ans4:effect of adding zero

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

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

ans3:poles and zeros

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 encoder

The incremental encoder, sometimes called a relative encoder, is simpler in design than the absolute encoder. It consists of two tracks and two sensors whose outputs are called channels A and B. As the shaft rotates, pulse trains occur on these channels at a frequency proportional to the shaft speed, and the phase relationship between the signals yields the direction of rotation. The code disk pattern and output signals A and B are illustrated in Figure 5. By counting the number of pulses and knowing the resolution of the disk, the angular motion can be measured. The A and B channels are used to determine the direction of rotation by assessing which channels "leads" the other. The signals from the two channels are a 1/4 cycle out of phase with each other and are known as quadrature signals. Often a third output channel, called INDEX, yields one pulse per revolution, which is useful in counting full revolutions. It is also useful as a reference to define a home base or zero position.


Figure 5 illustrates two separate tracks for the A and B channels, but a more common configuration uses a single track with the A and B sensors offset a 1/4 cycle on the track to yield the same signal pattern. A single-track code disk is simpler and cheaper to manufacture.

The quadrature signals A and B can be decoded to yield the direction of rotation as hown in Figure 6. Decoding transitions of A and B by using sequential logic circuits in different ways can provide three different resolutions of the output pulses: 1X, 2X, 4X. 1X resolution only provides a single pulse for each cycle in one of the signals A or B, 4X resolution provides a pulse at every edge transition in the two signals A and B providing four times the 1X resolution. The direction of rotation(clockwise or counter-clockwise) is determined by the level of one signal during an edge transition of the second signal. For example, in the 1X mode, A= with B =1 implies a clockwise pulse, and B=with A=1 implies a counter-clockwise pulse. If we only had a single output channel A or B, it would be impossible to determine the direction of rotation. Furthermore, shaft jitter around an edge transition in the single signal woudl result in erroneous pulses..



reference
mechatronics.mech.northwestern.edu/design

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 (a portmanteau of self and synchronizing).



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.


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.

1) Synchro systems were first used in the control system of the Panama Canal, to transmit lock gate and valve stem positions, and water levels, to the control desks.2)Fire-control system designs developed during World War II used synchros extensively, to transmit angular information from guns and sights to an analog fire control computer, and to transmit the desired gun position back to the gun location.3)Smaller synchros are still used to remotely drive indicator gauges and as rotary position sensors for aircraft control surfaces, where the reliability of these rugged devices is needed. Digital devices such as the rotary encoder have replaced synchros in most other applications.

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.