*Frequency response* describes the steady-state
response of a system to sinusoidal inputs.

For a linear system, a sinusoidal input of frequency ω:

$$u(t)={A}_{u}\mathrm{sin}\omega t$$

results in an output that is also a sinusoid with the same frequency,
but with a different amplitude and phase *θ*:

$$y(t)={A}_{y}\mathrm{sin}(\omega t+\theta )$$

Frequency response *G*(*s*)
for a stable system describes the amplitude change and phase shift
as a function of frequency:

$$\begin{array}{l}G(s)=\frac{Y(s)}{U(s)}\\ \left|G(s)\right|=\left|G(j\omega )\right|=\frac{{A}_{y}}{{A}_{u}}\\ \theta =\angle \frac{Y(j\omega )}{X(j\omega )}={\mathrm{tan}}^{-1}\left(\frac{\text{imaginarypartof}G(j\omega )}{\text{realpartof}G(j\omega )}\right)\end{array}$$

where *Y*(*s*)
and *U*(*s*) are the Laplace transforms
of *y*(*t*) and *u*(*t*),
respectively.

You can estimate the frequency response of a Simulink^{®} model
as a frequency response model (`frd`

object),
without modifying your Simulink model.

Applications of frequency response models include:

Validating exact linearization results.

Frequency response estimation uses a different algorithm to compute a linear model approximation and serves as an independent test of exact linearization. See Validate Linearization In Frequency Domain.

Analyzing linear model dynamics.

Designing controller for the plant represented by the estimated frequency response using Control System Toolbox™ software.

Estimating parametric models.

See Estimate Frequency Response Models with Noise Using System Identification Toolbox.

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