Chapter 6 State-Space Design
© National Instruments Corporation 6-31 Xmath Control Design Module
of the diagonal elements of the Schur-decomposed A and B matrices sum
to zero, a warning is given that the continuous equation solution may not
be unique. A similar warning appears for the discrete equation solution if
the product of any of the eigenvalues is 1.
To solve the special Lyapunov equation, use the following syntax:
lyapunov(A,C)
Example 6-10 Lyapunov Equation Solutions
The following examples each give results close to zero.
Discrete Lyapunov Equation
A = [1, 2;-3,.4];
C = [-1,3;6,2];
X = lyapunov(A,C, {discrete})
X (a square matrix) =
-0.0829686 0.946549
0.390993 -0.418771
norm(A*X*A'+C -X,1)
ans (a scalar) = 2.58127e-15
Continuous Lyapunov Equation
A = [1,-3;2,5];
B = [-4,3;2,1];
C = [1,3;-6,2];
X = lyapunov(A,B,C)
X (a square matrix) =
2.62963 -2.11111
-3.7037 2.22222
A*X + X*B + C;
norm(A*X + X*B + C,1)
ans (a scalar) = 0
AX×A'C+×X–=
AXXB×+×C–=