Koopman Bilinearization of Control Systems

• Koopman bilinearization is a lifting-based method that transforms nonlinear control-affine systems into bilinear forms using an observable space.
• It links operator-theoretic and data-driven approaches to enable tractable controller synthesis and rigorous safety analysis via convex optimization.
• This approach supports real-time implementation through quadratic programming and falsification loops, ensuring robust performance in safety-critical scenarios.

Koopman bilinearization is a lifting-based methodology for representing nonlinear control-affine systems as (potentially infinite-dimensional) bilinear systems in an appropriately selected observable space. This transformation enables powerful connections between nonlinear dynamics, operator-theoretic methods, and modern data-driven control synthesis, facilitating tractable controller design and robust safety analysis using classical linear or bilinear system methods.

1. Koopman Bilinearization: Mathematical Formulation

Consider a control-affine system

$$\dot{x} = f(x) + g(x)u, \quad x\in\mathbb{R}^n,\, u\in\mathbb{R}^m,$$

with $f$, $g$ globally Lipschitz on $X \times U$. The Koopman operator framework associates with such a system an evolution equation for a collection of smooth observables $\Psi(x) \in \mathbb{R}^N$ (possibly $N\gg n,\, N=\infty$ in the ideal case) as follows:

$$\frac{d}{dt}\Psi(x) = \nabla\Psi(x) f(x) + \nabla\Psi(x) g(x) u.$$

Bilinear closure assumption: If for each input channel $i$ there exists a constant matrix $C_i \in \mathbb{R}^{N\times N}$ such that

$\nabla\Psi(x) g_i(x) = C_i \Psi(x)\,,$

the system closes in the following bilinear ODE:

$f$0

with $f$1 and $f$2 (regarded as a linear operator on the span of the observables). Discretization (e.g., Euler with $f$3) yields:

$f$4

This structure is fundamentally distinct from classical Carleman linearization (polynomial expansion/linearization), as Koopman closure is algebraic and, for appropriate observables and classes of systems, can be exact for an infinite-dimensionally lifted system (Zhang et al., 2022).

2. Construction of Observables and Data-Driven Learning

The practical efficacy of Koopman bilinearization relies entirely on selecting a dictionary of observables $f$5 capturing the system's forward dynamics invariantly or approximating such an invariant space.

• Classical approaches: Use monomials, Fourier modes, radial basis functions, or Koopman eigenfunctions.
• Neural approaches: Parameterize $f$6 as a neural-network encoder, trained so that its image is approximately forward-invariant under the dynamics. Regularization, e.g., spectral normalization, is enforced for bounded Lipschitz constant $f$7 to control the error in the lifted model (Zinage et al., 2022).

EDMD (Extended Dynamic Mode Decomposition) regression: Given data $f$8, assemble:

• $f$9
• $g$0

The bilinear matrices are obtained by least squares:

$g$1

This procedure is differentiable and can be embedded in joint training with the encoder. Loss functions combine dynamics fitting, state reconstruction (using a decoder $g$2), and additional objectives (e.g., barrier constraints) (Zinage et al., 2022, Zinage et al., 2022).

3. Error Analysis and Validity of Bilinear Approximation

Koopman bilinearization accuracy is governed both by approximation/truncation errors (finite $g$3) and regularity of the observable map. The following bound holds (Zinage et al., 2022):

$g$4

where $g$5 is the maximum empirical residual over the data, $g$6 is the covering diameter of the training set, $g$7 is the Lipschitz constant for $g$8, and $g$9 is the spectral Lipschitz bound.

Implication: The barrier function $X \times U$0 computed on the lifted system with flow $X \times U$1 remains a valid CBF (Control Barrier Function) for the true nonlinear system provided this bound is sufficiently tight compared to the margin in the barrier constraint. This underpins the theoretical soundness for enforcing safety or stabilization in the original state space, even when the Koopman lift is purely data-driven (Zinage et al., 2022, Zinage et al., 2022).

4. Learner–Falsifier and Barrier Function Synthesis

Safety-critical control is addressed by searching for a CBF $X \times U$2 in a (potentially neural) function class, jointly trained with the lifted model. The minimization objective is a composite loss:

$X \times U$3

where $X \times U$4 penalizes model mismatch in lifted space, $X \times U$5 enforces invertibility ($X \times U$6), and $X \times U$7 penalizes violations of sampled barrier conditions.

A falsifier loop, typically using an SMT solver, tests whether there exist $X \times U$8 violating the CBF inequalities (for samples outside the training set). Counterexamples are iteratively added to the training data until the falsifier returns UNSAT, certifying empirical global satisfaction (Zinage et al., 2022).

5. Controller Synthesis and Online Implementation

After obtaining the lifted model and a valid CBF, controllers are synthesized via quadratic programs in the lifted (bilinear) coordinates. In online execution for safety, the input is computed at each time step as the solution to

$X \times U$9

where $\Psi(x) \in \mathbb{R}^N$0 is a nominal controller, and $\Psi(x) \in \mathbb{R}^N$1 are the bilinear operators in the lifted space. The argmin is over the admissible control set $\Psi(x) \in \mathbb{R}^N$2, and $\Psi(x) \in \mathbb{R}^N$3. This QP admits real-time implementation and can enforce safety constraints originating from the CBF (Zinage et al., 2022).

6. Illustrative Example: Differential-Drive Robot Collision Avoidance

In (Zinage et al., 2022), differential-drive rob ot dynamics

$\Psi(x) \in \mathbb{R}^N$4

are bilinearized using a 5-dimensional lifted state $\Psi(x) \in \mathbb{R}^N$5 (N=5), with safety constraints defined by a unit-disk obstacle at the origin. After learner–falsifier training, all of 50 simulated trajectories from random initial positions were collision-free—demonstrating the effectiveness and reliability of the Koopman bilinearized approach for data-driven, safety-critical control on nonlinear systems with unknown dynamics.

7. Significance and Research Implications

Koopman bilinearization provides a systematic algebraic framework to:

• Lift unknown nonlinear control-affine systems to higher-dimensional bilinear forms via operator-theoretic or data-driven means.
• Translate safety and stabilization controller synthesis (via CBFs or Lyapunov functions) to convex optimization problems in the lifted coordinates.
• Certify transfer of safety/stability properties to the true system under explicit error bounds, tunable through observable regularity and data coverage.
• Efficiently address challenging control objectives (guaranteed barrier satisfaction, region of attraction maximization) using neural networks, convex programming, and falsification loops.

This methodology has been validated for diverse systems and objectives, including stabilization and safety-critical tasks, without the need for explicit dynamical models (Zinage et al., 2022).

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References:

Neural Koopman Control Barrier Functions for Safety-Critical Control of Unknown Nonlinear Systems (Zinage et al., 2022) Neural Koopman Lyapunov Control (Zinage et al., 2022) Feedback Stabilization Using Koopman Operator (Huang et al., 2018)