Vai's Uni Notes

State Coordinate Transformations

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Re-expressing the state in a new basis, 𝐱=𝑃𝐳 (or 𝑃(𝑡)), generalizing the modal form: internal details change, input/output behavior does not.

Constant Transformations

  • Note: there is nothing special about the eigenvector matrix 𝑉 in the modal approach! We can do this with an arbitrary, nonsingular 𝑃 (any nonsingular matrix defines a basis), i.e.
    • 𝐱(𝑡)=∑𝑛𝑘=1𝑧𝑘(𝑡)𝐩𝑘=𝑃𝐳(𝑡)𝑃=[𝐩1,𝐩2,…,𝐩𝑛],{𝐩𝑘}𝑛𝑘=1linearly indep.
  • ⇒̇𝐳(𝑡)=𝐴𝑝𝐳(𝑡)+𝐵𝑝𝐮(𝑡)𝐲(𝑡)=𝐶𝑝𝐳(𝑡)+𝐷𝑝𝐮(𝑡)
  • with
    • 𝐴𝑝=𝑃−1𝐴𝑃,𝐵𝑝=𝑃−1𝐵,𝐶𝑝=𝐶𝑃,𝐷𝑝=𝐷

Time-Varying Transformations

  • This holds for a time-varying basis as well, i.e. {𝐩𝑘(𝑡)}𝑛𝑘=1 linearly independent at each 𝑡
  • Let 𝑃(𝑡)=[𝐩1(𝑡),…,𝐩𝑛(𝑡)] and
    • 𝐱(𝑡)=∑𝑛𝑘=1𝑧𝑘(𝑡)𝐩𝑘(𝑡)=𝑃(𝑡)𝐳(𝑡)
  • Then
    • ̇𝐱(𝑡)=̇𝑃(𝑡)𝐳(𝑡)⏟new term+𝑃(𝑡)̇𝐳(𝑡)
  • ⇒̇𝐳(𝑡)=𝐴𝑝(𝑡)𝐳(𝑡)+𝐵𝑝(𝑡)𝐮(𝑡)𝐲(𝑡)=𝐶𝑝(𝑡)𝐳(𝑡)+𝐷𝑝(𝑡)𝐮(𝑡)
  • where now
    • 𝐴𝑝(𝑡)=𝑃−1(𝑡)[𝐴𝑃(𝑡)−̇𝑃(𝑡)]𝐵𝑝(𝑡)=𝑃−1(𝑡)𝐵,𝐶𝑝(𝑡)=𝐶𝑃(𝑡),𝐷𝑝(𝑡)=𝐷
  • The above is the most general form for a coordinate transformation
  • Note: if 𝑃(𝑡) is in fact constant, then ̇𝑃(𝑡)=0, and these eq’ns collapse to the previous versions, 𝐴𝑝=𝑃−1𝐴𝑃
  • Also note, nothing changes here if 𝐴,𝐵, etc. are themselves time-varying, i.e.

General Coordinate Transformation

  • 𝐴𝑝(𝑡)=𝑃−1(𝑡)[𝐴(𝑡)𝑃(𝑡)−̇𝑃(𝑡)]
  • 𝐵𝑝(𝑡)=𝑃−1(𝑡)𝐵(𝑡),𝐶𝑝(𝑡)=𝐶(𝑡)𝑃(𝑡),𝐷𝑝(𝑡)=𝐷(𝑡)
  • Thus, the coordinate change game can be played equally well with LTI/LTV systems, and constant or time-varying basis vectors for the new coordinates
  • Time-varying 𝑃(𝑡) can even turn any 𝐴(𝑡) into a constant matrix: see Lyapunov Transformations

Input/Output Behavior Is Unchanged

  • Summary:
    • ̇𝐱=𝐴𝐱+𝐵𝐮̇𝐳=𝐴𝑝𝐳+𝐵𝑝𝐮𝐲=𝐶𝐱+𝐷𝐮𝐲=𝐶𝑝𝐳+𝐷𝑝𝐮
    • 𝐴𝑝=𝑃−1(𝐴𝑃−̇𝑃), 𝐵𝑝=𝑃−1𝐵, 𝐶𝑝=𝐶𝑃, 𝐷𝑝=𝐷
  • Block diagram: 𝐮→ [system (𝐱 or 𝐳)] →𝐲
    • 𝐮,𝐲 are the same in both models
    • ⇒ the I/O behavior of the system is the same regardless of the coordinates used for the states

Invariance of the Fundamental Matrix

  • Claim: let Ψ𝑥(𝑡) be any fundamental matrix for the 𝐱(𝑡) eq’ns, and similarly for Ψ𝑧(𝑡). Then
    • Ψ𝑥(𝑡)=𝑃(𝑡)Ψ𝑧(𝑡)
  • Pf: Ψ𝑥(𝑡)=[𝛙1(𝑡),…,𝛙𝑛(𝑡)], where each 𝛙𝑘(𝑡) satisfies the homogeneous dynamics for 𝐱(𝑡)
    • But any solution for 𝐱(𝑡) can be written as 𝐱(𝑡)=𝑃(𝑡)𝐳(𝑡) with 𝑃(𝑡) nonsingular
    • So 𝛙𝑘(𝑡)=𝑃(𝑡)𝛙𝑧,𝑘(𝑡) for some 𝛙𝑧,𝑘(𝑡)
    • Ψ𝑥=[(𝑃𝛙𝑧,1),(𝑃𝛙𝑧,2),…,(𝑃𝛙𝑧,𝑛)]=𝑃Ψ𝑧Ψ𝑧=[𝛙𝑧,1,𝛙𝑧,2,…,𝛙𝑧,𝑛]

Invariance of the State Transition Matrix

STM Under a Coordinate Change

  • Claim:
  • Φ𝑥(𝑡,𝜏)=𝑃(𝑡)Φ𝑧(𝑡,𝜏)𝑃−1(𝜏)
  • Pf:
    • Φ𝑥(𝑡,𝜏)=Ψ𝑥(𝑡)Ψ𝑥(𝜏)−1=[𝑃(𝑡)Ψ𝑧(𝑡)][𝑃(𝜏)Ψ𝑧(𝜏)]−1=𝑃(𝑡)Ψ𝑧(𝑡)Ψ𝑧(𝜏)−1⏟Φ𝑧(𝑡,𝜏)𝑃(𝜏)−1=𝑃(𝑡)Φ𝑧(𝑡,𝜏)𝑃−1(𝜏)

Invariance of the Impulse Response and TFM

  • Claim: the impulse response matrix is the same:
    • 𝐺𝑥(𝑡,𝜏)=𝐺𝑧(𝑡,𝜏)
  • Pf:
    • 𝐺𝑥(𝑡,𝜏)=𝐶(𝑡)Φ𝑥(𝑡,𝜏)𝐵(𝜏)+𝐷(𝑡)𝛿(𝑡−𝜏)=𝐶(𝑡)𝑃(𝑡)⏟𝐶𝑝(𝑡)Φ𝑧(𝑡,𝜏)𝑃−1(𝜏)𝐵(𝜏)⏟𝐵𝑝(𝜏)+𝐷(𝑡)⏟=𝐷𝑝(𝑡)𝛿(𝑡−𝜏)=𝐶𝑝(𝑡)Φ𝑧(𝑡,𝜏)𝐵𝑝(𝜏)+𝐷𝑝(𝑡)𝛿(𝑡−𝜏)=𝐺𝑧(𝑡,𝜏)
  • For LTI, this implies the TFM is invariant:
    • 𝐺𝑥(𝑠)=𝐺𝑧(𝑠)
  • Directly: 𝐺𝑥(𝑠)=𝐶[𝑠𝐼−𝐴]−1𝐵+𝐷, with 𝐴𝑝=𝑃−1𝐴𝑃 (LTI) ⇒𝐴=𝑃𝐴𝑝𝑃−1:
    • 𝐺𝑥(𝑠)=𝐶[𝑠𝐼−𝑃𝐴𝑝𝑃−1]−1𝐵+𝐷(note𝐼=𝑃𝑃−1)=𝐶[𝑃(𝑠𝐼−𝐴𝑝)𝑃−1]−1𝐵+𝐷=𝐶[𝑃(𝑠𝐼−𝐴𝑝)−1𝑃−1]𝐵+𝐷=𝐶𝑝(𝑠𝐼−𝐴𝑝)−1𝐵𝑝+𝐷𝑝=𝐺𝑧(𝑠)

Other Useful Transformations for LTI

Graph View

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