Page 69 - Servo Motors and Industrial Control Theory -

3.6 Controllability and Observability 61

ables cannot match that of the system. A forcing term must be added to the observer
to ensure that the observed variables reach the state variables. This can be achieved
by the following:
d X := AX + BuL yCX )+ ( − o (3.25)
0
o
dt
In Eq. (3.25), L is a constant matrix and for single output ( y) L is a vector. The term
in the right hand side introduces a disturbance when the output differs by that of the
observed output. C is the output matrix and again for a single output it is a vector.
Writing Eq. (3.25) in slightly different forms gives

d X :(= A LC)X + + (3.26)
−
o
o
dt Bu Ly
In Eq. (3.26), the matrix ( A − LC) is the new observer dynamic matrix and the val-
ues of L must be chosen such that the observer dynamics is faster than that of the
system. This means that the eigenvalues of observer must be chosen so that they are
located to the left of the eigenvalues of the system. The design procedure is exactly
the same as for selecting the gain vector K.
For very noisy output, the famous Kalman Filter or other form of estimator may
be used which is beyond the scope of this book.
The observer in reality is a piece of software that performs numerical solution to
the observer equation. This is the same as discussed in Eq. (3.18).
The above discussion is on full-order observer where all state variables are es-
timated. In some cases, a few state variables might be measurable and in this case
a reduced-order observer might be used. Those who are interested in design of ob-
server are referred to more advanced books. For servo control systems by some
intelligent choices of state variables, they all can be measured directly from the
system and use of observer must be avoided especially when there is significant
noise in transducers.

3.6 Controllability and Observability

Before a state variable feedback control or observer based state variable feedback
are used, it must be established that the system is controllable and observable. There
are various methods to establish the above concept but the modal analysis is used
in this chapter because it is more suitable with use of standard software than other
methods. The interested readers are referred to more advanced books on this sub-
ject.
If observer based control strategy has to be used, both controllability and observ-
ability must be investigated. Because an observable system might not be control-
lable and a controllable system might not be observable.