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Replotting the Nyquist Plot: A New Visualization Proposal Predrag Pejovi Introduction Nyquist stability criterion . . . started as a fix to Barkhausen criterion original derivation complicated . . . nowadays taught using

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Replotting the Nyquist Plot: A New Visualization Proposal

Predrag Pejović

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Introduction

◮ Nyquist stability criterion . . . ◮ started as a fix to Barkhausen “criterion” ◮ original derivation complicated . . . ◮ nowadays taught using Cauchy’s argument principle ◮ fundamental, fairly esoteric, deep math . . . but elegant! ◮ highly abstract topological criterion, reduces to:

  • 1. CW encirclement of −1 + j 0 adds one unstable pole
  • 2. CCW encirclement of −1 + j 0 removes one unstable pole

while closing the loop; topological and relative ◮ hard to teach, requires focused (and competent) students ◮ frequently hard to visualize due to imaginary axis poles, “closed” curve is not closed, but it “encloses” . . . ◮ and this is the point where our story begins . . .

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Nyquist Criterion Revisited: the tracking system

W(s) Σ + − x y e

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Nyquist Criterion Revisited: assumptions

the transfer function W(s) = N (s) D (s) let N(s) and D(s) be polynomials such that deg (N (s)) ≤ deg (D (s)) which is satisfied for systems without algebraic degeneration the problem is whether H (s) = W (s) 1 + W (s) is stable or not?

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SLIDE 5

A Word on Barkhausen . . .

poles at 1 + W (s) = 0 i.e. −W (s) = 1 + j 0 WRONG generalization and a NONSENSE: stable if

✭✭✭✭✭✭✭✭ ✭ ❤❤❤❤❤❤❤❤ ❤

−W (s) ≤ 1 + j 0

  • r

|W (s)| ≥ 1 BTW, which s? For oscillators s = j ω0, where ℑ (W (s)) = 0 . . .

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Nyquist Criterion Revisited: the first disaster, W 0(s) = 1 s, stable for sure

W 0(s) = 1 s W 0(s) 1 + W 0(s) = 1 1 + s pole at s = −1 + j 0, definitely stable

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SLIDE 7

Nyquist Criterion Revisited: the first disaster, W 0(s) = 1 s, straightforward

  • 2
  • 1

1 2

  • 1e+06
  • 500000

500000 1e+06

Real Axis Imaginary Axis Nyquist Diagram

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SLIDE 8

Nyquist Criterion Revisited: the first disaster, W 0(s) = 1 s, escape contour

−4 −2 2 4 ℜ (s) −4 −2 2 4 ℑ (s) j ωmin j ωmax −j ωmin −j ωmax

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SLIDE 9

Nyquist Criterion Revisited: the first disaster, W 0(s) = 1 s, here is what we need

−2 −1 1 2 ℜ (W(s)) −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 ℑ (W(s))

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What Do We Really Plot?

W (s) = Wr (s) + j Wi (s) where Wr (s) = ℜ (W (s)), and Wi (s) = ℑ (W (s)) in polar form r (s) =

  • (Wr (s))2 + (Wi (s))2

ϕ (s) = atan2 (Wi (s) , Wr (s)) and r (s) is the problem! idea: compress r (s) somehow?

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SLIDE 11

Compression Function Requirements

  • 1. ρ (r) in monotonic, to preserve topological properties of the

critical point encirclements,

  • 2. ρ (0) = 0, to keep the same base point where the phase is

irrelevant,

  • 3. ρ (1) = 1, to keep the critical point and visualization of the

phase margin,

  • 4. limr→∞ ρ (r) is finite, to confine the diagram in a finite

space.

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SLIDE 12

Compression Function

ρ (r) = 4 π arctan (r) “amplitude angle”

1 r

π 4 ρ

r = 1 ρ = 1 1 r → ∞ ρ = 2 1

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A Family of Compression Functions

ρ (r) = rmax rk rmax − 1 + rk rmax is the radius of the circle the plot is confined into, this degree of freedom might be of some value k > 1 is a parameter rmax = 2 and k = 4

π approximates the compression function

applied in this paper the best, almost the same function, within 1.5% of rmax (0.03)

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Bode Plot, W 0(s) = 1 s

10−6 10−4 10−2 100 102 104 106 −120 −80 −40 40 80 120 20 log (r) 10−6 10−4 10−2 100 102 104 106 ω −180 −90 90 180 ϕ

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Amplitude Compression, r, r in decibels, and ρ

r r [dB] ρ 1 1 10 20 1.8731 100 40 1.9873 1,000 60 1.9987 10,000 80 1.9999 100,000 100 2.0000

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Bode Plot Alternative, W 0(s) = 1 s

10−6 10−4 10−2 100 102 104 106 1 2 ρ 10−6 10−4 10−2 100 102 104 106 ω −180 −90 90 180 ϕ

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Alternative Nyquist Plot, W 0(s) = 1 s, strange, you’ve seen this graph before

−2 −1 1 2 ℜ (W(s)) −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 ℑ (W(s))

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1 2 of the Contour . . .

0 90 590 1090 1590 10−6 10−3 100 103 106 109 |s| 0 90 590 1090 1590 k 30 60 90 arg (s) [◦]

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Argument Increase, W 0(s) = 1 s

1 2 ρ−1 0 90 590 1090 1590 k −180 180 ϕ−1

no change in the number of unstable poles

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Example 1, W 1 (s) = 1 (s + 1)2

  • 1.1
  • 1.05
  • 1
  • 0.95
  • 0.9
  • 1
  • 0.5

0.5 1

Real Axis Imaginary Axis Pole-Zero Map

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Example 1, W 1 (s) = 1 (s + 1)2

  • 2
  • 1

1 2

  • 2
  • 1

1 2

Real Axis Imaginary Axis Nyquist Diagram

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Example 1, W 1 (s) = 1 (s + 1)2

−2 −1 1 2 ℜ (W(s)) −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 ℑ (W(s))

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Example 1, W 1 (s) = 1 (s + 1)2

10−6 10−4 10−2 100 102 104 106 1 2 ρ 10−6 10−4 10−2 100 102 104 106 ω −270 −180 −90 90 ϕ [◦]

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Example 1, W 1 (s) = 1 (s + 1)2

1 2 ρ−1 0 90 590 1090 1590 k −180 180 ϕ−1

no change in the number of unstable poles

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Example 2, W 2 (s) = 1 s (s + 1)2

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2
  • 1
  • 0.5

0.5 1

Real Axis Imaginary Axis Pole-Zero Map

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Example 2, W 2 (s) = 1 s (s + 1)2

  • 2
  • 1.5
  • 1
  • 0.5
  • 1e+06
  • 500000

500000 1e+06

Real Axis Imaginary Axis Nyquist Diagram

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Example 2, W 2 (s) = 1 s (s + 1)2

−2 −1 1 2 ℜ (W(s)) −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 ℑ (W(s))

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Example 2, W 2 (s) = 1 s (s + 1)2

10−6 10−4 10−2 100 102 104 106 1 2 ρ 10−6 10−4 10−2 100 102 104 106 ω −360 −270 −180 −90 ϕ [◦]

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Example 2, W 2 (s) = 1 s (s + 1)2

1 2 ρ−1 0 90 590 1090 1590 k −180 180 ϕ−1

no change in the number of unstable poles

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Example 2, W 2 (s) = 1 s (s + 1)2

closed loop, stable

  • 2
  • 1.5
  • 1
  • 0.5
  • 1
  • 0.5

0.5 1

Real Axis Imaginary Axis Pole-Zero Map

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Example 2a, W 2a (s) = 3 W 2 (s) = 3 s (s + 1)2

  • 6
  • 5
  • 4
  • 3
  • 2
  • 1
  • 3e+06
  • 2e+06
  • 1e+06

1e+06 2e+06 3e+06

Real Axis Imaginary Axis Nyquist Diagram

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Example 2a, W 2a (s) = 3 W 2 (s) = 3 s (s + 1)2

−2 −1 1 2 ℜ (W(s)) −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 ℑ (W(s))

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Example 2a, W 2a (s) = 3 W 2 (s) = 3 s (s + 1)2

10−6 10−4 10−2 100 102 104 106 1 2 ρ 10−6 10−4 10−2 100 102 104 106 ω −360 −270 −180 −90 ϕ [◦]

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Example 2a, W 2a (s) = 3 W 2 (s) = 3 s (s + 1)2

1 2 ρ−1 0 90 590 1090 1590 k −540 −360 −180 180 ϕ−1

+2 unstable poles

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Example 2a, W 2a (s) = 3 W 2 (s) = 3 s (s + 1)2

closed loop, unstable

  • 2.5
  • 2
  • 1.5
  • 1
  • 0.5

0.5

  • 1.5
  • 1
  • 0.5

0.5 1 1.5

Real Axis Imaginary Axis Pole-Zero Map

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Example 3, W 3 (s) = s + 1 s (0.1 s − 1)

  • 2

2 4 6 8 10

  • 1
  • 0.5

0.5 1

Real Axis Imaginary Axis Pole-Zero Map

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Example 3, W 3 (s) = s + 1 s (0.1 s − 1)

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2
  • 1e+06
  • 500000

500000 1e+06

Real Axis Imaginary Axis Nyquist Diagram

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Example 3, W 3 (s) = s + 1 s (0.1 s − 1)

−2 −1 1 2 ℜ (W(s)) −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 ℑ (W(s))

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Example 3, W 3 (s) = s + 1 s (0.1 s − 1)

10−6 10−4 10−2 100 102 104 106 1 2 ρ 10−6 10−4 10−2 100 102 104 106 ω 90 180 270 360 ϕ [◦]

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Example 3, W 3 (s) = s + 1 s (0.1 s − 1)

1 2 ρ−1 0 90 590 1090 1590 k −360 −180 180 ϕ−1

−1 unstable pole

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Example 3, W 3 (s) = s + 1 s (0.1 s − 1)

bypass ambiguity

−1 + j0 ∆ϕ = +π ∆ϕ = −π

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Example 3, W 3 (s) = s + 1 s (0.1 s − 1)

closed loop, on the stability boundary

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2

  • 4
  • 2

2 4

Real Axis Imaginary Axis Pole-Zero Map

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Example 3a, W 3a (s) = 2 W 3 (s) = 2 s + 1 s (0.1 s − 1)

  • 2
  • 1.5
  • 1
  • 0.5
  • 2e+06
  • 1e+06

1e+06 2e+06

Real Axis Imaginary Axis Nyquist Diagram

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Example 3a, W 3a (s) = 2 W 3 (s) = 2 s + 1 s (0.1 s − 1)

−2 −1 1 2 ℜ (W(s)) −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 ℑ (W(s))

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Example 3a, W 3a (s) = 2 W 3 (s) = 2 s + 1 s (0.1 s − 1)

10−6 10−4 10−2 100 102 104 106 1 2 ρ 10−6 10−4 10−2 100 102 104 106 ω 90 180 270 360 ϕ [◦]

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Example 3a, W 3a (s) = 2 W 3 (s) = 2 s + 1 s (0.1 s − 1)

1 2 ρ−1 0 90 590 1090 1590 k −360 −180 180 ϕ−1

−1 unstable pole

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Example 3a, W 3a (s) = 2 W 3 (s) = 2 s + 1 s (0.1 s − 1)

closed loop, stable

  • 8
  • 7
  • 6
  • 5
  • 4
  • 3
  • 2
  • 1
  • 1
  • 0.5

0.5 1

Real Axis Imaginary Axis Pole-Zero Map

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Example 3b, W 3b (s) = 1 2 W 3 (s) = 1 2 s + 1 s (0.1 s − 1)

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2
  • 400000
  • 200000

200000 400000

Real Axis Imaginary Axis Nyquist Diagram

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Example 3b, W 3b (s) = 1 2 W 3 (s) = 1 2 s + 1 s (0.1 s − 1)

−2 −1 1 2 ℜ (W(s)) −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 ℑ (W(s))

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Example 3b, W 3b (s) = 1 2 W 3 (s) = 1 2 s + 1 s (0.1 s − 1)

10−6 10−4 10−2 100 102 104 106 1 2 ρ 10−6 10−4 10−2 100 102 104 106 ω 90 180 270 360 ϕ [◦]

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Example 3b, W 3b (s) = 1 2 W 3 (s) = 1 2 s + 1 s (0.1 s − 1)

1 2 ρ−1 0 90 590 1090 1590 k −540 −360 −180 ϕ−1

+1 unstable pole

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Example 3b, W 3b (s) = 1 2 W 3 (s) = 1 2 s + 1 s (0.1 s − 1)

closed loop, unstable

  • 1

1 2 3 4

  • 1
  • 0.5

0.5 1

Real Axis Imaginary Axis Pole-Zero Map

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Example 4, W 4 (s) = (s + 10)2 s3

  • 10
  • 8
  • 6
  • 4
  • 2
  • 1
  • 0.5

0.5 1

Real Axis Imaginary Axis Pole-Zero Map

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Example 4, W 4 (s) = (s + 10)2 s3

  • 2e+13
  • 1.5e+13
  • 1e+13
  • 5e+12
  • 1e+20
  • 5e+19

5e+19 1e+20

Real Axis Imaginary Axis Nyquist Diagram

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SLIDE 55

Example 4, W 4 (s) = (s + 10)2 s3

−2 −1 1 2 ℜ (W(s)) −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 ℑ (W(s))

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SLIDE 56

Example 4, W 4 (s) = (s + 10)2 s3

10−6 10−4 10−2 100 102 104 106 1 2 ρ 10−6 10−4 10−2 100 102 104 106 ω 90 180 270 360 ϕ [◦]

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Example 4, W 4 (s) = (s + 10)2 s3

1 2 ρ−1 0 90 590 1090 1590 k −540 −360 −180 180 ϕ−1

+2 unstable poles

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SLIDE 58

Example 4, W 4 (s) = (s + 10)2 s3

closed loop, unstable

  • 10
  • 8
  • 6
  • 4
  • 2

2

  • 6
  • 4
  • 2

2 4 6

Real Axis Imaginary Axis Pole-Zero Map

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Example 4a, W 4a (s) = 10 W 4 (s) = 10 (s + 10)2 s3

  • 2e+13
  • 1.5e+13
  • 1e+13
  • 5e+12
  • 1e+20
  • 5e+19

5e+19 1e+20

Real Axis Imaginary Axis Nyquist Diagram

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SLIDE 60

Example 4a, W 4a (s) = 10 W 4 (s) = 10 (s + 10)2 s3

−2 −1 1 2 ℜ (W(s)) −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 ℑ (W(s))

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Example 4a, W 4a (s) = 10 W 4 (s) = 10 (s + 10)2 s3

10−6 10−4 10−2 100 102 104 106 1 2 ρ 10−6 10−4 10−2 100 102 104 106 ω 90 180 270 360 ϕ [◦]

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Example 4a, W 4a (s) = 10 W 4 (s) = 10 (s + 10)2 s3

1 2 ρ−1 0 90 590 1090 1590 k −360 −180 180 ϕ−1

no change in the number of unstable poles

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SLIDE 63

Example 4a, W 4a (s) = 10 W 4 (s) = 10 (s + 10)2 s3

closed loop, stable

  • 10
  • 8
  • 6
  • 4
  • 2
  • 15
  • 10
  • 5

5 10 15

Real Axis Imaginary Axis Pole-Zero Map

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Conclusions

◮ an alternative visualization for Nyquist plots ◮ affects only magnitude of the polar plot, not phase ◮ topological properties kept ◮ a compression function ρ (r) . . . ◮ . . . and a family of compression functions ◮ closed contour mapped into a closed contour ◮ problems with imaginary axis poles removed ◮ the plot confined into a finite space ◮ dynamics of the amplitude plot reduced ◮ a program that counts encirclements, ∆# of unstable poles ◮ three zones of loop gain identified ◮ numerous examples . . . ◮ this presentation is posted at Zenodo . . .