Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Learning the Integral Quadratic Constraints on Plant-Model Mismatch

Wentao Tang

TL;DR

The paper addresses learning a data-driven characterization of plant-model mismatch for nonlinear plants by expressing the mismatch as an integral quadratic constraint IQC and learning the corresponding dissipativity parameters from input–output trajectories using a one-class SVM. The method fixes the dynamic multiplier structure and estimates the symmetric matrix M via a soft OC-SVM formulation with block constraints to ensure dissipativity and stability, providing a generalization bound for unseen data. The approach is demonstrated through a SISO time-delay mismatch and a nonlinear two-phase reactor with a linear nominal model, successfully recovering frequency-domain uncertainties. The results enable robust controller design under mismatch without requiring a precise plant model, and point to extensions for nonlinear MPC.

Abstract

While a characterization of plant-model mismatch is necessary for robust control, the mismatch usually can not be described accurately due to the lack of knowledge about the plant model or the complexity of nonlinear plants. Hence, this paper considers this problem in a data-driven way, where the mismatch is captured by parametric forms of integral quadratic constraints (IQCs) and the parameters contained in the IQC equalities are learned from sampled trajectories from the plant. To this end, a one-class support vector machine (OC-SVM) formulation is proposed, and its generalization performance is analyzed based on the statistical learning theory. The proposed approach is demonstrated by a single-input-single-output time delay mismatch and a nonlinear two-phase reactor with a linear nominal model, showing accurate recovery of frequency-domain uncertainties.

Learning the Integral Quadratic Constraints on Plant-Model Mismatch

TL;DR

The paper addresses learning a data-driven characterization of plant-model mismatch for nonlinear plants by expressing the mismatch as an integral quadratic constraint IQC and learning the corresponding dissipativity parameters from input–output trajectories using a one-class SVM. The method fixes the dynamic multiplier structure and estimates the symmetric matrix M via a soft OC-SVM formulation with block constraints to ensure dissipativity and stability, providing a generalization bound for unseen data. The approach is demonstrated through a SISO time-delay mismatch and a nonlinear two-phase reactor with a linear nominal model, successfully recovering frequency-domain uncertainties. The results enable robust controller design under mismatch without requiring a precise plant model, and point to extensions for nonlinear MPC.

Abstract

While a characterization of plant-model mismatch is necessary for robust control, the mismatch usually can not be described accurately due to the lack of knowledge about the plant model or the complexity of nonlinear plants. Hence, this paper considers this problem in a data-driven way, where the mismatch is captured by parametric forms of integral quadratic constraints (IQCs) and the parameters contained in the IQC equalities are learned from sampled trajectories from the plant. To this end, a one-class support vector machine (OC-SVM) formulation is proposed, and its generalization performance is analyzed based on the statistical learning theory. The proposed approach is demonstrated by a single-input-single-output time delay mismatch and a nonlinear two-phase reactor with a linear nominal model, showing accurate recovery of frequency-domain uncertainties.

Paper Structure

This paper contains 9 sections, 2 theorems, 17 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Learning of IQC Parameters
  4. Problem Setting
  5. OC-SVM for IQC Learning
  6. Generalization Performance
  7. Numerical Example
  8. Application to a Chemical Process
  9. Conclusions

Key Result

Theorem 1

If $\Sigma$ is dissipative under the dynamic multiplier $\Psi$ with respect to the supply rate $\sigma(z)$ as specified in Definition def:dissipativity, then there exists a positive semidefinite function $V(x, \xi)$, defined for any $(x, \xi)$ that is reachable from the origin through some input tra holds for any trajectory on any time interval $[t_0, t_1]$. Such a function $V$ is called the stora

Figures (5)

  • Figure 1: The plant and dynamic multiplier
  • Figure 2: Frequency in the learned IQC of the example system with a delay mismatch.
  • Figure 3: The two-phase reactor.
  • Figure 4: Frequency response $\ell(j\omega)$ in the learned IQC for the two-phase reactor.
  • Figure 5: Comparison of actual and nominal responses under different frequencies.

Theorems & Definitions (10)

  • Definition 1
  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3: Choice of filters
  • Definition 2
  • Definition 3
  • Remark 4: Soft OC-SVM
  • Remark 5: Sampling the trajectories
  • Theorem 2