Residual Koopman MPC

• Residual Koopman MPC is a family of predictive control methods that uses lifted linear Koopman models with residual corrections to capture unmodeled dynamics.
• It integrates various formulations, including tube-based, adaptive, and dual-control schemes, to maintain linear or convex optimization structures while boosting prediction fidelity.
• Practical implementations in cart-pole stabilization, quadruped locomotion, and vehicle tracking demonstrate improved robustness, reduced estimation errors, and enhanced computational efficiency.

Searching arXiv for Residual Koopman MPC and closely related Koopman MPC variants to ground the article in the cited literature. Residual Koopman Model Predictive Control (RKMPC) denotes a family of data-driven predictive-control formulations in which Koopman-based lifted linear models are used to represent residual dynamics, residual prediction error, or residual control corrections inside a receding-horizon controller. In the literature, closely related labels include robust tube-based Koopman MPC or r-KMPC, RK-MPC, and RKMPC. Across these variants, the common objective is to improve prediction fidelity while preserving linear MPC or convex optimization structure, or to recover formal robustness and stability guarantees that are not automatic for finite-dimensional Koopman approximations (Zhang et al., 2021, Narayanan et al., 5 Apr 2026, Zhou et al., 22 Apr 2025, Fu et al., 24 Jul 2025, Pouladi, 16 May 2026). This suggests that RKMPC is best understood as a methodological family rather than a single canonical construction.

1. Conceptual development and scope

The 2021 robust tube-based formulation introduced r-KMPC for nonlinear discrete-time dynamical systems with additive disturbances and explicitly identified robustness of the closed-loop Koopman MPC under modeling approximation errors and possible exogenous disturbances as a crucial issue to be resolved. Its controller is composed of a nominal MPC using a lifted Koopman model and an off-line nonlinear feedback policy, and the approach does not assume the convergence of the approximated Koopman operator, which allows using a Koopman model with a limited order for controller design (Zhang et al., 2021).

Subsequent work broadened the meaning of “residual” within Koopman MPC. One direction treats the residual as a learned correction to a nominal mechanistic predictor. In quadruped locomotion, RK-MPC augments a nominal template model with a compact linear residual predictor learned from data in lifted coordinates, enabling systematic correction of model mismatch induced by contact variability and terrain disturbances with provable bounds on multi-step prediction error (Narayanan et al., 5 Apr 2026). In vehicle trajectory tracking, RKMPC uses two linear MPC architecture to calculate control inputs: a Linear Model Predictive Control computes the baseline control input based on the vehicle kinematic model, and a neural network-based RKMPC calculates the compensation input (Fu et al., 24 Jul 2025).

A second direction treats residuals as online decision variables or online learned unknown dynamics. The adaptive Koopman-MPC methodology develops a convex MPC formulation in which residual terms are incorporated directly in the lifted dynamics and regulated by quadratic penalties, while model parameters are adapted online through soft update of target networks (Uchida et al., 2024). The no-regret formulation studies simultaneous system identification and model predictive control for nonlinear systems with unknown residual dynamics that can be expressed by Koopman operators and proves sublinear dynamic regret (Zhou et al., 22 Apr 2025).

A third direction integrates residual dynamics into certified latent models. Stable Fiber-Koopman Residual Dynamics constructs an environment-conditioned Koopman operator for the dominant linear evolution and a contraction-constrained residual neural network for unmodeled nonlinear effects, then embeds the predictor in a sampling-based MPPI controller with an explicit input-to-state stability certificate (Pouladi, 16 May 2026).

2. Residual quantities and lifted models

The literature places the residual at different levels of the prediction stack: as aggregated disturbance in lifted coordinates, as one-step nominal-model error, as unknown additive dynamics, as a compensation control channel, or as a latent nonlinear correction. The following table organizes the principal formulations.

Formulation	Residual quantity	Lifted or corrected model
r-KMPC (Zhang et al., 2021)	Approximation error, disturbance, and reconstruction residual	$z_{k+1}=A z_k+B u_k+w_k,\; x_k=C z_k+v_k$
RK-MPC for quadrupeds (Narayanan et al., 5 Apr 2026)	One-step nominal error, in practice centroidal twist error	$$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$
Adaptive Koopman MPC (Uchida et al., 2024)	Embedded-space residual $r_k$	$\phi_{k+i+1}=K\phi_{k+i}+B u_{k+i}+r_{k+i}$
No-regret RKMPC (Zhou et al., 22 Apr 2025)	Unknown residual dynamics $w_t$	$z_{t+1}=A_* z_t+B_* u_t+\epsilon_t$
Vehicle RKMPC (Fu et al., 24 Jul 2025)	Residual control $\Delta U:=U_r-U_p$	$z_{t+1}=A z_t+B\Delta u_t,\; x_t=C z_t$
SFKD (Pouladi, 16 May 2026)	Residual neural correction $r_\theta$	$$\hat z_{k+1}=A(e_k)\hat z_k+B(e_k)u_k+r_\theta(\hat z_k,u_k,e_k)$$

In the robust tube-based setting, the starting point is a nonlinear, discrete-time system with additive disturbance,

$$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$0

together with a lifting map $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$1 and lifted state $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$2. After identifying $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$3, the perturbed linear predictor is

$$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$4

where $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$5 collects all approximation errors: the truncation of the true infinite-dimensional Koopman operator, the unknown disturbance $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$6, and any model-reconstruction residual (Zhang et al., 2021).

In quadruped locomotion, the nominal reduced-order physics template is a linearized single-rigid-body predictor

$$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$7

and the residual is the one-step nominal error

$$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$8

In practice RK-MPC corrects only the centroidal twist, so

$$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$9

With a lifted residual state $r_k$0, the learned residual dynamics are

$r_k$1

and the combined predictor becomes

$r_k$2

(Narayanan et al., 5 Apr 2026).

In the adaptive Koopman-operator formulation, the embedding model is

$r_k$3

and the true residual in embedded space is

$r_k$4

Over a horizon $r_k$5, the one-step dynamics with residual correction are written as

$r_k$6

(Uchida et al., 2024).

The no-regret formulation begins from

$r_k$7

defines the nominal step $r_k$8, and treats

$r_k$9

as unknown residual dynamics. With observables $\phi_{k+i+1}=K\phi_{k+i}+B u_{k+i}+r_{k+i}$0 and $\phi_{k+i+1}=K\phi_{k+i}+B u_{k+i}+r_{k+i}$1, the lifted residual evolves approximately as

$\phi_{k+i+1}=K\phi_{k+i}+B u_{k+i}+r_{k+i}$2

(Zhou et al., 22 Apr 2025).

In the vehicle-tracking formulation, the residual is not the state prediction error but the control compensation,

$\phi_{k+i+1}=K\phi_{k+i}+B u_{k+i}+r_{k+i}$3

where $\phi_{k+i+1}=K\phi_{k+i}+B u_{k+i}+r_{k+i}$4 is the LMPC’s predicted input and $\phi_{k+i+1}=K\phi_{k+i}+B u_{k+i}+r_{k+i}$5 the “true” input needed on-track. The lifted state is generated by a DNN,

$\phi_{k+i+1}=K\phi_{k+i}+B u_{k+i}+r_{k+i}$6

and the Koopman linear residual system is

$\phi_{k+i+1}=K\phi_{k+i}+B u_{k+i}+r_{k+i}$7

(Fu et al., 24 Jul 2025).

In SFKD, the residual is a contraction-constrained neural network added to an environment-conditioned Koopman backbone:

$\phi_{k+i+1}=K\phi_{k+i}+B u_{k+i}+r_{k+i}$8

Here the encoder maps $\phi_{k+i+1}=K\phi_{k+i}+B u_{k+i}+r_{k+i}$9 onto a fiber bundle latent manifold, and different environments occupy disjoint or smoothly separated submanifolds in the latent space (Pouladi, 16 May 2026).

3. MPC embeddings and computational structure

A central feature of RKMPC is that the residual model is inserted into a receding-horizon optimization without discarding the computational advantages of linear or convex predictive control. In the robust tube-based formulation, ignoring $w_t$0 and $w_t$1 yields the nominal lifted model

$w_t$2

and the nominal MPC solves

$w_t$3

subject to dynamics, tightened constraints, and a terminal set. The actual control law is

$w_t$4

so the online problem remains a nominal MPC, while residual robustness is handled by the tube and offline feedback (Zhang et al., 2021).

In quadruped locomotion, the finite-horizon controller is a convex Quadratic Program. The optimization minimizes a quadratic tracking-and-effort objective over the horizon subject to affine dynamics,

$w_t$5

with

$w_t$6

and inequality constraints

$w_t$7

Because all dynamics and constraints are affine in the decision variables, this is a convex Quadratic Program. The inequality constraints select only the contact forces of feet in stance and enforce a linearized friction pyramid plus unilaterality (Narayanan et al., 5 Apr 2026).

The adaptive Koopman-MPC formulation makes the residual itself a decision variable. The stacked lifted-state constraints are

$w_t$8

and the quadratic cost is

$w_t$9

Input bounds, state constraints via $z_{t+1}=A_* z_t+B_* u_t+\epsilon_t$0, and optional residual bounds $z_{t+1}=A_* z_t+B_* u_t+\epsilon_t$1 preserve convexity, so the optimization is a convex QP in $z_{t+1}=A_* z_t+B_* u_t+\epsilon_t$2 (Uchida et al., 2024).

The no-regret setting also uses finite-horizon receding-horizon MPC, but the residual dynamics are predicted by the currently learned Koopman model of the unknown disturbance. At each time $z_{t+1}=A_* z_t+B_* u_t+\epsilon_t$3, the controller solves

$z_{t+1}=A_* z_t+B_* u_t+\epsilon_t$4

subject to

$z_{t+1}=A_* z_t+B_* u_t+\epsilon_t$5

then applies $z_{t+1}=A_* z_t+B_* u_t+\epsilon_t$6, observes the next state, computes the realized residual, and updates the Koopman estimate (Zhou et al., 22 Apr 2025).

Vehicle RKMPC uses a dual-LMPC architecture. The baseline LMPC computes

$z_{t+1}=A_* z_t+B_* u_t+\epsilon_t$7

from a linearized kinematic bicycle model, while the residual Koopman MPC computes

$z_{t+1}=A_* z_t+B_* u_t+\epsilon_t$8

subject to lifted residual dynamics. The total control is

$z_{t+1}=A_* z_t+B_* u_t+\epsilon_t$9

The paper describes this as a design that preserves the reliability and interpretability of traditional mechanistic model while achieving performance optimization through residual modeling (Fu et al., 24 Jul 2025).

SFKD departs from the QP paradigm and embeds the residual Koopman predictor in a sampling-based MPPI controller. For each rollout, the latent state is propagated via the SFKD one-step map, decoded, and evaluated under a path-integral cost. This is still an RKMPC construction in the sense that the predictive model consists of an environment-conditioned Koopman linear part plus a residual correction, but the control optimizer is sampling-based rather than quadratic-program based (Pouladi, 16 May 2026).

4. Robustness, convergence, regret, and stability guarantees

The strongest formal robustness results in the cited literature appear in the tube-based r-KMPC formulation. With

$\Delta U:=U_r-U_p$0

the error dynamics are

$\Delta U:=U_r-U_p$1

Choosing a robust positively invariant set $\Delta U:=U_r-U_p$2 such that $\Delta U:=U_r-U_p$3 yields $\Delta U:=U_r-U_p$4 for all $\Delta U:=U_r-U_p$5. Tightened sets

$\Delta U:=U_r-U_p$6

guarantee hard constraint satisfaction. Under standard assumptions, the paper derives stabilizability and observability of the Koopman model and proves recursive feasibility, closed-loop robustness, and nominal point-wise convergence. The optimal value satisfies

$\Delta U:=U_r-U_p$7

so the nominal state and input converge to zero. If $\Delta U:=U_r-U_p$8 and the net gain

$\Delta U:=U_r-U_p$9

is Schur, then $z_{t+1}=A z_t+B\Delta u_t,\; x_t=C z_t$0, coupled with $z_{t+1}=A z_t+B\Delta u_t,\; x_t=C z_t$1, $z_{t+1}=A z_t+B\Delta u_t,\; x_t=C z_t$2, and $z_{t+1}=A z_t+B\Delta u_t,\; x_t=C z_t$3 (Zhang et al., 2021).

The quadruped formulation provides a multi-step prediction-error bound for the combined nominal-plus-residual predictor. Under mild Lipschitz assumptions on $z_{t+1}=A z_t+B\Delta u_t,\; x_t=C z_t$4 and a uniform bound $z_{t+1}=A z_t+B\Delta u_t,\; x_t=C z_t$5 on the residual-model one-step error,

$z_{t+1}=A z_t+B\Delta u_t,\; x_t=C z_t$6

and over $z_{t+1}=A z_t+B\Delta u_t,\; x_t=C z_t$7 steps,

$z_{t+1}=A z_t+B\Delta u_t,\; x_t=C z_t$8

Because $z_{t+1}=A z_t+B\Delta u_t,\; x_t=C z_t$9 is small, the paper states that multi-step error remains well-behaved (Narayanan et al., 5 Apr 2026).

The online adaptive Koopman-operator methodology emphasizes stabilization of model learning. It uses a “main” network and a “target” network and updates the target parameters by soft update,

$r_\theta$0

with $r_\theta$1 small. The stated stability remarks are that the target network in the QP prevents large swings in the controller and that one can choose to only adapt a subset of parameters online, such as only $r_\theta$2 and $r_\theta$3, to further improve numerical stability and reduce computation (Uchida et al., 2024).

The no-regret formulation provides a different type of guarantee. The Koopman matrices are updated online by ordinary least squares or online gradient descent on the per-step loss

$r_\theta$4

Dynamic regret is defined against the clairvoyant non-causal controller,

$r_\theta$5

and Theorem 1 gives

$r_\theta$6

hence $r_\theta$7 as $r_\theta$8. The paper interprets this as asymptotic convergence to the optimal non-causal controller (Zhou et al., 22 Apr 2025).

SFKD provides an explicit ISS certificate in latent space. With latent error $r_\theta$9, contraction bound

$$\hat z_{k+1}=A(e_k)\hat z_k+B(e_k)u_k+r_\theta(\hat z_k,u_k,e_k)$$0

and combined gain

$$\hat z_{k+1}=A(e_k)\hat z_k+B(e_k)u_k+r_\theta(\hat z_k,u_k,e_k)$$1

the one-step bound is

$$\hat z_{k+1}=A(e_k)\hat z_k+B(e_k)u_k+r_\theta(\hat z_k,u_k,e_k)$$2

which yields

$$\hat z_{k+1}=A(e_k)\hat z_k+B(e_k)u_k+r_\theta(\hat z_k,u_k,e_k)$$3

The corresponding ultimate bound is

$$\hat z_{k+1}=A(e_k)\hat z_k+B(e_k)u_k+r_\theta(\hat z_k,u_k,e_k)$$4

The paper further states a closed-loop tracking bound in physical coordinates when the nominal MPPI tracking error in latent space is $$\hat z_{k+1}=A(e_k)\hat z_k+B(e_k)u_k+r_\theta(\hat z_k,u_k,e_k)$$5 (Pouladi, 16 May 2026).

5. Reported implementations and empirical results

The empirical literature spans cart-pole stabilization, quadruped locomotion, vehicle trajectory tracking, and environment-switching autonomous vehicle path tracking. Reported metrics indicate that the residual construction is used either to recover performance lost by model mismatch or to reduce sensitivity to the choice of observables.

Domain	Configuration	Reported outcome
Cart-pole (Uchida et al., 2024)	Adaptive Koopman MPC	$$\hat z_{k+1}=A(e_k)\hat z_k+B(e_k)u_k+r_\theta(\hat z_k,u_k,e_k)$$6 s per 6 s rollout vs 0.70 s nominal MPC, 1.12 s RFF-MPC, 5.11 s GP-MPC
Cart-pole (Zhou et al., 22 Apr 2025)	Online learned residual Koopman MPC	Under 45% mismatch, only RKMPC succeeds in all trials
Quadruped locomotion (Narayanan et al., 5 Apr 2026)	Gazebo and Unitree Go1, 500 Hz	RK-MPC: 0.020 m/s linear-velocity RMSE, 0.095 rad/s angular-rate RMSE, 0.89 ms average QP solve time
Vehicle trajectory tracking (Fu et al., 24 Jul 2025)	Carsim–MATLAB and F1TENTH	Requires only 20% of KMPC training data; lateral error reduced by 11.7%-22.1%
Environment switching (Pouladi, 16 May 2026)	MPPI with SFKD	0.120 m RMSE in scenario S3 and 44% improvement in control smoothness

For quadruped locomotion, the open-loop comparison over 100-step rollouts on held-out data shows one-step RMSE for linear velocity $$\hat z_{k+1}=A(e_k)\hat z_k+B(e_k)u_k+r_\theta(\hat z_k,u_k,e_k)$$7 of SRB $$\hat z_{k+1}=A(e_k)\hat z_k+B(e_k)u_k+r_\theta(\hat z_k,u_k,e_k)$$8, EDMD-mono $$\hat z_{k+1}=A(e_k)\hat z_k+B(e_k)u_k+r_\theta(\hat z_k,u_k,e_k)$$9, EDMD-$$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$00, and Res-Koopman $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$01; for angular velocity $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$02, SRB $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$03, EDMD-mono $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$04, EDMD-$$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$05, and Res-Koopman $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$06. The paper states that $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$07 EDMD exhibits drift in the $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$08 geodesic error over horizon, while monomial EDMD blows up (Narayanan et al., 5 Apr 2026). In closed-loop simulation for a circular reference under randomized friction and rough terrain, linear-velocity RMSE is SRB $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$09 m/s, SE3-KMPC $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$10 m/s, and RK-MPC $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$11 m/s; angular-rate RMSE is SRB $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$12 rad/s, SE3-KMPC $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$13 rad/s, and RK-MPC $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$14 rad/s. Average QP solve times are SRB-MPC $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$15 ms, SE3-KMPC $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$16 ms, and RK-MPC $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$17 ms. On Unitree Go1 hardware with onboard NVIDIA Xavier NX, the controller runs at 500 Hz, achieves 9/10 successes over loose obstacles in debris traversal and push recovery, transfers from trot to crawl by simply changing contact-schedule $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$18, and is reported stable on grass, gravel, snow, and ice, with slip on ice beginning when the controller pushes normal forces to limit (Narayanan et al., 5 Apr 2026).

For vehicle dynamics, the simulation study on the Carsim–MATLAB platform reports the following table entries: LMPC lateral error $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$19 m, heading error $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$20 rad, steer-rate $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$21 rad/s, computation time mean $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$22 ms and max $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$23 ms; NMPC lateral error $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$24 m, heading error $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$25 rad, steer-rate $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$26 rad/s, computation time $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$27 ms; KMPC lateral error $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$28 m, heading error $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$29 rad, steer-rate $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$30 rad/s, computation time $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$31 ms; and RKMPC lateral error $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$32 m, heading error $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$33 rad, steer-rate $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$34 rad/s, computation time $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$35 ms. The paper states that RKMPC reduces lateral error by 11.2%, heading by 8.6%, steering jitter by 27.6% versus LMPC, while running well within a 50 ms cycle, and that KMPC’s performance suffers under small datasets while NMPC cannot meet real-time budgets (Fu et al., 24 Jul 2025). In physical 1:10 F1TENTH tests, LMPC yields lateral error $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$36 m and heading error $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$37 rad, KMPC is reported as uncontrollable, and RKMPC yields lateral error $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$38 m and heading error $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$39 rad with computation time $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$40 ms. The same paper states that RKMPC requires only 20% of the training data needed by traditional KMPC while delivering superior tracking performance (Fu et al., 24 Jul 2025).

For the adaptive cart-pole study, nominal MPC without residual or adaptation fails to stabilize quickly, whereas GP-MPC and RFF-MPC do stabilize but are 2–10× slower. The reported solve times are $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$41 s per 6 s rollout for RKMPC, 0.70 s for nominal MPC, 1.12 s for RFF-MPC, and 5.11 s for GP-MPC (Uchida et al., 2024). In the no-regret cart-pole study, the system has 4D state $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$42, scalar input $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$43, is discretized at 15 Hz with horizon $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$44, and uses cost weights $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$45, $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$46. Under 25% mismatch, RKMPC stabilizes fastest and outperforms nominal-MPC, GP-MPC, and RFF-MPC; under 45% mismatch, only RKMPC succeeds in all trials; and the estimation error in the Koopman fit decays quickly (Zhou et al., 22 Apr 2025).

For environment-constrained robust control, SFKD is evaluated on a bicycle model with varying road friction $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$47 and lateral wind $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$48 m/s. In scenario S3, mean $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$49 std over 200 runs gives Koopman MPC RMSE $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$50 m, Neural ODE RMSE $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$51 m, ICODE-MPPI RMSE $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$52 m, and SFKD RMSE $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$53 m. Control smoothness, measured as mean $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$54, is $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$55 rad/s for ICODE-MPPI and $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$56 rad/s for SFKD. The latent-stability violation study reports that uncertified baselines exceed the theoretical bound in 40–60% of trials, whereas SFKD stays within bound $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$57 of the time (Pouladi, 16 May 2026).

6. Interpretation, misconceptions, and active directions

A common misconception is that Residual Koopman MPC refers to one fixed algorithmic template. The literature does not support that reading. The label covers at least the following constructions: a tube-based lifted MPC with offline residual feedback (Zhang et al., 2021); a compact residual predictor correcting a nominal reduced-order model (Narayanan et al., 5 Apr 2026); a convex lifted QP with explicit residual decision variables and online parameter adaptation (Uchida et al., 2024); an online least-squares or OGD scheme for unknown residual dynamics with dynamic-regret guarantees (Zhou et al., 22 Apr 2025); a dual-MPC compensation architecture for vehicle tracking (Fu et al., 24 Jul 2025); and an environment-conditioned latent Koopman model with contraction-constrained residual network inside MPPI (Pouladi, 16 May 2026). This suggests that the defining feature is not one optimizer or one notion of residual, but the explicit insertion of a residual Koopman mechanism into predictive control.

A second misconception is that a Koopman lift alone yields robustness. The 2021 tube-based work states the opposite problem formulation: robustness of the closed-loop Koopman MPC under modeling approximation errors and possible exogenous disturbances is still a crucial issue to be resolved, and its solution requires tube-based robustification, tightened constraints, and offline feedback design (Zhang et al., 2021). The environment-conditioned SFKD results make a related point from a different angle: latent stability is enforced by spectral-norm constraints on $$z_{k+1}=A^{res} z_k+B^{res}u_k,\; \hat e_k=C^{res} z_k$$58 and a contraction constraint on the residual network, not by lifting alone (Pouladi, 16 May 2026).

A third misconception is that RKMPC is necessarily fully data-driven. Several implementations are explicitly hybrid. The quadruped controller combines a physics-based SRB template with a compact Koopman-learned residual (Narayanan et al., 5 Apr 2026). The vehicle controller keeps the well-understood kinematic baseline intact and uses the residual Koopman block only to correct unmodeled nonlinearities, which the paper presents as preserving interpretability and reliability (Fu et al., 24 Jul 2025).

Active directions are stated directly in the cited papers. For quadruped RK-MPC, future work includes online adaptation of residual operator by incremental least-squares or recursive EDMD, robust or tube-MPC extensions using residual uncertainty bounds, and application to other legged platforms and full-body hybrid dynamics (Narayanan et al., 5 Apr 2026). The adaptive Koopman-operator methodology discusses choosing only a subset of parameters as online updated parameters to improve numerical stability and reduce computation (Uchida et al., 2024). The no-regret formulation points toward continued integration of online identification and predictive control under explicit performance guarantees (Zhou et al., 22 Apr 2025). The SFKD formulation extends the residual Koopman paradigm toward environment-aware geometric consistency, latent-space stability certification, and bounded residual perturbation propagation under environment switching (Pouladi, 16 May 2026).