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Input
design for LTI Systems with Multiple Time Scales

Input design for
multiscale systems is much more challenging than the single scale systems due
to the presence of dynamics at multiple time scales. Conventional inputs such
as PRBS are persistently exciting and are proven to be excellent inputs for the
identification of LTI systems. In the case of systems with multiple scales,
there is very weak overlapping between the bandwidths of fast and the slow
subsystems unlike the single scale systems where there is significant overlap
between the subsystems. This weak overlapping calls for additional
identification friendly constraints on the input design unlike the single scale
system. In order to excite the all the subsystems, the sampling has to be commensurate
with the fast subsystem and the input should be designed such that it contains
the fast subsystem bandwidth (which is relatively quite large compared to the
slower one).  When a non-sequential input such as PRBS is used for such
systems, the slow subsystem is poorly excited as shown in Fig 1 unlike a single
scale system which is shown in Fig 2.  In Fig 1, we considered a system with
time constants 5 and 0.05 sec while a system with time constant 5 sec is
considered in Fig 2. The input bandwidth for the simulations is chosen based on
the individual system bandwidths.

The reason for such poor excitation in the case of
multiscale system is that the PRBS input excited all the frequencies in a non
sequential manner and has sparsely given importance to the frequencies in the
slow subsystem bandwidth. As a result, the observations obtained from such
excitation resulted in a poorer information on slow subsystems. This can also
be seen in an SNR perspective. In the case of single scale systems, including a
input bandwidth outside the system bandwidth results in poorer SNR  in the observations.
A similar issue of a slightly different nature prevails in multiscale systems
also. The contribution  of fast subsystem behaves like noise while identifying
the slow subsystem and vice versa along with the regular noise component. This
is the reason for the poor  SNR of slow subsystem  as shown  in Fig  1.

In this work, we propose an input that has
non-sequential nature within a frequency band but a sequential nature across
the frequency sub bands. The proposed input is more identification friendly for
multiscale systems and the identifiability of the slow and fast subsystems is
better than by using the conventional inputs.