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Approximate Model by Balanced Truncation at the Command Line

This example shows how to use balred to compute a reduced-order approximation of a model at the MATLAB ® command line.

balred removes the states with the lowest energy contribution to overall model behavior. Therefore, to use balred, you can begin by examining the energy contribution of the model states. You choose the approximation order based on the number of states that make a significant contribution to the overall model behavior.

For this example, load a high-order model. hplant is a 23rd-order SISO model.

load ltiexamples hplant
order(hplant)
ans = 23

Examine the relative amount of energy per state in hplant using a Hankel singular-value (HSV) plot.

hsvplot(hplant)

Figure contains an axes. The axes with title Hankel Singular Values and Approximation Error contains 3 objects of type bar, line. These objects represent Unstable modes, Stable modes, Absolute error bound.

Small Hankel singular values indicate that the associated states contribute little to the behavior of the system. The plot shows that two states account for most of the energy in the system. Therefore, try simplifying the model to just first or second order.

opts = balredOptions('StateElimMethod','Truncate');
hplant1 = balred(hplant,1,opts);
hplant2 = balred(hplant,2,opts);

The second argument to balred specifies the target approximation order, so that hplant1 is a first-order approximation and hplant2 is a second-order approximation of hplant. By default, balred discards the states with the smallest Hankel singular values, and alters the remaining states to preserve the DC gain of the system. Setting the StateElimMethod option to Truncate causes balred to discard low-energy states without altering the remaining states.

When working with reduced-order models, it is important to verify that the approximation does not introduce inaccuracies at frequencies that are important for your application. Therefore, compare the frequency responses of the original and approximated systems. For MIMO systems, use the sigmaplot command. For this SISO system, examine a Bode plot.

bodeplot(hplant,hplant2,hplant1)
legend('Original','2nd order','1st order')

Figure contains 2 axes. Axes 1 with title From: Robot Arm/Demux1 (pt. 2) To: Motorized Spindle1 (pt. 1) contains 3 objects of type line. These objects represent Original, 2nd order, 1st order. Axes 2 contains 3 objects of type line. These objects represent Original, 2nd order, 1st order.

The second-order approximation hplant2 matches the original 23rd-order system very well, especially at lower frequencies. The first-order system does not match as well.

In general, as you decrease the order of the approximated model, the frequency response of the approximated model begins to differ from the original model. Choose an approximation that is sufficiently accurate in the bands that are important to you. For example, in a control system you might want good accuracy inside the control bandwidth. Accuracy at frequencies far above the control bandwidth, where the gain rapidly rolls off, might be less important.

You can also validate the approximation in the time domain. For instance, examine the step responses of the original and reduced-order systems.

stepplot(hplant,hplant2,'r--',hplant1,'g--')
legend('Original','2nd order','1st order','Location','SouthEast')

Figure contains an axes. The axes with title From: Robot Arm/Demux1 (pt. 2) To: Motorized Spindle1 (pt. 1) contains 3 objects of type line. These objects represent Original, 2nd order, 1st order.

This result confirms that the second-order approximation is a good match to the original 23rd-order system.

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