1. Lec 6. Second Order Systems
Step response of standard 2nd order systems
Performance specifications
Reading: 5.3, 5.4
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2. 2nd Order Systems General second order system transfer function:
Two poles p1,p2 of H(s) are the two roots of denominator polynomial:
The locations of p1 and p2 have important implication in system responses.

3. Motivating Example Poles: By PFD: Zeros:
H(s) is the parallel connection of two 1st order systems.
Response of H(s) is the sum of the two 1st order system responses +

4. Motivating Example (cont.)
Step response:
Each pole p contributes a transient term
in the response
Transient will settle down if all poles are on the left half plane
Convergence to final value no longer monotone (overshoot)

5. Another Motivating Example
p1=-1+j2, p2=-1-j2
Poles:
Step response of H(s) is:
Oscillation in the step response

6. Standard 2nd Order Systems
Standard form of second order systems:
Represents only a special family of second order systems
Numerator polynomial is a constant
Denominator polynomial is 2nd order with positive coefficients
H(0)=1 (unit DC gain)
Standard form is completely characterized by two parameters , n
n: undamped natural frequency (n>0)
: damping ratio (>0)
Ex:

7. Example of 2nd Order Systems

8. Poles of Standard 2nd Order Systems
has two poles
Underdamped case (0<<1):
Two complex conjugate poles:
Critically damped case (=1):
Two identical real poles:
Overdamped case (>1):
Two distinct real poles:

9. Underdamped Case (0<<1)
has two complex poles
Undamped natural frequency n
is the distance of poles to 0
Damped natural frequency
Damping ratio  determines the angle 
As  increases from 0 to 1,  changes from 0 to 90 degree

10. Underdamped Case (0<<1)
has two complex poles
where
“dampled natural frequency”
Angle
As  increases from 0 to 1:
 increases from 0 to 90 degree
Undampled natural frequency n determines the distance of poles to origin
Damping ratio  determines the angle 

11. Some Typical 

12. Step Response: Underdampled Case
Step response of
steady state response
transient response
Pole p=-+jd contributes the term ept in the transient response
Transient responses are damped oscillations with frequency d, whose amplitude decay (or grow) exponentially according to e- t

13. Step Response: Underdampled Case
Step response of
steady state response
transient response
Intuitively, the pole p=-+jd contributes the term ept in the transient response
Transient responses are damped oscillation with frequency d, and amplitude will decay exponentially according to e- t

14. Step Responses: Underdamped Case (n is constant)

15. Summary of Underdamped Case
Overshoot and oscillation in the step response
(Negative of) real part =n of the poles determines the transient amplitude decaying rate
Imaginary part d of the poles determines the transient oscillation frequency
For a given undamped natural frequency n, as damping ratio  increases
 larger, poles more to the left, hence transient dies off faster
Transient oscillation frequency d decreases
Overshoot decreases
What if we fix  and increase n?

16. Critically Damped Case (=1)
Transfer function
has two identical real poles
Step response is
steady state response
transient response

17. Overdamped Case (>1)
Transfer function
has two distinct real poles
Step response is
steady state response
transient response

18. Step Responses for Different 

19. Remarks An overdamped system is sluggish in responding to inputs.
Among the systems responding without oscillation, a critically damped system exhibits the fastest response.
Underdamped systems with  between 0.5 and 0.8 get close to the final value more rapidly than critically dampled or overdampled system, without incurring too large an overshoot
Impulse response and ramp response of 2nd order systems can be obtained from the step responses by differentiation or integration.

20. Time Specifications
td: delay time, time for s(t) to reach half of s(1)
tr: rise time, time for s(t) to first reach s(1)
tp: peak time, time for s(t) to reach first peak
Mp: maximum overshoot
ts: settling time, time for s(t) to settle within a range (2% or 5%) of s(1)
A typical step response s(t)

21. Remarks Not all quantities are defined for certain step responses
Step response of a 2nd order critically damped or overdamped system
Step response of a 1st order system
Not defined: tr, tp, Mp

22. Time Specifications of 2nd Order Systems
A standard 2nd order system
is completely specified by the parameters  and n
What are the time specifications in terms of  and n?
Focus on the underdamped case (0<<1) as tr, tp, Mp are not defined for critically damped or overdamped systems

23. Time Specifications of Underdamped Systems
(0<<1)
Step response:
Numerical solution of a transcendental equation.
where
Delay time td: smallest positive solution of equation
Rise time tr: smallest positive solution of equation
Recall that Hence tr is smaller (faster rise) for larger n)

24. Step Responses for Fixed n and Different 
For fixed n, rise time tr is smallest when =0, and approaches 1 as  approaches 1

25. Peak time tp and Maximum Overshoot Mp
Peak time tp: smallest positive solution of equation
(tp decreases with n)
Maximum overshoot Mp:
“The smaller the damping ratio, the larger the maximum overshoot”

26. Step Responses for Fixed n and Different 
For fixed n, peak time tp increases to 1 as  increases from 0 to 1

27. Settling Time ts
Settling time ts: the smallest time ts such that |s(t)-1|< for all t>ts
Analytic expression of ts is difficult to obtain.
Idea: approximate s(t) by its envelope:
Settling time ts when =5%:
Settling time ts when =2%:
“The more to the left the poles are, the smaller the settling time”

28. Effect of Pole Locations on Responses of 2nd Order Systems
equa-z
equa-n
stable
unstable