Vector-Level Feedforward Control Framework

• Vector-level feedforward control framework is a design paradigm that computes controller inputs as multi-dimensional vectors or trajectory stacks rather than scalars.
• It integrates inverse dynamics, constrained optimization, and data-driven identification with feedback mechanisms to enhance robustness and disturbance rejection.
• Practical implementations in multibody systems, energy grids, and additive manufacturing show significant performance improvements in tracking and stability.

A vector-level feedforward control framework can be understood as a family of feedforward architectures in which the relevant quantities are treated as vectors or horizon-stacked trajectories rather than as scalar prefilters. In recent formulations, this viewpoint appears in multibody trajectory tracking, lifted finite-horizon feedforward design, predictive disturbance compensation, stochastic optimal control, and additive manufacturing: the manipulated input is a vector $u\in\mathbb R^m$ or a horizon vector $U_{\rm ff}$, the output or target is likewise vector-valued, and feedforward is computed either by inverse dynamics, constrained optimization, or data-driven identification, then combined with feedback for robustness, disturbance rejection, or constraint handling (Drücker et al., 2023, Bolderman et al., 2023, Guzmán et al., 5 Jun 2026, Berret et al., 5 Mar 2026, Kirschbaum et al., 16 Jul 2025).

1. Signal-level meaning of “vector-level”

In the multibody mechanical formulation, the plant is written directly with vector-valued generalized coordinates, velocities, inputs, and outputs,

$$\dot q(t)=v(t),\qquad M(q(t))\dot v(t)=f(q(t),v(t))+B(q(t))u(t),\qquad y(t)=h(q(t),v(t)),$$

with $q,v\in\mathbb R^n$ and $u,y\in\mathbb R^m$. The tracking objective is $y\to y_{\rm ref}$, and the servo-constraint construction imposes the entire vector constraint $h(q,v)-y_{\rm ref}=0\in\mathbb R^m$. The paper is explicit that, although its experiment is single-input/single-output, the formulation is genuinely multi-input/multi-output at the signal level; the vector character lies in the modeling and inversion framework itself (Drücker et al., 2023).

A second, equally important meaning appears in lifted trajectory formulations. In finite-horizon nonlinear feedforward design, the stacked reference, output, error, and feedforward are treated as trajectory vectors such as

$$R=[r(1)^\top,\ldots,r(N_k)^\top]^\top,\qquad Y=[y(1)^\top,\ldots,y(N_k)^\top]^\top,\qquad U_{\rm ff}=[u_{\rm ff}(1)^\top,\ldots,u_{\rm ff}(N_k)^\top]^\top,$$

and the plant is represented by a lifted map $Y=\Phi(x_0,R,U_{\rm ff})$. In that setting, “vector-level” refers not only to multi-channel signals but also to whole-horizon command vectors optimized as single objects (Bolderman et al., 2023).

The same interpretation appears in predictive control with measurable disturbances. There, the feedforward object is not a scalar compensator but a control vector over the prediction horizon. The proposed MPC augmentation computes two predictive control vectors, $\mathbf u_c$ and $U_{\rm ff}$0, then combines them into one applied move. The feedforward component is therefore explicitly vector-valued in the optimization space (Guzmán et al., 5 Jun 2026).

Application-specific uses of the term preserve this structure. In LPBF, “vector-level” refers to assigning one optimized laser power $U_{\rm ff}$1 to each scan vector $U_{\rm ff}$2, while the thermal state is itself stored as a temperature vector $U_{\rm ff}$3. The controller therefore acts on vectorized scan-path elements rather than on a single layerwise scalar setting (Kirschbaum et al., 16 Jul 2025).

A recurrent misconception is that vector-level necessarily implies experimentally demonstrated MIMO control. Several papers are explicit that their framework is vector-valued in formulation while the validation remains SISO or otherwise reduced: this is true for the funnel-plus-servo-constraint study, the LPV kernel-based identification method, and the physics-guided neural-network feedforward papers (Drücker et al., 2023, Haren et al., 2023, Kon et al., 2022).

2. Canonical architectural patterns

The most common structure is additive two-degree-of-freedom control. In the multibody trajectory-tracking architecture, the reference enters both an inverse model and a feedback controller; the inverse branch produces $U_{\rm ff}$4, the feedback branch produces $U_{\rm ff}$5, and the plant input is their sum,

$U_{\rm ff}$6

The feedforward term is responsible for steering the nominal motion, while the funnel-feedback term rejects disturbances and modeling errors (Drücker et al., 2023).

A closely related additive pattern appears in self-healing microgrids. There, the frequency-regulation input to each DG is decomposed into a feedback part and a switch-triggered feedforward part: $U_{\rm ff}$7 The feedforward signals $U_{\rm ff}$8 and $U_{\rm ff}$9 are generated from the anticipated topology-change disturbance caused by network reconfiguration, while the existing PFC, SFC, and inertial-response loops remain active (Kim, 2021).

Wind-farm wake redirection uses the same logic at another scale. A farm-level optimizer generates a yaw-angle vector for all turbines, and turbine-level feedback loops generate yaw-angle deviations from lidar-estimated wake-position errors. The deviations are added to the nominal yaw settings, yielding a hierarchical centralized-feedforward/decentralized-feedback design (Raach et al., 2021).

The additive decomposition is also central in constrained nonlinear flight control: $$\dot q(t)=v(t),\qquad M(q(t))\dot v(t)=f(q(t),v(t))+B(q(t))u(t),\qquad y(t)=h(q(t),v(t)),$$0 Here the learned policy provides a bounded vector-valued feedforward actuator command, while sliding-mode feedback enforces robustness and hard limits. The feedforward term is therefore not a supervisory reference but a matched control vector injected through the same channels as the robust controller (Sayyed et al., 19 Jan 2026).

Predictive control generalizes the same pattern to the horizon level. The MPC disturbance-rejection framework computes a tracking-oriented vector $$\dot q(t)=v(t),\qquad M(q(t))\dot v(t)=f(q(t),v(t))+B(q(t))u(t),\qquad y(t)=h(q(t),v(t)),$$1 and a feedforward-oriented vector $$\dot q(t)=v(t),\qquad M(q(t))\dot v(t)=f(q(t),v(t))+B(q(t))u(t),\qquad y(t)=h(q(t),v(t)),$$2, then applies the first combined move

$$\dot q(t)=v(t),\qquad M(q(t))\dot v(t)=f(q(t),v(t))+B(q(t))u(t),\qquad y(t)=h(q(t),v(t)),$$3

with constraints enforced on the sum. This is the predictive analogue of classical feedback-plus-feedforward superposition (Guzmán et al., 5 Jun 2026).

Taken together, these architectures suggest a shared organizational principle: feedforward supplies nominal inverse or anticipatory action, while feedback supplies residual correction, robustness, and constraint-aware stabilization.

3. Model-based inverse dynamics and constrained motion generation

One major branch of vector-level feedforward is based on inversion. In underactuated multibody systems, the output reference is embedded directly into the dynamics as a vector servo-constraint,

$$\dot q(t)=v(t),\qquad M(q(t))\dot v(t)=f(q(t),v(t))+B(q(t))u(t),\qquad y(t)=h(q(t),v(t)),$$4

which yields the differential-algebraic inverse model

$$\dot q(t)=v(t),\qquad M(q(t))\dot v(t)=f(q(t),v(t))+B(q(t))u(t),\qquad y(t)=h(q(t),v(t)),$$5

The unknowns are $$\dot q(t)=v(t),\qquad M(q(t))\dot v(t)=f(q(t),v(t))+B(q(t))u(t),\qquad y(t)=h(q(t),v(t)),$$6, $$\dot q(t)=v(t),\qquad M(q(t))\dot v(t)=f(q(t),v(t))+B(q(t))u(t),\qquad y(t)=h(q(t),v(t)),$$7, and $$\dot q(t)=v(t),\qquad M(q(t))\dot v(t)=f(q(t),v(t))+B(q(t))u(t),\qquad y(t)=h(q(t),v(t)),$$8 together, so feedforward is computed as constrained inverse motion rather than by explicit normal-form inversion. For minimum-phase systems, the DAE is integrated forward in time; for real-time implementation, the paper discretizes it by implicit Euler and solves the nonlinear system each sample with Newton’s method capped at 10 iterations (Drücker et al., 2023).

A second inversion-based line treats piecewise-affine systems. There, the plant is modeled as a state-switched discrete-time PWA system, explicit inverse formulas are derived for relative degrees $$\dot q(t)=v(t),\qquad M(q(t))\dot v(t)=f(q(t),v(t))+B(q(t))u(t),\qquad y(t)=h(q(t),v(t)),$$9, $q,v\in\mathbb R^n$0, and $q,v\in\mathbb R^n$1 under stated assumptions, and stable inversion is introduced for nonminimum-phase inverse dynamics. The stable modes are propagated forward, unstable modes backward, and the resulting bounded inverse trajectory is used to synthesize the feedforward signal

$q,v\in\mathbb R^n$2

The same paper then embeds this inverse map into iterative learning control, enabling feedforward refinement under model mismatch (Spiegel et al., 2024).

A third model-based route abandons explicit inversion and instead optimizes the whole feedforward trajectory in lifted space. The nonlinear closed-loop plant is represented by

$q,v\in\mathbb R^n$3

and the paper derives the tracking-error upper bound

$q,v\in\mathbb R^n$4

This motivates a control-oriented identification cost on lifted output trajectories and a finite-horizon optimization over the entire feedforward vector $q,v\in\mathbb R^n$5, which is especially suitable for nonlinear and nonminimum-phase systems (Bolderman et al., 2023).

These inversion-based frameworks differ in mechanics, but they share a defining property: feedforward is computed from a vector reference by solving for a coupled state-input evolution, not by static lookup or scalar gain scheduling.

4. Data-driven, kernel-based, and physics-guided identification

A second major branch treats vector-level feedforward as an identification problem. For LPV motion systems, the central difficulty is that the inverse depends dynamically on the scheduling sequence. The kernel-based LPV framework addresses this by introducing

$q,v\in\mathbb R^n$6

then parameterizing

$q,v\in\mathbb R^n$7

The coefficient functions $q,v\in\mathbb R^n$8 are learned nonparametrically in an RKHS from data, while the actual input inherits dynamic dependence on $q,v\in\mathbb R^n$9 through the outer differentiation. Although the validation is SISO, the paper explicitly interprets the basis-feature vector and scheduling-dependent coefficient vector as a natural vector-level feedforward map (Haren et al., 2023).

For Hammerstein systems with input nonlinearities, the learning objective is moved from forward-model fidelity to tracking relevance. The proposed feedforward has Wiener structure, with a linear inverse-dynamics block $u,y\in\mathbb R^m$0 followed by a static inverse nonlinearity $u,y\in\mathbb R^m$1, and is fitted using converged NOILC data by minimizing

$u,y\in\mathbb R^m$2

This identifies the inverse map actually needed for feedforward rather than a forward Hammerstein model optimized for output prediction (hulst et al., 2022).

Physics-guided neural-network frameworks add a learned residual branch to a model-based inverse branch. In the projection-based formulation,

$u,y\in\mathbb R^m$3

and the neural-network output is penalized in the model-output subspace through

$u,y\in\mathbb R^m$4

The explicit goal is to keep model coefficients identifiable and preserve extrapolation while allowing the network to capture only what the model cannot represent (Kon et al., 2022).

The later PGNN formulation with shared autoregressive dynamics makes the same point in a more structured way. Both the physics-based branch and the neural branch share the denominator $u,y\in\mathbb R^m$5, yielding

$u,y\in\mathbb R^m$6

Output-error minimization is then handled by Sanathanan–Koerner iterations, and the neural output is regularized in the model subspace through an orthogonal-projection term. The shared denominator keeps the dynamic structure interpretable and localizes the learned component to residual nonlinear exogenous effects (Kon et al., 2022).

Learning can also target the execution policy directly. In robot control robust to sensing failure, the feedforward policy is $u,y\in\mathbb R^m$7, the feedback policy is $u,y\in\mathbb R^m$8, and the executed action distribution is their mixture

$u,y\in\mathbb R^m$9

Here the feedforward term directly outputs action vectors, such as an 8-dimensional actuator stiffness vector on the snake robot, and learns sensor-independent nominal motion through cross-entropy regularization toward the feedback policy (Kobayashi et al., 2021).

5. Feedforward planned with awareness of uncertainty, latency, and constraints

A defining development in recent work is that feedforward is no longer planned as if feedback were ideal. In stochastic optimal control for partially observable sensorimotor systems, the admissible policy class is

$y\to y_{\rm ref}$0

where $y\to y_{\rm ref}$1 is a vector-valued nominal control trajectory. The nominal trajectory determines the linearization, which determines estimator covariance $y\to y_{\rm ref}$2 and the Kalman gain, so the feedforward plan explicitly shapes the future quality of feedback. The deterministic surrogate optimization propagates the nominal state, estimator covariance, and Riccati matrix jointly, and recovers the feedback gain as

$y\to y_{\rm ref}$3

This makes the feedforward trajectory a function-space object optimized with feedback uncertainty and latency already internalized (Berret et al., 5 Mar 2026).

In constrained flight control, the learned feedforward term is treated as a bounded matched input inside a robust sliding-mode design. The hybrid law is

$y\to y_{\rm ref}$4

and the Lyapunov analysis yields the gain condition

$y\to y_{\rm ref}$5

where $y\to y_{\rm ref}$6 bounds feedforward mismatch and $y\to y_{\rm ref}$7 bounds disturbance. The result is practical stability in an ultimate bound

$y\to y_{\rm ref}$8

The feedforward vector is therefore not trusted unconditionally; it is granted an admissible authority envelope enforced by the robust feedback law and by actuator saturation (Sayyed et al., 19 Jan 2026).

For measurable disturbances in MPC, the same design principle appears in predictive form. The feedforward-oriented vector is intentionally unpenalized: $y\to y_{\rm ref}$9 or, if $h(q,v)-y_{\rm ref}=0\in\mathbb R^m$0 is square and nonsingular,

$h(q,v)-y_{\rm ref}=0\in\mathbb R^m$1

This vector acts as a true disturbance-compensation action, while the regularized vector $h(q,v)-y_{\rm ref}=0\in\mathbb R^m$2 preserves nominal tracking behavior. Constraints are enforced on the combined action rather than on each branch separately (Guzmán et al., 5 Jun 2026).

These formulations suggest a common maturation of the field: feedforward is increasingly optimized jointly with estimator quality, robustness margins, or predictive feasibility rather than appended after the feedback design.

6. Applications, empirical evidence, and recurrent limitations

Experimental and simulation evidence spans mechanical systems, energy systems, manufacturing, transportation, and autonomous vehicles. In the torsional-oscillator study, pure feedforward was real-time computable at $h(q,v)-y_{\rm ref}=0\in\mathbb R^m$3, pure feedback required $h(q,v)-y_{\rm ref}=0\in\mathbb R^m$4 for acceptable behavior, and the combined funnel-plus-feedforward controller remained stable at $h(q,v)-y_{\rm ref}=0\in\mathbb R^m$5 for an aggressive funnel parameter set that made pure feedback unstable after a few seconds. The combined design also achieved the lowest stationary control variance and the lowest stationary tracking errors among the tested alternatives (Drücker et al., 2023).

In wind-farm wake redirection, the nine-turbine study reported total energy outputs of $h(q,v)-y_{\rm ref}=0\in\mathbb R^m$6 for baseline operation, $h(q,v)-y_{\rm ref}=0\in\mathbb R^m$7 for feedforward-only yaw optimization, and $h(q,v)-y_{\rm ref}=0\in\mathbb R^m$8 for the combined feedforward-feedback framework, corresponding to gains of $h(q,v)-y_{\rm ref}=0\in\mathbb R^m$9 and $$R=[r(1)^\top,\ldots,r(N_k)^\top]^\top,\qquad Y=[y(1)^\top,\ldots,y(N_k)^\top]^\top,\qquad U_{\rm ff}=[u_{\rm ff}(1)^\top,\ldots,u_{\rm ff}(N_k)^\top]^\top,$$0 relative to baseline. The architecture is explicitly farm-level vector feedforward plus turbine-level local feedback (Raach et al., 2021).

In self-healing microgrids, the proposed switch-triggered feedforward reduced peak frequency deviation from $$R=[r(1)^\top,\ldots,r(N_k)^\top]^\top,\qquad Y=[y(1)^\top,\ldots,y(N_k)^\top]^\top,\qquad U_{\rm ff}=[u_{\rm ff}(1)^\top,\ldots,u_{\rm ff}(N_k)^\top]^\top,$$1 to $$R=[r(1)^\top,\ldots,r(N_k)^\top]^\top,\qquad Y=[y(1)^\top,\ldots,y(N_k)^\top]^\top,\qquad U_{\rm ff}=[u_{\rm ff}(1)^\top,\ldots,u_{\rm ff}(N_k)^\top]^\top,$$2, RMS deviation from $$R=[r(1)^\top,\ldots,r(N_k)^\top]^\top,\qquad Y=[y(1)^\top,\ldots,y(N_k)^\top]^\top,\qquad U_{\rm ff}=[u_{\rm ff}(1)^\top,\ldots,u_{\rm ff}(N_k)^\top]^\top,$$3 to $$R=[r(1)^\top,\ldots,r(N_k)^\top]^\top,\qquad Y=[y(1)^\top,\ldots,y(N_k)^\top]^\top,\qquad U_{\rm ff}=[u_{\rm ff}(1)^\top,\ldots,u_{\rm ff}(N_k)^\top]^\top,$$4, and settling time from $$R=[r(1)^\top,\ldots,r(N_k)^\top]^\top,\qquad Y=[y(1)^\top,\ldots,y(N_k)^\top]^\top,\qquad U_{\rm ff}=[u_{\rm ff}(1)^\top,\ldots,u_{\rm ff}(N_k)^\top]^\top,$$5 to $$R=[r(1)^\top,\ldots,r(N_k)^\top]^\top,\qquad Y=[y(1)^\top,\ldots,y(N_k)^\top]^\top,\qquad U_{\rm ff}=[u_{\rm ff}(1)^\top,\ldots,u_{\rm ff}(N_k)^\top]^\top,$$6 for stepwise load variations. In the full restoration scenario, $$R=[r(1)^\top,\ldots,r(N_k)^\top]^\top,\qquad Y=[y(1)^\top,\ldots,y(N_k)^\top]^\top,\qquad U_{\rm ff}=[u_{\rm ff}(1)^\top,\ldots,u_{\rm ff}(N_k)^\top]^\top,$$7 fell from $$R=[r(1)^\top,\ldots,r(N_k)^\top]^\top,\qquad Y=[y(1)^\top,\ldots,y(N_k)^\top]^\top,\qquad U_{\rm ff}=[u_{\rm ff}(1)^\top,\ldots,u_{\rm ff}(N_k)^\top]^\top,$$8 to $$R=[r(1)^\top,\ldots,r(N_k)^\top]^\top,\qquad Y=[y(1)^\top,\ldots,y(N_k)^\top]^\top,\qquad U_{\rm ff}=[u_{\rm ff}(1)^\top,\ldots,u_{\rm ff}(N_k)^\top]^\top,$$9 and $Y=\Phi(x_0,R,U_{\rm ff})$0 from $Y=\Phi(x_0,R,U_{\rm ff})$1 to $Y=\Phi(x_0,R,U_{\rm ff})$2 (Kim, 2021).

In LPBF, vector-level laser-power scheduling reduced key-dimensional geometric inaccuracy by up to $Y=\Phi(x_0,R,U_{\rm ff})$3, overall porosity by $Y=\Phi(x_0,R,U_{\rm ff})$4, and photodiode variation by $Y=\Phi(x_0,R,U_{\rm ff})$5 on average, using separately calibrated thermal and melt-pool models and one optimized power value per scan vector (Kirschbaum et al., 16 Jul 2025).

A counterpoint comes from autonomous racing. There, the learning-based steering feedforward controllers achieved the lowest open-loop prediction errors, but closed-loop testing showed that this did not translate into the best path tracking or lap time. The proposed empirical handling-diagram method achieved the best overall closed-loop robustness and the fastest reported lap, $Y=\Phi(x_0,R,U_{\rm ff})$6, even though LSTM and MS-NN had better open-loop steering prediction metrics (Jank et al., 20 May 2026).

Several limitations recur across the literature. First, many frameworks are vector-valued in formulation but not in experimental realization; multivariable generality often remains strongest in the signal model or prediction space rather than in full MIMO hardware validation (Drücker et al., 2023, Haren et al., 2023, Kon et al., 2022). Second, many high-performance schemes rely on preview, offline optimization, or repeated-task structure, so they are not purely causal real-time feedforward laws in the classical sense (Spiegel et al., 2024, Bolderman et al., 2023, Kirschbaum et al., 16 Jul 2025). Third, robustness claims are often architectural or empirical rather than theorem-complete; several papers are explicit that closed-loop stability, recursive feasibility, or bias under closed-loop data remain open issues (Guzmán et al., 5 Jun 2026, Haren et al., 2023, Kobayashi et al., 2021).

Taken together, the current literature suggests that a vector-level feedforward control framework is best viewed not as a single algorithmic template but as a design stance: feedforward is computed over vectors of channels, coordinates, or future moves; it is usually paired with feedback in a nominal-plus-correction architecture; and its practical success depends as much on how it interacts with feedback, constraints, disturbance structure, and model mismatch as on the accuracy of the inverse map itself.