Data-driven feedback stabilization of nonlinear systems: Koopman-based model predictive control

I. INTRODUCTION

• The introduction frames Koopman theory as a data-driven alternative to complex state-space analysis, whose nonlinear control methods can require computationally intractable differential geometry.Koopman operators evolve observables along system trajectories, and their spectral structures encode global information about nonlinear dynamics.Source: §I · p. 1
• Stability certification for forced systems is difficult because Koopman prediction can be impaired unless manipulated inputs are properly accounted for.Prior approaches include defining the Koopman operator over states and inputs, modifying EDMD, and using a theoretically justified bilinear representation.Source: §I · p. 1
• Earlier Koopman-space MPC used Lyapunov constraints to characterize original-system stability, but its CLF required an explicit model and could be computationally demanding to derive.A prior stabilizing controller for the bilinear system did not solve an optimal control problem or account for explicit state and input constraints.Source: §I · p. 2
• The proposed controller combines a data-derived Koopman bilinear model with a CLF in eigenfunction coordinates to formulate Lyapunov-based MPC.The authors state that the design requires no a priori knowledge of the original system and that CLF search is more computationally affordable for the bilinear model.Source: §I · p. 2

A. Koopman Operator

• The Koopman operator advances observables—functions of the state—forward in time as a linear operator, even when the underlying system is nonlinear.The observable's evolution is governed by the operator acting on the function space.Source: A. Koopman Operator · p. 2
• A Koopman eigenfunction evolves according to its associated eigenvalue and can predict an observable's time evolution when that observable lies in the eigenfunctions' span.The eigenfunction is defined by its relationship to the Koopman operator.Source: A. Koopman Operator · p. 2
• The infinitesimal generator of the Koopman operator is the Lie derivative with respect to the vector field, and evolving observables satisfy a partial differential equation.The Lie derivative is denoted L_F = F · ∇.Source: A. Koopman Operator · p. 2
• A finite set of Koopman eigenfunctions can form an invariant subspace, enabling globally linear representations of strongly nonlinear systems.The representation is finite-dimensional when the selected observables span a subspace invariant under the operator.Source: A. Koopman Operator · p. 3

B. Modeling forced dynamics

• For control-affine dynamics, the observable evolution is governed by Lie derivatives associated with the uncontrolled vector field and each input vector field.The input vector fields encode how the control inputs affect the state dynamics.Source: B. Modeling forced dynamics · p. 3
• The resulting function-space dynamics are analogous to a bilinear system, with infinite-dimensional Lie derivative operators acting on observables.The vector fields are assumed locally Lipschitz continuous.Source: B. Modeling forced dynamics · p. 3
• If a finite observable subspace is invariant under the Koopman operator, its action can be represented by a finite-dimensional matrix.Koopman eigenfunctions are used as a basis for the finite-dimensional approximation.Source: B. Modeling forced dynamics · p. 3

C. Koopman bilinear system identification

• The method uses the Koopman canonical transform to express the system in Koopman eigenfunction coordinates, yielding a transformed controlled-dynamics model.The transformation is applied to the control-affine system.Source: C. Koopman bilinear system… · p. 3
• The finite-dimensional bilinear model uses a block-diagonal system matrix built from Koopman eigenvalues and control matrices associated with the eigenfunctions.The eigenfunction coordinates provide the finite-dimensional representation used for controller design.Source: C. Koopman bilinear system… · pp. 3–4
• The input matrices are represented in the eigenfunction span under the assumption that each input vector field's action on those eigenfunctions lies in that span.The matrices are obtained by equating coefficients, after which the Koopman bilinear system can be constructed for feedback design.Source: C. Koopman bilinear system… · pp. 3–4
• EDMD estimates the uncontrolled system matrix from snapshot data and a selected library of observables, followed by eigendecomposition to obtain eigenfunctions and eigenvalues.The eigenvalues are converted to continuous time, and the system matrix is assembled from them.Source: C. Koopman bilinear system… · pp. 3–4

• The section presents an integration of Koopman operator theory and Lyapunov-based predictive control for stabilizing system (7).Source: §III · p. 4

A. Lyapunov-based predictive control

• For the single-input setup, the stabilization problem seeks a state-dependent feedback law that stabilizes the uncontrolled system's unstable origin within a domain.The results are stated to generalize to multiple inputs.Source: A. Lyapunov-based predictive control · p. 4
• A CLF is a continuously differentiable positive definite function whose derivative can be made negative for every nonzero state in the domain.The derivative is LFV + uLGV < 0.Source: A. Lyapunov-based predictive control · p. 4
• LMPC adds a Lyapunov constraint to MPC, guaranteeing closed-loop stability from a characterized set of initial conditions irrespective of prediction horizon.Reducing the horizon can reduce the optimization problem's size; the proposed Koopman bilinear transformation makes CLF search easier by limiting it to quadratic functions.Source: A. Lyapunov-based predictive control · p. 4

B. Bounded explicit control h(z)

• Under the stabilizability assumption, the Koopman bilinear system admits a quadratic CLF, V(z) = z^T Pz, with P symmetric positive definite.The necessary and sufficient conditions for P are given in Proposition 1.Source: B. Bounded explicit control h(z) · p. 4
• Proposition 1 characterizes stabilizability through a condition on P that ensures the control term can make the CLF derivative negative when its uncontrolled term is nonnegative.Once the conditions hold, the passage introduces a Sontag formula for an explicit stabilizing law.Source: B. Bounded explicit control h(z) · p. 5
• The explicit law saturates at the input bounds, and within a level set of V its CLF derivative is negative definite, ensuring asymptotic stability.The control respects umin ≤ u(t) ≤ umax.Source: B. Bounded explicit control h(z) · p. 5
• The complete stability region is difficult to estimate in practice, even for simple systems.The guaranteed region is described using a level set contained within that complete region.Source: B. Bounded explicit control h(z) · p. 5

C. Koopman Lypaunov-based predictive control

• The LMPC optimizes a performance index over Koopman bilinear predictions while Lyapunov constraints preserve the sample-and-hold safe set Ωˆr ⊂ Ωr.The safe set is used to keep Ωr invariant despite discrete-time implementation.Source: C. Koopman Lypaunov-based predictive… · p. 5
• For z(0) ∈ Ωr and a sufficiently small sampling interval, the controller keeps z(t) in Ωr and ultimately bounds its norm by d.The guarantee applies when ∆ ∈ (0, ∆⋆].Source: C. Koopman Lypaunov-based predictive… · pp. 5–6
• The optimization is feasible throughout Ωr because the explicit stabilizing controller h(z) supplies a feasible trajectory, while the Lyapunov constraints preserve invariance.The constraints also allow the LMPC to inherit h(z)'s stability region without a terminal constraint.Source: C. Koopman Lypaunov-based predictive… · pp. 5–7
• In practice, Ωr's parameters are estimated through closed-loop simulations, and the estimated stability region need not cover the system's entire domain.The estimate may be improved by considering multiple CLFs.Source: C. Koopman Lypaunov-based predictive… · p. 7
• With a C1 inverse mapping and bounded prediction error, the controller guarantees Lyapunov stability of the original system, but not its asymptotic stability.The predicted state remains bounded near equilibrium, and the error between predicted and original states is bounded.Source: C. Koopman Lypaunov-based predictive… · pp. 7–8
• The C1 inverse assumption restricts basis selection; a fitted state map is an alternative only if its optimization error is certified bounded, and uncertainty-induced mismatch is unaddressed.Robust closed-loop stability under disturbances is left for future work.Source: C. Koopman Lypaunov-based predictive… · p. 8

• Both the Van der Pol oscillator and simple pendulum produced closed-loop results stable with respect to the original state space.Source: §IV · p. 8
• The examples demonstrate the proposed provably stable Lyapunov-based predictive controller designed in Koopman function space.Source: §IV · p. 8
• The numerical evaluation covered two illustrative systems: a Van der Pol oscillator and a simple pendulum.Source: §IV · p. 8

A. Van der Pol oscillator

• The unforced oscillator has a limit cycle and an unstable equilibrium at the origin, which is the control objective.Source: A. Van der Pol oscillator · p. 8
• The Koopman bilinear model used 21 lifted states from degree-5 monomials and data from unforced trajectories sampled every 0.01 s.Source: A. Van der Pol oscillator · p. 8
• The controller used a quadratic CLF with Sontag’s formula, identity state weighting, input weight 1, and a 1 s prediction horizon.Source: A. Van der Pol oscillator · pp. 8–9
• The closed-loop Van der Pol system was stabilized at the origin, as shown by the open- and closed-loop trajectory comparison.Source: A. Van der Pol oscillator · p. 9, Figure 2 · p. 9

B. Simple pendulum

• The control objective was stabilization at the origin, with model predictions spanning initial conditions from near the unstable equilibrium to the limit cycle.Source: B. Simple pendulum · p. 9
• The model was identified from unforced trajectories sampled every 0.01 s, using degree-5 monomials to form a 21-dimensional lifted state.Source: B. Simple pendulum · p. 9
• For the pendulum, the controller drove the closed-loop trajectory to the origin while keeping states inside the open-loop limit-cycle stability region.The limit cycle forms the boundary of the basin of attraction in this example.Source: B. Simple pendulum · p. 9
• The example used a simple quadratic-feedback choice for h(z), with k = 10, identity state weighting, input weight 1, and a 1 s horizon.Source: B. Simple pendulum · p. 9

V. CONCLUSIONS

• Koopman lifting yields bilinear models for LMPC with explicit Lyapunov stability constraints, and the bilinear structure makes quadratic CLF search tractable.Universal control approaches such as Sontag’s formula provide the explicit control law required by the LMPC formulation.Source: §V · p. 9
• The numerical examples successfully force unstable dynamics to the origin, with state-space stability supported when a continuously differentiable inverse mapping exists.The conclusion reports closed-loop results for the original state-space and conditions the stability transfer on that inverse mapping.Source: §V · p. 10
• Robustness in the presence of uncertainties remains a focus for future work.Source: §V · p. 10