Hi everyone my name is Claire Tomlin I'm
a professor of electrical engineering
and computer sciences at Berkeley and
this is a series of modules that we're
recording for the course eecs 221 a
which is linear system theory at
Berkeley this is the 30 second module
and in this module we're going to talk
about exponential stability of linear
time invariant systems okay so in a
previous module we talked about the
different definitions of stability
asymptotic stability and exponential
stability so now we're going to launch
right in and look at one of the cases
we're going to look at the linear time
invariant case so linear time invariant
system again we're going to ignore input
since we the internal stability depends
on the state matrix a and we're going to
ask about exponential stability or try
to characterize what are the properties
of the a matrix such that this system is
exponentially stable okay so the theorem
that we're going to prove here is the
following theorem which is exponential
stability of X dot equals ax so the
equilibrium at 0 of that system is
exponentially stable es for
exponentially stable if and only if all
of the eigenvalues of the matrix a are
in the open left half plane so the
spectrum of a is contained in the open
left half plane so it means all of the
eigenvalues have negative real parts
they're not allowed to be on the J Omega
axis they have to be in the open left
half plane alright so let's prove this
it's a very nice test because typically
you're given the a matrix it's not too
hard to calculate its eigenvalues and so
just by looking at where its eigenvalues
are you can ask whether or not that
system is exponentially stable ok so our
intuition behind proving the
is that we know that X of T for this
system is equal to e to the 80 times X 0
let's just suppose t0 is equal to 0 so
we know that the norm of X of T is going
to be less than or equal to the norm of
e to the 80 times the norm of X 0 and so
in order to prove exponential stability
we go back to the definition of
exponential stability which says that
the norm of X of T is bounded above by a
decaying exponential characterized by
parameters m and alpha multiplied by the
norm of the initial state ok so our goal
is to show that the norm of e to the 80
is less than or equal to M e to the
minus alpha T ok so if we can show this
then that allows us to show that the
state trajectory converges to the
equilibrium at 0 according to a decaying
exponential ok so so basically we're
asking if the eigenvalues show that like
the eigenvalues of a being in the open
left half plane is equivalent to the
that the norm of e to the 80 is bounded
is bounded above by this decaying
exponential ok all right so let's
actually think about what each of the 80
looks like ok so let's um let's erase
this what keep that in mind and prove
the theorem by writing out e to the 80
so we know that e to the 80 if we write
it out it's the following it's going to
be the summation from k equals 1 to N of
polynomials will call them pi k of t
times e to the lambda K T where the lamb
decays from 1 to n are the eigenvalues
of the matrix a okay so this this term
here I said it's a polynomial in time
it's indexed by K so each mode e to the
e to the lambda katie has its associated
polynomial and time pi k of t that pre
multiplies it okay so if we think about
how we compute e to the 80 it's the
inverse Laplace transform of si minus a
inverse and we've done that computation
before basically we compute si minus a
we compute its inverse we perform
partial fraction expansion to basically
compute the inverse Laplace transform of
that and so these polynomials basically
represent the the polynomials that
multiply the Exponential's in the
inverse Laplace transform so they're
going to be terms like K T or well let's
not use K because we're using K as our
index terms like some constant C T or
some constant C T squared or you know
just some constant so they're they're
basically the the coefficients of the
Exponential's in the inverse Laplace
transform of the individual terms when
we take the inverse Laplace transform of
e to the 80 so so this is our general
form these are its what's important here
is that these are polynomials in time
and these are Exponential's and we're if
we're thinking about proving the theorem
assuming this and proving this in terms
of the reverse direction proof we're
assuming that these lambda k's are all
have negative real part ok so these
correspond to decaying Exponential's so
with that representation of e to the 80
then we can
we can write it out as okay and actually
you know I didn't specify this so these
are if we think about this is an n-by-n
matrix these are scalars these are
actually not own they're not polynomials
they're matrices of polynomial so each
of the PI K of T is a matrix of
polynomials which are of the form matrix
of polynomials which are of the form
that we just mentioned so they're like
see CT CT squared etc okay so now if we
look at now the norm of e to the 80
that's going to be less than or equal to
the summation from k equals 1 up to n of
norm pi k of t e to the real part of
lambda k times T so the imaginary part
doesn't add anything to the norm okay
and so now that's less than or equal to
so we've got the norm of a matrix where
the entries in that matrix or polynomial
functions in time so we can replace that
with just some polynomial function p k
of t e to the real part of lambda K T
and now I'm going to include everything
in the summation and just found that by
some polynomial p of t e to the minus
mutti ok so these PK these PK of T's are
polynomials which bound the norm of the
matrix polynomial and then P of T is
just the summation of from k equals 1 to
N of all those individual pk's of T's so
it's just something that bounds this
above if we do that over the summation
mu is going to be the eigenvalue that
will be representative of the eigenvalue
that's closest to the j omega axis so
the slowest decaying eigenvalue so here
mu is going to be equal to negative of
the maximum
of the real part of lambda k for all
lambda k which are in the spectrum of
the matrix a okay so it gives us this is
the if we have a bunch of eigenvalues of
the matrix a so they're all in the open
left half plane by hypothesis of our
theorem then mu is this one here it's
negative so this is minus mu here okay
so mu is positive minus mu is negative
okay and that's the it's the real part
here so if this were if the if this were
imaginary it would just be the real part
of that eigenvalue okay so it's the it's
the slowest decaying eigenvalues it's a
number which is to the right of all of
the eigenvalues in the left half plane
but it still mu as a positive number
okay so that's enough to establish a
bound now so we have a bound on e to the
80 let's just rewrite that up here and
we can use that for establishing the
bound that we need for establishing
exponential stability okay so we've got
the norm of e to the 80 is less than or
equal to a polynomial in time e to the
minus mu T where mu is greater than 0
okay so that's where we are and now we
can do a few things this is a polynomial
function in time this is a decaying
exponential a polynomial grows faster
than any exponential grows more slowly
than an exponential does so a decaying
exponential will decay faster than that
polynomial is growing and it's that
relationship that we can use to
establish the required bound on the norm
of e to the or on e to the 80 okay so
basically what we can say because P of T
is a polynomial we can say that
are all epsilon there exists some M
which may depend on epsilon which is
positive such that P of T can be bounded
between 0 and this M epsilon e to the
epsilon T okay so and that should be P
absolute value of P of T so it's
basically saying that p of t is a
polynomial so it's not going to grow
faster than any growing exponential no
matter how small epsilon is and that's
true for all T greater than or equal to
t0 or in this case 0 okay so that tells
us that therefore for all epsilon there
exists some M of epsilon such that e to
the 80 is less than or equal to M e to
the so I'm going to replace p of t with
this exponential here and I'll just
choose this one e to the minus mu minus
epsilon T here so this is true for all T
greater than or equal to 0 all right so
we're almost done we basically just have
to show that mu minus epsilon is
positive this is our same m of epsilon
so we know that all the eigenvalues of a
are in the open left half plane so we
know that mu is greater than 0 this
statement is true for all epsilon so in
order to make this term here positive
and make that whole thing decay so mu
minus epsilon positive it's there's a
negative sign so this becomes a decaying
exponential we just have to choose
epsilon to be between 0 and mu so
epsilon has to be between 0 and mu mu is
given to us by the eigenvalues of the
matrix a and epsilon is something we can
choose and if this is the case then we
have that e to the 80
the norm of each dat is bounded less
than e to the minus alpha T where alpha
is equal to MU minus epsilon and it's a
positive number which guarantees that
the system is decaying so therefore we
have that X of T is going to be less
than or equal to M e to the minus alpha
T times norm X 0 and we get the result
that we need the system is exponentially
stable okay so what we've proven is that
if the eigenvalues of the matrix a are
all in the open left half plane then the
the system is exponentially stable okay
so to show the other direction is also
easy so if we have that now what we want
to prove is that if the system is
exponentially stable then the
eigenvalues they are all in the open
left half plane so exponential stability
implies I ghen values of a in the open
left half plane so the way I would prove
this is by contrapositive so we'll prove
it by showing that if the if at least
one of the eigenvalues of the matrix a
is not in the open left half plane then
the system is not exponentially stable
so we want to prove that if at least one
eigen value of a is not in the open left
half plane so c 0 minus then x e is not
exponentially stable okay so that's
equivalent to saying that exponential
stability implies that all of the
eigenvalues of a are in the open left
half plane okay so this is just the
other direction of the proof so we
finished proving one direction here and
now we'll go on to this next direction
well this direction is fairly
straightforward we've already done the
hard work if at least one of the
eigenvalues of the matrix
a is not in the open left half plane
then what can happen then the matrix e
to the 80 if we just compute the
solution to that which we know takes on
the general form K equals one to n of pi
k of t e to the lambda K T then for that
let's suppose we have a particular
lambda K which has zero or positive real
part then the solution corresponding
that part of that summation
corresponding to either e to the e to
the lambda k where lambda k has has real
part which is either zero or positive
that does not decay to 0 as T goes to
infinity and that means that the whole
solution is not going to decay to 0 as T
goes to infinity so all you need is one
eigen value of a not being in the open
left half plane to be able to get a
solution which doesn't decay to 0 as T
goes to infinity it's really that I ghen
value having negative real part which
takes over the overall solution and make
sure it decays to zero ok so so that
part is easy to prove and we basically
proven the result now which is that
exponential stability of the system X
dot equals ax is equivalent to saying
that the eigenvalues of a are in the
open left half plane okay so that's a
nice test it's sort of the easiest test
we can do just look at the eigenvalues
of the matrix a what happens if the
system can we come up with weaker
characterizations for internal stability
as we know for linear time invariant
systems exponential and asymptotic are
the same but what about stability so
we'll talk about tests for stability and
will relate this to bounded input
bounded output stability so the
relationship between internal and
bounded input bounded output stability
in our next module thanks very much
