S4-3.3 Stability & Root Locus

Work through these

• Routh-Hurwitz criterion including special cases

  Routh-Hurwitz answers whether a system is stable without ever solving for the roots, using a table built from the characteristic polynomial. The special cases are the ones examinations reach for, so work them deliberately.

• Root locus construction rules

  The root locus draws how the closed-loop poles move as the gain is turned, and its construction rules let you sketch those paths by hand in a minute. The branches begin at the open-loop poles and end at the zeros, and the sketch is for insight rather than precision.

• Analysis of typical systems using root locus

  Applying the sketching rules to the standard systems until the shapes become familiar. Recognising the shape of a locus tells you what a gain change will do before any calculation.

• Effect of root locations on system response

  Where the poles sit decides how the system behaves: well to the left is fast and calm, near the imaginary axis is oscillatory, and to the right is unstable. This mapping is the mental model the rest of the subject hangs on.

• Lab: plot a root locus in software and confirm your hand sketch

  Plot the locus in software and lay it against your hand sketch. The comparison is what tells you whether you have learned the rules or memorised a picture.