Solving matrix nearness problems via Hamiltonian systems, matrix factorization, and optimization

In these lectures notes, we review our recent works addressing various problems of finding the nearest stable system to an unstable one. After the introduction, we provide some preliminary background, namely, defining Port-Hamiltonian systems and dissipative Hamiltonian systems and their properties, briefly discussing matrix factorizations, and describing the optimization methods that we will use in these notes. In the third chapter, we present our approach to tackle the distance to stability for standard continuous linear time invariant (LTI) systems. The main idea is to rely on the characterization of stable systems as dissipative Hamiltonian systems. We show how this idea can be generalized to compute the nearest $\Omega$-stable matrix, where the eigenvalues of the sought system matrix $A$ are required to belong a rather general set $\Omega$. We also show how these ideas can be used to compute minimal-norm static feedbacks, that is, stabilize a system by choosing a proper input $u(t)$ that linearly depends on $x(t)$ (static-state feedback), or on $y(t)$ (static-output feedback). In the fourth chapter, we present our approach to tackle the distance to passivity. The main idea is to rely on the characterization of stable systems as port-Hamiltonian systems. We also discuss in more details the special case of computing the nearest stable matrix pairs. In the last chapter, we focus on discrete-time LTI systems. Similarly as for the continuous case, we propose a parametrization that allows efficiently compute the nearest stable system (for matrices and matrix pairs), allowing to compute the distance to stability. We show how this idea can be used in data-driven system identification, that is, given a set of input-output pairs, identify the system $A$.