Assessment of the control performance of fixed structure control for a process with limitations

Abstract: In this thesis, the assessment of performance of multivariable systems
(MIMO) that have time delays is presented. A time delay is defined as the
difference of time between a sent signal and the received one. Therefore,
time delays are in any dynamic system.A time delay imposes a serious
limitation on achievable control performance. The assessment of systems
that contain time delays are complex, since a delay is represented
mathematically as a irrational function. It can not be analysed with
classical methods, so approximations to rational functions are needed. The
problem is that this approximations are not accurate for systems with high
time delays. Moreover, the multivariable systems have interactions between
its elements, which make them more difficult to analyse.
Hence, for assess the performance of a multivariable system with time
delays, under the frequency domain, is necessary to apply special
mathematical techniques. Gershgorin circles will be used, for measuring the
interactions between the elements of a system. This circles are located on
the Nyquist diagram for all the frequencies and the radius is given by the
amplitude of the off-diagonal elements at each frequency. It has been
aplied a measure method (ultimate point) and it has been developed other
(modulus margin) for finding out what is the maximum delay that the system
can bear before getting instable. It can be also useful for finding out
what is the bandwidth of the system with delays. Thus, it can be seen how
the bandwidth is reduced because of the presence of time delays.
As a result, the delays that are in the off diagonal will not have
influence on the system performance, since the Gershgorin Circles depend on
the magnitude of those off diagonal elements and for the time delays is
unitary for all frequencies. From this point of view, the delays are only
important if they are in the diagonal elements.