Given a family of linear
systems: dx/dt = Aix, i=1,2,...,N, xÎRn. Two
types of systems can be defined:
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Type 1: dx/dt Î{Aix:
i=1,2,...N}
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Type 2: dx/dt = As(x)x,
s:
Rn ® {1,2,...N}
For type 1 system, the switch
among the linear systems is arbitrary or controlled by an
unknown force.
We have no idea which linear system will be chosen at any time
instant and we have to expect the
worst case.
It is a linear differential inclusion (LDI) and is also called a
switching system. A LDI can be
used to describe an uncertain nonlinear system. A LDI can be
unstable even if all linear systems are stable.
For type 2 system, the switch is
orchestrated by a controller and the feedback law
s(x) can
be designed
for optimal performances. This type of system is called switched
systems. A switched system can be
made stable via a proper switching law even if all linear
systems are unstable. Switched systems may naturally arise
from physical and engineering situations or be artificially
incorporated into a dynamic
system for the purpose of performance enhancement, especially,
in the presence of
various constraints.
Analysis and design of switching/switched systems have been well
explored via quadratic Lyapunov functions. For switching
systems, a quadratic function results in a linear control law.
Since quadratic functions may be too conservative, linear
control laws may not be able to bring out the best performances.
In our project, we investigate control design of these systems
by using non-quadratic Lyapunov functions,
in particular, the three composite
quadratic Lyapunov functions. In our recent works, we
developed
algorithms for the design of nonlinear control laws for
switching systems. Examples show significant performance
enhancement as compared to what can be achieved by linear
control laws.
For switched
systems, a difficult situation to deal with is the existence of
sliding mode. In our project,
sliding modes are carefully handled with the directional
derivatives of composite quadratic Lyapunov functions. More
details can be found in our recent papers and conference
presentations below.
Publications:
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T. Hu, L. Ma and Z. Lin, ``On several composite quadratic Lyapunov
functions
for switched systems,"
IEEE Conference on
Decision and Control, 2006.
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T. Hu, ``Nonlinear control design for linear differential inclusions via
convex hull of quadratics," Automatica, to appear.
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R. Goebel, A.R. Teel, T. Hu and Z. Lin, ``Conjugate convex Lyapunov functions for dual linear
differential equations," IEEE Transactions on Automatic Control, 51(4), pp.661-666, 2006.
Presentations:
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