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Constrained Koopman Control Parametrization: Convex Controller Synthesis for Tracking under Input Constraints

Taha Ondogan, Ran Jing, Andrew P. Sabelhaus, Roberto Tron

arXiv:2610.29984v1eess.SYcs.SY

TL;DR

Koopman-based control makes nonlinear control design more tractable, but incorporating input constraints for tracking remains challenging. The paper develops constrained KoCoP and a CLF-based tracking formulation; experiments report comparable tracking errors and faster control-problem runtimes than MPC baselines on two nonlinear systems.

  • Problem

    Koopman-based controller synthesis still faces challenges incorporating input constraints, which limits its use in constrained tracking tasks.

    Source: §I · p. 1

  • Method

    The paper extends KoCoP with convex conditions for polytopic or ellipsoidal input constraints and a CLF-based formulation for tracking time-varying references.

    Source: §I · p. 2

  • Results

    The CLF method has the fastest reported runtimes on both systems while achieving comparable MSE performance to MPC baselines.

    Source: D. Comparison of Computational Runtimes · p. 14

  • Takeaways & Limitations

    The experiments demonstrate constrained reference tracking on a double pendulum and a unicycle using the proposed approach.

    Source: §VIII · p. 14, D. Comparison of Computational Runtimes · p. 14

Abstract

from arXiv · show

Koopman operator theory represents nonlinear systems with linear and bilinear terms in the state and input observables, greatly simplifying control design. However, incorporating input constraints into Koopman-based controller synthesis remains challenging, limiting their applicability to constrained control tasks such as tracking. This paper develops a constrained Koopman Control Parametrization (cKoCoP) framework for synthesizing state-feedback controllers that satisfy prescribed input constraints with Lyapunov-certified stability of the identified lifted closed loop. The cKoCoP offers nonlinear state-feedback via linear-in-observables feedback, enabling linear controller design. The proposed controller is obtained through convex optimization: we incorporate ellipsoidal or polytopic input constraints with the S-procedure or Farkas-lemma-based conditions. We also introduce a Control Lyapunov Function (CLF) for trajectory tracking which incorporates input constraints. The approach is validated on two canonical nonlinear systems, a double pendulum and a unicycle, demonstrating data-driven tracking while respecting input constraints. Experiments validate the method's robustness to measurement noise, and runtime improvements over Koopman Model Predictive Control (KMPC) baselines. The results demonstrate that cKoCoP achieves competitive tracking performance with substantially reduced online computational cost, supporting its potential for real-time constrained tracking applications.

I. INTRODUCTION

Koopman methods offer linear-system tools for nonlinear control, but data-driven models, input constraints, and online MPC costs complicate constrained tracking. This paper extends KoCoP with constrained controller synthesis and feedforward and CLF-based tracking, against a background of Koopman preliminaries.

I. INTRODUCTION

  • Input constraints are difficult to enforce in lifted coordinates, where state-space constraints can become nonconvex, while constrained MPC adds online computational cost.Koopman model mismatch makes actuation limits especially important; the original KoCoP also omitted actuation limits and only stabilized the origin.Source: §I · p. 1
  • The paper’s contributions include constrained KoCoP, an Alternate Convex Search procedure, and feedforward and CLF-based tracking extensions.The constrained formulation uses Farkas’ Lemma for polytopic constraints and the S-procedure for ellipsoidal constraints, preserving convexity for a fixed Lyapunov matrix.Source: §I · pp. 1–2

II. NOTATION AND PRELIMINARIES

  • The notation distinguishes trajectory time t from dataset sample index n.Vector and matrix entries are denoted by v_i and [M]_ij, respectively.Source: §II · p. 2

A. Kronecker Product

  • The Kronecker product is defined by block entries A_i,j B, and the paper states that it distributes over matrix multiplication.A corollary gives a related identity for a ⊗ (Mb).Source: A. Kronecker Product · p. 2

B. Koopman Operator and Approximation of a Controlled Nonlinear System

  • Koopman observables map states into a higher-dimensional space, and finite-dimensional approximations are used in practice.For controlled systems, a purely linear model generally cannot capture state-input cross-coupling, motivating a bilinear Koopman representation.Source: B. Koopman Operator and Approximation… · p. 2

C. Lyapunov Stability and Schur Complement

The section links Lyapunov stability and invariant sublevel sets to convex containment tests, then describes a lifted Koopman model with linear and bilinear input effects and a feedback law linear in observables.

C. Lyapunov Stability and Schur Complement

  • A discrete-time linear system is asymptotically stable when a positive-definite Lyapunov matrix satisfies the stated decay condition, and its Lyapunov sublevel sets are positively invariant.The condition can also be written using a Schur complement.Source: C. Lyapunov Stability and Schur… · pp. 2–3

D. S-procedure and Farkas’ Lemma

  • The S-procedure certifies containment between centered ellipsoids, while Farkas’ lemma supplies a nonnegative-matrix condition for containment between polytopes.For a single quadratic constraint, the stated S-procedure condition is also necessary under its strict-feasibility condition.Source: D. S-procedure and Farkas’ Lemma · p. 3

III. KOOPMAN CONTROL PARAMETRIZATION

  • The Koopman model separates drift, linear input, and bilinear state-input effects, and uses feedback linear in observables to obtain a linear closed-loop form.The formulation makes the direct linear input term explicit, without requiring a constant observable as in the prior formulation.Source: §III · pp. 3–4

A. System Identification of the Koopman Control Parametrization

The identification procedure uses state-input data to fit the Koopman control model, separating the independent fit for C from the coupled selection and fitting of H and S.

A. System Identification of the Koopman Control Parametrization

  • The procedure identifies the Koopman matrices from state-input pairs collected under arbitrary controllers, using lifted and paired-lifted datasets.The matrices identified include Kxx, KB, Kxu, H, C, and S.Source: A. System Identification of the… · p. 4
  • C is identified independently by least squares, whereas H and S are coupled and cannot be solved directly through a least-squares fit.The method represents S with a binary selection vector s and elementwise multiplication.Source: A. System Identification of the… · p. 4
  • The procedure starts with all observables selected, then considers each selection-vector entry separately and turns entries off using a threshold before refitting H.After C, H, and S are identified, the remaining Koopman matrices are identified from the data.Source: A. System Identification of the… · p. 4

B. Lyapunov Function Selection and Control Design via Alternating Optimization for the Koopman Control Parametrization

The section develops Lyapunov-based controller design through alternating convex optimization and extends it to input constraints enforced over specified state or lifted-state regions. Ellipsoidal and polytopic constraints use different convex containment conditions.

B. Lyapunov Function Selection and Control Design via Alternating Optimization for the Koopman Control Parametrization

  • A quadratic Lyapunov candidate in the observables is used to formulate a stability-design problem, but its variables make the problem biconvex.The cited algorithm alternates between updating the Lyapunov matrix and controller.Source: B. Lyapunov Function Selection and… · pp. 4–5

1) Alternate Convex Search:

IV. CONTROL UNDER INPUT CONSTRAINTS

  • Input limits are guaranteed while the lifted state remains in a predefined set; ellipsoidal constraints use the S-procedure and polytopic constraints use Farkas’ lemma.The polytopic formulation yields linear constraints and is described as computationally faster; the ellipsoidal formulation adds a semidefinite constraint.Source: §IV · p. 5
  • The constraint construction can start from an allowable state-space set and guarantee input admissibility for all states in that set, rather than requiring a set specified in lifted coordinates.For nonlinear lifting, a state-space ellipsoid need not map to an ellipsoid in observable space.Source: §IV · p. 5

1) Ellipsoidal Constraints in the Embedding Space:

The section develops ellipsoidal and polytopic input-constraint formulations for sets specified in observable or state space, using data-based approximations when needed.

1) Ellipsoidal Constraints in the Embedding Space:

  • The S-procedure and Schur complements yield an ellipsoidal-constraint design problem, with biconvexity handled by alternating convex search.The procedure preserves convexity for a fixed Lyapunov matrix.Source: 1) Ellipsoidal Constraints in the… · p. 6

2) Ellipsoidal Constraints in State Space:

  • For state-space ellipsoids, the method lifts admissible dataset points and fits a minimum-volume ellipsoid that outer-approximates their lifted image.The approximation covers the lifted image over the available dataset.Source: 2) Ellipsoidal Constraints in State… · p. 6

1) Polytopic constraints in embedding space:

  • For polytopic input limits over an observable-space polytope, a nonnegative-matrix condition makes the constraint linear and convex in the design problem.Alternating optimization handles the remaining biconvexity.Source: 1) Polytopic constraints in embedding… · p. 6

2) Polytopic Constraints in the State Space via Box Approximation:

  • For state-space polytopes, the box-approximation approach bounds each observable feature using state bounds or a conservatively defined axis-aligned box.The supplied passages identify the special case of box-constrained states and a conservative box for general polytope data.Source: 2) Polytopic Constraints in the State… · p. 6

3) Polytopic Constraints in the State Space via Robust Optimization:

  • A robust linear-programming formulation uses data to obtain a tighter observable-space polytope than the conservative larger polytope, under approximate lifting consistency over data-point convex combinations.The consistency assumption is ψp ≈ ψ(Xp) for nonnegative convex-combination weights.Source: 3) Polytopic Constraints in the State… · pp. 6–7

V. STABILIZATION AROUND A NON-ZERO EQUILIBRIUM

The method extends KoCoP stabilization from the origin to a nonzero setpoint by adding a feedforward term and formulating dynamics in the lifted error coordinates.

V. STABILIZATION AROUND A NON-ZERO EQUILIBRIUM

  • A feedforward term extends KoCoP stabilization from the origin to an arbitrary setpoint by regulating the corresponding error dynamics.The controller is applied around a given reference setpoint.Source: §V · p. 7

A. Error Dynamics Around a Nonzero Equilibrium

  • The precomputed origin-stabilizing controller acts on the lifted error, while the feedforward term supplies the required steady-state input.The lifted error is ψu,t − ψref.Source: A. Error Dynamics Around a Nonzero… · p. 7

A. Error Dynamics Around a Nonzero Equilibrium:

  • The feedback-feedforward construction extends the standard linear-system decomposition to bilinear Koopman dynamics and yields an error equation.The passage describes the linear-system decomposition as standard and derives an analogous construction for bilinear Koopman systems.Source: A. Error Dynamics Around a Nonzero… · p. 7

B. Stabilization Around a Known Constant Equilibrium

For a constant reference, the method centers the lifted closed-loop dynamics on the reference and selects feedforward input to obtain a linear error relation.

B. Stabilization Around a Known Constant Equilibrium

  • The error dynamics are recentered around a constant reference using its steady-state Koopman dynamics and the feedforward controller.The formulation assumes the corresponding equilibrium input is known before extending to the unknown-input case.Source: B. Stabilization Around a Known… · pp. 7–8
  • For fixed reference and feedforward terms, the resulting closed-loop Koopman matrix expresses the error dynamics linearly in the error state.The passage notes that an unknown equilibrium input can be estimated from the inverse Koopman dynamics.Source: B. Stabilization Around a Known… · p. 8
  • Enforcing a known equilibrium exactly in the identified bilinear Koopman model eliminates equilibrium modeling error and reduces steady-state tracking error.The equilibrium constraint can be incorporated during model identification.Source: B. Stabilization Around a Known… · p. 8

C. Tracking of a Time-varying Trajectory with a CLF-based Method

The CLF formulation adapts feedforward selection to moving references, with a discrete-time stability condition and a control design linear in the feedforward term for fixed state and next error.

C. Tracking of a Time-varying Trajectory with a CLF-based Method

  • A CLF-based formulation changes the constant feedforward term into a time- and state-dependent choice for tracking a moving reference trajectory.The formulation uses V(ex) = e⊤Pe as its CLF.Source: C. Tracking of a Time-varying… · p. 8
  • Keeping the next reference observable explicit removes the static-equilibrium requirement, so the error formulation accommodates time-varying references.The discrete-time CLF condition includes a relaxation variable when the stability condition is infeasible at a timestep.Source: C. Tracking of a Time-varying… · p. 9
  • For fixed current observable and next-state error, bilinearity makes the tracking formulation linear in the feedforward term.This linearity facilitates control design.Source: C. Tracking of a Time-varying… · p. 9

1) CLF-based Tracking under Ellipsoidal Constraints:

The tracking controller adds a time-varying feedforward term computed online by a CLF optimization, with input constraints enforced for either ellipsoidal or polytopic sets. The procedure combines constrained controller synthesis with data-driven model identification and feedforward computation.

2) CLF-based Tracking under Polytopic Constraints:

VI. SUMMARY OF THE PROPOSED APPROACH:

  • The design workflow selects observable dictionaries, collects data under random control, identifies a bilinear Koopman model, synthesizes a stabilizing controller, and computes a setpoint or trajectory feedforward term.The feedforward computation uses the constant-setpoint formula or the CLF optimization, depending on the reference.Source: §VI · p. 9

1) CLF-based Tracking under Ellipsoidal Constraints:

  • The CLF tracking formulation computes a time- and state-dependent feedforward term, so the reference need not be a static equilibrium.The online CLF optimization computes the tracking input subject to the prescribed input constraints.Source: §VI · pp. 9–10

VII. EXPERIMENTS AND RESULTS ON DYNAMICAL SYSTEMS

  • The experiments evaluate tracking on dynamical systems and compare the proposed controllers with Koopman MPC baselines.The section also describes the models, datasets, lifting functions, and constraint sets used for evaluation.Source: §VII · p. 10

1) Inverted Double Pendulum:

The evaluation uses an inverted double pendulum and a unicycle, each with system-specific observables, datasets, and input constraints. In setpoint tracking, CLF-based online feedforward updates produce faster transients, while constant feedforward-feedback is computationally faster.

1) Inverted Double Pendulum:

  • The double-pendulum evaluation uses 900 trajectories of 100 samples at Ts = 0.02 s and 13 state observables with 6 feedback observables.The observables capture trigonometric terms and combinations of velocities; the input constraints are enforced over a state-space polytope.Source: 1) Inverted Double Pendulum · p. 10

2) Unicycle Model:

  • The unicycle dataset contains 60,000 samples from 600 initial conditions, with state observables for position, orientation, sinusoidal orientation functions, and heading error.Feedback observables use heading error and distance from the origin; the sampling time is Ts = 0.01 s.Source: 2) Unicycle Model · p. 10

A. Setpoint Tracking Performance

  • The CLF-based controller reaches a constant setpoint with a faster transient, while constant feedforward-feedback takes 0.38 ms per step versus 3.16 ms for CLF-based control.The CLF controller updates its feedforward term online and enforces a one-step Lyapunov decrease.Source: A. Setpoint Tracking Performance · p. 11

B. Reference Trajectory Tracking Performance

The CLF controller tracks prescribed trajectories across two systems and remains effective under tested noise, with error increasing gradually as noise rises. The authors report faster control-step runtimes than MPC baselines with comparable MSE, while noting that modeling errors remain unaddressed.

B. Reference Trajectory Tracking Performance

  • Across the double pendulum and unicycle, the CLF controller tracks references from same and different initial conditions while keeping forces within prescribed admissible sets.The evaluation includes a lemniscate reference for the double pendulum and a sinusoidal slalom reference for the unicycle.Source: B. Reference Trajectory Tracking… · p. 12, Figure 3 · p. 12

C. Trajectory Tracking Under Noise and Disturbance

D. Comparison of Computational Runtimes

VIII. CONCLUSION

  • The method addresses constrained stabilization and tracking, but does not directly address modeling errors between the Koopman and actual systems.The authors identify integral-based control terms as future work for compensating steady-state errors.Source: §VIII · pp. 14–15

APPENDIX

The appendix considers whether the quadratic Lyapunov condition remains valid when the structural relation used by the model holds only approximately. It gives a sufficient S-procedure condition for bounded mismatch.

A. Validity of the Lyapunov Condition under Model Mismatch

  • The appendix examines approximate satisfaction of the structural relation Sψx ⊗ψu = Hψx and bounds the resulting mismatch relative to the lifted state.The analysis concerns validity of the Lyapunov condition under this model mismatch.Source: A. Validity of the Lyapunov Condition… · p. 15, APPENDIX · p. 15
  • A sufficient condition for Lyapunov decrease under all admissible bounded mismatch signals is feasibility of an S-procedure condition with scalar τ ≥ 0.The nominal closed-loop matrix is used to describe the dynamics under the exact structural assumption.Source: A. Validity of the Lyapunov Condition… · p. 15
  • If the sufficient condition is feasible with P ≻ 0, λ ∈ [0, 1), and τ ≥ 0, P remains a valid Lyapunov matrix for the mismatched closed-loop dynamics.Source: A. Validity of the Lyapunov Condition… · p. 15

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