Zero-Pole

Model system by zero-pole-gain transfer function

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Libraries:
Simulink / 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

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

  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.

  

Examples

Prevent Block Windup in Multiloop Control

Use signal tracking to prevent block windup in a two-loop control system.

Open Model

Bumpless Control Transfer

This example shows how to use signal tracking to achieve bumpless control transfer in a system that switches between a PID Controller block and another controller.

Open Model

Ports

Input

Port_1 — Input signal

scalar

Input signal, specified as a scalar with data type double.

Data Types: double

Output

Port_1 — Output signal

scalar | vector

System modeled by a zero-pole gain transfer function, provided as a scalar or vector signal with data type double.

• When modeling a single-output system, the block outputs a scalar time-domain signal. For more information, see Modeling a Single-Output System.

• When modeling a multiple-output system, the block outputs a vector, where each element is an output of the system. For more information, see Modeling a Multiple-Output System.

Data Types: double

Parameters

Zeros — Matrix of zeros

[1] (default) | vector | matrix

Define the matrix of zeros.

• 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.

Programmatic Use

Block Parameter: Zeros

Type: character vector, string

Value: vector | matrix

Default: '[1]'

Poles — Vector of poles

[0 -1] (default) | vector

Define the vector of poles.

• 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.

Programmatic Use

Block Parameter: Poles

Type: character vector, string

Value: vector

Default: '[0 -1]'

Gain — Vector of gains

[1] (default) | vector

Define the vector of gains.

• 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.

Programmatic Use

Block Parameter: Gain

Type: character vector, string

Value: vector

Default: '[1]'

Parameter tunability — Tunable representation of block parameters in code

Auto (default) | Optimized | Unconstrained

Tunability level of the zeros, poles, and gains for accelerated simulation modes and simulations deployed using Simulink^® Compiler™. Set this parameter to Auto to allow Simulink to choose the appropriate level of parameter tunability.

Set this parameter to Optimized to generate an optimized representation zeros, poles, and gain in generated code for accelerated and deployed simulations.

Set this parameter to Unconstrained to support full tunability (between simulations) of zeros, poles, and gain parameters in accelerated and deployed simulations.

Programmatic Use

Block Parameter: ParameterTunability

Type: character vector, string

Values: 'Auto' | 'Optimized' | 'Unconstrained'

Default: 'Auto'

Absolute tolerance — Absolute tolerance for computing block states

auto (default) | scalar | vector

Absolute tolerance for computing block states, specified as a positive, real-valued, scalar or vector. To inherit the absolute tolerance from the Configuration Parameters, specify auto or -1.

• 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.

• 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.

Programmatic Use

Block Parameter: AbsoluteTolerance

Type: character vector, string

Values: 'auto' | '-1' | any positive real-valued scalar or vector

Default: 'auto'

State Name (e.g., 'position') — Assign unique name to each state

' ' (default) | 'position' | {'a', 'b', 'c'} | a | ...

Assign a unique name to each state. If this field is blank (' '), no name assignment occurs.

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

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

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

Limitations

• 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.

Programmatic Use

Block Parameter: ContinuousStateAttributes

Type: character vector, string

Values: ' ' | user-defined

Default: ' '

Block Characteristics

Data Types	double
Direct Feedthrough	yes
Multidimensional Signals	no
Variable-Size Signals	no
Zero-Crossing Detection	no

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using Simulink® Coder™.

Not recommended for production-quality code. Relates to resource limits and restrictions on speed and memory often found in embedded systems. The code generated can contain dynamic allocation and freeing of memory, recursion, additional memory overhead, and widely-varying execution times. While the code is functionally valid and generally acceptable in resource-rich environments, smaller embedded targets often cannot support such code.

In general, consider using the Simulink Model Discretizer to map continuous blocks into discrete equivalents that support production code generation. To start the Model Discretizer, in the Simulink Editor, on the Apps tab, under Apps, under Control Systems, click Model Discretizer. One exception is the Second-Order Integrator block because, for this block, the Model Discretizer produces an approximate discretization.

Version History

Introduced before R2006a