Chapter 6 State-Space Design
© National Instruments Corporation 6-33 Xmath Control Design Module
Example 6-11 rms( ) Response
Sys = system([-2.3,0.01,5.1;0,-0.35,-2;
0,2,-.35],[1,.25,.25]',[1.34,0,0],0);
w = logspace(0.01,1,50);
Uspec = pdm(ones(w),w);
[Ypsd,Yspec] = psd(Sys,Uspec);
Balancing a Linear System
Given a particular system model, the concept of model reduction centers
on finding a lower-order model with similar input-output response
characteristics. Typically this is assessed by comparing the impulse
responses of the two systems [Moo81]. The goal in balancing a linear
system is to find a state transformation that resolves the trade-off between
controllability and observability, returning a transformed system whose
states are equally controllable and observable. This raises the issue of
quantifying a system’s controllability or observability. You can do this
by considering the system singular values associated with the mappings
between the inputs and states, and those associated with the state-output
mappings.
These singular values can be obtained from decompositions of two
quantities referred to as the controllability and observability grammians.
These quantities are represented by W
c
and W
o
respectively, and defined by
the following equation for a system with an asymptotically stable A matrix.
(6-17)
For continuous systems, the controllability and observability grammians
satisfy the Lyapunov equations:
(6-18)
W
c
e
tA
BB'e
tA'
dt
0
∞
∫
=
W
o
e
tA'
C'Ce
tA
dt
0
∞
∫
=
AW
c
W
c
A' BB'++ 0=
A'W
o
W
o
AC'C++ 0=
