1. Stability of a system can be determined by observing its time response curve, with stable systems having oscillations that die out quickly or reach steady state fast. 2. Different types of stability include bounded input bounded output stability, asymptotic stability, absolute stability, and relative stability. 3. A system is stable if all poles are in the left half of the s-plane, marginally stable if poles are on the imaginary axis, and unstable if any poles are in the right half plane.

1. STABILITY (control system)
BY-VAIBHAV JINDAL

2. Physical Meaning of Stability

3. Introduction
Stability of any system is a very important characteristic of
any system .
In Control System STABILITY cab be judged by observing the
time response curve. (Which is basically depends on
location of poles)
Generally for a stable system oscillations must die out as
early as possible or steady state should be reached fast. (in
time response curve)

4. Time Response Curve

5. S-Plane and Time Response Curve
0<ζ<1 Under Damped System
ζ =0 Undamped System
ζ=1 Critical Damped
ζ>1 Over Damped System

6. Stability
Bounded Input, Bounded Output: Output must be bounded for
bounded input.
Asymptotic Stability: If system input is remove from the system, then
output of system is reduced to zero.
absolute Stability: A system is stable for all values of system
parameters for bounded output. (Define in terms of location of poles)
[Routh Hurwitz, Rout locus & Nyquist Plot]
Relative Stability: This is a quantitative measure of how fast system
oscillation die out with time and how fast steady state reached. (Define
in term of Damping Ratio, Gain Margin and Phase Margin)
System having poles away from the imaginary axis of S-Plane, in
negative direction, has higher stability.
[Bode Plot & Nyquist Plot]

7. Conclusion I
1. For a stable system, all roots of characteristic equation (Poles of the
system) must lie in negative half of s plane.
2. If roots lie on imaginary axis, system is called marginally stable or
limitedly stable.
3. If roots lie on the positive half side of s-plane, system if un-stable.
4. If there are repeated poles or roots on imaginary axis, then also
system is un-stable.
5. If poles moves away from imaginary axis towards the left of s-plane
the relative stability of system is improves.
Im
Re
Stable Un-Stable

8. System T(s)=
𝐶(𝑠)
𝑅(𝑠)
At least one of the
system pole is in
right half of s-plane
At least some of the system poles
are on the imaginary axis, the rest
being in left half of s-plane
All the system poles
are inside the left
half of the s-plane
UNSTABLE ASYMPTOTICALLY
STABLE
At least one of the system poles on the imaginary
axis is present in the form of a pole in R(s)
SYSTEM EXHIBITS INSTABILITY DUE TO RESONANCE
At least some of the
system poles are repeated
on the imaginary axis.
UNSTABLE
All poles on imaginary axis are
simple and none of these
poles are present in input R(s)
NON-ASYMPTOTICALLY STABLE
MARGINALLY STABLE

9. Routh Hurwitz Stability
For a stable system all elements of first column in Routh table should have same
sign(either “+” or “-”).
“If Characteristics Equation contain only even power of S “ or “elements of any odd
complete row of Routh table is zero” represents system will contain at least one pair
of poles on imaginary axis (System has undamped natural frequency), system is
marginally stable.
Note: Routh is only applicable for closed loop system and for implementation of
Routh table coefficient of characteristics equation. Should real

10. Bode Plot
Gain Margin Phase Margin Stability
ω 𝑔 < ω 𝑝 +ve dB +ve Stable
ω 𝑔 = ω 𝑝 0 0 Marginally Stable
ω 𝑔 > ω 𝑝 -ve dB -ve Un-Stable
Gain Margin= -[20 log|G(jω 𝑝).H(jω 𝑝)|]
Phase Margin=180+ ∠ G(jω 𝑔).H(jω 𝑔)
ω 𝑔= Gain Cross over frequency
ω 𝑝= Phase Cross over frequency
Bode plot is draw on Semi-log graph for
open and closed loop system both.

11. Nyquist Plot
Relative Stability
Gain =
1
|G(jω 𝑝).H(jω 𝑝)|
Gain Margin(dB)= -[20 log|G(jω 𝑝).H(jω 𝑝)|]
Phase Margin=180+ ∠ G(jω 𝑔).H(jω 𝑔)
ω 𝑔= Gain Cross over frequency
ω 𝑝= Phase Cross over frequency

12. Nyquist Plot
Relation between Open and Close Loop System
System Zeros Poles
Open System G(s)H(s)=
𝑺+𝟏
𝑺+𝟓
-1 -5
Close Loop
System
(-ve Feedback)
G(s)H(s)
1 + G(s)H(s)
=
𝑆 + 1
2𝑆 + 6
-1 -3
Characteristics
Equation
1 + G(s)H(s)=
2𝑆+6
𝑆+5 -3 -5

13. Nyquist Plot
Absolute Stability
𝑁 𝑖𝑠 𝑒𝑛𝑐𝑖𝑟𝑐𝑙𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝑝𝑜𝑖𝑛𝑡
(-1+j0).
N=-1
[For clockwise N is –ve and
anti-clockwise N is +ve]

14. Nyquist Plot
𝑁 = 𝑃+ − 𝑍+
Where,
N=Total no. of encirclement of point (-1+j0). [For clockwise N is –ve and
anti-clockwise N is +ve]
𝑃+=Positive poles of characteristic equation(Poles of open system) [Find
from given transfer function].
𝑍+=Positive zero of characteristics equation (Poles of closed loop
system) [Find from above equation]

15. K=3, Stable K=7, Unstable

16. The Relative Stability of Feedback CS
The verification of stability using the Routh-Hurwitz criterion provides
only a partial answer to the question of stability----whether the system
is absolutely stable.
In practice, it is desired to determine the relative stability.
- The relative stability of a system can be defined as the property that is
measured by the relative real part of each root or pair of roots.

17. THANK’S
MERCI