Page 126 - Modern Control of DC-Based Power Systems

90 Modern Control of DC-Based Power Systems

x 2

x(t )
0
x(t )
1

x 1
ðÞ.
Figure 3.2 Transition from state xðt 0 Þ to state x t 1

system. A system which is controllable in every state is, by definition,
completely state controllable. x 2
It can be shown that a linear system described by A n 3 n and with the
input matrix B n 3 m is completely state controllable if and only if the con-
trollability matrix W c as defined below is full rank:

2
W c 5 ½BABjA Bj .. . jA n21 BŠ (3.7)
j
If the rank of W c 5 n, i.e., if W c is full rank, it means that there are n
independent directions which are controllable in the state-space using the
input vector u tðÞ. Therefore, it is possible to go from any xðt 0 Þ to any
xðt 1 Þ in any time by simply adjusting the speed for each direction
properly.
For rank W c , n, at least one direction cannot be controlled. This
can be acceptable if the desired direction is controllable. If the natural
behavior of the system is not sufficient, there is the possibility to change
the input channel to the system and describe the system with a different
state-space model. It is important to note that the intrinsic nature of the
system is expressed by A but B is related to our input channel selection.
It should be also noted in that practice, there are maximum available
or allowable input values, for instance to avoid saturation of actuators.
Therefore, it is not possible to apply unconstrained inputs as mentioned
in the definition of controllability. To take into consideration such limita-
tions when dealing with real systems, optimal control approaches, which
allow embedding limitations on inputs in the mathematical formulation
of the problem, can be applied.

3.4.2 Observability
A system is observable at time t 0 if it is possible to determine the state
xðt 0 Þ of the system from the observation of its output over a finite time