balred

(Not recommended) Model order reduction

balred and balredOptions are not recommended. Use reducespec (Control System Toolbox) instead. (since R2023b) For more information on updating your code, see Version History.

Syntax

[rsys,info] = balred(sys,order)

[~,info] = balred(sys)

[___] = balred(___,opts)

balred(sys)

Description

[rsys,info] = balred(sys,order) computes a reduced-order approximation rsys of the LTI model sys. The desired order (number of states) is specified by order. You can try multiple orders at once by setting order to a vector of integers, in which case rsys is an array of reduced models. balred also returns a structure info with additional information like the Hankel singular values (HSV), error bound, regularization level and the Cholesky factors of the gramians.

[~,info] = balred(sys) returns the structure info without computing the reduced-order model. You can use this information to select the reduced order order based on your desired fidelity.

Note

When performance is a concern, avoid computing the Hankel singular values twice by using the information obtained from the above syntax to select the desired model order and then use rsys = balred(sys,order,info) to compute the reduced-order model.

[___] = balred(___,opts) computes the reduced model using the options set opts that you specify using balredOptions. You can specify additional options for eliminating states, using absolute vs. relative error control, emphasizing certain time or frequency bands, and separating the stable and unstable modes. See balredOptions to create and configure the option set opts.

balred(sys) displays the Hankel singular values and approximation error on a plot. Use hsvplot (Control System Toolbox) to customize this plot.

Examples

Reduced-Order Approximation with Offset Option

Compute a reduced-order approximation of the system given by:

$$G\left(s\right)=\frac{\left(s+0.5\right)\left(s+1.1\right)\left(s+2.9\right)}{\left(s+{10}^{-6}\right)\left(s+1\right)\left(s+2\right)\left(s+3\right)}.$$

Create the model.

sys = zpk([-0.5 -1.1 -2.9],[-1e-6 -2 -1 -3],1);

Exclude the pole at $$s=10^{-6}$$ from the stable term of the stable/unstable decomposition. To do so, set the Offset option of balredOptions to a value larger than the pole you want to exclude.

opts = balredOptions('Offset',0.001,'StateProjection','Truncate');

Visualize the Hankel singular values (HSV) and the approximation error.

balred(sys,opts)

The red HSV indicates that it is associated with an unstable mode.

Now, compute a second-order approximation with the specified options.

[rsys,info] = balred(sys,2,opts);
rsys

rsys =
 
  0.99113 (s+0.5235)
  -------------------
  (s+1e-06) (s+1.952)
 
Continuous-time zero/pole/gain model.

Notice that the pole at -1e-6 appears unchanged in the reduced model rsys.

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

bodeplot(sys,rsys,'r--')

Observe that the bode response of the original model and the reduced-order model nearly match.

Input Arguments

sys — Dynamic system

dynamic system model

Dynamic system, specified as a SISO or MIMO dynamic system model (Control System Toolbox). Dynamic systems that you can use can be continuous-time or discrete-time numeric LTI models, such as tf, zpk (Control System Toolbox), or ss (Control System Toolbox) models.

When sys has unstable poles, balred decomposes sys to its stable and unstable parts and only the stable part is approximated. Use balredOptions (Control System Toolbox) to specify additional options for the stable/unstable decomposition.

balred does not support frequency response data models, uncertain and generalized state-space models, PID models or sparse model objects.

order — Desired number of states

integer | vector of integers

Desired number of states, specified as an integer or a vector of integers. You can try multiple orders at once by setting order to a vector of integers, in which case rys is returned as an array of reduced models.

You can also use the Hankel singular values and error bound information to select the reduced-model order based on the desired model fidelity.

opts — Additional options for model reduction

options set

Additional options for model reduction, specified as an options set. You can specify additional options for eliminating states, using absolute vs. relative error control, emphasizing certain time or frequency bands, and separating the stable and unstable modes.

See balredOptions (Control System Toolbox) to create and configure the option set opts.

Output Arguments

rsys — Reduced-order model

dynamic system model | array of dynamic system models

Reduced-order model, returned as a dynamic system model or an array of dynamic system models.

info — Additional information about the LTI model

structure

Additional information about the LTI model, returned as a structure with the following fields:

• HSV — Hankel singular values (state contributions to the input/output behavior). In state coordinates that equalize the input-to-state and state-to-output energy transfers, the Hankel singular values measure the contribution of each state to the input/output behavior. Hankel singular values are to model order what singular values are to matrix rank. In particular, small Hankel singular values signal states that can be discarded to simplify the model.

• ErrorBound — Bound on absolute or relative approximation error. info.ErrorBound(J+1) bounds the error for order J.

• Regularization — Regularization level ⍴ (for relative error only). Here, sys is replaced by [sys,⍴*I] or [sys;⍴*I] that ensures a well-defined relative error at all frequencies.

• Rr, Ro — Cholesky factors of gramians.

Algorithms

1. balred first decomposes G into its stable and unstable parts:

   $$G=G_s+G_u$$

2. When you specify ErrorBound as absolute, balred uses the balanced truncation method of [1] to reduce G_s. This computes the Hankel singular values (HSV) σ_j based on the controllability and observability gramians. For order r, the absolute error $${\Vert G_s-G_r\Vert }_{\infty }$$ is bounded by $$2{\displaystyle \sum _{j=r+1}^n{\sigma }_j}$$. Here, n is the number of states in G_s.

3. When you specify ErrorBound as relative, balred uses the balanced stochastic truncation method of [2] to reduce G_s. For square G_s, this computes the HSV σ_j of the phase matrix $$F={\left(W\text{'}\right)}^{-1}G$$ where W(s) is a stable, minimum-phase spectral factor of GG’:

   $$W\text{'}(s)W(s)=G(s)G\text{'}(s)$$

   For order r, the relative error $${\Vert G_s{}^{-1}(G_s-G_r)\Vert }_{\infty }$$ is bounded by:

   $${\displaystyle \prod _{j=r+1}^H\left(\frac{1+{\sigma }_j}{1-{\sigma }_j}\right)}-1\approx 2{\displaystyle \sum _{j=r+1}^n{\sigma }_j}$$

   when, $$2{\displaystyle \sum _{j=r+1}^n{\sigma }_j}\ll 1$$.

Alternative Functionality

App

Model Reducer (Control System Toolbox)

Live Editor Task

Reduce Model Order (Control System Toolbox)

References

[1] Varga, A., "Balancing-Free Square-Root Algorithm for Computing Singular Perturbation Approximations," Proc. of 30th IEEE CDC, Brighton, UK (1991), pp. 1062-1065.

[2] Green, M., "A Relative Error Bound for Balanced Stochastic Truncation", IEEE Transactions on Automatic Control, Vol. 33, No. 10, 1988

Version History

Introduced before R2006a

R2023b: Not recommended

balred and balredOptions are not recommended. To perform balanced truncation model order reduction, use the reducespec (Control System Toolbox) function with the following syntax.

R = reducespec(sys,"balanced");

For the full workflow, see Task-Based Model Order Reduction Workflow (Control System Toolbox).

R2017b: MatchDC option honored when specified frequency or time intervals exclude DC

When you use balred for model reduction, you can use balredOptions (Control System Toolbox) to restrict the computation to specified frequency or time intervals. If the StateProjection option of balredOptions is set to 'MatchDC' (the default value), then balred attempts to match the DC gain of the original and reduced models, even if the specified intervals exclude DC (frequency = 0 or time = Inf).

Prior to R2017b, if you specified time or frequency intervals that excluded DC, balred did not attempt to match the DC gain of the original and reduced models, even if StateProjection = 'MatchDC'.