00:00:00
Sliding mode control
is a nonlinear control
00:00:03
law that has a few
nice properties,
00:00:05
like, robustness to
uncertainties and disturbances,
00:00:08
and it's relatively
straightforward to implement
00:00:11
and tune.
00:00:12
However, one drawback
is chattering,
00:00:14
where the system state and,
therefore, the actuators trying
00:00:17
to maintain that state jitter
back and forth quickly,
00:00:21
which might not be desirable.
00:00:23
But there are mitigations
for this chattering.
00:00:26
And overall, I think sliding
mode control is worth
00:00:28
understanding because, for one,
it's kind of a neat controller
00:00:33
and, for two, you
might find that it's
00:00:35
right for your application.
00:00:37
So this video is going to be
divided up into three parts.
00:00:40
In the first part, we'll walk
through a graphical explanation
00:00:42
of the controller and
try to get some intuition
00:00:45
into how it works.
00:00:46
For the second part, we'll
walk through a derivation of it
00:00:49
so that you understand
the mathematics.
00:00:51
And in the third part, we're
going to show a quick example
00:00:54
in Matlab and Simulink.
00:00:55
I think it's going to be really
interesting and informative,
00:00:58
so I hope you stick around.
00:01:00
I'm Brian, and welcome
to a Matlab Tech Talk.
00:01:04
To build up some intuition into
what sliding mode control is
00:01:07
and how it works,
we're going to look
00:01:09
at an extremely simple system,
just a mass and a spring.
00:01:13
Let's impart some initial
velocity onto the mass
00:01:16
and get the system moving.
00:01:18
And as you would expect
with a mass spring system,
00:01:20
it's just going to
oscillate forever
00:01:22
because there's no damping, and
so there's no energy being lost.
00:01:26
When we discuss sliding mode
control for second order
00:01:28
systems like this one,
it's helpful to visualize
00:01:32
the dynamics with a
phase plane diagram.
00:01:34
In a phase plane
diagram, the two states
00:01:37
are plotted on two
orthogonal axes.
00:01:39
And for this system, they
represent position and velocity.
00:01:44
So if I pause the system,
right here at this moment,
00:01:47
the position is zero,
and the velocity
00:01:50
is some large positive value.
00:01:52
So the current state of the
system in the phase plane
00:01:54
diagram is right here.
00:01:57
Now, as we move forward in
time, the velocity goes down,
00:02:01
and the position
increases to the right.
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Then the velocity
becomes negative,
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and the mass moves back.
00:02:06
And this eventually completes
a circle in the phase plane.
00:02:10
And like we said earlier,
there is no damping.
00:02:13
So we're just going to go
around this circle forever.
00:02:15
And this path that is traced
out is called a trajectory.
00:02:21
Now, one way to
change this trajectory
00:02:23
is by changing
the system itself.
00:02:25
Here, I'll add some damping
and then give the system
00:02:29
the same initial velocity.
00:02:30
And once again, the
mass moves to the right,
00:02:32
and the velocity reduces.
00:02:34
However, due to
damping, the system
00:02:36
now follows a trajectory
that spirals down
00:02:39
until the system comes to rest
at the zero position, zero
00:02:42
velocity state.
00:02:44
OK, let's just look at the phase
plane diagram going forward.
00:02:48
And instead of looking at
an individual trajectory,
00:02:51
we can plot arrows
representing the direction
00:02:54
of the entire phase plane field.
00:02:56
So hopefully with
these arrows, you
00:02:58
can start to visualize the
trajectory for any arbitrary
00:03:01
starting state.
00:03:02
For example, if we
start the system here,
00:03:05
you can see how the trajectory
will follow the arrows
00:03:08
and once again come
to rest at the origin.
00:03:12
This is the uncontrolled
behavior of the system,
00:03:16
you know, just its
natural response.
00:03:18
But we can also change this by
adding a feedback controller.
00:03:22
Feedback changes the
dynamics of the system,
00:03:25
and therefore, changes the
shape of the trajectories--
00:03:29
preferably to something
that is desired.
00:03:32
So with that being said, let's
talk about sliding mode control
00:03:35
and, specifically, how it
affects the trajectories
00:03:38
in a phase plane.
00:03:41
The mass spring damper
system is second order,
00:03:44
which means that, in
general, the trajectories can
00:03:47
bend and circle around in two
axes across the entire phase
00:03:50
plane like we see here.
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Now, sliding mode control tries
to constrain the trajectories
00:03:56
so that they can only exist
along a slice within the plane.
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And that's this
yellow line here.
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Now, the question
might be why could
00:04:05
we possibly want this kind
of constrained behavior?
00:04:08
Well, one reason is that the
controlled system will now
00:04:12
behave like a simple
first order linear system.
00:04:16
And we can see why
this is the case.
00:04:18
The equation for
this yellow line
00:04:19
can be written as bx1
plus x2 equals zero, where
00:04:23
x1 is position
and x2 is velocity
00:04:26
and b is the parameter that
adjusts the slope of the line.
00:04:29
And since x2 is just x1 dot,
we can rewrite this equation
00:04:33
as x1 dot equals minus bx1.
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This is a first order system.
00:04:39
So if the sliding
mode controller
00:04:41
ensures that the trajectory
stay along this line,
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then our second order
system will behave exactly
00:04:47
like this first order system.
00:04:49
And we can choose the
orientation of this yellow line
00:04:52
to describe the first order
response that we're looking for.
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This equation is called
the switching function, s,
00:04:59
and when we set s
equal to zero, we
00:05:01
get the sliding surface, which
is the yellow line itself.
00:05:05
And it's called this because the
states slide along that line.
00:05:09
Now, I want to be clear
that often when you
00:05:11
see sliding mode
control described,
00:05:13
it's done with second order
systems and a phase plane
00:05:15
diagram like I'm showing here.
00:05:17
But this is only because
second order systems
00:05:19
are easy to visualize and draw.
00:05:22
In general, we could
have higher order systems
00:05:24
and still describe a
lower order switching
00:05:27
function that cuts
through a cross section
00:05:29
of the original system.
00:05:31
The switching function can be
any function of system states,
00:05:34
even nonlinear.
00:05:36
However, it's most
common just to set it
00:05:38
to a linear
combination of states,
00:05:40
where the number of
rows of the matrix c
00:05:43
is the number of states in the
system and the number of columns
00:05:46
is the number of inputs
u into the system.
00:05:50
So for our mass spring
damper, c would be 2 by 1.
00:05:55
OK, so we now have
our sliding surface,
00:05:58
and the first job
of the controller
00:05:59
should be to keep the
states on that line.
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That's priority number one.
00:06:04
However, obviously, the
system could physically
00:06:06
be in any state at any time.
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There's no guarantee that
the system states are
00:06:10
initialized along this surface.
00:06:12
Therefore, the controller
has a second job.
00:06:15
It has to make sure
that the system can
00:06:17
reach the sliding surface.
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So, for example, for any
state above the yellow line,
00:06:22
the controller needs to adjust
the phase plane in such a way
00:06:25
that all of those arrows push
down towards the surface.
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And for any state
below the yellow line,
00:06:31
the controller has to drive
the states up to the surface.
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In this way, no matter
where the system starts,
00:06:38
the sliding mode controller
will first drive the states
00:06:41
to the sliding
surface during which
00:06:43
the system is going to behave
like a second order system.
00:06:46
And then once it
gets to the surface,
00:06:47
it's going to slide along it
and behave like a first order
00:06:51
system.
00:06:53
Now, it might not be
obvious at this point
00:06:55
why this is actually a
robust control strategy,
00:06:58
but with this diagram, we
can start to understand it.
00:07:01
Let's zoom into the phase plane
right at the sliding surface,
00:07:05
and let's look at these
arrows that are pointing in
00:07:08
towards the yellow line.
00:07:10
From their starting
point, both blue lines
00:07:12
are pointing in the correct
direction, just like we want.
00:07:15
But let's say that there's
some external disturbance
00:07:17
that we didn't account for.
00:07:19
That disturbance is going to
have some delta contribution
00:07:22
to the existing
phase plane, which
00:07:24
I'm showing here in orange.
00:07:26
And the combined effect
will be this green arrow.
00:07:29
So even with this disturbance,
the sliding mode controller
00:07:32
still causes the
states on either side
00:07:34
to tend towards the
switching surface.
00:07:37
And this is true no matter which
direction the disturbance is in.
00:07:42
Now, if the
disturbance is larger
00:07:45
than the controller can
handle, then the states
00:07:47
might not reach the
sliding surface.
00:07:49
In this case, if you
know ahead of time
00:07:51
that the disturbances are
going to be this large,
00:07:54
you can just increase the
gain in the controller,
00:07:56
which has the effect of
lengthening the blue line.
00:08:00
And now the combined
result, once again,
00:08:03
points in the correct direction.
00:08:05
Effectively, what we're doing
here is using very high gain
00:08:08
and switching back
and forth between two
00:08:10
different controllers
to make sure
00:08:12
that the system stays
along the sliding surface.
00:08:15
And with that high
gain, we get the benefit
00:08:17
of not being susceptible
to system uncertainties
00:08:20
and external disturbances
because the high gains will just
00:08:23
overpower them, which
is great for robustness.
00:08:28
However, the downside of this
switching high gain approach
00:08:32
is that we introduce
chattering into the system.
00:08:34
Chattering is when the system
states bounce back and forth
00:08:37
on either side of
the switching line,
00:08:39
and it occurs because of the
finite time between switching.
00:08:42
With high gain, this means
that for one time step,
00:08:46
the system is really
driving the state hard
00:08:48
in one direction, which
jumps it over the line.
00:08:52
And then the very
next time step it
00:08:54
wants to drive it hard
in the other direction.
00:08:57
And delay in the system only
exacerbates this chattering.
00:09:02
Now, if we lower the
gain, the controller
00:09:04
drives the system
less hard around this
00:09:07
sliding surface, which
will reduce chattering.
00:09:09
But as we saw earlier,
this affects the robustness
00:09:13
of the system.
00:09:14
So there's this trade
off between the two.
00:09:17
And since robustness
is important,
00:09:20
we want to find
other ways to address
00:09:22
chattering other than
just lowering the gain.
00:09:26
And one way to do this
is to define a boundary
00:09:29
layer near the
surface within which
00:09:31
you can change how the system
switches between the two
00:09:34
different controllers.
00:09:35
For example, one type
of boundary layer option
00:09:38
is saturation, where outside
of the boundary layer
00:09:41
you use full gain, but
inside the reaching log
00:09:45
gain is proportional to the
distance to the surface.
00:09:48
In this way, as long
as the system stays
00:09:51
within the boundary,
the control is smooth
00:09:53
and the chattering is reduced.
00:09:55
And it's only when the state
nears the edge of the boundary
00:09:58
does that heavy gain
really come into effect
00:10:01
and try to push the state back.
00:10:03
This is called saturation
because graphically it
00:10:06
looks like the symbol
for saturation,
00:10:08
where outside of the
boundary the gain
00:10:10
is capped at the gain
of the controller
00:10:12
and inside the boundary,
it's scaled linearly.
00:10:15
However, saturation is just
one boundary layer option
00:10:19
of many that are out there.
00:10:21
And so the point
I want to make is
00:10:22
that there are
methods for reducing
00:10:24
the chattering of a
sliding mode controller
00:10:26
without impacting
robustness too much.
00:10:30
OK, so that is, sort of,
the graphical introduction
00:10:33
to what sliding mode
control is doing.
00:10:35
Now to design one,
there are several things
00:10:37
we need to consider.
00:10:38
One, we need to define
a sliding surface.
00:10:41
This represents the
reduced order dynamics
00:10:43
that we want for our system.
00:10:45
Two, we need to come up
with a control scheme that
00:10:48
ensures that no matter
where the system starts,
00:10:50
it's always being pushed
towards that sliding surface.
00:10:53
And three, we need to come
up with a method for dealing
00:10:56
with chattering that is
acceptable for the hardware
00:10:59
and the system requirements
that you're working with.
00:11:01
So with all of that
being said, let's
00:11:04
move on to looking at
the actual equations
00:11:06
for sliding mode control and
try to figure out exactly how we
00:11:10
accomplish all of this.
00:11:13
OK, let's consider the
mass spring damper system
00:11:15
from earlier, which has
this differential equation.
00:11:19
And we can define the
state variables x1 and x2
00:11:21
and then rearrange this into
state space representation.
00:11:25
And so now this is in the form
x dot equals some state function
00:11:29
f plus some input
function g times
00:11:32
u, which is a
general form that can
00:11:35
be applied to many systems,
even nonlinear systems.
00:11:39
So for this
derivation, we're going
00:11:40
to use this equation
for the system model.
00:11:44
All right, let's design a
sliding mode controller for it.
00:11:47
And to begin, we need to
define a switching function,
00:11:50
s, and recall from
earlier that I
00:11:52
said that the switching function
is a linear combination of all
00:11:55
of the states.
00:11:55
So s equals c transpose x.
00:11:58
And when s equals 0, this
defines the sliding surface.
00:12:02
And if s is not 0, this means
that the system states are not
00:12:06
on that surface.
00:12:07
They could be anywhere
else in the phase plane.
00:12:09
And remember, we
want our controller
00:12:10
to ensure that the system
states always reach that line.
00:12:14
So how can we do that?
00:12:16
Well, let's think
about it this way.
00:12:17
If s is positive, then we
want s dot to be negative.
00:12:22
And if s is negative, we
want s dot to be positive.
00:12:25
If that is the case, then no
matter where the states are,
00:12:29
they're always going to be
moving towards s equals 0.
00:12:32
And this means
that we can define
00:12:34
the reachability
condition that s times
00:12:37
s dot must be less than zero.
00:12:39
Or another way of putting
it is that the sign of s
00:12:41
must always be opposite
of that of s dot.
00:12:45
It's a pretty simple idea.
00:12:48
All right, so this is the
reachability condition
00:12:50
that our controller
needs to satisfy.
00:12:52
So let's talk about
how we do that.
00:12:54
We know s.
00:12:55
It's c transpose x.
00:12:57
So to meet this
condition, we need
00:12:59
to define s dot so that
it's the opposite sign.
00:13:03
And the easiest way to do
that is to just say that's dot
00:13:06
is the negative sign of s.
00:13:08
Now, we're guaranteed that the
reachability condition is met.
00:13:12
And if we want to adjust
the speed of s dot--
00:13:15
that's the speed with which the
states approach s equals zero--
00:13:19
then we can just
add a constant eta.
00:13:22
Eta is that gain that changes
the length of the arrows
00:13:25
in the phase plane and is
responsible for that trade-off
00:13:27
between robustness
and chattering
00:13:29
that we talked about earlier.
00:13:32
All right, so this
particular equation
00:13:34
is called the constant
rate reaching law.
00:13:36
And another popular one is
the exponential reaching law.
00:13:40
But there are many,
many different kinds
00:13:42
of reaching laws.
00:13:43
Basically, anything
that satisfies
00:13:45
s times s dot less than 0
will work as a reaching law.
00:13:48
And choosing one
or another comes
00:13:50
down to how you want the system
to actually reach the switching
00:13:53
surface, how robust
you want it to be,
00:13:55
and what kind of
chattering you can handle.
00:13:58
All right, now, instead
of picking a specific one
00:14:01
for this derivation,
I'm just going
00:14:02
to set it to the function h.
00:14:05
Therefore, s dot needs to
equal the reaching law h.
00:14:10
But s dot also has to equal
the time rate of change of s,
00:14:14
the switching surface
that we defined.
00:14:16
And so if we take
the derivative of s,
00:14:18
we get the partial
derivative of s
00:14:20
with respect to x times the
derivative of x with respect
00:14:24
to t.
00:14:25
Well, the partial of
s with respect to x
00:14:27
is just c transpose,
and dxdt is just x dot.
00:14:31
And we know what that is from
the dynamics of the system.
00:14:34
So putting this all together,
we get c transpose times
00:14:37
the dynamics of the system has
to equal the reaching law h.
00:14:42
Now we just have to
ask what's the u that
00:14:45
makes this equation true, which
we can find by solving for u.
00:14:49
And this is the standard
sliding mode control compensator
00:14:52
formula.
00:14:54
And here's what I think
is interesting about it.
00:14:56
This one control law defines the
switching surface, c transpose.
00:15:00
And because of the
reaching law, h, it
00:15:03
ensures that the system
states reach that surface
00:15:05
from any direction.
00:15:06
And that's pretty cool, right?
00:15:08
But something to note
here is that in addition
00:15:10
to choosing h and
c, you also have
00:15:12
to have an estimate of
your system dynamics.
00:15:14
That's the f and g.
00:15:16
But the nice thing is that since
this is a robust controller,
00:15:20
it can handle some uncertainty
in the system model
00:15:23
because it's just going to
overpower it with high gain.
00:15:27
Now the third step is
to address chattering,
00:15:30
but it's taking care of also
within the reaching law.
00:15:33
Instead of using the sign
function, which switches harshly
00:15:37
between minus one
and positive one,
00:15:39
we can replace it with
a new function theta.
00:15:42
Now, for theta, we define
a boundary layer with phi
00:15:46
being the distance the boundary
is from the switching line.
00:15:49
And if s is outside
of the boundary layer,
00:15:52
then theta is just
1 or minus 1 like we
00:15:54
had with the sign function.
00:15:56
But if s is inside
the boundary layer,
00:15:59
then phi is proportional to the
distance to the switching line.
00:16:02
So our new reaching law
would be this little bit
00:16:05
of logic times negative eta.
00:16:09
Again, this particular
one is called saturation,
00:16:12
but there are numerous different
boundary layer approaches.
00:16:16
All right, so that's basically
sliding mode control.
00:16:19
Now let's jump over to
MATLAB and Simulink,
00:16:22
and I want to show
you a quick example
00:16:23
of controlling a mass spring
damper system with this control
00:16:27
law.
00:16:29
All right, I have this mass
spring damper subsystem
00:16:32
in Simulink, which takes in
input u, which is a force acting
00:16:36
on the mass, and then it
outputs the two states, position
00:16:39
and velocity.
00:16:40
And if I run the
model right now,
00:16:42
we can see the uncontrolled
response of this system.
00:16:45
And notice that it takes well
over 10 seconds for this system
00:16:48
to settle.
00:16:49
So let's fix that by using
the sliding mode controller
00:16:52
block in Simulink.
00:16:54
And if I click on the block,
I can set the sliding surface
00:16:57
and the reaching law.
00:16:58
Now, I'm going to keep this
in regulation mode, which
00:17:01
drives the system states to 0.
00:17:03
But there is the
option of tracking mode
00:17:05
where you define a reference,
and then the controller
00:17:07
drives the error to zero.
00:17:09
All right, so now I'm going
to set the sliding coefficient
00:17:12
matrix c to 1,1 which basically
means that a equals x1 plus x2.
00:17:18
And then on the
reaching law tab,
00:17:20
I'm going to keep the
constant rate reaching law
00:17:23
and set the gain eta to 1.
00:17:26
And for now, I'm just going to
keep the boundary layer just
00:17:28
as a simple sine function.
00:17:31
All right, so this controller
needs the system states as well
00:17:34
as the g and f functions.
00:17:36
Here f is the a matrix, which
is a function of the spring
00:17:41
mass and damper parameters,
times the state vector.
00:17:45
And with that, we're all set.
00:17:48
Now I'm going to run
this model again,
00:17:50
and you can see the
difference in the output.
00:17:52
At first, the control
input looks smooth,
00:17:55
and this is because
the system is
00:17:56
in the process of reaching
the sliding surface,
00:17:59
where it's going to behave
like a second order system.
00:18:02
And then notice at around two
seconds, it reaches the surface,
00:18:06
and then it begins
to slide along it.
00:18:08
And at this point, the response
looks like a first order system.
00:18:12
But as you can see,
the downside is
00:18:14
that there is a huge amount of
chattering on the control input.
00:18:18
So to address this, we can
go back to our controller
00:18:21
and change the boundary layer
from sine to saturation.
00:18:26
And I'm going to leave the
boundary layer fee at 0.01.
00:18:31
Now I'll rerun the simulation
again, and check this out.
00:18:35
The response looks
pretty much the same,
00:18:38
but we've reduced the chattering
in the system a great deal,
00:18:42
which is fantastic.
00:18:45
Now, this was just a
really quick overview
00:18:47
of using sliding mode
control in Simulink,
00:18:50
and I've left a link to a mass
spring damper example that
00:18:53
goes into much more
detail, including
00:18:55
showing how this controller
is robust to disturbances.
00:18:59
There is also a link
that I left to an example
00:19:02
showing a more practical
application of controlling
00:19:04
a robotic arm.
00:19:06
So I hope you check them
out, and I hope that you
00:19:08
found this explanation useful.
00:19:11
All right, now don't forget
to subscribe to this channel
00:19:13
so that you don't miss any
future tech talk videos.
00:19:16
And also you can find
all of the tech talks
00:19:18
across many different topics
nicely organized@mathworks.com.
00:19:21
And if you liked
this video, then you
00:19:23
might be interested to learn
about other robust controllers
00:19:26
like H-infinity
and Mu Synthesis.
00:19:29
All right, thanks for watching,
and I'll see you next time.
