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 Wc and Wo 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)
WcetABB'etA'dt
0
∞
∫
=
WoetA'C'CetAdt
0
∞
∫
=
AWcWcA'BB'++ 0=
A'WoWoAC'C++ 0=