JUNE 2014 Solved Question Paper

1 a: Explain with examples open loop and closed loop control systems. List merits and demerits

of both. Jun. 2014, 10 Marks

Open & Closed Loop System - Advantages & Disadvantages [Control System].

In control systems engineering, a system is actually a group of objects or elements capable of

performing individual tasks. They are connected in a specific sequence to perform a specific

function.

A system is of 2 types:

1. Open loop system which is also called as Manual control system.

2. Closed loop system which is also named as automatic control system.

In this post, we will be discussing various advantages and disadvantages of the 2 types of control

systems.

Open Loop System:

Advantages: 1. Simplicity and stability: they are simpler in their layout and hence

areeconomical and stable too due to their simplicity.

2. Construction: Since these are having a simple layout so are easier to construct.

Disadvantages: 1. Accuracy and Reliability: since these systems do not have a feedback

mechanism, so they are very inaccurate in terms of result output and hence they are

unreliable too.

2. Due to the absence of a feedback mechanism, they are unable to remove the

disturbances occurring from external sources.

Closed Loop System:

Advantages: 1. Accuracy: They are more accurate than open loop system due to their complex

construction. They are equally accurate and are not disturbed in the presence of non-

linearities.

2. Noise reduction ability: Since they are composed of a feedback mechanism, so

they clear out the errors between input and output signals, and hence remain unaffected to

the external noise sources.

Disadvantages: 1. Construction: They are relatively more complex in construction and hence it

adds up to the cost making it costlier than open loop system.

2. Since it consists of feedback loop, it may create oscillatory response of the system

and it also reduces the overall gain of the system.

3. Stability: It is less stable than open loop system but this disadvantage can be

striked off since we can make the sensitivity of the system very small so as to make the

system as stable as possible.

Open Loop Control System

A control system in which the control action is totally independent of output of the system then it

is called open loop control system. Manual control system is also an open loop control system.

Fig - 1 shows the block diagram of open loop control system in which process output is totally

independent of controller action.

Open-loop Motor Control

So for example, assume the DC motor controller as shown. The speed of rotation of the motor

will depend upon the voltage supplied to the amplifier (the controller) by the potentiometer. The

value of the input voltage could be proportional to the position of the potentiometer.

If the potentiometer is moved to the top of the resistance the maximum positive voltage will be

supplied to the amplifier representing full speed. Likewise, if the potentiometer wiper is moved

to the bottom of the resistance, zero voltage will be supplied representing a very slow speed or

stop.

Then the position of the potentiometers slider represents the input, θi which is amplified by the

amplifier (controller) to drive the DC motor (process) at a set speed N representing the

output,θo of the system. The motor will continue to rotate at a fixed speed determined by the

position of the potentiometer.

As the signal path from the input to the output is a direct path not forming part of any loop, the

overall gain of the system will the cascaded values of the individual gains from the

potentiometer, amplifier, motor and load. It is clearly desirable that the output speed of the motor

should be identical to the position of the potentiometer, giving the overall gain of the system as

unity.

However, the individual gains of the potentiometer, amplifier and motor may vary over time with

changes in supply voltage or temperature, or the motors load may increase representing external

disturbances to the open-loop motor control system.

But the user will eventually become aware of the change in the systems performance (change in

motor speed) and may correct it by increasing or decreasing the potentiometers input signal

accordingly to maintain the original or desired speed.

The advantages of this type of “open-loop motor control” is that it is potentially cheap and

simple to implement making it ideal for use in well-defined systems were the relationship

between input and output is direct and not influenced by any outside disturbances. Unfortunately

this type of open-loop system is inadequate as variations or disturbances in the system affect the

speed of the motor. Then another form of control is required.

Practical Examples of Open Loop Control System

1. Electric Hand Drier – Hot air (output) comes out as long as you keep your hand under the

machine, irrespective of how much your hand is dried.

2. Automatic Washing Machine – This machine runs according to the pre-set time

irrespective of washing is completed or not.

3. Bread Toaster - This machine runs as per adjusted time irrespective of toasting is

completed or not.

4. Automatic Tea/Coffee Maker – These machines also function for pre adjusted time only.

5. Timer Based Clothes Drier – This machine dries wet clothes for pre – adjusted time, it

does not matter how much the clothes are dried.

6. Light Switch – lamps glow whenever light switch is on irrespective of light is required or

not.

7. Volume on Stereo System – Volume is adjusted manually irrespective of output volume

level.

Closed Loop Control System

Control system in which the output has an effect on the input quantity in such a manner that the

input quantity will adjust itself based on the output generated is called closed loop control

system. Open loop control system can be converted in to closed loop control system by

providing a feedback. This feedback automatically makes the suitable changes in the output due

to external disturbance. In this way closed loop control system is called automatic control

system. Figure below shows the block diagram of closed loop control system in which feedback

is taken from output and fed in to input.

Closed-loop Motor Control

Any external disturbances to the closed-loop motor control system such as the motors load

increasing would create a difference in the actual motor speed and the potentiometer input set

point.

This difference would produce an error signal which the controller would automatically respond

too adjusting the motors speed. Then the controller works to minimize the error signal, with zero

error indicating actual speed which equals set point.

Electronically, we could implement such a simple closed-loop tachometer-feedback motor

control circuit using an operational amplifier (op-amp) for the controller as shown.

Closed-loop Motor Controller Circuit

Practical Examples of Closed Loop Control System

1. Automatic Electric Iron – Heating elements are controlled by output temperature of the

iron.

2. Servo Voltage Stabilizer – Voltage controller operates depending upon output voltage of

the system.

3. Water Level Controller– Input water is controlled by water level of the reservoir.

4. Missile Launched & Auto Tracked by Radar – The direction of missile is controlled by

comparing the target and position of the missile.

5. An Air Conditioner – An air conditioner functions depending upon the temperature of the

room.

6. Cooling System in Car – It operates depending upon the temperature which it controls.

Comparison of Closed Loop And Open Loop Control System

Sr. No. Open loop control system Closed loop control system

1 The feedback element is absent. The feedback element is always present.

2 An error detector is not present. An error detector is always present.

3 It is stable one. It may become unstable.

4 Easy to construct. Complicated construction.

5 It is an economical. It is costly.

6 Having small bandwidth. Having large bandwidth.

7 It is inaccurate. It is accurate.

8 Less maintenance. More maintenance.

9 It is unreliable. It is reliable.

10 Examples: Hand drier, tea maker Examples: Servo voltage stabilizer,

perspiration

1 b: Draw the electrical network based on torque-current analogy give all the perfor-mance

equations for Figure. 1 Jul. 2014, 10 Marks

T θ1 k1

θ2 k2

J1 J2

f1 f2

Figure 1:

3 a: Draw the transient response characteristics of a control system to a unit step input

and define the following: i) Delay time; ii) Rise time; iii) Peak time;

iv)Maximum overshoot; v) Settling time 5 Marks

3 b: Derive the expressions for peak time tp for a second order system for step input. 5 Marks

Jun. 2014, 4 Marks

3 c: The response of a servo mechanism is c(t) = 1 + 0.2e 60t

+1.2e 10t

when subjected

to a unit step input. Obtain an expression for closed loop transfer function. Determine

the undamped natural frequency and damping ratio. 6Marks Jun. 2014, 4 Marks

3 d: The open loop transfer function of a unity feedback system is given by G(s) =K/S(ST+1)

K

,

s(Ts + 1)

where K and T are positive constant. By what factor should the amplifier, gain ’K’ be re-

duced so that the peak overshoot of unit step response of the system is reduced from 75%

to 25%. 4 Marks Jun. 2014, 6 Marks

4 a: Explain Routh-Hurwitz criterion in stability of a control system. What are the disad-

vantages of RH criterion on stability of control system? Jun. 2014, 4+4 Marks

4 b: The characteristics equation for certain feedback control system is given below. Deter-mine

the system is stable or not and find the value of K for a stable system

s3 + 3Ks

2 + (K + 2)s + 4 = 0. Jun. 2014, 6 Marks

4 c: The open-loop TF of a unity negative feedback system is given by

G(s) =

K(s + 3)

s(s2 + 2s + 3)(s + 5)(s + 6)

Find the value of K of which the closed loop system is stable. Jun. 2014, 6 Marks

5 a: For a unity feedback system, the open-loop transfer function is given by

K

G(s) =

. Jun. 2014, 15 Marks

s(s + 2)(s2 + 6s + 25)

(i) Sketch the root locus for K .

(ii) At what value of K the system becomes unstable.

(iii) At this point of instability, determine the frequency of oscillation of the system.

5 b: Consider the system with G(s)H(s) =

K

using angle condition find

s(s + 2)(s + 4)

whether s = 0.75 and s = 1 + j4 are on the root locus or not. Jun. 2014, 5 Marks

(or)

s = 0.75 and s = 1 + j4 are not on the root locus.

6 a: Explain the procedure for investigating the stability using Nyquist criterion. Jun. 2014, 8 Marks, 6 Marks

Nyquist Stability Criterion The Nyquist plot allows us also 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.

The Nyquist diagram is basically a plot of G(j* w) where G(s) is the open-loop transfer function

and w is a vector of frequencies which encloses the entire right-half plane. In drawing the

Nyquist diagram, both positive and negative frequencies (from zero to infinity) are taken into

account. We will represent positive frequencies in red and negative frequencies in green. The

frequency vector used in plotting the Nyquist diagram usually looks like this (if you can imagine

the plot stretching out to infinity):

However, if we have open-loop

poles or zeros on the jw axis, G(s)

will not be defined at those points,

and we must loop around them

when we are plotting the contour.

Such a contour would look as

follows:

Please note that the contour loops

around the pole on the jw axis. As

we mentioned before, the Matlab

nyquist command does not take

poles or zeros on the jw axis into

account and therefore produces an

incorrect plot.

The Cauchy criterion

The Cauchy criterion (from complex analysis) states that when taking a closed contour in the

complex plane, and mapping it through a complex function G(s), the number of times that the

plot of G(s) encircles the origin is equal to the number of zeros of G(s) enclosed by the frequency

contour minus the number of poles of G(s) enclosed by the frequency contour. Encirclements of

the origin are counted as positive if they are in the same direction as the original closed contour

or negative if they are in the opposite direction.

When studying feedback controls, we are not as interested in G(s) as in the closed-loop transfer

function:

G(s)

---------

1 + G(s)

If 1+ G(s) encircles the origin, then G(s) will enclose the point -1.

Since we are interested in the closed-loop stability, we want to know if there are any closed-loop

poles (zeros of 1 + G(s)) in the right-half plane.

Therefore, the behavior of the Nyquist diagram around the -1 point in the real axis is very

important; however, the axis on the standard nyquist diagram might make it hard to see what's

happening around this point

To view a simple Nyquist plot using Matlab, we will define the following transfer function and

view the Nyquist plot:

0.5

-------

s - 0.5

Closed Loop Stability

Consider the negative feedback system

Remember from the Cauchy criterion that the number N of times that the plot of G(s)H(s)

encircles -1 is equal to the number Z of zeros of 1 + G(s)H(s) enclosed by the frequency contour

minus the number P of poles of 1 + G(s)H(s) enclosed by the frequency contour (N = Z - P).

Keeping careful track of open- and closed-loop transfer functions, as well as numerators and

denominators, you should convince yourself that:

the zeros of 1 + G(s)H(s) are the poles of the closed-loop transfer function

the poles of 1 + G(s)H(s) are the poles of the open-loop transfer function.

The Nyquist criterion then states that:

P = the number of open-loop (unstable) poles of G(s)H(s)

N = the number of times the Nyquist diagram encircles -1

clockwise encirclements of -1 count as positive encirclements

counter-clockwise (or anti-clockwise) encirclements of -1 count as negative

encirclements

Z = the number of right half-plane (positive, real) poles of the closed-loop system

The important equation which relates these three quantities is:

Z = P + N

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.

Throughout these tutorials, we will use a positive sign for clockwise encirclements.

Another way of looking at it is to imagine you are standing on top of the -1 point and are

following the diagram from beginning to end. Now ask yourself: How many times did I

turn my head a full 360 degrees? Again, if the motion was clockwise, N is positive, and if

the motion is anti-clockwise, N is negative.

Knowing the number of right-half plane (unstable) poles in open loop (P), and the number of

encirclements of -1 made by the Nyquist diagram (N), we can determine the closed-loop stability

of the system. If Z = P + N is a positive, nonzero number, the closed-loop system is unstable.

We can also use the Nyquist diagram to find the range of gains for a closed-loop unity feedback

system to be stable. The system we will test looks like this:

where G(s) is :

s^2 + 10 s + 24

---------------

s^2 - 8 s + 15

This system has a gain K which can be varied in order to modify the response of the closed-loop

system. However, we will see that we can only vary this gain within certain limits, since we have

to make sure that our closed-loop system will be stable. This is what we will be looking for: the

range of gains that will make this system stable in the closed loop.

Gain Margin

Gain Margin is defined as the change in open-loop gain expressed in decibels (dB), required at

180 degrees of phase shift to make the system unstable. Now we are going to find out where this

comes from. First of all, let's say that we have a system that is stable if there are no Nyquist

encirclements of -1, such as :

50

-----------------------

s^3 + 9 s^2 + 30 s + 40

Looking at the roots, we find that we have no open loop poles in the right half plane and

therefore no closed-loop poles in the right half plane if there are no Nyquist encirclements of -1.

Now, how much can we vary the gain before this system becomes unstable in closed loop? Let's

look at the following figure:

The open-loop system represented

by this plot will become unstable in

closed loop if the gain is increased past a certain boundary. The negative real axis area between -

1/a (defined as the point where the 180 degree phase shift occurs...that is, where the diagram

crosses the real axis) and -1 represents the amount of increase in gain that can be tolerated before

closed-loop instability.

Phase Margin

We have defined the phase margin as the change in open-loop phase shift required at unity gain

to make a closed-loop system unstable. Let's look at the following graphical definition of this

concept to get a better idea of what we

are talking about.

Let's analyze the previous plot and think

about what is happening. From our

previous example we know that this

particular system will be unstable in

closed loop if the Nyquist diagram

encircles the -1 point. However, we must

also realize that if the diagram is shifted by theta degrees, it will then touch the -1 point at the

negative real axis, making the system marginally stable in closed loop. Therefore, the angle

required to make this system marginally stable in closed loop is called the phase margin

(measured in degrees). In order to find the point we measure this angle from, we draw a circle

with radius of 1, find the point in the Nyquist diagram with a magnitude of 1 (gain of zero dB),

and measure the phase shift needed for this point to be at an angle of 180 deg.

6 b: Using Nyquist stability criterion, investigate the closed loop stability of a negative feedback

control system whose open loop transfer function is given by

G(s)H(s) =

K(sTa + 1) ; K, Ta > 0. Jun. 2014, 12 Marks

s3

7 b: List the limitations of lead and lag compensations. Jun. 2014, 5 Marks

Disadvantages of Phase Lead Compensation

Some of the disadvantages of the phase lead compensation -

1. Steady state error is not improved.

Effect of Phase Lead Compensation

1. The velocity constant Kv increases.

2. The slope of the magnitude plot reduces at the gain crossover frequency so that relative

stability improves & error decrease due to error is directly proportional to the slope.

3. Phase margin increases.

4. Response becomes faster.

Disadvantages of Phase Lag Compensation

Some of the disadvantages of the phase lag compensation -

1. Due to the presence of phase lag compensation the speed of the system decreases.

Effect of Phase Lag Compensation

1. Gain crossover frequency increases.

2. Bandwidth decreases.

3. Phase margin will be increase.

4. Response will be slower before due to decreasing bandwidth, the rise time and the settling

time become larger.

7 c: State the properties of state transition matrix and derive them. Jun. 2014, 5 Marks