balreal

Balanced state-space realization

Syntax

[sysb,g] = balreal(sys)

[sysb,g] = balreal(sys,Name=Value)

[sysb,g,TL,TR] = balreal(sys,___)

Description

[sysb,g] = balreal(sys) computes a balanced state-space realization of the LTI model sys. For stable models, sys 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. This balances the input-to-state and state-to-output energy transfers, and small entries of g indicate states that you can remove with xelim.

If sys has unstable poles, the function isolates the stable part, balances it, and adds back to the unstable part to form sysb. The entries of g corresponding to unstable modes are set to Inf.

example

[sysb,g] = balreal(sys,Name=Value) computes the balanced realization using options specified as one or more name-value arguments. (since R2023b)

[sysb,g,TL,TR] = balreal(sys,___) also returns the balancing state transformation matrices TL and TR. (since R2023b)

example

Examples

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Balanced Realization of Stable System

Open Live Script

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.

Obtain a state-space realization with balanced Gramians.

[sysb,g] = balreal(sys);

Examine the diagonal entries of the joint Gramians.

g'

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 using xelim.

sysr = xelim(sysb,[2 3]);

This yields a first-order approximation of the original system.

zpk(sysr)

ans =
 
  -0.00011597 (s-8638)
  --------------------
       (s+4.981)
 
Continuous-time zero/pole/gain model.

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

bp = bodeplot(sys,sysr,'r--');
bp.PhaseMatchingEnabled = "on";

MATLAB figure

The plots shows that removing the second and third states does not have much effect on system dynamics.

For faster and more accurate results, use reducespec for model reduction workflows.

Balanced Realization of Unstable System

Open Live Script

This example shows how to perform balanced realization of a system with unstable poles.

Consider the following system.

$H (s) = \frac{0.1}{(s + 2) (s + 1) (s - 1) (s + 0.001)}$

The system contains three stable and one unstable poles. One stable pole is really close to the stability boundary.

Create the model.

sys = ss(zpk(1,[-2 -1 1 -0.001],0.1));

Apply balreal to create a balanced realization.

[sysbal,g,TL,TR] = balreal(sys);
g

If sys has unstable poles, balreal isolates the stable part, balances the stable part, and adds back the unstable part to form the balanced realization. The unstable pole shows up as Inf in the vector g.

You can also use the offset option to shift the stability boundary to exclude nominally stable poles.

[sysbal,g,TL,TR] = balreal(sys,Offset=1e-2);
g

g = 4×1

       Inf
       Inf
    0.0393
    0.0018

For this realization, balreal treats the pole at s = –0.001 as unstable.

Input Arguments

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`sys` — Dynamic system
numeric LTI model

Dynamic system to decompose, specified as a numeric LTI model, such as a ss or tf model.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: [sys,g2] = balreal(sys,Algorithm="relative",FreqIntervals=[10,100]);

`Algorithm` — Balancing algorithm
`"absolute"` (default) | `"relative"`

Balancing algorithm, specified as either "absolute"' or "relative". Use this option to select the balancing algorithm and control the type of reduced-order model G_r.

"absolute" — Minimize the absolute error ${‖ G - G_{r} ‖}_{\infty}$ .
"relative" — Minimize the relative error ${‖ G^{- 1} (G - G_{r}) ‖}_{\infty}$ .

Relative error gives a better match across frequency while absolute error emphasizes areas with most gain.

`Regularization` — Regularization level
`"auto"` (default) | nonegative scalar

Regularization level, specified as "auto" or a nonnegative scalar value. When you set Algorithm to "relative", balreal balances the model [sys,α*I] instead of sys. Set this option to "auto" to let balreal pick a suitable regularization level α value. Specify a value α ≥ 0 to override this default.

This regularization value helps provide a well-defined relative error at all frequencies.

`FreqIntervals` — Frequency interval restriction of Gramians
`[]` (default) | two-column matrix

Frequency intervals for computing frequency-limited Hankel singular values, specified as a matrix with two columns. Each row specifies a frequency interval [fmin,fmax], where fmin and fmax are nonnegative frequencies, expressed in the frequency unit of the model. When identifying low-energy states to truncate, the software computes state contributions to system behavior in these frequency ranges only. For example:

To restrict the computation to the range between 3 rad/s and 15 rad/s, assuming the frequency unit of the model is rad/s, set FreqIntervals to [3,15].
To restrict the computation to two frequency intervals, 3-15 rad/s and 40-60 rad/s, use [3,15;40,60].
To specify all frequencies below a cutoff frequency fcut, use [0,fcut].
To specify all frequencies above the cutoff, use [fcut,Inf] in continuous time, or [fcut,pi/Ts] in discrete time, where Ts is the sample time of the model.

The default value, [], imposes no frequency limitation and is equivalent to [0,Inf] in continuous time or [0,pi/Ts] in discrete time. However, if you specify a TimeIntervals value other than [], then this limit overrides FreqIntervals = []. If you specify both a TimeIntervals value and a FreqIntervals value, then the computation uses the union of these intervals.

`TimeIntervals` — Time interval restriction of Gramians
`[]` (default) | two-column matrix

Time intervals for computing time-limited Hankel singular values, specified as a matrix with two columns. Each row specifies a time interval [tmin,tmax], where tmin and tmax are nonnegative times, expressed in the time unit of the model. When identifying low-energy states to truncate, the software computes state contributions to the system’s impulse response in these time intervals only. For example:

To restrict the computation to the range between 3 s and 15 s, assuming the time unit of the model is seconds, set TimeIntervals to [3,15].
To restrict the computation to two time intervals, 3-15 s and 40-60 s, use [3,15; 40,60].
To specify all times from zero up to a cutoff time tcut, use [0,tcut]. To specify all times after the cutoff, use [tcut,Inf].

The default value, [], imposes no time limitation and is equivalent to [0,Inf]. However, if you specify a FreqIntervals value other than [], then this limit overrides Timeintervals = []. If you specify both a TimeIntervals value and a FreqIntervals value, then the computation uses the union of these intervals.

`SepTol` — Maximum loss of accuracy in stable/unstable decomposition
`10` (default) | positive scalar

Maximum accuracy loss factor in stable and unstable decomposition, specified as a positive scalar. For models with unstable poles, balreal first extracts the stable dynamics using the transformations TL, TR. Use SepTol to control the decomposition accuracy. Increasing SepTol helps separate nearby stable and unstable modes at the expense of accuracy. For more information, see stabsep.

`Offset` — Offset for stable/unstable boundary
`1e-8` (default) | positive scalar

Offset for the stable/unstable boundary, specified as a positive scalar value. In the stable/unstable decomposition, the stable term includes only poles satisfying

Re(s) < -Offset × max(1,|Im(s)|) (continuous time)
|z| < 1 - Offset (discrete time)

Increase the value of Offset to treat poles close to the stability boundary as unstable.

Output Arguments

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`sysb` — Balanced state-space realization
state-space model

Balanced state-space realization, returned as a state-space model.

`g` — Hankel singular values
vector

Hankel singular values, returned as a vector.

The entries of g corresponding to unstable modes are set to Inf.

`TL` — Left-side matrix of transformation
matrix

Since R2023b

Left-side matrix of the balancing state transformation, returned as a matrix with dimensions Nx-by-Nx, where Nx is the number of states in the model sys.

The algorithm transforms the state-space realization (A, B, C, D, E) of a model to (A_b, B_b, C_b, D_b, E_b) given by:

For explicit state-space models
$\begin{array}{l} \begin{matrix} A_{b} = T_{L} A T_{R}, & B_{b} = T_{L} B, \end{matrix} \\ \begin{matrix} C_{b} = C T_{R}, & D_{b} = D, & E_{b} = T_{L} T_{R} = I \end{matrix} \end{array}$
For descriptor state-space models
$\begin{array}{l} \begin{matrix} A_{b} = T_{L} A T_{R}, & B_{b} = T_{L} B, \end{matrix} \\ \begin{matrix} C_{b} = C T_{R}, & D_{b} = D, & E_{b} = T_{L} E T_{R} \end{matrix} \end{array}$

The function returns an empty value [] for this argument when the input model sys is not a state-space model.

`TR` — Right-side matrix of transformation
matrix

Since R2023b

Right-side matrix of the balancing state transformation, returned as a matrix with dimensions Nx-by-Nx, where Nx is the number of states in the model sys.The algorithm transforms the state-space realization (A, B, C, D, E) of a model to (A_b, B_b, C_b, D_b, E_b) given by:

For explicit state-space models
$\begin{array}{l} \begin{matrix} A_{b} = T_{L} A T_{R}, & B_{b} = T_{L} B, \end{matrix} \\ \begin{matrix} C_{b} = C T_{R}, & D_{b} = D, & E_{b} = T_{L} T_{R} = I \end{matrix} \end{array}$
For descriptor state-space models
$\begin{array}{l} \begin{matrix} A_{b} = T_{L} A T_{R}, & B_{b} = T_{L} B, \end{matrix} \\ \begin{matrix} C_{b} = C T_{R}, & D_{b} = D, & E_{b} = T_{L} E T_{R} \end{matrix} \end{array}$

The function returns an empty value [] for this argument when the input model sys is not a state-space model.

Tips

For model order reduction purposes, use reducespec.

Algorithms

Consider the stable state-space model

$\begin{array}{l} E \dot{x} = A x + B u \\ y = C x + D u \end{array}$

with controllability and observability Gramians W_c and W_o.

The state coordinate transformation $x = T_{R} \bar{x}$ produces the equivalent model

$\begin{array}{l} T_{L} E T_{R} \dot{\bar{x}} = T_{L} A T_{R} \bar{x} + T_{L} B u \\ y = C T_{R} \bar{x} + D u \end{array}$

balreal transforms the Gramians to

$\begin{matrix} {\bar{W}}_{c} = T_{L} W_{c} T_{L}^{T}, & {\bar{W}}_{o} = T_{R}^{T} W_{o} \end{matrix} T_{R}$

such that

${\bar{W}}_{c} = {\bar{W}}_{o} = d i a g (g)$

If the model has unstable poles, the function isolates the stable part, balances it, and adds back to the unstable part to form the realization. The entries of g corresponding to unstable modes are set to Inf.

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

If you use the TimeIntervals or FreqIntervals options, 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].

References

[1] Laub, A., M. Heath, C. Paige, and R. Ward. “Computation of System Balancing Transformations and Other Applications of Simultaneous Diagonalization Algorithms.” IEEE Transactions on Automatic Control 32, no. 2 (February 1987): 115–22. https://doi.org/10.1109/TAC.1987.1104549.

[2] Moore, B. “Principal Component Analysis in Linear Systems: Controllability, Observability, and Model Reduction.” IEEE Transactions on Automatic Control 26, no. 1 (February 1981): 17–32. https://doi.org/10.1109/TAC.1981.1102568.

[3] Laub, Alan J. “Computation of ‘Balancing’ Transformations.” Joint Automatic Control Conference, no. 17 (1980): 84. https://doi.org/10.1109/JACC.1980.4232165.

[4] Gawronski, Wodek, and Jer-Nan Juang. “Model Reduction in Limited Time and Frequency Intervals.” International Journal of Systems Science 21, no. 2 (February 1990): 349–76. https://doi.org/10.1080/00207729008910366.

Version History

Introduced before R2006a

expand all

R2024b: Set balancing algorithm using `Algorithm` name-value argument, renamed from `Goal`

The Goal name-value argument is now called Algorithm. The behavior of this argument remains the same. There are no plans to remove support for existing references to Goal.

R2023b: Use simplified syntax

Starting in R2023b, you can specify the options directly as one or more name-value arguments. For example:

[sysb,g] = balreal(sys,Goal="relative",...
            Offset=1e-3,FreqIntervals=[10,100]);

As a result of this change, the syntax [sysb,g] = balreal(sys,opts) is not recommended. Here, opts was created using balredOptions.

balreal

Syntax

Description

Examples

Balanced Realization of Stable System

Balanced Realization of Unstable System

Input Arguments

sys — Dynamic system numeric LTI model

Name-Value Arguments

Algorithm — Balancing algorithm "absolute" (default) | "relative"

Regularization — Regularization level "auto" (default) | nonegative scalar

FreqIntervals — Frequency interval restriction of Gramians [] (default) | two-column matrix

TimeIntervals — Time interval restriction of Gramians [] (default) | two-column matrix

SepTol — Maximum loss of accuracy in stable/unstable decomposition 10 (default) | positive scalar

Offset — Offset for stable/unstable boundary 1e-8 (default) | positive scalar

Output Arguments

sysb — Balanced state-space realization state-space model

g — Hankel singular values vector

TL — Left-side matrix of transformation matrix

TR — Right-side matrix of transformation matrix

Tips

Algorithms

References

Version History

R2024b: Set balancing algorithm using Algorithm name-value argument, renamed from Goal

R2023b: Use simplified syntax

See Also

`sys` — Dynamic system
numeric LTI model

`Algorithm` — Balancing algorithm
`"absolute"` (default) | `"relative"`

`Regularization` — Regularization level
`"auto"` (default) | nonegative scalar

`FreqIntervals` — Frequency interval restriction of Gramians
`[]` (default) | two-column matrix

`TimeIntervals` — Time interval restriction of Gramians
`[]` (default) | two-column matrix

`SepTol` — Maximum loss of accuracy in stable/unstable decomposition
`10` (default) | positive scalar

`Offset` — Offset for stable/unstable boundary
`1e-8` (default) | positive scalar

`sysb` — Balanced state-space realization
state-space model

`g` — Hankel singular values
vector

`TL` — Left-side matrix of transformation
matrix

`TR` — Right-side matrix of transformation
matrix

R2024b: Set balancing algorithm using `Algorithm` name-value argument, renamed from `Goal`