Data-Driven Iterative Learning Control

• DDILC is a data-centric iterative control strategy that constructs nonparametric Hankel matrix models to capture dynamic system behavior for repetitive tasks.
• It integrates accelerated gradient methods, such as an inexact Nesterov scheme, to handle unknown LTI dynamics, output disturbances, and input constraints.
• A hybrid switching approach between accelerated and classical updates ensures fast initial error reduction with sustained monotonic convergence under structured uncertainties.

Data-Driven Iterative Learning Control (DDILC) is a class of iterative control methods designed for repetitive tasks in dynamical systems, exploiting directly measured input–output data without reliance on explicit parametric system identification. The DDILC approach leverages nonparametric data representations, typically via the Fundamental Lemma of Willems et al., to construct “lifted” models that govern system response across each trial. This methodology robustly addresses unknown LTI system dynamics, output disturbances, and input constraints, and achieves rapid convergence through integrated acceleration strategies. Recent DDILC frameworks also provide analytical convergence guarantees and robust performance under structured disturbances, motivating their application in precision robotics, motion systems, and beyond.

1. Nonparametric System Representation with Hankel Matrices

A central principle of DDILC is the avoidance of conventional model identification in favor of representing system behavior through input–output data trajectories. Following the Fundamental Lemma, an offline data matrix

$$w_d = \begin{bmatrix} u_d(0), y_d(0) \ \dots \ u_d(T), y_d(T) \end{bmatrix}$$

is used to form a Hankel matrix $H_{K+1}(w_d)$ of depth $K+1$, where $K = T_{\text{ini}} + N$, $N$ is the trial length, and $T_{\text{ini}}$ is the initialization horizon. Partitioning and boundary block elimination yield $\mathcal{H}_K(w_d)$: $$\mathcal{H}_K(w_d) = [U_p^\top, Y_p^\top, U_f^\top, Y_f^\top]^\top$$ If the input $u_d$ is persistently exciting of order $K+n$ and $\text{rank}(\mathcal{H}_K) = K+n$, every trajectory $(u_{\text{ini}}, y_{\text{ini}}, u_f, y_f)$ is captured via

$$\mathcal{H}_K(w_d) g = \begin{bmatrix} u_{\text{ini}} \ y_{\text{ini}} \ u_f \ y_f \end{bmatrix}$$

for some $g$. Eliminating $g$ yields the lifted SISO trial-domain model

$y_j = G u_j + c$

where $G$ is block-Toeplitz and $c$ depends on the unknown initial state.

Noisy datasets (output disturbance $d$ corrupting $y_d$) induce rank loss in $\mathcal{H}_K(w_d)$. The framework applies QR factorization and SVD-based low-rank reconstruction, ensuring a causal, lower-triangular model by projecting onto strictly lower-triangular matrices: $\tilde{G} = \mathcal{P}_0(\hat{Y}_f \Phi)$ resulting in a direct data-driven (DD) system approximation used in subsequent control synthesis.

2. Iterative Learning Control Law and Input Constraints

At each trial $j$, DDILC formulates the tracking problem as a constrained least-squares optimization using the lifted model: $$\min_{u \in \mathcal{Q}} J(u) = \frac{1}{2} \| r - (G u + c) \|_2^2$$ with box constraints $\mathcal{Q} = [u_{\min}, u_{\max}]^N$. Since only inexact gradient information is available due to model approximation and disturbances, the framework employs

$$\tilde{J}(u_j) = \frac{1}{2} \|e_j\|_2^2, \quad e_j = r - \hat{y}_j, \quad \tilde{\nabla} J(u_j) = -\tilde{G}^{\top} e_j$$

A projected gradient update is then applied: $$u_{j+1} = \Pi_{\mathcal{Q}}\left[u_j + \frac{1}{L} \tilde{G}^\top e_j\right]$$ where $L \geq \lambda_{\max}(G^\top G)$ ensures stability. Constraint handling reduces to simple Euclidean projection, making the scheme amenable to systems with box-bounded inputs.

3. Accelerated Iteration: Inexact Nesterov Gradient

To achieve rapid early-stage convergence, the algorithm incorporates Nesterov’s accelerated gradient strategy, adapted for inexact oracle conditions due to data-driven modeling and disturbances. At trial $j$: $$\begin{aligned} \mu_j & = \Pi_{\mathcal{Q}}[u_j - \frac{1}{L} g_j] \ \nu_j & = \arg\min_{\nu \in \mathcal{Q}}\left\{ \frac{1}{2}\|\nu - u_0\|_2^2 + \sum_{i = 0}^{j} \frac{i+1}{2L} \langle g_i, \nu - u_i \rangle \right\} \ u_{j+1} & = \frac{2}{j+3} \nu_j + \frac{j+1}{j+3} \mu_j \end{aligned}$$ where $g_j = \tilde{\nabla} J(u_j)$. This mirrors the inexact acceleration scheme of Devolder–Glineur–Nesterov (2014), yielding faster reduction of tracking error until approximation errors begin to accumulate.

4. Disturbance Robustness, Oracle Inexactness, and Switching Strategy

Structured (e.g. periodic or bounded) output disturbances $d_j$ propagate into model and gradient errors. Lemma 3.1 yields quantitative bounds,

$$|J(u) - \tilde{J}(u)| \leq \Delta_1, \qquad \|\nabla J(u) - \tilde{\nabla} J(u)\|_2 \leq \Delta_2$$

with $\Delta_1, \Delta_2$ as functions of disturbance magnitude $\bar{d}$, trial length $N$, and model mismatch $\|\tilde{G} - G\|$. The DDILC framework interprets these bounds as a $\delta$-inexact oracle, ensuring convergence even in the presence of bounded uncertainty.

For optimal performance, the algorithm deploys a hybrid acceleration scheme: accelerated DDILC is run until the error increases over a sliding window $W$, then switches to classical projected-gradient DDILC for guaranteed monotonic error decay.

5. Theoretical Convergence Properties

Convergence under relative degree 1 (invertible $G$) and bounded disturbance is established:

Mode	Error Bound	Convergence Rate
Classical ILC	$$J(u_j) - J(u^*) \leq \frac{L\|u^* - u_0\|^2}{4j} + \delta$$	$O(1/j)$ to $O(\delta)$ neighborhood
Fast ILC (Accelerated)	$$J(u_j) - J(u^*) \leq \frac{2L\|u^* - u_0\|_2^2}{(j+1)(j+2)} + \frac{(j+3)}{3}\delta$$	$O(1/j^2)$, but with $O(j)\delta$ error accumulation
Hybrid Switching	Explicit upper bound $j^*$ on trials for given accuracy $\tau>\delta$	Combined fast initial, monotonic tail

Accelerated DDILC achieves rapid initial error reduction, but accumulation of $\delta$ errors eventually dominates, justifying the hybrid switch for sustained monotonic convergence.

6. Empirical Demonstration and Real-World Case Studies

Numerical experiments validate the DDILC approach:

• Toy Example (4th order plant, $N=20$ trials): Direct DD modeling delivers relative modeling error $2.6 \cdot 10^{-5}$ (structured disturbance), significantly outperforming subspace ID ($4 \cdot 10^{-2}$).
• Hybrid ILC Performance: Hybrid DDILC converges as rapidly as fast ILC for the first $\sim$50 trials and then outpaces classical ILC, reaching low error with $\sim$200 fewer trials.
• Monte Carlo (100 randomized plants): Hybrid DDILC statistically outperforms alternatives for both early-stage ($1$–$50$ trials) and late-stage ($451$–$500$) error norms.
• Precision Robotics Case Study: On a 2-axis high-precision motion platform, the hybrid DDILC achieves sub-micron tracking accuracy within $\sim$10 trials, outperforming feedback-only and standard ILC variants, under output disturbance bounds of $0.1$ mm.

7. Advantages, Limitations, and Extensions

Key advantages:

• Eliminates explicit identification steps, operating solely from input–output data via Hankel filtering.
• Robust to structured (periodic) disturbances and systematic model mismatch.
• Achieves fast convergence in early trials, offering rigorous $O(1/j)$–$O(1/j^2)$ guarantees.
• Efficient constraint handling via projection.

Limitations:

• Under large unstructured (white) noise, parametric subspace identification may outperform DDILC.
• Sufficient PE (persistency of excitation) and large enough sampled data are required for Hankel construction.
• Hybrid mode/window for switching between accelerated and classical regimes remains heuristic.

Research directions:

• Development of theoretically grounded hybrid switching criteria.
• Extension of the DDILC methodology to MIMO systems and varying reference tasks.
• Integration with $\ell_1$ and nuclear-norm regularization for noise suppression and data-efficient learning.
• Adaptive tuning of Lipschitz constant $L$, oracle inexactness $\delta$, and step sizes.

DDILC constitutes an analytically rigorous, empirically validated methodology for repetitive learning control of unknown LTI systems with disturbances and constraints. Its theoretical and practical virtues motivate ongoing efforts for broader applicability, such as extension to general MIMO architectures, adaptive strategies for time-varying environments, and fusion with robust control and regularized machine learning paradigms (Wang et al., 2023).