Page 130 - Modern Control of DC-Based Power Systems

94 Modern Control of DC-Based Power Systems

B + ∫ C

A

B + ∫ C - +

A

G
Observer
Figure 3.4 Observer scheme.

variables. In practice, one or more of the state variables may not be mea-
sured for any of the following reasons:
• The State might not be measurable, e.g., magnetic flux;
• The measurement reliability may be low, e.g., encoders for speed in
electric machines;
• The sensor is not installed to reduce cost; or
• The state variable has no physical meaning.
The estimation of unmeasured state variables is referred to as observa-
tion or state estimation. A component which performs the estimation or
observation of the state variable, is called an observer. The necessary and
sufficient condition for the design of a state observer is that the observ-
ability condition, as described in Section 3.4.2, is satisfied.
As shown in Fig. 3.4, the state observer reconstructs the state variables
using the measurements of the control variables and the system outputs.
The model of the observer is basically identical to the plant model,
except for the additional term which includes the estimation error to
compensate for model inaccuracies and the lack of the initial conditions:

_ ^ x 5 A^x 1 Bu 1 Gy 2 C^xÞ 5 A 2 GCÞ^x 1 Bu 1 Gy (3.19)
ð
ð
where ^x is the observed or estimated state vector and G is the observer
gain matrix.
A^x 1 Bu is the term calculated based on our knowledge of the system
model and the previously estimated state. However, we know that this
term does not offer a perfect estimation of the system state. Therefore, a
compensation term Gy 2 C^xÞ is added to account for the inaccuracies
ð
associated with the system model, i.e., the matrices A and B as well as the
lack of knowledge about the initial system state. The larger the estimation