Koopman-Based Control: Framework & Applications

• Koopman-based control is a framework that lifts nonlinear systems into a higher-dimensional observable space where dynamics become approximately linear or bilinear, facilitating the use of linear control techniques.
• It employs operator-theoretic principles and identification strategies such as EDMD and deep learning to construct finite-dimensional surrogate models for controller synthesis.
• Applications in robotics, aerospace, and stochastic systems demonstrate its ability to achieve precise tracking and robust performance despite model uncertainties.

Searching arXiv for recent and foundational papers on Koopman-based control to ground the article in the literature. Koopman-based control denotes a family of methods that reformulate nonlinear control problems in a lifted space of observables, where the evolution is linear or bilinear and can therefore be addressed with linear-systems tools. In the literature, the starting point is typically a nonlinear system such as $x_{k+1}=f(x_k,u_k)$ or $\dot x=f(x)+\sum_{i=1}^m g_i(x)u_i$, together with a lifting $z=\phi(x)$ into a higher-dimensional space. Control is then synthesized on surrogate dynamics of the form $z_{k+1}=Az_k+Bu_k$, $z_{k+1}=Az_k+Bu_k+\sum_i u_i D_i z_k$, or $\dot z=Az+\sum_i u_i B_i z$, depending on the modeling assumptions and the treatment of control inputs (Tao et al., 2023, Zhang et al., 2022). The field now spans rigorous operator-theoretic bilinearization, EDMD-style identification, Lyapunov- and SOS-certified synthesis, model predictive control, robust optimal control, and learned latent-state controllers operating directly from pixels, lidar, or other high-dimensional observations (Strässer et al., 2024, Lyu et al., 2023).

1. Operator-theoretic basis

Koopman-based control rests on the observation, due to B. O. Koopman in 1931, that a nonlinear state evolution induces a linear evolution on scalar observables. For a discrete-time system, the Koopman operator acts by composition, $(K\phi)(x)=\phi(f(x,u))$, while in the autonomous case one writes $K\phi(x)=\phi(f(x))$. The standard properties emphasized in the control literature are linearity, infinite-dimensionality, and spectral decomposition through Koopman eigenfunctions (Tao et al., 2023).

For control-affine systems, the operator-theoretic picture becomes more structured. A rigorous derivation shows that the Koopman operator associated with a control system obeys an operator ODE of the form

$$\frac{d}{dt}U_t = U_t\,\mathcal L_f +\sum_{i=1}^m u_i(t)\,U_t\,\mathcal L_{g_i},$$

which yields an infinite-dimensional bilinear system on observables and directly leads to a global bilinearization of control-affine systems (Zhang et al., 2022). In parallel, generator-based formulations write

$L^u = L^0 + \sum_{i=1}^m u_i (L^{e_i}-L^0),$

making the control affinity explicit in lifted coordinates and enabling finite-dimensional bilinear surrogate models for synthesis (Strässer et al., 2024).

This operator-level viewpoint matters because it clarifies a recurring distinction in the literature: linearity is obtained in the space of observables, not in the original state coordinates. Finite-dimensional control designs are therefore approximations unless the chosen observables span an invariant subspace or the lifted representation is analytically closed.

2. Lifted models and identification strategies

The practical problem is to replace the infinite-dimensional Koopman evolution by a finite-dimensional surrogate. A common construction selects a dictionary of observables $\dot x=f(x)+\sum_{i=1}^m g_i(x)u_i$0, stacks them into $\dot x=f(x)+\sum_{i=1}^m g_i(x)u_i$1, and estimates matrices by least squares or EDMD-type regression. Representative models include a linear predictor $\dot x=f(x)+\sum_{i=1}^m g_i(x)u_i$2, a bilinear predictor $\dot x=f(x)+\sum_{i=1}^m g_i(x)u_i$3, and generator-based continuous-time bilinear forms (Tao et al., 2023, Zhang et al., 2022).

Classical identification uses polynomial, Fourier, or radial-basis dictionaries. The Sphero SPRK experiments used a basis containing $\dot x=f(x)+\sum_{i=1}^m g_i(x)u_i$4, a constant $\dot x=f(x)+\sum_{i=1}^m g_i(x)u_i$5, and monomials in $\dot x=f(x)+\sum_{i=1}^m g_i(x)u_i$6 up to degree $\dot x=f(x)+\sum_{i=1}^m g_i(x)u_i$7, while the cart- and VTOL-pendulum studies showed that Fourier observables can encode rotational structure more efficiently than plain polynomials (Abraham et al., 2017). For multi-robot utility optimization, the observables may include the nonlinear utility functions themselves, yielding a lifted state in which the original nonlinear program is converted into a linear or affine program (Tao et al., 2023).

A central identification issue is how to treat the control channel. The Control-Coherent Koopman Model introduces actuator states $\dot x=f(x)+\sum_{i=1}^m g_i(x)u_i$8 so that $\dot x=f(x)+\sum_{i=1}^m g_i(x)u_i$9 is affine in $z=\phi(x)$0, includes $z=\phi(x)$1 explicitly in the dictionary, removes the known input action before estimating the autonomous Koopman matrix $z=\phi(x)$2, and then reconstructs an exact lifted input matrix $z=\phi(x)$3 with no approximation in $z=\phi(x)$4 (Asada et al., 2024). This contrasts with prevailing DMDc-style fitting of $z=\phi(x)$5 by least squares on full input-plus-state data.

Recent work replaces hand-crafted dictionaries with learned embeddings. “Deep Koopman Operator with Control for Nonlinear Systems” learns $z=\phi(x)$6, the Koopman matrices $z=\phi(x)$7, and, when needed, an auxiliary control network $z=\phi(x)$8 that maps nonlinear state-dependent inputs into a latent control variable $z=\phi(x)$9 so that the lifted dynamics remain linear in $z_{k+1}=Az_k+Bu_k$0 (Shi et al., 2022). “Task-Oriented Koopman-Based Control with Contrastive Encoder” learns a probabilistic embedding $z_{k+1}=Az_k+Bu_k$1 with InfoNCE regularization, one-step model loss, and a task-cost objective inside an end-to-end RL loop; for pixels it uses a 4-layer convolutional encoder, for physical states a 3-layer MLP, and reports that $z_{k+1}=Az_k+Bu_k$2 is robust while $z_{k+1}=Az_k+Bu_k$3 degrades performance (Lyu et al., 2023).

A separate identification line combines Koopman lifting with Willems’ Fundamental Lemma. The resulting predictor is learned from Hankel data, can use differentiable regressors such as Bayesian neural networks, and incorporates model uncertainty through a Wasserstein-robust regression objective inside a data-driven predictive controller (Lian et al., 2021).

3. Control synthesis in Koopman coordinates

Once a lifted predictor is available, most Koopman-based controllers use standard linear or bilinear synthesis tools. The most direct case is lifted-space LQR. In the task-oriented latent framework, a differentiable infinite-horizon discrete LQR is embedded inside a Soft-Actor-Critic loop; the controller gain is obtained by Riccati iterations, $z_{k+1}=Az_k+Bu_k$4 and $z_{k+1}=Az_k+Bu_k$5 are learnable diagonal matrices, and gradients back-propagate through the Riccati recursion to the encoder and Koopman matrices (Lyu et al., 2023). In earlier deep Koopman work, LQR is solved on the learned linear latent model and the actual control input is recovered either directly or through an inverse of the learned control map (Shi et al., 2022).

Model predictive control is the most common synthesis layer for finite-dimensional lifted models. In the soft continuum manipulator, the lifted predictor $z_{k+1}=Az_k+Bu_k$6, $z_{k+1}=Az_k+Bu_k$7 is combined with a finite-horizon quadratic program solved in real time at $z_{k+1}=Az_k+Bu_k$8 Hz with Gurobi; the current load estimate is inserted into the lifted state through $z_{k+1}=Az_k+Bu_k$9 (2002.01407). The Control-Coherent Koopman Model is used with a standard convex quadratic MPC problem for robotic arms, with state and input constraints imposed directly in lifted coordinates (Asada et al., 2024). In virtually coupled train control, a closed-form observable lift produces an affine-parameter-varying predictor that is frozen along the shifted trajectory, condensed, and solved online as a QP, converting the original nonlinear, nonconvex MPC problem into a single convex quadratic program (Zhang et al., 10 Jun 2026).

Feedback linearization also appears in Koopman form. One rigorous development constructs a lifted observable map

$z_{k+1}=Az_k+Bu_k+\sum_i u_i D_i z_k$0

on a controllable submanifold and shows that, under an appropriate nonlinear feedback law, the dynamics in Koopman coordinates become a chain of integrators $z_{k+1}=Az_k+Bu_k+\sum_i u_i D_i z_k$1 (Zhang et al., 2022). A turbofan-engine study uses a Koopman model to derive both an adaptive Koopman-based MPC with disturbance observer and a Koopman-based feedback linearization controller; the same identified Koopman predictor is reused across spool-speed and EPR control formulations by switching the output map (Grasev, 2 Apr 2026).

Robust and optimal-control formulations are also present. Koopman-based policy iteration for robust optimal control rewrites the HJIE using the Koopman generator, projects the value function onto a finite basis, and alternates policy evaluation and policy improvement in coefficient space (Krolicki et al., 2022). For discrete-time nonlinear systems with communication constraints, Koopman-based event-triggered control designs both a data-driven state-feedback gain and an event-triggering threshold directly from data using LMIs and a Lyapunov argument in the lifted space (Manaa et al., 19 Apr 2025).

4. Guarantees, robustness, and closed-loop validity

A major theme in the literature is that predictive accuracy alone is not sufficient for control. The Control-Coherent Koopman work makes this point sharply: for a two-link horizontal robot arm, one-step-ahead prediction error histograms are nearly identical for CCK and DMDc, but the correct lifted input matrix $z_{k+1}=Az_k+Bu_k+\sum_i u_i D_i z_k$2 yields dramatically better closed-loop MPC performance (Asada et al., 2024). This is consistent with the broader shift from model-oriented fitting to control-oriented or certificate-oriented identification.

Several frameworks provide explicit closed-loop guarantees. SafEDMD-based methods estimate a bilinear surrogate together with proportional residual bounds of the form

$z_{k+1}=Az_k+Bu_k+\sum_i u_i D_i z_k$3

and then synthesize controllers by robust control theory and sum-of-squares optimization (Strässer et al., 2024). A related SOS formulation parametrizes a rational controller for the uncertain bilinear surrogate and proves exponential stability of the true nonlinear system, while reporting a larger region of attraction and improved data efficiency than LMI-based over-approximations of bilinearity (Strässer et al., 2024).

Neural Koopman Lyapunov Control addresses stabilizability directly by simultaneously learning the Koopman lifting, a bilinear lifted model, and a Control Lyapunov Function via a learner–falsifier loop. The resulting controller uses Sontag’s universal formula in lifted coordinates and provides asymptotic-stability guarantees for the Koopman bilinear representation (Zinage et al., 2022). Lyapunov-based MPC on Koopman eigenfunction coordinates uses an auxiliary CLF-based bounded controller as a constraint, and, under the existence of a continuously differentiable inverse map $z_{k+1}=Az_k+Bu_k+\sum_i u_i D_i z_k$4, translates stabilizability of the lifted bilinear system to the original nonlinear system (Narasingam et al., 2020).

Robustness has also been studied at the level of optimality. “Optimality Robustness in Koopman-Based Control” introduces a norm-bounded residual model, derives upper bounds on deviations of both the value function and the optimal controller, formulates a robustness-aware min–max control law, and proposes a policy-iteration algorithm whose well-posedness and convergence are established via vanishing viscosity regularization and elliptic PDE techniques (Lin et al., 7 Apr 2026). In stochastic settings, control-affine SDEs are lifted through the Koopman generator, approximated by gEDMD, and reduced to finite-dimensional bilinear optimal-control problems (Guo et al., 2024).

A practical robustness device is state augmentation for latent disturbances or loads. The soft continuum manipulator augments the lifted state with payload variables and estimates the unknown mass online from delayed measurements, while the turbofan-engine controller augments the Koopman estimator with a slowly varying disturbance on the outputs and chooses observer gains via a discrete-time Kalman-filter Riccati solve (2002.01407, Grasev, 2 Apr 2026).

5. Representative systems and reported performance

System	Koopman-control formulation	Reported result
4D CartPole Swing-up, 18D Cheetah Running, TurtleBot3 navigation	End-to-end task-oriented latent linear model with contrastive encoder and differentiable LQR (Lyu et al., 2023)	Within 10% of the SAC+curl reference cost on all three sim tasks; cost degrades by < 8% as model error grows from $z_{k+1}=Az_k+Bu_k+\sum_i u_i D_i z_k$5; zero-shot TurtleBot3 transfer after 40 Gazebo episodes navigates a curved narrow corridor without collisions
Two-link horizontal robot arm	Control-Coherent Koopman Model with exact $z_{k+1}=Az_k+Bu_k+\sum_i u_i D_i z_k$6 and MPC (Asada et al., 2024)	Mean tracking errors $z_{k+1}=Az_k+Bu_k+\sum_i u_i D_i z_k$7 cm versus DMDc $z_{k+1}=Az_k+Bu_k+\sum_i u_i D_i z_k$8 cm on three circular trajectories
Soft continuum manipulator under unknown payloads	Koopman-MPC with augmented load state and online observer (2002.01407)	Average RMSE $z_{k+1}=Az_k+Bu_k+\sum_i u_i D_i z_k$9 mm $\dot z=Az+\sum_i u_i B_i z$0 mm for KL-MPC versus $\dot z=Az+\sum_i u_i B_i z$1 mm $\dot z=Az+\sum_i u_i B_i z$2 mm for K-MPC and $\dot z=Az+\sum_i u_i B_i z$3 mm $\dot z=Az+\sum_i u_i B_i z$4 mm for L-MPC; payload estimate converges within $\dot z=Az+\sum_i u_i B_i z$5 s to within $\dot z=Az+\sum_i u_i B_i z$6 g; pick-and-place binning achieves 100% success in 2 trials
Virtually coupled train system	Analytical Koopman-based NMPC with frozen APV predictor and QP solve (Zhang et al., 10 Jun 2026)	Spacing deviation reduced from $\dot z=Az+\sum_i u_i B_i z$7 m to $\dot z=Az+\sum_i u_i B_i z$8 m; average solve time reduced by 40%–70% and worst-case time by up to 85%
Two-spool turbofan engine	Meta-heuristic EDMD with AKMPC + disturbance observer and Koopman feedback linearization (Grasev, 2 Apr 2026)	Single Koopman model predicts spool speeds and EPR with MAPE $\dot z=Az+\sum_i u_i B_i z$9; AKMPC is more robust than K-FBLC under varying flight conditions; EPR control improves thrust response
Double-well stochastic system	gEDMD-based Koopman-generator optimal control (Guo et al., 2024)	Under the optimal bias control the expected transition completes within $(K\phi)(x)=\phi(f(x,u))$0, achieving an order-of-magnitude acceleration of rare-event sampling

These examples show the breadth of the field rather than a single dominant recipe. Robotic locomotion, manipulation, navigation, soft robotics, rail systems, propulsion, and stochastic enhanced sampling are all represented. The common pattern is that the control problem is moved to a linear or bilinear lifted model, but the identification machinery and the synthesis layer vary substantially.

Other experimental programs emphasize basis design and data support. For cart- and VTOL-pendulum systems, increasing basis complexity improved swing-up and stabilization, and low-order Fourier bases were effective for rotational dynamics; on the Sphero SPRK robot, Koopman controllers outperformed a nominal linear controller on tarp and sand, but high-order monomials could become unstable when data coverage was insufficient (Abraham et al., 2017). Time-scale-structured Koopman models report RMS errors $(K\phi)(x)=\phi(f(x,u))$1 and $(K\phi)(x)=\phi(f(x,u))$2, a median running-cost reduction of $(K\phi)(x)=\phi(f(x,u))$3 when replacing a PI actuator loop by Koopman-LQR, and $(K\phi)(x)=\phi(f(x,u))$4 gains from optimizing the supervisory input in the combined hierarchical-and-separated setting (Bakker, 18 Jun 2025).

6. Limitations, misconceptions, and current directions

The first limitation is approximation error. Finite-dimensional lifts depend on the expressiveness of the chosen observables and the richness of the training data; this is stated explicitly for the soft manipulator, the Sphero experiments, and the certified SafEDMD frameworks (2002.01407, Abraham et al., 2017, Strässer et al., 2024). A related misconception is to treat Koopman linearization as an exact finite-dimensional transformation in generic settings. The literature is more careful: exactness is tied to invariant subspaces, special observable constructions, or analytic closed forms, whereas many practical controllers operate with residual terms, uncertainty bounds, or local validity regions.

The second limitation concerns the control input representation. Several papers argue that fitting $(K\phi)(x)=\phi(f(x,u))$5 by ordinary least squares can misinform the controller about the input structure in lifted space, and that one-step prediction error can therefore be a poor proxy for closed-loop quality (Asada et al., 2024). This has motivated actuator-aware constructions, auxiliary control networks for nonlinear input channels, and task-oriented training objectives that prioritize control cost over pure model fit (Shi et al., 2022, Lyu et al., 2023).

The third limitation is computational and algorithmic. End-to-end RL-based Koopman control may require careful tuning of auxiliary-loss weights and, like other off-policy loops, can be less data efficient than purely supervised model identification (Lyu et al., 2023). SOS- and PDE-based certified designs provide stronger guarantees but introduce semidefinite or elliptic-solver overhead (Strässer et al., 2024, Lin et al., 7 Apr 2026). Some feedback-linearization constructions also assume the existence of a continuously differentiable inverse mapping from Koopman coordinates back to state space (Narasingam et al., 2020).

Current directions widen the scope of the field. Event-triggered Koopman control reduces communication events while preserving exponential stability in Lyapunov sense (Manaa et al., 19 Apr 2025). Stochastic Koopman-generator control extends lifted modeling to control-affine SDEs and rare-event sampling (Guo et al., 2024). Structured Koopman models now incorporate hierarchical control and time-scale separation to quantify cross-scale interactions and derive fast- and slow-time-scale control policies (Bakker, 18 Jun 2025). High-dimensional sensory control has moved beyond low-dimensional states to pixels and lidar through contrastive encoders and latent LQR (Lyu et al., 2023).

An adjacent direction uses Koopman structure as an inductive bias inside generative control policies rather than as a classical lifted predictor. KoopmanFlow splits latent dynamics into slow and fast spectral branches using an RFFT-based prior, reports $(K\phi)(x)=\phi(f(x,u))$6-step inference latency of $(K\phi)(x)=\phi(f(x,u))$7 ms, and shows gains over state-of-the-art baselines in contact-rich manipulation tasks (Yao et al., 14 Mar 2026). This suggests that the term “Koopman-based control” is broadening: alongside EDMD, LQR, and MPC, the current literature also includes learned architectures that import spectral decoupling and linearizable latent structure into modern policy classes.

Taken together, the literature presents Koopman-based control as neither a single algorithm nor a purely spectral identification technique. It is a control framework in which nonlinear dynamics are lifted into observable coordinates, approximate linear or bilinear surrogates are identified with varying levels of structure and certification, and control is synthesized with tools ranging from LQR and MPC to CLF, SOS, policy iteration, and event-triggered design. The main technical questions are no longer only how to fit a Koopman model, but how to preserve the control channel, quantify residuals, guarantee stabilizability, and align the lifted representation with the actual closed-loop objective.