Derivative-Based Koopman Operators for Real-Time Control of Robotic Systems

I. INTRODUCTION

• Existing control approaches can be system-specific, structurally restricted, computationally prohibitive, or dependent on full dynamics knowledge.For underwater robots, underactuation, nonlinear behavior, and difficult, time-varying fluid environments further challenge real-time control.Source: §I · p. 1
• The paper selects Koopman basis functions from higher-order state derivatives and uses Taylor-series error bounds to choose their order for a desired accuracy.The dynamics need not be known, and the data-driven least-squares operator has a closed-form solution.Source: §I · p. 2

A. Koopman Operator

• The Koopman operator evolves functions of nonlinear states linearly, enabling linear optimal-control tools, but is generally infinite-dimensional.Practical use therefore motivates finite-dimensional approximations that retain acceptable accuracy.Source: §I · p. 1, A. Koopman Operator · p. 2, B. Koopman Invariant Subspaces · p. 2

B. Koopman Invariant Subspaces

• Finite-dimensional invariant Koopman subspaces have been found for only a few systems, leaving approximation accuracy dependent on choosing suitable basis functions.Existing data-driven regression assumes those basis functions are already known.Source: B. Koopman Invariant Subspaces · pp. 2–3, §I · p. 2

C. Data-driven Finite-dimensional Approximation to Koop-

• A finite-dimensional Koopman operator is fitted by least squares from measured initial states, next states, and corresponding control inputs.The observables may include both states and control terms.Source: C. Data-driven Finite-dimensional… · p. 3
• Training transitions must use a consistent time spacing.The operator estimate has a closed-form solution using the Moore-Penrose pseudoinverse.Source: C. Data-driven Finite-dimensional… · p. 3
• The Koopman framework identifies a linear state transformation useful for control rather than directly estimating nonlinear dynamics or unknown parameters.The operator can also be approximated with standard regression methods such as ridge or lasso.Source: C. Data-driven Finite-dimensional… · p. 3

D. LQR on Koopman Operator

• LQR feedback uses the Koopman model’s state-dependent observables, with gains calculated from its state and input dynamics.The objective penalizes deviations from desired observables and applied control; with fixed gains, control updates use only state observables.Source: D. LQR on Koopman Operator · p. 3

III. SYNTHESIS OF BASIS FUNCTIONS FOR ERROR-BOUNDED KOOPMAN REPRESENTATION

• Higher-order derivative observables increasingly capture local nonlinear dynamics, and exactly capture higher derivatives when a finite-dimensional invariant subspace exists.The representation uses derivatives of the nonlinear dynamics, which need not be known in advance.Source: §III · pp. 3–4

A. Error Bounds of Derivative-Based Koopman Operators

• Taylor-series error analysis extends beyond one-step local error to bound prediction errors over multiple time steps.The analysis models error in each propagated observable to bound error in the original function.Source: §III · p. 4, A. Error Bounds of Derivative-Based… · p. 5

III. SYNTHESIS OF BASIS FUNCTIONS FOR ERROR-BOUNDED KOOPMAN REPRESENTATION

• Bounds can select the lowest-order representation that meets a desired accuracy for a chosen prediction horizon.State derivatives may be numerically estimated from measurements when the dynamics are unknown.Source: §III · p. 4

A. Error Bounds of Derivative-Based Koopman Operators

• The error bound depends on the prediction horizon, derivative order, and magnitude of the next derivative.The theorem considers propagation of a function and its first n derivatives; its bound uses the (n+1)th time derivative.Source: A. Error Bounds of Derivative-Based… · p. 5

• The work provides prediction-error bounds for general nonlinear dynamics over arbitrary future time steps, beyond earlier single-step Taylor bounds.The authors describe this as the first such general-dynamics result to their knowledge.Source: where T ≜k∆t is the prediction time… · p. 5
• The state-and-derivative basis propagates each state independently, while control inputs can be handled as dummy states with their own derivatives.Source: where T ≜k∆t is the prediction time… · p. 6
• For a fixed prediction horizon, the error bound can guide the number of derivatives per state; alternatively, it can bound the horizon for a chosen derivative count.Derivative orders may differ across states.Source: where T ≜k∆t is the prediction time… · pp. 5–6
• The analytical error bound is not guaranteed for the data-driven operator, but remains relevant empirically when that operator resembles the Taylor-series structure.The bound can still indicate how amenable the dynamics are to a linear representation.Source: where T ≜k∆t is the prediction time… · p. 6

B. Error Bound Estimation Using Data-Driven Operator

• A data-driven operator can use the Taylor-based error formula when its structure is sufficiently close to the Taylor-series form.For data-derived operators, these are sound bound estimates rather than strictly enforced bounds.Source: B. Error Bound Estimation Using… · p. 6
• The training data should represent the state space: large local deviations from Taylor structure are discouraged by least squares, but not guaranteed to be bounded.The derivation uses a linear representation that approximates the Taylor-series structure.Source: B. Error Bound Estimation Using… · p. 6
• The maximum one-step training error estimates the magnitude of the first derivative order omitted from the basis, which feeds the multi-step error estimate.When no analytical dynamics model is available, that derivative magnitude can instead be estimated numerically from measurements.Source: B. Error Bound Estimation Using… · p. 7
• Koopman observables are populated with states and their derivatives, computed analytically when dynamics are known or estimated numerically from measurements otherwise.The process can continue to higher derivative orders to add basis functions.Source: C. Synthesis of Derivative-Based… · p. 7

D. Assessment of Error Bound Estimates

• The data-driven operator resembles the Taylor-series structure, supporting the relevance of the error estimates derived from that structure.The operator is compared with the Taylor-based matrix for the single pendulum.Source: D. Assessment of Error Bound Estimates · p. 7, Figure 2 · p. 8
• The data-driven error estimates reasonably predict maximum error, and both estimates and actual errors decline as derivative order increases.The estimate and actual error also show similar slopes over the prediction horizon.Source: D. Assessment of Error Bound Estimates · p. 8
• With noisy measurements, actual maximum error remains almost unchanged; estimates track it closely at low noise and become more conservative at high noise.The unknown-dynamics tests use numerically estimated derivatives and moving-average filtering.Source: D. Assessment of Error Bound Estimates · p. 8
• Error-bound estimates depend on prediction horizon and derivative magnitudes, and can guide basis-function selection for a desired accuracy-complexity balance.The paper does not argue that basis functions should always be augmented.Source: D. Assessment of Error Bound Estimates · p. 8

E. Comparison to alternative system identification methods

• The Koopman model uses the state and its first- and second-order derivatives, estimated numerically; the comparison also uses SINDy, NARX, and a state-only linear model.Each model is used to design MPC, while the Koopman model is also used for LQR feedback.Source: E. Comparison to alternative system… · pp. 8–9
• Koopman-LQR averaged error 6.00 (standard deviation 6.60), Koopman-MPC 6.05 (6.59), NARX 6.48 (7.00), SINDy 6.69 (7.18), and the linear model 7.44 (7.70).These averages are over 30 sampled initial conditions.Source: E. Comparison to alternative system… · p. 9
• Koopman-LQR delivers control performance comparable to Koopman-MPC and offers efficient control computation for online robotic control applications.The reported simulation comparison uses a short planning horizon because NARX is computationally expensive.Source: E. Comparison to alternative system… · p. 9

IV. DATA-DRIVEN CONTROL OF TAIL-ACTUATED ROBOTIC FISH

• The tail-beat frequency is fixed at 2π rad/s, and the modeled fish states include position, orientation, body-frame velocities, and angular velocity.Tail angle is actuated as a sinusoid with amplitude, bias, and frequency parameters.Source: §IV · p. 9
• Each additional derivative order improves the error-bound estimates for the robotic-fish dynamics over the selected prediction horizon.The estimates are computed using the error-bound formula (22).Source: Figure 6 · p. 10
• The computed control variables u1 and u2 are mapped to feasible tail amplitude and bias values using constrained global minimization before each update.The minimization uses Sequential Quadratic Programming.Source: §IV · pp. 10–11
• The fish cannot move backward, so feasible trajectories are expressed as an angle and forward-velocity profile rather than direct position tracking.A target behind the fish can otherwise require an infeasible negative tail amplitude.Source: §IV · p. 11

A. Simulation Results

• The simulation selected derivative order n = 2 as a balance between higher-derivative computation and model accuracy, using states and their first and second derivatives as observables.Separate derivative functions for individual dynamic terms support training across fish morphologies; the resulting model used 54 additional scalar functions.Source: A. Simulation Results · p. 11
• 3.35 was the Koopman-LQR cost over 120 s, while the data-driven linear model had 95% higher error.The fish tracked the desired figure-8 states using fixed LQR gains.Source: Figure 7 · p. 12

B. Experimental Results

• The learned Koopman model reasonably followed experimental fish data for at least five seconds, but single-step fitting can yield an unstable operator and inaccurate long-term predictions.The authors identify stability constraints on the operator as ongoing work.Source: 2) Training Phase · p. 12
• Experimental testing used the same LQR weights across trajectory tasks, with model and gain updates performed online in one implementation.The two implementations computed the model and gains once offline or updated them in real time.Source: 3) Testing Phase · pp. 12–13
• Koopman-LQR outperformed PID across experimental tasks, except that linear-trajectory angle tracking was similar; online updates improved performance under fluid flow.The reported comparison covers linear and circular trajectories, with and without fluid disturbance.Source: Figure 12 · p. 14, 3) Testing Phase · p. 14

Discussion and future work

• The method uses Taylor-series error bounds to select derivative orders for data-driven Koopman bases without requiring known dynamics.LQR is demonstrated as an efficient control design example; without online model updates, its gains are computed once.Source: §V · p. 15
• With numerically estimated derivatives, measurement noise can worsen error-bound estimates, yet moving-average filtering preserves performance at high noise levels.The estimates may still violate the true bounds, though they remain close; more complex dynamics and noise-reduction schemes merit study.Source: §V · p. 15
• Koopman modeling is comparable to SINDy and NARX, while Koopman-based LQR delivers performance comparable to MPC-based methods.Its linear form also supports more efficient controller computation.Source: §V · p. 15
• Underwater robotics is a particularly suitable application because environmental uncertainty, nonlinear dynamics, and limited computation coincide there.The authors consider the method less useful for low-dimensional systems with known dynamics and few nonlinearities.Source: §V · p. 15
• The robotic-fish average model is inaccurate because it omits tail-bias effects on angular velocity at zero amplitude during forward motion.The authors suggest rigid-body fluid equations or SINDy identification as alternatives.Source: §V · p. 15
• The authors propose studying whether time-delay observables and derivative-based observables are equivalent and whether the error bound extends to other Koopman applications.They relate the idea to numerical approximation of derivatives from past measurements and reported gains in predictive power with more delays [81].Source: §V · p. 15

Global error bounds

• The Taylor remainder depends on the n-th derivative evaluated within each time step.The appendix notes that the last error term comes from Lagrange’s remainder formula.Source: Appendix A · p. 16
• The bound concerns the p-th derivative, where p ranges from zero through n and n is the number of derivatives used in propagation.The prediction window is T = k · ∆t.Source: Appendix A · p. 16, Appendix B · p. 17
• The total prediction error combines derivative inaccuracies with Taylor-series expansion error accumulated at each step.The derivation assumes exact function values initially and, in the first iteration, exact derivatives through order n.Source: Appendix A · p. 16

Incremental Update of Koopman Operator

• The update combines P prior measurements with ∆P new measurements, for Ptotal = P + ∆P, and rewrites the operator computation to separate past and new data.The excerpt identifies this as an incremental computation of Anew.Source: A. Incremental Update of Koopman… · p. 18