System Level Synthesis (SLS)

• SLS is a framework for controller synthesis that parametrizes closed‐loop responses to disturbances, allowing convex optimization over achievable system behaviors under various constraints.
• It supports centralized and distributed architectures by incorporating locality, sparsity, actuator and state limits, which enables scalable and efficient controller design.
• SLS extends to nonlinear, data-driven, and continuous-time systems, providing robustness and safety guarantees while facilitating practical implementation in diverse applications.

System Level Synthesis (SLS) is a framework for controller synthesis that parametrizes the closed-loop system responses to disturbances rather than the controller itself. This approach enables direct convex optimization over achievable system behaviors under a wide range of convex constraints, including locality, sparsity, actuator and state limits, and robustness to modeling uncertainty. SLS supports both centralized and distributed architectures and is applicable to a variety of settings, including linear, nonlinear, and data-driven scenarios, as well as output-feedback and continuous-time systems.

1. Mathematical Foundations and Core Parameterizations

At the heart of SLS is the reparametrization of the controller design problem. For a discrete-time linear system

$x_{k+1} = A x_k + B u_k + w_k,$

the closed-loop responses from disturbance to state and input are given by transfer matrices

$$\Phi_x(z) = (z I - A - B K)^{-1},\quad \Phi_u(z) = K (z I - A - B K)^{-1},$$

which satisfy the affine achievability constraint

$$[zI-A\ -B] \begin{bmatrix} \Phi_x(z) \ \Phi_u(z) \end{bmatrix} = I,$$

with stability (strict properness, i.e., $\Phi_x,\,\Phi_u \in (1/z)\mathcal{RH}_\infty$). All internally stabilizing controllers are captured by feasible $(\Phi_x,\Phi_u)$, and a controller can be realized via $K = \Phi_u\Phi_x^{-1}$ (Anderson et al., 2019, Wang et al., 2016).

For finite-horizon or time-varying cases, the same principle holds: the system responses $(\Phi_x,\Phi_u)$ are block-lower-triangular operators which map disturbance sequences to state and input trajectories under a corresponding affine dynamical consistency constraint (Alonso et al., 2019).

The SLS paradigm extends to output-feedback, where the closed-loop behavior is parameterized by a set of four transfer matrices $(R, M, N, L)$, satisfying two affine constraints that generalize the Youla parameterization and recover all stabilizing controllers (Wang et al., 2016, Conger et al., 2021).

2. Convex Synthesis and System Level Constraints

Constrained and optimal control problems are naturally formulated as convex programs over the system response variables. General system level synthesis problems have the form: $$\begin{aligned} &\min_{\Phi_x,\,\Phi_u} \; J(\Phi_x, \Phi_u) \ &\text{subject to} \ \ [zI-A\ -B]\begin{bmatrix}\Phi_x \ \Phi_u\end{bmatrix} = I, \quad (\Phi_x,\Phi_u)\in \mathcal{C}, \end{aligned}$$ where $\mathcal{C}$ encodes convex system level constraints (SLCs): spatial/temporal locality, finite impulse response, actuator/sensor sparsity, polytopic state and input limits, positivity, performance constraints (H₂, H∞), and robustness to disturbances or model uncertainty. Convexity and Achievability are guaranteed by construction (Chen et al., 2019, Anderson et al., 2019, Wang et al., 2016).

The SLS structure admits row- and column-wise separable constraints, naturally enabling distributed and localized control by directly encoding information flow and locality in the optimization variables (Alonso et al., 2019, Alonso et al., 2020).

Key theoretical results show that the system-level subspaces with arbitrary convex constraints are strictly larger than those allowed by quadratic invariance, enabling synthesis of a broader class of constrained controllers (Wang et al., 2016).

3. Distributed and Scalable Controller Synthesis

Due to the coupling structure of the affine SLS constraints and the imposed locality/SLCs, both the formulation and solution of SLS problems can be distributed:

• Locality constraints restrict $$\Phi_x(z) = (z I - A - B K)^{-1},\quad \Phi_u(z) = K (z I - A - B K)^{-1},$$0 and $$\Phi_x(z) = (z I - A - B K)^{-1},\quad \Phi_u(z) = K (z I - A - B K)^{-1},$$1 to have nonzero blocks only within node neighborhoods or prescribed communication sets, ensuring that each subsystem’s behavior depends only on local disturbances and neighbors' information (Alonso et al., 2019, Li et al., 2020).
• Exploiting separability, primal-dual methods and ADMM-based decomposition yield parallelizable solutions with per-agent complexity depending only on local neighborhood size and FIR horizon, independent of the global system dimension (Chen et al., 2019, Alonso et al., 2019, Alonso et al., 2020).
• For SLS with box-type (polytopic) constraints, dual variables in the robust optimization dualization inherit the same sparsity, allowing efficient distributed primal-dual updates (Chen et al., 2019).
• Dynamic programming approaches, both for state-feedback (Tseng et al., 2020) and output-feedback (Conger et al., 2021), further accelerate solution of SLS programs, yielding order-of-magnitude speedups over generic convex solvers and scaling to high dimensions and long horizons.

4. Robustness, Uncertainty, and Safety Guarantees

SLS provides transparent robustness certificates via the affine sensitivity of closed-loop responses to model perturbation:

• Robust SLS designs guarantee bounded state and input trajectories under worst-case additive disturbances and structured model uncertainty using small-gain analysis (Anderson et al., 2019, Chen et al., 2019, Chen et al., 2021).
• For linear-fractional (LFT) model uncertainty, convex relaxations incorporating the SLS parametrization yield less conservative and more computationally tractable robust MPC than classical tube MPC (Chen et al., 2019).
• Chance-constrained and scenario-based SLS approaches address uncertainty due to multiplicative noise, with sample complexity guarantees and tractable convex formulations (Mazouchi et al., 2022).
• Distributionally robust SLS extends to output-feedback affine policies, utilizing Wasserstein-ambiguity sets and robust optimization to guarantee closed-loop constraint satisfaction against worst-case distributions and model mismatch (Li et al., 7 Aug 2025).
• SAFETY: SLS-based controllers have been empirically validated to produce 100% empirical safety (constraint satisfaction) across high-dimensional systems when robust reachability constraints are enforced (Fang et al., 8 Apr 2026, Srinivasan et al., 12 Feb 2026).

5. Extensions to Nonlinear, Data-Driven, and Continuous-Time Systems

SLS generalizes to several advanced domains:

• Nonlinear systems: Achievability and stabilization are characterized via operator equations over causal nonlinear maps; parameterization admits gradient-based learning over stable operator classes such as Recurrent Equilibrium Networks, ensuring robust stability by construction (Furieri et al., 2022, Conger et al., 2022).
• Data-driven SLS: For unknown plants, synthesis can be performed directly from informative trajectory data using behavioral theory and Hankel matrix constraints (Willems’ lemma), both in model-based and fully data-driven formulations, with explicit sub-optimality and sample complexity bounds (Xue et al., 2020, Schüepp et al., 2 Apr 2025).
• Affine control policies: Time-varying and constant-offset (affine) control structures can be exactly captured in SLS by augmenting system response maps; data-driven affine SLS achieves identical closed-loop trajectories as traditional MPC, with model-based equivalence and convexity preserved (Schüepp et al., 2 Apr 2025).
• Continuous-time SLS: Extension to continuous-time LTI plants uses Laplace-domain transfer matrices and partial fraction expansions, allowing convex $$\Phi_x(z) = (z I - A - B K)^{-1},\quad \Phi_u(z) = K (z I - A - B K)^{-1},$$2 or $$\Phi_x(z) = (z I - A - B K)^{-1},\quad \Phi_u(z) = K (z I - A - B K)^{-1},$$3 synthesis with structural constraints, again supporting scalable distributed design (Du et al., 2024).

6. Saturation, Actuator Limits, and Internal Model Control

SLS incorporates input saturation and actuator constraints through robust optimization and compensation structures:

• State/input constraints are imposed via robustification with primal-dual algorithms and the dual variables inherit primal sparsity, affording decomposition (Chen et al., 2019).
• Input saturation can be analyzed in the Internal Model Control (IMC) framework via small-gain conditions on the controller system response. If the IMC loop gain is less than one, global closed-loop stability is preserved (Chen et al., 2019).
• A saturation compensation scheme augments the SLS controller to account for cut-off disturbances due to saturation, leading to provably faster recovery and uniformly better state/input decay compared to naive truncation-based SLS (Chen et al., 2019).

7. Applications and Impact Across Disciplines

SLS has demonstrated effectiveness across a wide array of domains:

• Power systems: Layered SLS-based MPC achieves within 3% of centralized MPC performance at 20$$\Phi_x(z) = (z I - A - B K)^{-1},\quad \Phi_u(z) = K (z I - A - B K)^{-1},$$4 lower online computation and scales efficiently with problem size, even under tight locality and actuator constraints (Li et al., 2020).
• Biological circuits: SLS has been used to explain the structure and prevalence of internal feedback pathways in neural systems from first principles of distributed optimal control. The SLS "mesocircuit" paradigm unifies massive internal feedback, delay embedding, and local reflexes in brain-inspired architectures (Li, 2021).
• Large-scale robotics and cyber-physical systems: GPU-accelerated SLS pipelines for nonlinear, robust NMPC achieve millisecond-scale, safe receding-horizon control for humanoids and quadrupeds with $$\Phi_x(z) = (z I - A - B K)^{-1},\quad \Phi_u(z) = K (z I - A - B K)^{-1},$$5 variables, with empirical zero constraint violations (Fang et al., 8 Apr 2026).
• Learning-based and safety-critical control: SLS combined with conformal prediction enables robust, out-of-distribution MPC with theoretical finite-sample safety guarantees under distributional model shift (Srinivasan et al., 12 Feb 2026).

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