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# zero

Zeros and gain of SISO dynamic system

## Description

example

Z = zero(sys) returns the zeros of the single-input, single-output (SISO) dynamic system model, sys. The output is expressed as the reciprocal of the time units specified in sys.TimeUnit.

example

[Z,gain] = zero(sys) also returns the zero-pole-gain of sys.

example

[Z,gain] = zero(sys,J1,...,JN) returns the zeros and gain of the entries in the model array sys with subscripts J1,...,JN.

## Examples

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Compute the zeros of the following transfer function:

$sys\left(s\right)=\frac{4.2{s}^{2}+0.25s-0.004}{{s}^{2}+9.6s+17}$

sys = tf([4.2,0.25,-0.004],[1,9.6,17]);
Z = zero(sys)
Z = 2×1

-0.0726
0.0131

Calculate the zero locations and zero-pole gain of the following transfer function:

$sys\left(s\right)=\frac{4.2{s}^{2}+0.25s-0.004}{{s}^{2}+9.6s+17}$

sys = tf([4.2,0.25,-0.004],[1,9.6,17]);
[z,gain] = zero(sys)
z = 2×1

-0.0726
0.0131

gain = 4.2000

The zero locations are expressed in ${\mathrm{second}}^{-1},$ because the time unit of the transfer function (H.TimeUnit) is seconds.

For this example, load a 3-by-1 array of transfer function models.

size(sys)
3x1 array of transfer functions.
Each model has 1 outputs and 1 inputs.

Find the zeros and gain values of the models in the array.

[Z, gain] = zero(sys);
Z(:,:,1,1)
ans = 2×1

-0.0726
0.0131

gain(:,:,1,1)
ans = 4.2000

zero returns an array each for the zeros and the gain values respectively. Here, Z(:,:,1,1) and gain(:,:,1,1) corresponds to the zero and the gain value of the first model in the array, that is, sys(:,:,1,1).

## Input Arguments

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Dynamic system, specified as a SISO dynamic system model, or an array of SISO dynamic system models. Dynamic systems that you can use include continuous-time or discrete-time numeric LTI models such as tf, zpk, or ss models.

If sys is a generalized state-space model genss or an uncertain state-space model uss, zero returns the zeros of the current or nominal value of sys. If sys is an array of models, zero returns the zeros of the model corresponding to its subscript J1,...,JN in sys. For more information on model arrays, see Model Arrays (Control System Toolbox).

Indices of models in array whose zeros you want to extract, specified as a positive integer. You can provide as many indices as there are array dimensions in sys. For example, if sys is a 4-by-5 array of dynamic system models, the following command extracts the zeros for entry (2,3) in the array.

Z = zero(sys,2,3);

## Output Arguments

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Zeros of the dynamic system, returned as a column vector or an array. If sys is:

• A single model, then Z is a column vector of zeros of the dynamic system model sys

• A model array, then Z is an array containing the zeros of each model in sys

Z is expressed as the reciprocal of the time units specified in sys.TimeUnit. For example, zero is expressed in 1/minute if sys.TimeUnit = 'minutes'.

Zero-pole-gain of the dynamic system, returned as a scalar. In other words, gain is the value of K when the model is written in zpk form.

## Tips

• If sys has internal delays, zero sets all internal delays to zero, creating a zero-order Padé approximation. This approximation ensures that the system has a finite number of zeros. zero returns an error if setting internal delays to zero creates singular algebraic loops. To assess the stability of models with internal delays, use step or impulse.

• To calculate the transmission zeros of a multi-input, multi-output (MIMO) system, use tzero.