Designing should be such that damping should be increased to 0.7-0.9 from 0.2. Designing of controllers is also a vast research topic. Suppose a system has damping 0.2, then with the help of PD Controllers, Lead compensation, etc., damping can be increased. (In fact, if the damping is one, then it is the best system, but it is very difficult to achieve accurate damping. A good control system should have damping around 0.7-0.9.The system has two real roots both are real and unequal. You can find it has ‘ζ’= 1.5, ‘ω n’= 4 rad/sec. Transfer function = is an example of an overdamped system. Critically damped and overdamped systems don’t have oscillations. If the damping is more than one, then it is called overdamped system (i.e.The system has two real roots both at ‘-4’. You can find it has ‘ζ’= 1, ‘ω n’= 4 rad/sec. For example transfer function = is an example of a critically damped system. If the damping is one, then it is called critically damped system.System-1 is the example of an underdamped system. If the damping is between zero to one then poles of the closed-loop transfer function will be complex.In the present example damping is 0.2, hence the system is stable.

Stable systems have positive damping, marginally stable systems have zero damping and unstable systems have negative damping.Now, please note the following points regarding stability: So, with the above discussions, you might have understood about various time domain specifications. In the present case, the steady-state value of the input is ‘1’.įrom figure-3, it can be viewed that after 5 seconds oscillations are within two percent range of input value, hence settling time is five seconds. In the case of step input, the time required to decrease the oscillations within two percent range of input is called settling time. Oscillations may be continued after settling time.

Perhaps you may think that the time required until all the oscillations are completely damped out (vanished) is called settling time no, it is not true. In figure-3, y-axis is from 0.98 to 1.02 (2% range of final value). Time response shown in figure-2 is just reproduced in figure-3, but with a change in the y-axis. To understand the term settling time ‘t s’, refer figure 3. (If you will generate this graph from software’s, then you can zoom the graph and verify all the values such as %M p, ω d, t p, t r, t d, etc.). If you will calculate the time period of one cycle, reciprocate it, it will be frequency in Hertz, multiply it by 2π, now it is in rad/sec. The exact point is 1.5266 (not shown in the figure), hence the %M p is 52.66%.

The first peak can be viewed between 1.4 and 1.6. Theoretical values calculated are shown in figure-2. Time response of system-1, against unit step input, is as shown in figure-2,įrom figure -2 and definitions of Time response specifications such as delay time, rise time, peak time, etc. Figure: 1 – Calculated values in complex plane (s-plane) In figure-1, few of the above-calculated values are shown in a complex plane (s-plane), so you can understand, what the relation between them are. 0.8±j3.92 also called roots of characteristics equation. Poles of the transfer function (roots of the denominator): It is equal to the imaginary part of the poles of the transfer function. T p= 0.802 seconds Damped frequency of oscillations (ω d): The peak time is the time required for the response to reach the first peak of the overshoot. It is the percentage difference in the first peak of time response and steady-state output value. Percentage maximum peak overshoot (% M p): In this article formula and calculation of settling time is based on 2% tolerance band. In general, tolerance bands are 2% and 5%. It is the time required for the response to reach the steady state and stay within the specified tolerance bands around the final value. If you will compare the system-1 with standard form, you can find that damping ‘ζ’= 0.2 (damping is a unitless quantity), Natural frequency of oscillations ‘ω n’= 4 rad/sec. The standard form of a second-order transfer function is given by We will call it system-1 in the subsequent discussion. (Do you know why it is second order transfer function the reason is, the highest power of ‘s’ in the denominator is two). In this article we will explain you stability analysis of second-order control system and various terms related to time response such as damping (ζ), Settling time (t s), Rise time (t r), Percentage maximum peak overshoot (% M p), Peak time (t p), Natural frequency of oscillations (ω n), Damped frequency of oscillations (ω d) etc.ġ) Consider a second-order transfer function =.

Poles of the transfer function (roots of the denominator):.

Percentage maximum peak overshoot (% Mp):.

Closed loop transfer function