5 chapter 5 stability exploration methods

5 Stability Exploration Methods  2011 – Vo Tuong Quan Poles and Zeros Method Zeros and Poles plot: a graph which represents the position of poles and zeros in the complex s-plane...

SYSTEM DYNAMICS & CONTROL

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Stability Exploration Methods

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Examples

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Stability Exploration Methods

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Poles and Zeros Method

Zeros and Poles plot: a graph which represents the position

of poles and zeros in the complex s-plane

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Stability Exploration Methods

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Poles and Zeros Method

The stability of a system depends on the locations of Its poles:

- If all the poles of the system lie in the left-half s-plan then the system is stable

- If any of the poles of the systems lie in the right-half s-plane then the system is unstable

- If some of the poles of the system lie in the imaginary axis and the others lie in the left-half s-plane then the system is at the stability boundary

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Stability Exploration Methods

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Routh Method

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Stability Exploration Methods

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Routh Method

table have positive signs

The number of sign changes in the first column of the Routh table is equal the number of roots lying in the right-half s-plane

parameters

in the first column are

positive

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Stability Exploration Methods

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Routh Method

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Stability Exploration Methods

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Routh Method

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Stability Exploration Methods

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Routh Method

Find the codition of K for the following system to be stable

The characteristic equation of the system

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Stability Exploration Methods

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Routh Method

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Stability Exploration Methods

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Routh Method

Why ??

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Stability Exploration Methods

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Stability Exploration Methods

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Stability Exploration Methods

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Stability Exploration Methods

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Roots Locus Method – Using Matlab - Example

Consider the negative feedback system shown in Figure Plot root locus

0.12 0.24

0.38 0.5

0.64 0.78

0.88 0.97

1 2 3 4 5 6

1 2 3 4 5 6

0.12 0.24

0.38 0.5

0.64 0.78

0.88 0.97

6 Root-Locus Plot of G(s) = K/[s(s + 0.5)(s2 + 0.6s + 10)]

6 Root-Locus Plot of G(s) = K/[s(s + 0.5)(s2 + 0.6s + 10)]

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Stability Exploration Methods

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Roots Locus Method – Using Matlab - Example

Consider the negative feedback system shown in Figure Plot root locus

text(1.0, 0.55,'K = 12')

text(1.0,3.0,'K = 73')

text(1.0,4.15,'K = 154')

0.16 0.34

0.5 0.64

0.76 0.86

0.94 0.985

0.16 0.34

0.5 0.64

0.76 0.86

0.94 0.985

1 2

3 4

5 6