Nyquist Stability Criterion, Part 1

hello and welcome back to control system

lectures let's cover the highly

requested topic of the Nyquist stability

Criterion at least the first part this

is a topic that I can honestly say I

didn't really get as an undergraduate

student but it turns out that I was just

being lazy because the criterian is

actually pretty easy to understand with

just a small amount of concentrated

effort so let me explain it to you the

way I understand it and maybe with this

video and some other textbooks and other

videos on YouTube you'll be able to

create your own understand understanding

of the Criterion as well but first

things first let's talk about what we're

trying to accomplish by using the

Nyquist stability Criterion here is a

typical open loop block diagram of plant

G ofs and Control process H ofs you

generally have both of these transfer

functions for your system since you know

the plant that you're trying to control

and you get to pick the Control process

to control it the open loop transfer

function then just becomes G of s * h of

s but to save time in this video I'm

going to drop the of s's and just write

G * H when we close the loop this

physically adjusts the transfer function

for the entire system and you can reduce

the block diagram into a closed loop

transfer function G / 1 + G of

H so you can determine if your open loop

system is stable by finding the poles of

G * H the poles are values of s that

cause you to divide by zero in the

transfer function and so that the result

blows up to Infinity if there's a pole

in the right half plane then the open

loop transfer function is unstable for

the closed loop system we're still

looking for poles however you can see

from the transfer function that in order

to divide by 0o 1 + GH would have to

equal zero therefore we can check

stability of the closed loop system by

looking at the location of the zeros in

1 + GH this might get a little confusing

because I might accidentally say we need

to check for closed loop stability by

looking at the location of the zero

but what I really mean is that just the

locations of the zeros of 1 plus GH so

just keep that in mind as we move

forward so all we're really trying to do

is find where the zeros are of 1 plus GH

or take the open loop transfer function

add one to it and then find the location

of the zeros why is this so difficult

well for one thing if your open loop

system is high order let's say 50 poles

or something then finding the roots of

that large 50th order equation can be

difficult by hand of course now we have

computers so that isn't a big deal but

even so all you would have is stability

information and no other useful

information like stability margins so

let me show you something real quick in

mat

lab I'll just generate a random transfer

function called CIS I'll plot the

locations of the poles and zeros for

this open loop system and then right

next to it I'm going to plot the poles

and zeros of one plus the open loop

system so here you can can see how the

zeros in the right graph are not

obviously related to the poles in the

open loop system in the left graph

that's because when we add one to the

open loop system it can really move the

poles and zeros around however let's

generate the Nyquist plot for G * H and

1 plus G * H hey this kind of looks like

Mickey Mouse randomly anyway uh isn't

this much simpler looking all we're

doing is adding one and just shifting it

over so adding one to the open loop

transfer function and trying to find the

pole zero locations is hard graphically

but adding one to a nyis plot is really

easy to do now the only thing we need to

know is how to determine stability and

stability margins from this graph of

squiggly lines and once we know that

then we're golden luckily it's pretty

straightforward once you know what to

look for but before I tell you that I

first want to show you how this plot is

generated and in doing so I think

reading it and determining stability

will become obvious to start we need to

discuss kosh's argument principle let's

say that we have this arbitrary transfer

function s + 2/ S +1 and I'll put the S

plane on the left here and mark the

locations of the poles and zeros for our

transfer function there's a pole at Min

-1 and a zero at minus 2 now if we take

just a single point in the S plane let's

say s = -1 + J and plug it into our

transfer function which is s + 2/ S +1

you'll get a new complex number out 1us

J and let me plot this new complex

number on a new plane that I'm just

going to give an arbitrary letter W the

W plane this process of plugging in one

complex number in the S plane and

getting a new complex number in the W

plane is called mapping so the transfer

function Maps a point from the S plane

to a different point in the W plane and

if we add a second s s point near the

first we're going to get a second Point

somewhere else on the W plane and if we

plugged in more and more points on the S

plane as to form a continuous line then

this will form a continuous line in the

W plane and if you draw a continuous

line that connects back to itself we

call this a contour and it'll generate

some sort of continuous squiggly line

over in the W plane that also connects

back up with itself we're going to call

this line in the W plane a plot except

that's squiggly line isn't just random

gibberish it actually contains the phase

and magnitude information from each of

the systems poles and zeros let me

explain it in the following way using

just a small amount of math but really

concentrate on the idea rather than the

math if you find it confusing I'll

simplify our transfer function just a

bit for this example down to a single

zero at s = minus 2 and I'll map the

point s = -1 + J over to the W plane

using our transfer function s + 2 1 and

we find it to be 1 + J now here's the

really interesting part the phaser of

the point in the W plane is exactly the

same as the phaser between the zero and

the point we chose in the S plane and

this concept extends to multiple poles

and zeros as well and what's nice is

that you can figure out the mapping

between the S plane and the W plane

graphically I'll show you how in this

set of three poles and two zeros step

one is to pick the point in the S plane

that you want to map over to the W plane

step two is to draw the phasers from

each of the poles and zeros to that

point step three is to determine the

magnitude or the length of the phaser by

multiplying all of the zero phaser

magnitudes and dividing out all of the

pole phaser magnitudes this resulting

magnitude sets the length of the phaser

in the W plane

step four is to determine the phase by

adding the phase of all of the zeros and

subtracting the phase of all of the

poles so in our case we' add the two

blue angles subtract all of the orange

ones which just by eyeballing it looks

like the phase would be just greater

than 90° and so it would be here in the

W

plane so you might be wondering why I'm

telling you all this we aren't going to

be doing this mapping graphically in

general since it's usually easier just

to plug in the point and solve the

equation however understanding what is

happening in a graphical sense is going

to help us interpret the plot later on

let me show you what I mean using matlb

again I wrote a small program in mat lab

to give you a visualization of what I

just drew for you here we have a

transfer function with two poles and one

zero you see the exact function in the

title I've chosen a value of s right in

the middle at s = minus 2 and on the

right is the mapping of that point into

the W plane you can see very quickly

that if we add the phase of the zero

which is 180° and subtract the phases

from the poles which is positive and

negative of the same angle so they

cancel each other out that the phase

should be 180° on the point on the right

and it is now let me move this point to

a new spot in the S plane now the phase

from the zero is+ 90° and the phase from

the pole is Maybe around 60° or so and

therefore 90- 60 is about 30° and that's

again what we see of the point in the W

plane now if I sort of Trace out a

contour with this point but make sure

not to include any Pole or zero then the

addition and subtraction of the phases

will never go around

360° and you can see this very clearly

on the W plot that the phase just sort

of hovers between 150° and 210°

however if I do encircle zero then as we

go along the Contour in a clockwise

Direction we'll be adding 360° of phase

as we move around the zero this causes

the point in the W plane to rotate 360°

in a clockwise Direction like our

contour and if I Circle a pole then

we're subtracting 360° of phase as we

move around the pole causing a

counterclockwise rotation in the

plane now I've written a second demo

that might make this a little bit easier

to see here I have a single pole that I

encircle and the resulting plot circles

the origin one time in the

counterclockwise Direction since we're

subtracting 360° of phase now I'll add a

second pole and rerun it and as you can

see there's going to be two rotations

around the origin one for each pole

finally I'm going to add two zeros as

well and now the resulting plot doesn't

Circle the origin at all and that's

because we're adding 2 * 360° of phase

from the two zeros but at the same time

subtracting 2 * 360° of phase from the

two poles sure the plot on the right

which is the green line is still going

to have a bunch of circles and squiggles

in it but it never encircles the origin

which is really important to

note and that is really all you need to

know for koshy's argument principle that

you can tell the relative difference

between the number of poles and zeros

inside of a contour by how many times

the plot circles the origin and in which

direction so let's see if I gave you

this arbitrary contour and then told you

that the mapping in the W plane looked

like this what could you tell me about

what's inside well let's see it circles

the origin once in the clockwise

Direction so there must be one zero

inside the Contour right wrong well sort

of wrong remember remember all this

tells us is that there is one more zero

than poles inside the Contour so if

there are no poles then there is one

zero but what if there's three poles

then there must be four zeros now let's

take a new contour and if I said that

the mapping looked like this where it

encircled the origin twice in the

counterclockwise Direction well then you

would say that there are two more poles

than zeros and if we knew that there

were only two poles to begin with then

we could confidently say that there were

no zeros in this Contour in order to

figure out how this information helps us

let's get back to our problem we're

trying to figure out if there are any

zeros in the right half plane for 1 + GH

because if there are that means that the

closed loop system is unstable and now

we know how to use koshy's argument

principle to determine if there are any

zeros inside of a contour so at this

point it should be obvious that if we

want to see if there's any zeros in the

right half plane then we need a contour

that encloses the entire right half

plane and we do that by saying that the

Contour runs the entire length of the J

Omega axis up to positive J infinity and

then it sweeps around at Infinity to

enclose the entire right half plane and

then goes back up the negative J Omega

line back to zero now if there are any

zeros in the right half plane we're

going to know about them this is called

the Nyquist Contour all those other

Contours we drew didn't have a name

so maybe I'll call one like the Douglas

Contour or something but when you map

the Nyquist Contour into the W plane you

get What's called the nyis plot it's all

those squiggly graphs I plotted in mat

lab earlier and again you get those

plots by plugging in every single value

along the J Omega axis and then all the

points along Infinity in the right half

plane this is easier to do than you

might imagine but I'm going to cover

that in the next video for now we'll

rely on the computing power of mat lab

to generate those for us so now if we

take 1 + GH and use this function to map

the Nyquist Contour in the S plane into

a Nyquist plot in the W plane we can

instantly see how many times the origin

is circled and in which direction from

there we can determine how many more

poles or zeros lie within the Contour to

do this mapping though we need to find

the plot for GH and then shift the

entire plot to the right by one but this

is kind of difficult because there's

lots of Curves and circles so instead of

Shifting the plot to the right by one we

shift the origin to the left by one and

this is why we're concerned with how

many times minus one is circled instead

of how many times the origin is circled

so the steps are that we take the open

loop transfer function GH we make a

Nyquist plot by plugging in each point

on the Nyquist contour and count the

number of times minus one is in circled

and in which dire direction from that we

can determine how many more poles or Zer

are inside of the Nyquist Contour which

again is the entire right half plane but

to know for sure whether there's a zero

in the right half plane we first need to

know how many poles are in the right

half plane luckily we usually know

exactly how many poles are in the right

half plane of 1+ GH because it's the

exact same number of poles in the right

half plane as the open loop system GH

and since we usually have a good

understanding of our open loop plant

then we already have that information

and this drives the famous equation that

the number of Zer in the right half

plane Z is equal to the number of

clockwise encirclements of

minus1 plus the number of open loop

right half plane poles or another way of

saying this is that in order to

guarantee there are no zeros in the

right half plane then you better have

exactly one counterclockwise

encirclement for every open loop pole in

the right half plane any less than that

and you know you have at least one right

half plane zero messing things up so one

very important bit of information you

should be taking away from this is that

if someone gives you a nyis plot and

asks you to tell them how many zeros are

in the right half plane of 1 plus GH you

can't unless you know a little

information about the open loop system

GH namely how many unstable poles there

are now one thing that sets Nyquist

plots apart from Bodi plots in terms of

determining closed loop stability is

that with Nyquist you have the ability

to analyze a system with an unstable

open loop plant if you try to determine

closed loop stability of an unstable

open loop plant with Bodie you're going

to be out of luck because you'll

possibly Come Away with the wrong answer

and not know it there are also other

examples of other types of systems that

will confuse you in Bodi plot form but

will always work perfectly in Nyquist

form bodh plots are nice because they're

easy to sketch by hand however if you

have access to a computer you won't go

wrong if you always use Nyquist plots to

determine stability and stability

margins and then just use Bodi plots for

frequency response

analysis so this is all I wanted to

cover in part one of the Nyquist

stability Criterion in the next video

I'm going to explain what to do when

your open loop plant has a pole or two

on the J Omega axis and also how to

determine phase and gain margins

directly from the nyas plot plus I'll

give you a few examples now if you have

any questions or comments please leave

them below and I'll try to answer them

if I can subscribe so you don't miss any

future videos and as always thanks for

watching