight because of that heavy weight in the trunk. It allows the control policy to have two parts:

The Reaction: "If the car drifts left, steer right." (This is the linear part).
The Constant Push: "Also, always keep the steering wheel turned slightly right to fight that heavy weight." (This is the new "affine" part).

How They Did It (The Two Settings)

The paper shows this new method works in two different scenarios:

1. The "I Know the Rules" Scenario (Model-Based)
Imagine you have the car's manual and know exactly how heavy the trunk is.

What they did: They rewrote the math so the computer can plan a route that accounts for that constant weight while reacting to traffic.
The Result: They proved that this new way of planning is mathematically identical to the best possible traditional planning. It's just a different, more flexible way of writing the same instructions.

2. The "I Only Have a Video" Scenario (Data-Driven)
Imagine you don't have the car's manual. You don't know the weight of the trunk or the engine power. You only have a video of the car driving successfully in the past.

What they did: They created a trick using a "Hankel Matrix" (think of this as a giant, organized scrapbook of every move the car ever made). They showed that you can look at this scrapbook and figure out exactly how to drive the car in the future, including how to handle that constant weight, without ever knowing the weight's actual value.
The Result: The robot can learn to drive perfectly just by looking at past examples, even if the system has that constant "drift."

The "Aha!" Moment: Why This Matters

The authors ran a simulation using a model of a power grid generator (which naturally has friction, acting like that heavy weight in the trunk).

They compared three drivers:

The Traditional Driver: Uses the standard, complex math to plan the route.
The New "Drift-Aware" Driver (Model-Based): Uses their new method with the known physics.
The New "Drift-Aware" Driver (Data-Driven): Uses their new method with only past video data.

The Outcome: All three drivers arrived at the destination at the exact same time, took the exact same path, and used the exact same amount of fuel.

The Takeaway

This paper doesn't invent a new type of car or a new engine. Instead, it invents a new way of writing the instruction manual for the driver.

Old Manual: "Turn the wheel based on where you are." (Fails if there is a constant drift).
New Manual: "Turn the wheel based on where you are, plus a constant adjustment for the drift."

The authors proved that this new manual works just as well as the old, complex one, but it is much more flexible. It works whether you know the car's physics perfectly or if you are learning entirely from past video footage. This opens the door for controlling complex, real-world systems (like power grids or social networks) that don't behave in perfectly straight lines.

Technical Summary: System Level Synthesis for Affine Control Policies

Problem Formulation
Traditional Model Predictive Control (MPC) for systems with complex, non-linear dynamics often relies on linear approximations or piecewise affine solutions that are difficult to characterize globally. While System Level Synthesis (SLS) has emerged as a powerful framework for designing controllers for large-scale linear systems by parameterizing closed-loop responses, existing SLS approaches are restricted to linear systems and time-varying linear control policies. This limitation prevents SLS from capturing the inherent piecewise affine nature of optimal MPC solutions, particularly for systems with constant offsets (affine systems) and constraints. Furthermore, while data-driven control methods have advanced, integrating them with closed-loop formulations for affine policies under constraints remains an open challenge.

Methodology
The paper introduces a novel closed-loop parameterization that extends the SLS framework to accommodate time-varying affine control policies. The core methodology involves two primary settings:

Model-Based Setting:
The authors define a discrete-time varying affine system $x(t+1) = A(t)x(t) + B(t)u(t) + s + w(t)$ , where $s$ is a constant term. They propose an affine control policy of the form $u_t = \sum_{\tau=0}^t K_{t,\tau}x_\tau + u^s_t$ . By utilizing block-diagonal operators and the block-downshift operator $Z$ , they derive a closed-loop parameterization where the state and input trajectories ( $x, u$ ) are expressed as linear maps of the disturbance $w$ and a constant term (represented by a scalar 1).
- Theorem 1 establishes that the set of all possible closed-loop maps under affine policies is characterized by an affine subspace constraint involving the system matrices and the new response parameters ( $\Phi_x, \phi_x, \Phi_u, \phi_u$ ).
- Corollary 2 demonstrates that the standard MPC optimization problem (minimizing a quadratic cost subject to affine constraints) can be equivalently reformulated as an optimization over these closed-loop maps. This allows the MPC problem to be solved directly in the space of system responses rather than control inputs.
Data-Driven Setting:
Extending the approach to scenarios where system matrices $A, B, s$ are unknown, the authors leverage Willems' Fundamental Lemma adapted for affine systems.
- Theorem 2 shows that for time-invariant affine systems, the closed-loop system responses can be equivalently characterized using Hankel matrices constructed from a single persistently exciting (PE) trajectory of past state and input data.
- Corollary 3 reformulates the MPC problem entirely in terms of these data-driven Hankel matrices and auxiliary variables ( $G, \hat{g}$ ), eliminating the need for explicit system identification.

Key Contributions

Generalization of SLS: The work provides the first extension of the SLS framework to time-varying affine policies, allowing the framework to capture the affine behaviors inherent in constrained MPC problems.
Closed-Loop MPC Reformulation: It establishes an equivalent formulation of general MPC problems (with quadratic costs, affine prediction models, and polytopic constraints) using closed-loop maps, both in model-based and data-driven settings.
Data-Driven Affine SLS: The paper introduces a data-driven parameterization for affine systems that relies solely on past trajectory data, removing the requirement for explicit knowledge of the constant term $s$ or system matrices.

Results
The authors validate their approach through numerical experiments on a linearized swing dynamics model of a power system generator, which includes constant Coulomb friction (an affine term). The simulation compares three controllers:

Traditional Model-Based MPC.
Model-Based Affine SLS.
Data-Driven Affine SLS.

The results demonstrate that both the model-based and data-driven affine SLS formulations achieve performance on par with traditional model-based MPC. Specifically, the state and input trajectories generated by all three methods are identical, confirming that the proposed parameterizations can exactly recover the optimal MPC solution without the computational burden of solving the original non-convex or high-dimensional input-space problem.

Significance and Claims
The paper claims to be the first work to extend System Level Synthesis to affine policies in both model-based and data-driven settings. Its primary significance lies in bridging the gap between the theoretical advantages of SLS (such as distributed control and robustness) and the practical necessity of handling affine dynamics and constraints found in real-world MPC applications. By enabling an equivalent formulation of MPC using closed-loop maps, the approach facilitates the design of controllers for systems where linear policies are insufficient, while maintaining the computational and structural benefits of the SLS framework. The authors note that while the current work focuses on nominal control, extensions to distributed cases with stability guarantees are left for future work.