Koopman Control Family

• Koopman Control Family is defined by extending classical state observables to include constant control inputs, enabling a linear representation of inherently nonlinear systems.
• It provides an input–state separable lifted form that encapsulates linear, bilinear, and LPV models through observable-based lifting techniques.
• The framework supports data-driven identification and control design with applications in robotics, power electronics, epidemiology, and more.

The Koopman Control Family (KCF) unifies several generalizations of Koopman operator theory to handle controlled (input-driven) nonlinear dynamical systems. In contrast to the original autonomous Koopman setting, the control family formalism organizes the collection of all Koopman operators arising from different constant control signals and provides a rigorous functional-analytic foundation for modeling, identification, and control of nonlinear systems with actuation or exogenous inputs.

1. Concept and Definition

The Koopman Control Family is defined by augmenting the classical state-based observables to accommodate control inputs. Given a general discrete-time dynamical system

$x^+ = T(x, u),$

with $x$ the state and $u$ the input, each constant input $\bar{u}$ induces an associated autonomous system:

$x^+ = T_{\bar{u}}(x) := T(x, \bar{u}).$

To every such $T_{\bar{u}}$, one can associate a Koopman operator $K_{\bar{u}}$ acting on a function (observable) space $F$, typically by

$K_{\bar{u}} f = f \circ T_{\bar{u}}.$

The set $\{K_{\bar{u}} : \bar{u} \in U\}$ is called the Koopman Control Family. This construction allows the original input-driven, generally nonlinear evolution to be represented as a collection of linear (though infinite-dimensional) operators parameterized by constant inputs.

A function $g$ of both state and input, $g(x, u)$, becomes the basic observable, and the generalized Koopman operator with inputs acts as

$\mathcal{K} g(x, u) = g( T(x, u), * ),$

where $*$ is a secondary input argument with several natural choices, such as $u$ (propagating input alongside state for input-dynamical systems) or $0$ (resetting input for exogenous or open-loop disturbances) (Proctor et al., 2016). This formulation subsumes previous representations including Dynamic Mode Decomposition with Control (DMDc) as special cases.

2. Universal Input-State Separable Lifted Forms

A key theoretical result is that, when there exists a finite-dimensional subspace $L \subset F$ that is common-invariant under the KCF (i.e., $K_{\bar{u}}[L] \subseteq L$ for every $\bar{u}$), the system admits an input–state separable lifted representation:

$\Psi(T(x, u)) = A(u) \Psi(x)$

for a lifting (vector-valued observable) $\Psi: X \to \mathbb{C}^s$ and a matrix-valued map $A: U \to \mathbb{C}^{s \times s}$ (Haseli et al., 2023). In this form, $A(u)$ encodes the effect of the input on the evolution in the lifted space.

This universal representation encapsulates linear, bilinear, and switched-linear models as special cases. For instance:

• Lifted linear systems: $A(u)$ is linear in $u$;
• Bilinear systems: $A(u) = A_0 + \sum_{i} u_i A_i$;
• Linear parameter-varying (LPV) systems: $A(u)$ state- and input-dependent, possibly nonlinear in $u$.

When a finite-dimensional common invariant subspace does not exist, the closeness of an approximate subspace $S$ can be quantified by the invariance proximity,

$$I(S) = \sup_{f \in S, \|f\| \neq 0} \frac{\|f - P_S f\|}{\|f\|},$$

where $P_S$ is the orthogonal projection onto $S$ under the action of the family.

3. Functional and Spectral Theory: Set-Valued and Bilinear Extensions

For controlled systems, the input creates a non-uniqueness in flow; to fully encode this non-determinism, a set-valued extension of the Koopman operator is proposed:

$$\mathcal{K}_{\tau, t}(\phi) = \{ \phi \circ \Phi_{\tau, t}^u : u(\cdot) \in \mathcal{U} \}$$

with $\mathcal{U}$ the family of admissible input functions (Bonnet-Weill et al., 21 Jan 2024). This operator bundles all possible control-induced evolutions and connects with set-valued generalizations of the Liouville and Perron–Frobenius operators. The spectral mapping theorem adapts in this context via set-valued eigenvalue relations:

$$e^{(t-\tau)\lambda} \in \sigma_p(\mathcal{K}_{\tau, t})$$

for $\lambda$ in the spectrum of the (set-valued) generator $L$.

In the continuous-time control-affine case, the generator’s dynamics induce a bilinear operator evolution on the space of observables:

$$\frac{d}{dt} U_t = U_t \mathcal{L}_f + \sum_{i=1}^m u_i(t) U_t \mathcal{L}_{g_i}$$

where $\mathcal{L}_f$ and $\mathcal{L}_{g_i}$ are Lie derivatives with respect to the drift and control vector fields, respectively (Zhang et al., 2022). This global bilinearization enables powerful geometric approaches for controllability (e.g., via de Rham differentials and infinite-dimensional Lie algebra rank conditions).

4. Data-Driven and Algorithmic Implementations

KCF provides a natural setting for data-driven system identification, most notably via extensions of Dynamic Mode Decomposition such as DMDc (Proctor et al., 2016) and the Extended Dynamic Mode Decomposition (EDMD). For empirical modeling:

• Measurements are organized as snapshot pairs $(x_k, u_k), (x_{k+1}, u_{k+1})$;
• Observable vectors are evaluated at these data points;
• Koopman matrices $A(u)$ are estimated by least squares or regression, possibly augmented with sparsity-promoting approaches to select relevant nonlinear or mixed observables (e.g., $x^2$, $x u$, etc.).

When modeling actuated (e.g., epidemiological) systems, mixed observables such as the susceptible–infected coupling $SI$ in SIR models with vaccination as input allow for inclusion of nonlinear input–output characteristics and approximate predictive power in the lifted space.

For general nonlinear input–output systems, the Koopman LPV form is analytically derived as:

$\dot{\Phi}(x) = A \Phi(x) + B(x, u) u$

where $B(x, u)$ is generally state and input dependent (Iacob et al., 2022). Discrete-time analogues further clarify these dependencies and show that commonly used LTI approximations can yield significant and quantifiable errors except in special cases.

5. Connections to Control Design and Model Structures

In the KCF framework, controllers can be designed in the lifted space using classical linear or bilinear system techniques (LQR, MPC, SDRE), though care must be taken as the input matrix in the lifted space is typically state- or input-dependent. The MPC application in power electronics (Hanke et al., 2018) demonstrates practical performance: Koopman-based reduced-order surrogate models offer real-time implementation advantages and, with proper input mapping and regular updates, can closely match the performance of white-box models even as system complexity grows. For model structure selection and controller design, KCF provides a theoretical and practical rationale for when to enforce (or relax) time-invariance or “affinity” in the input action.

A key notion is dynamical consistency: the lifted model must exactly track the original system’s evolution. This leads to constraints on the choice of observables and the form of the lifted operator, especially for hybrid joint state–control observable architectures (Bakker et al., 2019).

6. Advanced Generalizations and Modern Directions

Recent efforts systematically extend KCF beyond control-affine contexts:

• Set-valued KCF enables a mathematically rigorous generalization to include all possible control policies and connections to operator-valued measures and adjoint theory (Bonnet-Weill et al., 21 Jan 2024).
• Neural and nonparametric approaches unify KCF with modern RKHS methods and neural network parameterizations, enabling invariant subspace discovery, parametric families of Koopman operators, and high-dimensional/nonlinear system modeling without fixed finite dictionaries (Bevanda et al., 12 May 2024, Guo et al., 2023).
• Multi-agent, multi-scale KCF extends the theory to coupled agent systems, hierarchical time-scale separation, and the game-theoretic analysis of distributed Nash equilibria versus social optimum (Bakker, 18 Jun 2025).
• Factorized Koopman representations introduce convexity in closed-loop controller design for control-affine systems using semi-definite programming, leveraging the bilinear structure of the closed-loop Koopman operator (Ondogan et al., 6 Oct 2025).
• Equivalence theory demonstrates that the infinite-input-sequence Koopman operator and the operator family approaches are functionally equivalent in terms of trajectory prediction and functional representation, for appropriately chosen function spaces (Haseli et al., 16 Oct 2025).

7. Applications and Impact

KCF enables robust modeling and data-driven control of complex nonlinear systems in a variety of domains:

• Epidemiological models with interventions such as vaccination schemes, where SIR with input observables elucidate nonlinear input–state coupling (Proctor et al., 2016).
• Engineering systems including electrical drives, power electronics, and robotics (e.g., robotic arms and multi-cable systems) where model predictive control benefits from physically accurate, control-coherent input matrices and real-time adaptation (Hanke et al., 2018, Asada et al., 24 Mar 2024).
• Adaptive control and observer design, leveraging recursive updates and forgetting factors to accommodate for time-variation and unpredictable system changes (Junker et al., 2022).

The KCF offers a rigorous foundation for both theoretical analyses and practical algorithms, bridges model-based and data-driven disciplines, and supports a wide variety of identification, estimation, and control architectures in nonlinear dynamical systems. As the framework matures, it continues to incorporate more sophisticated perspectives from spectral theory, infinite-dimensional geometry, operator-theoretic game theory, and modern machine learning, providing a comprehensive paradigm for the paper and control of complex systems with actuation and exogenous effects.