Varying Transfer Function

Transfer function with varying coefficients

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Libraries:
Control System Toolbox / Linear Parameter Varying

Description

This block implements a continuous-time transfer function with varying coefficients. The instantaneous transfer function is given by:

$$H\left(s\right)=\frac{b_0+b_1{s}^{-1}+\cdots +b_N{s}^{-N}}{1+a_1{s}^{-1}+\cdots +a_N{s}^{-N}}=\frac{b_0{s}^N+b_1{s}^{N-1}+\cdots +b_N}{s^N+a_1{s}^{N-1}+\cdots +a^N}.$$

N is number of poles, specified with the Transfer function order parameter. Feed the values of the coefficients b₀, b₁,…, b_N and a₁, a₂,…, a_N to the corresponding block input ports.

Note

The above expression for H(s) applies only to Varying Transfer Function blocks added to a model in R2023a or later. For information about Varying Transfer Function blocks created in R2022b or earlier, see Varying Transfer Function block formula changed.

Use this block and the other blocks in the Linear Parameter Varying library to implement common control elements with variable parameters or coefficients. For more information, see Model Gain-Scheduled Control Systems in Simulink.

Caution

Avoid making the transfer-function coefficients depend on the block output y. If you have such dependence, the resulting transfer function causes an algebraic loop, because computing the block output value requires knowing the block output value. This algebraic loop is prone to instability and divergence. Instead of the output, try expressing the coefficients in terms of the time t and the block input u.

Ports

Input

u — Transfer function input

scalar

Transfer function input signal.

b0, b1, ... — Numerator coefficients

scalar

Transfer function numerator coefficients. The number of coefficient ports is determined by the Transfer function order parameter.

For a zero-feedthrough transfer function, clear the Enable feedthrough checkbox. Clearing this checkbox disables the b0 input port. (since R2023a)

a1, a2, ... — Denominator coefficients

scalar

Transfer function denominator coefficients. The number of coefficient ports is determined by the Transfer function order parameter. The coefficient of the lowest-order term (a₀) is fixed to 1.

Before R2023a: The denominator coefficients are a0, a1,…. The coefficient a_N is fixed to 1. For more information, see Varying Transfer Function block formula changed.

Output

y — Transfer function output

scalar

Transfer function output signal.

Parameters

Transfer function order — Number of poles N

1 (default) | positive integer

Transfer function order N, specified as a positive integer. This parameter determines the number of coefficient input ports on the block.

Enable feedthrough — Add input port for feedthrough term

on (default) | off

Since R2023a

Enable the b0 input port for a transfer function with direct feedthrough. For a zero-feedthrough transfer function, clear this checkbox. Disabling the port for zero-feedthrough models is numerically more reliable than feeding a zero-constant into the port.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2017b

R2023a: Varying Transfer Function block formula changed

In R2023a, the Varying Transfer Function block formula was changed to make the polynomial coefficient definitions consistent with the Discrete Varying Transfer Function block. Block diagrams to which the block was added in R2022b or earlier are unaffected by this change.

• A Varying Transfer Function block added to a Simulink^® model in R2023a or later has an instantaneous transfer function described by:

  $$H\left(s\right)=\frac{b_0+b_1{s}^{-1}+\cdots +b_N{s}^{-N}}{1+a_1{s}^{-1}+\cdots +a_N{s}^{-N}}=\frac{b_0{s}^N+b_1{s}^{N-1}+\cdots +b_N}{s^N+a_1{s}^{N-1}+\cdots +a^N}.$$

• A block added in R2022b or before has an instantaneous transfer function described by:

  $$H\left(s\right)=\frac{b_0+b_{1}s+\cdots b_N{s}^N}{a_0+a_{1}s+\cdots a_{N-1}{s}^{N-1}+s^N}.$$

  Note that replacing a pre-R2023a block with the new block requires rewiring the coefficient inputs. For instance, consider the following second-order transfer function:

  $$H_{22b}\left(s\right)=\frac{10+9s+8s^2}{7+6s+s^2}.$$

  Using a block added in R2022b or before, you wire up the block as shown in the following illustration. The block fixes a2 to 1.

  

  To replace the block in R2023a or after, rewrite the transfer function as follows:

  $$H_{23a}\left(s\right)=\frac{8+9s^{-1}+10s^{-2}}{1+6s^{-1}+7s^{-2}}.$$

  Rewire the block as shown. Now, the block fixes a0 to 1.

  

R2023a: Enable feedthrough checkbox added

For a zero-feedthrough transfer function, clear the Enable feedthrough checkbox. Clearing this checkbox disables the b0 input port. Disabling the port for zero-feedthrough models is numerically more reliable than feeding a zero-constant into the port.