Control Nyquist (N)

8/6/2019 Control Nyquist (N)

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Stability Analysis in Frequency DomainStability Analysis in Frequency Domain

Stability and relative (degree of) stability:Stability and relative (degree of) stability:-- time responsetime response

-- root locationsroot locations

-- frequency responsefrequency response

Frequency response around feedback loop:Frequency response around feedback loop:

-- relationship betweenrelationship between T(jT(j) and) and GH(jGH(j))-- NyquistNyquist stability criterionstability criterion

-- measures of relative stabilitymeasures of relative stabilityUse in adjusting control parametersUse in adjusting control parameters


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Polar Plot of Frequency ResponsePolar Plot of Frequency Response

System:System: r(tr(t)) c(tc(t))G(sG(s))

r(tr(t) =) = Asin(Asin(tt))c(tc(t) =) = A[A[(()cos()cos(tt) +) +jj(()sin()sin(tt)])]G(jG(j) =) = (() +) +jj(())

(() is the real part) is the real part(() is the imaginary part) is the imaginary partobtain byobtain by

G(jG(j) replacing) replacing ssjj inin G(sG(s))

Polar plot:Polar plot: (() vs.) vs. (() as) as variesvaries


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F. R. of 1F. R. of 1stst--Order SystemOrder System

G(s) = 1s + 1G(j) = 1j + 1G(j) = 11+ ()2 +j 1+ ()2G(j) = () +j()

where

() = 11+ ()2 () = 1+ ()2


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Polar Plot for 1Polar Plot for 1stst--Order SystemOrder System

00 0.10.1 0.20.2 0.30.3 0.40.4 0.50.5 0.60.6 0.70.7 0.80.8 0.90.9 11

--0.80.8

--0.60.6

--0.40.4

--0.20.2

00

0.20.2

0.40.4

0.60.6

Real AxisReal Axis

Imag.Imag.

AxisAxis

= 0= 0== --

= += +G=G=--4545

--0.50.5

< 0< 0

> 0> 0|G|=0.707|G|=0.707

increasingincreasing

increasingincreasing = 1

1s + 1


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Polar Plot for IntegrationPolar Plot for Integration

--0.50.5 0.50.5

--55

--44

--33

--22

--11

00

11

22

33

44

55


00

== -- = += +

00

increasingincreasing < 0< 0

> 0> 000

Real AxisReal Axis

--9090

Imag.Imag.

AxisAxis

G(s) = 1s

G(j) =1

j

G(j) = 0 j1


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Polar Plot for 1Polar Plot for 1stst--Order + IntegrationOrder + Integration

--11 --0.50.5 00 0.50.5

--55

--44

--33

--22

--1100

11

22

33

44

55

Real AxisReal Axis

Imag.Imag.

AxisAxis

00

== -- = += +


00


1

s s + 1) < 0< 0 > 0> 0


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Utility of Polar F. R. PlotsUtility of Polar F. R. Plots

Limitations:Limitations:

-- frequencyfrequency does not appear on an axisdoes not appear on an axis-- effect of each pole or zero is not obviouseffect of each pole or zero is not obvious

-- effects of poles and zeros are not additiveeffects of poles and zeros are not additive

-- addition of a pole or zero requiresaddition of a pole or zero requires

recalculation and replottingrecalculation and replotting

-- logarithmic plots are more commonlogarithmic plots are more common

Useful for analysis of stabilityUseful for analysis of stability

-- the plot for 1+GH is of particular interestthe plot for 1+GH is of particular interest


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Analysis of Stability using F. R.Analysis of Stability using F. R.

++--

G(s)G(s)

H(s)H(s) 1 + GH(s) = 01 + GH(s) = 0

CharacteristicCharacteristicequation:equation:

The stability of the closedThe stability of the closed--loop system canloop system can

be determined by analyzing its openbe determined by analyzing its open--looplooptransfer function GH(s).transfer function GH(s).

A mapping is made between contours inA mapping is made between contours inthe sthe s--plane and the 1+GH plane.plane and the 1+GH plane.

The contours allow determination of theThe contours allow determination of thenumber of unstable roots in 1+GH(s).number of unstable roots in 1+GH(s).


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Poles and Zeros of 1+GHPoles and Zeros of 1+GH

O.L.T.F. : G(s)H(s) = GH(s) = O.L. zerosO.L. polesC.L.T.F. :

G(s)

1+ GH(s) = C.L. zerosC.L. polesC.L.C.E. : 1+ GH(s) = 0

1+ O.L. zerosO.L. poles

= 0O.L. poles)+ O.L. zeros)

O.L. poles= 0

1+GH poles = O.L.T.F. poles!1+GH poles = O.L.T.F. poles!

C.L.T.F. polesC.L.T.F. poles= 1+GH zeros!= 1+GH zeros!


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j0j0


Conformal MappingConformal Mapping

F(s) = 1s +1F(j) = 1j + 1

00 0.20.2 0.40.4 0.60.6 0.80.8 11--0.80.8

--0.60.6

--0.40.4

--0.20.2

00

0.20.2

0.40.4

0.60.6

jj

--0.50.5 0.50.5--44

--22

00

22

44

00

jj

j0j0 Re(s)Re(s)

Im(s)Im(s) Im[F(jIm[F(j)])]

Re[F(jRe[F(j)])]

s = js = j



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Closed ContoursClosed Contours Closed ContoursClosed Contours(from complex variable theory)(from complex variable theory)

j0j0

00 0.20.2 0.40.4 0.60.6 0.80.8 11--0.80.8

--0.60.6

--0.40.4--0.20.2

00

0.20.20.40.4

0.60.6

--jj

--0.50.5

--44

--22

00

22

44

jj

j0j0

Im(s)Im(s) Im[F(jIm[F(j)])]

Re[F(jRe[F(j)])]jj

Re(s)Re(s)

00

--jj

0.50.5

F(s) = 1s + 1

aa

aa

bb

cc

cc


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F(s)F(s) 1 + F(s)1 + F(s)

j0j0

11 1.21.2 1.41.4 1.61.6 01.801.8 22--0.80.8

--0.60.6--0.40.4

--0.20.2

000.20.2

0.40.4

0.60.6

--jj

--0.50.5

--44

--22

00

22

44

j

j

j0j0

Im(s)Im(s) Im[1+F(jIm[1+F(j)])]

Re[1+F(jRe[1+F(j)])]jj

Re(s)Re(s)

00

--jj

0.50.5

aa

aa

bb

cc

cc

(Shift of scale on real axis!)(Shift of scale on real axis!)

1+ F(s) = 1+ 1s + 1

C h ThC h Th


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Cauchys TheoremCauchys Theorem

If a contourIf a contourss in the sin the s--plane encircles (toplane encircles (toright of contour traversal) Z zeros and Pright of contour traversal) Z zeros and Ppoles of F(s) and does not pass throughpoles of F(s) and does not pass through

any poles or zeros of F(s) with traversal inany poles or zeros of F(s) with traversal in

the clockwise direction,the clockwise direction,

then the corresponding contourthen the corresponding contourff in thein theF(s) plane encircles the origin of the F(s)F(s) plane encircles the origin of the F(s)

plane Zplane Z--P times in the clockwise direction.P times in the clockwise direction.Use: GH(jUse: GH(j) polar plot to determine the) polar plot to determine thenumber of zeros of 1+GH(s) in the right 1/2number of zeros of 1+GH(s) in the right 1/2plane (# of unstable poles of the C.L.T.F.).plane (# of unstable poles of the C.L.T.F.).

U f N i t C tU f N i t C t


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Use of Nyquist ContourUse of Nyquist Contour

The Nyquist contour encircles the right 1/2The Nyquist contour encircles the right 1/2of the sof the s--plane in the clockwise direction.plane in the clockwise direction.

The number of clockwise encirclements ofThe number of clockwise encirclements of

the origin of the 1+GH plane is N = Zthe origin of the 1+GH plane is N = Z -- PP

where:where: Z = # of zeros of 1+GHZ = # of zeros of 1+GH

P = # of poles of 1+GHP = # of poles of 1+GH

The origin 0+0j of the 1+GH plane can beThe origin 0+0j of the 1+GH plane can bereplaced by pointreplaced by point --1+j0 in the GH plane.1+j0 in the GH plane.

P = # of unstable poles of O.L.T.F. (known)P = # of unstable poles of O.L.T.F. (known) Z = N + P = # of unstable poles of C.L.T.F.Z = N + P = # of unstable poles of C.L.T.F.

N i t St bilit C it iN i t St bilit C it i


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Nyquist Stability CriterionNyquist Stability Criterion

1)1) Determine the number of unstable polesDetermine the number of unstable poles

in GH(s) (often known to be zero).in GH(s) (often known to be zero).

2)2) Substitute jSubstitute j for s in GH(s).for s in GH(s).3)3) Map the Nyquist contour in the sMap the Nyquist contour in the s--plane toplane to

the GH plane.the GH plane.

4)4) Determine the number of clockwiseDetermine the number of clockwise

encirclements of the pointencirclements of the point --1+j0.1+j0.

5)5) # of unstable poles in the C.L.T.F.# of unstable poles in the C.L.T.F.

= # of clockwise encirclements= # of clockwise encirclements

+ # of unstable poles in GH(s)+ # of unstable poles in GH(s)


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O.L.T.F.:O.L.T.F.:

--1010--44

--22

00

22

44

jj

Im(s)Im(s)

00

--jj

1010

GH(s) = 100s + 1) 0.1s + 1)

aa bb

cc

ss--planeplane

--22 00 22 44 66 88 1010--66

--44

--22

00

22

44

66

--11

Im[GH(jIm[GH(j)])]

Re[GH(jRe[GH(j)])]

GH planeGH plane

area encircledarea encircled

clockwiseclockwise

=0=0== --= += +

aa

cc

Re(s)Re(s)xxxx

O.L. PolesO.L. Poles

0 clockwise encirclements of0 clockwise encirclements of--1+0j,1+0j,0 unstable OL poles, 0 unstable CL poles0 unstable OL poles, 0 unstable CL poles


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O.L.T.F.:O.L.T.F.:

--11--44

--22

00

22

44

jj

Im(s)Im(s)

00

--jj11

GH(s) = Ks s + 1)2

aa bb

cc

ss--planeplaneIm[GH(jIm[GH(j)])]GH planeGH plane

--22 --11 00 11

--11

00

11

Re[GH(jRe[GH(j)])]cc dd

aa

=0=0++= += +

== --

, K = 0.5=0=0--

Re(s)Re(s)xxxx

O.L. PolesO.L. Poles

dd bbInfinitisimalInfinitisimal

detour!detour!



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O.L.T.F.:O.L.T.F.:

--11--44

--22

00

22

44

jj

Im(s)Im(s)

--jj11


--22 --11 00 11

--11

00

11

Re[GH(jRe[GH(j)])]

=0=0++= += +

== --=0=0--

Re(s)Re(s)

--1+0j is intersected by the contour1+0j is intersected by the contour(C.L.T.F. therefore is marginally stable!)(C.L.T.F. therefore is marginally stable!)

xxxx

O.L. PolesO.L. Poles GH(s) = Ks s + 1)2

, K = 2

00

K


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O.L.T.F.:O.L.T.F.:

--11--44

--22

00

22

44

jj

Im(s)Im(s)

--jj11


--22 --11 00 11

--11

00

11

Re[GH(jRe[GH(j)])]

=0=0++ = += +

== --=0=0--

Re(s)Re(s)xxxx

O.L. PolesO.L. Poles GH(s) = Ks s + 1)2

, K = 3

00


I t f N i t St bilit C it iI t f N i t St bilit C it i


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Importance of Nyquist Stability CriterionImportance of Nyquist Stability Criterion

If it is known that O.L.T.F. has no unstableIf it is known that O.L.T.F. has no unstablepoles (common situation), orpoles (common situation), or

the poles of the O.L.T.F. are known,the poles of the O.L.T.F. are known,then the Nyquist method can determinethen the Nyquist method can determine

the stability of the closedthe stability of the closed--loop system.loop system.Modern computational tools (root finding,Modern computational tools (root finding,

etc.) diminish the need for this test.etc.) diminish the need for this test.More importantly, pointMore importantly, point --1+0j serves as a1+0j serves as a

reference for measuring relative stability.reference for measuring relative stability.

-- measure the distance away frommeasure the distance away from --1+j01+j0

Control Nyquist (N)

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