Zero to a Control System

What would 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+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 = – p₂ and s = –p₁.

Here, the system can have only real poles. Hence only one value for K to 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

What do the poles and zeros contribute to in the control system ?

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:

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).

Incremental Encoders

What are incremental encoders? Are they useful to us in any way?
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.

Whenever mechanical rotary motions have to be monitored, the encoder is the most important interface between the mechanics and the control unit.

Syncro and its relation to stepper

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.
Synchros play a very important role in the operation of Navy equipment. Synchros are found in just about every weapon system, communication system, underwater detection system, and navigation system used in the Navy. The importance of synchros is sometimes taken lightly because of their low failure rate.
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). U.S. Naval terminology used the term "synchro" exclusively (possible exception: steering gear -- info. needed).

With their rugged construction and high reliability, Synchros have been used since World War II as the “angle” transducer of choice for Military, Space and Aviation applications, where only the best will do.

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.

Syncro and its relation to Stepper

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.
Synchros play a very important role in the operation of Navy equipment. Synchros are found in just about every weapon system, communication system, underwater detection system, and navigation system used in the Navy. The importance of synchros is sometimes taken lightly because of their low failure rate.
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). U.S. Naval terminology used the term "synchro" exclusively (possible exception: steering gear -- info. needed).

With their rugged construction and high reliability, Synchros have been used since World War II as the “angle” transducer of choice for Military, Space and Aviation applications, where only the best will do.

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.

What are incremental encoders? Are they useful to us in any way?
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.

Whenever mechanical rotary motions have to be monitored, the encoder is the most important interface between the mechanics and the control unit.
What do the poles and zeros contribute to in the control system ?

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:

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).

What would 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+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 = – p₂ and s = –p₁.

Here, the system can have only real poles. Hence only one value for K to 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. [1]

References:

1. http://www.palgrave.com/

A New Beginning

Servomechanism is an  automatic device for the control of a large power output by means of a small power input or for maintaining correct operating conditions in a mechanism. It is a type of feedback control system. The constant speed control system of a DC motor is a servomechanism that monitors any variations in the motor's speed so that it can quickly and automatically return the speed to its correct value. Servomechanisms are also used for the control systems of guided missiles, aircraft, and manufacturing machinery.

A servomechanism is unique from other control systems because it controls a parameter by commanding the time-based derivative of that parameter.

Servomechanism may or may not use a servomotor.

  The common type of servo provides position control. Servos are commonly electrical or partially electronic in nature, using an electric motor as the primary means of creating mechanical force. Other types of servos use hydraulics, pneumatics, or magnetic principles. Usually, servos operate on the principle of negative feedback, where the control input is compared to the actual position of the mechanical system as measured by some sort of transducer at the output. Any difference between the actual and wanted values (an "error signal") is amplified and used to drive the system in the direction necessary to reduce or eliminate the error. An entire science known as control theory has been developed on this type of system.

Small R/C servo mechanism is shown in the figure above the parts as labelled are:-
1. electric motor
2. position feedback potentiometer
3. reduction gear
4. actuator arm

When at I2It pune whe learned about the wiring and working of a servo motor .

Servo Wiring 
All servos have three wires: 
Black or Brown is for ground. 
Red is for power (~4.8-6V). 
Yellow, Orange, or White is the signal wire (3-5V).

Servomechanisms were first used in military fire-control and marine navigation equipment. Today servomechanisms are used in automatic machine tools, satellite-tracking antennas, remote control airplanes, automatic navigation systems on boats and planes, and antiaircraft-gun control systems. Other examples are fly-by-wire systems in aircraft which use servos to actuate the aircraft's control surfaces, and radio-controlled models which use RC servos for the same purpose. Many autofocus cameras also use a servomechanism to accurately move the lens, and thus adjust the focus. A modern hard disk drive has a magnetic servo system with sub-micrometre positioning accuracy.

Thier only disadvantage is that these motors are much more costlier than stepper or dc motors.

Cincinati Milacron T3 Arm 

 

The Cincinnati Milacron T3-776 robot is shown in .the picture. It is apparent that the
geometry of this manipulator is very similar to that of the Puma robot, i.e. the second
and third joint axes are parallel and the last three joint axes intersect at a point. The
detailed solution will therefore be very similar to that for the Puma robot. This solution
will be followed by a general discussion of a geometric solution that does not require that the hypothetical closure link be determined.

This robot is a more classically designed industrial robot. Designed as a healthy compromise between dexterity and strength this robot was one of the ground breakers, in terms of success, in factory environments. However, while this robot was a success in industry its inflexible interfacing system makes it difficult to use in research. 

The Cincinnati Milacron T3-786 robot was the workhorse of the robotics industry for many years. Almost indestructable, thousands of these machines are still in production in hostile, 24 hour manufacturing environments. Many manufacturers have come to depend on these machines, and are reluctant to give them up.

This robot is a more classically designed industrial robot. Designed as a healthy compromise between dexterity and strength this robot was one of the ground breakers, in terms of success, in factory environments. However, while this robot was a success in industry its inflexible interfacing system makes it difficult to use in research.