We define two physically meaningful specifications for second-order systems: Natural Frequency (Wn) and Damping Ratio (ζ).
Introduction
Now that we have become familiar with second-order systems and their responses, we generalize the discussion and establish quantitative specifications defined in such a way that the response of a second-order system can be described to a designer without the need for sketching the response. We define two physically meaningful specifications for second-order systems. These quantities can be used to describe the characteristics of the second-order transient response just as time constants describe the first-order system response.
Natural Frequency, Wn
The natural frequency of a second-order system is the frequency of oscillation of the system without damping. For example, the frequency of oscillation of a series RLC circuit with the resistance shorted would be the natural frequency.
Damping Ratio, 
We have already seen that a second-order system’s underdamped step response is characterized by damped oscillations. Our definition is derived from the need to quantitatively describe this damped oscillations regardless of the time scale.Thus, a system whose transient response goes through three cycles in a millisecond before reaching the steady state would have the same measure as a system that went through three cycles in a millennium before reaching the steady state. For example, the underdamped curve in Figure 4.10 has an associated measure that defines its shape. This measure remains the same even if we change the time base from seconds to microseconds or to millennia.
A viable definition for this quantity is one that compares the exponential decay frequency of the envelope to the natural frequency. This ratio is constant regardless of the time scale of the response. Also, the reciprocal, which is proportional to the ratio of the natural period to the exponential time constant, remains the same regardless of the time base.
We define the damping ratio, , to be:
Consider the general system:
Without damping, the poles would be on the jw-axis, and the response would be an undamped sinusoid. For the poles to be purely imaginary, a = 0. Hence:
Assuming an underdamped system, the complex poles have a real part, 
, equal to -a/2. The magnitude of this value is then the exponential decay frequency described in Section 4.4. Hence,
from which
Our general second-order transfer function finally looks like this:
Now that we have defined and Wn, let us relate these quantities to the pole location. Solving for the poles of the transfer function in Eq. (4.22) yields:
From Eq. (4.24) we see that the various cases of second-order response:
Underdamped Second-Order System
Now that we have generalized the second-order transfer function in terms of and Wn, let us analyze the step response of an underdamped second-order system.
Not only will this response be found in terms of and Wn, but more specifications
indigenous to the underdamped case will be defined. The underdamped second order system, a common model for physical problems, displays unique behavior that
must be itemized; a detailed description of the underdamped response is necessary
for both analysis and design. Our first objective is to define transient specifications
associated with underdamped responses. Next we relate these specifications to the
pole location, drawing an association between pole location and the form of the
underdamped second-order response. Finally, we tie the pole location to system
parameters, thus closing the loop: Desired response generates required system
components.
Let us begin by finding the step response for the general second-order system of Eq. (4.22). The transform of the response, C(s), is the transform of the input times the transfer function, or:
where it is assumed that < 1 (the underdamped case). Expanding by partial fractions, using the methods described, yields:
Taking the inverse Laplace transform, which is left as an exercise for the student, produces:
where:
A plot of this response appears in Figure 4.13 for various values of , plotted along a time axis normalized to the natural frequency.
We now see the relationship between the value of and the type of response obtained: The lower the value of , the more oscillatory the response.
The natural frequency is a time-axis scale factor and does not affect the nature of the response other than to scale it in time.
Other parameters associated with the underdamped response are rise time, peak time, percent overshoot, and settling time. These specifications are defined as follows (see also Figure 4.14):
- Rise time, Tr. The time required for the waveform to go from 0.1 of the final value to 0.9 of the final value.
- Peak time, TP. The time required to reach the first, or maximum, peak.
- Percent overshoot, %OS. The amount that the waveform overshoots the steady-state, or final value at the peak time, expressed as a percentage of the steady-state value.
- Settling time, Ts. The time required for the transient’s damped oscillations to reach and stay within 2% of the steady-state value.
All definitions are also valid for systems of order higher than 2, although analytical expressions for these parameters cannot be found unless the response of the higher-order system can be approximated as a second-order system.
Rise time, peak time, and settling time yield information about the speed of the transient response. This information can help a designer determine if the speed and the nature of the response do or do not degrade the performance of the system.
For example, the speed of an entire computer system depends on the time it takes for a hard drive head to reach steady state and read data; passenger comfort depends in part on the suspension system of a car and the number of oscillations it goes through after hitting a bump.
Evaluation of Tp
Tp is found by differentiating c(t) in Eq. (4.28) and finding the first zero crossing after t = 0.
Evaluation of %OS.
From Figure 4.14 the percent overshoot, %OS, is given by:
Evaluation of Ts
In order to find the settling time, we must find the time for which c(t) in Eq. (4.28) reaches and stays within ₎±2% of the steady-state value, C final.
Evaluation of Tr
A precise analytical relationship between rise time and damping ratio cannot be found. However, using a computer and Eq. (4.28), the rise time can be found. Let us look at an example.
We now have expressions that relate peak time, percent overshoot, and settling time to the natural frequency and the damping ratio. Now let us relate these quantities to the location of the poles that generate these characteristics. The pole plot for a general, underdamped second-order system is reproduced in Figure 4.17.
Now, comparing Eqs. (4.34) and (4.42) with the pole location, we evaluate peak time and settling time in terms of the pole location. Thus:
where
is the imaginary part of the pole and is called the damped frequency of oscillation, and 
is the magnitude of the real part of the pole and is the exponential damping frequency part.
At this point, we can understand the significance of Figure 4.18 by examining the actual step response of comparative systems. Depicted in Figure 4.19(a) are the step responses as the poles are moved in a vertical direction, keeping the real part the same. As the poles move in a vertical direction, the frequency increases, but the envelope remains the same since the real part of the pole is not changing.
Let us move the poles to the right or left. Since the imaginary part is now constant, movement of the poles yields the responses of Figure 4.19(b). Here the frequency is constant over the range of variation of the real part. As the poles move to the left, the response damps out more rapidly.
Moving the poles along a constant radial line yields the responses shown in Figure 4.19(c). Here the percent overshoot remains the same. Notice also that the responses look exactly alike, except for their speed. The farther the poles are from the origin, the more rapid the response.
Sources:
- Control Systems Engineering, Nise
- Sistemas de Control Automatico Benjamin C Kuo
- Modern_Control_Engineering, Ogata 4t
Literature Review by: Larry Francis Obando – Technical Specialist
Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, Caracas.
Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, Valle de Sartenejas.
Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.
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Sources:
Control Systems Engineering, Norman Nise
-
- Introduction Chapter 4 pp 162 (162)
- Poles and Zeros 4.1 pp 162 –
- First Order System 4.3 pp 165-168
- Second Order System 4.4 pp 168-177
- Underdamped Second-Order System 4.6 pp 177-186
- Modern_Control_Engineering__4t
- Introduction Chapter 5 pp 219 (232)
- First Order Systems 221 (234)-224
- Second Order System pp 224 (237)-234
Literature Review, Martes 14 noviembre 2017, 05:07 am – Caracas, Quito, Guayaquil.
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Figure 7
Figure 7
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