National Instruments 370753C-01 Manual De Usuario

The new state-update equation in the 

 section and the 

 section, the time-update equation in the 

 section, along with the corresponding block 

diagrams in Figures 6-1 and 6-2, indicate how you can move the 
eigenvalues, or poles, of a minimal system through the choice of a feedback 
gain K (or L). Given the system’s state-space representation and any 
desired set of closed-loop poles, you can solve an eigenvalue problem to 
find the gain that yields these poles for the complete system. Although the 
poles of a minimal system can be moved to any value, this approach does 
not guarantee that the resulting gain is small or physically practical—just 
that it is finite. 

The similarity between the new state-update equation (Equation 6-1) 
and the time-update equation (Equation 6-4) for controller and observer 
feedback brings up the principle of duality with respect to the 
controllability and observability of a system. Briefly, for a given state-space 
system 

Sys

1

 with system matrices {A,B,C,D}, there exists a dual system 

Sys

2

 described by {A*,C*,B*,D}, using * to denote a complex conjugate 

transpose. If 

Sys

1

 is controllable, 

Sys

2

 will be observable, and vice versa. 

This can be quickly verified by constructing the controllability and 
observability matrices for both. Thus, the gain value that yields a set of 
desired closed-loop poles for feedback control of a system also yields an 
observer with the same pole locations for the system’s dual. 

K = poleplace(A,B,poles)

The 

poleplace( )

 function solves the problem 

eig(A – B * K) = poles

 for single-input systems. This is essentially 

the problem posed in Figure 6-1 and the new state-update equation in the 

 section. If you know where you want the system poles to 

be located, 

poleplace( )

 returns the value of the gain vector 

K

 that will 

move the closed-loop poles to the desired locations.

The syntax 

poleplace(A',C',poles)

 can be used for regulator 

problems, or by duality, for estimator problems. In general, the system 
{A,B} must be reachable (all unstable poles controllable) for controller 
design and the system {A',C'} must be stabilizable (all unstable poles 
observable) for estimator design. The current 

poleplace( )

 

implementation is limited to single-input systems.