## Customize Model Display

### Configure Transfer Function Display Variable

This example shows how to configure the MATLAB® command-window display of transfer function (`tf`) models.

You can use the same steps to configure the display variable of transfer function models in factorized form (`zpk` models).

By default, `tf` and `zpk` models are displayed in terms of `s` in continuous time and `z` in discrete time. Use the `Variable` property change the display variable to `'p'` (equivalent to `'s'`), `'q'` (equivalent to `'z'`), `'z^-1'`, or `'q^-1'`.

1. Create the discrete-time transfer function $H\left(z\right)=\frac{z-1}{{z}^{2}-3z+2}$

with a sample time of 1 s.

` H = tf([1 -1],[1 -3 2],0.1)`
```H = z - 1 ------------- z^2 - 3 z + 2 Sample time: 0.1 seconds Discrete-time transfer function. ```

The default display variable is `z`.

2. Change the display variable to `q^-1`.

`H.Variable = 'q^-1'`
```H = q^-1 - q^-2 ------------------- 1 - 3 q^-1 + 2 q^-2 Sample time: 0.1 seconds Discrete-time transfer function. ```

When you change the `Variable` property, the software computes new coefficients and displays the transfer function in terms of the new variable. The `num` and `den` properties are automatically updated with the new coefficients.

Tip

Alternatively, you can directly create the same transfer function expressed in terms of `'q^-1'`.

`H = tf([0 1 -1],[1 -3 2],0.1,'Variable','q^-1');`

For the inverse variables `'z^-1'` and `'q^-1'`, `tf` interprets the numerator and denominator arrays as coefficients of ascending powers of `'z^-1'` or `'q^-1'`.

### Configure Display Format of Transfer Function in Factorized Form

This example shows how to configure the display of transfer function models in factorized form (`zpk` models).

You can configure the display of the factorized numerator and denominator polynomials to highlight:

• The numerator and denominator roots

• The natural frequencies and damping ratios of each root

• The time constants and damping ratios of each root

See the `DisplayFormat` property on the `zpk` reference page for more information about these quantities.

1. Create a `zpk` model having a zero at s = 5, a pole at s = –10, and a pair of complex poles at s = –3 ± 5i.

`H = zpk(5,[-10,-3-5*i,-3+5*i],10)`
```H = 10 (s-5) ---------------------- (s+10) (s^2 + 6s + 34) Continuous-time zero/pole/gain model. ```

The default display format, `'roots'`, displays the standard factorization of the numerator and denominator polynomials.

2. Configure the display format to display the natural frequency of each polynomial root.

` H.DisplayFormat = 'frequency'`
```H = -0.14706 (1-s/5) ------------------------------------------- (1+s/10) (1 + 1.029(s/5.831) + (s/5.831)^2) Continuous-time zero/pole/gain model. ```

You can read the natural frequencies and damping ratios for each pole and zero from the display as follows:

• Factors corresponding to real roots are displayed as (1 – s/ω0). The variable ω0 is the natural frequency of the root. For example, the natural frequency of the zero of `H` is 5.

• Factors corresponding to complex pairs of roots are displayed as 1 – 2ζ(s/ω0) + (s/ω0)2. The variable ω0 is the natural frequency, and ζ is the damping ratio of the root. For example, the natural frequency of the complex pole pair is 5.831, and the damping ratio is 1.029/2.

3. Configure the display format to display the time constant of each pole and zero.

`H.DisplayFormat = 'time constant'`
```H = -0.14706 (1-0.2s) ------------------------------------------- (1+0.1s) (1 + 1.029(0.1715s) + (0.1715s)^2) Continuous-time zero/pole/gain model. ```

You can read the time constants and damping ratios from the display as follows:

• Factors corresponding to real roots are displayed as (τs). The variable τ is the time constant of the root. For example, the time constant of the zero of `H` is 0.2.

• Factors corresponding to complex pairs of roots are displayed as 1 – 2ζ(τs) + (τs)2. The variable τ is the time constant, and ζ is the damping ratio of the root. For example, the time constant of the complex pole pair is 0.1715, and the damping ratio is 1.029/2.