Tempus Responsionis Systematis Controlis Secundi Ordinis (Exemplum Elaboratum)

Ordo systematis controlis determinatur per potentiam ‘s’ in denominatore functionis transferentiae eius.

Si potentia s in denominatore functionis transferentiae systematis controlis est 2, tunc systema dicitur systema controlis secundi ordinis.

Expressio generalis functionis transferentiae systematis controlis secundi ordinis data est ut

Hic, ζ et ωn sunt ratio amortizandi et frequens naturalis systematis, respectiviter (de his duobus terminis discemus in detalius ulterius).

Rearranging the formula above, the output of the system is given as

If we consider a unit step function as the input of the system, then the output equation of the system can be rewritten as

Taking the inverse Laplace transform of above equation, we get

The above expression of output c(t) can be rewritten as

The error of the signal of the response is given by e(t) = r (t) – c(t), and hence.

From the above expression it is clear that the error of the signal is of oscillation type with exponentially decaying magnitude when ζ < 1.

The frequency of the oscillation is ωd and the time constant of exponential decay is 1/ζωn.

Where, ωd, is referred as damped frequency of the oscillation, and ωn is natural frequency of the oscillation. The term ζ affects that damping a lot and hence this term is called damping ratio.

There will be different behaviors of output signal, depending upon the value of damping ratio and let us examine each of the cases, one by one.

Using this as a base, we will analyze the time response of a second order control system. We’ll do this by analyzing the unit step response of a second order control system in the frequency domain, before converting it into the time domain.

Tempus Responsus Systematis Controlis Secundi Ordinis

When damping ratio is zero, we can rewrite the above expression of output signal as

As in this expression, there is no exponential term, the time response of the control system is un-damped for unit step input function with zero damping ratio.

Page 137. Figure 6.4.3. of the book automatic control system by Hasan.

Now let us examine the case when damping ratio is unity.



In this expression of output signal, there is no oscillating part in subjective unit step function. And hence this time response of second-order control system is referred as critically damped.

Now we will examine the time response of a second order control system subjective unit step input function when damping ratio is greater than one.

Taking inverse Laplace transform of both sides of the above equation we get,


In the above expression, there are two time constants.

For the value of ζ comparatively much greater than one, the effect of faster time constant on the time response can be neglected and the time response expression finally comes as

Figure 6.4.5 of the page 139 of the book automatic control system by Hasan.

Tempus Responsus Systematis Controlis Critice Amortiti

The time response expression of a second order control system subject to unit step input function is given below.

The reciprocal of constant of negative power of exponential term in the error part of the output signal is actually responsible for damping of the output response.

Here in this equation it is ζωn. The reciprocal of constant of negative power of exponential term in error signal is known as time constant.

We have already examined that when the value of ζ (also know as damping ratio) is less than unity, the oscillation of the response decays exponentially with a time constant 1/ζωn. This is called under damped response.

On the other hand. when ζ is greater than unity, the response of the unit step input given to the system, does not exhibit oscillating part in it.

This is called over damped response. We have also examined the situation when damping ratio is unity that is ζ = 1.

In that situation the damping of the response is governed by the natural frequency ωn only. The actual damping at that condition is known as critical damping of the response.

As we have already seen in the associated expressions of time response of control system subject to input step function, the oscillation part is present in the response when damping ratio (ζ) is less than one and it is not present in the response when damping ratio is equal to one.

That means the oscillation part of the response just disappears when the damping ratio becomes unity. That is why damping of the response at ζ = 1, is known as critical damping.

More precisely, when damping ratio is unity, the response is critically damped and then the damping is known as critical damping.