AN IMPROVED ALGORITHM FOR SET INVERSION

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AN IMPROVED ALGORITHM FOR SET INVERSION

• By
• P. S. V. Nataraj

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Objectives

• Set inversion Problem
• Basic Set inversion Algorithm
• Proposed Set inversion Algorithm
• Test examples and results
• Conclusions

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Set Inversion Problem

• Let be a nonlinear
  function from . Then the set
  inversion can be posed as a problem of
  characterization of
• In this work we address the problem of
  characterizing S defined by a set of nonlinear
  inequalities

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Set Inversion Problem

• Here, we propose some improvements to the
  algorithm of Jaulin et al. (Applied Interval
  Analysis, Springer, 2001) for characterization of
  S via set inversion.
• We shall assume that f is continuously
  differentiable on X.
• The proposed improvements are based on exploiting
  the property of monotonicity in two ways
• The powerful monotonicity test form is used as an
  inclusion function for f .
• If f is found to be monotonically increasing or
  decreasing in every component direction on a
  given box, then the part of box where the
  inequality fjgt0 is certainly infeasible is found
  and discarded.

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Set inversion problem

• Further, only one box is processed in each
  iteration of the algorithm of Jaulin et al. Such
  processing is inherently slow, due to its
  sequential nature. On the other hand, in the
  proposed algorithm, all boxes present in the list
  at each iteration are processed concurrently
  using vectorization
• We then test and compare the performances of the
  proposed and existing algorithms on two robust
  stability problems, including a case study of
  speed control of a jet engine.

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Monotonicity Test Form

• The monotonicity test form (Moore 79) is given as
  
  
• where is the set of integers i such
  that
• (If is empty, f is monotonic in all
  directions), and

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MTF Example
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Basic Set Inversion Algorithm

• Let be the initial box . The set
  inversion algorithm encloses the portion S
  contained in X0, between two partitions Kin and
  Kout in the sense that
  
• Here, and Ke is the
  list of indeterminate boxes.

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Basic Set Inversion Algorithm (contd.)

• The algorithm of Jaulin et al. is as follows.
• Inputs The initial box X0, natural inclusion
  function F, and an accuracy parameter X.
• Outputs A list Kin of all boxes guaranteed to
  belong to S, and a list Kout K? ? Kin.
• BEGIN Algorithm
• Initialize XX0, Kin , Kout , LX
• Remove the first box X from list L and evaluate
  F(X).
• If inf F(X)gt0 for all i1,,m then deposit X in
  lists Kin and Kout, and go to step 2.

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Basic Set Inversion Algorithm (contd.)

• If sup Fi(X)lt0 for any i1,,m then go to step
  2.
• If w(X)lt?x then deposit X in list Kout and go to
  step 2.
• Bisect X in maximum width coordinate direction k,
  getting boxes V1, V2 such that XV1?V2. Deposit
  these subboxes in L.
• If the list L is empty, EXIT algorithm. Else go
  to step 2.
• END Algorithm

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Proposed Set Inversion Algorithm

• BEGIN Algorithm
• Initialize XX0, Kin, Kout, LX
• Remove all boxes from list L and evaluate FMT
  over all the boxes.
• Deposit all boxes for which
  inf F(X)gt0 for all i1,,m in the
  lists Kin and Kout.
• Discard all those remaining boxes for which
  sup Fi (X)lt0 for any i1,,m.
• Deposit all those remaining boxes for which
  w(X)lt?x in list Kout

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Proposed Set Inversion Algorithm (Contd)

• Find all those boxes for which Fi is
  monotonically increasing or decreasing in every
  direction for, i1,. ,m. Apply Algorithm TPB to
  discard infeasible parts of these boxes.
• Bisect all remaining boxes in the maximum width
  coordinate direction k, getting subboxes V1, V2
  such that . Deposit all these
  subboxes in L
• If the list L is empty, EXIT algorithm. Else, go
  to step 2.
• END Algorithm

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Throwing part boxes Algo. (TPB)

• If f is monotonically increasing/decreasing in
  every direction on X, then algorithm TPB locates
  the subbox C on which the inequality
  fgt0 is certainly infeasible, and outputs a list
  LX of boxes whose union is the complement of C in
  X,
• Algorithm TPB parameterizes the line joining
  points
• in terms of single parameter ?.
• Use Newton-Raphson to find ?, such that f(?)0

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TPB (Contd)

• Construct a subbox on which fgt0
  is certainly infeasible.
• Using the box complementation algorithm in
  (Kearfotts book, 1996), find the complement of
  box C in X to get a list LX such that

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TPB (Contd)
X\C
X
C
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Algorithm TPB

• Inputs The function f and box X. It is assumed
  that f is monotonically increasing in every
  component direction on X.
• Outputs A list LX such that ?W?LX WX \C.
• Check for the trivial cases
• If f(X1,,Xl) gt0 then the inequality fgt0 is
  certainly feasible on entire X. Set LX X and
  RETURN.
• If f(X1,Xl)? 0, then the inequality fgt0 is
  certainly infeasible on entire X. Set LX and
  RETURN.
• Parameterize the line joining
  in terms of single parameter
  

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Algorithm TPB (contd.)

• Using a nonlinear solver such as Newton -
  Raphson, find such that
• Construct a subbox C?X which inequality fgt0 is
  certainly infeasible
• Using the box complementation algorithm in
  (Kearfott 1996), find the complement of the box C
  in X to get a list LX such that
• END Algorithm

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Test example (1)

• A polynomial stability problem
• Enforcing positivity in the first column of the
  Routh table gives the set of inequalities
• The initial box is X-10,10,-10,10

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Test example (1) (Contd.)

• Computations on a Sun 440 MHZ ultra Sparc 10
  machine, with 1 GB RAM using FORTRAN 95.
• Accuracy ?x 10 3
• Performance Metrics
• Computational time in seconds
• Number of boxes in list K? for which system
  stability in indeterminate
• Maximum length of list L in the algorithm

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Test example (1) - results

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Zoomed plot of Kin
Shaded region shows the extra region covered by
the proposed algorithm
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Zoomed plot of K?
Shaded region is the extra region covered by the
existing algorithm
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Test example (2)

• A SISO Jet engine interval plant with input as
  fuel flow and output as acceleration of
  compressor speed, Ndot

Desired Ndot
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Test example (2) Contd

• The transfer function is
• Where
• Compensator of the form is
  required to be designed.

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Test example (2) Contd

• A compensator of first order of structure
  is designed as follows.
• Set up sixteen Routh tables (Barmish etal, 1992)
  for closed loop polynomials associated with each
  extreme plant.
• Enforce positivity for each of the first column
  entries which are functions of a, b and c. This
  leads to set of inequalities involving a, b and
  c.
• Solve the inequalities for the sixteen extreme
  plants by the proposed set inversion algorithm
  and obtain the set of stabilizing gains in the
  given initial bounds.

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Test example (2) - results

• The initial domain is a? 0.7, b ? 0.1,1, c ?
  0.01,8

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Zoomed plot of Kin
Shaded region shows the extra region covered by
the proposed algorithm
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Zoomed plot of K?
Shaded region is the extra region covered by the
existing algorithm
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Test example (2) - results

• Compensator parameters a 0.7, b 0.5074 and c
  0.1062 are selected from the feasible region
  given by the algorithm, assures robust stability
  of the interval plant.

Responses of the compressor speed control system
to a unit step in the compressor speed command
input.
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Conclusions

• The improved algorithm is based on the powerful
  tool of monotonicity.
• For two robust stability problems, including a
  case study of control of a jet engine, we found
  that the proposed algorithm encloses the
  stability domain more accurately, requires
  smaller list lengths, but takes more
  computational time.

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Thank you