1
 An Interval Analysis Approach to Quantitative
Feedback Theory

• Prof. P. S. V. Nataraj
• Systems and Control
  Engg. IITB

2
Contributions

• A systematic procedure
• to generate QFT bounds of desired accuracy for
  RS RSR spec. at a given ?.
• to choose frequencies in the QFT design for
  robust stability.

3
Abstract of the work

• IQFT to solve QFT problems
• Provides vigor rigor to QFT
• Systematic procedure to choose
• phase intervals
• Template points
• Frequency intervals
• Produces reliable results of desired accuracy
• Computationally efficient
• Two reliable template generation algorithms.
• An exact boundary extraction algthm.
• Two unified procedures to solve QFT open
  problems

4
Choice of IA to QFT

• IA is a mathematical tool for analyzing uncertain
  data.
• QFT is a design technique for uncertain systems.

5
Problems addressed

• For a LTI SISO system
• At a given frequency, generate reliable
• RSR Gain-Phase margin bounds to
• A desired accuracy for a given system
• Over a given range of frequencies
• Choose frequency intervals such that
• Adjacent bounds are spaced as desired
• Each bound accuracy is specified

6
Template generation

• Template mapping of the uncertainty into Nichols
  plane.
• Existing methods
• Point methods Except grid method, restricted
  application.
• Interval methods Either slow or large no. of
  boxes.

7
PA Algorithm for template generation
Main steps

• 1. Initial non-uniform subdivision.
• . Angle-magnitude evaluation.
• . Collect satisfied rectangles
• . Any more unsatisfied rectangles?
• A. If yes,
  B. If no, go to
  step 5.
• a. Collect required parameter
  combinations,
• b. Subdivide along the longest
  direction,
• c. Go to step 2.
• 5. Output the template.

8
Comparison of Features of PA and existing IATG
algorithms
9
Performance comparison of the PA and existing
IATG algorithm.
10
Performance comparison of the PA and existing
IATG algorithm contd.
11
Motivation for a new algorithm

• PA algorithm
• Combines advantages of existing IATG algorithms
• Comparison shows drastic reduction in time
• Fails in some of the examples
• Always bisected along the longest direction.
• Function widths may not be reduced.
• Still large number of rectangles.

12
Summary of Template Generation algorithms

• Generate reliable templates of desired accuracy
• Savings in QFT design time
• as high as 99 by PA
• PABT
• further reduces the number of boxes

13
Comparison of interval and point methods on
template generation
14
Boundary Extraction Algorithm

• Existing method only an overbounding boundary
• Proposed method
• Extracts the exact boundary
• Can extract the four side boundaries
• Computationally efficient

15
Procedure for upper boundary
16
Performance comparison of various boundary
extraction algorithm.
17
Per formance
comparison of various boundary
extraction algorithm Contd.
18
A typical template its boundary
19
An arbitrary shape and its boundaries
upper
right
left
lower
Final
20
Extracted boundary and typical grid points
21
Extracted boundary and typical grid points
22
Reliable Bound generation

• Existing method
• Importance First method to generate reliable
  bounds.
• Drawbacks
• Fixed phase interval selection.
• Lack of a priori error estimates.
• No relationship bw. template and bound
  accuracies.

23
Open Problems in QFT

• At a given frequency
• how many template points?
• what should be the phase grid size?
• How accurate the bounds are?
• From a given range of frequencies,
• how to choose frequency points ?
• how many should be chosen?

24

Effect of choice of no.of template points on
bound accuracy
30 points
255 points
25
Solution to open problems

• Class A
• Identification of working phase interval
• Adaptive subdivision of the working interval
• Unifies template accuracy and bound accuracy
• Computationally efficient algorithm
• Class B
• Frequency intervals are chosen to satisfy bound
  spacing
• Over each frequency interval, properties same as
  Class A problem

26

Answers to open problems

• Class A
• At given design frequency
• generate reliable controller bounds
• to the specified accuracy
• for a given uncertain system.
• Class B
• over a given design frequency
• choose frequencies such that
• bound spacing is satisfied
• bw.adjacent frequencies.
• Each bound accuracy is as in class A. accuracy.
• Choose the phase template intervals
  automatically.

27
Class A procedure

• Find out the working phase interval.
• Split the required accuracy of the bounds into ?G
  ?Q.
• subdivide the working interval Y, till w(Q)? ?G
  .
• Let wˆ min(width(Ysubdivided)).
• Generate templates to accuracy wˆ and ?G .
• Generate bounds.

28
Generation of bounds

• Shift the values of Q by lower boundary of
  template magnitude.
• Shift the values of Q- by upper boundary of
  template magnitude.
• Extract the final boundary, the bound.

29
Comparison of bounds
30
Comparison of interval and point methods on bound
generation
31
Class B procedure for Robust Stability

• Choose the value of ?, magnitude spacing.
• Find out wˆ and ?G .
• Find out the value of ?, overestimation in
  magnitude of template.
• Subdivide the frequency range till spacing is
  satisfied.
• Obtain the frequency intervals.
• Generate templates over these frequency
  intervals.
• Generate bounds.

32
Controller bounds generated over the frequency
intervals
0.5,.75 0.75,.88 0.88,1
1,1.12 1.12,1.25 1.25,1.5
33
Remarks

• The controller bounds plotted have
• Accuracy within the specified limits
• Varying phase interval widths
• Accuracy visible from the width of the bounds
• The frequency widths varying
• Bound spacing satisfied between adjacent
  intervals
• Over each frequency interval,
• the controller bounds have the properties of
  discrete frequency bounds

34
Conclusions

• Application of IA lends rigor and reliability to
  Horowitzs basic QFT.
• Solves some of the open problems in QFT.
• Systematic procedure to select design
  frequencies, controller phases and parameter
  combinations.

35
Conclusions (contd)

• For a given uncertain system, a systematic
  procedure
• to generate reliable bounds
• Of specified accuracy,
• At a given frequency.
• Over a given frequency range,
• to generate reliable bounds
• of specified accuracy
• with desired spacing bw. adjacent frequencies.

36

Thank You
Nataraj