System Level Synthesis (SLS) Parameterization

• SLS parameterization is a unified framework that maps disturbances to closed-loop responses through direct affine mappings.
• It enables convex optimization of controller design while enforcing performance, sparsity, delay, and robustness constraints.
• The approach is practical for distributed, large-scale systems, offering scalable, finite-dimensional solutions for robust feedback control.

System Level Synthesis (SLS) parameterization is a unified framework for characterizing, synthesizing, and constraining the closed-loop behaviors of feedback-controlled dynamical systems. SLS parameterization generalizes classical methods such as state-space and Youla, and enables direct convex optimization of closed-loop system responses—including locality, delay, sparsity, and robustness constraints—thus facilitating controller design for large-scale, distributed, and structurally constrained systems (Wang et al., 2016). By focusing on achievable system responses rather than controller laws, SLS enables the simultaneous enforcement of complex structural constraints and performance metrics in a computationally tractable manner.

1. Core Principles of SLS Parameterization

The foundation of SLS parameterization is the direct affine mapping of disturbances to state and control signals in the closed-loop system. For a linear time-invariant (LTI) plant in state-space form,

$x[t+1] = A x[t] + B_2 u[t] + w[t]$

with control law $u = Kx$, SLS defines transfer matrices $R$ (state response) and $M$ (control response) such that

$x = R w \quad\text{and}\quad u = M w$

with $R, M \in (1/z)RH_\infty$ (strictly proper and stable). The central SLS constraint is an affine equation coupling $R$ and $M$: $$\begin{bmatrix} zI - A & -B_2 \end{bmatrix} \begin{bmatrix} R \ M \end{bmatrix} = I$$ This affine structure generalizes for output feedback (involving four system responses $R, M, N, L$ between state/measurement disturbances and state/control signals), yielding

$$\begin{aligned} & \begin{bmatrix} zI - A & -B_2 \end{bmatrix} \begin{bmatrix} R & N \ M & L \end{bmatrix} = [I \ 0] \ & \begin{bmatrix} R & N \ M & L \end{bmatrix} \begin{bmatrix} zI-A \ -C_2 \end{bmatrix} = \begin{bmatrix} I \ 0 \end{bmatrix} \end{aligned}$$

Controllers achieving these system responses are recovered via $K = M R^{-1}$ (state feedback) or $K = L - M R^{-1}N$ (output feedback).

This parameterization generalizes prior approaches by internally characterizing all stabilizing controllers via the achievable closed-loop maps (Wang et al., 2016, Zheng et al., 2019). Unlike Youla, SLS parameterization does not require doubly-coprime factorizations and provides direct access to closed-loop responses, which is essential for enforcing structural and performance constraints convexly in large-scale settings.

2. System Level Constraints (SLCs)

After parameterizing achievable closed-loop maps, constraint sets—termed System Level Constraints—are imposed directly on the system responses rather than on the controller parameters. Classes of SLCs include:

• Sparsity and Locality: Constraints restricting the support of $R, M, N, L$, often reflecting physical or communication neighborhood relations (e.g., requiring $R_{ij}=0$ if nodes $i$ and $j$ are not neighbors).
• Delay and FIR Constraints: Imposing a finite impulse response horizon ($R, M, N, L \in \mathcal{F}_T$), transforming infinite-dimensional constraints to finite convex programs.
• Performance and Robustness: Constraints on norm-based performance (e.g., $H_2$, $H_\infty$), expressed as $g(R, M, N, L) \leq \gamma$.
• Arbitrary Convex Structural Properties: Unlike classical Youla approaches, SLS enables convex imposition of constraints not requiring quadratic invariance (QI), thus vastly expanding the class of tractable constrained controller synthesis problems (Wang et al., 2016, Zheng et al., 2019).

$$\text{Example SLC:}\quad \begin{bmatrix} R & N \ M & L \end{bmatrix} \in \mathcal{L}$$

for a linear subspace $\mathcal{L}$ encoding locality structure.

3. Advantages and Generalizations

The SLS framework yields several fundamental advantages and generalizations beyond prior parameterizations:

• Convexifiability Beyond QI: While Youla parameterization requires QI for convex structural constraints, SLS admits convex synthesis problems under much broader structural constraint classes.
• Transparent Tradeoff Analysis: By parameterizing the closed-loop response directly, one can explicitly trade off closed-loop performance, robustness, and implementation complexity. FIR and sparsity constraints explicitly yield localized, distributed, and low-complexity designs.
• Expansion to Robust, Data-Driven, and Nonlinear Settings: SLS generalizes to operator (infinite dimensional) systems for robust/distributed control (Matni et al., 2019), supports data-driven synthesis via Hankel data matrices (Xue et al., 2020), and extends even to nonlinear systems with appropriate base control/policy structure (Conger et al., 2022, Furieri et al., 2022).
• Affine/Convex Equivalence among Perspectives: Explicit affine mappings translate between SLP, Youla, and input–output parameterizations, allowing any convex controller synthesis problem to be equivalently formulated in any of these domains (Zheng et al., 2019).

4. Computational Scalability and Distributed Control

SLS parameterization leads to scalable synthesis methods for large-scale systems:

• Finite-Dimensionality via FIR Approximation: By imposing FIR constraints, infinite-dimensional problems are reduced to finite convex programs.
• Decomposability and Parallelization: Structural SLCs (e.g., locality) induce sparsity patterns that decouple optimization variables, enabling distributed and localized computation and implementation (Alonso et al., 2019).
• Efficient Algorithms: SLS permits efficient solution via primal–dual (Chen et al., 2019) or dynamic programming and vectorization (Tseng et al., 2020, Conger et al., 2021) approaches, sharply reducing computation time relative to CVX or Lagrange methods.

Implementation Strategy	Key Feature	Scalability/Notes
FIR SLS + ADMM	Localized, distributed	Per-subsystem complexity ∼ $O(d^2 T)$, global N independent (Alonso et al., 2019)
DP/Vectorized SLS (e.g. output)	Non-separable/multi-sided constraints	Up to 7× faster than CVX, scalable to large FIR (Conger et al., 2021)

5. Robustness, Feasibility, and Nonlinear/Adaptive Extensions

SLS provides a foundation for robust controller synthesis:

• Direct Robust Constraints: Additive and parametric uncertainties are embedded as affine/convex perturbations to the SLS achievability constraints. Robust feasibility is guaranteed through explicit bounds on performance as a function of model mismatch and disturbance structure (Matni et al., 2019, Chen et al., 2019, Chen et al., 2021).
• Distributionally Robust/Finite-Sample Guarantees: Data-driven SLS approaches (with Wasserstein ambiguity sets) provide finite-sample, distributionally robust synthesis for unknown or stochastic disturbance distributions (Micheli et al., 28 May 2024, Li et al., 7 Aug 2025).
• Nonlinear and Learning-Based Extensions: By leveraging stabilizing base controllers and representing stable “correction” policies (e.g., via REN-based DNNs), SLS can be used for learning all stabilizing policies for nonlinear settings, rigorously preserving closed-loop stability (Furieri et al., 2022). Taylor–series/PID–like polynomial approximations further support robust SLS controller design for nonlinear systems without requiring explicit Lyapunov function construction (Conger et al., 2022).

6. Applications and Illustrative Examples

SLS parameterization is applicable to a wide array of real and theoretical contexts:

• Distributed Large-Scale Control: Power grids, chain-structured systems, and networks with explicit communication/sparsity constraints benefit from SLS-enabled localized control architectures (Wang et al., 2016, Alonso et al., 2019, Du et al., 10 Oct 2024).
• Biological Neural Control: SLS models can accurately reflect temporal delays, locality, and abundance of internal feedback observed in neurobiological systems (Li, 2021).
• Constrained MPC and Tube-Based Robust Control: SLS supports convex formulations for robust/constrained MPC, outperforming traditional tube–MPC in computational feasibility and reduced conservatism (Chen et al., 2019, Chen et al., 2021).
• Identification and Adaptive Control: Dual SLS parameterization allows direct identification of plant models from closed-loop data without requiring inverting or factorizing plant models (Srivastava et al., 2023).

7. Theoretical and Practical Impact

The SLS framework moves controller synthesis from an actuator-centric to a system-centric viewpoint—designing the “movie” of closed-loop behavior rather than the “actor” of the controller. This paradigm enables transparent, convex, and scalable synthesis of high-performance, robust, and structured controllers, bridging fundamental theoretical advances in parameterization with pressing practical needs in distributed, data-driven, and uncertain environments. The generality of the SLS approach—its convexity, equivalence with Youla/input–output methods, and capacity to accommodate complex structural and performance constraints—marks it as a foundational tool for modern control synthesis (Wang et al., 2016, Zheng et al., 2019, Chen et al., 2019, Chen et al., 2019).