Control System Toolbox™

ssbal - Balance state-space model using diagonal similarity transformation

Syntax

[sysb,T] = ssbal(sys,condT)
ssbal

Description

Given a state-space model sys with matrices ,

[sysb,T] = ssbal(sys)

computes a diagonal similarity transformation and a scalar such that

has approximately equal row and column norms. ssbal returns the balanced model sysb with matrices

and the state transformation where is the new state.

[sysb,T] = ssbal(sys,condT) specifies an upper bound condT on the condition number of . Since balancing with ill-conditioned can inadvertently magnify rounding errors, condT gives control over the worst-case roundoff amplification factor. The default value is condT=Inf.

ssbal returns an error if the state-space model sys has varying state dimensions.

Example

Consider the continuous-time state-space model with the following data.

a = [1 1e4 1e2;0 1e2 1e5;10 1 0];
b = [1;1;1];
c = [0.1 10 1e2];
sys = ss(a,b,c,0)

Balance this model with ssbal by typing

ssbal(sys)
a = 
                        x1           x2           x3
           x1            1         2500      0.39063
           x2            0          100       1562.5
           x3         2560           64            0
 
 
b = 
                        u1
           x1        0.125
           x2          0.5
           x3           32
 
 
c = 
                        x1           x2           x3
           y1          0.8           20        3.125
 
 
d = 
                        u1
           y1            0
 
Continuous-time system.

Direct inspection shows that the range of numerical values has been compressed by a factor 100 and that the and matrices now have nearly equal norms.

Algorithm

ssbal uses the MATLAB^® function balance to compute and .

See Also

balreal, ss2ss

ss2ss

ssdata