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

`[`

* sysb*,

`g`

`sys`

`[``sysb`

,`g`

,`T`

,`Ti`

]
= balreal(`sys`

)

[___] = balreal(

`sys`

`opts`

`[`

computes a
balanced realization * sysb*,

`g`

`sys`

`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`

.

also
returns the vector `[`

* sysb*,

`g`

`T`

`Ti`

`sys`

`g`

containing the diagonal of
the balanced Gramian, the state similarity transformation `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.

`[___] = balreal(`

computes
the balanced realization using options that you specify using * sys*,

`opts`

`hsvdOptions`

. 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 `hsvdOptions`

for
details. Consider the model

$$\begin{array}{l}\dot{x}=Ax+Bu\\ y=Cx+Du\end{array}$$

with controllability and observability Gramians * W_{c}* and

$$\begin{array}{l}\dot{\overline{x}}=TA{T}^{-1}\overline{x}+TBu\\ y=C{T}^{-1}\overline{x}+Du\end{array}$$

and transforms the Gramians to

$$\begin{array}{cc}{\overline{W}}_{c}=T{W}_{c}{T}^{T},& {\overline{W}}_{o}={T}^{-T}{W}_{o}\end{array}{T}^{-1}$$

The function `balreal`

computes a particular
similarity transformation * T* such that

$${\overline{W}}_{c}={\overline{W}}_{o}=diag(g)$$

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

If you use the `TimeIntervals`

or `FreqIntervals`

options
of `hsvdOptions`

, then `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 [4].

[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.

[4] 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.

`balred`

| `gram`

| `hsvd`

| `hsvdOptions`

| `modred`

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