Vai's Uni Notes

Matrix Exponential

4 min read

The state transition matrix of an LTI system, Φ(𝑡,𝜏)=𝑒(𝑡−𝜏)𝐴.

Definition

  • By analogy with the scalar exponential 𝑒𝑚=∑∞𝑘=0𝑚𝑘𝑘!, the Peano-Baker series (when 𝐴(𝑡) commutes with its integral) defines the matrix exponential function:
    • exp[𝑀]=∑∞𝑘=01𝑘!𝑀𝑘
    • “exp” means this formal power series
  • For constant 𝐴:
    • 𝑒𝐴𝑡≝∑∞𝑘=0𝑡𝑘𝑘!𝐴𝑘𝑒(𝑡−𝜏)𝐴=∑∞𝑘=0(𝑡−𝜏)𝑘𝑘!𝐴𝑘

Must be careful

  • This looks comfortingly familiar, but for example, if
    • 𝑀=[𝑎𝑐𝑏𝑑]
    • 𝑒𝑀≠[𝑒𝑎𝑒𝑐𝑒𝑏𝑒𝑑]No!
  • So what does 𝑒𝑀 look like? (see below and the Laplace approach)

Properties for Constant A

  • 𝑒𝐴𝑡=Φ(𝑡,0) when 𝐴 is constant
  • 𝑒𝐴𝑡|𝑡=0=Φ(0,0)=𝐼
  • 𝑒𝐴𝑡 is nonsingular for any finite 𝑡
  • 𝑒𝐴(𝑡1+𝑡2)=𝑒𝐴𝑡1𝑒𝐴𝑡2
  • [𝑒𝐴𝑡]−1=𝑒−𝐴𝑡
  • [𝑒𝐴𝑡]𝖳=𝑒𝐴𝖳𝑡
  • 𝑒(𝐴+𝐵)𝑡≠𝑒𝐴𝑡𝑒𝐵𝑡

WARNING

  • 𝑒(𝐴+𝐵)𝑡≠𝑒𝐴𝑡𝑒𝐵𝑡 unless 𝐴,𝐵 commute: 𝐴𝐵=𝐵𝐴

What do these look like?

  • Consider
    • 𝐴1=[−1−22−1]𝐴2=[0−51−2]
  • 𝑒𝐴1𝑡 and 𝑒𝐴2𝑡 will both have entries that are combinations of the scalar functions
    • 𝑒−𝑡cos2𝑡,𝑒−𝑡sin2𝑡
  • How to generally deduce facts like this, and to systematically construct 𝑒𝐴𝑡 for a given 𝐴? → next lecture (common forms, Laplace, eigenvectors)

Common Matrix Exponentials

  1. Diagonal:
    • 𝐴=[𝑎100𝑎2]=diag{𝑎1,𝑎2}⇒𝑒𝐴𝑡=[𝑒𝑎1𝑡00𝑒𝑎2𝑡]=diag{𝑒𝑎1𝑡,𝑒𝑎2𝑡}
    • 𝐴=diag{𝑎1,…,𝑎𝑛}⇒𝑒𝐴𝑡=diag{𝑒𝑎𝑘𝑡}𝑛𝑘=1
  2. Rotation-like:
    • 𝐴=[𝑎−𝑏𝑏𝑎]⇒𝑒𝐴𝑡=𝑒𝑎𝑡[cos𝑏𝑡−sin𝑏𝑡sin𝑏𝑡cos𝑏𝑡]
  3. Jordan-like:
    • 𝐴=[𝑎01𝑎]⇒𝑒𝐴𝑡=𝑒𝑎𝑡[10𝑡1]

Jordan Block, n x n

  • Case 3 extends to 𝑛×𝑛:
    • 𝐴=[𝑎0⋮001𝑎……01⋱……⋱𝑎000⋮1𝑎]
  • Then
    • 𝑒𝐴𝑡=𝑒𝑎𝑡[10⋮00𝑡1……12𝑡2𝑡⋱……⋱101(𝑛−1)!𝑡𝑛−11(𝑛−2)!𝑡𝑛−2⋮𝑡1]
    • (the same entry repeats along each diagonal)
  • See also Jordan Normal Form

Block-Diagonal Matrices

  • Additional property: if
    • 𝐴=blkdiag{𝐴1,…,𝐴𝑟}
    • each 𝐴𝑘 is 𝑛𝑘×𝑛𝑘, with ∑𝑟𝑘=1𝑛𝑘=𝑛, i.e. square blocks 𝐴1,𝐴2,…,𝐴𝑟 down the diagonal and zeros elsewhere
  • Then
    • 𝑒𝐴𝑡=blkdiag{𝑒𝐴1𝑡,…,𝑒𝐴𝑟𝑡}

Block-Diagonal Matrix Exponential (5×5)

  • 𝐴=[−200001−200000−1−30003−100000−3]=blkdiag{𝐴1,𝐴2,𝐴3}
    • 𝐴1=[−201−2] (Jordan-like, case 3), 𝐴2=[−1−33−1] (rotation-like, case 2), 𝐴3=−3
  • 𝑒𝐴𝑡=blkdiag{𝑒𝐴1𝑡,𝑒𝐴2𝑡,𝑒𝐴3𝑡}=[𝑒−2𝑡0000𝑡𝑒−2𝑡𝑒−2𝑡00000𝑒−𝑡cos3𝑡−𝑒−𝑡sin3𝑡000𝑒−𝑡sin3𝑡𝑒−𝑡cos3𝑡00000𝑒−3𝑡]

Systematic Construction

  • More systematically, 𝑒𝐴𝑡 can be computed:
  • All show that the entries of 𝑒𝐴𝑡 are linear combinations of the scalar exponentials 𝑒𝜆𝑘𝑡, 𝜆𝑘 eigenvalues of 𝐴 (times powers of 𝑡 when eigenvalues repeat)

Graph View

Backlinks