1. Salman Bin Abdulaziz University
EE3511: Automatic Control Systems
Concept of Stability
Dr. Ahmed Nassef
EE3511_L10
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2. Salman Bin Abdulaziz University
Learning Objective
To be able to
Define BIBO stability
Explain the connection between stability and pole locations
Recognize the importance of stability
Classify a system as being stable, unstable or marginally stable
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3. Salman Bin Abdulaziz University
Stability Concept
Stable Marginally stable unstable
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4. Salman Bin Abdulaziz University
Introduction
Stability is the most important issue for control systems
If a system is unstable then it is not useful (in general)
If a system is unstable, our first concern in controller design is to stabilize it.
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Introduction
R (s)
Y(s)
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6. Salman Bin Abdulaziz University
BIBO stability
A system is BIBO stable if
The response to any bounded input is bounded
(Bounded Input Bounded Output stability)
BIBO
Stable
System
Bounded
Output
Bounded
input
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7. BIBO stability of Linear Systems
For linear systems, BIBO stability depends on the location of the poles of the system’s transfer function
Linear Systems
If ALL poles of the transfer function of a system have negative real parts then the system is BIBO stable.
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8. Poles Location and the response
Systems in (a) and (c) are stable, however systems in (b) and (d) are unstable.
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A stable System
r (t)
y (t)
Ch. Eqn is: (s+1)(s+3)=0, So s = -1 and s = -3
This is a BIBO stable system
If the input r(t) is bounded then y(t) is bounded
This is true for all bounded inputs r(t)
Unit step
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10. Salman Bin Abdulaziz University
Unstable System
r (t)
y (t)
Ch. Eqn is: (s+1)(s-4)=0, So s = -1 and s = 4
This is an unstable system
There are some bounded functions that results in unbounded outputs
Unit step
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Stable Systems
G(s) is BIBO stable if ALL poles have negative real part, i.e ALL poles are in LEFT hand side.
stable x
unstable x x x
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12. Salman Bin Abdulaziz University
Marginal Stability
G(s) is marginally stable if
It has distinct poles with zero real part i.e the poles are on the imaginary axis
It has no pole with positive real part.
stable
unstable x x x x x
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13. Salman Bin Abdulaziz University
Unstable Systems
G(s) is unstable if one of the following is true:
It has at least one pole with positive real part.
It has repeated poles with zero real part.
stable
unstable x x x x x x x x x
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14. Are these system stable?
Unstable
Marginally stable
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15. Are these system stable?
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16. Are these system stable?
Unstable
Marginally
Stable
Unstable
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17. Why do we study stability?
In general, systems that are unstable are not useful
Many physical systems are unstable without feedback
We need to design controllers to stabilize them.
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18. Salman Bin Abdulaziz University
Open loop System
u (t):step
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19. Relative stability_ single pole case
More stable
More unstable X X X X
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Is this system stable?
This system consists of two blocks that are unstable
Is the feedback system stable?
Yes, the feedback system is stable? But this is not always true.
E(s)
Y(s) + _
Ch. Eq. = 1 + GH = 0 +
E(s)
Y(s) _
Y(s)
Poles :-0.5 ± j0.866
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21. How Do We Check Stability?
Nyquist Theorem
Routh Hurwitz Test
Find poles
of closed loop
system and check at least one has positive
real part
Lyapunov Theorems
State Space Tests
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22. Salman Bin Abdulaziz University
Stability Test
Poles of a system are the roots of the characteristic equation.
If any coefficient of the characteristic polynomial is –ve then there is at least one root with (positive) real part, and then the system is unstable.
If all the pole are in the LHS of the S-plane, then all the coefficients must be > 0.
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23. Routh Hurwitz Stability Test
For higher order systems, it is too difficult to solve the characteristic equation.
Some tricks allows us to determine stability without determining the values of the poles.
Determining stability is reduced to determining the signs of the roots of the characteristic polynomial.
Routh-Hurwitz stability test is used to count the RHS poles, without solving the Ch. Eqn.
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24. Routh Array (Routh-Hurwitz table)
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25. Salman Bin Abdulaziz University
Stability test
The number of positive roots = The number of sign changes in the first column
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26. Example of Stability test
First two rows are generated from the polynomial coefficients
No sign changes, so the system is STABLE
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27. Example of Stability test
First two rows are generated from the polynomial coefficients
2 sign changes  roots with positive real part. UNSTABLE system
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28. Example of Stability test
The second order polynomial
has no roots with positive real part if and only if all coefficients have same sign.
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29. Special cases (a row begins with zero element)
A row begins with zero element and some of the elements in that row are not zero.
A zero in the first column indicates that there is a pair of purely imaginary roots.
Replace zero by ε (a small positive number)
Continue computation of the table by taking the limit as ε = 0.
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30. Special cases (a row begins with zero element)
First element in a row is zero but some other elements in the row are nonzero.
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31. Salman Bin Abdulaziz University
Special case
Two sign changes, So two poles are in the right half plane, hence the system is UNSTABLE.
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32. Special case (All-zero row)
Row of zeros
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Special cases
All-zero row
Obtain the auxiliary equation and use its derivative to replace the all-zero row.
Continue computation of the table
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Special case_ zero row
All-zero row
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35. Salman Bin Abdulaziz University
Auxiliary Equations
The order of the auxiliary polynomial is equal to the number of roots symmetrically located about the origin
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36. Salman Bin Abdulaziz University
Auxiliary Polynomial
Replace the zero row by the derivative of the auxiliary polynomial
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37. The auxiliary equation
The auxiliary polynomial comes before a zero row.
The auxiliary polynomial is a factor of the original polynomial
Roots of the auxiliary polynomial are roots of the original polynomial
The order of the auxiliary polynomial is even
The order of the auxiliary polynomial indicates the number of symmetrically located poles about the origin.
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38. When does Zero Row OccurX X X X X X X X
- Second Order auxiliary polynomial
- Two pure imaginary roots symmetrically located about origin
- Fourth Order auxiliary polynomial
- four complex roots symmetrically located about origin
- Second Order auxiliary polynomial
- Two real roots symmetrically located about origin
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39. When does Zero Row Occur
- Fourth Order auxiliary polynomial
- Four real roots symmetrically located about origin X X X X X X
- Fourth Order auxiliary polynomial
- Two pure imaginary roots symmetrically located about origin X X
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40. Robust stability Using Routh Hurwitz
E(s)
Y(s) _
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41. Designing for Stability
Consider the following feedback control system:
What are the values of K that produce stable systems?
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42. Designing for Stability
Solution
Compute the transfer function
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43. Designing for Stability
Apply Routh-Hurwitz test to
for all positive coefcients: K >0
Routh Array:
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44. Designing for Stability
Routh Array:
To have all positive elements in column 1, we must have 0 < K < 1.
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45. Stability and Zero Position
For what values of a is this system stable?
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46. Stability and Zero Position
Solution
Compute the transfer function
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47. Stability and Zero Position
Apply Routh-Hurwitz test to
All coefficients must be positive, so a > 0.
Routh Array:
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48. Stability and Zero Position
Routh Array:
Thus, if 0 < a < , then all the entries in the first column will be positive, indicating that the system is stable.
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