Page 127 - Modern Control of DC-Based Power Systems

Background 91

interval t 0 # t # t 1 . Complete observability is given if the system is
observable in every state. This means that each state transition affects all
output variables after some time. This is important, especially in state
feedback control, since not all state variables might be accessible from
outside and they have to be estimated from the available output variables.
It can be shown that the complete state observability of a linear system
described by A n 3 n and the output matrix C m 3 n can be determined by its
observability matrix which is defined as:
2 3
C
CA 7
W o 5 6 7 (3.8)
6
^
4 5
CA n21
The system is completely observable if and only if the rank of
W o 5 n.
If rank W o , n, the system is not completely observable, which is not
acceptable for control purposes, whereas it could be fine if complete con-
trollability was not given. To solve this issue, it should be checked if there
are alternatives for placing the measurement sensors in the system. In this
way, the output matrix C can be changed.

3.4.3 Pole Placement

For a completely state-controllable system, it is possible to place the
closed-loop poles in arbitrary locations using the state feedback technique.
In this way, by means of pole placement, it can be defined exactly how
the system reacts to a specific input or disturbance. In contrast to tradi-
tional control design, where the damping ratio and undamped natural
frequency of the dominant closed-loop poles is shaped, pole placement
directly adjusts all closed-loop poles.
The main idea in the pole placement technique is to feed the states of
the system back to the input with some properly selected coefficients k.
Therefore, this technique is also referred to as the state feedback method.
The state feedback matrix K composed of state feedback gains k is calcu-
lated in such a way that the resulting system has the desired eigenvalues.
In the presence of state feedback, the control signal is given by:
u 52 Kx (3.9)