Time Domain Analysis of a Control System
Introduction
Time domain gives the view of how the state of dynamic system changes with respect to time when a specific input is given.
A control system may comprise of several energystoring elements. Moreover, energy storing elements are generally inductors and capacitors in the case of an electrical system. Whenever the energy state of these elements is disturbed, a certain time is required by the element to change from one energy state to the other. Thus, this exact time is known as the transient time and the values and patterns exhibited by the voltage during this period is termed as the transient response.
A transient response is normally associated with an oscillation, which may be sustained or decaying in nature. Furthermore, the exact nature of the system depends upon the parameters of the system. Any system can be represented with a linear differential equation. The solution of this linear differential equation gives the response of the system. The representation of a control system by linear differential equation of functions of time and its solution is collectively called the time domain analysis of a control system.
The time response of linear system is the addition of transient response that is dependent on initial conditions and the steadystate response that is derived from the input of the system.

Transient Response
It is that component of the entire time response that changes with time and can also be defined as the time period the system takes to reach equilibrium or the steady state.

SteadyState Response
It is that component of the system response that is achieved when time approaches to infinity. Therefore, the response of the system after transient response is essentially called the steady state response.
Transient response does not depend upon the input of systems and can therefore be analyzed using step input. Steady state, however, depends upon the input and the dynamics of a system and can be determined using different test signals with the help of the final value theorem.
Time Domain Specifications
 Delay Time : It is the time required for the response to reach half of its final value from the zero instant.
 Rise Time : It is the time required for the response to rise from 0% to 100% of its final value. This is applicable for the underdamped systems. For the overdamped systems, consider the duration from 10% to 90% of the final value.
 Peak Time : It is the time required for the response to reach the peak value for the first time.
 Peak Overshoot : It is defined as the deviation of the response at peak time from the final value of response. It is also called the maximum overshoot.
 Settling Time: It is the time required for the response to reach the steady state and stay within the specified tolerance bands around the final value. In general, the tolerance bands are 2% and 5%.
Follow these steps to do the time domain analysis of a given control system using MATLAB:
 Open MATLAB. Go to the Command Window.
2. Create a Transfer Function that describes the control system you want to analyze.
3. Plot the step response for the give system with the use of the command stepplot.
The output for the above command is as shown below.
4. Obtain the time domain specifications for the step response for the system described above using the function step info. This is illustrated in the picture below.
5. Plot the impulse response for the given system using the function impulseplot. This is shown below:
The output for the above command is as follows:
6. We can also plot a time response to a particular system with respect to a customized signal and time using the lsimplot function. This is shown below :
The output to the above command is as shown below:
7. Obtain the time domain specifications for the response of the system described above using the function lsiminfo. This is illustrated in the picture below.
Conclusion
Time domain analysis was done for a given control system and its specifications was also calculated for the different responses obtained.