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VIDYA COLLEGE OF ENGG AND TECH,VIRUDHUNAGAR COURSE MATERIAL (LECTURE NOTE

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IC 6501 CONTROL SYSTEMS UNIT-I Page 1

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UNIT I SYSTEM AND THEIR REPRESENTATION

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Root Locus Plots

It gives complete dynamic response of the system. It provides a

measure of sensitivity of roots to the Variation in the parameter beingconsidered. It is applied for single as well as multiple loop system. It

can be defined as follows,

It is the plot of the loci of the root of the complementary equation

when one or more parameters of the open-loop Transfer function arevaried, mostly the only one variable available is the gain K The

negative gain has no physical significance hence varying K from 0 to , the plot is obtained called the Root Locus Point.

Rules for the Construction of Root Locus

(1) The root locus is symmetrical about the real axis.

(2) The no. of branches terminating on equals the no. of open-looppole-zeroes.

(3) Eachbranch of the root locus originates from an open-looppole at

K = 0 &terminates at open-loop zero corresponding to K = .(4) A point on the real axis lies on the locus, if the no. of open-loop

poles & zeroes on the real axis to the right of this point is odd.

(5) The root locus branches that tend to , do so along thestraightline.

(6) The asymptotes cross the real axis at a point known as Centroid.

(7) The break away or the break in points [Saddle points] of the rootlocus or determined from the roots of the equation dk/ds =0

(8) The intersection of the root locus branches with the imaginary axiscan be determined by the use of Routh-Hurwitz criteria or by putting

s=j in the characteristic equation & equating the real part andimaginary tozero. To solve for & K i.e., the value of is

intersection point on the imaginary axis & K is the value ofgain at theintersection point.

(9) The angle of departure from a complex open-loop pole(d) is given

by,

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There are basically two methods of analysis of systems to determine certain properties, so that

design procedures can be developed, using these properties as performance measures. If a time

signal like step and ramp are used to excite the system and its time response is obtained , we call

it a time response analysis or time domain analysis. On the other hand, if a sinusoidal signal of

variable frequency is used to excite the system and the magnitude and phase of the steady state

output from the system is measured, we call it frequency response analysis or frequency domain

analysis. Both the methods have their own advantages and disadvantages.

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.Both signal filtering and control systems applications are described (briefly) later in this chapter.

The definition of the frequency response which will be given in the next section applies

only to linear models, but this linear model may very well be the local linear model about some

operating point of a non-linear model. The frequency response can found experimentally or from

a transfer function model. It can be presented graphically or as a mathematical function.

UNIT III Frequency Response

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Polar plot

To sketch the polar plot ofG(j) for the entire range of frequency , i.e., from 0 to infinity,

there are four key points that usually need to be known:

(1) the start of plot where = 0,

(2) the endof plot where = ,

(3) where the plot crosses the real axis, i.e., Im(G(j)) = 0,and

(4) where the plot crosses the imaginary axis, i.e., Re(G(j)) = 0.

BASICS OF POLAR PLOT

The polarplot of a sinusoidal transfer function G(j) is a plot of the magnitude of G(j) Vs

the phase of G(j) on polar co-ordinates as is varied from 0 to .

(ie) |G(j)| Vs G(j) as 0 to .

Polar graph sheet has concentric circles and radial lines.

A concentric circle represents the magnitude.

A radial line represents the phase angles.

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In polar sheet

+ve phase angle is measured in ACW from 0

-ve phase angle is measured in CW from 0

PROCEDURE

Express the given expression of OLTF in (1+sT) form.

Substitute s = j in the expression for G(s)H(s) and get G(j)H(j).

Get the expressions for | G(j)H(j)| & G(j)H(j).

Tabulate various values of magnitude and phase angles for different values of

ranging from 0 to .

Usually the choice of frequencies will be the corner frequency and around corner

frequencies.

Choose proper scale for the magnitude circles.

Fixall the points in the polar graph sheet and join the points by a smooth curve.

Write the frequency corresponding to each of the point of the plot.

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Time Domain Analysis Vs Frequency Domain Analysis

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In the following a comparison of time domain and frequency domain analysis is given.

(i) Variable frequency, sinusoidal signal generators are readily available and precision

measuring instruments are available for measurement of magnitude and phase angle. The

time response for a step input is more difficult to measure with accuracy.

(ii) It is easier to obtain the transfer function of a system by a simple frequency domaintest. Obtaining transfer function from the step response is more tedious.

(iii) If the system has large time constants, it makes more time to reach steady state at

eachFrequency of the sinusoidal input. Hence time domain method is preferred over

frequency domain method in such systems.

(iv) In order to do a frequency response test on a system, the system has tobe isolated

and the sinusoidal signal has to be applied to the system. This may not be possible in

systems Which can not be interrupted. In such cases, a step signal or an impulse signal may

be given to the system to find its transfer function. Hence for systems which cannot be

interrupted, time domain method is more suitable.

(v) The design ofa controller is easily done in the frequency domain method than in time

domain method. For a given set of performance measures in frequency domain. the

parameters of the open loop transfer function can be adjusted easily using techniques to be

discussed later in this chapter.

(vi) The effect of noise signals can be assessed easily in frequency domain rather than time

domain.

(vii) The most important advantage of frequency domain analysis is the ability to obtain the

relative stability of feedback control systems. The correlation between time and frequency

response can be easily established so that the time domain performance measures can

be obtained from the frequency domain specifications and vice versa.

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Frequency response of lead-lag compensator

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