Frequency Response

Frequency response is an important way to evaluate a Control System. Some of these notes use content from the Labview tutorials.

A simple rule of thumb to remember is that:

+3 dB Power Gain = twice (ratio)

+6 dB Power Gain = four times (ratio)

+6 dB Voltage gain = twice (ratio)

+12 dB Voltage gain = four times (ratio)

Let us remind ourselves the calculation of the Decibel scale, which is logarithmic:

for voltage we have 20 * log (V_out/V_in)

for power we have 10 * log (P_out/P_in)

1) Frequency Response

DEFINITION: The frequency response is a representation of the system's response to sinusoidal inputs at varying frequencies. The output of a linear system to a sinusoidal input is a sinusoid of the same frequency but with a different magnitude and phase. The frequency response is defined as the magnitude (deciBel) and phase (deg) differences between the input and output sinusoids. This is typically captured in a Bode Plot.

To plot the frequency response, we create a vector of frequencies (varying between zero or "DC" and infinity) and compute the value of the plant transfer function at those frequencies.

Bode Plot = plot of Magnitude and Phase

where frequency in x-axis is on logarithmic scale

phase is given in degrees

magnitude is given as the gain in decibels: 1 dB = 20*log10 ( |G(j*w)| ) .

Plot code:

num = 50;

den = [1 9 30 40];

sys = tf(num,den);

bode(sys)

2) Gain and Phase Margin

DEFINITION: The Gain Margin is defined as the change in open loop gain required to make the system unstable. Systems with greater gain margins can withstand greater changes in system parameters before becoming unstable in closed loop. Keep in mind that unity gain in magnitude is equal to a gain of zero in dB.

DEFINITION: The Phase Margin is defined as the change in open loop phase shift required to make a closed loop system unstable. The phase margin also measures the system's tolerance to time delay.

Let us explain Gain Margin and Phase Margin in the plot:

The Gain Margin is the difference between the magnitude curve and 0dB at the point corresponding to the frequency that gives us a phase of -180 deg (the phase cross over frequency, Wpc).

The Phase Margin is the difference in phase between the phase curve and -180 degrees, at the point corresponding to the frequency that gives us a gain of 0dB (the gain cross over frequency, Wgc).

3) Closed Loop Frequency Bandwidth

DEFINITION: The bandwidth of a closed-loop control system is defined as the frequency range where the magnitude of the closed loop gain does not drop below −3 dB. The frequency ωB is defined as the cutoff frequency.

We would like to have a very large bandwidth so that we can reproduce the input signal (external forcing function) very accurately. Unfortunately, very large bandwidths pass high-frequency noise which can be a detriment to the control-system performance. Therefore, there are conflicting requirements on the bandwidth and cutoff frequency.

3) Nyquist plot

DEFINITION: A parametric plot of frequency response used in automatic control and signal processing. The most common use of Nyquist plots is for assessing the stability of a system with feedback. In Cartesian coordinates, the real part of the transfer function is plotted on the X axis. The imaginary part is plotted on the Y axis. The frequency is swept as a parameter, resulting in a plot per frequency.

Although the frequencies are not indicated on the curve, it can be inferred that the zero-frequency point is on the right, and the curve spirals toward the origin at high frequency. This is because gain at zero frequency must be purely real (on the X axis) and is commonly non-zero, while most physical processes have some amount of low-pass filtering, so the high-frequency response is zero.

Stability is determined by looking at the number of encirclements of the point at (−1,0). The range of gains over which the system will be stable can be determined by looking at crossings of the real axis.

The Nyquist plot can provide some information about the shape of the transfer function. For instance, the plot provides information on the difference between the number of poles and zeros of the transfer function by the angle at which the curve approaches the origin.