A Koopman Set-Membership Approach for Nonlinear Data-Driven Control with Stability Guarantees

Abstract: This paper proposes a data-driven controller design method for unknown nonlinear systems based on a Koopman bilinear realization. Using Koopman operator theory, the nonlinear system can be represented as a bilinear discrete-time system with a residual error term. The residual error is proportionally bounded by the norm of the lifted state and input, while the system matrices of the bilinear model are unknown. Assuming that bounds on the residual error are available, the unknown system matrices are characterized via a set-membership representation using the collected input-state data pairs of the nonlinear system. A data-driven controller design method is proposed to ensure stability for all bilinear systems within this set-membership description and for all admissible residual errors. More specifically, we design a rational state-feedback controller that stabilizes the bilinear model with residual error and, consequently, the original nonlinear system, by solving a sum-of-squares (SOS) program. The effectiveness of the proposed approach is demonstrated through numerical examples.

• The paper develops a direct data-driven method that combines Koopman bilinear models, proportional residual bounds, and set-membership quadratic matrix inequalities to characterize every model consistent with sampled data.
• The SOS-based synthesis program designs rational state-feedback controllers and guarantees exponential stability for all consistent lifted systems and admissible residuals, with the convergence rate tunable through an optimization variable.
• A scalar nonlinear example using 200 data pairs demonstrates exponential convergence, while increasing controller degree improves performance but raises computation time from 15.24 to 39.85 seconds; guarantees remain regional and generally probabilistic.

Overview

This paper addresses the problem of stabilizing an unknown continuous-time nonlinear control-affine system using only sampled input–state data, without an intermediate system identification step. The approach combines three ingredients: a Koopman bilinear realization of the nonlinear dynamics with a proportionally bounded residual error, a set-membership (data informativity) characterization of all system matrices consistent with the data, and a sum-of-squares (SOS) program that synthesizes a rational state-feedback controller robust to both matrix uncertainty and residual errors. The main result is an exponential stability guarantee for the original nonlinear closed-loop system under a sampled version of the synthesized controller.

Problem setup

The plant is an unknown control-affine system $\dot{x} = f(x) + \sum_i g_i(x)u_i$ with $f(0)=0$. Via Koopman operator theory and a known lifting function $\Phi \in \mathcal{C}^1$ with $\Phi(0)=0$, the sampled dynamics are represented as a discrete-time bilinear model

$$\Phi(x_{t+1}) = A_s\Phi(x_t) + B_su_t + \tilde{B}_s(u_t\otimes\Phi(x_t)) + r(x_t,u_t),$$

where $A_s$, $B_s$, $\tilde{B}_s$ are unknown. The key structural assumption is that the residual satisfies a proportional bound $\|r(x,u)\| \le c_x\|\Phi(x)\| + c_u\|u\|$ over compact sets $\mathbb{X}$ and $f(0)=0$0, with known coefficients $f(0)=0$1. This type of bound is available from existing Koopman error analyses such as SafeDMD and EDMD-based frameworks, which the authors cite as the source of the coefficients; the paper itself assumes these bounds are given rather than deriving them.

The data consist of $f(0)=0$2 input–state pairs collected from the nonlinear system, lifted through $f(0)=0$3. The goal is to design a controller directly from this data that exponentially stabilizes the origin of the continuous-time nonlinear system.

Set-membership characterization via quadratic matrix inequalities

The set $f(0)=0$4 collects all triples $f(0)=0$5 for which some admissible residual $f(0)=0$6 explains each data point. Building on the quadratic-matrix-inequality machinery of van Waarde et al., the authors show that $f(0)=0$7 admits an exact characterization: $f(0)=0$8 if and only if

$f(0)=0$9

for all SOS multipliers $\Phi \in \mathcal{C}^1$0, where each data point contributes a matrix $\Phi \in \mathcal{C}^1$1 built from the lifted data and the squared residual bound $\Phi \in \mathcal{C}^1$2. This extends prior bilinear informativity results in two ways that matter for the subsequent design: the uncertainty enters through a state-dependent (proportional) rather than constant noise bound, and the model lives in the lifted Koopman coordinates rather than the physical state space.

Rational controller synthesis by SOS programming

A central technical point motivates the choice of controller class: citing Vatani et al., the authors note that unstable discrete-time bilinear systems may be stabilizable by rational feedback but not by any linear or polynomial feedback law. Accordingly, the synthesis targets a rational law $\Phi \in \mathcal{C}^1$3 with polynomial numerator $\Phi \in \mathcal{C}^1$4 and strictly SOS denominator $\Phi \in \mathcal{C}^1$5.

The main theorem states that feasibility of a single SOS condition — linear in the decision variables $\Phi \in \mathcal{C}^1$6, $\Phi \in \mathcal{C}^1$7, $\Phi \in \mathcal{C}^1$8, $\Phi \in \mathcal{C}^1$9, and $\Phi(0)=0$0 once $\Phi(0)=0$1 and the degree $\Phi(0)=0$2 are fixed — guarantees exponential stabilization of every bilinear system in $\Phi(0)=0$3 under every admissible residual. The proof proceeds by two Schur complements and the dualization lemma to convert the SOS condition into a Lyapunov decay inequality of the form

$\Phi(0)=0$4

where the residual terms are dominated using the proportional bound and the data consistency constraint is enforced through the multiplier $\Phi(0)=0$5. The decay rate $\Phi(0)=0$6 can be optimized directly by minimizing $\Phi(0)=0$7 within the same convex program, or fixed to prescribe a desired convergence rate.

The extension to the nonlinear plant follows because the true matrices $\Phi(0)=0$8 belong to $\Phi(0)=0$9: the Lyapunov decay holds for the bilinear realization, and invoking a corollary from the SafeDMD framework yields exponential stability of the continuous-time nonlinear system under the zero-order-hold sampled controller $$\Phi(x_{t+1}) = A_s\Phi(x_t) + B_su_t + \tilde{B}_s(u_t\otimes\Phi(x_t)) + r(x_t,u_t),$$0.

Two caveats deserve emphasis. First, the proportional residual bound can generally only be certified over suitable compact sets $$\Phi(x_{t+1}) = A_s\Phi(x_t) + B_su_t + \tilde{B}_s(u_t\otimes\Phi(x_t)) + r(x_t,u_t),$$1, $$\Phi(x_{t+1}) = A_s\Phi(x_t) + B_su_t + \tilde{B}_s(u_t\otimes\Phi(x_t)) + r(x_t,u_t),$$2 with certain probability, so closed-loop stability of the nonlinear system holds probabilistically and only for initial conditions inside a subset of $$\Phi(x_{t+1}) = A_s\Phi(x_t) + B_su_t + \tilde{B}_s(u_t\otimes\Phi(x_t)) + r(x_t,u_t),$$3. Second, feasibility depends on the chosen denominator $$\Phi(x_{t+1}) = A_s\Phi(x_t) + B_su_t + \tilde{B}_s(u_t\otimes\Phi(x_t)) + r(x_t,u_t),$$4 and degree $$\Phi(x_{t+1}) = A_s\Phi(x_t) + B_su_t + \tilde{B}_s(u_t\otimes\Phi(x_t)) + r(x_t,u_t),$$5; the authors suggest increasing $$\Phi(x_{t+1}) = A_s\Phi(x_t) + B_su_t + \tilde{B}_s(u_t\otimes\Phi(x_t)) + r(x_t,u_t),$$6, changing $$\Phi(x_{t+1}) = A_s\Phi(x_t) + B_su_t + \tilde{B}_s(u_t\otimes\Phi(x_t)) + r(x_t,u_t),$$7, or enlarging the dataset when the program is infeasible, but no completeness guarantee is provided.

Relation to prior work

Compared with the authors' earlier bilinear min–max MPC scheme, the present setting differs in three respects: the disturbance is proportionally bounded rather than constantly bounded, the model is expressed in lifted Koopman coordinates, and consequently both the data characterization and the stability proof differ. Compared with identify-then-design Koopman approaches based on EDMD, the contribution is directness — the controller is synthesized from raw data without first estimating $$\Phi(x_{t+1}) = A_s\Phi(x_t) + B_su_t + \tilde{B}_s(u_t\otimes\Phi(x_t)) + r(x_t,u_t),$$8, avoiding identification error compounding while retaining rigorous guarantees.

Numerical results

The method is demonstrated on the scalar system $$\Phi(x_{t+1}) = A_s\Phi(x_t) + B_su_t + \tilde{B}_s(u_t\otimes\Phi(x_t)) + r(x_t,u_t),$$9 with lifting function $A_s$0, sampling time $A_s$1 s, and regions $A_s$2, $A_s$3. The residual bounds are small ($A_s$4, $A_s$5), and $A_s$6 data pairs construct $A_s$7. With degree $A_s$8 and denominator $A_s$9, the closed-loop trajectories converge exponentially from $B_s$0.

Adding the objective $B_s$1 substantially accelerates convergence, and raising the degree to $B_s$2 further improves the decay rate at a quantified computational cost: solve times increase from 15.24 s ($B_s$3) to 39.85 s ($B_s$4). This illustrates a concrete trade-off between closed-loop performance and tractability governed by the controller degree.

Limitations and open questions

The framework inherits several assumptions that bound its applicability. The lifting function must be known a priori and chosen appropriately; the residual bounds $B_s$5, $B_s$6 must be available over the operating sets, and since such bounds typically hold only probabilistically, the stability certificate for the nonlinear system is correspondingly probabilistic and regional. Feasibility of the SOS program is not guaranteed for arbitrary $B_s$7 and $B_s$8, and scalability to systems with high-dimensional lifted states remains untested — the numerical study is limited to a scalar example. Open questions include how to select the denominator $B_s$9 systematically, how dataset size affects conservatism of $\tilde{B}_s$0, and whether the approach extends to output-feedback or constrained settings.

Conclusion

The paper provides a direct data-driven stabilization method for unknown nonlinear control-affine systems, coupling Koopman bilinearization with proportional residual bounds, a QMI-based set-membership characterization, and SOS-based rational feedback design. Its principal contribution is a convex program whose feasibility certifies exponential stability of all consistent bilinear models and hence of the underlying nonlinear system under sampled control, with an tunable Lyapunov decay rate. The guarantees are regional and, in general, probabilistic, and validation is limited to low-dimensional examples.