An algorithm for robust stability analysis

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An algorithm for robust stability analysis

• By
• Prof. P. S. V. Nataraj

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Objectives

• Introduction
• Proposed algorithm
• Application Mass spring damper system
• Conclusion

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Introduction

• Let P be a family of real nth order polynomials
• where s is the complex variable, q??l is a
  vector of physical parameters occurring
  nonlinearly in the coefficients of polynomial in
  P and Q0 ??l is a bounding box for q, given as
• We wish to find if the above polynomial family is
  robustly stable, that is, if

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Introduction (Contd)

• Most of the current robust analysis methods are
  for the case of the polynomial coefficients
  having affine or multilinear dependencies on the
  parameters q
• More recently, Garloff and co-workers proposed
  Bernstein polynomial based robust stability
  analysis methods for the case of polynomial
  parametric dependencies.
• Here we present an algorithm to analyze the
  robust stability of polynomials with nonlinear
  parametric dependencies.
• The proposed algorithm is based on the powerful
  tool of interval analysis.

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Introduction (Contd)

• At least one member of the polynomial family
  should be stable for the test.
• Let ??? denote the frequency variable.
• Expressing the polynomial p(j?,q) in terms of its
  even and odd parts gives
• p(j?,q) pe(?2,q) j?p0(?2,q) .
• By the continuous dependency of the zeros of a
  polynomial on its coefficients, the entire
  polynomial family in p is robustly stable if and
  only if
• 0?( pe(?2,q),p0(?2,q)), q ?Q0, ??? .

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Introduction (Contd)

• Let x (?2,q), f(x) (pe(x), p0(x))
• X denotes the interval vector corresponding to x,
  with X0 ? x Q0
• If F is any inclusion function of f, then it
  follows from the inclusion property of interval
  analysis that, for any X ? X0 , the zero
  exclusion test is sufficient to check for robust
  stability.
• i.e.,
• 0? F(X) ? 0?f(x), x ? X

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Proposed Algorithm

• Input An inclusion function F of f, the initial
  box
• X0 ? I(?l 1)
• Output The message the system is robustly
  stable' or the system is not robustly stable'.
• BEGIN Algorithm
• Set X X0, initialize list L X
• Remove all boxes from list L and evaluate F over
  all the boxes
• (interval zero exclusion test) Discard all boxes
  for which 0? F(X)
• For any box, if the width of F(X) is within
  machine precision, then print the system is not
  robustly stable' and EXIT algorithm.

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Proposed Algorithm contd

• Bisect all boxes in coordinate direction k,
  getting subboxes V1,V2 such that X V1?V2
• Deposit all these subboxes in L.
• If the list L is empty, then print the system is
  robustly stable' and EXIT algorithm. Else, go to
  step 2.
• EXIT Algorithm
• Remark In the proposed algorithm, function
  evaluations, zero exclusion checks, width checks,
  and bisections on all boxes in an iteration are
  performed concurrently using vectorized interval
  arithmetic operations.
• Remark In the proposed algorithm, step 5, we
  have used randomized bisection to find out the
  coordinate direction K

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Application Mass spring damper system

• This example is given by Ackerman and Sienel
• where u is a force accelerating the mass m2,
  position y1 is the measured variable.

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Application contd

• The values of the parameters vary over the
  intervals
• Using the pole-placement technique for the
  nominal plant, a minimum-phase and stable third
  order compensator of the form is designed as

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Application contd

• ?10 variation in the coefficients of the
  compensator are assumed
• The characteristic polynomial of the complete
  feedback system is given by

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Application contd

• This polynomial has thirteen parameters with
  multilinear parametric dependencies where
• The initial frequency interval is 0,5000.
• Computations are done on a Pentium III, 800MHZ
  machine with 500MB RAM using INTLAB
• The algorithm is able to report in 0.14 seconds
  that the system is robustly stable.
• Convex Hull Bernstein algorithm of Garloff et al
  took about 80 seconds on a HP 9000/755 work
  station.

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Conclusions

• A simple and direct method is presented for
  robust stability analysis of polynomials having
  nonlinear parametric dependencies.
• We demonstrated the effectiveness of the proposed
  algorithm on a physical example with thirteen
  uncertain parameters.
• By considering the difficult and general class of
  nonlinear parametric dependencies, we believe
  that the proposed method fills in a significant
  gap in the robust stability analysis literature

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Thank You