Final answer:
To construct the state-space representation of the system represented by the given transfer function, we need to convert it into a set of first-order differential equations and find the values of the state matrix A, input matrix B, output matrix C, and the direct transmission matrix D. In this case, the state matrix A is [[0, 1, 0], [0, 0, 1], [-6, -11, -6]], the input matrix B is [0, 0, 6]ᵀ, the output matrix C is [6, 0, 0], and the direct transmission matrix D is 0.
Step-by-step explanation:
To construct the state-space representation of the system, we need to convert the transfer function into a set of first-order differential equations. The general form of a state-space representation is:
x_dot = Ax + Bu
y = Cx + Du
Where x is the state vector, u is the input vector, and y is the output vector. In this case, the transfer function is Y(s)/R(s) = T(s) = 6/s³+6s²+11s+6. To convert it to a state-space representation, we can rewrite it as:
x_dot = Ax + Bu
y = Cx + Du
Where x = [x₁, x₂, x₃]ᵀ, A is a 3x3 matrix, B is a 3x1 vector, C is a 1x3 vector, and D is a scalar. By comparing coefficients, we can find the values of A, B, C, and D. In this case, A = [[0, 1, 0], [0, 0, 1], [-6, -11, -6]], B = [0, 0, 6]ᵀ, C = [6, 0, 0], and D = 0.