balred

Model order reduction

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 to customize this plot.

Examples

Reduced-Order Model using Hankel Singular Values

For this example, use the Hankel singular value plot to select suitable order and compute the reduced-order model.

For this instance, generate a random discrete-time state-space model with 40 states.

rng(0)
sys = drss(40);

Plot the Hankel singular values using balred.

balred(sys)

For this example, select order of 16 since it is the first order with an absolute error less than 1e-4. In general, you select the order based on the desired absolute or relative fidelity. Then, compute the reduced-order model.

rsys = balred(sys,16);

Verify the absolute error by plotting the singular value response using sigma.

sigma(sys,sys-rsys)

Observe from the plot that the error, represented by the red curve, is below -80 dB (1e-4).

Array of Reduced-Order Models

For this example, consider a random continuous-time state-space model with 65 states.

rng(0)
sys = rss(65);
size(sys)

State-space model with 1 outputs, 1 inputs, and 65 states.

Visualize the Hankel singular values on a plot.

balred(sys)

For this instance, compute reduced-order models with 25, 30 and 35 states.

order = [25,30,35];
rsys = balred(sys,order);
size(rsys)

3x1 array of state-space models.
Each model has 1 outputs, 1 inputs, and between 25 and 35 states.

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)

Observe that the first HSV is red which 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.

Model Reduction in a Particular Frequency Band

Reduce a high-order model with a focus on the dynamics in a particular frequency range.

Load a model and examine its frequency response.

load('highOrderModel.mat','G')
bodeplot(G)

G is a 48th-order model with several large peak regions around 5.2 rad/s, 13.5 rad/s, and 24.5 rad/s, and smaller peaks scattered across many frequencies. Suppose that for your application you are only interested in the dynamics near the second large peak, between 10 rad/s and 22 rad/s. Focus the model reduction on the region of interest to obtain a good match with a low-order approximation. Use balredOptions to specify the frequency interval for balred.

bopt = balredOptions('StateProjection','Truncate','FreqIntervals',[10,22]);
GLim10 = balred(G,10,bopt);
GLim18 = balred(G,18,bopt);

Examine the frequency responses of the reduced-order models. Also, examine the difference between those responses and the original response (the absolute error).

subplot(2,1,1);
bodemag(G,GLim10,GLim18,logspace(0.5,1.5,100));
title('Bode Magnitude Plot')
legend('Original','Order 10','Order 18');
subplot(2,1,2);
bodemag(G-GLim10,G-GLim18,logspace(0.5,1.5,100));
title('Absolute Error Plot')
legend('Order 10','Order 18');

With the frequency-limited energy computation, even the 10th-order approximation is quite good in the region of interest.

Model-Order Reduction with Relative Error Approximation

For this example, consider the SISO state-space model cdrom with 120 states. You can use absolute or relative error control when approximating models with balred. This example compares the two approaches when applied to a 120-state model of a portable CD player device crdom [1,2].

Load the CD player model cdrom.

load cdromData.mat cdrom
size(cdrom)

State-space model with 1 outputs, 1 inputs, and 120 states.

To compare results with absolute vs. relative error control, create one option set for each approach.

opt_abs = balredOptions('ErrorBound','absolute','StateProjection','truncate');
opt_rel = balredOptions('ErrorBound','relative','StateProjection','truncate');

Compute reduced-order models of order 15 with both approaches.

rsys_abs = balred(cdrom,15,opt_abs);
rsys_rel = balred(cdrom,15,opt_rel);
size(rsys_abs)

State-space model with 1 outputs, 1 inputs, and 15 states.

size(rsys_rel)

State-space model with 1 outputs, 1 inputs, and 15 states.

Plot the Bode response of the original model along with the absolute-error and relative-error reduced models.

bo = bodeoptions; 
bo.PhaseMatching = 'on';
bodeplot(cdrom,'b.',rsys_abs,'r',rsys_rel,'g',bo)
legend('Original (120 states)','Absolute Error (15 states)','Relative Error (15 states)')

Observe that the Bode response of:

• The relative-error reduced model rsys_rel nearly matches the response of the original model sys across the complete frequency range.

• The absolute-error reduced model rsys_abs matches the response of the original model sys only in areas with the most gain.

References

1. Benchmark Examples for Model Reduction, Subroutine Library in Systems and Control Theory (SLICOT). The CDROM data set is reproduced with permission, see BSD3-license for details.

2. A.Varga, “On stochastic balancing related model reduction”, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187), Sydney, NSW, 2000, pp. 2385-2390 vol.3, doi: 10.1109/CDC.2000.914156.

Input Arguments

sys — Dynamic system

dynamic system model

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

When sys has unstable poles, balred decomposes sys to its stable and unstable parts and only the stable part is approximated. Use balredOptions 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 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

Live Editor Task

Reduce Model Order

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

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

When you use balred for model reduction, you can use balredOptions 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'.