User`s guide

balreal
11-16
11balreal
Purpose Input/output balancing of state-space realizations
Syntax sysb = balreal(sys)
[sysb,g,T,Ti] = balreal(sys)
Description sysb = balreal(sys) produces a balanced realization sysb of the LTI model
sys with equal and diagonal controllability and observability gramians (see
gram foradefinitionofgramian).balreal handlesbothcontinuousanddiscrete
systems. If
sys is not a state-space model, it is first and automatically
converted to state space using
ss.
[sysb,g,T,Ti] = balreal(sys) also returns the vector g containing the
diagonal ofthe balancedgramian,the state similaritytransformation
used to convert
sys to sysb, and the inverse transformation Ti = .
If the system is normalized properly, the diagonal
g of the joint gramian can be
used to reduce the model order. Because
g reflects the combined controllability
and observability of individual states of the balanced model, you can delete
thosestateswitha small
g(i) whileretainingthemostimportantinput-output
characteristics of the original system. Use
modred to perform the state
elimination.
Example Consider the zero-pol e-gain model
sys = zpk([–10 –20.01],[–5 –9.9 –20.1],1)
Zero/pole/gain:
(s+10) (s+20.01)
----------------------
(s+5) (s+9.9) (s+20.1)
A state-space realization with balanced gramians is obtained by
[sysb,g] = balreal(sys)
The diagonal entries of the joint gramian are
g'
ans =
1.0062e–01 6.8039e–05 1.0055e–05
x
b
Tx
=
T
1
–