Resolvent Matrix

A systematic way to compute the matrix exponential: (LTI only).

Laplace Transform Review

• One approach to computing more systematically: use Laplace!
• Recall important properties:
  • Linearity: if are constant,
  • Derivative rule:

WARNING

• Linearity does not hold if are themselves time-varying

Laplace of Vectors

• Apply to vectors as well (component by component):
• Derivative:
  • with
• Linearity holds in the linear algebra sense:

Homogeneous Equation

• Apply the LT to the homogeneous eq’n :
  • : the “resolvent matrix”
• But we know

Matrix Exponential via Laplace

Structure of the Resolvent

• Note:
  • : matrix of polynomials in ; the max degree of each polynomial is
  • : scalar polynomial in of degree
• each entry of is a ratio of polynomials in
• each ratio has the same denominator
• find by inverse transforming each of these rational functions use partial fractions

Matrix Exponential by Partial Fractions

• Look at the entry:
  • entry of here is
• Repeat for the other entries (exercise!):
• every entry of is a linear combination of the same 2 scalar exponential functions
  • b/c each inverse transform has the same denominator
• The same is found with eigenvectors in Dyadic Expansion

Possible slip in notes

• Where the full is written out, the notes label the matrix ; the result shown is that of from the start of the example, so the label looks like a slip

Poles and Eigenvalues

• Conclude: generally each entry of will be a linear combination of the scalar exponential functions , where are the roots of the char. poly
  • these roots are the poles of the system
• But note
  • also defines the eigenvalues of : equivalently , eigenvalues of

Repeated poles

• We will also get terms like if poles/eigenvalues are repeated, like
• will treat the repeated pole case separately later (Jordan Normal Form)

• The forced response (with ) leads to the Transfer Function Matrix