Introduction to System Stability and Control Video Lecture - Electrical

welcome back to control system lectures
in this video I'm going to present an
introduction to system stability and
control before we begin let's start by
defining stability stability is a
measure of the tendency of a systems
response to return to zero after being
disturbed if a system is exposed say to
an impulse function for example you hit
it with a hammer and the system response
equals zero as time approaches infinity
then this is a system that is considered
stable if instead the response blows up
to infinity then the system is unstable
another way you might see this explained
is that for a stable system a bounded
input produces a bounded output or bibo
stability bounded-input bounded-output
which means that the only way to get an
infinite output is to put in an infinite
input I think the easiest way to grasp
this concept is with a classic example
take a system that consists of a small
ball that sits at the bottom of a valley
the ball is initially at rest at the
equilibrium point at the bottom of the
valley
if the ball is disturbed say with a
force that pushes it away from its
equilibrium point then the ball will
start to oscillate and if there's
friction eventually come to rest back at
the equilibrium point this is a stable
system however if you were to take the
ball and place it on top of a mountain
and perform the same experiment you'll
see that this is an unstable system the
ball is initially at rest at the
equilibrium point at the top of the
mountain however once it's disturbed
from this point it will roll down the
hill away from the equilibrium this is
very much similar to balancing something
on its end like an egg it can be done
but since it's unstable it is
susceptible to disturbances and will
fall over as you can imagine there is
some delineation between having a system
that is stable and one that is unstable
in the case of the ball we can visualize
this as a flat table that goes on for
infinity or halfway between these two
different scenarios we can see that if
the table slopes up ever so slightly on
each side then the ball will event
we come back to rest at the starting
point after it's been disturbed this is
a stable system if the table ever so
slightly slopes down then when the ball
is disturbed it will continue off into
infinity and this is an unstable system
we'll get back to this idea of a point
between stable and unstable systems
later on in this video for now let's
take a second example which i think is a
little bit more practical than just a
ball sitting at the bottom of a valley I
want to show you why there's an inherent
stability in the shape of a dart a
traditional dart has a lot of mass at
the tip which skews the center of
gravity towards the front of the dart
and it also has flights on the back end
of the dart these are just like feathers
on an arrow and this skews the center of
pressure towards the aft end if we look
at this arrow in equilibrium during
flight that is when it's flying straight
and level to the air there are no
aerodynamic forces perpendicular to the
dart shaft you can visualize this easier
by looking at the dart head-on you can
see that both the darts shaft and the
flights on the aft end are both aligned
with the air and don't produce any
torque about the center of gravity but
what if the dart is disturbed off its
equilibrium flight path and this can be
either from an initial rotation given by
the thrower or from wind gusts as it
flies through the air then the dart
would be in a position relative to its
flight path that looks something a bit
like this in this case the air pressure
acting on the flights will generate a
restoring torque about the center of
gravity in the direction that causes the
back end of the dart to line back up
with the air currents and the dart will
oscillate back and forth about this
equilibrium point
until the oscillations are damped from
the loss of energy through friction this
is similar to a weather vane that
rotates back and forth to align itself
with the wind
this system is stable because the torque
generated by the air pressure tends to
rotate the dart back to a starting
equilibrium point alternatively we can
see that we can make the dart unstable
by moving the center of gravity further
towards the back of the dart
moving the center of pressure forward to
the front of the dart this can be
demonstrated by trying to throw a dart
backwards this causes an unstable
situation because this puts the center
of pressure in front of the center of
gravity which I'll show graphically here
shortly the dart though still has an
equilibrium point and that's when the
velocity vector is parallel to the dart
and you can still see this from the
head-on view that shows that the flights
are still lined up with the air currents
the difference comes though when the
dart is disturbed off its equilibrium
point now the air pressure on the
flights tend to generate a diverging
torque about the center of gravity which
causes the front end of the dart to flip
away from its starting equilibrium
position and that's why attempting to
throw a dart backwards causes it to flip
around in the air but just like with the
ball lying on a flat table there is a
point where you could place the flights
that would delineate between a stable
system and an unstable system for a dart
this is when the center of pressure and
the center of gravity coincide in this
case no aerodynamic torque is generated
regardless of the orientation of the
dart relative to its flight path
practically speaking though if we move
the flights so that the center of
pressure is ever so slightly aft of the
center of gravity then the dart will be
stable and the reason is because this
will still generate a restoring torque
although very small but this brings up
an interesting question how stable is
stable enough as an engineer you aren't
just concerned with building a stable
system you also want to ensure that the
system has enough stability margin to be
able to handle small parameter changes
in the systems transfer function now
these small changes might come about for
a variety of reasons and I'll use this
dart example to illustrate two of them
when manufacturing darts you might not
be able to reproduce each one with the
exact same center of gravity or center
of pressure therefore there's going to
be some uncertainty in where those
parameters exist on any given dart also
the characteristics of the dart could
change over time for example dropping
the dart could bend the flights which
will change the center of pressure or
chip off part of the mass causing
the center of gravity to change in order
to have a robust system that can handle
variations in these parameters you have
to add margin to your design but there's
another reason adding stability to your
design also decreases the impact that
disturbances have on the system we can
see this with the ball example here by
increasing the steepness of the valley
or adding stability the same disturbance
force will result in a smaller magnitude
of the oscillations so you can imagine
that if you flicked each of these three
balls in these three different cases
they might have responses that look
similar to these so now you might be
thinking well why not design the system
to have huge stability margins if a
little stability is good then a lot of
stability might be great well this might
not be a good solution either the reason
is that just like a stable system won't
respond much to disturbances it's also
not going to respond much to actual
commands either for example a very
stable airplane seems like a good idea
however the more stable and airplane is
the slower it will respond to pilot
commands because it won't want to move
away from its equilibrium point so far
what we've looked here open-loop
responses of a system that is the
response of the system due to its
inherent passive design and an open-loop
system can be inherently stable like in
the case of the ball inside the valley
but what happens if the inherent design
is unstable like the case of the ball
sitting on top of a mountain well the
system can be made stable by active
control means and this is one of the
jobs of the control engineer we'll spend
a lot of time on this and other videos
so I just want to quickly touch on it
here if you were given this unstable
open-loop system you can make it stable
by adding feedback control
I'll start by drawing a block diagram
for a typical control system the plant
consists of the ball and the mountain
and the output of the plant is the
horizontal position of the ball I want
to measure the position of the ball with
a sensor and then feedback that measured
position my reference position is zero
since I want the ball to stay right at
the top of the mountain
I can generate an error term which is
fed into a controller and the controller
can create commands that are fed into
some actuator that affects the balls
position so let's go ahead and populate
this to measure the position of the ball
I think I'd use maybe a radar gun that
would bounce the signal off the ball and
measure the time it takes to return I'll
need one of these on each side since the
ball can roll down either side of the
mountain and I can write the transfer
function for the radar gun in my block
diagram now for the actuator maybe I
could use variable speed fans one on
either side that I can control the
amount of wind that each one produces
which I can use to generate a force on
the ball to blow it back up the hill and
I'll put the transfer function for the
fan in the spot for the actuator now for
the controller I could just use a
computer and it would take inputs from
the radar gun and then feed the error
term through a PID controller that will
generate commands to go to the fan don't
worry too much about PID controllers
we'll talk about those in a future
lecture so now when the ball is
disturbed it'll start to roll down one
of the sides of the mountain the radar
guns will measure the motion and through
the controller activate the fans to
resist and reverse the motion of the
ball and if I designed this control loop
correctly not only will this system be
stable but we'll have enough stability
margin to account for any variations in
the system and to reject unwanted
disturbances but it won't be so stable
that the ball can't move at all
now this is just one solution of many
different designs that can accomplish
this very same thing and I'll admit that
this one's fairly silly and probably a
bit impractical but that's one of the
great things about being a control
systems engineer you aren't forced to
stay with the same design for every
single problem in fact you can tackle
the same problem in different ways as
sensors and actuators and materials
change over time for example instead of
a fan we could have used Springs
attached to the ball that will impart a
force back up the mountain and instead
of a radar gun maybe there's pressure
sensors in the mountain itself that can
tell where the ball is
so that's my question for you what
sensors and actuators would you use in
this situation to create a stable system
I'd love to hear your ideas so please
leave a comment in the section below
also don't forget to subscribe because I
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