Effect Of Adding Zero:Answer4

EFFECTS OF ADDING A ZERO ON THE ROOT LOCUS FOR A SECOND-ORDER SYSTEM

We can put the zero at three different positions with respect to the poles:

1. To the right of s = –p1

2. Between s = –p2 and s = –p1

3. To the left of s = –p2

We now discuss the effect of changing the gain K on the position of closed-loop poles

and type of responses.

(a) The zero s = –z1 is not present.

For different values of K, the system can have two real poles or a pair of complex

conjugate poles. This means that we can choose K for the system to be overdamped,

critically damped or underdamped.

(b) The zero s = –z1 is located to the right of both poles, s = – p2 and s = –p1.

In this case, the system can have only real poles and hence we can only find a value

for K to make the system overdamped. Thus the pole–zero configuration is even more

restricted than in case (a). Therefore this may not be a good location for our zero,

since the time response will become slower.

(c) The zero s = –z1 is located between s = –p2 and s = –p1.

This case provides a root locus on the real axis. The responses are therefore limited to

overdamped responses. It is a slightly better location than (b), since faster responses

are possible due to the dominant pole (pole nearest to jaxis) lying further from the j

axis than the dominant pole in (b).

(d) The zero s = –z1 is located to the left of s = –p2.

This is the most interesting case. Note that by placing the zero to the left of both

poles, the vertical branches of case (a) are bent backward and one end approaches the

zero and the other moves to infinity on the real axis. With this configuration, we can

now change the damping ratio and the natural frequency (to some extent). The

closed-loop pole locations can lie further to the left than s = –p2, which will provide

faster time responses. This structure therefore gives a more flexible configuration for

control design.

We can see that the resulting closed-loop pole positions are considerably influenced by

the position of this zero. Since there is a relationship between the position of closed-loop

poles and the system time domain performance, we can therefore modify the behaviour of

closed-loop system by introducing appropriate zeros in the controller.

Reference:

Web.mit.edu

www.wikipedia.com

www.palgrave.com

Poles And Zeros:Answer3

POLES AND ZEROS

Poles and Zeros of a transfer function are the frequencies for which the value of the transfer function becomes infinity or zero respectively. The values of the poles and the zeros of a system determine whether the system is stable, and how well the system performs. Control systems, in the most simple sense, can be designed simply by assigning specific values to the poles and zeros of the system.
Physically realizable control systems must have a number of poles greater than or equal to the number of zeros. Systems that satisfy this relationship are called proper. We will elaborate on this below.
Let's say we have a transfer function defined as a ratio of two polynomials:

H(s)=N(s)/D(s)
Where N(s) and D(s) are simple polynomials. Zeros are the roots of N(s) (the numerator of the transfer function) obtained by setting N(s) = 0 and solving for s.
Poles are the roots of D(s) (the denominator of the transfer function), obtained by setting D(s) = 0 and solving for s. Because of our restriction above, that a transfer function must not have more zeros then poles, we can state that the polynomial order of D(s) must be greater then or equal to the polynomial order of N(s).

Effects of Poles and Zeros

As s approaches a zero, the numerator of the transfer function (and therefore the transfer function itself) approaches the value 0. When s approaches a pole, the denominator of the transfer function approaches zero, and the value of the transfer function approaches infinity. An output value of infinity should raise an alarm bell for people who are familiar with BIBO stability. Tthe locations of the poles, and the values of the real and imaginary parts of the pole determine the response of the system. Real parts correspond to exponentials, and imaginary parts correspond to sinusoidal values.
The stability of a linear system may be determined directly from its transfer function. An nth order linear system is asymptotically stable only if all of the components in the homogeneous response from a finite set of initial conditions decay to zero as time increases.In order for a linear system to be stable, all of its poles must have negative real parts.



Reference:
Web.mit.edu

incremental encoders:answer2

Incremental Encoders

Incremental encoder produce an output which 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 rotary(or translator )disc

A stationary mask

A sensor (photodiode

The disc has alternate opaque and transparent sectors of equal width which is etched by means of a photographic process on to a plastic disc(slots are cut out if it is a metal disc).As the disc rotates during half of the increment cycle the transparent sectors of rotating and stationary discs come in alignment permitting the light from the LED to reach the sensor and thereby generating an electrical pulse.For fine resolution encoders ,multi-slit mask is often used to maximize the reception of shutter light.

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 OPAMP and comparator.Alternate transparent/opaque sectors of the disc and the square wave pulse form (obtained after signal processing) in synchronous with the disc is shown in figure.The resolution of such an incremental encoder is given as:

Basic resolution=360/N

N:number of sectors of disc;each sector is half transparent and half opaque.

In a dual channel encoder two optoelectronic channels are employed.These are installed in the same rotating disc and the mask but displaced at 900 to each other such that the two pulse output signals have a relative times phase displacement of 900electrical.A circuit that senses the relative time phase of the outputs of the two channels determines the direction of rotation of the disc or the encoder shaft.

The output of the encoder is fed to a counter which counts the number of pulses;the count being the measure of angle(or translation)through which the encoder shaft has rotated.By sampling the counter at regular intervals by means of clock pulses it is possible to compute the speed of encoder shaft.

Reference:

Control System And Engineering(Nagarath And Gopal).

www.wikipedia.com

synchro

SYNCHRO

A synchro is an electromagnetic transducer commonly used to convert angular position of shaft into an electrical signal.It is commercially known as a selsyn or an autosyn.It basically consists of a synchro transmitter (generator) and a synchro receiver(control transformer).

Schematic of Synchro Transducer The complete circle represents the rotor. The solid bars represent the cores of the windings next to them. Power to the rotor is connected by slip rings and brushes, represented by the circles at the ends of the rotor winding. As shown, the rotor induces equal voltages in the 120° and 240° windings, and no voltage in the 0° winding. [Vex] does not necessarily need to be connected to the common lead of the stator star windings.

A synchro or "selsyn" is a type of rotary electrical transformer that is used for measuring the angle of a rotating machine such as an antenna platform. In its general physical construction, it is much like an electric motor . The primary winding of the transformer, fixed to the rotor, is excited by a sinusoidal electric current (AC), which by electromagnetic induction causes currents to flow in three star-connected secondary windings fixed at 120 degrees to each other on the stator. The relative magnitudes of secondary currents are measured and used to determine the angle of the rotor relative to the stator, or the currents can be used to directly drive a receiver synchro that will rotate in unison with the synchro transmitter. In the latter case, the whole device (in some applications) is also called a selsyn (a portmanteau of self and synchronizing).

stepper motor

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.We can use an open step servo system with the same accuracy as that of a closed loop analog system.

The relation between a synchro and stepper motor is that the stepper motor is just a special type of the synchro. A stepper motor is designed to rotate through a specific angle (called a step) for each electrical pulse received from its control unit.

If we need to operate the stepper motor in closed loop(positional feedback)mode,we need to use synchros for error detection.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:

Control System And Engineering(Nagarath And Gopal).

www.wikipedia.com