Coarse-ID Control: Robust Feedback with Limited Data

• Coarse-ID Control is a framework that uses low-resolution or quantized system information for robust controller design, ensuring finite-time convergence under minimal information.
• It encompasses methods from sign-based formation control to data-driven equation-free approaches, with rigorous performance guarantees and explicit sample complexity bounds.
• The approach demonstrates practical effectiveness in multi-agent and high-dimensional systems by handling estimation errors and model uncertainties in a computationally efficient manner.

Coarse-ID control encompasses a class of control methodologies that leverage coarse, quantized, or data-driven system identification to enable robust feedback control when fine-grained models are unavailable, expensive to compute, or inapplicable. The approach is characterized by leveraging minimal or low-resolution information for both model estimation and controller design, often aiming for non-asymptotic, end-to-end performance guarantees. Methodologies span quantized guidance for multi-agent formations, finite-data control synthesis via system-level synthesis (SLS), and recent equation-free feedback design based on neural operator surrogates for high-dimensional distributed parameter systems.

1. Control with Coarse-Quantized Information

Early instances of Coarse-ID control arise in the context of formation control and multi-agent systems under stringent information constraints. In the one-dimensional rigid formation setting, agents with scalar positions $x_i\in\mathbb{R}$ for $i=1,\dots,n$ must achieve prescribed inter-agent distances $|x_i - x_{i+1}| = d_i$, fixing shape up to translation. The guidance system supplies each agent with at most four bits: for each active neighbor, the sign of the relative position $\mathrm{sgn}(x_i-x_j)$ and the sign of the distance error $\mathrm{sgn}(|x_i-x_j|-d)$ (Persis et al., 2010).

The resulting discontinuous, sign-based feedback law for each agent (with gains $k_i>0$) is of the form:

• For interior agents:

$$\dot{x}_i = \mathrm{sgn}(x_{i-1}-x_i)\mathrm{sgn}(|x_{i-1}-x_i| - d_{i-1}) - k_i \mathrm{sgn}(x_i-x_{i+1})\mathrm{sgn}(|x_i-x_{i+1}| - d_i)$$

• End-agents use only the terms associated with their single neighbor.

Non-smooth Lyapunov analysis, carried in the Krasowskii sense, confirms global convergence to the prescribed rigid shape (up to reflection and translation) for all but a measure-zero set of initial conditions. Robustness is guaranteed in the presence of arbitrary positive distance specifications and initial orderings; convergence occurs in finite time due to the nature of the constant-rate sign feedback. Sliding modes may be addressed with dynamic quantization or hysteresis (Persis et al., 2010).

2. Finite-Bit and Binary Formation Control in Multi-Agent Networks

Formation control with even more restrictive information — namely, binary (sign-only) relative state measurements — further exemplifies coarse-information-driven feedback. Here, agents (potentially with general strictly passive dynamics) exchange only the sign of each relative position error with their neighbors, i.e., $\mathrm{sign}(z_k-z_k^*)\in\{\pm 1\}^p$ for each edge $k$ in the interaction graph (Jafarian et al., 2013).

The closed-loop, finite-valued control is

$u = - (B\otimes I_p)\ \mathrm{sign}(z-z^*)$

where $B$ is the incidence matrix of an undirected, connected graph and $z$ the vector of all relative positions. Under weak assumptions (agent strict passivity, full-rank input matrices, connected network), non-smooth Lyapunov analysis shows exact, asymptotic convergence of all Krasowskii solutions to the desired formation. The approach extends to dynamic scenarios such as reference tracking and matched disturbance rejection by augmenting the scheme with appropriate internal model observers (Jafarian et al., 2013). This demonstrates that exact shape-keeping, trajectory tracking, and disturbance rejection can all be achieved under minimal, quantized information.

3. Coarse-ID Control Pipeline for Data-Driven Robust Feedback Synthesis

Complementing quantized-control approaches, Coarse-ID control orthogonally refers to robust feedback design pipelines that use only coarse-grained or finite-data system identification. In the context of unknown, time-invariant linear systems (e.g., discrete-time linear quadratic regulator/LQR),

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

where $A,B$ are unknown but multiple finite-length rollouts are available, the Coarse-ID pipeline proceeds as follows (Dean et al., 2017, Boczar et al., 2018, Tu et al., 2017):

1. Model Estimation: Estimate system matrices $\widehat A, \widehat B$ via ordinary least squares from experimental trajectories.
2. Error Certification: Use non-asymptotic random matrix theory to compute high-probability operator-norm error bounds $\epsilon_A$, $\epsilon_B$ on $(\widehat A,\widehat B)$.
3. Robust Controller Synthesis: Formulate a robust SLS-based optimization that synthesizes static output-feedback $K$ to minimize worst-case quadratic cost over all systems within the error bounds. The SLS parameterization enforces internal stability and accommodates model uncertainty directly in the synthesis.
4. End-to-End Performance Guarantee: Provided small-gain conditions are met, the synthesized feedback law $\hat K$ achieves a relative cost gap

$$\frac{J(A,B,\;\hat K)\;-\;J_\star}{J_\star} \le c \cdot (\epsilon_A+\epsilon_B\|K_\star\|) \; \chi(A,B,Q,R)$$

where $$\chi(A,B,Q,R) = \|(zI-A-BK_\star)^{-1}\|_{H_\infty}$$ and $K_\star$ is the true LQR gain (Dean et al., 2017).

This sample complexity is minimax-optimal up to log factors, with required rollouts $N=O((n+p)\ln(1/\delta)/\gamma^2)$ to achieve a relative cost gap $\le \gamma$ at confidence $1-\delta$. This framework is robust to identification error and always certifies stability, in contrast to naive certainty-equivalent designs which may destabilize the true system (Dean et al., 2017, Tu et al., 2017, Boczar et al., 2018).

4. Robust Control Synthesis from Coarse FIR/Black-Box Models

For SISO stable linear time-invariant systems, coarse FIR (finite impulse response) identification from I/O data is sufficient for robust loop-shaping control under performance guarantees. The methodology approximates the plant $G$ by an $r$-tap FIR $\hat G_r(z) = \sum_{k=0}^{r-1} \hat g_k z^{-k}$ via least squares, splits identification error into truncation plus estimation, and sets the truncation length $r$ using bounds on tail decay (e.g., $|g_k|\le \|G(\gamma z)\|_\infty \gamma^k$ for $0<\rho<\gamma<1$).

Given I/O constraints (e.g., $\ell_2$ or $\ell_\infty$ energy/excitation), Theorems 2.1–2.3 (Tu et al., 2017) establish minimax-optimal sample complexity per desired uniform error $O(\sigma^2 r^2/\epsilon^2)$ or $O(\sigma^2 r/\epsilon^2)$ in $\ell_2$ or $\ell_\infty$ input norms, respectively. Standard robust $H_\infty$ loop-shaping or LMI-based synthesis then guarantees, via the small-gain theorem, that the true plant achieves robust performance margins within $O(\epsilon)$. Empirical results confirm that coarse, low-order identifications suffice for robust control, and collecting further data past the robust-margin threshold yields diminishing returns (Tu et al., 2017).

5. Data-Driven Equation-Free Coarse Control in High Dimensions

Recent developments extend Coarse-ID principles to distributed parameter systems and high-dimensional fields, adopting an equation-free, data-driven paradigm. When neither a closed-form coarse PDE nor analytic Jacobian is available (e.g., in large-scale agent-based or mesoscopic simulators), local neural operators are trained on simulated or experimental trajectory pairs to serve as mesh-independent, short-time coarse timesteppers $\mathcal{N}_\theta[x]\approx x(t+\Delta t)$ (Fabiani et al., 28 Sep 2025). Feedback design then proceeds as follows:

• Use the neural operator in matrix-free Newton–Krylov and Arnoldi Arnoldi subspace routines to extract fixed points and dominant slow-mode eigenspaces.
• Approximate the Jacobian and actuation matrix via directional finite differences on the surrogate.
• Project the dynamics onto the slow subspace to obtain a low-dimensional linear system $y_{n+1}=F y_n + D z_n$.
• Synthesize a classical controller (e.g., discrete-time LQR or pole placement) on this reduced model, then "lift" the resulting feedback to the full system.

Empirical studies for nonlinear PDEs confirm the approach's efficacy: reduced controllers based on neural-operator surrogates stabilize unstable equilibria and achieve convergence rates matching those of traditional (but far costlier) equation-based methods. The surrogate-based approach obviates assembly of full coarse models and Jacobians, providing substantial computational gains at the cost of requiring representative training data (Fabiani et al., 28 Sep 2025).

6. Performance Guarantees, Limitations, and Extensions

Coarse-ID methodologies are marked by explicit, non-asymptotic, end-to-end performance guarantees linking finite sample size, model error, and closed-loop cost. For quantized-formation control, global convergence holds except for a measure-zero set of initial conditions; for FIR and LQR settings, performance degrades only $O(1/\sqrt{N})$ or $O(1/N)$ in sample count given standard problem regularity.

Limitations include requirements for multiple independent experiments (to prevent statistical dependence in least-squares estimators), monotone decay or knowledge of the stability radius (for FIR truncation), and, in some cases, Gaussian noise. Extensions to partially observed systems, MIMO settings, and online/closed-loop identification remain active research areas. The methodology has also been applied for unknown disturbance rejection, adaptive control, and output-feedback synthesis. Data-driven equation-free approaches specifically call for uncertainty quantification, extension to nonlinear output feedback, and online adaptation for dynamic environments (Boczar et al., 2018, Fabiani et al., 28 Sep 2025).

7. Summary of Representative Methodologies and Guarantees

Approach	System/Setting	Information Used	Guarantee/Result
Sign-based formation control (Persis et al., 2010)	1D rigid formations	$\leq$4 bits/agent	Finite-time exact convergence except measure-zero set
Binary multi-agent formation (Jafarian et al., 2013)	General passive agents	Binary relative errors	Exact convergence, trajectory tracking, dist. rejection
FIR-based robust SLS (Tu et al., 2017, Boczar et al., 2018)	SISO LTI, output noise	Short noisy trajectories	$O(1/N)$ or $O(1/\sqrt{N})$ cost gap, robust $H_\infty$ margin
System matrix LQR Coarse-ID (Dean et al., 2017)	Unknown linear state-space	State measurements	End-to-end near-optimality, sample-optimal cost gap
NN operator equation-free (Fabiani et al., 28 Sep 2025)	High-dimensional DPS	Simulated data pairs	Empirical stabilization, speed-up, robust to missing PDE

This comprehensive spectrum of Coarse-ID control demonstrates the efficacy of control synthesis under coarse, quantized, or finite-data information in diverse problem domains, yielding strong theoretical guarantees and significant practical value.