- Second Order System Definition: A second order control system is defined by the power of ‘s’ in the transfer function’s denominator, reflecting the system’s complexity and behavior.
- Step Response Analysis: Analyzing the step response of such systems helps in understanding how they react to sudden changes in input.
- Damping Ratio Impact: The damping ratio (ζ) critically influences whether the system’s response will oscillate, critically damp, or overdamp.
- Settling Time Equation: The settling time equation is vital as it defines the time taken for the system to stabilize within a specific range, crucial for ensuring performance stability.
- Practical Examples Provided: Worked examples illustrate how to apply theoretical formulas to real-world scenarios, enhancing understanding of system behaviors.
The time response of a second-order control system is the output c(t) after a known input, most often a unit step. The order itself is the highest power of ‘s’ in the denominator of its transfer function.
A control system is second order when that highest power of s is 2. Its unit-step shape is then set by the damping ratio ζ and the natural frequency ωn.
The general expression of the transfer function of a second order control system is given as
ζ is the damping ratio and ωn is the undamped natural frequency. Together they set the poles and the shape of c(t).
Rearranging the formula above, the output of the system is given as
The rest of the page finds that unit-step response in the s-domain and then takes the inverse Laplace transform to reach c(t).
Step Response of Second Order System
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 is e(t) = r(t) – c(t), so
The error in the signal shows an oscillating pattern with a magnitude that decreases exponentially, observed when the damping ratio ζ is less than 1.
The frequency of the oscillation is ωd and the time constant of exponential decay is 1/ζωn.
ωd is the damped frequency of the oscillation and ωn is the undamped natural frequency. ζ scales the real-part decay, so it is called the damping ratio.
The shape of c(t) changes with ζ. The cases below are ζ = 0, ζ = 1 and ζ > 1.
When damping ratio is zero, we can rewrite the above expression of output signal as
That expression has no decaying exponential, so the unit-step response is undamped when the damping ratio is zero.
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.
That output has no oscillating term for a unit-step input. The response is therefore called critically damped.
Now take a unit-step input when the damping ratio is greater than one.
The inverse Laplace transform of both sides is
That expression has two time constants.
When ζ is much greater than one, the faster time constant can be dropped and the response reduces to
Figure 6.4.5 of the page 139 of the book automatic control system by Hasan.
Critical Damping Time Response of Control System
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.
When ζ (the damping ratio) is less than one, the oscillation decays exponentially with time constant 1/ζωn. That case is underdamped.
When ζ is greater than one, the unit-step response has no oscillating term.
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.
The ratio of time constant of critical damping to that of actual damping is known as damping ratio. As the time constant of time response of control system is 1/ζωn when ζ≠ 1 and time constant is 1/ωn when ζ = 1.
Second Order System Transfer Function
The general equation for the transfer function of a second order control system is given as
If the denominator of the expression is zero,
These two roots of the equation or these two values of s represent the poles of the transfer function of that system. The real part of the roots represents the damping and imaginary part represents damped frequency of the response.
The location of the roots of the characteristics equation for various values of ζ keeping ωn fixed and the corresponding time response for a second order control system is shown in the figure below.
Figure 8.4.7 page 140 of the book automatic control system by Hasan.
Transient response specifications of second-order control system.
The performance of the control system can be expressed in the term of transient response to a unit step input function because it is easy to generate.
Let us consider a second-order control system in which a unit step input signal is given and it is also considered that the system is initially at rest. That is all initial conditions of the system are zero. The time response characteristics of the system at under damped condition is drawn below.
Figure 2.17 page 92 of the book automatic control system by Hasan.
Several common terms define the transient response characteristics, including:
- Delay time (td) is the time required to reach at 50% of its final value by a time response signal during its first cycle of oscillation.
- Rise time (tr) is the time required to reach at final value by a under damped time response signal during its first cycle of oscillation. If the signal is over damped, then rise time is counted as the time required by the response to rise from 10% to 90% of its final value.
- Peak time (tp) is simply the time required by response to reach its first peak i.e. the peak of first cycle of oscillation, or first overshoot.
- Maximum overshoot (Mp) is straight way difference between the magnitude of the highest peak of time response and magnitude of its steady state. Maximum overshoot is expressed in term of percentage of steady-state value of the response. As the first peak of response is normally maximum in magnitude, maximum overshoot is simply normalized difference between first peak and steady-state value of a response.
- Settling time (ts) is the time required for a response to become steady. It is defined as the time required by the response to reach and steady within specified range of 2 % to 5 % of its final value.
- Steady-state error (e ss ) is the difference between actual output and desired output at the infinite range of time.
Rise Time Formula
The underdamped second-order unit-step response is
At rise time the output first reaches the final value, so c(t) = 1 and
Peak Time Formula
As per definition at the peak time, the response curve reaches to its maximum value. Hence at that point,
The maximum overshoot occurs at n = 1.
Maximum Overshoot Formula
If we put the expression of peak time in the expression of output response c(t), we get,
Settling Time Formula
Settling time is the time after which the response stays inside a stated band around the final value. The 2% band is the usual teaching choice, so the envelope is then about 98% of the final value.
That 2% envelope is about four time constants. The time constant of this second-order system is 1/ζ ωn, so the 2% settling time is
