Overview

Electronic systems operate on two types of signals.

1. Continuous-Time (CT) signals or discrete-time (DT) signals. A continuous-time system is one in which the input signals are defined along a continuum of time, such as an analogue signal which “continues” over time producing a continuous-time signal. But a continuous-time signal can also vary in magnitude or be periodic in nature with a time period T. Generally, most of the signals present in the physical world which we can use tend to be continuous-time signals. For example, voltage, current, temperature, pressure, velocity, etc.
2. Discrete-time system is one in which the input signals are not continuous but a sequence or a series of signal values defined in “discrete” points of time. This results in a discrete-time output generally represented as a sequence of values or numbers. A continuous-time signal, x(t) can be represented as a discrete set of signals only at discrete intervals or “moments in time”. Discrete signals are not measured versus time, but instead are plotted at discrete time intervals, wheren is the sampling interval. As a result discrete-time signals are usually denoted as x(n) representing the input and y(n)representing the output.

         

           In feedback systems, the output signal is “fed back” and either added to or subtracted from the original input signal. The result is that the output of the system is continuously altering or updating its input with the purpose of modifying the response of a system to improve stability. A feedback system is also commonly known as a Closed-loop System. 

If the feedback loop reduces the value of the original signal, the feedback loop is known as “negative feedback”. If the feedback loop adds to the value of the original signal, the feedback loop is known as “positive feedback”.

Transfer Functions:

            Transfer Function is defined as the ratio of the Laplace transform of the output variable to the Laplace transform of the input variable, with all zero initial conditions.

A transfer function has the following properties:

·  The transfer function is defined only for a linear time-invariant system. It is not defined for nonlinear                systems.

·  The transfer function between a pair of input and output variables is the ratio of the Laplace transform of        the output to the Laplace transform of the input.

·  All initial conditions of the system are set to zero.

·  The transfer function is independent of the input of the system.

To derive the transfer function of a system, we use the following procedures:

1. Develop the differential equation for the system by using the physical laws, e.g. Newton’s laws and Kirchhoff’s laws.
2. Take the Laplace transform of the differential equation under the zero initial conditions.
3. Take the ratio of the output Y(s) to the input U(s). This ratio is the transfer function.

The part in the denominator of the transfer function which indicates us the position of poles in the system is known as Characteristic Equation.

Root Locus:

              Path of  poles of a close loop transfer function of characteristic equation as a function if gain.

         For Example:

          Transfer function  

         Locus on Real Axis

We have n=3 poles at s = 0, -3, -2.

         Cross Imag. Axis

Locus crosses imaginary axis at 2 values of K. Locus crosses where K = 0, 30.2, corresponding to crossing imaginary axis at s=0, ±2.45j, respectively.

    For Example 2:

    Transfer function

       Completed Root Locus

We have n=2 poles at s = 2, -1.  We have m=1 finite zero at s = -3.

         Cross Imag. Axis

Locus crosses where K = 0.646, 1, corresponding to crossing imaginary axis at s=0, ±0.994j, respectively.

Response of Control System

Steady state occurs after the system becomes settled and at the steady system starts working normally. Steady state response of control system is a function of input signal and it is also called as forced response. 
The transient state response of control system gives a clear description of how the system functions during transient state and steady state response of control system gives a clear description of how the system functions during steady state.

Steady State Error : It can be defined as the difference between the actual output and the desired output as time tends to infinity.

It is clear that the steady state response of control system depends only on the time constant ‘T’ and it is decaying in nature. 

Let us consider the block diagram of the second order system.

Now we will see the effect of different values of ζ on the response. We have three types of systems on the basis of different values of ζ.

1. Under damped system : A system is said to be under damped system when the value of ζ is less than one. In this case roots are complex in nature and the real parts are always negative. System is asymptotically stable. Rise time is lesser than the other system with the presence of finite overshoot.
2. Critically damped system : A system is said to be critically damped system when the value of ζ is one. In this case roots are real in nature and the real parts are always repetitive in nature. System is asymptotically stable. Rise time is less in this system and there is no presence of finite overshoot.
3. Over damped system : A system is said to be over damped system when the value of ζ is greater than one. In this case roots are real and distinct in nature and the real parts are always negative. System is asymptotically stable. Rise time is greater than the other system and there is no presence of finite overshoot.
4. Oscillations : A system is said to be sustain damped system when the value of zeta is zero. No damping occurs in this case.

PID:

PID control stands for proportional plus derivative plus integral control. PID control is a feedback mechanism which is used in control system. This type of control is also termed as three term control. By controlling the three parameters – proportional, integral and derivative we can achieve different control actions for specific work.
let's take a look at how the PID controller works in a closed-loop system using the schematic shown above. The variable () represents the tracking error, the difference between the desired input value () and the actual output (). This error signal () will be sent to the PID controller, and the controller computes both the derivative and the integral of this error signal. The control signal () to the plant is equal to the proportional gain () times the magnitude of the error plus the integral gain () times the integral of the error plus the derivative gain () times the derivative of the error.