Bilinear Koopman Realizations

• Bilinear Koopman realizations are lifted-state representations for control-affine nonlinear systems where observables evolve bilinearly in state and control.
• The structure offers a natural framework for integrating Lyapunov analysis, MPC, and DeePC by preserving key state-input interactions beyond linear predictors.
• Data-driven methods like EDMD help construct these models, though finite-dimensional exact realizations remain challenging and often approximate.

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 (Dony, 13 May 2025, Xiong et al., 6 May 2025, Bruder et al., 2020).

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

$z=\Psi(x)$0

where $z=\Psi(x)$1 is assembled from Koopman eigenvalues, including real $z=\Psi(x)$2 blocks for complex-conjugate pairs after conversion to real coordinates (Dony, 13 May 2025).

The discrete-time formulation used in predictive control and data-enabled control typically writes a Koopman bilinear realization as

$z=\Psi(x)$3

where $z=\Psi(x)$4, $z=\Psi(x)$5, and $z=\Psi(x)$6 is the Kronecker product. This form makes explicit that the model is linear in $z=\Psi(x)$7 for fixed $z=\Psi(x)$8, linear in $z=\Psi(x)$9 for fixed $\dot z=\Lambda z+\sum_{i=1}^{m}u_i B_i z.$0, and bilinear in the pair $\dot z=\Lambda z+\sum_{i=1}^{m}u_i B_i z.$1 (Xiong et al., 6 May 2025).

The distinction from a lifted LTI model is structural rather than cosmetic. In the literature, a lifted LTI predictor has the form $\dot z=\Lambda z+\sum_{i=1}^{m}u_i B_i z.$2, whereas a bilinear realization preserves state-input cross terms. This matters because terms such as $\dot z=\Lambda z+\sum_{i=1}^{m}u_i B_i z.$3 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 (Bruder et al., 2020).

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 $\dot z=\Lambda z+\sum_{i=1}^{m}u_i B_i z.$4, a realization is bilinear if and only if each derivative $\dot z=\Lambda z+\sum_{i=1}^{m}u_i B_i z.$5 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 (Bruder et al., 2020).

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 (Dony, 13 May 2025).

A stronger exact statement appears in the discrete-time RKHS setting. There, states and inputs are lifted separately into RKHSs with canonical features $\dot z=\Lambda z+\sum_{i=1}^{m}u_i B_i z.$6 and $\dot z=\Lambda z+\sum_{i=1}^{m}u_i B_i z.$7, and the dynamics admit an exact representation

$\dot z=\Lambda z+\sum_{i=1}^{m}u_i B_i z.$8

where $\dot z=\Lambda z+\sum_{i=1}^{m}u_i B_i z.$9 is a bounded linear operator from $$z_{k+1}=Az_k+Bu_k+H(z_k\otimes u_k),\qquad x_k=Cz_k.$$0 to $$z_{k+1}=Az_k+Bu_k+H(z_k\otimes u_k),\qquad x_k=Cz_k.$$1. 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 $$z_{k+1}=Az_k+Bu_k+H(z_k\otimes u_k),\qquad x_k=Cz_k.$$2 and the state/input domains (Morris et al., 9 Jun 2026).

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 $$z_{k+1}=Az_k+Bu_k+H(z_k\otimes u_k),\qquad x_k=Cz_k.$$3 are not formally addressed, even though stabilizability assumptions and CLF conditions are analyzed in detail (Dony, 13 May 2025).

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 $$z_{k+1}=Az_k+Bu_k+H(z_k\otimes u_k),\qquad x_k=Cz_k.$$4, forms

$$z_{k+1}=Az_k+Bu_k+H(z_k\otimes u_k),\qquad x_k=Cz_k.$$5

and computes

$$z_{k+1}=Az_k+Bu_k+H(z_k\otimes u_k),\qquad x_k=Cz_k.$$6

Eigenvalues and eigenvectors of $$z_{k+1}=Az_k+Bu_k+H(z_k\otimes u_k),\qquad x_k=Cz_k.$$7 provide approximate Koopman eigenfunctions; a generator approximation and the difference

$$z_{k+1}=Az_k+Bu_k+H(z_k\otimes u_k),\qquad x_k=Cz_k.$$8

yield the control operator, which is then transformed into Koopman eigenfunction coordinates. In that development, the bilinear model is written as

$$z_{k+1}=Az_k+Bu_k+H(z_k\otimes u_k),\qquad x_k=Cz_k.$$9

and the approximation error in $$\dot{x}=F(x)+\sum_{i=1}^{m}G_i(x)u_i,\qquad x\in X\subseteq \mathbb{R}^n.$$0 and $$\dot{x}=F(x)+\sum_{i=1}^{m}G_i(x)u_i,\qquad x\in X\subseteq \mathbb{R}^n.$$1 is stated to decay as $$\dot{x}=F(x)+\sum_{i=1}^{m}G_i(x)u_i,\qquad x\in X\subseteq \mathbb{R}^n.$$2 with data length $$\dot{x}=F(x)+\sum_{i=1}^{m}G_i(x)u_i,\qquad x\in X\subseteq \mathbb{R}^n.$$3 (Huang et al., 2019).

For partially observed systems, the lifted state can be treated as latent. One paper writes the observable dynamics as a bilinear hidden Markov model,

$$\dot{x}=F(x)+\sum_{i=1}^{m}G_i(x)u_i,\qquad x\in X\subseteq \mathbb{R}^n.$$4

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 (Otto et al., 2022).

A further generalization lifts states and inputs independently and couples them through Khatri-Rao or tensor features. In that formulation,

$$\dot{x}=F(x)+\sum_{i=1}^{m}G_i(x)u_i,\qquad x\in X\subseteq \mathbb{R}^n.$$5

with $$\dot{x}=F(x)+\sum_{i=1}^{m}G_i(x)u_i,\qquad x\in X\subseteq \mathbb{R}^n.$$6 and $$\dot{x}=F(x)+\sum_{i=1}^{m}G_i(x)u_i,\qquad x\in X\subseteq \mathbb{R}^n.$$7. The associated multi-step estimator stacks time-shifted Khatri-Rao feature blocks over trajectory windows and solves a single ridge regression

$$\dot{x}=F(x)+\sum_{i=1}^{m}G_i(x)u_i,\qquad x\in X\subseteq \mathbb{R}^n.$$8

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 (Lazar, 1 Jun 2026).

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

$$\dot{x}=F(x)+\sum_{i=1}^{m}G_i(x)u_i,\qquad x\in X\subseteq \mathbb{R}^n.$$9

or its Koopman-eigenfunction variant with $\tilde g(t,x)$0, a quadratic control Lyapunov function

$\tilde g(t,x)$1

is used. The main stabilizability condition is identical across several papers: the system is quadratically stabilizable if and only if there exists $\tilde g(t,x)$2 such that, for every nonzero $\tilde g(t,x)$3,

$\tilde g(t,x)$4

This ensures that whenever the drift term does not decrease $\tilde g(t,x)$5, the bilinear control term has nonzero authority (Huang et al., 2018, Huang et al., 2019).

The search for $\tilde g(t,x)$6 is commonly posed as a convex semidefinite program. A representative formulation is

$\tilde g(t,x)$7

subject to

$\tilde g(t,x)$8

This LMI-based procedure is used to construct Koopman-space CLFs and then stabilizing feedback laws (Huang et al., 2018).

Several feedback laws are considered. The simplest are sign or proportional laws based on

$\tilde g(t,x)$9

while a more structured choice uses Sontag’s formula in Koopman coordinates. In one formulation,

$$\frac{\partial}{\partial t}\tilde g=L_F\tilde g+\sum_{i=1}^{m}u_i L_{G_i}\tilde g,$$0

followed by saturation to enforce input bounds. In the earlier stabilization papers, the same idea is given an inverse-optimal interpretation with cost

$$\frac{\partial}{\partial t}\tilde g=L_F\tilde g+\sum_{i=1}^{m}u_i L_{G_i}\tilde g,$$1

for a suitable state penalty $$\frac{\partial}{\partial t}\tilde g=L_F\tilde g+\sum_{i=1}^{m}u_i L_{G_i}\tilde g,$$2 (Huang et al., 2019, Dony, 13 May 2025).

Koopman Lyapunov MPC embeds this CLF directly into prediction. For the lifted bilinear model

$$\frac{\partial}{\partial t}\tilde g=L_F\tilde g+\sum_{i=1}^{m}u_i L_{G_i}\tilde g,$$3

the continuous-time optimization minimizes

$$\frac{\partial}{\partial t}\tilde g=L_F\tilde g+\sum_{i=1}^{m}u_i L_{G_i}\tilde g,$$4

subject to the bilinear dynamics, input bounds, and Lyapunov constraints such as $$\frac{\partial}{\partial t}\tilde g=L_F\tilde g+\sum_{i=1}^{m}u_i L_{G_i}\tilde g,$$5 and

$$\frac{\partial}{\partial t}\tilde g=L_F\tilde g+\sum_{i=1}^{m}u_i L_{G_i}\tilde g,$$6

A corresponding sampled-data stability result states that for sufficiently small $$\frac{\partial}{\partial t}\tilde g=L_F\tilde g+\sum_{i=1}^{m}u_i L_{G_i}\tilde g,$$7, trajectories starting in a stability region $$\frac{\partial}{\partial t}\tilde g=L_F\tilde g+\sum_{i=1}^{m}u_i L_{G_i}\tilde g,$$8 remain in $$\frac{\partial}{\partial t}\tilde g=L_F\tilde g+\sum_{i=1}^{m}u_i L_{G_i}\tilde g,$$9 and satisfy $z=\Psi(x)=[\psi_1(x),\dots,\psi_N(x)]^\top$0. Mapping back to the original state through a smooth inverse $z=\Psi(x)=[\psi_1(x),\dots,\psi_N(x)]^\top$1, the modeling error obeys

$z=\Psi(x)=[\psi_1(x),\dots,\psi_N(x)]^\top$2

yielding practical stability of the original nonlinear system (Dony, 13 May 2025).

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 (Zinage et al., 2022).

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=\Psi(x)=[\psi_1(x),\dots,\psi_N(x)]^\top$3

the lifted bilinear signal

$z=\Psi(x)=[\psi_1(x),\dots,\psi_N(x)]^\top$4

is introduced, and a Hankel matrix built from $z=\Psi(x)=[\psi_1(x),\dots,\psi_N(x)]^\top$5 replaces the classical LTI Hankel matrix. This leads to a DeePC formulation in lifted coordinates with constraints

$z=\Psi(x)=[\psi_1(x),\dots,\psi_N(x)]^\top$6

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 (Xiong et al., 6 May 2025).

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 $z=\Psi(x)=[\psi_1(x),\dots,\psi_N(x)]^\top$7 larger than that of bilinear and nonlinear Koopman MPC, while nonlinear Koopman MPC is more than $z=\Psi(x)=[\psi_1(x),\dots,\psi_N(x)]^\top$8 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 (Bruder et al., 2020).

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

$z=\Psi(x)=[\psi_1(x),\dots,\psi_N(x)]^\top$9

and the generalized bilinear model is

$L_{G_i}\Psi=B_i\Psi,\qquad i=1,\dots,m,$0

where $L_{G_i}\Psi=B_i\Psi,\qquad i=1,\dots,m,$1 and $L_{G_i}\Psi=B_i\Psi,\qquad i=1,\dots,m,$2. 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 $L_{G_i}\Psi=B_i\Psi,\qquad i=1,\dots,m,$3 values of $L_{G_i}\Psi=B_i\Psi,\qquad i=1,\dots,m,$4 in simulation and $L_{G_i}\Psi=B_i\Psi,\qquad i=1,\dots,m,$5 in experiments, outperforming both lifted LTI and naive bilinear alternatives (Yahagi et al., 17 Feb 2026).

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 (Peitz, 2018).

In power systems, the modularized Koopman Bilinear Form represents each distributed energy resource by a local bilinear lifted model,

$L_{G_i}\Psi=B_i\Psi,\qquad i=1,\dots,m,$6

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 (Jiang et al., 2022).

Bilinear lifted models also support nonstandard optimal-control formulations. One paper defines a value bilinear form

$L_{G_i}\Psi=B_i\Psi,\qquad i=1,\dots,m,$7

for a Koopman lift of the closed-loop dynamics, proves that the minimal value function admits a sum-of-squares expansion

$L_{G_i}\Psi=B_i\Psi,\qquad i=1,\dots,m,$8

and derives a Riccati-like quadratic operator equation for the corresponding nuclear operator $L_{G_i}\Psi=B_i\Psi,\qquad i=1,\dots,m,$9. 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 (Breiten et al., 24 Sep 2025).

Robustness to approximation error is an active concern. A set-membership approach models the sampled nonlinear system as

$B_i$0

with residual bound

$B_i$1

and characterizes all matrices $B_i$2 consistent with the data and residual bound. A rational feedback law

$B_i$3

is then synthesized by SOS programming so as to stabilize every bilinear model in the consistent set and every admissible residual (Xie et al., 31 May 2026).

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 (Bruder et al., 2020, Xiong et al., 6 May 2025, Morris et al., 9 Jun 2026).