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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

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

Lyapunov Equation

- Linear dynamic systems
- For Hurwitz (stable) A and negative definite Q,

P is positive definite and

is a valid energy function

Solving the Lyapunov Equation

- Bartels and Stuart, 1972

(Schur factorization)

becomes

where

Solving the Lyapunov Equation

Solving the Lyapunov Equation

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

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

Application to System Theory

- Hybrid systems involve discrete and continuous

components - Example modern power distribution system
- Continuous dynamics inertial machines
- Discrete components breakers, switches

A Quick Example

Linearization

Linear System

But what matrix Q?

QIdentity

Problem Rotation

T is an orthogonal matrix of desired eigenvectors

Problem Rotation

Minimize off-diagonal elements of ? subject to

row dominant with positive diagonal terms

Example Continued

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