Gramian-based input/output balancing of state-space realizations
[ computes a
sysb for the stable portion
of the LTI model
both continuous and discrete systems. If
not a state-space model, it is first and automatically converted to
state space using
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
reduce the model order).
sys has unstable poles, its stable part
is isolated, balanced, and added back to its unstable part to form
The entries of
g corresponding to unstable modes
are set to
additional options for the stable/unstable decomposition. See the
stabsep reference page for more information
about these options. The default values are
ATOL = 0,
= 1e-8, and
ALPHA = 1e-8.
the condition number of the stable/unstable decomposition. Increasing
separate close by stable and unstable modes at the expense of accuracy.
returns the vector
g containing the diagonal of
the balanced gramian, the state similarity transformation xb = Tx used
the inverse transformation Ti = T-1.
If the system is normalized properly, the diagonal
the joint gramian can be used to reduce the model order. Because
the combined controllability and observability of individual states
of the balanced model, you can delete those states with a small
retaining the most important input-output characteristics of the original
modred to perform
the state elimination.
the balanced realization using the options specified in the
Consider the following zero-pole-gain model, with near-canceling pole-zero pairs:
sys = zpk([-10 -20.01],[-5 -9.9 -20.1],1)
sys = (s+10) (s+20.01) ---------------------- (s+5) (s+9.9) (s+20.1) Continuous-time zero/pole/gain model.
A state-space realization with balanced gramians is obtained by
[sysb,g] = balreal(sys);
The diagonal entries of the joint gramian are
ans = 0.1006 0.0001 0.0000
This 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');
This yields the following first-order approximation of the original system.
ans = 1.0001 -------- (s+4.97) Continuous-time zero/pole/gain model.
Compare the Bode responses of the original and reduced-order models.
The plots shows that removing the second and third states does not have much effect on system dynamics.
Create this unstable system:
sys1=tf(1,[1 0 -1]) Transfer function: 1 ------- s^2 - 1
balreal to create a balanced gramian
[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
 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.
 Moore, B., "Principal Component Analysis in Linear Systems: Controllability, Observability, and Model Reduction," IEEE Transactions on Automatic Control, AC-26 (1981), pp. 17-31.
 Laub, A.J., "Computation of Balancing Transformations," Proc. ACC, San Francisco, Vol.1, paper FA8-E, 1980.