How it works
The standard state-space form is ẋ(t) = Ax(t) + Bu(t) and y(t) = Cx(t) + Du(t), where x is the n×1 state vector, u is the input, y is the output, A is the n×n system matrix, B is n×1 input matrix, C is 1×n output matrix, and D is the direct transmission term (usually 0 for physical systems). The transfer function relates to state-space by G(s) = C(sI−A)⁻¹B + D. Eigenvalues of the A matrix are the system poles — same as roots of det(sI−A) = 0, the characteristic polynomial.
Key points to remember
State transition matrix Φ(t) = e^(At) = L⁻¹[(sI−A)⁻¹], giving the zero-input response x(t) = Φ(t)x(0). For a 2×2 A matrix, e^(At) is computed using the Cayley-Hamilton theorem or Laplace inverse of the resolvent matrix. The system is stable if all eigenvalues of A have negative real parts. Similarity transformation x = Pz transforms A to PAP⁻¹ without changing eigenvalues. Diagonal canonical form (if eigenvalues are distinct) decouples the state equations. Phase variable canonical form places the characteristic polynomial coefficients directly in the A matrix bottom row, making it easy to implement from a transfer function.
Exam tip
The examiner always asks you to convert a given transfer function into phase variable (controllable canonical) form and write the A, B, C, D matrices — the last row of A contains the negative of the denominator coefficients in ascending order.