[Fourier analysis of Control System]

[Fourier analysis of Control System]

Submitted to:
Dr. S. K. Raghuwanshi

Submitted By:
Rishi Kant Sharan
Semester: V
Branch: Electronics & Communication Engineering

Submitted to:
Dr. S. K. Raghuwanshi

Submitted By:
Rishi Kant Sharan
Adm. No: 2010JE1117
Semester: V
Branch: Electronics & Communication Engineering

Abstract
The assignment focuses on the Fourier analysis of Control System. Which leads to frequency domain analysis of control system. The scope of estimation and controlling the behavior a system by means of Fourier transformation of its transfer function and analyzing its frequency response.

Abstract
The assignment focuses on the Fourier analysis of Control System. Which leads to frequency domain analysis of control system. The scope of estimation and controlling the behavior a system by means of Fourier transformation of its transfer function and analyzing its frequency response.

ACKNOWLEDGEMENT

There is an old adage that says that you never really learn a subject until you teach it. I now know that you learn a subject even better when you write about it.
Preparing this term paper has provided me with a wonderful opportunity to unite my love of concept in CONTROL SYSTEM. This term paper is made possible through the help and support from everyone, including: professor, friends, parents, family, and in essence, all sentient beings.
Especially, please allow me to dedicate my acknowledgment of deepest gratitude towards Dr. S. K. Raghuwanshi for his continuous support, encouragement and foremost assigning me the topic for this term paper. Apart from the subject of my paper, I learnt a lot from him, which I am sure, will be useful in different stages of my life.
I would also like to thank Mr. Santosh kumar to read my paper and to provide valuable advices to make my paper more and more effective.
Finally, I sincerely thank to my friends, parents, family, and all others who provided me support and suggestions at different steps of the paper. The product of this term paper would not be possible to come in light without all of them.
……………..

Rishi Kant Sharan

CONTENTS 1. Introduction………………………………………………………………………..3 2. Control System Response…………………………………………………………4 a. Time Response……………………………………………………………...4 b. Frequency Response………………………………………………………...5 3. Fourier Analysis of Control System………………………………………………6 c. Introduction…………………………………………………………………6 d. Analysis……………………………………………………………………...7 4. Methods & Applications of analysis in frequency domain……………………...10 a. Application……………………………………………………………...…..10 b. Bode Plot……………………………………………………………..……..11 c. Nyquist Plot…………………………………………………...…………….16 d. Nichol’s Chart………………………………….……………………………21 5. References………….……………………………………………………………….23

INTRODUCTION
Engineering is concerned with understanding and controlling the materials and forces of nature for the benefit of humankind. Control system engineers are concerned with understanding and controlling segments of their environment, often called systems, to provide useful economic products for society. The twin goals of understanding and controlling are complementary because effective systems control requires that the systems be understood and modelled. Furthermore, control engineering must often consider the control of poorly understood systems such as chemical process systems. The present challenge to control engineers is the modelling and control of modern, complex, interrelated systems such as traffic control systems, chemical processes, and robotic systems. Simultaneously, the fortunate engineer has the opportunity to control many useful and interesting industrial automation systems. Perhaps the most characteristic quality of control engineering is the opportunity to control machines and industrial and economic processes for the benefit of society.
Control engineering is based on the foundations of feedback theory and linear system analysis, and it integrates the concepts of network theory and communication theory. Therefore control engineering is not limited to any engineering discipline but is equally applicable to aeronautical, chemical, mechanical, environmental, civil, and electrical engineering. For example, a control system often includes electrical, mechanical, and chemical components. Furthermore, as the understanding of the dynamics of business, social and political systems increases, the ability to control these systems will also increase.
A control system is a device or set of devices to manage, command, direct or regulate the behavior of other devices or system. Modern control engineering practice includes the use of control design strategies for improving manufacturing processes, the efficiency of energy use, and advanced automobile control (including rapid transit, among others).
The input–output relationship represents the cause-and-effect relationship of the process, which in turn represents a processing of the input signal to provide an output signal variable, often with power amplification. An open-loop control system utilizes a controller or control actuator to obtain the desired response.
An open-loop control system utilizes an actuating device to control the process directly without using feedback. (Figure 2)

Control system response
Time response: Time response of a control system is a study on how the output variable changes with time when a typical test input signal is given to the system. The commonly test input signals are those of step functions, impulse functions, ramp functions, parabolic function and sinusoidal functions. The time response of a control system consists of two parts: the transient response and the steady-state response. Transient response is the manner in which the system goes form initial state to the final (desired) state. Steady-state response is the behavior in which the system output behaves as the time approaches infinity. Thus the system response y(t) may be written as: y(t) = yt(t) + yss(t) where yt (t) denotes the transient response, and yss(t) denotes the steady-state response which are shown in the Figure 2.

Output Response Time

Figure 2: basic representation of time response

Frequency response: Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input. The frequency response is characterized by the magnitude of the system's response, typically measured in decibels (dB) or as a decimal, and the phase, measured in radians or degrees, versus frequency in radians/sec or Hertz (Hz). The frequency response of a system is a frequency dependent function which expresses how a sinusoidal signal of a given frequency on the system input is transferred through the system. Time-varying signals — at least periodical signals — which excite systems, as the reference (set point) signal or a disturbance in a control system or measurement signals which are inputs signals to signal filters, can be regarded as consisting of a sum of frequency components. Each frequency component is a sinusoidal signal having certain amplitude and a certain frequency. (The Fourier series expansion or the Fourier transform can be used to express these frequency components quantitatively.) The frequency response expresses how each of these frequency components is transferred through the system. Some components may be amplified, others may be attenuated, and there will be some phase lag through the system. The frequency response is an important tool for analysis and design of signal filters (as low pass filters and high pass filters), and for analysis, and to some extent, design, of control systems. To plot the frequency response, we create a vector of frequencies (varying between zero to infinity) and compute the value of transfer function at those frequencies. If G(s) is the open loop transfer function of a system and w is the frequency vector ,we first replace S of transfer function with jw then plot G(jw) versus w. since G(jw) is a complex number, we can plot both its magnitude and phase (the Bode plot) or its position in the complex plane(the Nyquist plot). We can use the frequency response of a system to locate poles and zeros of a system. Two applications of frequency response analysis are related but have different objectives. For an audio system, the objective may be to reproduce the input signal with no distortion. That would require a uniform (flat) magnitude of response up to the bandwidth limitation of the system, with the signal delayed by precisely the same amount of time at all frequencies. That amount of time could be seconds, or weeks or months in the case of recorded media. In contrast, for a feedback apparatus used to control a dynamical system, the objective is to give the closed-loop system improved response as compared to the uncompensated system. The feedback generally needs to respond to system dynamics within a very small number of cycles of oscillation (usually less than one full cycle), and with a definite phase angle relative to the commanded control input. For feedback of sufficient amplification, getting the phase angle wrong can lead to instability

Fourier Analysis of control system
Introduction: The Fourier Transform is used to break a time-domain signal into its frequency domain components. Hence Fourier analysis can be performed by directly analyzing frequency domain analysis of the System. The Fourier Transform is very closely related to the Laplace Transform, and is only used in place of the Laplace transform when the system is being analyzed in a frequency context. In Laplace transform we work in space domain. In which the symbol
S= (σ+jω). When the term σ becomes zero the Laplace transform converts to Fourier transform.
In control systems engineering, frequency domain is the domain for analysis of signals with respect to frequency, rather than time. Put simply, 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. A frequency-domain representation can also include information on the phase shift that must be applied to each sinusoid in order to be able to recombine the frequency components to recover the original time signal.
A given function or signal can be converted between the time and frequency domains with a pair of mathematical operators called a transform. An example is the Fourier transform, which decomposes a function into the sum of a (potentially infinite) number of sine wave frequency components. The 'spectrum' of frequency components is the frequency domain representation of the signal.
The Fourier Transform is defined as:
Fјω=Fft=0∞f(t)e-јωtdt
By the term frequency response, we mean the steady-state response of a system to a sinusoidal input. 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.
Consider a system with sinusoidal input . The steady-state output may be written as, . The magnitude and the phase relationship between the sinusoidal input and the steady-state output of a system is called frequency response. The frequency response test is performed by keeping the amplitude A fixed and determining B and for a suitable range of frequencies. Whenever it is not possible to obtain the transfer function of a system through analytical techniques, frequency response test can be used to compute its transfer function.
The design and adjustment of open-loop transfer function of a system for specified closed-loop performance is carried out more easily in frequency domain. Further, the effects of noise and parameter variations are relatively easy to visualize and assess through frequency response. The Nyquist criteria is used to extract information about the stability and the relative stability of a system in frequency domain.

Correlation between time and frequency response :
Time Domain

Frequency domain

Analysis

The transfer function of a standard second-order system can be written as,

Substituting s by j we obtain,

Where, is the normalized signal frequency. From the above equation we get,

.

The steady-state output of the system for a sinusoidal input of unit magnitude and variable frequency is given by,

.

It is seen from the above equation that when,

The magnitude and phase angle characteristics for normalized frequency u for certain values of are shown in figure.
The frequency where M has a peak value is called resonant frequency. At this point the slope of the magnitude curve is zero. Setting we get,

Solving, or, resonant frequency . ………… (01)
The resonant peak is given by, resonant peak, . ………… (02) * For, , the resonant frequency does not exist and M decreases monotonically with increasing u.

* For , the resonant frequency is always less than and the resonant peak has a value greater than 1.

From equation (01) and (02) it is seen that The resonant peak of frequency response is indicative of damping factor and the resonant frequency is indicative of natural frequency for a given and hence indicative of settling time.
For , M decreases monotonically. The frequency at which M has a value of is called the cut-off frequency . The range of frequencies over which M is equal to or greater than is defined as bandwidth, .
The bandwidth of a second-order system is given by, ………….(03)

Figure below shows the plot of resonant peak of frequency response and the peak overshoot of step response as a function of ω.

Methods & application of Analysis in Fourier domain:
Two applications of frequency response analysis are related but have different objectives. For an audio system, the objective may be to reproduce the input signal with no distortion. That would require a uniform (flat) magnitude of response up to the bandwidth limitation of the system, with the signal delayed by precisely the same amount of time at all frequencies. That amount of time could be seconds, or weeks or months in the case of recorded media. In contrast, for a feedback apparatus used to control a dynamical system, the objective is to give the closed-loop system improved response as compared to the uncompensated system. The feedback generally needs to respond to system dynamics within a very small number of cycles of oscillation (usually less than one full cycle), and with a definite phase angle relative to the commanded control input. For feedback of sufficient amplification, getting the phase angle wrong can lead to instability for an open-loop stable system, or failure to stabilize a system that is open-loop unstable. Digital filters may be used for both audio systems and feedback control systems, but since the objectives are different, generally the phase characteristics of the filters will be significantly different for the two applications.
These response measurements can be plotted in three ways: by plotting the magnitude and phase measurements on two rectangular plots as functions of frequency to obtain a Bode plot; by plotting the magnitude and phase angle on a single polar plot with frequency as a parameter to obtain a Nyquist plot; or by plotting magnitude and phase on a single rectangular plot with frequency as a parameter to obtain a Nichols plot. For audio systems with nearly uniform time delay at all frequencies, the magnitude versus frequency portion of the Bode plot may be all that is of interest. For design of control systems, any of the three types of plots [Bode, Nyquist, Nichols] can be used to infer closed-loop stability and stability margins (gain and phase margins) from the open-loop frequency response, provided that for the Bode analysis the phase-versus-frequency plot is included.

BODE PLOT
Engineer Hendrik Wade Bode (1905–1982), while working at Bell Labs in the United States in the 1930s, devised a simple but accurate method for graphing gain and phase-shift plots. These bear his name, Bode gain plot and Bode phase plot. A Bode plot is a graph of the transfer function of a linear, time-invariant system versus frequency, plotted with a log-frequency axis, to show the system's frequency response. It is usually a combination of a Bode magnitude plot, expressing the magnitude of the frequency response gain, and a Bode phase plot, expressing the frequency response phase shift. In magnitude plot the magnitude axis of the Bode plot is usually expressed as decibels of power, i.e. by the 20 log rule. If the transfer function is a rational function with real poles and zeros, then the Bode plot can be approximated with straight lines. These asymptotic approximations are called straight line Bode plots or uncorrected Bode plots. At low frequencies, the output is able to respond to the slow varying input disturbances with only a small attenuation (AR (amplitude ratio) close to 1). However, at higher frequencies, the AR (amplitude ratio) decreases rapidly, approaching an asymptote with a slope of –1 in the log-log graph shown for the first-order self-regulating process of the tank. Note that this system acts as a low pass filter, which removes the high-frequency inputs. The Bode plot method gives a graphical procedure for determining the stability of a control system based on sinusoidal frequency response. The transfer function of a system for sinusoidal input response can be obtained by substituting j in place of S. therefore, if the open loop transfer function of a system is G(s)*H(s), the Corresponding sinusoidal open-loop transfer function is G(j )*H(j ) which can be expressed in the form of magnitude and phase angle. Two relative stability indicators "Gain Margin" and "Phase Margin" can be easily obtained from Bode Plots as shown in Figure.

The transfer function is represented by, .
Taking natural logarithm of both sides, …………………(04)
The unit of real part is called neper.
Similarly, …………..…………………..(05)
The standard procedure is to plot and phase angle vs. log . The unit of magnitude is decibel. These two plots are called Bode plots in honor of HW Bode. Drawing Bode Diagram Bode magnitude plot:
For magnitude plots the Y- axis is typically expressed in terms of dB and the X-axis in terms of frequency in logarithmic scale. 1. 20 log10(|A|), where A is a real number.

2. 20 log10(|j.ω.τ1|), where τ1 is a real number:

3. 20 log10(|1±j.ω.τ2|), where τ2 is a real number.

Bode Phase Plots

Example of Bode Plot in Electronic Circuit:
Here is an example of doing Bode Plots with Matlab.
+
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+
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Vin
Rs
Rg
Cgs
+
Vgs
-
gmVgs
Rout
rd
Vout
+

-
+
-
+
-
Vin
Rs
Rg
Cgs
+
Vgs
-
gmVgs
Rout
rd
Vout
+

-

Assume we have a small signal circuit like this. Now we have to find out Vout/Vin.
Assume we have the following parameters and try to draw the bode plots.
Steps:

1. First, we need to find out Vout/Vin.

2. Plot the Bode Plot with MatLab
Assume that you wanted to use Matlab in order to obtain Bode magnitude and phase plots for the following transfer function. The s is jw (j omega). H(s) = Z(s)/P(s), where Z(s) = [2.5329e-14*s^2 + 1.5915e-4*s + 1] P(s) = [2.5329e-20*s^2 + 1.5915e-7*s + 1]
We would need to type the following:
>> num = [2.5329e-14 1.5915e-4 1];
>> den = [2.5329e-20 1.5915e-7 1];
>> sys = tf(num,den)
>> bode(sys,{1,1e15})
>> num = 5.82e6; %%the numerator den = [4e-5 8.1e5]; %%the denominator sys = tf(num, den) %%the transfer function bode(sys,{1,1e15}) %%plot the magnitude and phase of the transfer function %% The frequency range is 1 rad/s to 1e15 rad/s

Transfer function: %Output 5.82e006
--------------------------
4e-005 s + 810000
>> num = 5.82e6; %%the numerator den = [4e-5 8.1e5]; %%the denominator sys = tf(num, den) %%the transfer function bode(sys,{1,1e15}) %%plot the magnitude and phase of the transfer function %% The frequency range is 1 rad/s to 1e15 rad/s

Transfer function: %Output 5.82e006
--------------------------
4e-005 s + 810000

NYQUIST PLOT

A Nyquist plot is a polar plot of the frequency response function of a linear system. It is used in automatic control and signal processing. Nyquist Plots were invented by Nyquist - who worked at Bell Laboratories, the premiere technical organization in the U.S. at the time. He was interested in designing telephone amplifiers to be placed in ocean-floor cables. In those days, between the first and second world wars, undersea cables were the only reliable means of intercontinental communication. It is based on the complex analysis result known as Cauchy’s principle of argument. Note that the system transfer function is a complex function. The Nyquist plot allows us to predict the stability and performance of a closed-loop system by observing its open-loop behavior. The Nyquist criterion can be used for design purposes regardless of open-loop stability (remember that the Bode design methods assume that the system is stable in open-loop). Therefore, we use this criterion to determine closed-loop stability when the Bode plots display confusing information. By applying Cauchy’s principle of argument to the open-loop system transfer function, we will get information about stability of the closed-loop system transfer function and arrive at the Nyquist stability criterion. The importance of Nyquist stability lies in the fact that it can also be used to determine the relative degree of system stability by producing the so-called phase and gain stability margins. These stability margins are needed for frequency domain controller design techniques. We present only the essence of the Nyquist stability criterion and define the phase and gain stability margins. The Nyquist method is used for studying the stability of linear systems with pure time delay. For a SISO feedback system the closed-loop transfer function is given by
M(s) = G(s)1+G(s)H(s)
Where G(s) represents the system and H(s) is the feedback element. Since the system poles are determined as those values at which its transfer function becomes infinity, it follows that the closed-loop system poles are obtained by solving the following equation
1+G(s)H(s) = 0 = (s)
Which, in fact, represents the system characteristic equation.
In the following we consider the complex function
D(s) = 1+G(s) H(s)
Whose zeros are the closed-loop poles of the transfer function? In addition, it is easy to see that the poles of D(s) are the zeros of M(s). At the same time the poles of D(s) are the open-loop control system poles since they are contributed by the poles of H(s) G(s), which can be considered as the open-loop control system transfer function—obtained when the feedback loop is open at some point. The Nyquist stability test is obtained by applying the Cauchy principle of argument to the complex function D(s). First, we state Cauchy’s principle of argument.
Cauchy’s Principle of Argument
Let F(s) be an analytic function in a closed region of the complex Plane s given in Figure 4 except at a finite number of points (namely, the poles of F(s)). It is also assumed that F(s) is analytic at every point on the contour. Then, as s travels around the contour in the s-Plane in the clockwise direction, the function F(s) encircles the origin in the (Re{F(s)}, Im{F(s)}) -plane in the same direction times N times (see figure 4), with N given by
N = Z – P
Where Z and P stand for the number of zeros and poles (including their multiplicities) of the function F(s) inside the contour.

Figure 4: Cauchy’s principle of argument
The Nyquist plot is a polar plot of the function D(s) = 1+G(s) H(s) when s travels around the contour given in Figure 5.

Figure 5: Contour in the s-plane

The contour in this figure covers the whole unstable right half plane of the complex plane s, R→∞ . Since the function D(s), according to Cauchy’s principle of argument, must be analytic at every point on the contour, the poles of D(s) on the imaginary axis must be encircled by infinitesimally small semicircles.
Nyquist Stability Criterion

It states that the number of unstable closed-loop poles is equal to the number of unstable open-loop poles plus the number of encirclements of the origin of the Nyquist plot of the complex function D(s).
This can be easily justified by applying Cauchy’s principle of argument to the function D(s) with the s-plane contour given in Figure 5. Note that Z and P represent the numbers of zeros and poles, respectively, Of D(s) in the unstable part of the complex plane. At the same time, the zeros of D(s) are the closed-loop system poles, and the poles of D(s) are the open-loop system poles (closed-loop zeros).
The above criterion can be slightly simplified if instead of plotting the
Function D(s) = 1+G(s) H(s) , we plot only the function G(s) H(s) and count encirclement of the Nyquist plot of G(s) H(s) around the point (-1, j0), so that the modified Nyquist criterion has the following form.
The number of unstable closed-loop poles (Z) in the right half-plane is equal to the number of unstable open-loop poles (P) plus the number of encirclements (N) of the point (-1, j0), of the Nyquist plot of G(s) H(s), that is
Z = P+N.
According to Nyquist stability criterion clockwise encirclements of -1 count as positive encirclements and counter-clockwise encirclements of -1 count as negative encirclements.
Note: This is only one convention for the Nyquist criterion. Another convention states that a positive N counts the counter-clockwise or anti-clockwise encirclements of -1. The P and Z variables remain the same. In this case the equation becomes Z = P - N.
Phase And Gain Stability Margins
Two important notions can be derived from the Nyquist diagram: phase and gain stability margins. The phase and gain stability margins are presented in Figure 6.

Figure 6: phase and gain stability margins
They give the degree of relative stability; in other words, they tell how far the given system is from the instability region. Their formal definitions are given by

pm = 180o +arg{ G(јωcg)H(јωcg)} gm(dB) = 20log1|G(јωcp)H(јωcp)|
Where ωcg and ωcp stand for, respectively, the gain and phase crossover frequencies, which from Figure 6 are obtained as
|G(јωcg)H(јωcg)| = 1 ⇒ ωcg arg{G(јωcp)H(јωcp } = 180o ⇒ ωcp

Example and Analysis of control system By nyquist plot :
The open loop transfer function of a unity-feedback (negative sign) system is:

Sketch the Nyquist plot of for to and determine the range of K for the closed loop system to be stable for:
a) n =2 b) n =3 c) n =4

solution:
: Note that for all value of n the system is minimum phase. a) For n=2 the characteristic equation is . So one must draw the simplified Nyquist plot of for to ( set K=1)

- for positive value of K the closed loop system is stable since,

- for negative value of K:

So the system is stable for

b) For n=3 the characteristic equation is . So one must draw the simplified Nyquist plot of for to (set K=1).

- for positive value of K:

- for negative value of K:

So the system is stable for .

c) For n=4 the characteristic equation is . So one must draw the simplified Nyquist plot of for to ( set K=1).

- for positive value of K:
- for negative value of K:

So the system is stable for

NICHOL’S CHART
Nichols creates a Nichols chart of the frequency response. A Nichols chart displays the magnitude (in dB) plotted against the phase (in degrees) of the system response. Nichols charts are useful to analyze open-and closed-loop properties of SISO systems, but offer little insight into MIMO control loops. Use ngrid to superimpose a Nichols chart on an existing SISO Nichols chart.
In the case of Bode plot or Nyquist plot, we need to rapidly sketch the open-loop frequency response but not the closed-loop frequency response. We need to separately sketch both the plots. But in case of Nichol’s chart, we can combine the ease of sketching the open-loop response with the transient response information contained in the closed-loop response. In Nichol’s chart, graph is obtained by plotting the magnitude of G(jw)H(jw) in dB against the phase of G(jw )H(jw) in degrees on the Nichol’s chart with as a changing parameter from zero to infinity. From the curve plotted for G(jw )H(jw) besides from the open-loop system performance criterion obtained, closed-loop system performance criterion is also obtained simultaneously. Simply, by using open-loop system of G(jw)H(jw) , we can determine the information which relates the performance of an open-loop system with the closed-loop system in frequency domain. Therefore, information such as phase and gain margin, peak resonance, resonance frequency and bandwidth can be easily obtained simultaneously.
Terms in Nichol’s chart * The critical point is the intersection of the 0-db axis and the -180o axis. * The phase crossover frequency is where the locus intersects the -180o axis. * The gain crossover frequency is where the locus intersects the 0-db axis. * The gain margin is the vertical distance in dB measured from the phase crossover frequency to the critical point. * The phase margin is the horizontal distance measured in degrees from the gain crossover frequency to the critical point. * The bandwidth of the closed-loop system is the frequency at which the G(jw ) curve intersects the M=0.707 or M= -3dB locus. * The resonant peak Mr is found by locating the smallest of the constant-M locus (M≥ 1) that is the tangent to the G(jw) curve from the above. * The resonant frequency is the frequency of G(jw) at the point of tangency.

References

* Modern Control Systems: Richard C. Dorf, Robert H. Bishop * Linear control systems: Yi Guo * Frequency domain analysis: En Muhammad Nasiruddin Mahyuddin * Automated control system: Benjamin C. Kuo and Farid Golnaraghi * Frequency response: Finn Haugen * Internet * www.me.umn.edu * http://www.ece.rutgers.edu/~gajic/psfiles/nyquist.pdf * http://www.cds.caltech.edu/~murray/courses/cds101/fa04/caltech/am04_ch6-3nov04.pdf * Frequency response: www.wikipedia.com * Control system/frequency response methods bode.htm

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