ee301-assignment 4

What is the effect of adding a zero to a control system?
Consider second order closed loop transfer function of the form given by
C(s)/R(s) =ωn2/ s2 +2ζωns+ωn2
ζ=damping factor(or damping ratio)
ωn =undamped natural frequency

Let a zero at s=-z be added to this transfer function .Then we have
C(s)/R(s)= [(s+z)/z]ωn2 / s2 +2ζωns+ωn2 →→→→(1)
Note that multiplication term in numerator of above expression has been adjusted so that steady state gain C/R(0) of system is unity .This gives steady state value of output Css=1 When input is unit step.Thus the system will track the step input with zero steady error .

From equation we have
C(s)/R(s)= ωn2/ s2 +2ζωns+ωn2 +s/z{ωn2/ s2 +2ζωns+ωn2 }
Let Cz(t) be the response of the system with a zero at s=-z.Then from the above equation we have
Cz(t)=c(t) + 1/z ( dc(t)/dt) (a)

The effect of added derivative term may be seen by examining fig (1) where a case for a typical value of ζ(less than one )is considered.We dee from this figure that the effect of zero is to contribute a pronounced early peak to the system’s response wherby peak overshoot may increase appreciably.From equation(a) and fig (1)it is seen that smaller the value of z ie closer is the value of zero to the origin,the more pronounced is the peaking phenomenon.On account of this fact the zeros of real axis near the origin are generally avoided in design.However ,in asluggish system artful introduction of a zero at the proper position can improve the transient response.

We further observe from equation(a)that as z increases ie zero moves farther into left half of s plane ,its effect become less pronounced.For sufficiently large values of z,the effect of zero on transient response may become negligible.
For closed loop transfer function of eq(1) peak percentage overshoot for a unit step input can be read from log –log graphs of fig (1) as function of z/ζωn for various values of ζ<=1

Reference: (1)Control Systems Engineering-I.J Nagarath,M.Gopal

(2)Control Systems Engineering-S.N Sivanandam

ee301 assignment 3

What do the poles and zeros contribute in the control system?
The open loop transfer function of a general linear feedback system can be written as
G(s)H(s)=k(s+x 1)(s+x2)…………/(s+p1)(s+p2)……..→→→→→→→→→→→→(1)
Where - x 1 ,-x 2, -x 3…………..are the zeros and -p1 ,-p2 ,-p3………..are the poles of open-loop transfer function.
The characterstic equation in terms of open loop transfer function is formed as
1+G(s)H(s)= 1 + k(s+x 1)(s+x2)…………/(s+p1)(s+p2)……..
= (s+p1)(s+p2 )…k(s+x1)(s+x2)…/(s+p1)(s+p2)….→→→→→→→(2)
Equation (2) can also be written as
1+G(s)H(s)= k(s+z 1)(s+z2)………/(s+p1)(s+p2)……..→→→→→→→→→(3)
The roots of charactersatic equation are obtained from the expression
1+G(s)H(s)=0
Hence ,for this condition equation (3)yields that
(s+z 1)(s+z2)……=0
These roots are (-z1 ,-z2 ,-z3 .........................) called as zeros of the characterstic equation .It canbe observed that the poles of open loop transfer function are also the poles of characterstic equation (2)
Criterion for stability:The linear feedback control system represented by characterstic equation (1+G(s)H(s))will be stable only if zeros - z1 ,-z2 ,-z3……………….lie in left half of s plane ie roots of the characterstic equation must contain negative real part.This indicates that no zero of charactrstic equation should lie in right half of the s plane for the system to be stable


Reference:
(1)Control Systems Engineering-I.J Nagarath,M.Gopal
(2)Control Systems Engineering-S.N Sivanandam

ee301 assignment 2

What are incremental encoders ?Are they useful to us in any way?
Increment encoders are optical encoders that are frequently used in control systems to convert linear or rotory displacement into digital code or pulse signals. Their output is a pulse for each increment of resolution but these make no distinction between increments.
An incremental encoder typically has four parts:a light source(LED),a rotory (or translatory) disc ,a stationary mask and a sensor (photodiode) .The disc has alternate opaque and transparent sectors which are etched by means of photographic process on to a plastic disc.(slots are cut out in case a metal disc is used).As disc rotates during half of the increment cycle the transparent sectors of rotating and stationary discs come in alignment permitting the light from LED to reach the sensor thereby generating an electric pulse .
The waveform of the sensor output of an encoder is generally triangular or sinusoidal depending upon the resolution required.Square wave signal compatible with digital logic are obtained from it by means of linear OPAM and comparator.The resolution of incremental encoders are given by

Basic resolution =360/N where N=number of sectors of disc:each sector is half transparent and half opaque.

Reference:
(1)Control Systems Engineering-I.J Nagarath,M.Gopal
(2)Control Systems Engineering-S.N Sivanandam

ee301 assignment 1

What is synchro?Is it relalated in any way to stepper motor?
Synchros are electromagnetic devices which are used to transmit angular displacement ,velocity e.t.c.It is commercially known as selsyn or autosyn.The types of synchros according to operations are given below:
1.synchro transmitter: This is an electromechanical unit in which the input ,the mechanical signal is given to the rotor is converted into an electrical signal in the stator.
2.Synchro receiver.:This is also known as synchro control transformer,in which the input electrical signal given to the stator is converted to the mechanical signal[angular displacement] in the rotor.
3.Synchro differential transmitter and differential receiver:The differential transmitter is an electro mechanical unit in which the rotor is driven mechanically to modify the received signal and to transmit an electrical signal which is the sum of difference of impressed signal and modified signal.

There are two distinctly different ways of using stepper motors in control systems.One is the open loop mode and other is the closed loop mode.
The stepper motor is a digital device whose output in shaft angular displacement is completely determined by the number of input pulses.Consequently,there is no need for a feedback device to determine the position of motor shaft and ,therefore,of the load connected to the motor shaft.This means that an open –loop step servo system can be designed to yield the the same accuracy as that of a closed loop analog system.figure(1) shows use of stepper motor in open loop mode. Use of steppermotor in closedloop mode requires synchros as error detector.Here the motor is used like conventional servomotor.A signal from the output is fed back and is used to operate a gate controlling the pulses from a pulse generator.This is shown in the figure below:

Reference:
(1)Control Systems Engineering-I.J Nagarath,M.Gopal
(2)Control Systems Engineering-S.N Sivanandam