Page 124 - Modern Control of DC-Based Power Systems

88 Modern Control of DC-Based Power Systems

The phase margin and the gain margins both have positive values,
which indicate that the closed-loop system is stable. If a very large gain
(more precisely, a gain larger than 32 dB) is added to the open-loop trans-
fer function, then the magnitude diagram will be shifted upwards while
the phase diagram remains untouched. In this case, depending on the
value of the gain, it can be that the phase margin becomes negative, thus
leading to an unstable closed-loop behavior.
The very interesting aspect of using Bode (and Nyquist) diagrams is
that they provide a very intuitive insight into the impact of each factor in
the transfer function of the system on the overall system behavior. This is
particularly useful when designing a control system as the designer would
have a qualitative understanding of the pros and cons of adding each type
of controller to the system by considering their contribution to the fre-
quency response of the system.
Another important point is that when using frequency response
approaches, the user will have to translate the desired behavior usually in
terms of damping ratio, phase margin, and gain margin and then select
the controller to satisfy these requirements. With this effort, the user
hopes to move the location of the closed-loop poles in such a way that
the desired overall system is fulfilled. However, there is no direct control
on the location of the closed-loop poles during this procedure. In the
state-space approach, the situation is different and it is possible to place
the closed-loop poles in arbitrary locations if certain preconditions are
met. These points will be discussed in more detail in the next sections.

3.4 LINEAR STATE-SPACE
In traditional control theory, the assumption is that the output has
enough information to control the system. This may, however, not be the
case since the output could also contain redundant or too little informa-
tion. In contrast, modern control theory considers the states of the con-
trolled system. The idea is that if the state of the system can be fed back,
the amount of information that is needed to control the system is mini-
mized. This becomes clear with the definition of the term state:

The state of a dynamic system is the smallest set of variables (called state vari-
ables) such that knowledge of these variables at t 5 t 0 , together with knowl-
edge of the input for t $ t 0 , completely determines the behavior of the system
for any time t $ t 0 [1].