Introduction
As we discussed earlier we have two methods of analyzing the working and functioning of a control system named as:- Time domain analysis
- Frequency domain analysis
- Transient response
- Steady state response
- Impulse
- Step
- Ramp
- Parabolic
Second Order Systems
The order of a differential equation is the highest degree of derivative present in that equation. A system whose input-output equation is a second order differential equation is called Second Order System.
There are a number of factors that make second order systems important. They are simple and exhibit oscillations and overshoot. Higher order systems are based on second order systems. In case of mechanical second order systems, energy is stored in the form of inertia whereas in case of electrical systems, energy can be stored in a capacitor or inductor.
Standard form of second order system is given by:
Where:
- ωn Is the natural frequency
- is the damping ratio
- If 0< <1 as="" damped="" is="" li="" named="" system="">
- If < =1, system is named as Critically Damped System
- If < >1, system is named as Over Damped System 1>
Response of a Second order system
We analyze the responses in second order systems in undamped, under damped, critically damped and over damped cases. Let us have a look on these:- 1. Step response of Second-order systems:
Two poles are equal. That means:
In a unit step input, we have:
And output is:
Steady-state error: e (∞) = 0
Over Damped case: ( > 1)
We can write the transfer function of a second-order system by factoring the denominator as:
Taking the inverse Laplace transform yields the time response:
The unit-step time response is:
Damped case: ( < 1)
For a damped case in which 0 < < 1 time response is given by:
- 1. Ramp response of a second-order system:
= 1, critically damped case
> 1, over damped case
0 < < 1, under damped case
The Laplace transform of a unit-ramp input is R(s) = 1/s^2
The output is given by:
- 2. Impulse response of a second-order system:
The unit impulse response is give by:
Conclusion
In this tutorial we have discussed second order systems and their responses. As we are now done with the order of systems we will move on to Transient Response Analysis of Control Systems. Stay tuned for more insight on control systems.
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