Engineers try to find out techniques for systems to become more efficient and reliable. There are two methods of controlling the systems. One is open loop system and other is feedback control system. In open loop system, input proceed to the given process and produces output. There is no feedback which tells whether desired output comes or not. Other is feedback control system which has the ability to check whether desired output is achieved and if not, it gives feedback to the controller who controls the system.
If the input is unit step, the output is step response. The step response yields a clear vision of the system’s transient response. We have two types of system, first order system and second order system, which are representative of many physical systems. First order of system is defined as first derivative with respect to time and second order of system is second derivative with respect to time. Our main topic which we discuss here is First Order Control System.
In theory, first order system is a system which has one integrator. As number of order increases, number of integrator in a system also increases. In mathematically, first derivative of given function with respect to time.
We have different techniques to solve system equations using differential equations or Laplace Transform but engineers find the ways to minimize the technique of solving equations for abrupt output and work efficiency. The total response of the system is the sum of forced response and natural response.
The forced response is also called the steady state response or particular equation. The natural response is also called the homogeneous equation.
Before proceeding to the topic, reader should be encouraged about the fundamental concept of poles, zeros and transfer function and fundamental concepts of feedback control system. Here, remind your memory with fundamental concepts of feedback control system.
It is defined as the ratio of output and input.
Poles of a Transfer Function
The poles of transfer function are the value of Laplace Transform variable(s), that cause the transfer function becomes infinite.
The denominator of a transfer function is actually the poles of function.
Zeros of a Transfer Function
The zeros of transfer function are the values of Laplace Transform variable(s), that cause the transfer function becomes zero.
The nominator of a transfer function is actually the zeros of function
First Order Control System
Here we discuss the first order control system without zeros. First order control system tell us the speed of the response that what duration it reaches to the steady state.
If the input is unit step, R(s) = 1/s so the output is step response C(s). The general equation of 1st order control system is , i.e is transfer function.
There are two poles, one is input pole at the origin s = 0 and other is system pole at s = -a, this pole is at negative axis of pole plot. We can find the pole and zeros in MATLAB SOFTWARE by using command pzmap means pol zero map.
We now taking the inverse transform so total response become which is sum of forced response and natural response. Due to the input pole at the origin, produces the forced response as name describe by itself that giving forced to the system so it produces some response which is forced response and the system pole at -a produces natural response which is due to the transient response of the system.
After some calculation, here General form of first order system is C(s) = 1-e-at that is equal to forced response which is “1” and natural response which is equal to “e-at”. The only thing which is needed to find is the parameter “a”.
Many techniques like differential equation or inverse Laplace Transform, these all solves the total response but these are time consuming and laborious. The use of poles, zeros and their some fundamental concept gives us the qualitative information to solve the problems and due to these concepts, we can easily tell the speed of response and the time of a system to reach the steady-state point. Let us describe the three transient response performance specifications, time constant, rise time and settling time.
It can be defined as the time it takes for the step response to rise up to 63% or 0.63 of it’s final value. We call it as t = 1/a. If we take reciprocal of time constant, its unit is 1/seconds, or frequency. We call the parameter “a” the exponential frequency. Because the derivative of e-at is -a at t = 0. So the time constant is considered as transient response specification for first order control system.
We can control the speed of response by setting the poles. Because the farther the pole from imaginary axis, the faster the transient response is. So, we can set poles farther from the imaginary axis to speed up the whole process.
Rise time is defined as the time for waveform to go from 0.1 to 0.9 or 10% to 90% of its final value. For the equation of rise time, we put 0.1 and 0.9 in general first order system equation respectively.
For t = 0.1
For t = 0.9
Taking the difference between 0.9 and 0.1
Here the equation of rise time. If we know the parameter of a, we can easily find the rise time of any given system by putting “a” in equation.
It is defined as the time for the response to reach and stay within 2% of its final value. We can limit the percentage up to 5% of its final value. Both percentages are consideration.
The equation of settling time is given by: Ts = 4/a.
By using these three transient response specification, we can easily compute the step response of given system that’s why this qualitative technique is useful for order systems equations.
Conclusion of First Order Control System
After learning all things related to 1st order control system, we come to the following conclusions:
- A pole of the input function generates the form of the forced response. It is because of pole at the origin which generates a step function at output.
- A pole of the transfer function generates the natural response. It the pole of the system.
- A pole on the real axis generates an exponential frequency of the form e-at. Thus, the farther the pole to the origin, the faster the exponential transient response will decay to zero.
- Using poles and zeros, we can speed up the performance of system and get the desired output.