User`s guide

lyap
11-135
11lyap
Purpose Solve continuous-time Lyapunov equations
Syntax X = lyap(A,Q)
X = lyap(A,B,C)
Description lyap solves the special and g eneral forms of the L yapunov matrix equation.
Lyapunov equations arise in several areas of control, including stability theor y
and the study of the RMS behavior of syst ems.
X = lyap(A,Q) solves the Lyapunov equation
where and are square matrices of identical sizes. The solution
X is a
symmetric matrix if is.
X = lyap(A,B,C) solves the generalized Lyapunov equation (also called
Sylvester equation).
The ma trices must have compatible dimensions but need not be
square.
Algorithm lyap transforms the and matrices t o complex Schur form, computes the
solution of the resulting triangular system, and transforms this solution back
[1].
Limitations The continuous Lyapunov equation has a (unique) solution if the eigenvalues
of and of satisfy
If this condition is violated,
lyap produces the error message
Solution does not exist or is not unique.
See Also covar Covariance of system response to white noise
dlyap S olv e discrete Lyapunov equations
AX XA
T
Q++0=
A
Q
Q
AX XB C
++
0
=
ABC
,,
A
B
α
1
α
2
...
α
n
,,,
A
β
1
β
2
...
β
n
,,,
B
α
i
β
j
0
≠+
for all pairs ij
,()