Vai's Uni Notes

Modes and Poles

3 min read

Two true facts about linear systems, starting with the SISO (Single Input, Single Output) case.

Fundamental Solutions (SISO)

  • The output of a linear system can always be written as a free response plus a forced response:
    • 𝑦(𝑡)=∑𝑛𝑘=1𝑐𝑘Ψ𝑘(𝑡,𝑡0)+∫𝑡𝑡0𝑔(𝑡,𝜏)𝑢(𝜏)d𝜏,𝑔(𝑡,𝜏)=∑𝑛𝑘=1𝛾𝑘Ψ𝑘(𝑡,𝜏)
  • There exists a set of 𝑛 fundamental solutions {Ψ𝑘(𝑡)}𝑛𝑘=1 such that all solutions for 𝑦(𝑡) can be computed from these
    • The free response is a combination of them (weights 𝑐𝑘 set by the initial conditions)
    • The impulse response 𝑔(𝑡,𝜏) is built from them as well
  • This is the scalar preview of the fundamental matrix and general solution developed for MIMO/LTV systems

Corrected from the notes

  • The quick notes wrote the convolution as ∫𝑡0 and the kernel as 𝑔(𝑡); since the free response starts at 𝑡0 and the kernel depends on both 𝑡 and 𝜏, they are written here as ∫𝑡𝑡0 and 𝑔(𝑡,𝜏)

Modes and Poles (LTI)

  • Fact: for LTI systems specifically, the fundamental solutions are complex exponentials:
    • Ψ𝑘(𝑡)=𝑒𝑝𝑘𝑡,𝑝𝑘∈ℂ,𝑝𝑘=𝜎𝑘+𝑗𝜔𝑘(𝑗2=−1)
    • The 𝑒𝑝𝑘𝑡 are the modes of the system
    • The 𝑝𝑘 are the poles of the system
  • Generically, splitting a mode into its real and imaginary parts:
    • 𝑒𝑝𝑡=𝑒(𝜎+𝑗𝜔)𝑡=𝑒𝜎𝑡𝑒𝑗𝜔𝑡=𝑒𝜎𝑡[cos(𝜔𝑡)+𝑗sin(𝜔𝑡)]
    • with 𝜎=Re(𝑝) and 𝜔=Im(𝑝)
    • 𝜎 sets the growth/decay envelope, 𝜔 sets the oscillation frequency

Corrected from the notes

  • The quick notes wrote Ψ𝑘(𝑡)=𝑒𝑃𝑘𝑡, the expansion’s last line as 𝑒𝜔𝑡[cos(𝜔𝑡)+𝑗sin(𝜔𝑡)], and 𝜔=Im{𝜔}; these are written here as 𝑒𝑝𝑘𝑡, 𝑒𝜎𝑡[…], and 𝜔=Im(𝑝)

Stability of a Fundamental Solution

  • Fact: a fundamental solution (for a linear system) is stable if
    • lim𝑡→∞|Ψ𝑘(𝑡)|=0
  • For LTI, Ψ(𝑡)=𝑒𝑝𝑡=𝑒𝜎𝑡𝑒𝑗𝜔𝑡, so
    • |Ψ(𝑡)|=|𝑒𝜎𝑡|⋅|𝑒𝑗𝜔𝑡|
    • and |𝑒𝑗𝜔𝑡|=1 for all 𝑡
  • Thus

LTI Mode Stability

  • |Ψ(𝑡)|=|𝑒𝜎𝑡|→0if𝜎<0,i.e. ifRe(𝑝)<0
  • A mode is stable exactly when its pole lies in the open left half of the complex plane

Speed of Convergence

  • How fast does a stable mode die out? Use the 2% criterion (settling time 𝑡𝑠):
    • 𝑒𝜎𝑡𝑠=0.02𝑡𝑠=ln(50)|𝜎|≈3.9|𝜎|⇒𝑡𝑠≈4|𝜎|
    • Settling time is inversely proportional to |𝜎|: poles further left decay faster

Damping Ratio

  • The damping ratio of a pole compares its decay rate to its distance from the origin:
    • 𝜁=|Re(𝑝)||𝑝|=|𝜎|√𝜎2+𝜔2
  • Equivalently:
    • 𝜁=1√1+(𝜔𝜎)2
    • 𝜁→0 as 𝜔𝜎→∞ (lightly damped, oscillation dominates)
    • 𝜁→1 as 𝜔𝜎→0 (real pole, no oscillation)
  • The MIMO version of “the response is a sum of modes” is the modal decomposition, where the poles become the eigenvalues of $A$

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