Something seemed fishy to me, so I went back to this and found a mistake. I need your help to see if what I have done this time is correct and legal.
I'll start from the top again.
-- Show that x(k) goes to zero in the limit.
Let ϵ>0. Now, we know that 0.5k and u(k) both go to zero(by...
-- Show that x(k) goes to zero in the limit.
Let \epsilon > 0 . Now, we know that 0.5^{k} and u(k) both go to zero, therefore, for all \epsilon_{1} > 0 , there exists k_{1}>0, such that if k > k_{1}, then | 0.5^{k} | < \epsilon_{1} and | u(k) | < \epsilon_{1}. Now
x(k) = | 0.5^{k}...
Hi. Thanks a lot for replying.
If I replace 0.5^{i}u(k-i) by M where M is it's max value, then I can get an upper bound on x(k). But this upper bound might blow up as k tends to infinity (for the case where M>1). I won't be able to conclude what happens to x(k) as k tends to infinity.
Homework Statement
Hello. I am trying to prove a result that I have been making use of, but never really proved. Consider the recurrence equation
x(k+1) = 0.5 x(k) + u(k),
where u(k) is a bounded sequence. For this problem, assume that u(k) goes to zero. I want to prove that x(k) goes to...
Homework Statement
Let x_{n} = y_{n} + z_{n}
Also, x_{n}>0 , y_{n}>0 , z_{n}>0 . We also know that x_{n} converges.
Prove that y_{n} converges.
Homework Equations
I want to use the Cauchy criterion because the limits are not given. So start with an [tex] \epsilon >0 [/itex]...
Hi,
I had a question about understanding some basic thing about the Hausdorff dimension. Specifically, I'm trying to understand why the two dimensional Hausdorff dimension of a 1-d line is zero.
In terms of the two dimensional Lebesgue measure, I can see that I can cover the line by a...
I think your system has two degrees of freedom. You should have two differential equations, one for acceleration of x_{1} and one for acceleration of x_{2}.
Also, It seems from the problem statement that you have one input f, and two outputs x_{1} and x_{2}. Which means that you should have...
Thanks a lot. I guess now I will move on to the next page in the book. I don't know if I should open a new thread or not but here goes.
This is Proposition 1.3 in Folland. I am interested more in the notation than in the proof given in the book. The proposition is
If A is countable, then the...
Homework Statement
I should mention beforehand that I do not come from a math background so I may ask some trivial questions.
I am reading the book "Real Analysis" by Folland for a course I am taking and am attempting to understand a definition of product sigma algebra. It is stated in the...
Homework Statement
I think I will start with the figure below:
The wire of infinite length rotates about the point "a" with constant angular speed. The bead starts out at rest. There is no friction or gravity.
I have already derived the equations of motion for this system (I used...
Homework Statement
I have set up this problem for myself.
Let P be a system of the form
x' = Ax + Bu
y = Cx + Du
The definition of a "state" is:
"x(t) is a state for a system P if knowledge of x at some initial time t_{0} and the input u(t), t \geq t_{0} is sufficient to uniquely determine...
Homework Statement
This is not a homework problem. I encountered this while working with total least squares for the first time. Ultimately a point is reached where Az=0 must be solved. z is of the form [x,1]^{T}. Let A be nxm, z be mx1.
Suppose A is rank deficient by one. So the SVD of A...