Documentation

# impulse

Impulse response plot of dynamic system; impulse response data

## Syntax

```impulse(sys) impulse(sys,Tfinal) impulse(sys,t) impulse(sys1,sys2,...,sysN) impulse(sys1,sys2,...,sysN,Tfinal) impulse(sys1,sys2,...,sysN,t) [y,t] = impulse(sys) [y,t] = impulse(sys,Tfinal) y = impulse(sys,t) [y,t,x] = impulse(sys) [y,t,x,ysd] = impulse(sys) ```

## Description

`impulse` calculates the unit impulse response of a dynamic system model. For continuous-time dynamic systems, the impulse response is the response to a Dirac input δ(t). For discrete-time systems, the impulse response is the response to a unit area pulse of length `Ts` and height `1/Ts`, where `Ts` is the sample time of the system. (This pulse approaches δ(t) as `Ts` approaches zero.) For state-space models, `impulse` assumes initial state values are zero.

`impulse(sys)` plots the impulse response of the dynamic system model `sys`. This model can be continuous or discrete, and SISO or MIMO. The impulse response of multi-input systems is the collection of impulse responses for each input channel. The duration of simulation is determined automatically to display the transient behavior of the response.

`impulse(sys,Tfinal)` simulates the impulse response from `t = 0` to the final time ```t = Tfinal```. Express `Tfinal` in the system time units, specified in the `TimeUnit` property of `sys`. For discrete-time systems with unspecified sample time (`Ts = -1`), `impulse` interprets `Tfinal` as the number of sampling periods to simulate.

`impulse(sys,t) ` uses the user-supplied time vector `t` for simulation. Express `t` in the system time units, specified in the `TimeUnit` property of `sys`. For discrete-time models, `t` should be of the form `Ti:Ts:Tf`, where `Ts` is the sample time. For continuous-time models, `t` should be of the form `Ti:dt:Tf`, where `dt` becomes the sample time of a discrete approximation to the continuous system (see Algorithms). The `impulse` command always applies the impulse at `t=0`, regardless of `Ti`.

To plot the impulse responses of several models `sys1`,..., `sysN` on a single figure, use:

`impulse(sys1,sys2,...,sysN)`

`impulse(sys1,sys2,...,sysN,Tfinal)`

`impulse(sys1,sys2,...,sysN,t)`

As with `bode` or `plot`, you can specify a particular color, linestyle, and/or marker for each system, for example,

```impulse(sys1,'y:',sys2,'g--') ```

See "Plotting and Comparing Multiple Systems" and the `bode` entry in this section for more details.

When invoked with output arguments:

`[y,t] = impulse(sys)`

`[y,t] = impulse(sys,Tfinal)`

`y = impulse(sys,t)`

`impulse` returns the output response `y` and the time vector `t` used for simulation (if not supplied as an argument to impulse). No plot is drawn on the screen. For single-input systems, `y` has as many rows as time samples (length of `t`), and as many columns as outputs. In the multi-input case, the impulse responses of each input channel are stacked up along the third dimension of `y`. The dimensions of `y` are then

For state-space models only:

`[y,t,x] = impulse(sys)`

(length of t) × (number of outputs) × (number of inputs)

and `y(:,:,j)` gives the response to an impulse disturbance entering the `j`th input channel. Similarly, the dimensions of `x` are

(length of t) × (number of states) × (number of inputs)

`[y,t,x,ysd] = impulse(sys)` returns the standard deviation `YSD` of the response `Y` of an identified system `SYS`. `YSD` is empty if `SYS` does not contain parameter covariance information.

## Examples

### Impulse Response Plot of Second-Order State-Space Model

Plot the impulse response of the second-order state-space model

`$\begin{array}{c}\left[\begin{array}{l}{\stackrel{˙}{x}}_{1}\\ {\stackrel{˙}{x}}_{2}\end{array}\right]=\left[\begin{array}{cc}-0.5572& -0.7814\\ 0.7814& 0\end{array}\right]\left[\begin{array}{l}{x}_{1}\\ {x}_{2}\end{array}\right]+\left[\begin{array}{cc}1& -1\\ 0& 2\end{array}\right]\left[\begin{array}{l}{u}_{1}\\ {u}_{2}\end{array}\right]\\ y=\left[\begin{array}{cc}1.9691& 6.4493\end{array}\right]\left[\begin{array}{l}{x}_{1}\\ {x}_{2}\end{array}\right]\end{array}$`
```a = [-0.5572 -0.7814;0.7814 0]; b = [1 -1;0 2]; c = [1.9691 6.4493]; sys = ss(a,b,c,0); impulse(sys)``` The left plot shows the impulse response of the first input channel, and the right plot shows the impulse response of the second input channel.

You can store the impulse response data in MATLAB® arrays by

`[y,t] = impulse(sys);`

Because this system has two inputs, `y` is a 3-D array with dimensions

`size(y)`
```ans = 1×3 139 1 2 ```

(the first dimension is the length of `t`). The impulse response of the first input channel is then accessed by

```ch1 = y(:,:,1); size(ch1)```
```ans = 1×2 139 1 ```

### Impulse Data from Identified System

Fetch the impulse response and the corresponding 1 std uncertainty of an identified linear system .

```load(fullfile(matlabroot, 'toolbox', 'ident', 'iddemos', 'data', 'dcmotordata')); z = iddata(y, u, 0.1, 'Name', 'DC-motor'); set(z, 'InputName', 'Voltage', 'InputUnit', 'V'); set(z, 'OutputName', {'Angular position', 'Angular velocity'}); set(z, 'OutputUnit', {'rad', 'rad/s'}); set(z, 'Tstart', 0, 'TimeUnit', 's'); model = tfest(z,2); [y,t,~,ysd] = impulse(model,2); % Plot 3 std uncertainty subplot(211) plot(t,y(:,1), t,y(:,1)+3*ysd(:,1),'k:', t,y(:,1)-3*ysd(:,1),'k:') subplot(212) plot(t,y(:,2), t,y(:,2)+3*ysd(:,2),'k:', t,y(:,2)-3*ysd(:,2),'k:')```

## Limitations

The impulse response of a continuous system with nonzero D matrix is infinite at t = 0. `impulse` ignores this discontinuity and returns the lower continuity value Cb at t = 0.

## Tips

You can change the properties of your plot, for example the units. For information on the ways to change properties of your plots, see Ways to Customize Plots.

## Algorithms

Continuous-time models are first converted to state space. The impulse response of a single-input state-space model

`$\begin{array}{l}\stackrel{˙}{x}=Ax+bu\\ y=Cx\end{array}$`

is equivalent to the following unforced response with initial state b.

`$\begin{array}{cc}\stackrel{˙}{x}=Ax,& x\left(0\right)=b\\ y=Cx& \end{array}$`

To simulate this response, the system is discretized using zero-order hold on the inputs. The sample time is chosen automatically based on the system dynamics, except when a time vector `t = 0:dt:Tf` is supplied (`dt` is then used as sample time).