Time behavior of the system is very important.  When we design the system, the time behavior is the most important aspect of the system.  Points you might worry about include the following.

 The parameters in a first order system determine aspects of various kinds of responses. Whether we are talking about impulse response, step response or response to other inputs.

x(t)  is response of the system, 
u(t)  is input to the system, 
t  is  system time constant, 
Gdc  =  The DC Gain of the System.

Every system will have an input which we can call u(t), and a response we will denote by x(t).  Each system will also have a time constant and a DC gain.

  • t, the time constant, will determine how quickly the system moves toward steady state.
  • Gdc, the DC gain of the system, will determine the size of steady state response when the input settles out to a constant value.

Time Response

 The time response or the transient response which is also called as the naturalresponse is the response of the system to find the change in the system from the state of equilibrium. The time response is the portion of the response which approaches zero after the sufficiently long time we can say that as time approaches infinity. This one of the three types of damping that defines the output with the steady-state value.

Under damped

The under damped response is that which oscillates within a decaying envelope. Here damping ratio should be <1.

Critically damped

The critically damped response is the response which reaches at the steady-state value without being the under damped. Here the damping ratio should be equal to one. There should be no oscillation about the steady state value in the ideal case.

Over damped

The over damped response is the response which does not oscillate about the steady-state value but it takes longer to reach than the critically damped system. Here damping ratio is >1

The second order system time response

Rise time

The rise time refers to the time which required for the signal to change from the specified low value to the specified high value. These values are 10% and 90% of the step height.


Overshoot occurs when a signal exceeds its target. It is often associated with ringing.

Settling time

Thesettling time is the time which elapsed from the application of the ideal instantaneous step input to the time at which output has entered and remained within the specified error band.

Delay time

The delay time is the time required to the response to reach half  of the final value the very first time.


Peak time

The peak time is the time required for the response to reach the first peak of the overshoot.

Steady-state error

The steady state error is the difference between the desired final output and the actual one when the system reaches a steady state.




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