Vai's Uni Notes

Notation

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Prof. Sanner’s conventions in the ENAE 641 notes, and how they are typeset here.

Vectors and Matrices

  • Vectors are underlined in the handwritten notes (𝑥); here they are bold upright: 𝐱, 𝐮, 𝐲, 𝐳
  • Matrices are plain capitals: 𝐴,𝐵,𝐶,𝐷,𝐸,𝐹,𝐺,𝑃,𝑄,𝑉
  • Identity: 𝐼 (written 𝕀 in the notes)
  • Transpose: 𝐴𝖳; a row vector is a transposed column vector, 𝐯𝖳=[𝑣1,…,𝑣𝑛]
  • Row partition of a matrix uses Greek or transposed rows (𝛂𝖳𝑖, 𝐜𝖳𝑖, 𝐰𝖳𝑘); column partition uses Latin columns (𝐚𝑗, 𝐛𝑗, 𝐯𝑘)
  • diag{⋅}, blkdiag{⋅}: diagonal and block-diagonal matrices

Physical vs. Deviation Variables

QuantityPhysical (nonlinear model)Deviation (linearized model)
State𝐳(𝑡)∈ℝ𝑛𝐱(𝑡)=𝐳(𝑡)−𝐳∗(𝑡)
Input𝐰(𝑡)∈ℝ𝑚𝐮(𝑡)=𝐰(𝑡)−𝐰∗(𝑡)
Output𝛈(𝑡)∈ℝ𝑝𝐲(𝑡)=𝛈(𝑡)−𝛈∗(𝑡)
  • (⋅)∗: reference (nominal) solution used for linearization
  • 𝚪: higher-order Taylor terms (H.O.T.) neglected by linearization
  • In the modal / coordinate-change material 𝐳 is reused for the new coordinates, 𝐱=𝑃𝐳 (State Coordinate Transformations)

System Matrices

  • Implicit linear state eq’n: 𝟎=𝐸̇𝐳+𝐹𝐳+𝐺𝐰 (𝐸,𝐹,𝐺 Jacobians of 𝐟)
  • Standard form: ̇𝐱=𝐴(𝑡)𝐱+𝐵(𝑡)𝐮, 𝐲=𝐶(𝑡)𝐱+𝐷(𝑡)𝐮
  • Transformed system: 𝐴𝑝,𝐵𝑝,𝐶𝑝,𝐷𝑝; modal: 𝐴𝑚,𝐵𝑚,𝐶𝑚,𝐷𝑚

Solution Objects

  • Ψ(𝑡): fundamental matrix, columns 𝛙𝑘(𝑡)
  • Φ(𝑡,𝜏)=Ψ(𝑡)Ψ−1(𝜏): state transition matrix
  • 𝑒𝐴𝑡: matrix exponential (LTI STM)
  • Φ(𝑠)=(𝑠𝐼−𝐴)−1=𝑄(𝑠)/𝑟(𝑠): resolvent matrix
  • 𝐺(𝑡,𝜏), 𝐺(𝑠): impulse response matrix, transfer function matrix
  • ℒ︀{⋅}, ℒ︀−1{⋅}: Laplace transform and inverse
  • 𝐗(𝑠), 𝐘(𝑠), 𝐔(𝑠): Laplace-domain vectors, written here as capitals (Sanner writes them as underlined lowercase 𝐱(𝑠), 𝐲(𝑠), 𝐮(𝑠), telling them apart from the time functions only by the argument 𝑠)
  • 𝑉, Λ: eigenvector matrix, diag{𝜆𝑘}
  • 𝐯𝑘, 𝐰𝖳𝑘: right and left eigenvectors
  • 𝐽𝑖(𝜆): Jordan block

Overloaded symbols

  • Φ(𝑠) (resolvent) vs. Φ(𝑡,𝜏) (STM)
  • 𝑄(𝑠)=Adj(𝑠𝐼−𝐴) (Laplace) vs. 𝑄 in the Lyapunov equation vs. the monodromy matrix 𝑄 in Floquet theory
  • 𝑃: any coordinate transformation, and the Lyapunov matrix in 𝑉=𝐱𝖳𝑃𝐱
  • 𝐳: physical state (linearization) and transformed/modal coordinates
  • 𝜂: physical output, and the lower bound min𝑡𝜆min(𝑃(𝑡)) in Lyapunov arguments

Norms and Stability

  • ‖𝐱‖2,‖𝐱‖1,‖𝐱‖∞,‖𝐱‖𝑝: vector norms; ‖𝐴‖𝑖: induced matrix norm
  • 𝜆min(𝑀), 𝜆max(𝑀): smallest / largest eigenvalue of a symmetric 𝑀
  • Lyapunov constants: 𝜌=max𝑡𝜆max(𝑃), 𝜂=min𝑡𝜆min(𝑃), 𝜈=min𝑡𝜆min(𝑄)
  • Abbreviations: i.s.L. (in the sense of Lyapunov), AS, UAS, ES, UES, QLF (quadratic Lyapunov function), LTI, LTV, LTP (linear time periodic), TFM, STM, IC, B/C, H.O.T.

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