ee301 answer4

What would be the effect of adding a zero to a control system?

Consider the second-order system given by

G(s) =1/(s+p1)(s+p2)

p1 > 0, p2 > 0

The poles are given by s = –p1 and s = –p2 and the simple root locus plot for this system is

shown in Figure.

When we add a zero at s = –z1 to the controller, the open-loop

transfer function will change to:

G1(s) =K( s+ z)/(s +p1)( s+ p2)

z1 > 0

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

1. To the right of s = –p1

Figure (b)

2. Between s = –p2 and s = –p1 Figure (c)

3. To the left of s = –p2

Figure (d)

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 jwaxis) lying further from the jw

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:

http://www.palgrave.com/science/engineering/wilkie/sample/0333_77129Xcha13sample.pdf

Control Systems Engineering, Nagrath & Gopal

ee301 answer1

What is a Synchro? Is it related in any way to a stepper motor?

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 (See below.) 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 .

On a practical level, synchros resemble motors, in that there is a rotor, stator, and a shaft. Ordinarily, slip rings and brushes connect the rotor to external power. A synchro transmitter's shaft is rotated by the mechanism that sends information, while the synchro receiver's shaft rotates a dial, or operates a light mechanical load. Single and three-phase units are common in use, and will follow the other's rotation when connected properly. One transmitter can turn several receivers; if torque is a factor, the transmitter must be physically larger to source the additional current. In a motion picture interlock system, a large motor-driven distributor can drive as many as 20 machines, sound dubbers, footage counters, and projectors.

A different type of receiver, called a control transformer (CT), is part of a position servo that includes a servo amplifier and servo motor. The motor is geared to the CT rotor, and when the transmitter's rotor moves, the servo motor turns the CT's rotor and the mechanical load to match the new position. CTs have high-impedance stators and draw much less current than ordinary synchro receivers when not correctly positioned.

Synchro transmitters can also feed synchro to digital converters, which provide a digital representation of the shaft angle.

Reference:

en.wikipedia.org

ee301 answer2

What are incremental encoders? Are they useful to us in any way?
Optical encoders are devices that convert a mechanical position into a representative electrical signal by means of a patterned disk or scale, a light source and photosensitive
elements. With proper interface electronics, position and speed information can be derived.

Incremental encoders

The disk of an incremental encoder is patterned with a single track of lines around its periphery. The disk count is defined as the number of dark/light linepairs that occur per revolution ("cycles / revolution" or "c/r").As a rule, a second track is added to generate a signal that occurs once per revolution (index signal), which can be used to indicate an absolute position.
To derive direction information, the lines on the disk are read out by two different photo-elements that "look" at the disk pattern with a mechanical shift of 1/4 the pitch of a linepair between them. This shift is realized with a "reticle" or "mask" that restricts the
view of the photo-element to the desired part of the disk lines. As the disk rotates, the two photo-elements generate signals that are shifted 90° out of phase from each other. These are commonly called the quadrature "A" and "B" signals. The clockwise direction for most encoders is defined as the "A" channel going positive before the "B" channel.




If the readout of the disk is obtained by a single photoelement for each of the A and the B channels, it is called a "single-ended" readout. This type of readout generates signals that are very susceptible to disk runout ("wobble"), slight imperfections in disk etching, etc. A much more effective and accurate readout system is called "push-pull" where the A and B channels are generated by two photo elements for each channel.

Incremental encoders are used to track motion and can be used to determine position and velocity. This can be either linear or rotary motion. Because the direction can be determined, very accurate measurements can be made.
Rotary encoders are often used to track the position of the motor shaft on permanent magnet brushless motor, which are commonly used on CNCmachines,robots, and other industrial equipment. In these applications, the feedback device (encoder) plays a vital role in ensuring that the equipment operates properly. The encoder synchronizes the relative rotor magnet and stator winding positions to the current provided by the drive.

Reference:
en.wikipedia.org

ee301 answer3

What do the poles and zeros contribute to in the control system?
The transfer function provides a basis for determining important system response characteristics without solving the complete differential equation. As defined, the transfer function is a rational function in the complex variable s = σ + jω.

It is often convenient to factor the polynomials in the numerator and denominator, and to write
the transfer function in terms of those factors:

H(s) = N(s)/D(s)
= K(s − z1)(s − z2) . . . (s − zm−1)(s − zm)/((s − p1)(s − p2) . . . (s − pn−1)(s − pn) )
where the numerator and denominator polynomials, N(s) and D(s), have real coefficients defined
by the system’s differential equation . Here,
the zi’s are the roots of the equation
N(s) = 0
and are defined to be the system zeros, and the pi’s are the roots of the equation
D(s) = 0
and are defined to be the system poles. The factors in the numerator and denominator
are written so that when s = zi the numerator N(s) = 0 and the transfer function vanishes, that is

lim H(s) = 0.
s→zi

and similarly when s = pi the denominator polynomial D(s) = 0 and the value of the transfer
function becomes unbounded,

lim H(s) = ∞.
s→pi

All of the coefficients of polynomials N(s) and D(s) are real, therefore the poles and zeros must
be either purely real, or appear in complex conjugate pairs. In general for the poles,
either pi = σi, or else pi, pi+1 = σi±jωi. The existence of a single complex pole without a corresponding conjugate pole would generate complex coefficients in the polynomial D(s). Similarly, the system zeros are either real or appear in complex conjugate pairs.

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.The locations of the poles, and the values of the real and imaginary parts of the pole determine the response of the system.

Reference:
Control Systems Engineering, Nagrath & Gopal