Stable Second Order System:

Standard Form:

Further, consider the case where

( 0 ) = 0
Then the characteristic equation is :

whose roots are given by:

and
Case 1    Both roots are real and negative ( and thus stable)

Using initial condtions,

or
Case 2:   Repeated real roots

Using initial conditions,

or

Case 3:  Roots are complex conjugates
and
where,
and
Thus,

Using initial conditions,

Alternatively, we can use the sin of sums formula,
comparing right hand side with,
then
and
from which,

so, an alternative form for x(t) is given by

where,         is the frequency
and         is the phase shift
Case 4: z=0   Pure imaginary roots
Case 5: z<0  Unstable roots

Derivation of Overshoot, Decay Ratio and Frequency (Underdamped Systems) :

a) Overshoot:
First, calculate the time when the peaks occur. To do so, obtain dx/dt, and then find the value of t which makes it zero.

for upper peaks,

The overshoot occurs at the first upper peak, i.e.

but,

so the overshoot is given by

b) Decay ratio
Compare the value at the first peak and the second upper peak. The second upper peak occurs at tc3
thus, the decay ratio is given by,
c) Frequency
The period of oscillation is the amount of time between two upper peaks

(in cycles per second or Hz)

(in rads per second)

Overshoot =
Decay Ratio=
Frequency= ( in rads/sec )

This page is maintained by Tomas B. Co (tbco@mtu.edu). Last revised 12/2//99.

Tomas B. Co
Associate Professor
Department of Chemical Engineering
Michigan Technological University
1400 Townsend Avenue
Houghton, MI 49931-1295