1. Unit 4 STABILITY ANALYSIS
Subject Name: Control Systems
Subject Code:10ES43
Prepared By: M.Brinda, Sreepriya Kurup, Robina Gujral
Department: ECE
Date:30/3/2015
Unit 4 STABILITY ANALYSIS

2. UNIT 4
STABILITY ANALYSIS

3. Necessary conditions for stability Routh Stability Criterion
Topics
Concepts of stability
Necessary conditions for stability
Routh Stability Criterion
Relative stability analysis
More on Routh Stability Criterion

4. The Concept of Stability
The concept of stability can be illustrated by a cone placed on a plane horizontal surface.
A necessary and sufficient condition for a feedback system to be stable is that all the poles of the system transfer function have negative real parts.
A system is considered marginally stable if only certain bounded inputs will result in a bounded output.

5. Stable system Response or output is predictable.
A system is said to be stable if for a bounded disturbing input signal, the output vanishes ultimately as t infinity.
A system is unstable if for a bounded disturbing input signal the output is of infinite amplitude or oscillatory.
For a bounded i/p, it produces unbounded o/p.
In the absence of i/p, o/p may not return to zero. It shows certain o/p without i/p.

6. STABLE SYSTEM

7. UNSTABLE SYSTEM

8. UNCONTROLLABLE RESPONSE

9. Definitions of stability
Bounded Input, Bounded Output (BIBO) Stability:
A system is said to be BIBO Stable if
For Bounded i/p, we have Bounded o/p; o/p – Controllable.
In the absence of i/p, o/p must tend to zero irrespective of initial conditions.
Relaxed system: A System in which initial conditions are zero.

10. Definitions of stability
Critically or Marginally Stable system:
for a bounded i/p, o/p oscillates with constant frequency and amplitude. Such oscillations are called Damped or sustained oscillations.
Conditionally Stable system:
o/p is bounded only for certain condition. If this condition is violated, o/p is unbounded. Stability depends on condition of parameter of the system.

11. CRITICALLY OR MARGINALLY STABLE SYSTEM

12. Definitions of stability
Zero input stability:
If the zero input response of the system subjected to finite initial conditions, reaches to zero as tinfinity, then the system is zero input stable.
Asymptotic Stability:
As magnitude of zero input response reaches zero as t approaches infinity, then zero input stability is also called asymptotic stability.
If in the absence of i/p, the o/p tends to zero or to the equilibrium state irrespective of initial conditions.

13. Definitions of stability
Absolutely Stable system:
If the system o/p is stable for all variations of its parameters then the system is called absolutely stable system.

14. Transfer Function
When order of the denominator polynomial is greater than the numerator polynomial the transfer function is said to be ‘proper’.
Otherwise ‘improper’

15. Stability of Control System
Roots of denominator polynomial of a transfer function are called ‘poles’.
And the roots of numerator polynomials of a transfer function are called ‘zeros’.

16. Stability of Control System
Poles of the system are represented by ‘x’ and zeros of the system are represented by ‘o’.
System order is always equal to number of poles of the transfer function.
Following transfer function represents nth order plant.

17. Stability of Control System
Poles is also defined as “it is the frequency at which system becomes infinite”. Hence the name pole where field is infinite.
And zero is the frequency at which system becomes 0.

18. Stability of Control System
Poles is also defined as “it is the frequency at which system becomes infinite”.
Like a magnetic pole or black hole.

19. Stability of Control Systems
The poles and zeros of the system are plotted in s-plane to check the stability of the system.
s-plane
LHP
RHP

20. Stability of Control Systems
If all the poles of the system lie in left half plane the system is said to be Stable.
If any of the poles lie in right half plane the system is said to be unstable.
If pole(s) lie on imaginary axis the system is said to be marginally stable.
s-plane
LHP
RHP
If all the poles

21. Stability of Control Systems
For example
Then the only pole of the system lie at
s-plane
LHP
RHP X -3

22. LOCATION OF ROOTS ON S PLANE

23. LOCATION OF ROOTS ON S PLANE

24. LOCATION OF ROOTS ON S PLANE

25. LOCATION OF ROOTS ON S PLANE

26. CONCLUSIONS BASED ON THE LOCATION OF ROOTS OF CHARACTERISTIC EQUATION
Roots-LHS – negative real parts – Response – Bounded- BIBO Stable.
Roots-RHS – Positive real parts –Response – Unbounded- Unstable.
Repeated roots on Imaginary axis –Response – Unbounded- unstable.
Single root at origin – Bounded –Unstable.
Repeated roots at origin –Unbounded, unstable.
Non repeated roots on imaginary axis or single pole at origin- Limitedly or marginally stable system.

27. All the co efficients –Positive => roots –LHS
Observations
All the co efficients –Positive => roots –LHS
If any co efficient is zero=> roots- Imaginary axis or RHS
If any co efficient is negative => atleast one root -RHS

28. Necessary conditions for stability
All the co efficients of a characteristic polynomial be positive.
If any co efficient is zero or negative, we can immediately say that the system is unstable.
But not sufficient condition
s3+ s2+2s+8 = (s+2) (s- 0.5 – 1.93j) (s j)
Co efficients –positive but roots –RHS
So s/m – Unstable.

29. Routh Hurwitz Criterion
Sufficient conditions for stability.
Hurwitz – investigated stability interms of determinants.
Routh – in terms of array formulation.
Routh Stability criterion:
The necessary and sufficient condition for stability is that all the elements in the first column of the routh array must be positive. If this condition is not met, the system is unstable and the no of sign changes in the elements of the first column of the routh array corresponds to the no of roots of characteristic equation in RHS of s plane.