Matrices: A control system consists of an “output” variable Xn and a “control” variable Yn ....?

which are updated in discrete steps labelled by n = 1,2,3… In each step, the output variable is first updated according to the rule





Xn+1 = Xn + kYn,





Where k is a constant. Once the variable is updated in this way, the control variable is updated according to the rule





Yn+1 = Yn + Xn+1 .





Then the cycle starts over again to produce Xn+2 and Yn+2 from Xn+1 and Yn+1 , and so on.





How do I write Yn+1 in terms of Xn and Yn and hence find a matrix A such that





(Xn+1) = A( Xn)


(Yn+1) (Yn)








Sorry if the question is unclear...I'm not sure how to layout matrices properly on here!





Thanks for your help :)Matrices: A control system consists of an “output” variable Xn and a “control” variable Yn ....?
X(n+1) = X(n) + kY(n)


Y(n+1) = Y(n) + X(n+1)





First, replace X(n+1) in the second equation:





Y(n+1) = Y(n) + X(n) + kY(n) = X(n) + (k+1)Y(n).





Suppose A is 2x2 and is [a b | c d] (';|'; indicates a new row).





What is A[X(n) | Y(n)]?


On the one hand, it is [X(n+1) | Y(n+1)] (replaced by the recursive formula). On the other, each coordinate is a combination involving X(n), Y(n), a, b, c, and d. Treat X(n) and Y(n) as independent variables, and match up the coefficients.





Good luck.