National Instruments 370753C-01 Manual De Usuario

or the following for the discrete case:

(6-16)

These results derive from the Lyapunov method of stability analysis for 
linear systems. Steady state means that at some point the states no longer 
change. The derivative term   approaches zero in the large for continuous 
systems, and x

k+1

= x

 for discrete-time ones. The state value vector x for 

which this is true is defined as the equilibrium state. As stated in [Oga70], 
a unique equilibrium state exists for systems with a nonsingular A matrix, 
whereas infinitely many equilibrium states exist if A is singular. A system 
is described as asymptotically stable if the state values approach the 
equilibrium state over time, no matter what value of x one started with. 
Such systems will always satisfy the following: for any positive-definite 
matrix Q, a positive definite matrix X can be found satisfying in the 
X equation (Equation 6-15) for the continuous case and Y equation 
(Equation 6-16) for the discrete case. 

Lyapunov equations also can be used to compute system controllability 
and observability grammians, which play an important role in internal 
balancing and model reduction. This application will be discussed further 
in the 

 section.

X = lyapunov(A,B,{C, discrete})

The 

lyapunov( )

 function provides a solution to both the discrete and 

continuous-time Lyapunov equations. When called with three inputs 
(A,B,C), it solves the general continuous Lyapunov equation 
(Equation 6-10); when called with two inputs (A,C), it solves the special 
Lyapunov equation (Equation 6-11). When called with two inputs (A,B) 
and the 

{discrete}

 keyword, it solves the discrete Lyapunov equation 

(refer to Equation 6-12). For examples of discrete, continuous, and special 
Lyapunov equation solutions, refer to Example 6-10.

The algorithm for

 lyapunov( )

 uses the Schur decomposition to convert 

A and B to upper triangular form, then finds the Lyapunov equation 
solution a column at a time by solving. 

lyapunov( )

 warns the user if the 

eigenvalues of (A + 

eye

(A)) are close to –1, in which case singularity may 

occur and cause the function to terminate. Furthermore, if any combination 

+

=

CXC' DQD'

+

=

x·