MIMO Dual Iterative Learning Control

• The paper introduces a dual iterative learning scheme that simultaneously updates the system model and control input, ensuring monotonic convergence in both tracking and model errors.
• It employs lifting operators and self-parametrized gain selection to create an autonomous, plug‐and‐play controller for complex MIMO dynamics.
• Experimental validations show rapid reduction of tracking errors in robotic systems, achieving significant performance improvements within 10–20 trials without manual tuning.

Searching arXiv for the cited DILC and related MIMO ILC papers to ground the article in current literature. MIMO Dual Iterative Learning Control (DILC) is a data-driven iterative learning scheme for repetitive Multiple-Input Multiple-Output (MIMO) systems that simultaneously learns the tracking control input and the plant model, without requiring any prior system knowledge or manual parameter tuning. In the formulation reported in "Dual Iterative Learning Control for Multiple-Input Multiple-Output Dynamics with Validation in Robotic Systems" (Ewering et al., 23 Sep 2025), the method is designed for repetitive MIMO systems, integrates with established iterative learning control methods, and provides monotonic convergence conditions for both reference tracking error and model error in linear time-invariant systems. Within the broader MIMO ILC literature, MIMO DILC belongs to a family of trial-domain learning methods concerned with interaction, uncertainty, scalability, and convergence; related work includes robust convergence analysis for nonsquare MIMO ILC via system equivalence transformation (Meng et al., 2019), multivariable ILC design procedures ranging from decentralized to centralized synthesis (Blanken et al., 2018), and data-driven stochastic-gradient MIMO ILC methods (Aarnoudse et al., 2021).

1. Definition and problem setting

MIMO DILC addresses the problem of autonomously solving repetitive motion tasks on unknown MIMO dynamics. The underlying setting in (Ewering et al., 23 Sep 2025) is a repetitive, unknown MIMO system with inputs $u_j(n) \in \mathbb{R}^O$, outputs $y_j(n) \in \mathbb{R}^O$, trial index $j$, sample index $n$, and $N$ samples per trial. Under the linear time-invariant assumption, the dynamics are written as $y_j = P u_j$, where the plant matrix $P$ is unknown and must be learned. The objective is to find input trajectories $u_j$ such that $y_j$ tracks a reference trajectory $r$, while avoiding prior model identification and manual tuning (Ewering et al., 23 Sep 2025).

The adjective “dual” refers to the coupling of two learning processes: model learning and control learning. Rather than treating system identification and trial-domain control refinement as separate procedures, DILC intertwines them on every trial. This distinguishes it from single-loop ILC formulations that update only the feedforward input. In the language of (Ewering et al., 23 Sep 2025), the framework is simultaneously a tracking controller and a model learner.

The MIMO setting is central rather than incidental. Multivariable interaction can undermine stability and performance if it is ignored, and the broader MIMO ILC literature has emphasized a spectrum of design procedures that trade modeling effort against achievable performance (Blanken et al., 2018). DILC is positioned at the data-driven end of that spectrum: it is intended to cope with unknown dynamics and to eliminate the need for manual parameter tuning, including in complex robotic systems (Ewering et al., 23 Sep 2025).

2. Dual learning architecture and lifted representation

The DILC algorithm in (Ewering et al., 23 Sep 2025) is organized around three steps per trial: model learning via Iterative Model Learning (IML), control input learning via model-based ILC, and self-parametrization.

In the model-learning stage, two lifting operators are introduced: one arranges the model as a large parameter vector, and the other constructs a Toeplitz matrix for input, exploiting system structure. The unknown plant matrix $y_j(n) \in \mathbb{R}^O$0 is reshaped into a vector $y_j(n) \in \mathbb{R}^O$1, and the model estimate $y_j(n) \in \mathbb{R}^O$2 into $y_j(n) \in \mathbb{R}^O$3. The model prediction is written as

$y_j(n) \in \mathbb{R}^O$4

where $y_j(n) \in \mathbb{R}^O$5 is built from $y_j(n) \in \mathbb{R}^O$6 via the lifting operation (Ewering et al., 23 Sep 2025).

The prediction error is

$y_j(n) \in \mathbb{R}^O$7

and the model update law is

$y_j(n) \in \mathbb{R}^O$8

The learning gain $y_j(n) \in \mathbb{R}^O$9 is generated automatically using standard ILC design functions adapted to the model-learning context. For gradient IML,

$j$0

and for norm-optimal IML,

$j$1

with $j$2 and $j$3 positive definite weights (Ewering et al., 23 Sep 2025).

In the control-learning stage, the latest model estimate $j$4 is used to construct the input update. With tracking error

$j$5

the update law is

$j$6

For gradient ILC,

$j$7

and for norm-optimal ILC,

$j$8

The paper describes these gains as self-parametrized: all gain matrices are computed automatically, based on current model sensitivity, to ensure fast and stable convergence and to adapt to scale and coupling variations typical in MIMO systems (Ewering et al., 23 Sep 2025).

Algorithmically, the procedure is trial-by-trial. One initializes $j$9 and $n$0, applies $n$1, measures $n$2, updates $n$3, computes $n$4, determines $n$5, and updates $n$6. The paper characterizes this as plug-and-play and purely data-driven (Ewering et al., 23 Sep 2025).

3. Convergence properties and theoretical guarantees

The theoretical analysis in (Ewering et al., 23 Sep 2025) separates model convergence, prediction error convergence, and tracking error convergence.

For model learning, the stated goal is monotonic convergence of the model error $n$7. Under sufficient excitation and appropriately designed gains, the main condition is

$n$8

together with $n$9 full column rank. Under these conditions,

$N$0

with strict decrease after enough trials (Ewering et al., 23 Sep 2025).

For prediction, the requirement is exponential contraction of the prediction error: $N$1 provided

$N$2

This gives a direct statement that the model-induced output prediction becomes progressively more accurate over trials (Ewering et al., 23 Sep 2025).

For tracking, the control-learning stage is said to achieve exponential monotonic tracking convergence: $N$3 The key statement is that once the model estimate enters a bounded neighborhood of the true plant, the control update ensures monotonic tracking convergence. The paper further states that this is independent of initial model quality and requires no tuning (Ewering et al., 23 Sep 2025).

An important technical qualification is that these monotonic convergence conditions are provided for linear time-invariant systems. The same paper validates the method on nonlinear high-fidelity simulation and multiple nonlinear real-world MIMO systems. This suggests that the empirical operating regime extends beyond the formal LTI analysis, although that extension is experimental rather than a stated theorem (Ewering et al., 23 Sep 2025).

4. Relation to MIMO ILC theory, interaction, and nonsquare systems

MIMO DILC is naturally read against earlier MIMO ILC theory on interaction and convergence. In multivariable ILC, one standard lifted update law is

$N$4

with robust convergence characterized by

$N$5

and monotonic convergence by

$N$6

The work in (Blanken et al., 2018) develops a range of procedures—from independent SISO ILC to robust multi-loop SISO, decentralized robust MIMO ILC using Gershgorin-type bounds and the structured singular value, and centralized MIMO ILC—precisely to address interaction without or with increasing degrees of MIMO model knowledge.

A more specific theoretical issue arises in general MIMO ILC convergence analysis for nonsquare systems. "System Equivalence Transformation: Robust Convergence of Iterative Learning Control with Nonrepetitive Uncertainties" (Meng et al., 2019) identifies a contradiction between convergence conditions for the output tracking error and for the input signal or input error. In the summary provided for that paper, indirect input-based analysis typically requires full column rank, whereas direct output-error-based analysis typically requires full row rank; for nonsquare MIMO systems, both cannot hold unless the system is square and invertible. The proposed system equivalence transformation (SET) transforms a general nonsquare MIMO tracking problem into an equivalent square MIMO one, updates only the minimum required $N$7 input channels for a $N$8-channel reference, and yields a unified convergence condition for boundedness of all signals and robust convergence of the tracking error (Meng et al., 2019).

A plausible implication is that DILC design and convergence analysis can use SET-type reasoning when nonsquare MIMO structure or nonrepetitive uncertainties create contradictions between input-domain and output-domain conditions. The summary for (Meng et al., 2019) explicitly notes that DILC design and convergence analysis can use the SET technique for easy stability checks in the transformed domain. This does not make SET part of the DILC algorithm in (Ewering et al., 23 Sep 2025); rather, it situates DILC within a broader theoretical apparatus for MIMO learning control.

5. Experimental validation in robotic and nonlinear systems

The experimental validation reported in (Ewering et al., 23 Sep 2025) spans both simulation and real-world systems.

In a 6-DOF industrial robot simulation (UR10e, MuJoCo), the dynamics are described as highly nonlinear and strongly coupled, with aggressive, discontinuous references. The reported outcome is convergence to small tracking error within 10–100 trials without model knowledge or tuning, while tracking all six joint references well (Ewering et al., 23 Sep 2025).

In a two-link real-world robot with friction and backlash, three different references are tested, and all converge within 50 trials. In a three-wheeled inverted pendulum robot, the experiments involve balancing, non-repetitive initial states, and strong coupling; for three complex reference tasks, the reported reductions are more than 50% error reduction in 10 trials and more than 80% in 20 trials (Ewering et al., 23 Sep 2025).

Across all systems and reference trajectories, the paper states that validation is autonomous, requires no tuning and no task-specific parameters, applies to both gradient-based and norm-optimal update policies, and is robust to nonlinearities, time-variance, and modeling errors, provided the system is repetitive. The abstract also states that many reference tracking tasks are solved within 10–20 trials, and even complex motions are learned in less than 100 iterations (Ewering et al., 23 Sep 2025).

These experimental results are significant because the theoretical development is framed in terms of LTI systems, whereas the validation includes nonlinear simulation and nonlinear real-world MIMO systems. The intended interpretation is not that the LTI proofs directly cover all tested scenarios, but that the dual learning mechanism and self-parametrization are empirically effective in settings with coupling, measurement noise, and plant complexity (Ewering et al., 23 Sep 2025).

6. Distinctions from related learning-control frameworks

MIMO DILC is adjacent to, but distinct from, several other families of iterative learning methods.

First, it differs from model-free stochastic-gradient MIMO ILC methods whose primary object is feedforward optimization rather than simultaneous model learning. In "Conjugate gradient MIMO iterative learning control using data-driven stochastic gradients" (Aarnoudse et al., 2021), the cost is written as

$N$9

with true gradient

$y_j = P u_j$0

and an unbiased stochastic gradient estimate constructed from randomized experiments. The method develops conjugate search directions and optimal step-size selection using dedicated experiments. That paper explicitly does not mention or discuss Dual Iterative Learning Control specifically (Aarnoudse et al., 2021). The contrast is useful: DILC unites control learning and model learning, whereas stochastic conjugate-gradient ILC addresses data-driven input learning with experimental efficiency.

Second, MIMO DILC should not be conflated with two-loop adaptive ILC frameworks in which “dual” refers to coupled physical subsystems or coordinated learning loops rather than simultaneous model and control learning. The segment-based two-loop adaptive ILC framework for spacecraft position and attitude tracking combines position and attitude errors into a dual-number representation, employs two learning loops that interact through a dual-number representation of tracking errors, and uses a segment-based dynamic projection mechanism to keep parameter estimates and control inputs bounded (Zhang et al., 16 Feb 2026). That framework targets unknown but repeatable parameters and disturbances in strongly coupled rigid-body proximity operations. The shared theme is coordinated multivariable learning under coupling; the mechanism and terminology are different (Zhang et al., 16 Feb 2026).

A recurring misconception is that high-performance MIMO learning control necessarily requires either a full prior model or extensive manual tuning. The DILC formulation in (Ewering et al., 23 Sep 2025) is explicitly intended to reject that premise: it is data-driven, self-parametrizing, and designed to operate without prior system knowledge or manual parameter tuning. A second misconception is that DILC is merely a rebranding of ordinary model-based ILC. The defining feature of DILC in (Ewering et al., 23 Sep 2025) is the simultaneous update of the model estimate and the control input on every trial.

7. Significance and research outlook

Within the MIMO ILC literature, MIMO DILC represents an overview of autonomous model learning and autonomous control learning in the trial domain. Its reported advantages are plug-and-play deployment, compatibility with established ILC design functions, scalability to systems with many inputs and outputs, and validation on both simulated and real robotic systems (Ewering et al., 23 Sep 2025).

Its significance is sharpened by the background problems identified elsewhere in the literature. Multivariable interaction can force trade-offs between user effort and performance (Blanken et al., 2018); nonsquare systems and nonrepetitive uncertainties can create contradictory convergence conditions for input and output analyses (Meng et al., 2019); and purely model-free gradient estimation methods can require careful experimental design for efficiency and robustness (Aarnoudse et al., 2021). MIMO DILC addresses a different but complementary axis of the problem by making model acquisition itself an iterative learning process coupled to tracking improvement (Ewering et al., 23 Sep 2025).

A plausible implication is that future work will continue to connect these lines: autonomous DILC-type model/control co-learning, interaction-aware robust MIMO ILC design, and transformed-domain convergence analysis for nonsquare or uncertain systems. The available sources support a narrower, factual conclusion: MIMO DILC, as formulated in (Ewering et al., 23 Sep 2025), is a data-driven, self-parametrized framework for simultaneous model learning and trial-domain tracking improvement in repetitive MIMO systems, with monotonic convergence conditions in the LTI case and experimental validation on high-dimensional robotic platforms.