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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'`

.

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`

.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'`

.

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.

Create a

`zpk`

model having a zero at*s*= 5, a pole at*s*= –10, and a pair of complex poles at*s*= –3 ± 5*i*.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.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*/*ω*_{0}). The variable*ω*_{0}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 – 2

*ζ*(*s*/*ω*_{0}) + (*s*/*ω*_{0})^{2}. The variable*ω*_{0}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.

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 – 2

*ζ*(*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.