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Model Updating for Nonlinear Systems with Stability Guarantees

Farhad Ghanipoor, Carlos Murguia, Peyman Mohajerin Esfahani, Nathan van de Wouw

TL;DR

This work tackles learning uncertainties in nonlinear systems by augmenting a known physics-based model with a learnable uncertainty term while guaranteeing stability. It develops two stability frameworks—invariant-set guarantees for locally Lipschitz and input-to-state stability for globally Lipschitz systems—and solves the resulting non-convex learning problems via convex approximations using two SDP-based approaches and sequential convex programming. A robust uncertainty-state estimator is then designed to extract uncertainty and state trajectories from input-output data, with guarantees on estimation performance. A roll-plane vehicle case study demonstrates improved predictive accuracy and practical viability, underscoring the framework's value for stable, data-driven model updating in complex nonlinear systems.

Abstract

To improve the predictive capacity of system models in the input-output sense, this paper presents a framework for model updating via learning of modeling uncertainties in locally (and thus also in globally) Lipschitz nonlinear systems. First, we introduce a method to extend an existing known model with an uncertainty model so that stability of the extended model is guaranteed in the sense of set invariance and input-to-state stability. To achieve this, we provide two tractable semi-definite programs. These programs allow obtaining optimal uncertainty model parameters for both locally and globally Lipschitz nonlinear models, given uncertainty and state trajectories. Subsequently, in order to extract this data from the available input-output trajectories, we introduce a filter that incorporates an approximated internal model of the uncertainty and asymptotically estimates uncertainty and state realizations. This filter is also synthesized using semi-definite programs with guaranteed robustness with respect to uncertainty model mismatches, disturbances, and noise. Numerical simulations for a large data-set of a roll plane model of a vehicle illustrate the effectiveness and practicality of the proposed methodology in improving model accuracy, while guaranteeing stability.

Model Updating for Nonlinear Systems with Stability Guarantees

TL;DR

This work tackles learning uncertainties in nonlinear systems by augmenting a known physics-based model with a learnable uncertainty term while guaranteeing stability. It develops two stability frameworks—invariant-set guarantees for locally Lipschitz and input-to-state stability for globally Lipschitz systems—and solves the resulting non-convex learning problems via convex approximations using two SDP-based approaches and sequential convex programming. A robust uncertainty-state estimator is then designed to extract uncertainty and state trajectories from input-output data, with guarantees on estimation performance. A roll-plane vehicle case study demonstrates improved predictive accuracy and practical viability, underscoring the framework's value for stable, data-driven model updating in complex nonlinear systems.

Abstract

To improve the predictive capacity of system models in the input-output sense, this paper presents a framework for model updating via learning of modeling uncertainties in locally (and thus also in globally) Lipschitz nonlinear systems. First, we introduce a method to extend an existing known model with an uncertainty model so that stability of the extended model is guaranteed in the sense of set invariance and input-to-state stability. To achieve this, we provide two tractable semi-definite programs. These programs allow obtaining optimal uncertainty model parameters for both locally and globally Lipschitz nonlinear models, given uncertainty and state trajectories. Subsequently, in order to extract this data from the available input-output trajectories, we introduce a filter that incorporates an approximated internal model of the uncertainty and asymptotically estimates uncertainty and state realizations. This filter is also synthesized using semi-definite programs with guaranteed robustness with respect to uncertainty model mismatches, disturbances, and noise. Numerical simulations for a large data-set of a roll plane model of a vehicle illustrate the effectiveness and practicality of the proposed methodology in improving model accuracy, while guaranteeing stability.
Paper Structure (30 sections, 5 theorems, 74 equations, 8 figures, 1 table)

This paper contains 30 sections, 5 theorems, 74 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Problem Settings
  4. Cost Function
  5. Locally Lipschitz Model Class
  6. Globally Lipschitz Model Class
  7. Cost Modification Approach
  8. Locally Lipschitz Model Class
  9. Globally Lipschitz Model Class
  10. Constraint Modification Approach
  11. Locally Lipschitz Model Class
  12. Globally Lipschitz Model Class
  13. Sequential Convex Program Approach
  14. Uncertainty and State Estimation
  15. Ultra Local Uncertainty Representation
  16. ...and 15 more sections

Key Result

Theorem 3.1

(Stable Locally Lipschitz Model Learning with Modified Cost) Consider system eq:sys, a given data-set $D$ of input, estimated uncertainty, and state realizations. In addition, consider given ellipsoidal sets $\mathcal{E}_{sys}$ and $\mathcal{E}_{u}$, as introduced in eq:system_set and eq:input_set w with the involved matrices defined as follows: for given positive scalars $\bar{l}_{h_x}, \beta, \

Figures (8)

  • Figure 1: Overview of the MUNSyS methodology.
  • Figure 2: Illustration of different (ellipsoidal) sets for the system state $\mathcal{E}_{sys}$, input $\mathcal{E}_{u}$, and model state $\mathcal{E}_{inv}$.
  • Figure 3: Roll plane system schematic.
  • Figure 4: Histogram for comparison of system outputs and the outputs of different models with cubic basis function (i.e., $h = (V_\eta x)^{\circ 3}$) for the test data.
  • Figure 5: Histogram for comparison of system outputs and the outputs of different models with basis function $h = [(V_\eta x)^{\circ 2^\top} , (V_\eta x)^{\circ 3^\top}]^\top$ for the test data.
  • ...and 3 more figures

Theorems & Definitions (11)

  • Remark 1
  • Definition 1
  • Theorem 3.1
  • Theorem 3.2
  • Remark 2
  • Theorem 4.1
  • Theorem 4.2
  • Remark 3
  • Remark 4
  • Proposition 1
  • ...and 1 more