# Create Tunable Second-Order Filter

This example shows how to create a parametric model of the second-order filter:

$F\left(s\right)=\frac{{\omega }_{n}^{2}}{{s}^{2}+2\zeta {\omega }_{n}s+{\omega }_{n}^{2}},$

where the damping $\zeta$ and the natural frequency ${\omega }_{n}$ are tunable parameters.

Define the tunable parameters using realp.

wn = realp('wn',3);
zeta = realp('zeta',0.8);

wn and zeta are realp parameter objects, with initial values 3 and 0.8, respectively.

Create a model of the filter using the tunable parameters.

F = tf(wn^2,[1 2*zeta*wn wn^2]);

The inputs to tf are the vectors of numerator and denominator coefficients expressed in terms of wn and zeta.

F is a genss model. The property F.Blocks lists the two tunable parameters wn and zeta.

F.Blocks
ans = struct with fields:
wn: [1x1 realp]
zeta: [1x1 realp]

You can examine the number of tunable blocks in a generalized model using nblocks.

nblocks(F)
ans = 6

F has two tunable parameters, but the parameter wn appears five times - Twice in the numerator and three times in the denominator.

To reduce the number of tunable blocks, you can rewrite F as:

$F\left(s\right)=\frac{1}{{\left(\frac{s}{{\omega }_{n}}\right)}^{2}+2\zeta \left(\frac{s}{{\omega }_{n}}\right)+1}.$

Create the alternative filter.

F = tf(1,[(1/wn)^2 2*zeta*(1/wn) 1]);

Examine the number of tunable blocks in the new model.

nblocks(F)
ans = 4

In the new formulation, there are only three occurrences of the tunable parameter wn. Reducing the number of occurrences of a block in a model can improve the performance of calculations involving the model. However, the number of occurrences does not affect the results of tuning the model or sampling it for parameter studies.