THESIS
2016

v, viii, 105 pages : illustrations ; 30 cm

**Abstract**
Semidefinite cone-invariant (SCI) systems are defined as a class of linear
time-invariant (LTI) systems which possess the spectrahedral cone-invariance
property. Such systems have a rich structure and include a large class of LTI
systems, e.g., positive systems, as special cases. However, they have not received
enough attention from the control community. In this thesis, we make some
preliminary attempts to study the theory and applications of SCI systems.

The first part of this thesis is the study of the theory of SCI systems.
We define weak notions of spectrahedral cone-stability, spectrahedral cone-observability
and spectrahedral cone-detectability of SCI systems and some
criteria of these properties are given in terms of the distinguished eigenvalues
and linear matrix in...[

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Semidefinite cone-invariant (SCI) systems are defined as a class of linear
time-invariant (LTI) systems which possess the spectrahedral cone-invariance
property. Such systems have a rich structure and include a large class of LTI
systems, e.g., positive systems, as special cases. However, they have not received
enough attention from the control community. In this thesis, we make some
preliminary attempts to study the theory and applications of SCI systems.

The first part of this thesis is the study of the theory of SCI systems.
We define weak notions of spectrahedral cone-stability, spectrahedral cone-observability
and spectrahedral cone-detectability of SCI systems and some
criteria of these properties are given in terms of the distinguished eigenvalues
and linear matrix inequalities. More importantly, we investigate the SCI
realization problem, which was our initial motivation to study SCI systems. For
discrete-time LTI systems with nonnegative impulse responses, much research
has been devoted to studying their positive realization. However, the eigenvalues of nonnegative matrices, which obey the Perron-Frobenius theorem and Karpelevich
theorem, cannot cover all possible modes of a nonnegative impulse response.
The limitations in the eigenvalue positions suggest that positive systems are not
adequately powerful as a modeling tool. Hence we propose to employ the more
powerful systems, namely, SCI systems, to do the SCI realization of nonnegative
impulse responses. This is a novel idea and has not previously appeared in the
literature. At present, we can find SCI realizations for a large class of systems
with nonnegative impulse responses, whose positive realizations may not exist.
Moreover, a minimal SCI realization with the dimension equal to the order of
the transfer function is obtained for a subclass of these systems. The current
results indeed demonstrate the potential power of SCI realization. We have
faith that an SCI realization of any nonnegative impulse response, even with
the dimension equal to the order of the transfer function, can be obtained.

The second part of this thesis is the study of the linear quadratic (LQ)
optimal control problem of discrete-time networked control systems (NCSs) with
random input gains. It is shown that the solvability of this LQ optimal control
problem depends on the existence of a mean-square stabilizing solution to a
modified algebraic Riccati equation (MARE). With the help of the theory of SCI
systems, we provide a necessary and sufficient condition, which is given directly
in terms of the system parameters, to ensure the existence of such a mean-square
stabilizing solution. Such a condition is derived for the very first time,
and it indicates that the common assumption or condition of the observability or
detectability of certain stochastic systems is unnecessary. The other highlight
is that we put the problem under a channel/controller co-design framework,
which differentiates our work from a certain pure stochastic optimal controller
design problem. The controller designer has the freedom to participate in the
channel design by allocating the given overall channel capacity, as desired, to
the individual input channels. Under this framework, the stabilization issue
involved can be analytically solved.

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