An overview of Koopman-based control: From error bounds to closed-loop guarantees

• Accurate first-principles models are often difficult to obtain for high-dimensional, uncertain, or complex systems, motivating control methods learned from observed trajectories.Source: §1 · p. 1
• Koopman methods lift nonlinear dynamics into a higher-dimensional observable space with linear evolution, while EDMD provides a tractable data-driven approximation.Source: §1 · p. 1
• Despite extensive Koopman theory and applications, a unified rigorous account linking approximation-error bounds to closed-loop stability and performance guarantees remains needed.Source: §1 · p. 2
• The overview reviews Koopman foundations and error analysis, then examines linear and bilinear EDMD control, feedback and MPC, and robust designs for closed-loop guarantees.It also identifies open issues and future research challenges.Source: §1 · p. 2

2.1. Koopman operator for dynamical systems

• Nonlinear state evolution induces linear evolution of observables through a Koopman semigroup; its generator is generally unbounded and can be evaluated from a model or estimated from state-derivative data.The generator and semigroup play roles analogous to a matrix and its exponential for linear time-invariant systems.Source: §2.1 · pp. 2–3

2.2. Eigenfunctions

• Koopman eigenfunctions can yield exact finite-dimensional representations on invariant subspaces, supporting stability analysis and long-term prediction, but the operator’s infinite-dimensional spectrum complicates analysis and computation.The spectrum may include continuous components associated with irreducible nonlinear dynamics such as chaos.Source: §2.2 · p. 4

2.3. Extended dynamic mode decomposition

• EDMD fits a finite-dictionary linear surrogate from successor data, estimating the Galerkin projection of the Koopman operator; spectral guarantees and numerical stability motivate specialized variants.Related approaches include ResDMD and rigged DMD for spectral approximation, and truncated-SVD pseudoinverses or sparsity-promoting regularization to improve regression stability.Source: §2.3 · p. 5

2.4. Finite-data error bounds

• Finite-data EDMD error reflects both dictionary projection and estimation; one bound requires d = O(M^2/(δε^2)), while existing full-error bounds are generally least-squares L2 bounds.The stated sample requirement also depends linearly on δ^-1; the ergodic-sampling result relies on exponential stability of the Koopman semigroup.Source: §2.4 · pp. 5–6

2.5. Kernel EDMD

• Kernel EDMD chooses data-informed kernel features; reproducing-property evaluation supports uniform L∞ estimation bounds, and Wendland kernels preserve Sobolev regularity under a C^m flow map with m > n/2.This invariance supports uniform L∞ approximation-error bounds for deterministic systems.Source: §2.5 · p. 6
• Kernel EDMD uses approximation-theoretic bounds based on fill distance to estimate error for the infinite-dimensional Koopman operator and arbitrary observables.Complexity-reduction options include random Fourier features and Nyström sketching; trajectory-sampling analyses also give error bounds under mixing conditions that ensure ergodicity.Source: §2.5 · p. 7

2.6. Delay embeddings, reprojections, and PDEs

• Because data typically provides only output observables, delay coordinates can address dictionary restrictions and support Koopman analysis when invariant subspaces or EDMD applicability are difficult to obtain.Delay-based methods include Hankel views, latent EDMD, and deep delay autoencoders; reported applications include flow prediction, with initial extensions toward PDEs and control.Source: §2.6 · p. 7
• EDMD predictions can violate relations preserved by the true Koopman operator, so coordinate reprojection or weighted reprojection can restore consistency on the lifted-state manifold.Coordinate functions may be included in the dictionary for reprojection; differential-geometric weighted methods can instead use detectability to avoid including essentially the whole state.Source: §2.6 · pp. 7–8, Figure 4 · p. 7
• Finite-data error bounds for Koopman-based PDE methods remain an open problem, despite work on PDE spectral approximation, control, parabolic problems, and symmetry-based transfer learning.Such bounds are identified as crucial groundwork for data-driven control of distributed parameter systems.Source: §2.6 · p. 8

3.1. The linear EDMDc scheme

• EDMDc fits lifted dynamics and input influence from controlled-system snapshots to approximate the action of the controlled Koopman operator.The method extends EDMD by solving a least-squares problem for the lifted dynamics and input influence simultaneously.Source: §3.1 · p. 8
• A nonlinear controlled system generally cannot be accurately represented by a finite-dimensional linear Koopman model; nonlinear lifts can make the input term state-dependent.A linear representation is possible when the lift is linear in each input-actuated component; accurate representations need at least a bilinear model.Source: §3.1 · p. 9
• Linear EDMDc has practical applications with LQR or MPC, but its surrogate offers no closed-loop guarantees for the nonlinear system and may fail in practice.The lifted linear model enables prediction and controller design methods from linear systems theory.Source: §3.1 · p. 9

3.2. Extensions of linear EDMDc

• Eigenfunction-based lifting can bring the observable space closer to invariance and reduce approximation error, but lacks complete error bounds for robust closed-loop guarantees.The cited extensions include learning autonomous eigenfunctions before regressing the input matrix and defining eigenfunctions on a concatenated state-input space.Source: §3.2 · p. 9
• Control-coherent modeling preserves the actuated states’ linear input dependence, while finite-dimensional truncation leaves error in unactuated states; feedback linearization requires derivative measurements and an invariant finite dictionary.Source: §3.2 · p. 10
• IQC-based residual-error characterization supports frequency-domain controller design with stability and performance guarantees, conditional on capturing the true residual error.For unstable systems, the approach requires a given pre-stabilizing controller; the true-error-capture assumption generally cannot be validated.Source: §3.2 · p. 10

4.1. Bilinear EDMDc

• Finite-dimensional bilinear surrogates still incur residual error, so controller design and stability analysis must explicitly account for it to establish closed-loop guarantees.Plain bilinear EDMDc is described as more accurate than linear EDMDc, but without guaranteed error bounds.Source: §4 · p. 10, §4.1 · p. 11

4.2. SafEDMD: Structured bilinear surrogate models with error bounds

• SafEDMD combines autonomous Koopman estimates into a structured bilinear model and bounds finite-data learning error, but projection error must also be bounded for an overall guarantee.Its learning-error bound is probabilistic; the projection-error bound holds trivially under dictionary invariance, a property the passage says dynamical systems hardly satisfy.Source: §4.2 · p. 12

4.3. Kernel-based EDMD with error bounds

• Kernel-based EDMD constructs a bilinear control-affine predictor with a proportional bound on the full residual, covering learning and projection error.The constants approach zero as the sampling interval tends to zero and the data size increases, with the interval shrinking sufficiently fast; the approach also requires dense data including the origin and sufficiently exciting inputs.Source: §4.3 · pp. 12–13

4.4. Discussion

• Both SafEDMD and kernel-based EDMD yield the same bilinear structure, enabling common controller design, but their bounds and lifting choices have different trade-offs.SafEDMD permits user-chosen liftings but lacks a rigorous projection-error bound; kernel-based EDMD gives a deterministic full-residual bound but its data-dependent lifting dimension scales with data size. The section identifies noisy output measurements in rigorous error bounds as an open challenge.Source: §4.4 · pp. 13–14
• Proportional residual bounds that vanish at the origin let robust controllers incorporate surrogate error and provide closed-loop guarantees for the underlying nonlinear system.The section presents bilinear EDMD with control as a link between data-driven modeling and robust control.Source: §4.4 · p. 14

5. Controller design based on Koopman models with closed-loop guarantees

• Feedback design and MPC use bilinear Koopman surrogates with error bounds to seek rigorous stability and performance guarantees for the underlying nonlinear system.The section focuses on robust control of uncertain bilinear models derived from data.Source: §5 · pp. 14–15

5.1. Feedback design

• Linear EDMDc lacks closed-loop guarantees for the nonlinear system, while earlier bilinear feedback work did not explicitly account for finite-data and finite-dictionary residual error.A later robust design over-approximates bilinearity within a user-chosen region to represent the model as an LFR.Source: §5.1 · p. 15
• LMI-based designs couple proportional residual bounds to Koopman learning-error bounds and yield nonlinear controllers with exponential stability guarantees on a Lyapunov sublevel set.Continuous-time designs require state-derivative measurements; a discrete-time surrogate offers a derivative-free alternative, and both LMI feasibility problems scale as O(M^6).Source: §5.1 · p. 15
• SOS optimization avoids over-approximating the bilinear term and can enlarge the feasible region of attraction, but its error-bound guarantees are limited to the sampling region.The discrete-time controller is exponentially stabilizing on a maximized Lyapunov sublevel set; the continuous-time design also uses a polynomial control law and a sublevel set.Source: §5.1 · pp. 15–16
• SOS-based feedback is extended to closed-loop quadratic performance specifications, including an L2-gain bound, while extensions to uncertainties such as noise remain prospective.The approach is framed using robust-control techniques.Source: §5.1 · p. 16
• Tighter learned IQC error bounds may improve closed-loop performance, but they are typically valid only in the infinite-data limit and may not preserve guarantees for the nonlinear system.The survey identifies reducing controller-design conservatism without losing rigorous guarantees as an open problem.Source: §5.1 · p. 16

5.2. Model predictive control

• Some bilinear Koopman MPC approaches neglect approximation errors and validate closed-loop behavior only numerically; investigating MPC with bilinear kernel-based surrogates remains future work.The survey describes the combination of a proportional error bound and robust controller design as a route to guarantees.Source: §5.2 · pp. 17–18
• Data-driven MPC schemes using SafEDMD or kernel-based predictors establish asymptotic stability, while a Koopman-based min–max robust MPC scheme guarantees stability and feasibility via LMI constraints.The kernel-based predictor has a proportional error bound and uses a uniform Lipschitz bound to establish cost controllability.Source: §5.2 · p. 18

Conclusions

• Koopman-based control has promising results, but the field remains early and faces fundamental challenges to robust and scalable application.The survey emphasizes methods that provide closed-loop guarantees.Source: §6 · p. 18

Future research challenges

• Open challenges include extending guarantees to input-output data, reducing conservative finite-data bounds, choosing control-relevant dictionaries, and scaling methods beyond restrictive assumptions such as noise-free data.The dictionary must balance expressivity and tractability, while preserving features needed for closed-loop guarantees.Source: §6 · p. 18

Conclusions and future research challenges

• Practical progress calls for principled input-output learning, dictionary design, scalable synthesis with reduced conservatism, and robust guarantees for nonlinear systems in changing environments.The survey presents these directions as important to broader use in complex real-world applications.Source: §6 · p. 18