Linear systems

Problem 3 (State space representation - 8 points). Consider a system with input u(t) and output y(t) which can be described using the following set of differential equations: Zi (t) = 21(t) + zz(t) + i(t) 22 (t) = zi(t) + zz(t) + u(t) y(t) = zi(t) a) (4 points) Define the states of the system such that it can be represented as an 3-dimensional LTI system, i.e., as the following: i(t) = Ar(t) + Bu(t) y(t) = Cx(t) + Du(t) where A, B, C, D are constant matrices. b) (4 points) Consider T defined below, as a new basis for the state space and let a(t) be the representation of x(t) with respect to the basis T. Compute A, B, C, D in the new representation of the system with respect to T: i(t) = Ax(t) + Bu(t) y(t) = Ci(t) + Du(t)