# Frequency Domain Analysis of a Control System

#### Introduction

The frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Simply speaking, a time-domain graph shows how a signal changes over time, whereas a frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies. This is also illustrated in the picture below.

Industrial control systems are often designed using frequency response methods. Many techniques are available in the frequency response methods for the analysis and design of control systems.

Some of those techniques are:

1. Bode Plot : Bode Plot is a visualization of the frequency response of a system.The Bode plot for a linear, time-invariant system with a given transfer function H(s) with ‘s’  being the complex frequency in the Laplace domain consists of a magnitude plot and a phase plot.The Bode magnitude plot is the graph of the function |H(s = jw)| of frequency w (with j being the imaginary unit). The w -axis of the magnitude plot is logarithmic and the magnitude is given in decibels.The Bode phase plot is the graph of the phase of the transfer function arg(H(s = jw)) as a function of w , commonly expressed in degrees. The phase is plotted on the same logarithmic w -axis as the magnitude plot, but the value for the phase is plotted on a linear vertical axis.
2. Nyquist Plot: The Nyquist contour is a closed contour in the s-plane which completely encloses the entire right hand half of s-plane. In order to enclose the complete RHS of s-plane a large semicircle path is drawn with diameter along jω axis and centre at origin. The radius of the semicircle is treated as infinity.
3. Nichols’ Chart: The chart consisting of constant-magnitude loci and constant phase-angle loci in the log-magnitude versus phase diagram is called Nichols chart.The Nichols chart contains curves of constant closed-loop magnitude and phase angle. The designer can graphically determine the phase margin, gain margin, resonant peak magnitude, resonant peak frequency, and bandwidth of the closed loops system from the plot of the open-loop locus. The Nichols chart is symmetric about -180 degree axis.

#### Frequency Domain Specifications

1. Resonant peak : Maximum value of M (jw) when w is varied from 0 to infinite. The magnitude of resonant peak gives the information about the relative stability of the system. A large value of resonant peak implies undesirable transient response.
2. Resonant frequency : The frequency at which resonant peak  occurs. If resonant frequency is large, then the time response is fast.
3. Cut-off frequency : The frequency at which M (jw) has a value (1/2) ^1/2. It is the frequency at which the magnitude is 3 dB below its zero frequency value.
4. Bandwidth : It is the range of frequencies in which the magnitude of a closed-loop system is (1/2) ^1/2 times of Mr or the magnitude of the closed loop doesn’t drop -3 dB.
5. Cut-off rate: It is the slope of the log magnitude curve near cut-off frequency.
6. Gain Margin: Gain Margin is defined as the margin in gain allowable by which gain can be increased till system reaches on the verge of instability. Phase cross-over frequency is the frequency at which phase plot crosses -180°.
7. Phase Margin: The additional phase lag can be introduced without affecting the magnitude plot for gain. So, Phase margin can be defined as the amount of additional phase lag which can be introduced in the system till the system reaches on the verge of instability. Gain cross-over frequency is the frequency at which gain or magnitude plot crosses 0 dB line.

Let us observe the steps to carry out frequency domain analysis of a control system in MATLAB.

Step 1. Open MATLAB.

Step 2. Define a transfer function for the system you want to analyse.

Step 3. Use the MATLAB function ‘bodeplot’ to view the Bode Plot for the given system.

The output for the above command is as follows:

Step 4. Next, use the MATLAB command ‘nyquist plot’ to view the frequency response of the given system using the Nyquist Contour. This is shown below:

The output for the aforementioned command comes out to be as follows:

Step 5. Furthermore, use the command ‘Nichols’ to view the Nichols Chart of the system defined by the given transfer function. This is illustrated below:

The output comes out be as follows:

Step 6. Now, in order to calculate the frequency domain specifications such as bandwidth, DC gain, Gain Crossover Frequency and Peak Gain of the given system, use the following commands. The output is also displayed.

#### Conclusion

The frequency domain analysis of a system defined by a given transfer function was done and its frequency response using various methods was viewed. The frequency domain specifications were also calculated for the same system.

##### Shivani Singh

A final year engineering student from Bharati Vidyapeeth's College of Engineering, New Delhi with great zeal and passion for writing and technology. I have great love for food, nature, reading and am a true Delhiite at heart !

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