Lecture 14 Stability & the Jordan Form-4 PDF

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LECTURE 14: Stability & the Jordan Form ESE 210 : R. GHRIST

OVERVIEW: We can get the explicit, analytic solution to a linear system in continuous or discrete time, at least, when the eigenvalues are real and distinct. But what about those "other" cases? Well, it's time to deal with those, through a wonderful result from linear algebra called the Jordan Canonical Form. Once we get all those technicalities worked out, what do we do with it all? We will commence with a general treatment of stability in terms of eigenvalues: the Stability Criterion.

RECALL: The explicit, analytical solutions for a linear system are: for a matrix…

So, all we have to do is compute powers and exponentials of matrices. But if we don't care about the explicit dependence on the initial conditions, and if the eigenvalues are distinct, then we have some nice general solutions in terms of a linear combination of eigenvectors and basis solutions.

POWERS & EXPONENTIALS OF JORDAN BLOCKS: This is straightforward…mostly… Block matrices behave nicely with respect to powers and exponentiation; all you need to know are the powers and exponentials of the blocks. Let's go! TYPE 1: real eigenvalue with eigenvector linearly independent of others.

TYPE 2: real, repeated eigenvalues with one eigenvector.

This explains why the basis solutions to a higher order ODE with repeated roots of the characteristic equation are etc. Of course, as basis solutions, you don't bother putting the factorial coefficients out front, but you can clearly see where they come

from: the matrix exponential. So, how do you compute this? Same idea that we used in the 2-by-2 case. You split this matrix into , and show that is nilpotent: , so that the powers (or the exponential series) eventually terminates. What do those nonzero powers of look like? You may wish to recall the ever-useful Binomial Theorem:

TYPE 3: complex non-repeated eigenvalues . this gets a bit complicated… taking a detour through the complex plane allows us to do the following: place the eigenvalues = in polar form. Then, I claim that we can express

not gonna lie -- that is a complicated computation, and I'm not 100% sure it's right… fortunately, the continuous-time case is a lot simpler:

The method here is that that of type 2. You split this into

and then take powers & exponentiate. Not so bad when exponentiating, since sin & cos come to the rescue.

TYPE 4: complex repeated eigenvalues . Nope. Nooope. Nope nope. Noooope nope nope nope. STABILITY: So, what? Who cares about the Jordan form? Let's be honest, you don't really use it in numerical computations, and it's not the easiest thing to compute. However, it really helps with understanding stability of linear systems. Since powers and exponentials are by blocks, and we can analyze the behavior of the block solutions, we get the following criteria on eigenvalues for stability. CONTINUOUS TIME: An eigenvalue is called stable if

.

An eigenvalue is called unstable if An eigenvalue is called neutral if

.

The equilibrium at zero is called stable if all eigenvalues are stable. The equilibrium at zero is called unstable if any eigenvalue is unstable. DISCRETE TIME: An eigenvalue is called stable if . An eigenvalue is called unstable if An eigenvalue is called neutral if

. .

The equilibrium at zero is called stable if all eigenvalues are stable. The equilibrium at zero is called unstable if any eigenvalue is unstable.

LET'S DRAW SOME PICTURES: What I care about is that you know what a Jordan form looks like and how it encodes all the important information about the solution to a linear system. CONTINUOUS TIME: First, work with the 1-by-1 blocks in the Jordan form; these tell you about one-dimensional eigenspaces. The dynamics in these directions are either stable, unstable, or neutral. The hard part, visually, is to compose all of these independent 1-d systems into a global flow. You are used to doing it in 2-d with sources, sinks, and saddles; now do it in 3-d & beyond. (see video material for this lecture)

Next comes the complex 2-by-2 blocks in the Jordan form. These do not yield a 1-d eigenspace but, like their block size, a 2-d eigenspace, within which the dynamics is a spiral (or center, if the real part vanishes). Composing spiral motion with the other eigenspace dynamics is a delightful challenge to one's visual imagination. The following examples are not exactly good drawings, but it's what one finds on the internet…

Most difficult (I think) is dealing with the repeated-root blocks in the Jordan form. They give you -dimensional subspaces with a 1-d straight-line eigenspace, and the other dimensions in the subspace "trying to do the same thing" -- sorry, I can't think of another way to put it. The good news is that, since they all the same eigenvalue, the net behavior is either a big sink or a

big source… …or a big neutral. And that is the tricky part… DISCRETE TIME: Pretty much everything we've drawn for continuous time holds in discrete with a few exceptions. • First, modify the eigenvalue conditions via exponentiation, as we have learned. • Second, think of the discrete time dynamics as "hopping" along a curve. • Third, be careful of negative eigenvalues, which involve a change in orientation along the eigenspace.

AND A BIT OF COMPLEX GEOMETRY: For understanding the stability criteria, you may wish to review a bit of the basic geometry of the complex plane. Recall: given a complex number

, the modulus is

.

Also, as to why the continuous and discrete-time stability criteria seem so different, do you remember the relationship between the discrete and continuous time evolution operators? Recall:

So: consider the stability criteria for the continuous-time operator . In the complex plane, the stable eigenvalues live on the left hand side; the unstable on the right; the neutral along the imaginary axis. What happens when you exponentiate these regions of the complex plane? Recall, once again, Euler's Formula, which states

Ahha…...