Koopman-Based Control

• Koopman-based control is a data-driven framework that lifts nonlinear dynamics into a linear observable space using a finite set of basis functions.
• The approach integrates nonlinear system identification with traditional linear control methods, facilitating robust control of systems like robots and electrical drives.
• Empirical studies demonstrate that careful basis function selection improves model fidelity and controller performance, balancing computational cost with accuracy.

Koopman-based control is a data-driven framework for synthesizing controllers for nonlinear dynamical systems by exploiting the linearity of the Koopman operator in function space. Rather than operating directly in physical state space, the system’s nonlinear evolution is recast as a (possibly infinite-dimensional) linear or bilinear system in the space of observables—functions of the state. When approximated by a finite but sufficiently rich dictionary of basis functions, this yields a linearizable model suitable for model-based control synthesis. The approach effectively bridges nonlinear system identification and traditional linear control design, offering a unified and scalable way to control complex systems—including robotic platforms, electrical drives, and systems with unmodeled or hard-to-characterize nonlinearities—using measured data in lieu of hand-derived first-principles models (Abraham et al., 2017).

1. Koopman Operator Theory and Lifting

The Koopman operator 𝒦 is a linear operator—potentially infinite-dimensional—governing the evolution of scalar observables of a nonlinear system. For a discrete-time nonlinear system

$x_{k+1} = F(x_k),$

and any observable $g : \mathbb{R}^n \to \mathbb{R}$, the Koopman operator acts as

$[\mathcal{K}g](x) = g(F(x)).$

Crucially, while the map $F$ may be nonlinear in state space, $\mathcal{K}$ is linear when acting on observables—elements of a suitable function space such as $L^2$. This linearity forms the basis for model-based control since Koopman’s approach allows the nonlinear dynamics to be described by a (potentially infinite) system of linear equations.

In practical applications, a finite set of basis functions $\Psi(x) = [\psi_1(x), ..., \psi_N(x)]$ is chosen, and the evolution is approximated as

$\Psi(x_{k+1}) \approx \Psi(x_k) K,$

where $K \in \mathbb{R}^{N \times N}$ is estimated from measured trajectories using, e.g., least-squares minimization over data. When designing controllers, the control input $u$ is incorporated by augmenting the observables to $\Psi(x, u)$, leading to a lifted dynamics model

$x_{k+1} \approx \hat{K}^T \Psi(x_k, u_k)^T.$

Linearization in the lifted space is achieved using the chain rule, enabling transformation into a suitable form for linear control methods such as sequential action control or model predictive control (Abraham et al., 2017).

2. Model Synthesis and Basis Function Choice

Implementation of Koopman-based control heavily depends on the selection of basis functions used to approximate the infinite-dimensional operator. The chosen dictionary must be rich enough to capture key nonlinearities but not so complex as to induce excessive computational overhead or overfitting to the sampled trajectories. The paper examines polynomial bases (monomials up to order $Q$) and (for systems with rotational or periodic states) Fourier bases exploiting problem structure. For instance:

• In the cart-pendulum system, a Fourier basis exploits the natural periodicity of angular coordinates, yielding effective control performance even at low order ($Q=1$).
• Polynomial bases require raising the order $Q$ to increase model fidelity, at the cost of increased data requirements and numerical ill-conditioning.

An essential part of the procedure is tuning $Q$ or selecting function families so that the Koopman approximation produces a model whose error is sufficiently small for effective controller synthesis.

3. Control Design Integration

The data-driven, lifted, linear(izable) model output by the Koopman operator approximation becomes the basis for control synthesis:

• The finite-dimensional Koopman model

$x_{k+1} \approx A(x_k, u_k)x_k + B(x_k, u_k)u_k$

(with $A$ and $B$ determined via the chosen basis and chain rule) is used in classical model-based control strategies requiring a (possibly state- or input-varying) linearization.

• This enables the deployment of optimization-based controllers such as sequential action control or model predictive control in nonlinear, poorly modeled, or data-rich domains.

For open-loop control, the trajectory optimization cost is formulated as

$$J = \sum_k \left[ \frac{1}{2}(x_k - \tilde{x}_k)^T P (x_k - \tilde{x}_k) + \frac{1}{2}u_k^T R u_k \right]$$

where $\tilde{x}_k$ is a reference trajectory.

4. Empirical Validation on Nonlinear Robotic Systems

Simulation studies on a cart-pendulum and VTOL-pendulum system show that Koopman-based identification and model-based control accurately stabilize challenging nonlinear systems—even those with configuration spaces on nonlinear manifolds, such as SO(n). The basis function choice is central: for the pendulum, a Fourier basis leverages system periodicity for immediate inversion and stabilization, while with polynomial bases, control performance improves gradually with the order. For a VTOL-pendulum, the Koopman operator is used to model the coupling between nominal (known) and residual (unknown) dynamics, augmenting model-based control with learned components.

On robotic hardware, the framework is demonstrated with the Sphero SPRK robot. Experimental results show:

• In open-loop, increasing basis function complexity (Q) reduces drift and improves sinusoidal reference tracking.
• In closed-loop, richer basis sets yield better capture of the robot’s nonlinear internal dynamics, thus improving controller effectiveness on various terrains, including sand. Enhanced robustness is observed under challenging ground conditions, especially with higher-order bases.

5. Limitations and Trade-offs

Several key limitations and trade-offs are highlighted:

• Computational Cost vs. Model Richness: Choosing high-dimensional or high-order function sets increases computational burden and risks overfitting, particularly with sparse data in complex regimes.
• Data Requirements: Model accuracy improves with the basis complexity, but this necessitates dense coverage of state space in the collected data to avoid extrapolation artifacts.
• Physical Interpretability: The approximation’s utility is contingent upon the physical relevance of basis functions. Periodic bases are superior for manifolds such as SO(n), but generic choices may require significant tuning.
• Practical Modeling: For robotic systems or systems with significant unmodeled or environment-induced nonlinearities, including terrain or load variations, accuracy and adaptability are governed by the richness and relevance of the basis set.

6. Mathematical Summary

The workflow involves the following key equations, as presented in the original work:

• Koopman operator definition:

$[\mathcal{K}g](x) = g(F(x)).$

• Finite-dimensional lifting (least-squares estimate of $K$):

$\Psi(x_{k+1}) = \Psi(x_k) K + r(x_k);$

$$K = G^\dagger A, \ \ \text{with} \ G = \frac{1}{P}\sum_p \Psi(x_p)^T \Psi(x_p), \ \ A = \frac{1}{P}\sum_p \Psi(x_p)^T \Psi(x_{p+1}).$$

• Augmented observable with control input:

$$\Psi(x, u) = [x^T, u^T, \psi_1(x,u), ..., \psi_N(x,u)], \quad x_{k+1} \approx \hat{K}^T \Psi(x_k, u_k)^T,$$

and (via linearization)

$x_{k+1} \approx A(x_k, u_k)x_k + B(x_k, u_k)u_k.$

7. Practical Implications and Outlook

Koopman-based control facilitates data-driven adaptation, direct use of linear control strategies on nonlinear systems, and real-world applicability in scenarios with hard-to-model dynamics or environmental variations (Abraham et al., 2017). The methodology is particularly advantageous for robotic and mechatronic systems where physical modeling is intractable or incomplete, and for applications requiring adaptation to external disturbances (e.g., movement on sand).

The dominant challenge remains balancing model expressivity against computational tractability and data availability. The methodology demonstrates that careful basis selection and complexity tuning significantly impact performance. As shown empirically, increased basis richness enhances tracking and stabilization, but also imposes higher demands on data coverage and computation.

In summary, Koopman-based control offers a unified framework for leveraging data to control nonlinear systems by constructing linearizable models in lifted spaces. With appropriate basis selection and data, this strategy enables model-based control synthesis—such as optimal control, tracking, and stabilization—across a range of real-world nonlinear systems.