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 the figure(a).


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

We can put the zero at different positions and see the effects.

(a) The zero s = –z1 is not present.

Here, we can choose K for the system to be over damped, critically damped or under damped.

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

Here we can only find a value for K to make the system over damped.

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

The responses here are limited to over damped responses but, with faster responses.

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

Here, we can change the damping ratio and the natural frequency. This structure gives a more flexible configuration for control design.

Hence we conclude that there is a relationship between the positions of closed-loop poles and the system time domain performance. We can therefore modify the behavior of closed-loop system by introducing appropriate zeros in the controller.

The effect of the zero is to contribute a pronounced early peak to the system’s response whereby the peak overshoot may increase appreciably. The smaller the value of z,the closer the zero to origin, the more pronounced is the peaking phenomenon. Thus, the zeros on the real axis near the origin are generally avoided in design. However in a sluggish system the introduction of a zero at proper position can improve the transient response.


Control Systems Engineering, Nagrath & Gopal



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

A system can be characterized by its poles and zeros since they allow reconstruction of the input/output differential equation. 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 provide a complete description of the system.

If you take a transfer function in terms of time G(t) = O(t) / I(t), and do a Laplace transform on it, you’ll get G(s) = O(s) / I(s), where s is a complex variable.
Now instead of solving partial differential equations in time, you can solve algebraic equations in terms of s, and then convert back to time. If you factor numerator and denominator, there will be a value for each factor that make it 0

Consider G(s) = (s-2) (s+3) / [(s-1) (s+4)].
Any value of s that makes the numerator 0, is called a zero, so in the above, you get zeros when s=+2, and s=-3.
Likewise, any value of s that makes the denominator 0, is called a pole. So in the above, s=+1 and s=-3 are poles.

The stability of a linear system may be determined directly from its transfer function. In a stable system, all components of the homogeneous response must decay to zero as time increases. If any pole has a positive real part, there is a component in the output that increases without bound, causing the system to be unstable.

Because the transfer function completely represents a system differential equation, its poles and zeros effectively define the system response. In particular, the system poles directly define the components in the homogeneous response. The location of the poles and zeros provide qualitative insights into the response characteristics of a system.

By carefully choosing or plotting poles and zeros, you can control how fast a system oscillates, if the oscillations grow or disappear, and other such related stability issues.



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

Incremental encoders are a type of Optical encoders which are used in control systems to convert linear or rotary displacement into digital code or pulse signals. The speciality of incremental encoders is that their output is a pulse for each increment of resolution but these makes no distinction between increments as compared to Absolute encoders( the other type of optical encoder), whose output is a digitally coded signal with distinct digital code indicative of each particular least significant increment of resolution.

The incremental encoder consists of

  • a rotary disc having alternate opaque and transparent sectors
  • a light source (LED)
  • a stationary mask
  • a sensor (photo diode)


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 thereby generating an electrical pulse.

Rotary encoders are used to track the position of the motor shaft on permanent magnet brushless motors, which are commonly used on CNC machines, robots, and other industrial equipment. In these applications, the encoder (feedback device) plays a vital role in ensuring that the equipment operates properly.

Control Systems Engineering, Nagrath & Gopal

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

The term “synchro” is an abbreviation of the word “synchronous.” It is the name given to a variety of rotary, electromechanical, position-sensing devices.

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.

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. It is commercially known as a selsyn or an autosyn.


Synchros can be thought of as “variable transformers” .When an AC voltage applied to the rotor shaft winding, it causes a change in the synchro’s Stator output voltage. 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 the 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 (self and synchronizing). When several synchros are correctly connected, all of the rotors will align themselves in the same angular position. This is useful, since when the angular position of one synchro is forced to change, it can drive another synchro to indicate the angular change.

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.