Effect of Adding a Zero

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+z₁) / ((s+p₁)(s+p₂)) , z₁>0

Effect of adding a zero to a second-order system root locus.

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)

(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. Thus K for the system can be overdamped, critically damped or underdamped.

(b) The zero s = –z₁ is located to the right of both poles, s = – p₂and s = –p₁.

Here, the system can have only real poles. Hence only one value for Kto make the system overdamped exists. 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 jω-axis) lying further from the jω-axis than the dominant pole in (b).

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

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 . 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

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. 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. [1]

The poles and zeros are properties of the transfer function, and therefore of the differential equation describing the input-output system dynamics. Together with the gain constant K they completely characterize the differential equation, and provide a complete description of the system. A system is characterized by its poles and zeros in the sense that they allow reconstruction of the input/output differential equation. In general the system dynamics may be represented graphically by plotting their locations on the complex s-plane, whose axes represent the real and imaginary parts of the complex variable s (pole-zero plots). For the stability of a linear system, all of its poles must have negative real parts,that is they must all lie within the left-half of the s-plane. A system having one or more poles lying on the imaginary axis of the s-plane has non-decaying oscillatory components in its homogeneous response, and is defined to be marginally stable. [2]

Incremental encoders !!!!!!!!!

Incremental encoders are position feedback devices that provides incremental counts. Thus, incremental encodersprovide relative position, where the feedback signal is always referenced to a start or home position. Forincrementalencoders, each mechanical position is uniquely defined. The current position sensed is only incremental from the last position sensed. Incremental encoders are also non-contacting optical, rotary, quadrature output device. •Theseincremental encodersare also called optical encoders or optical incremental encoders because they utilizes optical technology. Optical incremental encoders are highly sort after as position feedback devices due to its durability and ability to achieve high resolution. Avago’s optical incremental encoders are exceptionally recognized for its reliability and accuracy.

Synchro?? how is it Related to stepper motor!!!!!

A SYNCHRO is a motor like device containing a rotor and a stator and capable of converting an angular position into an electrical signal, or an electrical signal into an angular position. A Synchro can provide an electrical output (at the Stator) representing its shaft position or it can provide a mechanical indication of shaft position in response to an applied electrical input to its stator winding.

STEPPER MOTOR

A stepper motor is a “digital” version of the electric motor. The rotor moves in discrete steps as commanded, rather than rotating continuously like a conventional motor. When stopped but energized, a stepper (short for stepper motor) holds its load steady with a holding torque. Wide spread acceptance of the stepper motor within the last two decades was driven by the ascendancy of digital electronics. Modern solid state driver electronics was a key to its success. And, microprocessors readily interface to stepper motor driver circuits.