In real-world applications, one uses mathematical models to simulate, control or optimize a system or process. This mathematical model is infinite-dimensional (e.g., the determination of the electric or magnetic field associated with an electronic device) and is approximated by a finite element or finite difference model. The model is non-linear, and linearization is used to obtain a linear model. The model may also be obtained by a realization or system identification or result from a model order reduction procedure. These mathematical models, therefore, are typically inexact and contain uncertainties. Thus it is vital to study the following question:

“How robust is a property of a dynamical system under perturbations of the coefficient matrices?”

In the literature, the nearness problems have been studied usually with respect to the Frobenius norm or the spectral norm of the system matrices. However, for practical purposes, studying the nearness problems with respect to system norms, like H_2 or H_infinity is more appropriate. The classical methods to address these nearness problems (e.g., H_infinity control problem) involve solving a sequence of algebraic Riccati inequalities/equations which can be quite challenging, particularly if the state-space dimension is large.

Therefore, this project aims to develop direct optimization methods for solving nearness problems for system norms using the structure of port-Hamiltonian (PH) systems. PH systems generalize the classical Hamiltonian systems and recently have received much attention in energy-based modeling. One of the significant advantages of PH modelling is that system properties are encoded algebraically, and they are robust under structured perturbations. The algebraic structure of PH systems guarantees that the system is automatically stable and passive.

This motivates us to propose the two aspects of the project:

The first component (Theoretical aspects of nearness problems): This project component is devoted to developing tools to reformulate the H∞ nearness problems in a direct optimization problem using the PH structure. For example, the development of a port-Hamiltonian H_infinity controller.

The second component (Algorithmic aspects of nearness problems): This component is devoted to proposing new algorithms to compute approximate solutions to the nearness problems studied in the first component.

The results of the project would be useful in system identification, where one needs to identify a system with a given property (stability, passivity, etc.) from observations. The results will also be helpful in the robustness analysis of system properties.

Basic knowledge of Linear Algebra and Optimization

Basic Knowledge of Control Systems, Matlab Knowledge,

Master degree in Mathematics or Electrical Engineering