Control System Toolbox™

balreal - Gramian-based input/output balancing of state-space realizations

Syntax

[sysb,g] = balreal(sys) [sysb,g,T,Ti] = balreal(sys)

Description

[sysb,g] = balreal(sys) computes a balanced realization sysb for the stable portion of the LTI model sys. balreal handles both continuous and discrete systems. If sys is not a state-space model, it is first and automatically converted to state space using ss.

For stable systems, sysb is an equivalent realization for which the controllability and observability Gramians are equal and diagonal, their diagonal entries forming the vector G of Hankel singular values. Small entries in G indicate states that can be removed to simplify the model (use modred to reduce the model order).

If sys has unstable poles, its stable part is isolated, balanced, and added back to its unstable part to form sysb. The entries of g corresponding to unstable modes are set to Inf. You can specify additional options for the stable/unstable decomposition:

[sysb,g] = balreal(sys,... 
                   'AbsTol',ATOL,'RelTol',RTOL,'Offset',ALPHA)

See stabsep for more details on these options. The default values are ATOL=0, RTOL=1e-8, and ALPHA=1e-8.

Use balreal(sys,condmax) to control the condition number of the stable/unstable decomposition. Increasing condmax helps separate close by stable and unstable modes at the expense of accuracy. By default condmax=1e8.

[sysb,g,T,Ti] = balreal(sys) also returns the vector g containing the diagonal of the balanced gramian, the state similarity transformation used to convert sys to sysb, and the inverse transformation .

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 those states with a small g(i) while retaining the most important input-output characteristics of the original system. Use modred to perform the state elimination.

There are also overloaded methods available. Type

help ss/balreal
help lti/balreal
help idmodel/balreal

for more information.

Example 1

Consider the zero-pole-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 =

    0.1006    0.0001    0.0000

which indicates that the last two states of sysb are weakly coupled to the input and output. You can then delete these states by

sysr = modred(sysb,[2 3],'del')

to obtain the following first-order approximation of the original system.

zpk(sysr)

Zero/pole/gain:
 1.0001
--------
(s+4.97)

Compare the Bode responses of the original and reduced-order models.

bode(sys,'-',sysr,'x')

Example 2

Create this unstable system:

sys1=tf(1,[1 0 -1])
 
Transfer function:
   1
-------
s^2 - 1

Apply balreal to create a balanced gramian realization.

[sysb,g]=balreal(sys1)
 
a = 
       x1  x2
   x1   1   0
   x2   0  -1
 
 
b = 
           u1
   x1  0.7071
   x2  0.7071
 
 
c = 
            x1       x2
   y1   0.7071  -0.7071
 
 
d = 
       u1
   y1   0
 
Continuous-time model.

g =

       Inf
    0.2500

The unstable pole shows up as Inf in vector g.

Algorithm

Consider the model

with controllability and observability gramians and . The state coordinate transformation produces the equivalent model

and transforms the gramians to

The function balreal computes a particular similarity transformation such that

See [1], [2] for details on the algorithm.

References

[1] Laub, A.J., M.T. Heath, C.C. Paige, and R.C. Ward, "Computation of System Balancing Transformations and Other Applications of Simultaneous Diagonalization Algorithms," IEEE^® Trans. Automatic Control, AC-32 (1987), pp. 115-122.

[2] Moore, B., "Principal Component Analysis in Linear Systems: Controllability, Observability, and Model Reduction," IEEE Transactions on Automatic Control, AC-26 (1981), pp. 17-31.

[3] Laub, A.J., "Computation of Balancing Transformations," Proc. ACC, San Francisco, Vol.1, paper FA8-E, 1980.