# evalfr

Evaluate system response at specific frequency

## Syntax

## Description

## Examples

### Evaluate Discrete-Time Transfer Function

Create the following discrete-time transfer function.

$$H\left(z\right)=\frac{z-1}{{z}^{2}+z+1}$$

H = tf([1 -1],[1 1 1],-1);

Evaluate the transfer function at `z = 1+j`

.

z = 1+j; evalfr(H,z)

ans = 0.2308 + 0.1538i

### Evaluate Frequency Response of Identified Model at Given Frequency

Create the following continuous-time transfer function model:

$$H\left(s\right)=\frac{1}{{s}^{2}+2s+1}$$

sys = idtf(1,[1 2 1]);

Evaluate the transfer function at frequency 0.1 rad/second.

w = 0.1; s = j*w; evalfr(sys,s)

ans = 0.9705 - 0.1961i

Alternatively, use the `freqresp`

command.

freqresp(sys,w)

ans = 0.9705 - 0.1961i

### Frequency Response of MIMO State-Space Model

For this example, consider a cube rotating about its corner with inertia tensor `J`

and a damping force `F`

of 0.2 magnitude. The input to the system is the driving torque while the angular velocities are the outputs. The state-space matrices for the cube are:

$$\begin{array}{l}A=-{J}^{-1}F,\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}B={J}^{-1},\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}C=I,\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}D=0,\\ where,\phantom{\rule{0.2777777777777778em}{0ex}}J=\left[\begin{array}{ccc}8& -3& -3\\ -3& 8& -3\\ -3& -3& 8\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}and\phantom{\rule{0.2777777777777778em}{0ex}}F=\left[\begin{array}{ccc}0.2& 0& 0\\ 0& 0.2& 0\\ 0& 0& 0.2\end{array}\right]\end{array}$$

Specify the `A`

, `B`

, `C`

and `D`

matrices, and create the continuous-time state-space model.

J = [8 -3 -3; -3 8 -3; -3 -3 8]; F = 0.2*eye(3); A = -J\F; B = inv(J); C = eye(3); D = 0; sys = ss(A,B,C,D); size(sys)

State-space model with 3 outputs, 3 inputs, and 3 states.

Compute the frequency response of the system at 0.2 rad/second. Since `sys`

is a continuous-time model, express the frequency in terms of the Laplace variable `s`

.

w = 0.2; s = j*w; frsp = evalfr(sys,s)

`frsp = `*3×3 complex*
0.3607 - 0.9672i 0.3197 - 0.5164i 0.3197 - 0.5164i
0.3197 - 0.5164i 0.3607 - 0.9672i 0.3197 - 0.5164i
0.3197 - 0.5164i 0.3197 - 0.5164i 0.3607 - 0.9672i

Alternatively, you can use the `freqresp`

command to evaluate the frequency response using the scalar value of the frequency directly.

H = freqresp(sys,w)

`H = `*3×3 complex*
0.3607 - 0.9672i 0.3197 - 0.5164i 0.3197 - 0.5164i
0.3197 - 0.5164i 0.3607 - 0.9672i 0.3197 - 0.5164i
0.3197 - 0.5164i 0.3197 - 0.5164i 0.3607 - 0.9672i

## Input Arguments

`sys`

— Dynamic system

dynamic system model | model array

Dynamic system, specified as a SISO or MIMO dynamic system model or array of dynamic system models. Dynamic systems that you can use include:

Generalized or uncertain state-space models such as

`genss`

or`uss`

(Robust Control Toolbox) models. (Using uncertain models requires Robust Control Toolbox™ software.)For tunable control design blocks, the function evaluates the model at its current value to evaluate the frequency response.

For uncertain control design blocks, the function evaluates the frequency response at the nominal value and random samples of the model.

Identified state-space models, such as

`idss`

(System Identification Toolbox) models. (Using identified models requires System Identification Toolbox™ software.)

For a complete list of models, see Dynamic System Models.

`x`

— Point in complex plane

complex scalar

Point in complex plane at which to evaluate system response, specified as a complex
scalar. For continuous-time `sys`

, the point `x`

is in the plane of the continuous-time Laplace variable *s*. For
discrete-time `sys`

, `x`

is in the plane of the
discrete-time Laplace variable *z*.

To evaluate the response of the system at a particular frequency, specify the
frequency in terms of the appropriate Laplace variable. For instance, if you want to
evaluate the frequency response of a system `sys`

at a frequency
value of `w`

rad/s, then use:

`x = j*w`

, for continuous-time`sys`

.`z = exp(j*w*Ts)`

, for discrete-time`sys`

, where`Ts`

is the sample time.

## Output Arguments

## Version History

**Introduced before R2006a**

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