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In The Name of Allah The Most Beneficent The Most Merciful

In The Name of Allah The Most Beneficent The Most Merciful. ECE 4545: Control Systems Lecture: Stability. Engr. Ijlal Haider UoL, Lahore. Definitions of stability in various domains.

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In The Name of Allah The Most Beneficent The Most Merciful

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  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

  4. 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.

  5. 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.

  6. 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

  7. 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

  8. The Routh-Hurwitz rule • 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

  9. 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. Control Systems

  10. Equivalent closed-looptransfer function Control Systems

  11. Initial layout for Routh table Control Systems

  12. Ordering the coefficients of the characteristic equation. Control Systems

  13. The Routh-Hurwitz array Control Systems

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

  15. Completed Routh table Control Systems

  16. 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

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

  18. 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

  19. 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

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

  21. 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.

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

  23. 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

  24. Root r2 is relatively more stable than roots r1 and r1’. Control Systems

  25. Axis shift Control Systems

  26. 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

  27. 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

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

  29. Thank You!

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