ELEC4631 Chap.3 Solving the State Equation: Matrix Exponential, Modes and Free Response

Only when A is diagonal in the coordinates being used. In general, matrix powers mix coordinates and the exponential is defined by the matrix power series or an equivalent spectral/resolvent method. Entrywise exponentiation usually fails Phi(0)=I or Phi-dot=A Phi.

Use diagonalisation for a complete easy eigenbasis, a Jordan/nilpotent split for defective structure, Cayley–Hamilton for low-order matrices whose powers reduce cleanly, and the resolvent when Laplace inversion is natural. The methods must agree; choose the one that makes structure and verification clearest.

A size-two Jordan block is lambda I+N with N^2=0, so e^{Jt}=e^{lambda t}(I+tN). In the resolvent, the same structure appears as a repeated factor (s-lambda)^{-2}, whose inverse transform is t e^{lambda t}.

Confirm the forced term is zero at the initial time, differentiate the complete expression and recover Ax+Bu, and—when A is Hurwitz and the input is constant—compare the long-time limit with x_ss=-A^{-1}Bu_ss. Include y=Cx+Du if the question asks for output.

State the spectral structure before the final matrix. If A is diagonalisable, list eigenvalues/eigenvectors, form V and V^-1 and write Ve^{Lambda t}V^-1. If it is defective, identify the nilpotent part or repeated resolvent pole and show where t, t^2/2 or higher factors arise. If eigenvalues are complex, combine conjugate terms into a real sine/cosine expression. Then perform Phi(0)=I and Phi-dot=A Phi checks. Multiply by x0 for free response; add the correctly oriented convolution for input; finally form y=Cx+Du when required. End with modal or stability interpretation only after considering real parts and Jordan multiplicities. This visible sequence gives recoverable credit even if one entry has a sign error, whereas an unexplained calculator matrix does not.

Taking the Laplace transform gives X(s)=(sI-A)^{-1}x0+(sI-A)^{-1}BU(s). The inverse transform of the resolvent is e^{At}; multiplication by U(s) becomes convolution. Simple resolvent poles yield scalar exponentials, repeated poles yield polynomial-times-exponential terms and conjugate poles yield real oscillations. Multiplication by C and B produces the transfer function only for the zero-state input-output route, so an internal mode may cancel even though it remains in e^{At}x0. Practise moving between these views: derive the same second-order response once by diagonalisation and once through partial fractions, compare entries, and use any disagreement to locate an eigenvector normalisation, inverse or sign error.

Change one structural feature at a time. Start with a diagonal stable A and constant input; then add an off-diagonal coupling, replace distinct roots with a Jordan block, move a conjugate pair across the imaginary axis and introduce a nonzero D in the output. Before recalculating, predict whether e^{At} will contain pure exponentials, polynomial factors or sine/cosine terms and whether the steady state should exist. An AI tutor can generate the matrix and check your final residual, but require it to preserve your chosen route and ask only for the first error. This develops method selection rather than recognition of a stored answer.