[Paper Review] Physics-Based Causal Lifting Linearization of Nonlinear Control Systems Underpinned by the Koopman Operator

Nicholas Selby, Filippos E. Sotiropoulos+1 more|arXiv (Cornell University)|Aug 24, 2021

Model Reduction and Neural Networks4 citations

TL;DR

This paper proposes two physics-informed methods to resolve the causality problem in Koopman-based lifting linearization of nonlinear control systems: (1) replacing anticausal observables with their integral variables, and (2) augmenting the system with small inertial or capacitive elements to break input dependency. The methods enable accurate, causal, data-driven modeling using physically measured observables, significantly outperforming state-of-the-art approaches in numerical benchmarks.

ABSTRACT

Methods for constructing causal linear models from nonlinear dynamical systems through lifting linearization underpinned by Koopman operator and physical system modeling theory are presented. Outputs of a nonlinear control system, called observables, may be functions of state and input, $\\phi(x,u)$. These input-dependent observables cannot be used for lifting the system because the state equations in the augmented space contain the time derivatives of input and are therefore anticausal. Here, the mechanism of creating anticausal observables is examined, and two methods for solving the causality problem in lifting linearization are presented. The first method is to replace anticausal observables by their integral variables $\\phi^*$, and lift the dynamics with $\\phi^*$, so that the time derivative of $\\phi^*$ does not include the time derivative of input. The other method is to alter the original physical model by adding a small inertial element, or a small capacitive element, so that the system's causal relationship changes. These augmented dynamics alter the signal path from the input to the anticausal observable so that the observables are not dependent on inputs. Numerical simulations validate the effectiveness of the methods.

Motivation & Objective

• Address the fundamental causality problem in Koopman-based lifting linearization when observables depend on exogenous inputs.
• Identify the physical origin of anticausal observables in nonlinear control systems, particularly in energy-dissipative elements.
• Develop causality-respecting lifting methods that preserve physical interpretability and leverage real sensor measurements.
• Enable accurate, data-driven system identification for nonlinear control systems without relying on synthetic, noise-sensitive observables.
• Bridge the gap between Koopman-based modeling of autonomous systems and practical control system design using measured inputs and outputs.

Proposed method

• Replace input-dependent observables φ(x,u) with their integral variables φ* = ∫φ dt, ensuring the time derivative of φ* does not involve ẋ or Ẇ, thus restoring causality.
• Augment the physical system with a small inertial element (mass) or capacitive element (spring) to alter signal paths, decoupling the anticausal observable from future input dependence.
• Use the resulting augmented dynamics to derive a causal lifted state space model under the Koopman operator framework.
• Formulate the lifted system using observable functions derived from physical laws, ensuring compatibility with dynamic mode decomposition (DMD) and system identification.
• Apply the method to nonlinear spring-damper systems with nonlinear constitutive laws, validating performance against KSOS and AL2/IL2 baselines.
• Demonstrate robustness to measurement noise by comparing models built from real observables (DFL) versus synthetic observables (KSOS), showing superior noise resilience in the proposed method.

Experimental results

Research questions

• RQ1 Why do input-dependent observables in nonlinear control systems lead to anticausal lifted models under Koopman-based lifting?
• RQ2 How can the causality problem be traced to the physical structure of energy-dissipative elements in control systems?
• RQ3 Can physical system augmentation (e.g., adding mass or capacitance) restore causality in lifted models without altering system behavior?
• RQ4 Does replacing anticausal observables with their integral counterparts eliminate causality violations while preserving model accuracy?
• RQ5 How does the proposed method compare in accuracy and noise robustness to existing Koopman-based lifting techniques using synthetic observables?

Key findings

• The integral-based method (AL2) outperforms the inertial augmentation method (IL2) in median error (1.0 vs. 1.6) on a nonlinear spring-damper system with random inputs.
• The IL2 method achieves lower 90th percentile error (2.1 vs. 2.4), indicating more consistent performance across diverse input signals.
• The proposed methods significantly reduce model error compared to KSOS, especially under noisy conditions, where KSOS degrades rapidly due to noise amplification in synthetic observables.
• DFL (Directly using physical observables) shows superior noise resilience compared to KSOS, confirming the advantage of measuring real signals over computing synthetic ones.
• Improving basis functions in KSOS (e.g., higher-order polynomials or Fourier bases) yields minimal performance gain without physical insight into system connectivity.
• The causality problem arises specifically from energy-dissipative elements whose outputs depend on exogenous inputs, while energy-storing elements (e.g., springs) do not introduce anticausality.