1. In The Name of Allah The Most Beneficent The Most Merciful

2. ECE 4545: Control Systems Lecture: Stability
Engr. Ijlal Haider
UoL, Lahore

3. Definitions of stability in various domains
Ecological stability, measure of the probability of a population returning quickly to a previous state, or not going extinct
Social stability, lack of civil unrest in a society
Quotes: “Every time I try to define a perfectly stable person, I am appalled by the dullness of that person.”
− J. D. Griffin

5. Stability
Examples of stable and unstable systems are the spring-mass and spring-mass-damper systems.
These two systems are shown next. The spring-mass system (Figure 1(a)) is unstable since if we pull the mass away from the equilibrium position and release, it is going to oscillate forever (assuming there is no air resistance). Therefore it will not go back to the equilibrium position as time passes.
On the other hand, the spring-mass-damper system (Figure 1(b)) is stable since for any initial position and velocity of the mass, the mass will go to the equilibrium position and stops there when it is left to move on its own.

6. Stability
Conceptually, a stable system is one for which the output is small in magnitude whenever the applied input is small in magnitude.
In other words, a stable system will not “blow up” when bounded inputs are applied to it.
Equivalently, a stable system’s output will always decay to zero when no input is applied to it at all.
However, we will need to express these ideas in a more precise definition.

14. Characteristic root locations criterion
A system is stable if all the poles of the transfer function have negative real parts.
Stability in the s-plane.
Control Systems

15. Stable system
A necessary and sufficient condition for a feedback system to be stable is that all the poles of the transfer function have negative real parts.
This means that all the poles are in the left-hand s-plane.
Control Systems

16. The Routh-Hurwitz rule
If any of the coefficients ai, i = 0,1,2,…,n-1
are zero or negative, the system is not stable.
It can be either unstable or neutrally stable.
Control Systems

17. Routh-Hurwitz Criterion
The Routh-Hurwitz criterion states that the number of roots of q(s) with positive real parts is equal to the number of changes in sign of the first column of the Routh array.
This criterion requires that there be no changes in sign in the first column for a stable system.
Is a necessary and sufficient criterion for the stability of linear systems.
Control Systems

18. Equivalent closed-loop transfer function
Control Systems

19. Initial layout for Routh table
Control Systems

20. Ordering the coefficients of the characteristic equation.
The R-H criterion is based on ordering the coefficients of the characteristic equation into an array
Further rows of the schedule are completed as :
Control Systems

21. The Routh-Hurwitz array
Further rows of the array
Control Systems

22. The algorithm for the entries in the array
Control Systems

23. Completed Routh table
Control Systems

24. Four distinct cases of the first column array
1. No element in the first column is zero.
2. There is a zero in the first column and in this row.
There is a zero in the first column and the other zero
in this row.
4. As in the previous case, but with repeated roots on the j-axis.
Control Systems

25. The Routh-Hurwitz Stability Criterion
Case One: No element in the first column is zero

26. The Routh-Hurwitz Stability Criterion
Case Two: Zeros in the first column while some elements of the row containing a zero in the first column are nonzero

27. The Routh-Hurwitz Stability Criterion
Case Three: Zeros in the first column, and the other elements of the row containing the zero are also zero

28. The auxiliary polynomial
The equation that immediately precedes the zero entry in the Routh array.
Control Systems

29. The Routh-Hurwitz Stability Criterion
Case Four: Repeated roots of the characteristic equation on the jw-axis.
With simple roots on the jw-axis, the system will have a marginally stable behavior. This is not the case if the roots are repeated. Repeated roots on the jw-axis will cause the system to be unstable. Unfortunately, the routh-array will fail to reveal this instability.

30. Absolute Stability
A closed loop feedback system which could be characterized as either stable or not stable.
Control Systems

31. Relative Stability
Further characterization of the degree of stability of a stable closed loop system.
Can be measured by the relative real part of each root or pair of roots.
Control Systems

32. Root r2 is relatively more stable than roots r1 and r1’.
If the system satisfies the R_H criterion and is absolutely stable we ca talk about the relative stability.Can also be defined in terms of the relative damping coefficients of each complex root pair, in terms of the speed of response and overshoot instead of settling time.
Root r2 is relatively more stable than roots r1 and r1’.
Control Systems

33. Axis shift Control Systems
There are roots on the shifted imaginary axis that can be obtained from the auxiliary polynomial.
Shifted variable sn= s+1
The correct magnitude to shift the vertical axis must be obtained on a trial -and -error basis.
Control Systems

34. Stability versus Parameter Range
Consider a feedback system such as:
The stability properties of this system are a function of the proportional feedback gain K. Determine the range of K over which the system is stable.
Control Systems

35. Stability versus Two Parameter Range
Consider a Proportional-Integral (PI) control such as:
Find the range of the controller gains so that the PI feedback system is stable.
Control Systems

36. Example 4 (cont’d)
The characteristic equation for the system is given by:
Control Systems

39. Thank You!