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
˜
=