Bilinear EDMD Control Schemes

• Bilinear EDMD-based control schemes are data-driven methods that lift control-affine nonlinear systems into a higher-dimensional space using a bilinear structure.
• They leverage finite-dimensional approximations of the Koopman operator to construct surrogate models that facilitate robust controller synthesis with certified error bounds.
• Applications span robotics, quantum control, and large-scale networks, addressing scalability challenges while ensuring high-performance control under model uncertainties.

Bilinear Extended Dynamic Mode Decomposition (EDMD)-based control schemes are data-driven methods for representing and controlling nonlinear, control-affine dynamical systems by lifting their evolution into a higher-dimensional space of observables where the influence of the control input is modeled in a bilinear form. This class of techniques leverages finite-dimensional approximations of the infinitesimal Koopman operator to create surrogate models that robustly capture state-input interactions, address the limitations of purely linear Koopman formulations, and facilitate tractable, high-performance control algorithms—often with certified closed-loop guarantees—even in the presence of model uncertainty and finite-data effects.

1. Mathematical Foundations of Bilinear Koopman Realizations

The framework is built upon the Koopman operator, a linear (infinite-dimensional) operator acting on observables of the state. For a control-affine nonlinear system

$$\dot{x}(t) = f_0(x(t)) + \sum_{j=1}^{m} f_j(x(t)) u_j(t)$$

the classical Extended Dynamic Mode Decomposition (EDMD) lifts $x$ to a vector of observables $z = \Psi(x)$, leading to an evolution

$$\Psi(x_{k+1}) \approx A \Psi(x_k) + B_0 u_k + \sum_{j=1}^m u_{j,k} B_j \Psi(x_k)$$

where $A, B_0, B_j$ are matrices estimated from data, with the bilinear terms $u_j B_j \Psi(x_k)$ encoding the interaction between lifted state and input.

This bilinear structure arises naturally when finite-dimensional projections of the control-affine Koopman generator onto an observable space are formed. Finite approximations rarely admit a purely linear structure with respect to the control, but, provided the observables are sufficiently rich, they always admit a bilinear realization for control-affine systems (Bruder et al., 2020, Schaller et al., 2022, Strässer et al., 2 Sep 2025).

A necessary and sufficient condition for a set of observables $\{z_i\}$ to admit a bilinear Koopman realization is

$$\frac{\partial z_i}{\partial x} \cdot F(x,u) \in \mathrm{span} \big( \{z_j\} \cup \{u_j\} \cup \{z_k u_j\} \big)$$

for all $i$ (Bruder et al., 2020). This allows the time evolution of each observable to be written as a combination of lifted state, control, and their products, yielding the core bilinear form.

2. System Identification and Data-Driven Model Construction

Bilinear EDMD-based models are typically identified via regression using snapshot pairs (state and input evolution data). For each input direction, data is collected with the control held at either zero or a canonical value, and multiple autonomous Koopman operators or generators are estimated (Schaller et al., 2022, Chatzikiriakos et al., 26 Sep 2024, Strässer et al., 2 Sep 2025):

• For each autonomous subsystem (i.e., at $u=0$ and $u=e_j$), the Koopman generator/operator $L^0$ and $L^{e_j}$ is estimated.
• The generator for arbitrary $u$ is constructed as a bilinear combination:

$L^u = L^0 + \sum_{j=1}^{m} u_j (L^{e_j} - L^0)$

ensuring the bilinear dependence is retained while maintaining model parsimony and mitigating the curse of dimensionality (Schaller et al., 2022, Guo et al., 22 Aug 2024).

This approach preserves the low input complexity of the original system and allows the construction of a global data-driven surrogate without state-control dimensional augmentation.

3. Error Bounds, Robustness, and Closed-Loop Guarantees

Finite data and projection onto finitely many observables introduce a structured model error, typically expressed as a residual $r(x,u)$: $$\Psi(x_{k+1}) = A \Psi(x_k) + B_0 u_k + \sum_{j=1}^m u_{j,k} B_j \Psi(x_k) + r(x_k, u_k)$$ Recent advances provide proportional, uniform error bounds of the form (Schaller et al., 2022, Strässer et al., 6 Nov 2024, Strässer et al., 15 Nov 2024, Strässer et al., 2 Sep 2025): $$\| r(x,u) \|_2 \leq c_x \|\Psi(x)\|_2 + c_u \|u\|_2$$ where $c_x, c_u$ are functions of the data richness, regularity of the dictionary, dynamics, and number of samples $m=O(M^2/(\epsilon^2\delta))$ for accuracy $\epsilon$ and confidence $1-\delta$.

Controllers are designed to account for these errors, embedding them directly into robust control designs (such as sum-of-squares (SOS) or LMI-based optimization). Feasibility of the resulting convex program establishes rigorous closed-loop certificates (e.g., exponential stability within a certified region of attraction) given explicit $c_x, c_u$ bounds (Strässer et al., 6 Nov 2024, Strässer et al., 15 Nov 2024). In min-max MPC formulations, a rational state-feedback (SOS-parametrized) law achieves robust constraint satisfaction and cost bounds (Xie et al., 7 Apr 2025).

4. Controller Synthesis: Algorithmic Strategies

Controller design is performed on the bilinear surrogate with explicit error bounds embedded. Two principal strategies appear:

• Convex/robust control synthesis: LMI-based (or SOS-based) programs are constructed to robustly stabilize

$$\Psi(x_{k+1}) = A\Psi(x_k) + B_0 u_k + \sum_j u_{j,k} B_j \Psi(x_k) + r(x_k, u_k)$$

with $r(x,u)$ bounded as above. The classic synthesis yields controllers of the form (Strässer et al., 6 Nov 2024, Strässer et al., 15 Nov 2024):

$$u(x) = [I - L_w (\Lambda^{-1} \otimes \Psi(x))]^{-1} L P^{-1} \Psi(x)$$

or, in rational-SOS form,

$$u(x) = \frac{L_n(\Psi(x)) P^{-1} \Psi(x)}{u_d(\Psi(x))}$$

with $L_n$ and $u_d$ SOS polynomials ensuring global or semi-global robust stability.

• Predictive control (MPC/DeePC): Bilinear Koopman surrogates enable efficient implementation of predictive controllers. In model predictive control, the surrogate model replaces the true dynamics in the horizon updates, allowing for convex (when locally linearized) or bilinear constrained optimization (Bruder et al., 2020, Xiong et al., 6 May 2025). In DeePC frameworks, the bilinear structure is exploited directly in the data, removing explicit model identification and using Hankel data and bilinear consistency constraints (Xiong et al., 6 May 2025).

When properly accounting for $r(x,u)$, such controllers guarantee constraint satisfaction and closed-loop stability within robustness margins set by the data-derived error bound (Strässer et al., 6 Nov 2024, Xie et al., 7 Apr 2025).

5. Scalability, Curse of Dimensionality, and Modularization

Bilinear models circumvent the bulk of the curse of dimensionality: unlike state-augmented EDMDc approaches, the model's dimension depends linearly on the number of state variables and affine in the number of control channels (Schaller et al., 2022). Modularized schemes for interconnected systems further decompose the problem: local Koopman generators for each subsystem are learned individually on lower-dimensional state spaces, then globally coupled through known network topology (Guo et al., 22 Aug 2024). This supports transfer learning, network adaptation, and error propagation analysis via graph-theoretic tools.

Empirical illustrations confirm that modularized EDMD-based surrogates require $\mathcal{O}(s \cdot e^{n_0})$ data (with $s$ subsystems each of dimension $n_0$) versus $\mathcal{O}(e^{s n_0})$ for a monolithic EDMD approach (Guo et al., 22 Aug 2024).

6. Applications and Limitations

Bilinear EDMD-based schemes have been applied to:

• Robotics and mechatronic systems: Real-time trajectory-following and MPC of robots; high-accuracy predictive control with computational efficiency (Bruder et al., 2020, Junker et al., 2021, Rosenfelder et al., 11 Nov 2024).
• Quantum control: Optimal control pulse synthesis and system identification leveraging known Hamiltonian structure (Goldschmidt et al., 2020).
• Modular large-scale networks: Coupled oscillators and multi-component physical systems, where scalability and transfer learning are critical (Guo et al., 22 Aug 2024).
• Data-driven inverse optimal control: Estimation of cost and dynamics via bilinear Koopman representations, enabling a tractable extension of the classical inverse LQR solution to unknown nonlinear systems (Fernandez-Ayala et al., 30 Jan 2025).

However, precise dictionary selection remains crucial; improper choice reduces model accuracy or increases conservatism in robust control (Strässer et al., 2 Sep 2025). For systems with geometric structure (e.g., nonholonomic robots), domain-specific knowledge must inform the selection of observables and cost functions to achieve high-performance closed-loop behavior (Rosenfelder et al., 11 Nov 2024). The data requirements implied by theory may be conservative. Scaling computational tools (especially SOS or LMI-based synthesis) to high-dimensional lifted spaces remains an area for future progress.

7. Theoretical Implications and Future Directions

Bilinear Koopman realizations provide a systematic, universally applicable surrogate for control-affine systems, bridging data-driven system identification and robust controller synthesis. This structurally enforces that, as dictionary richness and data volume increase, model fidelity and control performance improve monotonically (in contrast to the stagnation of purely linear lifts) (Bruder et al., 2020, Schaller et al., 2022, Strässer et al., 2 Sep 2025).

Future directions include:

• Automated/learned observable selection—potentially using neural networks or adaptive dictionaries—to optimize tradeoffs between expressivity and computational tractability.
• Extension to output-only (partial observation) settings and the development of observer architectures with guaranteed error bounds.
• Integrated data-driven robust performance synthesis with joint consideration of state/input constraints, performance indices, and model uncertainty.
• Scalable algorithms for high-dimensional, real-time applications—especially regarding the efficient solution of large-scale SOS or LMI programs.
• Hybrid pipelines combining deep learning with operator-theoretic error quantification and geometric insights for systems with inherent structure.

Bilinear EDMD-based control thus provides an algorithmically flexible, theoretically grounded platform for robust, data-driven nonlinear control with quantifiable performance under realistic data and computational constraints.