Learning Stable and Robust Linear Parameter-Varying State-Space Models

TL;DR

This work introduces two direct, Cayley-transform-based parameterizations for stable and robust LPV-SS models that enforce contraction or a prescribed $\gamma$-Lipschitz bound across all scheduling values $p$. By expressing the system matrices through structured, Lyapunov-based forms and mapping scheduling-dependent components via unconstrained neural or linear mappings, the approach enables unconstrained training while guarantees stability. Theoretical results provide constructive representations that characterize when an LPV-SS model is contracting or $\gamma$-Lipschitz, and extend to non-square Lipschitz extensions. An empirical example compares the proposed approach to a linear fractional representation baseline, highlighting the Lipschitz model’s superior extrapolation safety, which is valuable for industrial applications with limited excitation range.

Abstract

This paper presents two direct parameterizations of stable and robust linear parameter-varying state-space (LPV-SS) models. The model parametrizations guarantee a priori that for all parameter values during training, the allowed models are stable in the contraction sense or have their Lipschitz constant bounded by a user-defined value $γ$. Furthermore, since the parametrizations are direct, the models can be trained using unconstrained optimization. The fact that the trained models are of the LPV-SS class makes them useful for, e.g., further convex analysis or controller design. The effectiveness of the approach is demonstrated on an LPV identification problem.

Figures (7)

• Figure 1: The LPV state-space model and its parameterized scheduling dependency $\psi_\theta$.
• Figure 2: Training data-set
• Figure 3: Validation data-set
• Figure 4: Test-a data-set
• Figure 5: Test-b data-set

Theorems & Definitions (14)

• Definition 1
• Definition 2
• Remark 1
• Proposition 1
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Remark 2