Chapter 6 State-Space Design
Xmath Control Design Module 6-6 ni.com
Figure 6-2. General Observer Block Diagram
If the observability matrix is nonsingular, you will be able to put the
eigenvalues (pole locations) of (A – LC), shown in Equation 6-4, anywhere
you want. Thus, you can choose them to make decay to zero as quickly
as possible.
(6-4)
The problem of finding the eigenvalues of (A – LC) can be equivalently
posed as that of finding the eigenvalues of (A'–C'L'). This statement can
be recognized as equivalent to that of the pole-placement problem for a
state-feedback controller (refer to the new state-update equation in the
Controllability section), with A, B, and K replaced by A', C', and L',
respectively. Notice that these two representations correspond to a
state-space system and its transpose. This illustrates the principle of duality
between the controller and estimator forms. For more information, refer to
the Duality and Pole Placement section.
observable( )
[SysO,T,nuo] = observable(Sys,{tol})
The observable( ) function is the analogue to controllable( ).
As described in the Controllability section, if a system {A,B,C,D} is
controllable, its transpose {A',C',B',D'} is observable.
observable( )
returns the observable partition of a state-space system, the number of
unobservable states in the original system, and a linear transformation
matrix which can be used to partition the states into observable and
unobservable sets. For an example of how to use the
observable( )
function, refer to Example 6-2.
observable( ) uses the staircase algorithm, which is described in more
detail in the stair( ) section.
uy
Cx
+
–
x = Ax + Bu
x
y
y
Cx
L
x
x = A + Bu + Ly
x
˜
x
˜
·
ALC–()x
˜
=
