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Root Locus

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Most Slides from the Root Locus Method by Brian Douglas

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Motivation for Root Locus

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Motivation for Root Locus

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Root Locus

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Definition: Root Locus

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Root Locus in MATLAB

  • The basic form for drawing the root locus

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  • In MATLAB, rlocus(G(s))
    • The same denominator system is

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Standard Form for Root Locus

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Graphical Representation of Closed Loop Poles

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Pole Locations for Closed Loop

  • So why should we care about this?

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  • Now that we understand how pole locations affect the system

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How to Draw Root Locus

  • Question: how do we draw root locus ?
    • for more complex system and
    • without calculating poles

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  • We will be able to make a rapid sketch of the root locus for higher-order systems without having to factor the denominator of the closed-loop transfer function.
  • You might not use an exact sketch very often in practice, but you will use an approximated one!
  • What does closed loop root locus look like from open loop?

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  • The closed loop system is

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How to Draw Root Locus

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8 Rules for Root Locus

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Rule 1

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Rule 2 (1/2)

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Rule 2 (2/2)

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Rule 3

  • Rule 3: When roots are complex, they occur in conjugate pairs (= symmetric about real axis)

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Rule 4

  • Rule 4: At no time will the same root cross over its path

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Rule 5 (1/2)

  • Rule 5: The portion of the real axis to the left of an odd number of open loop poles and zeros are part of the loci
    • which parts of real line will be a part of root locus?

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Rule 5 (2/2)

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  • For real zeros or poles

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Rule 6 and Rule 7

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Rule 8 (1/3)

  • Rule 8 : Lines go to infinity along asymptotes

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    • The angles of the asymptotes

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    • The centroid of the asymptotes on the real axis

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Rule 8 (2/3)

  • Lines go to infinity along asymptotes

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Rule 8 (3/3)

  • The centroid of the asymptotes on the real axis

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Rule 8

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Rule 8

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Rule 8

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Rule 8

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Break-away, Break-in Points

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Break-away, Break-in Points

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Break-away, Break-in Points

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Find Angles of Departure/Arrival for Complex Poles/Zeros

  • Loot at a very small region around the departure point

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Rule 6: Lines leave (break out) and enter (break in) the real axis at 90°

  • Revisit

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Root Locus for Stability

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Root Locus for Stability Evaluation

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Root Locus for Stability Evaluation

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Root Locus for Stability Evaluation

  • How can we make this stable?

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Root Locus in MATLAB

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Root Locus in MATLAB

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Root Locus in MATLAB

  • Example 1: Lines leave the real axis at 90 degrees

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Root Locus in MATLAB

  • Example 2: Asymptotes

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Root Locus in MATLAB

  • Example 3: determining the breakaway points

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Root Locus in MATLAB

  • Example 4: Departure angle

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Root Locus in MATLAB

  • Example 4: Departure angle

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