Chapter 6 State-Space Design
© National Instruments Corporation 6-7 Xmath Control Design Module
Example 6-2 Observability of a System
A system is described by:
A = [1,0,0.01;0,1,0;0,0,1];
B = [1,0,0]';C = [0.6,0.8,0];D = 0;
Sys = system(A,B,C,D);
Performing,
[SysO,T,nuo] = observable(Sys);
The system has 1 unobservable state
This example indicates that one state of the system’s states corresponds to
an unobservable mode, but that the other two are observable.
Similarly, if you form the observability matrix for this system,
[,states] = size(A);
Obs = C;
For i = 1:states-1;
Obs = [C; Obs*A];
endFor
det(Obs)
ans (a scalar) = 0
you see that the observability matrix is singular (its determinant is zero),
confirming the results you saw from observable( ).
Minimal Realizations
All state-space systems have an infinite number of realizations. All systems
have a minimum number of states needed to express the system dynamics,
but can be described using any number of states greater than or equal to this
minimum number. If a system has more states than are needed to express a
given transfer function, it will have unobservable and/or uncontrollable
modes corresponding to eigenvalues of the A matrix that are not poles of
the transfer function.
All minimal realizations of the same system are related by a coordinate
transformation.