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Let the state-space representation of an LTI system be ẋ(t) = A x(t) + B u(t), y(t) = C x(t) + d u(t) where A, B, C are matrices, d is a scalar, u(t) is the input to the system, and y(t) is the output. Let B = [0 0 1]^T and d = 0. Which one of the following options for A and C will ensure that the transfer function of this LTI system is H(s) = 1/(s³ + 3s² + 2s + 1)?

1. A = [[0,1,0],[0,0,1],[-1,-2,-3]] and C = [1 0 0]
2. A = [[0,1,0],[0,0,1],[-3,-2,-1]] and C = [1 0 0]
3. A = [[0,1,0],[0,0,1],[-1,-2,-3]] and C = [0 0 1]
4. A = [[0,1,0],[0,0,1],[-3,-2,-1]] and C = [0 0 1]

Correct answer: A = [[0,1,0],[0,0,1],[-1,-2,-3]] and C = [1 0 0]

Solution

This option correctly represents the system dynamics and output configuration needed to achieve the specified transfer function. The matrix A defines the system's characteristic polynomial, which matches the desired transfer function, while the output matrix C ensures that the output is taken from the appropriate state variable.

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