Gramian-based input/output balancing of state-space realizations
[___] = balreal(
[ 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
Hankel singular values. Small entries in
states that can be removed to simplify the model (use
modred to 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
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.
[___] = balreal(
computes the balanced realization using options that you specify using
balredOptions. Options include offset and tolerance options for
computing the stable-unstable decompositions. The options also allow you to limit the
Gramian computation to particular time and frequency intervals. See
balredOptions for details.
Balanced Realization of Stable System
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 = 1×3 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.
Balanced Realization of Unstable System
Create an unstable system.
sys = tf(1,[1 0 -1])
sys = 1 ------- s^2 - 1 Continuous-time transfer function.
balreal to create a balanced-gramian realization.
[sysbal,g] = balreal(sys)
sysbal = 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 state-space model.
g = 2×1 Inf 0.2500
The unstable pole shows up as
Inf in the vector
Consider the model
with controllability and observability Gramians Wc and Wo. The state coordinate transformation produces the equivalent model
and transforms the Gramians to
balreal computes a particular
similarity transformation T such that
If you use the
FreqIntervals options of
balreal bases the balanced realization on time-limited or
frequency-limited controllability and observability Gramians. For information about
calculating time-limited and frequency-limited Gramians, see
gram and .
 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.
 Gawronski, W. and J.N. Juang. “Model Reduction in Limited Time and Frequency Intervals.” International Journal of Systems Science. Vol. 21, Number 2, 1990, pp. 349–376.