koopman-stability-preserving-id

name

koopman-stability-preserving-id

description

Stability-preserving system identification using Koopman operator lifting with ISS-LMI constraints. Enables data-driven modeling of nonlinear systems (especially Persidskii-class and electromechanical systems) while guaranteeing input-to-state stability. Use when: (1) identifying nonlinear system models from trajectory data, (2) needing stability guarantees in learned models, (3) working with Persidskii systems or sector-bounded nonlinearities, (4) designing robust observers for partially-measured systems, (5) combining data-driven identification with model predictive control.

Stability-Preserving Koopman System Identification

Overview

Methodology for data-driven system identification that embeds stability guarantees as convex constraints during parameter regression. Combines Koopman operator theory with ISS-LMI analysis for safe learning-based control.

Based on: Pouladi, "Stability Analysis and Data-Driven State Estimation for Generalized Persidskii Systems with Time Delays" (arXiv:2604.27509, 2026)

Core Concept

Persidskii System Class

Systems of the form: ẋ = Ax + Σ f_i(M_i x) + Bw

where f_i are sector-bounded nonlinearities. This class captures:

• Electromechanical systems (motors, actuators)
• Neural networks as dynamical systems
• Systems with saturation/nonlinear feedback

Stability-Preserving Koopman Lifting

Standard Koopman lifting linearizes nonlinear dynamics in a lifted space: z = Θ(x) (lifting functions) ż = Kz + Lw (linear dynamics in lifted space)

Key innovation: Embed ISS-LMI constraints as convex side conditions during Koopman matrix identification:

min ||ż - Kz - Lw||² subject to P > 0, LMI_ISS(K, P) ≤ 0

This ensures the identified model is provably stable.

Workflow

Step 1: Choose Lifting Functions

Select observable functions Θ: ℝⁿ → ℝᴺ:

• Polynomial basis: [x₁, x₂, x₁², x₁x₂, x₂², ...]
• Fourier basis: [sin(ωx), cos(ωx), ...]
• Radial basis functions: Gaussian kernels
• Domain-specific: Physical energy functions, activation functions

Guideline: N should be large enough to capture dynamics but small enough for tractable LMI.

Step 2: Collect Trajectory Data

Generate or collect data tuples (x(t), ẋ(t), w(t)):

• Persist data as matrices X, Ẋ, W
• Ensure sufficient excitation for identifiability
• For Persidskii systems: include data near sector boundaries

Step 3: Solve Constrained Koopman Identification

# Optimization problem:
# min_{K,L} ||Ẋ - K*Θ(X) - L*W||_F²
# s.t. P > 0 (positive definite)
#      AᵀP + PA + ... ≤ 0 (ISS-LMI constraint)
#
# This is a convex SDP if LMI is convex in (K, P)

Key LMI condition for ISS:

[AᵀP + PA + Q   PB]
[     BᵀP       -γ²I] ≤ 0

Step 4: Design Robust Observer

For partially-measured systems (y = Cx):

Observer: ż̂ = Kẑ + Lw + G(y - ŷ) ŷ = CΘ⁻¹(ẑ)

Observer gain G designed via H∞ synchronization criterion: min ||e||_2 / ||w||_2 ≤ γ

Step 5: Deploy in MPC

Use identified model for Model Predictive Path Integral (MPPI) control:

• Prediction horizon uses K, L from identified Koopman model
• Stability guaranteed by ISS-LMI constraint
• Handles time-varying delays via Lyapunov-Krasovskii terms

Implementation Patterns

Pattern 1: EDMD with Stability Constraint

# Extended Dynamic Mode Decomposition with ISS constraint
import cvxpy as cp

# Data matrices
Z = lifting_functions(X)  # N x M
Z_dot = lifting_functions(X_dot)  # N x M

# Variables
K = cp.Variable((N, N))
P = cp.Variable((N, N), PSD=True)

# Objective: minimize prediction error
obj = cp.sum_squares(Z_dot - K @ Z)

# ISS-LMI constraint
constraints = [
    P >> 0,
    K.T @ P + P @ K + Q << 0,  # Simplified ISS condition
]

prob = cp.Problem(cp.Minimize(obj), constraints)
prob.solve()

Pattern 2: Persidskii-Specific Lifting

For Persidskii systems ẋ = Ax + Σ f_i(M_i x):

• Lifting: Θ(x) = [x, f₁(M₁x), ..., f_k(M_k x)]
• Koopman matrix has structure: K = [[A, I, ..., I], [0, 0, ..., 0], ...]
• Exploit sector bounds: α_i s² ≤ s·f_i(s) ≤ β_i s²

Pattern 3: Delay-Aware Identification

For time-delay systems:

• Augment state with delayed terms: z_aug = [z(t), z(t-τ)]
• Use Lyapunov-Krasovskii functional with Persidskii integral terms
• LMI includes delay-dependent terms

Key Formulas

ISS-LMI for Persidskii Systems

[AᵀP + PA + Σ λ_i(M_iᵀM_i)   PB]
[           BᵀP              -γ²I] ≤ 0

where λ_i are sector bounds of nonlinearities.

Observer Error Dynamics

ė = (K - GC)e + Lw

H∞ performance: ||e||₂ ≤ γ||w||₂ guaranteed when:

[(K-GC)ᵀP + P(K-GC) + I   PL]
[          LᵀP            -γ²I] ≤ 0

Validation Metrics

1. Prediction accuracy: RMSE between predicted and actual trajectories
2. Stability margin: Largest γ satisfying ISS-LMI
3. Observer performance: Estimation RMSE vs. benchmark (EKF)
4. Control performance: Tracking accuracy vs. baseline (FOC)

Expected results (from paper): 35% RMSE reduction vs EKF, 67% tracking improvement vs FOC.

When to Use

• Use this method: Nonlinear systems with sector-bounded nonlinearities, need stability guarantees
• Avoid: Strongly chaotic systems where Koopman spectrum is ill-conditioned
• Alternative: Neural ODEs for systems without sector structure (but no stability guarantee)

Related Methods

• Extended Dynamic Mode Decomposition (EDMD)
• Neural ODEs / Neural SSMs
• Linear Parameter-Varying (LPV) identification
• Set-membership identification