Measure Frequency Response by Using Complex Data in Stateflow
This example shows how to emulate a spectrum analyzer that measures the frequency response of a continuous-time system driven by a complex sinusoidal signal. In this Simulink® model, the Plant block describes a second-order resonant system of the form
with a natural frequency of (or 150 Hz) and a damping ratio of . Because the system is underdamped (), the system has two complex conjugate poles. In a real-world application, replace this block with a subsystem composed of xPC D/A and A/D blocks that measure the response of a device under test (DUT).
The spectrum analyzer consists of these components:
The Sinusoidal Generator block, which produces a complex sinusoidal signal of increasing frequency. Inside the block, a Stateflow® chart uses temporal logic to iterate over a range of frequencies.
The Stateflow chart Analyzer, which calculates the frequency response (magnitude and phase angle) of the system at a specified frequency. The chart registers changes in the frequency by using change detection logic.
The Stateflow chart Unwrap, which processes the measured phase angle and unwraps the result so that there are no sharp jumps between and .
The spectrum analyzer displays the measured frequency response as a pair of discrete Bode plots.
Generate Sinusoid Signal
The Sinusoid Generator block has two outputs:
fthat represents the current frequency.
A complex signal
ythat has a frequency of
You can control the behavior of the block by modifying its mask dialog box parameters. For example, the default values specify a sinusoidal signal with frequencies between 10Hz and 1000Hz. The block holds each frequency value for 0.2 seconds, and then increments the frequency by a step of 10Hz.
To control the timing of the signal generator, the block contains a Stateflow chart that applies absolute-time temporal logic.
During simulation, the chart goes through these stages:
Initialize Frequency: The default transition sets the signal frequency
fto the value of the parameter
fstart. Specify the value of
fstartin the Sinusoid Generator mask dialog box.
Generate Signal: While the
Generatestate is active, the chart produces the complex signal
y = exp(2*pi*f*t*1i)based on the frequency
fand the time
tsince the last change in frequency. To determine the elapsed time (in seconds) since the state became active, the chart calls the temporal logic operator
Update Frequency: After the
Generatestate is active for
delayseconds, the chart transitions out of the state, increases the frequency
fstep, and returns to the
Generatestate. To determine the timing of the frequency updates, the chart calls the temporal logic operator
afterand checks the sign of the imaginary part of the signal before and after the current time step. The sign checks ensure that the output signal completes a full cycle before the frequency is updated, preventing large changes in the output signal. Specify the values of
delayin the Sinusoid Generator mask dialog box.
Stop Simulation: When the frequency
freaches the value of the parameter
fstop, the chart transitions to the
Stoppedstate. The active state output
Stoppedtriggers a Stop Simulation (Simulink) block and the simulation ends. Specify the value of
fstopin the Sinusoid Generator mask dialog box.
Calculate Frequency Response
The Analyzer chart takes the complex signal
u and the output
yp from the Plant block and computes the magnitude and phase of the plant output.
For each frequency value, the chart maintains two running sums:
yapproximates the integral of the product of the plant output
yp(a real number) and the complex signal
ynapproximates the integral of the product of the plant input
imag(u)and the complex signal
u. This integral represents the accumulation of a hypothetical plant with a unit transfer function.
To detect a change in frequency, the chart uses the operator
hasChanged to guard the self-transition on the state
Analyze. When the frequency changes, the actions on this transition compute the magnitude and phase of the plant output by normalizing
y with respect to
Unwrap Measured Phase Angle
The Unwrap chart prevents the measured phase angle from changing by more than radians in a single time step.
In this chart, the transitions test the change in the input
u before computing the new output value
If the input increases by more than radians, the chart offsets the output by radians.
If the input decreases by more than radians, the chart offsets the output by radians.
Examine Simulation Results
When you simulate the model, the scope block shows the frequency response of the system (magnitude and phase) as a function of simulation time.
In the magnitude plot, the maximum value at indicates the response of the Plant block to a resonant frequency. The peak amplitude is approximately 1.7.
In the phase plot, the angle changes from 0 to radians. Each complex pole of the system adds radians to the phase angle.
To determine the measured resonant frequency, plot the measured magnitude and phase against the plant input frequency. During simulation, the model saves these values in a signal logging object
logsout in the MATLAB workspace. You can access the logged values by using the
method. For example, to create the Bode plot for the measured frequency response of the system and draw a red cursor at 150Hz, enter:
figure; subplot(211); plot(get(logsout,"frequency").Values.Data, ... get(logsout,"magnitude").Values.Data); line([150 150],[0 2],Color="red",LineStyle="--"); grid on; title("Magnitude (Output Amplitude/Input Amplitude)");
subplot(212); plot(get(logsout,"frequency").Values.Data, ... get(logsout,"phase").Values.Data*180/pi) grid on; title("Phase (Degrees)");
The Bode plot shows that the measured resonant frequency is approximately 150Hz, matching the value predicted by the plant dynamics.