Bilinear EDMD with Control

• Bilinear EDMD with Control is a data-driven operator-theoretic method that lifts nonlinear control-affine systems into a higher-dimensional observable space to form bilinear surrogate models.
• It employs regression on snapshot data to identify matrices and bilinear terms, yielding quantitative error bounds that support both offline controller design and online predictive control.
• The approach scales to complex domains—such as quantum, robotics, and cyber-physical systems—by leveraging modularization and robust optimization techniques to ensure system stability and performance.

Bilinear EDMD with Control refers to the class of data-driven operator-theoretic methods that approximate controlled nonlinear dynamical systems by bilinear models in a lifted (observable) space, using the extended dynamic mode decomposition (EDMD) framework or its controlled derivatives. This methodology, grounded in the Koopman operator theory, allows nonlinear control-affine systems to be represented as bilinear systems, facilitating control synthesis, analysis, and guarantees through linear-algebraic and robust optimization techniques. The resultant surrogate models support both offline controller design and online predictive control in high-dimensional, potentially unknown, and data-rich systems.

1. Theoretical Foundations and Bilinear Koopman Realizations

Bilinear EDMD with control operates by lifting the original state $x \in \mathbb{R}^n$ into a space of observables $\Psi(x) \in \mathbb{R}^M$, where the nonlinear state evolution and its control influence can be approximated by a surrogate model of the form

$$\Psi(x_{k+1}) = A \Psi(x_k) + B_0 u_k + \sum_{i=1}^m u_k^i B_i \Psi(x_k) + r(x_k, u_k)$$

where $A$, $B_0$, and $B_i$ are matrices obtained from data, $u_k$ is the control input, and $r(x_k, u_k)$ is a residual capturing truncation and estimation errors (Strässer et al., 2 Sep 2025). The bilinear term arises naturally when lifting control-affine dynamics: $\dot{x}(t) = f(x(t)) + G(x(t)) u(t)$ to observables $\Psi(x)$, as the chain rule $$\frac{d}{dt}\Psi(x) = \nabla\Psi(x)f(x) + \nabla\Psi(x)G(x)u$$ generically induces a multiplicative dependence on both state and input. Thus, while linear representations are insufficient except in trivial cases, bilinear models capture the leading-order structure for generalized Koopman operator approaches (Bruder et al., 2020).

For control-affine systems, there always exists an (infinite-dimensional) bilinear representation in the observable space, but an exact linear realization occurs only under restrictive conditions (Bruder et al., 2020). Consequently, approximate bilinear surrogates constructed from generic bases systematically improve as the dictionary size grows (Bruder et al., 2020).

2. System Identification, Lifting Design, and Approximation Guarantees

Bilinear EDMD is implemented via regression on a data set of snapshots $\{(x_k, u_k, x_{k+1})\}$ by selecting a dictionary $\Psi$, forming data matrices in the lifted space, and least-squares fitting $A$, $B_0$, $B_i$: $$\min_{A,B_0,B_i} \sum_k \|\Psi(x_{k+1}) - A\Psi(x_k) - B_0 u_k - \sum_i u_k^i B_i \Psi(x_k)\|^2$$ (Bruder et al., 2020, Junker et al., 2021). Inclusion of terms like $\Psi(x) \otimes u$ or $\Psi(x) \cdot u$ is often critical. Extensions such as modularized EDMD leverage networked dynamics and decouple the approximation by exploiting subsystem-level dictionaries, reducing effective sample complexity and accommodating topology changes (Guo et al., 22 Aug 2024).

The choice of observable functions is central. Physics-informed, kernel-based, or Hankel-based dictionaries are advocated to capture nonlinearity, memory, and compatibility with control directions (Sakib et al., 13 Aug 2024, Strässer et al., 2 Sep 2025). Using prior knowledge, such as inclusion of trigonometric functions for rotational variables or geometric moments in robotics, is crucial when the underlying geometry restricts feasible motions (Rosenfelder et al., 11 Nov 2024).

Quantitative error estimates for bilinear EDMD surrogates take the form

$\|r(x,u)\|_2 \leq c_x \|\Psi(x)\|_2 + c_u \|u\|_2$

with $c_x$, $c_u$ combining finite-data regression error and dictionary truncation bounds. Both probabilistic (Schaller et al., 2022, Strässer et al., 15 Nov 2024) and deterministic (Strässer et al., 2 Sep 2025) estimates ensure that, near equilibrium or for small input, the residual is negligible, a property critical for robust control guarantees.

3. Control Design and Stability Certification

The surrogate bilinear system forms the basis for a spectrum of controller design strategies. Typical approaches include:

• State-feedback synthesis: Robust stabilizing controllers are synthesized via:
  • Sum-of-squares (SOS) programs that assign a polynomial (or rational) controller and Lyapunov certificate to the bilinear system with structured proportional uncertainty, maximizing the certified region of attraction and minimizing conservatism (Strässer et al., 6 Nov 2024, Strässer et al., 15 Nov 2024). Representative conditions are polynomial matrix inequalities that must belong to the SOS cone.
  • Linear matrix inequalities (LMIs), especially for linear fractional representations including neural feedback loops, guaranteeing local exponential stability and quantifiable robustness to parametric or model uncertainties (Shah et al., 30 May 2025).
• Model Predictive Control (MPC): For predictive control, the bilinear Koopman model is embedded within a receding-horizon optimization. Because the surrogate is bilinear, the resulting MPC problem is typically at worst quadratic and thus convex or efficiently solvable under horizon-wise linearization (Bruder et al., 2020, Bold et al., 2023). Explicit error bounds are leveraged to tighten constraints or ensure practical asymptotic stability (Bold et al., 2023, Strässer et al., 2 Sep 2025).
• Data-Enabled Predictive Control (DeePC): Recent advances bypass explicit model identification by extending the Fundamental Lemma of Willems to the bilinear setting, using trajectory-level data. Here, predictions and control synthesis are achieved directly in lifted data space, subjected to trajectory consistency conditions that incorporate (lifted) state-input cross terms (Xiong et al., 6 May 2025).

Certification of the closed-loop system relies on the proportionality of the residual error: robust controller synthesis incorporates these bounds, and the origin (or target state) remains an exact equilibrium. These results guarantee, under mild conditions, exponential stability or practical asymptotic stability over a quantifiable domain (Chatzikiriakos et al., 26 Sep 2024, Strässer et al., 15 Nov 2024, Strässer et al., 6 Nov 2024, Bold et al., 2023).

4. Scalability, Modularization, and Data Efficiency

Curse of dimensionality is addressed by several strategies:

• Bilinear lifting avoids explicit state–input augmentation (which increases the observable dimension combinatorially), confining the complexity to the observable dictionary dependent on state dimension and number of control directions (Schaller et al., 2022, Strässer et al., 2 Sep 2025).
• Modularized EDMD decomposes interconnected or networked systems, identifying subsystem surrogates individually and leveraging topological sparsity. This allows for scalable training, transfer learning, and efficient reparameterization upon changes in the system’s structure (Guo et al., 22 Aug 2024).
• Sample efficiency is further enhanced by methods such as Jacobian-regularized DMD, which leverages approximate prior model derivatives during training (in addition to snapshot data), ensuring that both value and local sensitivity (critical for MPC) match, substantially reducing training trajectory requirements (Jackson et al., 2022).

Iterative data augmentation—in which the control policy is continually updated using newly collected closed-loop trajectories—improves performance and coverage of critical state-space regions, particularly under non-stationary or noise-prone conditions (Sakib et al., 13 Aug 2024).

5. Practical Impact and Domain-Specific Applications

Bilinear EDMD with control has demonstrated efficacy in a wide array of domains:

• Quantum control: Physics-informed regression methods such as biDMD provide data-efficient and interpretable models for real-time optimal control of quantum systems, distinguishing drift and control effects and leveraging stroboscopic reconstruction (Goldschmidt et al., 2020).
• Robotic motion and mechatronics: Integration of bilinear EDMD models in predictive controllers for nonholonomic robots and high-DOF manipulators yields real-time feasible and precise tracking. However, studies indicate that control performance fundamentally depends on embedding geometric insight (e.g., sub-Riemannian metrics) into the cost structure and observable design, as data-driven surrogates cannot compensate for intrinsic system constraints solely via increased data (Rosenfelder et al., 11 Nov 2024).
• Cyber-physical and neural-feedback loops: When bilinearities and neural networks are present in the loop (e.g., due to integrated learned components), robust controller design is possible by abstracting neural nonlinearities with quadratic constraints in an LFR framework, with LMIs providing certificates of stability and safety (Shah et al., 30 May 2025).
• Data-driven inverse optimal control: Bilinear EDMD with modified learning and PMP analysis enables identification of both dynamics and quadratic costs from optimal trajectory data, supporting applications from human intent prediction in robotics to general inverse optimal control (Fernandez-Ayala et al., 30 Jan 2025).

Experimental and simulation studies consistently show that bilinear surrogates improve the balance of accuracy (close to nonlinear models) and computational efficiency (close to linear models) for both prediction and control when compared with purely linear or highly nonlinear surrogates.

6. Challenges and Future Directions

A number of open issues remain:

• Rigorous characterization of minimal sample requirements for a given model and control objective remains an active area, with existing error bounds sometimes being conservative relative to empirical findings (Strässer et al., 2 Sep 2025).
• Selection or automatic learning of control-relevant dictionaries remains unsolved; while deep-learning-based observable selection is being explored, theoretical analysis and practical guides are lacking.
• Extending proportional error-bounded bilinear EDMD to settings with partial state observation (input–output data) is theoretically nontrivial due to hidden state reconstruction ambiguities.
• Scalability in high or infinite-dimensional state spaces, such as in distributed or multi-agent systems, is being partially tackled through modularization but demands further research.
• The balance between conservatism (for guarantees) and achievable closed-loop performance is an ongoing focus, particularly in switching from LMI-based to polynomial/SOS-based controller synthesis (Strässer et al., 6 Nov 2024, Strässer et al., 15 Nov 2024).
• Incorporation of safety constraints and certified regions of attraction is being advanced via robust SOS programs and explicit region characterization (Chatzikiriakos et al., 26 Sep 2024, Strässer et al., 6 Nov 2024).

A plausible implication is that integration of bilinear EDMD with advances in reinforcement learning, online adaptive control, distributed optimization, and geometric model selection will further expand the reach and rigor of data-driven control for complex nonlinear systems.

7. Summary Table: Core Ingredients and Objectives

Methodological Component	Purpose in Bilinear EDMD with Control	Representative Reference
Bilinear Koopman surrogate	Accurate, tractable surrogate for lifted control-affine dynamics	(Bruder et al., 2020, Strässer et al., 2 Sep 2025)
Finite-data error quantification	Certified proportional residual bounds for robust controller synthesis	(Schaller et al., 2022, Strässer et al., 15 Nov 2024)
Controller synthesis (LMI/SOS)	Systematic robust control design with closed-loop exponential/practical stability	(Strässer et al., 6 Nov 2024, Shah et al., 30 May 2025)
Modularization	Scalability for interconnected/networked systems, transfer learning	(Guo et al., 22 Aug 2024)
Data-enabled control (DeePC)	Direct trajectory-based control via lifted data constraints, avoids model ID	(Xiong et al., 6 May 2025)
Geometry-aware design	Integrates physical/structural constraints for improved control performance	(Rosenfelder et al., 11 Nov 2024)

This synthesis highlights the rigorous, data-driven, and robust nature of bilinear EDMD with control as a unifying paradigm in modern system identification and control theory, bridging operator-theoretic insights, quantitative guarantees, and practical engineering applications.