Nyquist Stability Criterion, Part 2

welcome back to control system lectures

this is part two of our explanation of

the nyas plot in the first part of this

series I showed you the method of

mapping from the S domain to the W

domain using the transfer function I

also explained how to use koshy's

argument principle to our benefit what

the nyas Contour was and finally how to

interpret the resulting nyis plot so now

let me continue from there and in this

video I'm going to show you a simple way

to estimate a nyis plot by hand I'll

explain how to handle open loop poles

and zeros on the imaginary axis and then

walk through an example that illustrates

the power of the nyas plot over similar

methods like the bod

plot so let's get to it we'll start with

estimating a nyis plot by hand let's say

that I have this s domain transfer

function 1 / s^2 + 3 S + 2 which has

these two poles s = -1 and S = -2 now

recall from part one that I need to map

the nyis Contour to some other plane

that I've arbitrarily called The W plane

using this transfer function so how do

we do that well the most straightforward

if not silly way is to take a a point in

the S plane on the Contour and plug it

into the transfer function and then plot

the resulting complex number in the W

plane just like I'm doing here in this

example where this one point in the S

plane maps to this other point in the W

plane

and we can continue this for every

single point around the entire nyas

Contour but I'm sure you can see a

problem with this not only is this a

very math intensive way but it also

causes us to miss out on some very

important intuition about how the

Nyquist Contour relates to the nyas

plot so let me illustrate a better way

to plot this first we're going to divide

the nyas Contour up into two pieces and

then we'll address both piece separately

the first piece can be represented by J

Omega since it's just the entire J Omega

line where Omega goes from negative

Infinity to positive infinity and we can

map this entire segment all at once by

plugging in J Omega for s in our

transfer function and then sweep through

all Omega by plotting the resulting real

and imaginary

components also because the symmetry of

the poles and zeros about the real line

the negative portion of the J Omega line

is just the reflection of the positive

portion

and so we really only need to calculate

the positive frequencies but I'm going

to explain more about this later the

second piece starts at Infinity on the

imaginary axis loops around to Infinity

on the real axis and then finally

negative Infinity on the imaginary axis

again now I think this segment is pretty

interesting because despite the fact

that it is infinitely long the entire

segment Maps into a single point in the

W plane pretty crazy huh but there is a

catch it maps to a single Point only if

the denominator of your transfer

function is higher order than the

numerator and this is called a strictly

proper transfer function or if it's

equal order to the numerator and this is

called a proper transfer function if the

numerator is of higher order than the

denominator then this is not a proper

transfer function and the second segment

maps to a single point for only the

first two and not the last but that's

okay because all physically realizable

systems are either proper transfer

functions or strictly proper and that's

why I'm going to focus on them in this

video you can show mathematically why

that second segment Maps into a single

point in the W plane and there's a lot

of great videos that do that so instead

I'm going to quickly explain in a

graphical sense why this is true

remember from part one that the phase in

the W plane is just the sum of the

angles of the zero phasers minus the sum

of the angles of the pole

phasers and then for the gain didn't

cover that really very well in the first

video because it's a bit misleading

graphically but for our following

example it's acceptable to say that the

gain is proportional to the magnitude of

the zero phasers divided by the

magnitude of the pole phasers so we're

going to use that definition going

forward let's take the case where there

are more poles than zeros or a strictly

proper system it's easy to see that if

you pick a point way out at Infinity

since the pole phasers outnumber the the

zero phasers the gain will become a very

large number in the numerator for the

zeros divided by more very large numbers

in the denominator for the poles and

that's going to put the gain really

really close to zero and it's going to

reach zero as that point goes out

actually out to infinity and since the

gain is zero we don't need to worry

about the phase because the point is

always at the origin in the W plane and

phase would just be akin to spinning

your pencil in place at the origin it's

still the same point and so we don't

really have to worry about

it for proper transfer functions where

you have the order of the numerator and

denominator equal you have this

situation now along this Infinity line

we get a very large number from the zero

divided by pretty much that exact same

large number as you approach Infinity

this puts the gain at one which remember

I said was just proportional to the real

gain so all I can say graphically is

that the gain is a nonzero but finite

value so somewhere out here but what

about the phase because it could exist

anywhere on this circle so now phase is

important well the angle for both of

these phasers approach the exact same

value as s goes to Infinity so when you

subtract the pole angle from the zero

angle you always get 0° which means that

this entire line maps to a single

nonzero point on the real axis so it

maps to zero for strictly proper systems

and a nonzero but positive real number

for nonstrictly proper

systems now to simplify the nyas Contour

since the second segment of the nyas

Contour all maps to the same point in

the W domain then we don't need to worry

about it when we're plotting the point

is taken care of with the Positive

Infinity in the J Omega axis and because

of the reflection we don't need the

negative omegas so we really only need

to plot the positive J Omega part now at

the risk of confusing I want to explain

what happens if you don't have a proper

transfer function in this situation on

the infinity portion of the nyis Contour

the gain also goes to Infinity since you

have more zeros than poles and as you

loop around that portion of the Contour

the phase is also changing as you do

that so in this case the portion of the

ny's Contour doesn't all map to a single

point however this is really only a

concern to mathematicians and not

Engineers since Engineers deal with

physically realizable systems but keep

this in mind in case you come across a

non-proper transfer function

someday so after all of that let's go

through the steps to generate a nyis

plot first replace S by J Omega in your

transfer function Second Sweep Omega

from zero to infinity and plot the

resulting complex numbers in the W plane

and then finally without picking up your

pencil draw the reflection about the

real axis to account for the negative

omegas and don't forget don't forget to

keep in mind the direction of the

Contour remember from part one that we

traced the Nyquist Contour in the

clockwise Direction so if we start at

Omega equals 0 then we Trace up to Omega

equals infinity and then continue around

in the clockwise direction from

there I know what you're thinking I've

been rambling on for 8 minutes and I

haven't actually told you anything about

how to plot all I've done is explain why

we only need to worry about plotting the

positive J Omega section of the nyas

Contour rather than the whole thing but

how do we estimate the plot after we

replace S by J Omega well the way I

prefer is actually explained very well

by professor gopal at the Indian

Institute of Technology he has a great

YouTube video on it and I've placed a

link in the description if you'd like to

watch the technique being explained from

a professional Professor but I'm going

to summarize it here for simple transfer

functions there are only four points

that you need to solve for and then from

there you can deduce the shape of the

entire plot

the first point is when Omega equals 0

the second point is when Omega equals

infinity these determine the starting

and midpoint of the ny's plot in the W

plane the third point is where the plot

crosses the imaginary axis and the

fourth point is where the plot crosses

the real

axis so let's try this with our original

transfer function and see what we get

for the first point where Omega equals z

we can basically just plug in zero for

each of the S's and we're going to get

1/2 for our starting point for Omega

equals infinity we just plug in Infinity

for each of the S's and we're going to

get one over a really large number which

is zero which is what we expected since

this is a strictly proper transfer

function so the midpoint is at the

origin the third and fourth points are

slightly more involved but you can solve

for them by setting s equals J Omega in

the transfer function and then

separating out the real and imaginary

components

I've done that very thing here albeit

really

quickly at this point to solve for the

imaginary intercept set the real part to

zero and then solve for Omega then plug

that Omega into the imaginary part and

solve for the value and that's where it

crosses the imaginary axis it's a

similar story to get the real intercept

but this time you set the imaginary

component to zero and solve for Omega in

this case it's only zero when Omega

equals z or Omega equals infinity and

since we've already plotted both of

those points in step one and two there's

really nothing else to plot it only

crosses the real line at the beginning

and at the midpoint so now that we have

this information what does it tell us

well we can start with our pencil at the

starting point and Trace through the

imaginary crossover and then through the

real crossover if there is one and then

end at the midpoint and then we can

continue on by reflecting this about the

real line and we would have an

approximate drawing of the nyas plot and

from this plot we can see that the

closed loop system is stable since

remember there were no open loop poles

in the right half plane and there are no

encirclements of minus one of course

this method doesn't tell us whether the

plot looks like this or this but it does

help us to determine closed loop

stability I'm going to plot this in mat

lab and see how close we

got the first thing I'm going to do is

set our systems transfer function and

then I'll plot the Nik was plot and here

it is and we did all right it doesn't

look exactly the same but you can see

that it starts at point5 on the real

axis Loops down crosses over the

imaginary axis and then ends at the

origin and the system is stable like we

predicted now a more complex transfer

function is difficult to do in this

method because you'll have several

imaginary and real Crossings but if

you're trying to assess a complex

transfer function you'd be better off

using a computer to draw it rather than

estimate it by hand I'm going to show

you an example of this at the end of

this video so now the question becomes

what happens if you have an open loop

pole on the imaginary axis well let's

start with a single pole at the origin 1

/ s I'll draw the nyis Contour and pick

a point to analyze somewhere up here and

we're going to do a thought experiment

what if we bring this point on our

Contour closer and closer to the pole

what happens well the phase stays the

same at 90° but the magnitude of the

phaser gets smaller and smaller

and since we're dividing by a smaller

and smaller number the magnitude in the

W plane tends towards

infinity and once we're right on top of

the pole the gain goes towards Infinity

since we're essentially dividing by zero

the length of the phaser and the phase

becomes what well it's undefined it's

sort of all angles at once and yet no

angle at all there is no phase since

this phaser basically disappears when

the point is on top of the pole in the W

plane we basically have a point that

lies out at Infinity but with some

undefined phase so we're out of luck

with understanding what that means we

can't use our original nyis Contour with

poles on the imaginary axis so what do

we do the solution I think is ingenious

let me zoom in on the pole here so you

can understand what we do we know that

the Contour can't lie directly on the

pole otherwise the solution is undefined

so why not move the Contour over

slightly just in that one spot

since we just moved in an infinite tmal

amount we've accomplished two things the

first is that we're still guaranteeing

that we haven't accidentally let another

close by pole escape the Contour and the

second thing is that we no longer have

this indeterminate solution the gain

stays at Infinity since we're dividing

by a really small number still but the

phase for our phaser very clearly starts

at - 90° and loops around to plus 90° so

now we can draw the Nyquist plot for

this system when we start at Omega

equals 0 you can see visually that the

gain is infinity since the length of the

phaser is zero and the phase is 0° and

then as we loop around the pole the gain

stays at Infinity but the phase in the W

plane sweeps down to minus 90 then as

the Contour continues up the J Omega

axis the gain in the W plane goes to

zero since the length of the phaser goes

to infinity and finally we can complete

the nyas plot by just drawing the

reflection about the real axis and we

can say that the closed loop system is

stable because there are no open loop

poles in the right half plane and no

encirclements of minus one but since you

now understand the nyis Criterion pretty

well you can easily see that this

adjustment of the nyis Contour works by

expanding it in the other direction also

if we move it slightly to the left

instead of to the right the gain is

still Infinity but now the phase starts

at- 180° and then just like before Loops

down to minus 90° phase then up to the

origin and then finally reflected about

the real axis and even though this nyis

plot looks different than the one we

drew before we can still say that the

system is stable because we have one

open loop pole in the right half plane

or inside our contour and one

counterclockwise encirclement of minus

one therefore zero closed loop poles in

the right half plane both methods work

as long as you keep track of it it's

pretty awesome

huh so what about an open loop zero on

the imaginary axis well you don't really

have to worry about that because even

though the phaser is still undefined in

Phase it turns out that we don't really

care because the gain is zero and like I

said before when gain is zero the point

lies at the origin in the W plane and so

the phase doesn't really change that you

can do the loop around thing like we did

before but you're going to find you're

going to get the exact same answer as

not doing it so for open loop zeros

proceed as normal and for open loop

poles on the imaginary axis do this

fancy adjusting of the

Contour let me show you what happens in

mat lab when you try to plot the Nyquist

plot for a system with open loop poles

on the imaginary axis you get this

vertical line but you can see that mat

lab doesn't really know how to handle

plotting Infinity so it just doesn't

even try in order to determine stability

you have to mentally complete the plot

swiping either clockwise or

counterclockwise at infinity and this

can be tough to visualize but luckily

even though mat lab can't plot it well

it will still tell you if the Clos loop

system is stable and in addition to that

there's this awesome plotting tool by

Tron andren called Nik logm his script

will handle infinti very nicely plus it

tells you the Clos Loop stability and

other bits of information you can find

this script at mathw works' website and

I've also included a link in the

description if you'd like to download

it so for this final example let me show

you where the nyas plot really shines

and that is when the open loop plant has

either poles or zeros in the right half

plane fighter jets are required to

maneuver very quickly this gives them

the edge in dog fights but a really

stable airplane doesn't want to turn

very quickly therefore Engineers have

found the best way to accomplish this is

by making the open loop airplane

slightly unstable or making it want to

turn and pitch on its own and then

stabilizing it using fly by wire

feedback control the unstable plant for

the pitch system of an F-16 fighter is

the following where the input is the

angle of the elevator and the output is

the pitch angle of the aircraft now

you've been tasked with designing a

pitch tracking autopilot system for the

F-16 and you want to see if Unity

feedback alone is capable of stabilizing

the system you can tell that the open

loop system is unstable just by

inspection by looking to see that there

is a negative in the characteristic

equation but I can use matlb to see

where the open loop poles are for this

system

and that's by finding the roots of the

characteristic equation and you can see

that there's one just barely in the

right half plane like we suspected so

the open loop system is

unstable let me first plot the bod plot

of this system and see what closed loop

stability information we can get from it

and it's a bit hard to read even if I

show stability margins in the normal

sense you can't easily tell if the

closed loop system is stable or not and

in some open loop unstable cases you

can't tell at all therefore in this

situation you'd be safer just to plot

the Nikas plot instead and if we do this

you can see very easily that there are

two clockwise encirclements of minus one

and since there's one open loop unstable

pole this means that there is a total of

three closed loop unstable poles now

with unity feedback so that's not a

great design so a little bit more work

needs to be done but we were able to see

that very clearly using the nyis plot

all right well that's all I want to

cover for now I hope you're starting to

see the benefit of Nyquist plots and

hopefully they aren't as impossible to

understand as you might have previously

thought in future videos I'll cover gain

and phase margins with Nyquist bod and

the root Locus methods don't forget to

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videos and thanks for watching