THESIS
2014

vii leaves, ix-xv, 87 pages : illustrations ; 30 cm

**Abstract**
Classical small gain theorem is a powerful tool in robust control theory
to deal with the stability of closed-loop systems with uncertainty. When the
uncertainty is deterministic norm-bounded nonlinearity or stochastic gain with
known mean and variance, the theorem asserts that the robust stability, i.e.
the simultaneous stability for all possible uncertainty, is true if and only if
the corresponding small gain condition holds. The theorem is very useful in
analyzing stability of uncertain systems, however its use is limited to the case
when only one type of uncertainty exists.

In recent years, the research on networked control systems (NCSs) has drawn
considerable attention from the community. In particular, we are interested in
the state feedback stabilization of a linear ti...[

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Classical small gain theorem is a powerful tool in robust control theory
to deal with the stability of closed-loop systems with uncertainty. When the
uncertainty is deterministic norm-bounded nonlinearity or stochastic gain with
known mean and variance, the theorem asserts that the robust stability, i.e.
the simultaneous stability for all possible uncertainty, is true if and only if
the corresponding small gain condition holds. The theorem is very useful in
analyzing stability of uncertain systems, however its use is limited to the case
when only one type of uncertainty exists.

In recent years, the research on networked control systems (NCSs) has drawn
considerable attention from the community. In particular, we are interested in
the state feedback stabilization of a linear time-invariant (LTI) system over
quantized fading channels. This leads to the introduction of both deterministic
norm-bounded and stochastic wide-sense-stationary (WSS) multiplicative uncertainties
into the closed-loop. Therefore to analyze the stability of the NCS, we need a small gain theorem being able to handle both uncertainty at the same
time, which has not yet appeared in the literature.

This thesis seeks for establishing such a small gain theorem such that both
types of uncertainty are allowed to exist simultaneously in the closed-loop. In
particular, we extend the l

_{2}-based input-output theory for deterministic systems
to stochastic case. We first extend the definitions of l

_{2} signals, spaces and
norms to the stochastic case. Then they are used to induce the system stability
and norms. The space structure, as well as the important properties of these
definitions are discussed.

Then the new small gain theorem is derived in the sense of these new definitions
of stability and norms. A general sufficient condition is shown first. The
condition holds for any given systems and uncertainty satisfying the small gain
condition, but the stability is not in the robust sense. The theorems concerning
robust stability are then established and the small gain condition is proved to be
necessary and sufficient for robust stability. Both unstructured and structured
cases are worked out.

Finally the theorems are applied to the NCS problem which motivates the
research. The quantization is modeled as a nonlinearity with norm bound and
the fading is modeled as a multiplicative stochastic noise, which fits the setup
of the new small gain theorems exactly. Hence the necessary and sufficient
stabilizability conditions of the NCS are derived by using the theorems, both
for single-input and multi-input cases. It is shown that the proposed NCS is
stabilizable if and only if the channel capacity, as defined in this thesis instead of
in classical communication theory, is strictly larger than the topological entropy
of the given LTI plant.

In summary, this thesis extends the l

_{2} input-output theory to the stochastic
setup and establishes a new small gain theorem under such a setup. The
application of the theorem is then shown by using it to solve an NCS problem
with mixed channel uncertainty. The theorem may be useful to more problems
which is also the future research direction of the author.

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