Analysis of the Lyapunov Equation:

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Analysis of the Lyapunov Equation

• Tom Rosenwinkel and Johnson Carroll
• CAM 383C Numerical Analysis Linear Algebra
• Professor Inderjit Dhillon
• Course Project Fall 2005

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Purpose Energy Functions

• Given a system of differential equations, find an
  energy function such that the energy decreases as
  the system evolves.
• From the energy function, bounds on convergence
  time and regions of convergence are much more
  easily found than by integrating the system
  equations directly

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Lyapunov Equation

• Linear dynamic systems
• For Hurwitz (stable) A and negative definite Q,
  P is positive definite and
  is a valid energy function

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Solving the Lyapunov Equation

• Bartels and Stuart, 1972

(Schur factorization)
becomes
where
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Solving the Lyapunov Equation

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Solving the Lyapunov Equation

• Cost of Solving Lypunov Equation Lxb
• Without Schur factorization
• With Schur factorization

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Solving Lyapunov Equation
if abs(H(k,k-1)) gt 2H1(k)
convergeconverge1 end if
converge n-1 stop1
break end H1(k)abs(H(k,k-1))
end if stop1 l,converge
break end if l gt 99998 H
end end TH

• function Q Tmyschur(A)
• m nsize(A)
• Q Hmyhess(A)
• stop0Hmmax(max(H)) H17000ones(n-1)'
• for l110000
• c s Rmyhqr(H)
• VmygivensformQ(c,s)
• HRV
• QQV
• converge0
• for k2n
• if abs(H(k,k-1)) lt 1e-18Hm
• convergeconverge1
• end

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Application to System Theory

• Hybrid systems involve discrete and continuous
  components
• Example modern power distribution system
• Continuous dynamics inertial machines
• Discrete components breakers, switches

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A Quick Example
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Linearization
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Linear System
But what matrix Q?
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QIdentity
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Problem Rotation
T is an orthogonal matrix of desired eigenvectors
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Problem Rotation
Minimize off-diagonal elements of ? subject to
row dominant with positive diagonal terms
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Example Continued
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