What are Transfer Function Models?

Definition of Transfer Function Models

Transfer function models describe the relationship between the inputs and outputs of a system using a ratio of polynomials. Themodel orderis equal to the order of the denominator polynomial. The roots of the denominator polynomial are referred to as the modelpoles. The roots of the numerator polynomial are referred to as the modelzeros.

传递函数模型的参数s poles, zeros and transport delays.

Continuous-Time Representation

In continuous-time, a transfer function model has the form:

$Y (s) = \frac{n u m (s)}{d e n (s)} U (s) + E (s)$

Where,Y(s),U(s) andE(s) represent the Laplace transforms of the output, input and noise, respectively.num(s) andden(s) represent the numerator and denominator polynomials that define the relationship between the input and the output.

Discrete-Time Representation

In discrete-time, a transfer function model has the form:

$\begin{array}{l} y (t) = \frac{n u m (q^{- 1})}{d e n (q^{- 1})} u (t) + e (t) \\ n u m (q^{- 1}) = b_{0} + b_{1} q^{- 1} + b_{2} q^{- 2} + \dots \\ d e n (q^{- 1}) = 1 + a_{1} q^{- 1} + a_{2} q^{- 2} + \dots \end{array}$

The roots ofnum(q^-1) andden(q^-1) are expressed in terms of the lag variableq^-1.

If you take the Z-transform, the transfer function has the form:

$\begin{array}{l} Y (z^{- 1}) = \frac{n u m (z^{- 1})}{d e n (z^{- 1})} U (z^{- 1}) + E (z^{- 1}) \\ n u m (z^{- 1}) = b_{0} + b_{1} z^{- 1} + b_{2} z^{- 2} + \dots \\ d e n (z^{- 1}) = 1 + a_{1} z^{- 1} + a_{2} z^{- 2} + \dots \end{array}$

Where,Y(z^-1),U(z^-1) andE(z^-1) represent the Z-transforms of the output, input and noise, respectively.z^-1is the Z-transform of the lag operator.

Delays

In continuous-time, input and transport delays are of the form:

$Y (s) = \frac{n u m (s)}{d e n (s)} e^{- s τ} U (s) + E (s)$

Whereτrepresents the delay.

In discrete-time:

$y (t) = \frac{n u m}{d e n} u (t - τ) + e (t)$

wherenumanddenare polynomials in the lag operatorq^(-1).

Multi-Input Multi-Output Models

A single-input single-output (SISO) continuous transfer function has the form $G (s) = \frac{n u m (s)}{d e n (s)}$ . The corresponding transfer function model can be represented as:

$Y (s) = G (s) U (s) + E (s)$

A multi-input multi-output (MIMO) transfer function contains a SISO transfer function corresponding to each input-output pair in the system. For example, a continuous-time transfer function model with two inputs and two outputs has the form:

$\begin{array}{l} Y_{1} (s) = G_{11} (s) U_{1} (s) + G_{12} (s) U_{2} (s) + E_{1} (s) \\ Y_{2} (s) = G_{21} (s) U_{1} (s) + G_{22} (s) U_{2} (s) + E_{2} (s) \end{array}$

Where,G_ij(s)is the SISO transfer function between thei^thoutput and thej^thinput.E₁(s)andE₂(s)are the Laplace transforms of the noise corresponding to the two outputs.

The representation of discrete-time MIMO transfer function models is analogous.