Data-Assisted Control (DAC)

• Data-Assisted Control (DAC) is a hybrid architecture that augments conventional model-based controllers with data-driven corrections to compensate for uncertainties and disturbances.
• DAC employs a clear structural decomposition by isolating a physically grounded core from an uncertain residual map, ensuring stability even under damage or unmodeled effects.
• This approach enhances control performance in aerospace and robotics by effectively blending model predictions with real-time data estimation to reduce tracking errors.

Data-Assisted Control (DAC) is a hybrid control architecture in which a conventional model-based nonlinear controller is continuously augmented by a data-driven correction or override when the baseline model no longer provides acceptable performance. In the aerospace and robotic formulations currently associated with the term, DAC preserves a physically grounded core dynamics model and uses data to estimate, learn, or realize the uncertain part of the closed loop, including damage effects, dissipation, actuator mismatch, contact forces, friction, and other unmodeled interactions. Recent work has recast this assistance in explicitly structured forms—most notably as a decomposition connected by a virtual port variable—so that stability arguments remain attached to the modeled subsystem while learning is confined to a lower-complexity residual map (Eslami et al., 2023, Eslami et al., 2023, Eslami et al., 15 Sep 2025, Eslami et al., 11 Sep 2025, Eslami et al., 8 Jun 2025).

1. Core definition and problem setting

The 2023 aerospace formulation defines DAC as a hybrid control architecture in which a conventional, model-based nonlinear controller is continuously augmented by a data-driven correction or override when the baseline model no longer provides acceptable performance, for example after damage or under large unmodeled effects. Its stated objectives are to preserve the rigorous stability and performance guarantees of a Lyapunov-based, model-based controller over the nominal flight envelope, to estimate the true force-moment behavior in real time when unexpected dynamics appear, and to switch or convexly blend between pure model-based control and data-corrected control while retaining closed-loop stability (Eslami et al., 2023).

A related 2023 flight-control variant, Sequential Data-Assisted Control (SDAC), classifies uncertainties into three types: known-predictable, known-unpredictable, and unknown. It decouples the full equations of motion into internal dynamics, where only known-predictable uncertainties appear, and external dynamics, collecting known-unpredictable and unknown effects. In that setting, a model-based nonlinear controller handles the internal dynamics and provides the desired momentum to a data-based controller responsible for the external dynamics (Eslami et al., 2023).

Across later robotic formulations, the same general theme persists but is stated more abstractly. In tensor-invariant multibody control, the system is decomposed into a structurally certain, physically grounded part and an uncertain, empirical, and interaction-focused part, mediated by a virtual port variable. In port-Hamiltonian DAC, a physically meaningful observable links conservative dynamics to all actuation, dissipation, and disturbance channels, and learning is confined to the simplest part of the dynamics (Eslami et al., 15 Sep 2025, Eslami et al., 11 Sep 2025). This suggests that DAC is best understood not as end-to-end replacement of first-principles control, but as a controlled interface between known physics and learned residual behavior.

2. Structural decomposition and virtual-port formulations

A central development in recent DAC research is the introduction of an explicit decomposition between a modeled subsystem and an uncertain subsystem. In the tensor-invariant multibody formulation, the unreduced dynamics are written as a differential-algebraic system

$K\cdot \xi=b,$

with

$K=\begin{bmatrix}\mathcal M & J^\top\ J & 0\end{bmatrix}, \qquad \xi=\begin{bmatrix}D^I\nu\-F_J\end{bmatrix}, \qquad b=\begin{bmatrix}\mathcal F\ \gamma\end{bmatrix}.$

This closed-form, non-recursive Newton–Euler representation keeps the internal constraint forces $F_J$ explicit, which the paper identifies as essential for stability proofs. After rearrangement, all known tensor-mechanics terms are placed on the Left-Hand Side and the unknown interaction terms are collected on the Right-Hand Side through a generalized interaction force

$\tau\in\mathbb R^{6N},$

yielding the symbolic relation

$$\mathrm{LHS}(\nu,D^I\nu,F_J)=\mathrm{Port}(\tau)=\mathrm{RHS}(F_B,\mathrm{Friction},\dots).$$

The unknown mapping is then written as $\tau=\mathcal F_\theta(\mathrm{state})$ (Eslami et al., 15 Sep 2025).

Port-Hamiltonian DAC adopts an analogous structure. For a system

$\dot x=(J(x)-R(x))\nabla\mathcal H(x)+g(x)u,$

a virtual port variable $\Pi$ is introduced so that

$$\dot x-J(x)\nabla\mathcal H(x)=\Pi=-R(x)\nabla\mathcal H(x)+g(x)u.$$

For mechanical systems, only the momentum block is affected, so $\Pi=[\,0^\top\;\tau^\top]^T$. The papers emphasize that $K=\begin{bmatrix}\mathcal M & J^\top\ J & 0\end{bmatrix}, \qquad \xi=\begin{bmatrix}D^I\nu\-F_J\end{bmatrix}, \qquad b=\begin{bmatrix}\mathcal F\ \gamma\end{bmatrix}.$0 or $K=\begin{bmatrix}\mathcal M & J^\top\ J & 0\end{bmatrix}, \qquad \xi=\begin{bmatrix}D^I\nu\-F_J\end{bmatrix}, \qquad b=\begin{bmatrix}\mathcal F\ \gamma\end{bmatrix}.$1 is not merely an algebraic convenience: it is the physically meaningful observable linking stored energy to actuators, dissipation, disturbances, or interaction forces (Eslami et al., 11 Sep 2025, Eslami et al., 8 Jun 2025).

The significance of this decomposition is stated directly in several ways. The tensor-mechanics paper argues that the port cleanly separates known physics from learned uncertainty, increases explainability and interpretability, and provides a naturally ideal input for data-efficient, frame-invariant learning algorithms. The port-Hamiltonian formulations state that the structured design confines learning to the simplest part of the dynamics, enhances data efficiency, and preserves physical interpretability (Eslami et al., 15 Sep 2025, Eslami et al., 11 Sep 2025). A plausible implication is that DAC’s architectural identity lies less in a particular estimator or learner than in this decomposition principle.

3. Control synthesis and stability mechanisms

DAC formulations typically assign the modeled subsystem a stabilizing controller with an explicit Lyapunov or passivity proof. In the tensor-invariant multibody setting, pure velocity control uses the sliding error $K=\begin{bmatrix}\mathcal M & J^\top\ J & 0\end{bmatrix}, \qquad \xi=\begin{bmatrix}D^I\nu\-F_J\end{bmatrix}, \qquad b=\begin{bmatrix}\mathcal F\ \gamma\end{bmatrix}.$2, with $K=\begin{bmatrix}\mathcal M & J^\top\ J & 0\end{bmatrix}, \qquad \xi=\begin{bmatrix}D^I\nu\-F_J\end{bmatrix}, \qquad b=\begin{bmatrix}\mathcal F\ \gamma\end{bmatrix}.$3, and the port command

$K=\begin{bmatrix}\mathcal M & J^\top\ J & 0\end{bmatrix}, \qquad \xi=\begin{bmatrix}D^I\nu\-F_J\end{bmatrix}, \qquad b=\begin{bmatrix}\mathcal F\ \gamma\end{bmatrix}.$4

With the Lyapunov candidate $K=\begin{bmatrix}\mathcal M & J^\top\ J & 0\end{bmatrix}, \qquad \xi=\begin{bmatrix}D^I\nu\-F_J\end{bmatrix}, \qquad b=\begin{bmatrix}\mathcal F\ \gamma\end{bmatrix}.$5, the derivative satisfies

$K=\begin{bmatrix}\mathcal M & J^\top\ J & 0\end{bmatrix}, \qquad \xi=\begin{bmatrix}D^I\nu\-F_J\end{bmatrix}, \qquad b=\begin{bmatrix}\mathcal F\ \gamma\end{bmatrix}.$6

because $K=\begin{bmatrix}\mathcal M & J^\top\ J & 0\end{bmatrix}, \qquad \xi=\begin{bmatrix}D^I\nu\-F_J\end{bmatrix}, \qquad b=\begin{bmatrix}\mathcal F\ \gamma\end{bmatrix}.$7 implies $K=\begin{bmatrix}\mathcal M & J^\top\ J & 0\end{bmatrix}, \qquad \xi=\begin{bmatrix}D^I\nu\-F_J\end{bmatrix}, \qquad b=\begin{bmatrix}\mathcal F\ \gamma\end{bmatrix}.$8, and $K=\begin{bmatrix}\mathcal M & J^\top\ J & 0\end{bmatrix}, \qquad \xi=\begin{bmatrix}D^I\nu\-F_J\end{bmatrix}, \qquad b=\begin{bmatrix}\mathcal F\ \gamma\end{bmatrix}.$9 is skew. The same construction is extended to position-attitude-velocity tracking and to end-effector task-space control with null-space obstacle avoidance (Eslami et al., 15 Sep 2025).

In learning-based port-Hamiltonian DAC for free-floating space manipulators, the LHS controller is a sliding-mode primary controller acting on the momentum dynamics. With

$F_J$0

the desired port torque is

$F_J$1

which yields exponentially stable error dynamics

$F_J$2

The same architecture adds a high-gain decentralized integrator on the RHS to drive the port-tracking error $F_J$3 rapidly to zero. The design requires the Decentralized Integral Controllability condition $F_J$4 and a strict time-scale separation in which RHS decay rates exceed the LHS sliding dynamics by a factor parameterized through $F_J$5. The stated purpose is to ensure that subsequent learning cannot destabilize the primary dynamics (Eslami et al., 11 Sep 2025).

The earlier GTM framework achieves continuity between model-based and data-assisted modes by introducing a decision factor $F_J$6. When $F_J$7, the pure baseline controller is used; as $F_J$8, the data-derived estimate of the generalized force-moment and its associated linear model take over. The blended controller replaces nominal quantities such as $F_J$9 by convex combinations of nominal and estimated values, and $\tau\in\mathbb R^{6N},$0 is updated by minimizing a short-horizon tracking cost $\tau\in\mathbb R^{6N},$1. The sufficient conditions stated for a stable transition are that $\tau\in\mathbb R^{6N},$2 varies slowly and that the DUKF has already driven $\tau\in\mathbb R^{6N},$3 so that the mismatch vanishes (Eslami et al., 2023).

SDAC in flight uses robust sliding mode control for the internal dynamics. With

$\tau\in\mathbb R^{6N},$4

the control law is

$\tau\in\mathbb R^{6N},$5

and the robust term

$\tau\in\mathbb R^{6N},$6

is introduced to eliminate chattering while preserving the inequality

$\tau\in\mathbb R^{6N},$7

The data-based outer loop then regulates momentum error through an LQR synthesized from a Koopman-identified linear model (Eslami et al., 2023).

4. Learning, estimation, and identification layers

DAC does not prescribe a single data-driven mechanism. Instead, different formulations attach different learners or estimators to the uncertain side of the decomposition.

Formulation	Target quantity	Method
GTM DAC (Eslami et al., 2023)	Fixed parameters $\tau\in\mathbb R^{6N},$8, pseudo-observed generalized force moments $\tau\in\mathbb R^{6N},$9, and linear force-moment map	Dual Unscented Kalman Filter; Koopman least-squares estimator
SDAC in flight (Eslami et al., 2023)	Momentum dynamics $$\mathrm{LHS}(\nu,D^I\nu,F_J)=\mathrm{Port}(\tau)=\mathrm{RHS}(F_B,\mathrm{Friction},\dots).$$0 for $$\mathrm{LHS}(\nu,D^I\nu,F_J)=\mathrm{Port}(\tau)=\mathrm{RHS}(F_B,\mathrm{Friction},\dots).$$1	Koopman-DMDc, followed by LQR recomputation
Tensor-invariant DAC (Eslami et al., 15 Sep 2025)	$$\mathrm{LHS}(\nu,D^I\nu,F_J)=\mathrm{Port}(\tau)=\mathrm{RHS}(F_B,\mathrm{Friction},\dots).$$2 or $$\mathrm{LHS}(\nu,D^I\nu,F_J)=\mathrm{Port}(\tau)=\mathrm{RHS}(F_B,\mathrm{Friction},\dots).$$3	Equivariant network, including graph-equivariant or tensor-equivariant MLP
DAC-pH for space manipulators (Eslami et al., 11 Sep 2025)	$$\mathrm{LHS}(\nu,D^I\nu,F_J)=\mathrm{Port}(\tau)=\mathrm{RHS}(F_B,\mathrm{Friction},\dots).$$4	Physics-informed neural network trained online with ADAM
Generalized DAC-pH (Eslami et al., 8 Jun 2025)	RHS policy mapping $$\mathrm{LHS}(\nu,D^I\nu,F_J)=\mathrm{Port}(\tau)=\mathrm{RHS}(F_B,\mathrm{Friction},\dots).$$5	Reinforcement learning, including Soft-Actor-Critic

In the GTM framework, the DUKF estimates the augmented state $$\mathrm{LHS}(\nu,D^I\nu,F_J)=\mathrm{Port}(\tau)=\mathrm{RHS}(F_B,\mathrm{Friction},\dots).$$6, where $$\mathrm{LHS}(\nu,D^I\nu,F_J)=\mathrm{Port}(\tau)=\mathrm{RHS}(F_B,\mathrm{Friction},\dots).$$7 and $$\mathrm{LHS}(\nu,D^I\nu,F_J)=\mathrm{Port}(\tau)=\mathrm{RHS}(F_B,\mathrm{Friction},\dots).$$8 follows a 3rd-order Gauss-Markov process. The subsequent Koopman estimator constructs

$$\mathrm{LHS}(\nu,D^I\nu,F_J)=\mathrm{Port}(\tau)=\mathrm{RHS}(F_B,\mathrm{Friction},\dots).$$9

for the mapping $\tau=\mathcal F_\theta(\mathrm{state})$0. The paper explicitly notes that richer observables could also be used, including products, higher-order monomials, extended DMD, or SINDy (Eslami et al., 2023).

In SDAC, the external dynamics are expressed around a trim through the momentum deviation $\tau=\mathcal F_\theta(\mathrm{state})$1, then discretized and identified from data using DMDc under the observable $\tau=\mathcal F_\theta(\mathrm{state})$2. Every data window $\tau=\mathcal F_\theta(\mathrm{state})$3, the pair $\tau=\mathcal F_\theta(\mathrm{state})$4 is re-estimated and the LQR gain is recomputed, provided the controllability matrix has full rank (Eslami et al., 2023).

Tensor-invariant DAC makes frame invariance an explicit design principle. The learner receives tensor inputs such as twists, forces, joint states, and $\tau=\mathcal F_\theta(\mathrm{state})$5, and, when implemented with $\tau=\mathcal F_\theta(\mathrm{state})$6-equivariant layers or another equivariant architecture, commutes with rotations and translations of the entire mechanism. The stated consequence is that training data collected in one orientation or workspace generalizes to new frames without re-learning gravitational or inertial couplings (Eslami et al., 15 Sep 2025).

The free-floating space-manipulator formulation uses a PINN with two subnetworks for $\tau=\mathcal F_\theta(\mathrm{state})$7 and $\tau=\mathcal F_\theta(\mathrm{state})$8, using monomials and trigonometric functions of $\tau=\mathcal F_\theta(\mathrm{state})$9 as input features and enforcing positive dissipation through a $\dot x=(J(x)-R(x))\nabla\mathcal H(x)+g(x)u,$0-parameterized diagonal output for $\dot x=(J(x)-R(x))\nabla\mathcal H(x)+g(x)u,$1. The loss $\dot x=(J(x)-R(x))\nabla\mathcal H(x)+g(x)u,$2 is minimized online with ADAM, while the surrounding integrator and switching term are designed to keep learning transients bounded (Eslami et al., 11 Sep 2025).

5. Applications and reported empirical results

Several papers provide simulation studies intended to validate distinct DAC variants rather than a single standardized benchmark.

Application	Setup	Reported results
NASA Generic Transport Model (Eslami et al., 2023)	5.5% scale GTM; damage at $\dot x=(J(x)-R(x))\nabla\mathcal H(x)+g(x)u,$3 s; added high-frequency yaw-moment term at $\dot x=(J(x)-R(x))\nabla\mathcal H(x)+g(x)u,$4 s	Pure model-based control yields velocity error jump to $\dot x=(J(x)-R(x))\nabla\mathcal H(x)+g(x)u,$5 ft/s; under DAC, velocity error returns below $\dot x=(J(x)-R(x))\nabla\mathcal H(x)+g(x)u,$6 ft/s within 5 s; Koopman estimate error remains $\dot x=(J(x)-R(x))\nabla\mathcal H(x)+g(x)u,$7; $\dot x=(J(x)-R(x))\nabla\mathcal H(x)+g(x)u,$8 rises from 0 to 1 over $\dot x=(J(x)-R(x))\nabla\mathcal H(x)+g(x)u,$9 s after damage
SDAC in flight (Eslami et al., 2023)	GTM cruise trim; uncertainty in $\Pi$0 at $\Pi$1 s; identification window $\Pi$2 s	$\Pi$3 estimation error drops below 5% after one window; RMS attitude error reduced by $\Pi$4; RMS velocity error reduced by $\Pi$5; peak momentum overshoot cut by $\Pi$6
Tensor-invariant multibody DAC (Eslami et al., 15 Sep 2025)	5-body serial chain; open-loop and end-effector tracking tests	Total linear/angular momentum error $\Pi$7; constraint violation $\Pi$8; max position error $\Pi$9 mm; max orientation error $$\dot x-J(x)\nabla\mathcal H(x)=\Pi=-R(x)\nabla\mathcal H(x)+g(x)u.$$0; settling time $$\dot x-J(x)\nabla\mathcal H(x)=\Pi=-R(x)\nabla\mathcal H(x)+g(x)u.$$1 s post-disturbance; equivariant network converged within 500 samples vs. 5,000 for a non-equivariant baseline
Free-floating space manipulator DAC-pH (Eslami et al., 11 Sep 2025)	2-DOF planar manipulator with link masses $$\dot x-J(x)\nabla\mathcal H(x)=\Pi=-R(x)\nabla\mathcal H(x)+g(x)u.$$2 kg and lengths $$\dot x-J(x)\nabla\mathcal H(x)=\Pi=-R(x)\nabla\mathcal H(x)+g(x)u.$$3 m	By end of phase 2, relative $$\dot x-J(x)\nabla\mathcal H(x)=\Pi=-R(x)\nabla\mathcal H(x)+g(x)u.$$4 estimation errors dropped to 63% for $$\dot x-J(x)\nabla\mathcal H(x)=\Pi=-R(x)\nabla\mathcal H(x)+g(x)u.$$5 and 30% for $$\dot x-J(x)\nabla\mathcal H(x)=\Pi=-R(x)\nabla\mathcal H(x)+g(x)u.$$6; DAC vs. model-based gives 77.5% reduction in attitude tracking error, 16% increase in control effort, and full rejection of high-impact disturbance
Pendulum DAC-pH (Eslami et al., 8 Jun 2025)	$$\dot x-J(x)\nabla\mathcal H(x)=\Pi=-R(x)\nabla\mathcal H(x)+g(x)u.$$7 s, $$\dot x-J(x)\nabla\mathcal H(x)=\Pi=-R(x)\nabla\mathcal H(x)+g(x)u.$$8 kg, $$\dot x-J(x)\nabla\mathcal H(x)=\Pi=-R(x)\nabla\mathcal H(x)+g(x)u.$$9 m, $\Pi=[\,0^\top\;\tau^\top]^T$0 N·m·s	10 random-IC trials show stable convergence $\Pi=[\,0^\top\;\tau^\top]^T$1, $\Pi=[\,0^\top\;\tau^\top]^T$2; moving-average reward declines monotonically over $\Pi=[\,0^\top\;\tau^\top]^T$3 episodes; without integrator $\Pi=[\,0^\top\;\tau^\top]^T$4 rad; with integrator $\Pi=[\,0^\top\;\tau^\top]^T$5 exactly and $\Pi=[\,0^\top\;\tau^\top]^T$6

Taken together, these case studies support a consistent empirical picture. DAC is used where a purely model-based controller degrades under damage, contact, actuation uncertainty, or unmodeled disturbances, while the addition of a structured estimator or learner restores tracking, reduces residual error, or reduces sample complexity. The tensor-invariant results add a specific geometric claim: when the known physics is factored out and the learner respects symmetry, sample complexity can drop by an order of magnitude on the reported benchmark (Eslami et al., 15 Sep 2025).

6. Terminological scope and common ambiguities

The acronym “DAC” is not used uniformly across adjacent literatures. In adaptive online non-stochastic control, “DAC” denotes a disturbance action controller with policy

$\Pi=[\,0^\top\;\tau^\top]^T$7

embedded in an AdaFTRL-C update for the matrices $\Pi=[\,0^\top\;\tau^\top]^T$8. That work proves a data-adaptive policy-regret bound

$\Pi=[\,0^\top\;\tau^\top]^T$9

and explicitly remarks that the matrices $K=\begin{bmatrix}\mathcal M & J^\top\ J & 0\end{bmatrix}, \qquad \xi=\begin{bmatrix}D^I\nu\-F_J\end{bmatrix}, \qquad b=\begin{bmatrix}\mathcal F\ \gamma\end{bmatrix}.$00 adaptively “learn” a model of how past disturbances affect future costs, complementing the robust baseline $K=\begin{bmatrix}\mathcal M & J^\top\ J & 0\end{bmatrix}, \qquad \xi=\begin{bmatrix}D^I\nu\-F_J\end{bmatrix}, \qquad b=\begin{bmatrix}\mathcal F\ \gamma\end{bmatrix}.$01 (Mhaisen et al., 2023).

In a different community, DAC denotes Dynamic Algorithm Configuration. There, the problem is formulated as an MDP $K=\begin{bmatrix}\mathcal M & J^\top\ J & 0\end{bmatrix}, \qquad \xi=\begin{bmatrix}D^I\nu\-F_J\end{bmatrix}, \qquad b=\begin{bmatrix}\mathcal F\ \gamma\end{bmatrix}.$02 with a factored action space $K=\begin{bmatrix}\mathcal M & J^\top\ J & 0\end{bmatrix}, \qquad \xi=\begin{bmatrix}D^I\nu\-F_J\end{bmatrix}, \qquad b=\begin{bmatrix}\mathcal F\ \gamma\end{bmatrix}.$03, and the CANDID benchmark studies Coupled Action Dimensions with Importance Differences through sequential policies of the form

$K=\begin{bmatrix}\mathcal M & J^\top\ J & 0\end{bmatrix}, \qquad \xi=\begin{bmatrix}D^I\nu\-F_J\end{bmatrix}, \qquad b=\begin{bmatrix}\mathcal F\ \gamma\end{bmatrix}.$04

The reported result is that sequential policies outperform independent learning of factorized policies in those action spaces (Bordne et al., 2024).

This suggests that “Data-Assisted Control” should be disambiguated carefully from other established expansions of the same acronym. Within robotics, aerospace, and port-Hamiltonian control, DAC refers to hybrid physics-plus-data architectures centered on a modeled subsystem, an uncertain subsystem, and an interface variable such as $K=\begin{bmatrix}\mathcal M & J^\top\ J & 0\end{bmatrix}, \qquad \xi=\begin{bmatrix}D^I\nu\-F_J\end{bmatrix}, \qquad b=\begin{bmatrix}\mathcal F\ \gamma\end{bmatrix}.$05, $K=\begin{bmatrix}\mathcal M & J^\top\ J & 0\end{bmatrix}, \qquad \xi=\begin{bmatrix}D^I\nu\-F_J\end{bmatrix}, \qquad b=\begin{bmatrix}\mathcal F\ \gamma\end{bmatrix}.$06, $K=\begin{bmatrix}\mathcal M & J^\top\ J & 0\end{bmatrix}, \qquad \xi=\begin{bmatrix}D^I\nu\-F_J\end{bmatrix}, \qquad b=\begin{bmatrix}\mathcal F\ \gamma\end{bmatrix}.$07, or $K=\begin{bmatrix}\mathcal M & J^\top\ J & 0\end{bmatrix}, \qquad \xi=\begin{bmatrix}D^I\nu\-F_J\end{bmatrix}, \qquad b=\begin{bmatrix}\mathcal F\ \gamma\end{bmatrix}.$08. In online control and algorithm-configuration literatures, the same initials may denote different formal objects and different problem classes.