Time Delays in Linear Systems

使用以下模型属性表示线性系统中的时间延迟。

InputDelay,OutputDelay— Time delays at system inputs or outputs
ioDelay,InternalDelay— Time delays that are internal to the system

在离散时间模型中，这些属性被约束在代表样品时间整数倍数表示延迟的整数值中。要近似延迟的离散时间模型，该模型是样本时间的分数倍数，请使用thiran。

First Order Plus Dead Time Model

This example shows how to create a first order plus dead time model using theInputDelay或者OutputDelay属性tf。

要以2.1 s的时间延迟创建以下一阶传输函数：

$G (s) = e^{- 2.1 s} \frac{1}{s + 10},$

enter:

G = tf(1,[1 10],'InputDelay',2.1)

whereInputDelayspecifies the delay at the input of the transfer function.

Tip

您可以使用InputDelaywithZPK与与之相同的方式tf:

G = zpk([],-10,1,'InputDelay',2.1)

For SISO transfer functions, a delay at the input is equivalent to a delay at the output. Therefore, the following command creates the same transfer function:

G = tf(1,[1 10],'OutputDelay',2.1)

利用dot notation to examine or change the value of a time delay. For example, change the time delay to 3.2 as follows:

G.OutputDelay = 3.2;

To see the current value, enter:

G.Outputdelay ans = 3.2000

Tip

An alternative way to create a model with a time delay is to specify the transfer function with the delay as an expression ins:

Create a transfer function model for the variables。
```
s = tf('s');
```
SpecifyG(s）作为表达式s。
```
G = exp(-2.1*s)/(s+10);
```

州空间模型的输入和输出延迟

此示例显示了如何使用输入和输出的延迟创建状态空间模型InputDelay或者OutputDelay属性ss。

创建一个描述以下一输入两输出系统的状态空间模型：

$\begin{array}{l} \frac{d x (t)}{d t} = - 2 x (t) + 3 u (t - 1.5) \\ y (t) = [\begin{matrix} x (t - 0.7) \\ - x (t) \end{matrix}] 。 \end{array}$

该系统的输入延迟为1.5。第一个输出的输出延迟为0.7，第二个输出不会延迟。

Note

In contrast to SISO transfer functions, input delays are not equivalent to output delays for state-space models. Shifting a delay from input to output in a state-space model requires introducing a time shift in the model states. For example, in the model of this example, definingT=t– 1.5和X(T) =x(T+ 1.5）results in the following equivalent system:

$\begin{array}{l} \frac{d X (T)}{d T} = - 2 X (T) + 3 u (T) \\ y (T) = [\begin{matrix} X (T - 2.2) \\ - X (T - 1.5) \end{matrix}] 。 \end{array}$

All of the time delays are on the outputs, but the new state variableXis time-shifted relative to the original state variablex。因此，如果您的状态具有物理意义，或者您已经知道状态初始条件，请在转移输入和输出之间的时间延迟之前仔细考虑。

To create this system:

Define the state-space matrices.
```
A = -2; B = 3; C = [1;-1]; D = 0;
```

Create the model.

g = ss（a，b，c，d，'inputdelay'，1.5，'oppoteDelay'，[0.7; 0]）

Gis ass模型。

Tip

利用delayssto create state-space models with more general combinations of input, output, and state delays, of the form:

$\begin{array}{l} \frac{d x}{d t} = A x (t) + B u (t) + \sum_{j = 1}^{N} (A j x (t - t j) + B j u (t - t j)) \\ y (t) = C x (t) + D u (t) + \sum_{j = 1}^{N} (C j x (t - t j) + D j u (t - t j)) \end{array}$

MIMO转移功能的运输延迟

此示例显示了如何为每个输入输出（I/O）对创建具有不同传输延迟的MIMO传输函数。

Create the MIMO transfer function:

$H (s) = [\begin{matrix} e^{- 0.1} \frac{2}{s} & e^{- 0.3} \frac{s + 1}{s + 10} \\ 10 & e^{- 0.2} \frac{s - 1}{s + 5} \end{matrix}] 。$

Time delays in MIMO systems can be specific to each I/O pair, as in this example. You cannot useInputDelay和OutputDelayto model I/O-specific transport delays. Instead, useioDelayto specify the transport delay across each I/O pair.

To create this MIMO transfer function:

Create a transfer function model for the variables。
```
s = tf('s');
```
利用the variables指定的传输功能Hwithout the time delays.
```
H = [2/s (s+1)/(s+10); 10 (s-1)/(s+5)];
```
指定ioDelayproperty ofHas an array of values corresponding to the transport delay for each I/O pair.
```
H.IODelay = [0.1 0.3; 0 0.2];
```

His a two-input, two-outputtf模型。每个I/O对H有时间延迟由相应的条目指定tau。

离散时间传输功能随时间延迟

打开实时脚本

This example shows how to create a discrete-time transfer function with a time delay.

在离散时间模型,一个采集的延迟ng period corresponds to a factor of $z^{- 1}$ in the transfer function. For example, the following transfer function represents a discrete-time SISO system with a delay of 25 sampling periods.

$H (z) = z^{- 25} \frac{2}{z - 0 。 95} 。$

要代表MATLAB中离散时间系统中的整数延迟，请设置'InputDelay'模型对象的属性为整数值。例如，以下命令创建一个tfmodel representing $H (z)$ with a sampling time of 0.1 s.

h = tf（2，[1 -0.95]，0.1，，'InputDelay',25)

H = 2 z^(-25) * -------- z - 0.95 Sample time: 0.1 seconds Discrete-time transfer function.

如果系统的时间延迟不是采样时间的整数倍数，则可以使用thirancommand to approximate the fractional portion of the time delay with an all-pass filter. SeeTime-Delay Approximation。

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