---
title: Bilinear Koopman Realizations
url: https://www.emergentmind.com/topics/bilinear-koopman-realizations
type: topic
---

# Bilinear Koopman Realizations

Searching arXiv for recent and foundational papers on bilinear Koopman realizations.
[arXiv search] Query: "bilinear Koopman realization control-affine Lyapunov MPC DeePC"
Bilinear Koopman realizations are lifted-state representations of nonlinear control systems in which the dynamics of selected observables evolve bilinearly in the lifted state and the control. In the cited literature, the canonical continuous-time form is obtained from a control-affine system
\[
\dot{x}=F(x)+\sum_{i=1}^{m}G_i(x)u_i
\]
by choosing observables \(z=\Psi(x)\) such that the control Lie derivatives close on their span, yielding
\[
\dot z=\Lambda z+\sum_{i=1}^{m}u_i B_i z.
\]
A discrete-time counterpart is
\[
z_{k+1}=Az_k+Bu_k+H(z_k\otimes u_k),\qquad x_k=Cz_k.
\]
Across recent work, these realizations are presented as the natural Koopman form for control-affine dynamics: more expressive than lifted LTI predictors, but still structured enough for Lyapunov analysis, MPC, DeePC, and operator-theoretic optimal control [2505.08139] [2505.03346] [2010.09961].

## 1. Formal structure of bilinear Koopman realizations

A standard starting point is a control-affine nonlinear system
\[
\dot{x}=F(x)+\sum_{i=1}^{m}G_i(x)u_i,\qquad x\in X\subseteq \mathbb{R}^n.
\]
For an observable \(\tilde g(t,x)\), the controlled Koopman evolution satisfies
\[
\frac{\partial}{\partial t}\tilde g=L_F\tilde g+\sum_{i=1}^{m}u_i L_{G_i}\tilde g,
\]
which is already “analogous to a bilinear system” in an infinite-dimensional function space. A finite-dimensional realization is obtained by choosing observables or Koopman eigenfunctions
\[
z=\Psi(x)=[\psi_1(x),\dots,\psi_N(x)]^\top
\]
and imposing the closure condition
\[
L_{G_i}\Psi=B_i\Psi,\qquad i=1,\dots,m,
\]
with constant matrices \(B_i\). Under this assumption, the lifted dynamics become
\[
\dot z=\Lambda z+\sum_{i=1}^{m}u_i B_i z,
\]
where \(\Lambda\) is assembled from Koopman eigenvalues, including real \(2\times 2\) blocks for complex-conjugate pairs after conversion to real coordinates [2505.08139].

The discrete-time formulation used in predictive control and data-enabled control typically writes a Koopman bilinear realization as
\[
z_{k+1}=Az_k+Bu_k+H(z_k\otimes u_k),\qquad x_k=Cz_k,
\]
where \(z=\Psi(x)\), \(H\in\mathbb{R}^{n_z\times n_un_z}\), and \(z_k\otimes u_k\) is the Kronecker product. This form makes explicit that the model is linear in \(z_k\) for fixed \(u_k\), linear in \(u_k\) for fixed \(z_k\), and bilinear in the pair \((z_k,u_k)\) [2505.03346].

The distinction from a lifted LTI model is structural rather than cosmetic. In the literature, a lifted LTI predictor has the form \(z_{k+1}=Az_k+Bu_k\), whereas a bilinear realization preserves state-input cross terms. This matters because terms such as \(u_k^i z_k^j\) are the generic lifted signature of control-affine nonlinear systems, and several papers explicitly contrast bilinear realizations with purely linear lifted predictors on that basis [2010.09961].

## 2. Existence, exactness, and realization-theoretic scope

A central theoretical result is that bilinear realizations are the generic Koopman structure for control-affine systems, whereas linear realizations are not. Over a chosen set of state-only observables \(\{z_i\}_{i=1}^N\), a realization is bilinear if and only if each derivative \(\frac{\partial z_i}{\partial x}F(x,u)\) lies in the span of state observables, input projections, and their products; it is linear if and only if the same derivatives lie in the smaller span of state observables and pure inputs alone. From this, the literature concludes that every smooth control-affine system admits an infinite-dimensional bilinear realization over a basis of state-only observables, but does not necessarily admit a linear one [2010.09961].

This point is repeatedly used to correct a common misconception: Koopman lifting does not, in general, turn nonlinear controlled dynamics into an exact lifted LTI system. Rather, for control-affine dynamics, bilinearity is the natural exact infinite-dimensional structure. Finite-dimensional realizations are exact only when an invariant finite-dimensional observable subspace exists; otherwise they are approximations obtained by truncation or regression. One paper states this explicitly: Assumption 1 postulates a finite-dimensional invariant subspace for the control Lie derivatives, and exact finite-dimensional existence is not proved there; EDMD is then used to obtain an approximation [2505.08139].

A stronger exact statement appears in the discrete-time RKHS setting. There, states and inputs are lifted separately into RKHSs with canonical features \(\phi_x\) and \(\varphi_u\), and the dynamics admit an exact representation
\[
\phi_{x^+}
=
M
\begin{bmatrix}
\phi_x\\
\varphi_u\\
\phi_x\otimes \varphi_u
\end{bmatrix},
\]
where \(M\) is a bounded linear operator from \(\mathcal{H}_x\oplus\mathcal{H}_u\oplus(\mathcal{H}_x\otimes\mathcal{H}_u)\) to \(\mathcal{H}_x\). This construction uses linear-radial product kernels so that the origin maps to the zero feature and Lyapunov functions become kernel sum-of-squares objects. In that framework, the bilinear realization is exact under smoothness, compactness, and regularity assumptions on \(f\) and the state/input domains [2606.10344].

The cited control papers are less expansive on realization-theoretic notions such as minimality, observability, or controllability in Koopman coordinates. One of them states directly that minimal realization, observability, and controllability of \((\Lambda,B_i,\Psi)\) are not formally addressed, even though stabilizability assumptions and CLF conditions are analyzed in detail [2505.08139].

## 3. Construction from data and partially observed trajectories

A standard data-driven construction uses EDMD to approximate Koopman operators or generators. For the control-affine single-input setting, one approach collects time-series data under two constant inputs \(s\in\{0,1\}\), forms
\[
G^s=\frac{1}{M}\sum_{m=1}^M \Psi(x_m^s)\Psi(x_m^s)^\top,\qquad
A^s=\frac{1}{M}\sum_{m=1}^M \Psi(x_m^s)\Psi(y_m^s)^\top,
\]
and computes
\[
U^s=(G^s)^\dagger A^s.
\]
Eigenvalues and eigenvectors of \(U^0\) provide approximate Koopman eigenfunctions; a generator approximation and the difference
\[
\bar B=\frac{U^1-U^0}{\Delta t}
\]
yield the control operator, which is then transformed into Koopman eigenfunction coordinates. In that development, the bilinear model is written as
\[
\dot z=\Lambda z+uBz,
\]
and the approximation error in \(\Lambda\) and \(B\) is stated to decay as \(\mathcal{O}(1/\sqrt{T})\) with data length \(T\) [1901.07678].

For partially observed systems, the lifted state can be treated as latent. One paper writes the observable dynamics as a bilinear hidden Markov model,
\[
\begin{bmatrix}1&z_{l+1}\end{bmatrix}
=
\left(I+\Delta t\sum_{k=0}^{\dim u}[u_l]_k V_k\right)^\top
\begin{bmatrix}1&z_l\end{bmatrix}
+
\begin{bmatrix}0&w_l\end{bmatrix},
\qquad
y_l=c_0+\begin{bmatrix}I&0\end{bmatrix}z_l+v_l,
\]
and learns the parameters with EM. The E-step uses a Kalman filter and Rauch smoother; the M-step yields a closed-form update for the generator matrices that resembles control-affine dynamic mode decomposition for the Koopman generator [2209.09977].

A further generalization lifts states and inputs independently and couples them through Khatri-Rao or tensor features. In that formulation,
\[
z_{t+1}=K(z_t\otimes v_t),
\]
with \(z_t=\Psi_x(x_t)\) and \(v_t=\Psi_u(u_t)\). The associated multi-step estimator stacks time-shifted Khatri-Rao feature blocks over trajectory windows and solves a single ridge regression
\[
\hat K=W\Phi^\top(\Phi\Phi^\top+\gamma I)^{-1},
\]
thereby exposing the bilinear operator to evolved snapshots rather than isolated one-step pairs. The same work develops a structured SVD reduction of the lifted state and input spaces that preserves the Khatri-Rao realization [2606.02938].

## 4. Stability, Lyapunov functions, and feedback synthesis

A major theme in the literature is that bilinear Koopman realizations make nonlinear stabilization amenable to quadratic CLF design in lifted coordinates. For the single-input bilinear system
\[
\dot z=Az+uBz,
\]
or its Koopman-eigenfunction variant with \(A=\Lambda\), a quadratic control Lyapunov function
\[
V(z)=z^\top P z,\qquad P=P^\top\succ0
\]
is used. The main stabilizability condition is identical across several papers: the system is quadratically stabilizable if and only if there exists \(P\succ0\) such that, for every nonzero \(z\),
\[
z^\top(PA+A^\top P)z\ge0
\quad\Longrightarrow\quad
z^\top(PB+B^\top P)z\neq0.
\]
This ensures that whenever the drift term does not decrease \(V\), the bilinear control term has nonzero authority [1810.00089] [1901.07678].

The search for \(P\) is commonly posed as a convex semidefinite program. A representative formulation is
\[
\min_{t>0,\;P=P^\top}\; t-\gamma\,\mathrm{Trace}(PB)
\]
subject to
\[
tI-(PA+A^\top P)\succeq0,\qquad
c^{\max}I\succeq P\succeq c^{\min}I.
\]
This LMI-based procedure is used to construct Koopman-space CLFs and then stabilizing feedback laws [1810.00089].

Several feedback laws are considered. The simplest are sign or proportional laws based on
\[
z^\top(PB+B^\top P)z,
\]
while a more structured choice uses Sontag’s formula in Koopman coordinates. In one formulation,
\[
b(z)=
\begin{cases}
-\dfrac{L_\Lambda V+\sqrt{L_\Lambda V^2+L_BV^4}}{L_BV}, & L_BV\neq0,\\[6pt]
0,&L_BV=0,
\end{cases}
\]
followed by saturation to enforce input bounds. In the earlier stabilization papers, the same idea is given an inverse-optimal interpretation with cost
\[
J=\int_0^\infty \big(q(z)+u^\top u\big)\,dt
\]
for a suitable state penalty \(q(z)\) [1901.07678] [2505.08139].

Koopman Lyapunov MPC embeds this CLF directly into prediction. For the lifted bilinear model
\[
\dot z(t)=\Lambda z(t)+u(t)Bz(t),
\]
the continuous-time optimization minimizes
\[
\int_{t_k}^{t_{k+N_p}}
\big(z^\top(\tau)Wz(\tau)+u^\top(\tau)Ru(\tau)\big)\,d\tau
\]
subject to the bilinear dynamics, input bounds, and Lyapunov constraints such as \(V(z(t))\le \hat r\) and
\[
\dot V(z(t_k),u(t_k))\le \dot V(z(t_k),h(z(t_k))).
\]
A corresponding sampled-data stability result states that for sufficiently small \(\Delta\), trajectories starting in a stability region \(\Omega_r\) remain in \(\Omega_r\) and satisfy \(\lim_{t\to\infty}\|z(t)\|\le d\). Mapping back to the original state through a smooth inverse \(x=\xi(z)\), the modeling error obeys
\[
\|e(t)\|\le \frac{\nu}{l_x}(e^{l_x t}-1),
\]
yielding practical stability of the original nonlinear system [2505.08139].

Neural Koopman Lyapunov Control replaces hand-crafted observables with a jointly learned encoder, decoder, bilinear Koopman model, and CLF. It enforces Lyapunov conditions during learning and uses a learner-falsifier loop with dReal to search for counterexamples. In that framework, the goal is not merely to fit a bilinear model, but to identify a stabilizable bilinear realization together with a CLF that supports Sontag feedback [2201.05098].

## 5. Predictive control, DeePC, and input-output generalizations

One major development is the extension of Willems’ Fundamental Lemma to nonlinear systems that admit an exact finite-dimensional Koopman bilinear realization. For a discrete-time KBR
\[
z_{k+1}=Az_k+Bu_k+H(z_k\otimes u_k),\qquad x_k=Cz_k,
\]
the lifted bilinear signal
\[
v_k=z_k\otimes u_k
\]
is introduced, and a Hankel matrix built from \((z,u,v)\) replaces the classical LTI Hankel matrix. This leads to a DeePC formulation in lifted coordinates with constraints
\[
v=Vg,\qquad v_i=z_i\otimes u_i,
\]
so that dynamic consistency is enforced directly from data rather than through identified matrices. The same paper emphasizes that this approach bypasses the EDMD identification step and can be more robust when a finite exact Koopman realization does not exist [2505.03346].

A complementary line of work compares linear, bilinear, and nonlinear Koopman predictors inside MPC. For a simulated 3-link planar robot arm, bilinear Koopman MPC linearizes the bilinear term around the initial lifted state, producing a convex QP of complexity comparable to linear Koopman MPC. In the reported trajectory-following task, the mean tracking error of linear Koopman MPC is more than \(15\times\) larger than that of bilinear and nonlinear Koopman MPC, while nonlinear Koopman MPC is more than \(500\times\) slower than the linear and bilinear QP-based variants. This comparison is used to argue that bilinear realizations often provide the favorable compromise between predictive accuracy and computational cost [2010.09961].

For partially observed industrial systems, an input-output generalized bilinear Koopman realization can be built from delay-embedded outputs and exogenous inputs. In the diesel-engine airpath application, the embedded variable is
\[
\zeta_k=
\begin{bmatrix}
(\zeta_k^y)^\top & (\zeta_k^w)^\top
\end{bmatrix}^\top,
\]
and the generalized bilinear model is
\[
z_{k+1}=Az_k+B_0 w_k+B(z_{w,k}\otimes w_k),
\]
where \(z_k=\psi_x(\zeta_k)\) and \(z_{w,k}=\phi_w(\zeta_k)\). A structural theorem is used to justify restricting nonlinear state liftings to output delays while relegating input-dependent nonlinearities to the bilinear scheduling lift. With RBF liftings whose centers are optimized by particle swarm optimization on a multi-step loss, the generalized bilinear model achieves long-term test \(R^2\) values of \(0.922/0.949\) in simulation and \(0.936/0.911\) in experiments, outperforming both lifted LTI and naive bilinear alternatives [2602.15422].

## 6. Applications, operator-theoretic extensions, and limitations

Bilinear Koopman realizations appear in a wide range of applications. For nonlinear PDE control, one approach approximates several Koopman operators corresponding to constant control inputs and interpolates them by convex combination, yielding a low-dimensional bilinear reduced-order model. Under linear dependence of the PDE on the control and a linear observation map, the reduced and full objectives coincide in the limit of EDMD convergence; in more general cases, the method is still demonstrated empirically on flow-control problems [1801.06419].

In power systems, the modularized Koopman Bilinear Form represents each distributed energy resource by a local bilinear lifted model,
\[
\dot z = A z + \sum_{i=1}^m u_i B_i z,
\]
and couples subsystems through a network model. This decomposition is presented as a way to avoid handling the full high-dimensional microgrid in a single global lifting, and is used for transient prediction with plug-and-play modeling of subsystem modules [2205.03214].

Bilinear lifted models also support nonstandard optimal-control formulations. One recent paper defines a value bilinear form
\[
\langle P_u\phi,\psi\rangle
=
\int_0^\infty
\langle C_u S_u(t)\phi,\; C_u S_u(t)\psi\rangle_{\ell_2}\,dt
\]
for a Koopman lift of the closed-loop dynamics, proves that the minimal value function admits a sum-of-squares expansion
\[
v(z)=\sum_{i=1}^\infty \sigma_i p_i(z)^2,
\]
and derives a Riccati-like quadratic operator equation for the corresponding nuclear operator \(\Sigma\). In that framework, the bilinear realization is infinite-dimensional and operator-valued, and it is used to connect HJB and Riccati theory rather than to fit a finite state-space predictor [2509.20122].

Robustness to approximation error is an active concern. A set-membership approach models the sampled nonlinear system as
\[
z_{t+1}=A_s z_t+B_s u_t+\tilde B_s(u_t\otimes z_t)+r(x_t,u_t),
\]
with residual bound
\[
\|r(x,u)\|\le c_x\|\Phi(x)\|+c_u\|u\|,
\]
and characterizes all matrices \((A_s,B_s,\tilde B_s)\) consistent with the data and residual bound. A rational feedback law
\[
u(z)=\frac{1}{d(z)}K(z)z
\]
is then synthesized by SOS programming so as to stabilize every bilinear model in the consistent set and every admissible residual [2606.01378].

Several limitations recur across the literature. Finite-dimensional exact bilinear realizations are exceptional unless a suitable invariant observable space is known; otherwise model quality depends heavily on dictionary choice, lifting richness, and data coverage. Bilinear MPC and DeePC formulations are generally nonconvex, even when they admit useful linearizations or convex surrogates. Scalability remains an issue because EDMD, SOS programs, and high-order dictionaries can become large-scale rapidly. These points suggest that “bilinear Koopman realization” is best understood not as a single algorithm, but as a family of structured lifted models whose theoretical appeal lies in their alignment with control-affine dynamics, and whose practical success depends on how that structure is approximated, reduced, and regularized in a given application [2010.09961] [2505.03346] [2606.10344].

Source: https://www.emergentmind.com/topics/bilinear-koopman-realizations