# Zero-Pole

Model system by zero-pole-gain transfer function

Continuous

## Description

The Zero-Pole block models a system that you define with the zeros, poles, and gain of a Laplace-domain transfer function. This block can model single-input single output (SISO) and single-input multiple-output (SIMO) systems.

### Conditions for Using This Block

The Zero-Pole block assumes the following conditions:

• The transfer function has the form

`$H\left(s\right)=K\frac{Z\left(s\right)}{P\left(s\right)}=K\frac{\left(s-Z\left(1\right)\right)\left(s-Z\left(2\right)\right)\dots \left(s-Z\left(m\right)\right)}{\left(s-P\left(1\right)\right)\left(s-P\left(2\right)\right)\dots \left(s-P\left(n\right)\right)},$`

where Z represents the zeros, P the poles, and K the gain of the transfer function.

• The number of poles must be greater than or equal to the number of zeros.

• If the poles and zeros are complex, they must be complex-conjugate pairs.

• For a multiple-output system, all transfer functions must have the same poles. The zeros can differ in value, but the number of zeros for each transfer function must be the same.

 Note:   You cannot use a Zero-Pole block to model a multiple-output system when the transfer functions have a differing number of zeros or a single zero each. Use multiple Zero-Pole blocks to model such systems.

### Modeling a Single-Output System

For a single-output system, the input and the output of the block are scalar time-domain signals. To model this system:

1. Enter a vector for the zeros of the transfer function in the Zeros field.

2. Enter a vector for the poles of the transfer function in the Poles field.

3. Enter a 1-by-1 vector for the gain of the transfer function in the Gain field.

### Modeling a Multiple-Output System

For a multiple-output system, the block input is a scalar and the output is a vector, where each element is an output of the system. To model this system:

1. Enter a matrix of zeros in the Zeros field.

Each column of this matrix contains the zeros of a transfer function that relates the system input to one of the outputs.

2. Enter a vector for the poles common to all transfer functions of the system in the Poles field.

3. Enter a vector of gains in the Gain field.

Each element is the gain of the corresponding transfer function in Zeros.

Each element of the output vector corresponds to a column in Zeros.

### Transfer Function Display on the Block

The Zero-Pole block displays the transfer function depending on how you specify the zero, pole, and gain parameters.

• If you specify each parameter as an expression or a vector, the block shows the transfer function with the specified zeros, poles, and gain. If you specify a variable in parentheses, the block evaluates the variable.

For example, if you specify Zeros as `[3,2,1]`, Poles as `(poles)`, where `poles` is `[7,5,3,1]`, and Gain as `gain`, the block looks like this:

• If you specify each parameter as a variable, the block shows the variable name followed by `(s)` if appropriate.

For example, if you specify Zeros as `zeros`, Poles as `poles`, and Gain as `gain`, the block looks like this:

## Data Type Support

The Zero-Pole block accepts real signals of type `double`. For more information, see Data Types Supported by Simulink in the Simulink® documentation.

## Parameters

### Zeros

Define the matrix of zeros.

#### Settings

Default: `[1]`

#### Tips

• For a single-output system, enter a vector for the zeros of the transfer function.

• For a multiple-output system, enter a matrix. Each column of this matrix contains the zeros of a transfer function that relates the system input to one of the outputs.

#### Command-Line Information

 Parameter: `Zeros` Type: vector Value: `'[1]'` Default: `'[1]'`

### Poles

Define the vector of poles.

#### Settings

Default: `[0 -1]`

#### Tips

• For a single-output system, enter a vector for the poles of the transfer function.

• For a multiple-output system, enter a vector for the poles common to all transfer functions of the system.

#### Command-Line Information

 Parameter: `Poles` Type: vector Value: `'[0 -1]'` Default: `'[0 -1]'`

### Gain

Define the vector of gains.

#### Settings

Default: `[1]`

#### Tips

• For a single-output system, enter a 1-by-1 vector for the gain of the transfer function.

• For a multiple-output system, enter a vector of gains. Each element is the gain of the corresponding transfer function in Zeros.

#### Command-Line Information

 Parameter: `Gain` Type: vector Value: `'[1]'` Default: `'[1]'`

### Absolute tolerance

Specify the absolute tolerance for computing block states.

#### Settings

Default: `auto`

• You can enter `auto`, –1, a positive real scalar or vector.

• If you enter `auto` or –1, then Simulink uses the absolute tolerance value in the Configuration Parameters dialog box (see Solver Pane) to compute block states.

• If you enter a real scalar, then that value overrides the absolute tolerance in the Configuration Parameters dialog box for computing all block states.

• If you enter a real vector, then the dimension of that vector must match the dimension of the continuous states in the block. These values override the absolute tolerance in the Configuration Parameters dialog box.

#### Command-Line Information

 Parameter: ` AbsoluteTolerance` Type: character vector, scalar, or vector Value: `'auto'` | `'-1'` | any positive real scalar or vector Default: ` 'auto'`

### State Name (e.g., 'position')

Assign a unique name to each state.

#### Settings

Default: `' '`

If this field is blank, no name assignment occurs.

#### Tips

• To assign a name to a single state, enter the name between quotes, for example, `'velocity'`.

• To assign names to multiple states, enter a comma-delimited list surrounded by braces, for example, `{'a', 'b', 'c'}`. Each name must be unique.

• The state names apply only to the selected block.

• The number of states must divide evenly among the number of state names.

• You can specify fewer names than states, but you cannot specify more names than states.

For example, you can specify two names in a system with four states. The first name applies to the first two states and the second name to the last two states.

• To assign state names with a variable in the MATLAB® workspace, enter the variable without quotes. A variable can be a character vector, cell array, or structure.

#### Command-Line Information

 Parameter: `ContinuousStateAttributes` Type: character vector Value: `' '` | user-defined Default: `' '`

## Characteristics

 Data Types Double Sample Time Continuous Direct Feedthrough Only if the lengths of the Poles and Zeros parameters are equal Multidimensional Signals No Variable-Size Signals No Zero-Crossing Detection No Code Generation Yes

Discrete Zero-Pole