Newton-Raphson Controller for Predictive Tracking

• Newton-Raphson controller is a prediction-based variable gain integrator that adjusts inputs so the predicted future output matches a reference signal.
• It combines output prediction, sensitivity via the inverse Jacobian, and a speedup factor to improve tracking accuracy and ensure closed-loop stability.
• Applications span autonomous vehicles, aerial robots, and distributed optimization, offering computational efficiency and robust control under resource constraints.

The Newton-Raphson controller is a control technique for output tracking of reference signals in continuous-time dynamical systems. In its classical form, it combines three elements: output prediction, a controller based on an integrator with variable gain, and a speedup of the control action for enhancing tracking accuracy and, in some cases, guaranteeing closed-loop stability. In the control literature it is also described as a prediction-based variable gain integral controller: the input is adjusted continuously so that a future predicted output matches a future reference. The framework has been developed for linear and nonlinear systems, including differentially flat systems, autonomous vehicles, aerial robots, and data-driven Koopman-lifted models; the same Newton-Raphson principle has also appeared in distributed optimization and quantum optimal control under related but not identical meanings of “controller” (Wardi et al., 2019, Zhou, 2023).

1. Classical formulation and control law

For a memoryless plant written as

$g(u(t)) = r(t),$

the Newton-Raphson flow is the continuous-time analogue of Newton’s method: $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$ A speedup factor $\alpha$ yields

$$\dot{u}(t) = \alpha \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big),$$

so that the controller acts as an integrator with a variable gain determined by the inverse Jacobian of the input-output map. For dynamical systems,

$\dot{x}(t)=f(x(t),u(t)), \qquad y(t)=h(x(t)),$

the controller is constructed around a lookahead predictor $g(x(t),u(t))$ for the output at time $t+T$. The resulting tracking law takes the form

$$\dot{u}(t)=\alpha\left(\frac{\partial g}{\partial u}(x(t),u(t))\right)^{-1}\big(r(t+T)-g(x(t),u(t))\big),$$

which seeks, at each instant, to solve the prediction equation $g(x(t),u(t))=r(t+T)$ by a continuous Newton-Raphson iteration. In the nonlinear flat-systems literature, the finite-$\alpha$ law is termed the dynamical Newton-Raphson controller (DNRC), while the limit $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$0 gives the statical Newton-Raphson controller (SNRC), written as $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$1 (Shivam et al., 2018, Niu et al., 18 Aug 2025).

This formulation makes the controller fundamentally predictive rather than purely reactive. The control objective is not to drive the instantaneous output $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$2 directly to $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$3, but to drive the predicted future output to the future reference. That structural distinction is central in later stability and performance results.

2. Output prediction, Jacobians, and controller speedup

For nonlinear state-space plants, the prediction step is defined by holding the current input $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$4 constant over a horizon $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$5 and integrating

$$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$6

The predicted output is then

$$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$7

The Jacobian $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$8 required by the Newton-Raphson flow is obtained by differentiating the predictor dynamics: $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$9 with boundary condition $\alpha$0. In the autonomous-vehicle formulation with linear output map, this gives

$\alpha$1

Prediction and Jacobian can be implemented numerically in real time, for example by Euler integration, and this algorithmic simplicity is repeatedly emphasized in the tracking literature (Shivam et al., 2018).

A more elaborate version augments the basic Newton-Raphson flow with feedforward terms: $\alpha$2 Here the variable gain still comes from $\alpha$3, but the extra terms explicitly compensate plant dynamics and reference variation. The same body of work identifies $\alpha$4 as a controller speedup parameter: increasing $\alpha$5 makes the controller more aggressive, reduces the ultimate tracking error, and in some cases permits stable operation with smaller prediction horizons $\alpha$6 (Wardi et al., 2019).

A plausible implication is that the Newton-Raphson controller should be viewed less as a single feedback formula than as a design pattern built from three coupled ingredients: predictor, sensitivity model, and speedup law.

3. Stability, convergence, and error structure

Under perfect prediction and in the feedforward-enhanced formulation, the Lyapunov function

$\alpha$7

satisfies

$\alpha$8

which yields exponential convergence of predicted output to reference. When prediction is imperfect, the asymptotic tracking error is characterized explicitly. If $\alpha$9, then

$$\dot{u}(t) = \alpha \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big),$$0

where $$\dot{u}(t) = \alpha \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big),$$1. If there is also an additive error $$\dot{u}(t) = \alpha \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big),$$2 in the feedforward terms, then

$$\dot{u}(t) = \alpha \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big),$$3

and therefore

$$\dot{u}(t) = \alpha \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big),$$4

This decomposition isolates prediction error from controller-induced error and shows how the speedup factor $$\dot{u}(t) = \alpha \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big),$$5 suppresses the latter (Wardi et al., 2019).

For differentially flat nonlinear systems, later work gives explicit local stability conditions. In output regulation, the closed-loop system is locally asymptotically stable on a neighborhood $$\dot{u}(t) = \alpha \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big),$$6 if the corresponding linear flat system is asymptotically stable under the same DNRC law, and the region of attraction satisfies

$$\dot{u}(t) = \alpha \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big),$$7

For tracking, if

$$\dot{u}(t) = \alpha \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big),$$8

is Hurwitz, then the SNRC uniquely tracks the reference in predicted-output coordinates, and for the finite-$$\dot{u}(t) = \alpha \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big),$$9 DNRC,

$\dot{x}(t)=f(x(t),u(t)), \qquad y(t)=h(x(t)),$0

The same analysis bounds predictor mismatch by terms linear in the prediction horizon $\dot{x}(t)=f(x(t),u(t)), \qquad y(t)=h(x(t)),$1: both the true predictor error $\dot{x}(t)=f(x(t),u(t)), \qquad y(t)=h(x(t)),$2 and the flat-space predictor error $\dot{x}(t)=f(x(t),u(t)), \qquad y(t)=h(x(t)),$3 satisfy bounds of the form $\dot{x}(t)=f(x(t),u(t)), \qquad y(t)=h(x(t)),$4 for sufficiently small $\dot{x}(t)=f(x(t),u(t)), \qquad y(t)=h(x(t)),$5 (Niu et al., 18 Aug 2025).

The 2026 $\dot{x}(t)=f(x(t),u(t)), \qquad y(t)=h(x(t)),$6-stability analysis sharpens this viewpoint. It defines $\dot{x}(t)=f(x(t),u(t)), \qquad y(t)=h(x(t)),$7-stability through boundedness of the augmented state $\dot{x}(t)=f(x(t),u(t)), \qquad y(t)=h(x(t)),$8,

$\dot{x}(t)=f(x(t),u(t)), \qquad y(t)=h(x(t)),$9

and shows that, for sufficiently large $g(x(t),u(t))$0, smoothness and invertibility assumptions imply

$g(x(t),u(t))$1

For flat outputs requiring up to $g(x(t),u(t))$2 derivatives in the inverse transformation, the sufficient polynomial condition for $g(x(t),u(t))$3-stability is automatically satisfied because the relevant roots lie in the left half-plane (Sabry et al., 28 May 2026).

As a mathematical backdrop, scalar Newton-Raphson theory has also been extended with interval conditions that guarantee convergence even when the initial point is far from the root. In particular, if $g(x(t),u(t))$4 and the appropriate sign condition on $g(x(t),u(t))$5 holds on an interval with a unique root, then the Newton-Raphson sequence converges monotonically to that root from any initial value in the interval (Khandani, 2021). This suggests a parallel between classical predictable root-finding regions and the controller literature’s emphasis on verifiable predictor and Jacobian conditions.

4. Vehicle and mobile-robot tracking

A major application area is trajectory control for autonomous vehicles. For a dynamic bicycle model, the controller predicts the state at $g(x(t),u(t))$6, computes the Jacobian through coupled ODEs, and updates the input through the Newton-Raphson flow. In curve-tracking simulations at $g(x(t),u(t))$7, $g(x(t),u(t))$8, and $g(x(t),u(t))$9, with prediction horizon $t+T$0, step $t+T$1, and $t+T$2, the reported peak lateral tracking errors after transients were about $t+T$3 cm, $t+T$4 cm, and $t+T$5 cm, while heading error remained below $t+T$6. In lane-change simulations at $t+T$7, $t+T$8, and $t+T$9, the peak lateral errors were $$\dot{u}(t)=\alpha\left(\frac{\partial g}{\partial u}(x(t),u(t))\right)^{-1}\big(r(t+T)-g(x(t),u(t))\big),$$0 cm, $$\dot{u}(t)=\alpha\left(\frac{\partial g}{\partial u}(x(t),u(t))\right)^{-1}\big(r(t+T)-g(x(t),u(t))\big),$$1 cm, and $$\dot{u}(t)=\alpha\left(\frac{\partial g}{\partial u}(x(t),u(t))\right)^{-1}\big(r(t+T)-g(x(t),u(t))\big),$$2 cm, and peak heading error remained around $$\dot{u}(t)=\alpha\left(\frac{\partial g}{\partial u}(x(t),u(t))\right)^{-1}\big(r(t+T)-g(x(t),u(t))\big),$$3-$$\dot{u}(t)=\alpha\left(\frac{\partial g}{\partial u}(x(t),u(t))\right)^{-1}\big(r(t+T)-g(x(t),u(t))\big),$$4. In a four-robot platoon experiment, the control error for each robot’s tracking point approached $$\dot{u}(t)=\alpha\left(\frac{\partial g}{\partial u}(x(t),u(t))\right)^{-1}\big(r(t+T)-g(x(t),u(t))\big),$$5 mm after transients and the spacing between robots converged to $$\dot{u}(t)=\alpha\left(\frac{\partial g}{\partial u}(x(t),u(t))\right)^{-1}\big(r(t+T)-g(x(t),u(t))\big),$$6. The same work stresses that the controller does not require real-time optimization, unlike MPC, and reports tracking performance that matched or exceeded that of an MPC-based controller from the literature (Shivam et al., 2018).

Flatness-based developments extend these results to closed-form control policies. For the kinematic unicycle, the flat outputs are $$\dot{u}(t)=\alpha\left(\frac{\partial g}{\partial u}(x(t),u(t))\right)^{-1}\big(r(t+T)-g(x(t),u(t))\big),$$7, the assigned flat dynamics are a double integrator, and the actual controls $$\dot{u}(t)=\alpha\left(\frac{\partial g}{\partial u}(x(t),u(t))\right)^{-1}\big(r(t+T)-g(x(t),u(t))\big),$$8 are recovered algebraically from the endogenous transformation. For the dynamic bicycle, the flat outputs are again $$\dot{u}(t)=\alpha\left(\frac{\partial g}{\partial u}(x(t),u(t))\right)^{-1}\big(r(t+T)-g(x(t),u(t))\big),$$9, but the assigned flat dynamics are higher order and the actual controls $g(x(t),u(t))=r(t+T)$0 are reconstructed from $g(x(t),u(t))=r(t+T)$1, and $g(x(t),u(t))=r(t+T)$2. Under stated assumptions, the paper proves equivalence between applying the Newton-Raphson controller in flat-output coordinates and applying it directly to the original nonlinear dynamics, and demonstrates the approach on both kinematic unicycle and dynamic bicycle models (Sabry et al., 28 May 2026).

A separate nonlinear-systems stability study uses the inverted pendulum for output regulation and the kinematic bicycle for tracking. For the cart-pole example, the DNRC regulates the pole upright from different initial conditions, with smaller prediction horizons giving faster convergence but more aggressive inputs, and larger horizons smoothing the control action at the cost of slower convergence. For the kinematic bicycle under SNRC, short horizons such as $g(x(t),u(t))=r(t+T)$3 yielded tracking error below $g(x(t),u(t))=r(t+T)$4 cm but larger input transients, whereas longer horizons such as $g(x(t),u(t))=r(t+T)$5 and $g(x(t),u(t))=r(t+T)$6 increased tracking error while smoothing the control input and reducing saturation (Niu et al., 18 Aug 2025).

5. Data-driven and resource-constrained implementations

The model-based Newton-Raphson controller requires two objects that are often difficult to obtain in practice: a future-output predictor and the derivative of that predictor with respect to the input. A data-driven alternative uses Koopman operator theory to reconstruct a lifted linear model from data. In this formulation, a nonlinear system is lifted to

$g(x(t),u(t))=r(t+T)$7

using EDMD on collected snapshots, and the output predictor and its Jacobian are computed directly from the linear model. For the lifted system, the input-to-predicted-output derivative is

$g(x(t),u(t))=r(t+T)$8

Three nonlinear examples were reported: for the Van der Pol oscillator the Koopman-based Newton-Raphson controller achieved mean squared error $g(x(t),u(t))=r(t+T)$9 versus $\alpha$0 for the classical Newton-Raphson controller; for the overhead crane the corresponding values were $\alpha$1 and $\alpha$2; for the differentially driven car they were $\alpha$3 and $\alpha$4, with reduced input magnitudes (Zhou, 2023).

For aggressive quadrotor tracking, the Newton-Raphson flow controller has been implemented on a Holybro x500v2 quadrotor with a Pixhawk 6X flight controller and a Raspberry Pi 4 companion computer, using OptiTrack motion capture and the ROS2 API for PX4. The predictor is a small-angle linearized hover model with zero-order hold, so $\alpha$5 is constant and computed offline. With integral control barrier functions added for actuator limits, the reported average computation time for the controller plus I-CBF was $\alpha$6 seconds per iteration. Reported RMSE values were consistently below those of the native PX4 controller: for example, $\alpha$7 versus $\alpha$8 on a vertical circle, $\alpha$9 versus $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$00 on a horizontal circle, $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$01 versus $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$02 on a horizontal lemniscate, $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$03 versus $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$04 on a vertical short lemniscate, and $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$05 versus $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$06 on a vertical tall lemniscate (Morales-Cuadrado et al., 2024).

Experimental work on a miniature blimp and a mid-size quadrotor emphasizes the controller’s computational profile under deployment constraints. In that study the Newton-Raphson flow-based tracker was compared with feedback linearization for the blimp and nonlinear MPC for both platforms using RMSE, computation time, and CPU energy consumption. On the blimp, the Newton-Raphson controller was reported at less than $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$07 per update, compared with approximately $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$08 for feedback linearization and $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$09-$$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$10 for nonlinear MPC; the corresponding CPU energy consumptions were $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$11, $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$12, and $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$13. On the quadrotor, the Newton-Raphson controller required $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$14-$$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$15 per update versus $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$16-$$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$17 for nonlinear MPC, with CPU energy consumption $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$18 versus $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$19. The study also reports that the Newton-Raphson controller remained stable on all tested trajectories, including cases where nonlinear MPC caused crashes because of computation overruns (Morales-Cuadrado et al., 19 Aug 2025).

These results address a recurrent misconception that prediction-based Newton-Raphson control is necessarily computationally heavy. In the cited hardware studies, the controller is presented precisely as a lightweight alternative to online optimization-based schemes.

6. Broader variants and related meanings of the term

The label “Newton-Raphson controller” does not refer to a single architecture across all literatures. In quantum optimal control, Newton-Raphson appears inside the GRAPE family as a second-order update for time-discretized control amplitudes in the Hamiltonian

$$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$20

with update

$$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$21

A central observation is that, in GRAPE, the Hessian has the same asymptotic computational scaling as the fidelity functional and gradient, which makes true Newton methods practical. The 2015 work reports that Newton-Raphson with an RFO-regularized Hessian requires fewer system trajectory evaluations than any other algorithm in the GRAPE family, and the 2022 acceleration reformulates the Hessian computation from $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$22 to $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$23, with speedups of $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$24-$$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$25 times for auxiliary matrices and $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$26-$$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$27 times for ESCALADE on ensembles of $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$28-level systems (Goodwin et al., 2015, Goodwin et al., 2022).

In distributed convex optimization, Newton-Raphson consensus and continuous-time modified Newton-Raphson algorithms use Newton structure as a coordination mechanism across agents rather than as a predictor-based tracking law. The original Newton-Raphson Consensus algorithm combines average consensus and separation of time scales and is proved, under suitable hypotheses, to be globally convergent to the true minimizer. An asynchronous and lossy-network extension provides sufficient conditions for local exponential convergence of node states to the global centralized minimizer even in the presence of packet losses. A related continuous-time modified Newton-Raphson algorithm, HISO, replaces the inverse of the sum of Hessians with the sum of local Hessian inverses, so that only local gradients need to be shared; the reported communication cost per agent is $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$29 rather than $$\dot{u}(t) = \left(\frac{\partial g}{\partial u}(u(t))\right)^{-1}\big(r(t)-g(u(t))\big).$$30, with convergence rate no worse than that of the classical continuous-time Newton-Raphson method (Varagnolo et al., 2015, Carli et al., 2017, Moradian et al., 2021).

This broader usage suggests that “Newton-Raphson controller” is best understood as a family resemblance term. In output-tracking control it denotes a prediction-based variable gain integrator; in distributed optimization it denotes a Newton-structured coordination law; in quantum control it denotes a second-order optimal-control update. The common thread is the embedding of Newton-Raphson structure into feedback, coordination, or control-field synthesis, but the surrounding dynamics, guarantees, and computational trade-offs are domain specific.