Chapter 6 State-Space Design
Xmath Control Design Module 6-32 ni.com
Special Lyapunov Equation
A = [-4,10;2,7];
C = [.3,6;2,9];
X = lyapunov(A,C)
X (a square matrix) =
1.1816 -0.209028
1.12431 -0.773611
A*X + X*A' + C;
norm(A*X + X*A' + C,1)
ans (a scalar) = 5.4956e-15
rms( )[Yrms,Ycov] = rms(Sys,Ucov)
The rms( ) function computes the root-mean-square response (average
power at the system output) and the output covariance of a dynamic system
driven by zero-mean white noise input. You can specify the intensity of the
noise with the optional input covariance parameter Ucov, which defaults to
identity.
For a continuous system, the covariance of the states is given by X, where
X is the differential Lyapunov solution (shown in Equation6-14) with
equal to zero for steady-state. Thus, for a system with output Y defined by:
the output covariance matrix (Ycov) is expressed as:
The output covariance for a discrete system follows analogously,
with X being the solution to Equation 6-12 in this case. Thus, a call to
lyapunov( ) forms the core of rms( ).
The diagonal terms of the covariance matrix correspond to the expected
values of the squares of the power at each output. Taking the square root of
these diagonal terms, you obtain the rms (root mean square) power at each
output. For an example of rms( ) responses, refer to Example 6-11.
AXXA'×+×C–=
X
·
YCxDu+=
Ycov CXC′=DUcovD′+