A control system consists of an 鈥渙utput'; variable x_n & a 鈥渃ontrol'; variable y_n, wich?

A control system consists of an 鈥渙utput'; variable x_n %26amp; a 鈥渃ontrol'; variable y_n, wich


r updated n discrete steps labeled by n = 1; 2; 3 鈥? in each step, d output variable


is 1st updated accordn 2 d rule


x_(n+1) = x_n + 渭y_n;


where 渭 is a constant. Once d output variable is updated in dis way, d control variable is updated accordn 2 d rule


y(n+1) = yn + x(n+1):


Then d cycle starts over again 2 produce x_(n+2) and y_(n+2) 4rm x_(n+1) and y_(n+1), %26amp; so on.


(a) Write y_(n+1) in terms of x_n and y_n and hence find a matrix A such dat


(x_(n+1)) A (x_n)


(y_(n+1)) = (y_n )





(b) Find d eigenvalues and eigenvectors of d matrix A wen 渭 = 1/2 .


(c) Show dat if a matrix M has an eigenvector x correspondn 2 d eigenvalue 位, then


x is also an eigenvector of d matrix M^2 %26amp; state d correspondn eigenvalue.


Write an analogous result 4 M^n.


(d) Hence show dat if 渭 = 1/2 %26amp; d initial values of d output %26amp; control variables


are x_0 = 1 and y_0 = 2 respctivly, then


X_n = 2^nA control system consists of an 鈥渙utput'; variable x_n %26amp; a 鈥渃ontrol'; variable y_n, wich?
Sorry, George, I got about two sentences in before I quit. It's extremely difficult to read when you write in internet speak, especially because of all the variable names involved.





Try resubmitting your question in English.