---
title: Koopman Bilinearization of Control Systems
url: https://www.emergentmind.com/topics/koopman-bilinearization-of-nonlinear-control-systems
type: topic
---

# Koopman Bilinearization of Control Systems

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:
$$
\dot{z} = K z + \sum_{i=1}^m C_i z\, u_i
$$
with $z = \Psi(x)$ and $K := \nabla\Psi(x)f(x)$ (regarded as a linear operator on the span of the observables). Discretization (e.g., Euler with $\Delta t$) yields:
$$
z_{k+1} = K_d z_k + \sum_{i=1}^m D_i z_k u_{k,i},\qquad K_d = I + K\Delta t,\ D_i = C_i \Delta t.
$$
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 [2211.07112].

## 2. Construction of Observables and Data-Driven Learning

The practical efficacy of Koopman bilinearization relies entirely on selecting a dictionary of observables $\Psi(x)$ 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 $\Psi(x;\theta)$ 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 $\|\Psi\|_{\mathrm{Lip}} \le K_\Psi$ to control the error in the lifted model [2209.07685].

**EDMD (Extended Dynamic Mode Decomposition) regression:** Given data $\mathcal{D} = \{(x_k, u_k, x_{k+1})\}$, assemble:
- $\eta = \big[(1;u_1)\otimes \Psi(x_1),\, \dots\big] \in\mathbb{R}^{(m+1)N \times (N_d-1)}$
- $\Gamma = [\Psi(x_2),\dots] \in\mathbb{R}^{N\times(N_d-1)}$

The bilinear matrices are obtained by least squares:
$$
[K_d,\ D_1,\ldots, D_m] = \Gamma \eta^\top (\eta \eta^\top)^{-1}.
$$
This procedure is differentiable and can be embedded in joint training with the encoder. Loss functions combine dynamics fitting, state reconstruction (using a decoder $\Phi^{-1}$), and additional objectives (e.g., barrier constraints) [2209.07685, 2201.05098].

## 3. Error Analysis and Validity of Bilinear Approximation

Koopman bilinearization accuracy is governed both by approximation/truncation errors (finite $N$) and regularity of the observable map. The following bound holds [2209.07685]:
$$
\|\nabla\Psi(x)F(x,u) - \psi(\Psi(x),u) \| \leq K_\Psi K_F \delta + 2 K_\Psi K_F \tau + \mu + K_\psi \delta,
$$
where $\mu$ is the maximum empirical residual over the data, $\delta$ is the covering diameter of the training set, $K_F$ is the Lipschitz constant for $(f,g)$, and $K_\Psi$ is the spectral Lipschitz bound.

**Implication:** The barrier function $h(x)$ computed on the lifted system with flow $\psi$ 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 [2209.07685, 2201.05098].

## 4. Learner–Falsifier and Barrier Function Synthesis

Safety-critical control is addressed by searching for a CBF $h(x;\varphi)$ in a (potentially neural) function class, jointly trained with the lifted model. The minimization objective is a composite loss:
$$
L = \beta_1 L_{\mathrm{dyn}} + \beta_2 L_{\mathrm{recons}} + \beta_3 L_{\mathrm{barr}},
$$
where $L_\mathrm{dyn}$ penalizes model mismatch in lifted space, $L_\mathrm{recons}$ enforces invertibility ($x \approx \Phi^{-1}(\Psi(x))$), and $L_\mathrm{barr}$ penalizes violations of sampled barrier conditions.

A **falsifier** loop, typically using an SMT solver, tests whether there exist $(x, u)$ 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 [2209.07685].

## 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
$$
u^* = \arg\min \|u - k_l(x)\|^2
\quad\text{s.t.}\quad
\nabla h(x)\cdot (K z + \sum_i C_i z\, u_i) \geq -\lambda h(x),
$$
where $k_l(x)$ is a nominal controller, and $(K, C_i)$ are the bilinear operators in the lifted space. The argmin is over the admissible control set $U$, and $z = \Psi(x)$. This QP admits real-time implementation and can enforce safety constraints originating from the CBF [2209.07685].

## 6. Illustrative Example: Differential-Drive Robot Collision Avoidance

In [2209.07685], differential-drive rob ot dynamics
$$
\dot{x} = r \sin\theta, \quad \dot{y} = r \cos\theta, \quad \dot{\theta} = (r/L) \omega, \; u=\omega\in[-1,1],
$$
are bilinearized using a 5-dimensional lifted state $z$ (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 [2209.07685].

---

**References:**  
Neural Koopman Control Barrier Functions for Safety-Critical Control of Unknown Nonlinear Systems [2209.07685]  
Neural Koopman Lyapunov Control [2201.05098]  
Feedback Stabilization Using Koopman Operator [1810.00089]

Source: https://www.emergentmind.com/topics/koopman-bilinearization-of-nonlinear-control-systems