Stability from Open Loop Bode Plot
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## 1 Introduction to Deterimining Stability from the Open Loop Bode Plot

Determining stability from the open loop bode plot is possible for most systems. The system conditions are such that the open loop magnitude can cross magnitude = 1 (the open loop crossover at 0 dB) only once. The criteria are based on an examination of the root locus and bode plot. However, the article reduces the analysis into how you determine stability from the open loop bode plot.

## 2 Open Loop Bode Plot Stability Criteria[1]

Another way to determine closed loop stability is to evaluate the frequency response of the open loop transfer function KG(jω)and then perform a test on that response.

Suppose that we have a system where in stability results if K >α where

K is the control gain and
α is the neutral stability point (i.e. the closed loop roots are on the imaginary axis)

At the point of neutral stability the following root locus conditions hold

 $LaTeX: \left | KG\left(j\omega \right) \right |=1$ (6.22a)

 $LaTeX: \angle G\left(j\omega \right)=180^\circ$ (6.22b)

A Bode plot of a system that is neutrally stable will satisfy the conditions of Eqn. (6.22a & b). Does increasing K (from the neutral stability point) increase or decrease the system's stability? At the frequency ω where the phase {{Eqn eqnValue=$LaTeX: \angle G\left(j\omega \right)=-180^\circ$, | number=|}} the magnitude

 $LaTeX: \left | KG\left(j\omega \right) \right | < 1.0$ for stable values of K

and

 $LaTeX: \left | KG\left(j\omega \right) \right | > 1.0$ for unstable values of K

The following stability criteria are only valide when the open loop gain crosses magnitude = 1 only once.

### 2.1 For systems where an increasing K leads to instability

Stability conditions are

 $LaTeX: \left | KG\left(j\omega \right) \right | < 1$ at $LaTeX: \angle G\left(j\omega \right)=-180^\circ$

### 2.2 For systems where an increasing K leads from instability to stability

Stability conditions are

 $LaTeX: \left | KG\left(j\omega \right) \right | > 1$ at $LaTeX: \angle G\left(j\omega \right)=-180^\circ$

## 3 References

• Franklin, G. F., Emami-Naeini, A., and Powell, J. D. 1993 Feedback Control of Dynamic Systems. 3rd. Addison-Wesley Longman Publishing Co., Inc. ISBN 0201527472

### 3.1 Notes

1. Franklin, pp. 359-361