evalfr

Evaluate system response at specific frequency

Syntax

frsp = evalfr(sys,x)

Description

evalfr is a simplified version of freqresp meant for quick evaluation of the system response at any point in the complex plane. To evaluate system response over a set of frequencies, use freqresp. To obtain the magnitude and phase data as well as plots of the frequency response, use bode.

frsp = evalfr(sys,x) evaluates the dynamic system model sys at the point x in the complex s plane (for continuous-time sys) or z plane (for discrete-time sys).

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:

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

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:

• LTI models such as ss (Control System Toolbox), tf (Control System Toolbox), and zpk (Control System Toolbox) models.

• Sparse state-space models, such as sparss (Control System Toolbox) or mechss (Control System Toolbox) models.

• Generalized or uncertain state-space models such as genss (Control System Toolbox) 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 models.

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

frsp — Frequency response

complex scalar | complex array

Frequency response of the system at the point x, returned as a complex scalar (for SISO sys) or a complex array (for MIMO sys). For MIMO systems, the array dimensions correspond to the I/O dimensions of sys.

Version History

Introduced in R2012a