Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $A$.

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The categories of this paper are: alg-geom
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The categories of this paper are: q-bio
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The categories of this paper are: econ
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The categories of this paper are: funct-an
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The categories of this paper are: quant-ph
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The categories of this paper are: q-alg
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The categories of this paper are: hep-ex
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The categories of this paper are: nucl-ex
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The categories of this paper are: supr-con
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The categories of this paper are: adap-org
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The categories of this paper are: gr-qc
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The categories of this paper are: eess
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The categories of this paper are: math-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: physics
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The categories of this paper are: comp-gas