Bounding L2 Gain System Error Generated by Approximations of the Nonlinear Vector Field

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Bounding L2 Gain System Error Generated by
Approximations of the Nonlinear Vector Field

• Kin Cheong Sou
• Alexandre Megretski
• Luca Daniel
• Massachusetts Institute of Technology

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Nonlinear MOR Projection Framework
The full order model
For any projection matrix
The reduced order model
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Nonlinear MOR Nonlinear Vector Field
Approximation
q ltlt n, the number of states is reduced
Desirable to simplify the reduced vector field
For example

• Trajectory piecewise linear approximations
  Rewienski, White, Linear Algebra and its
  Applications 2005
• Trajectory piecewise polynomial approximations
  Dong, Roychowdhury, DAC 2003
• Kernel methods Philips et. al., ICCAD 2003

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Nonlinear MOR Nonlinear Vector Field
Approximation
The original system
The approximated system
How does the difference in ? affect the
difference in y?
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Background
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Error System as a Feedback
Error system
u
e
original
-
approxi- mated
Small gain from u to e ? Small system error
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Error Metric

• L2 gain from u to e measures system error
• L2 gain has energy multiplier interpretation

• Incremental L2 gain

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Small Gain Theorem
?G
?
u
e
G
u
e
G
w
z
??
?
?
z
w
Small Gain Theorem
Small gain theorem can be conservative if ?? is
very small
If ?G ?? lt 1, then ? lt ?G
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A linear Error Bound
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Scaling Feedback Loop

• Insert scaling parameters to tighten error bound
• Consider scalar parameter a ? 0

Ga
e
u
G
w
y
?
?a

• Scaled setup is the same as the original setup
• The small gain theorem is now applied to Ga and
  ?a

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Scaling Feedback Loop (1)
1. How does error bound ?Ga change with a?
2. How does the condition ?Ga ??a lt 1 change with
a?
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Main Theorem Linear Error Bound
where
e
u
G
w
u
z
G
w
then
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Numerical Procedure for Tighter Bound
Small Gain Theorem (scaled setup)

• Compute L2 gain upper bound numerically
• Quadratic Lyapunov function/IQC combined with LMI
  optimization
• LMI optimization efficient for small scale
  problems (i.e. reduced model related problems)

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Numerical Experiment
Nonlinear transmission line with diodes

• Desired system error lt 1
• ?Ga lt 1
• Maximum a allowed (by small gain thm) is 2e-5
• Corresponding a/?Ga 1e-3
• To make sure (???Ga)/a lt 1
• ?? lt 1e-3 0.1

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Conclusion

• Feedback setup for vector field system error
  analysis
• Extended the applicability of the small gain
  theorem through scaling parameters
• System error L2 gain is linear with respect to
  vector field difference L2 gain, asymptotically
• LMI/IQC numerical procedure can be applied to
  provide less conservative error bound

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Scaling I/O
e
u
G
?

• Scaling factors d and 1/d do not change closed
  loop L2 gain
• Gains in the Lemma setup are changed

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Scaling I/O (1)
Consider the systems
?1 is the incremental L2 gain from g to y
?2 be the L2 gain from uw to x2
?2(d) is monotonically non-decreasing, but
saturates when d is too small
where
Revised Lemma
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Numerical Procedure
1. Choose a set ?a1, a2,?
2. For k 1,2,
Compute ?k ? ?Gak
4. Search for
Output system error upper bound ?i
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Error System as a Feedback
The error system
Small L2 gain from u to e means small system error
u
e
The block diagram
G
w
z
?
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Small Gain Theorem Discussion

• Can bound closed loop L2 gain (i.e. system
  error) using open loop L2 gains
• Relatively easy to compute open loop L2 gains
• Open loop L2 gain (?G) is independent of ??
• Error bound not meaningful if ?? is very small

Small Gain Theorem
If ?G ?? lt 1, then ? lt ?G