INTERVAL QFT IQFT A MATHEMATICAL AND COMPUTATIONAL ENHANCEMENT OF QFT This work was nominated for th

1
INTERVAL - QFT (IQFT) A MATHEMATICAL AND
COMPUTATIONAL ENHANCEMENT OF QFT(This work
was nominated for the Ramon E. Moore prize at
SIAM Workshop on Validated Computing, Toronto,
2002) P. S. V. Nataraj
2
Quantitative Feedback Theory

• Robust control system design using Horowitz's
  quantitative feedback theory (QFT) approach is
  popular among control designers.
• The QFT approach has a collection of techniques
  for dealing with several classes of uncertain
  plants
• single-loop and multiple-loop,
• single input-output and multi input-output,
• linear and nonlinear,
• time-invariant and time-varying,
• lumped and distributed

3
The Two DOF Structure
Controller
Plant
Setpoint

G(s)
C(s)
F(s)
-
Filter
Fig Two DOF Feedback Structure
4
The Main QFT Steps

• Choose frequency range for performing the design,
  and select the design frequencies from this
  range.
• Generate the template of the given uncertain
  plant at each design frequency.
• A plant template is a set of complex numbers
  representing the frequency response, at a given
  fixed frequency, of an uncertain plant transfer
  function.

5
Main QFT Steps (Contd.)

• Translate the robust closed loop specs into
  bounds on the controller magnitude, at various
  selected controller phases.
• Synthesize a controller transfer function that
  satisfies the bounds various design frequencies.
  
• Synthesize a filter transfer function to meet
  the tracking performance requirements at each
  design frequency.

6
What is a Template?

• Consider a system represented by the transfer
  function G(s,?), where ? is a real vector of the
  fundamental system parameters.
• Suppose the parameters vary independently over
  given real intervals, so that we have a box ? of
  system parameters.
• Denote the phase angle and magnitude functions of
  G(s,?) as
• fang(?)arg G(sj?,?) fmag(?) G(sj?,?)

7
Whats a Template? (Contd.)

• Define the angle-magnitude function f as
• f(?) (fang(?), fmag(?))
• The set f(?), ? e ? defines a region in the
  angle-magnitude plane (i.e., in the Nichols
  chart).
• This region is called the template of G(s,?) at
  the given ?.
• Several methods are available for generating
  plant templates.

8
Example of a Plant Template
9
Bound Generation

• Translate the robust closed loop specs into
  bounds on the controller magnitude, at various
  selected controller phases from the controller
  phase range -2p,0. (Use Nichols chart).
• Several bound generation methods exist
• search,
• geometric, and
• quadratic inequality

10
Bound Generation Using Quadratic Inequalities
(Chait and Yaniv, 1993)

• The Robust Sensitivity Reduction Spec is
• 1C(j?) G(j?, ?)-1 ws(?), for all ? e ?
• Use polar forms Ckej? , G gejf in above spec
  gives
• g2 k2 2 g k cos(? f) zs 0
• where, zs 1- 1/ws. Solving for the roots gives
• kuroot, klroot ( - cos(x) sqrt(cos2(x)
  zs) ) / g
• x ? f
• Thus for plant G, the allowable magnitude range
  of
• controller for satisfying spec at is
• 0, klroot U kuroot,, 8)

11
Example of a Bound
12
Problems with Existing Template Generation Methods

• Existing methods are restricted in scope to
• Particular kind of parametric dependencies
  (independent, affine, multi-linear, special
  nonlinear), or
• Particular form of transfer functions
  (tree-structured, rational)
• Grid based methods can be applied to any form of
  uncertain transfer function
• But they are potentially risky, as critical
  points could be missed out.
• Hence, the reliability and accuracy of generated
  QFT templates is not guaranteed !

13
Problems with Existing Bound Generation Methods

• Bounds generated at arbitrarily selected
  controller phases
• In QFT toolbox, these are 5 deg apart, from 0 to
  -360 deg.
• Bounds at other phases are not computed, but
  interpolated
• Bounds computed using only finite set of
  arbitrarily generated template points.
• So, bounds may not be valid for entire plant
  family, see next figure.
• Thus, the reliability and accuracy of generated
  QFT bounds is not guaranteed !

14
Effect of Number of Template Points on Bound
Accuracy

• Top Bound generated based on plant template of
  30 points.
• Bottom Bound generated based on plant template
  of 255 points.

15
Open Problems in QFT

• From a given design frequency range,
• how to choose frequency points?
• how many should be chosen?
• At a given frequency,
• how many which template points?
• what should be controller phase grid size?
• How reliable accurate are the obtained bounds?

16
Classes of QFT design problems

• Class A
• At a given design frequency, generate
• the plant template,
• controller magnitude bounds at controller phase
  intervals
• so that that the bounds are
• Guaranteed to be reliable, and
• Guaranteed to have prescribed accuracy.

17
Classes of QFT design problems (Contd.)

• Class B
• Given the range of design frequencies, generate
• the frequency intervals,
• the plant templates,
• controller phase intervals with their
  corresponding controller bounds,
• So that the bounds are guaranteed to be
• reliable,
• of prescribed accuracy, and
• have prescribed spacing.

18
Interval QFT - Objectives

• To provide an improved QFT technique that
• Automatically finds design frequencies from
  design frequency range
• Guarantees the reliability and accuracy of the
  generated plant templates and controller bounds
• Automatically finds controller phases for
  efficient bound generation
• Note that available methods to design QFT based
  controllers lack these features.

19
The Approach of IQFT

• Problems due to point QFT approach can be
  resolved using interval mappings based on
  Interval Analysis (IA).
• Idea is to design Algorithms based on IA, which
  in a single computation do the approximation and
  rigorous error analysis.
• With interval methods, one can deal directly with
  sets (intervals) containing infinitely many
  points, and perform set operations.

20
Basic Idea of Interval Template Generation

• Natural interval extension F(?,?), of magnitude
  and phase expressions are obtained from the given
  TF.
• One evaluation of F(?,?) encloses the actual
  template, but with overestimation.
• Repeated subdivision of the parameter box ? gives
  the angle and magnitude rectangles of prescribed
  accuracy. Union of all these small rectangles
  gives the desired template.

21
Various Interval Template Algorithms

• Uniform Subdivision, IEEE CDC 1997
• Adaptive Subdivision, Automatica, 2000
• Vector- Adaptive Subdivision, ASME, 2002
• Randomized Subdivision, Reliable Computing
• Others
• Using Favorable Direction Selection,
• Finding Rectangular Templates
• Boundary of the interval template can also be
  extracted.

22
IA based Template Algorithms

• Interval Template Generation Algorithms (IATG)
• offer several key guarantees
• The generated template is of prescribed
  accuracy.
• The generated template is reliable, despite all
  kind of computational errors.
• The generated template encloses all actual
  template points, hence no loss of robustness due
  to missing critical points.

23
Basic Idea of Interval Bound Generation

• Recall Robust sensitivity reduction spec at
  given ? is
• 1 C(j?) G(j?,?) -1 ? ws (?), ?? ? ?0
• Convert to Quadratic Constraints
• g2k2 2gk cos(x) zs 0
• Solving for the roots, gives
• kuroot, klroot (-cos(x) ? sqrt (cos2(x) -zs)) /
  g
• x controller phase angle plant phase angle

24
Basic Idea of Interval Bound Generation (Contd.)

• Interval Version of Quadratic Constraints is
• Kuroot, Klroot (-cos(X) ? sqrt (cos2(X) -zs)) /
  G
• Working phase interval Y is found as
• Y ( - arccos (sqrt(zs)) - ?, -
  arccos(-sqrt(zs) ) )
• We need to also define
• Q 20 log10 -cos(X) sqrt (cos²(X) - zs)

25
Basic Idea of Interval Bound Generation (Contd.)

• IATG algorithm is used generate the interval
  template at the given frequency.
• The boundary template rectangles are extracted
  using boundary extraction algorithm.
• Working phase interval Y is calculated and
  subdivided in to several phase subintervals.
• The root intervals are computed at each sub
  intervals and the bound intervals are composed.
• Bounds are composed from these bound intervals.

26
Various Interval Bound Generation Algorithms

• Robust Sensitivity Reduction and Robust
  Gain-Phase Margin, ASME 2000
• Robust Tracking, Automatica 2002.

27
IA based Bound Algorithms

• IA based Bound Generation Algorithm (IABG)
  provides the following guarantees
• Guaranteed to be robust against template
  inaccuracies.
• Guaranteed to be robust against phase
  discretization.
• Guaranteed to be reliable, despite all kinds of
  computational errors.
• Also, error estimates are readily available from
  the bounds.

28
Comparison of Bounds
29
Solving Design Problems Classes A B

• Class A and B problems can be solved using
  Unified Procedures.
• Unified procedure combines IATG and IABG steps
  for generating bounds of prescribed accuracy
• These procedures also determine
• Appropriate controller phases
• Appropriate design frequencies (only for class B)
• These are determined automatically in the
  procedures.

30
Solving Class A and B Problems

• Class A
• At given design frequency, generate controller
  bounds
• that are reliable, and
• that are of specified accuracy
• Class B
• Over a given design frequency range, choose
  frequencies such that bound spacing is satisfied
  between adjacent frequencies.
• Each bound is as given for class A.
• Generate appropriate templates and controller
  phase intervals automatically.

31
Spacing between bounds

• Different criteria can be used to select
  frequency intervals from the given design
  frequency range.
• We use a simple criterion for arriving at the
  design frequencies
• Find design frequencies such that
• The distance between the maximums of bounds for
  adjacent frequency intervals is no more than a
  prescribed value,
• and likewise for the minimums of bounds.
• The prescribed value is called adjacent bound
  spacing

32
Next, Some Theory

• We next see some theoretical results required to
• develop our procedure to solve class B problems
• (for robust gain phase margin specification)

33
Error Bound is Width of Bound Interval

• Generally, an interval template is an
  overestimation of the actual template.
• Due to this overestimation, the computed bounds
  also turn out to be overestimated.
• The maximum possible error in the computed bounds
  can be found using the following result.

34
Error Bound is Width of Bound Interval (Contd.)

• Theorem 1
• Let Y be the working phase interval corresponding
  to the
• given specification wm.
• Let X1, , Xl be any partition of Y.
• Assume that root intervals, bound intervals, and
  bounds
• are computed.
• Then, for a given template, the error in the
  computed upper
• bound over the entire controller phase range is
  at most
• equal to the maximum width of all upper bound
  intervals.
• Similarly, for the error in the computed lower
  bound.

35
Error bound is Width of Q ? plus Template
Magnitude Interval

• For any given partition of Y,
• The maximum possible error in the computed bounds
  can
• be found in terms of the elements of the
  partition and
• magnitude of template rectangles, using the
  following
• result.
• First, define the quantities
• Q?j 20 log10 -cos(Xi) ? cos2(Xi) - zm
  
• GdB ? 20 log10 G zm,dB20 log10 zm
• Kuroot,dB 20 log10 Kuroot Klroot,dB
  20 log10 Klroot

36
Error bound is Width of Q ? plus Template
Magnitude Interval (Contd.)

• Let Y be the working phase interval corresponding
  to the given specification wm
• Let X1, , Xl be an arbitrary partition of Y.
• Theorem 2 The error in the computed upper bound
  over the entire controller phase range is at most
  equal to
• maxw(Q,i),i 1, , l max w(GdB)
• where, the max in the second term is taken over
  all template rectangles in the given ITP.

37
Size of Template Rectangles for Specified Bound
Accuracy

• Theorem 3 Suppose the following conditions hold
  
• max w(GdB) lt ?, where, ? is any positive real
  number, and the max is taken over all template
  rectangles in the given ITP.
• The partition of Y is such that
  maxw(Q?,i), i 1, , l ? ? -
  max w(GdB)
• max w(?) lt w , where, the max is taken over all
  template rectangles in the given ITP, and w is
  defined as
• w min w(Xi), i 1, , l
• Then, the error in the computed upper and lower
  bounds over the entire controller phase range is
  at most equal to ?.

38
Size of Template Rectangles for Specified Bound
Accuracy (Contd.)

• Conditions (1) and (3) in Theorem 3 specify
  suitable magnitude and phase widths for the
  template rectangles in the ITP.
• Condition (2) describes a suitable partition of Y
• Fulfillment of all these conditions guarantees
  that the computed bounds are of prescribed
  accuracy ?.

39
Size of Partition for Specified Bound Accuracy

• Define ?G max w(Gdb) ?Q ? - ?G
• Then, by Theorem 3, bounds of prescribed accuracy
  ? can be obtained by fulfilling the following
  conditions
• Every template rectangle has its magnitude width
  at most equal to some positive quantity ?G lt ?.
• The partition of Y is such that over every
  element Xi of the partition, the widths of the
  functions Q? are at most equal to ?Q.
• Every template rectangle has its phase width at
  most equal to w where w is the minimum width of
  the elements of the partition of Y created above.

40
Size of Partition for Specified Bound Accuracy
(Contd.)

• The partitioning of Y can be executed via a
  recursive procedure.
• Starting with Y, a phase subinterval is
  successively bisected till the resulting widths
  of the function Q? is at most ?Q.
• This process eventually gives a partition of Y
  having phase subintervals of unequal widths.

41
How to Split Bound Accuracy?

• A method of splitting the given bound accuracy ?
  into ?Q and ?G is suggested below.
• Suppose the Lipschitz constants LQ and LG of the
  functions Q? and Fmag,dB are known over the
  respective domains Y and ?, where Fmag,dB
  20 log10 (Fmag)
• Then, split ?Q and ?G as per the ratio of the
  Lipschitz constants, i.e., as
• ?Q/?GLQ/LG Since ?Q ?G ?, this gives
  
  
• ?G ? / (LQ / LG 1) ?Q (LQ/LG) ?G

42
Dividing Frequency Range into Frequency Intervals

• Let ?i, ?i1 be any pair of adjacent frequency
  intervals of a given partition of the design
  frequency range ?0
• Let ? denote the prescribed adjacent bound
  spacing.
• Let ? be the maximum overestimation in
• min Fmag,dB(?i,?0) and min Fmag,dB(?i1,?0).
• Theorem 4 Suppose the following condition holds
• min Fmag,dB(?i,?0) min Fmag,dB(?i1,?0) ? ?
  ?
• Then, the distance between the adjacent bound
  maximums at ?i, ?i1 is at most equal to ?.

43
Dividing Frequency Range into Frequency Intervals
(Contd.)

• Suppose it is desired that all the adjacent bound
  maximums resp. minimums are to be at most ? dB
  away from each other.
• Then, by Theorem 4, this can be achieved by
  successively bisecting the design frequency range
  till the previous condition holds at all adjacent
  frequency intervals.
• Note Theorem 3 can be extended for guaranteeing
  the accuracy of the bounds in the frequency
  interval case

44
Procedure - Capabilities

• Based on earlier Theorems, a procedure for
  obtaining robust gain-phase margin bounds can be
  presented.
• The proposed procedure has the following
  capabilities
• It finds appropriate frequency intervals, from
  the given design frequency range ?0.
• On every such frequency interval, it generates
  bounds of prescribed accuracy ?.
• It finds controller phase intervals whose widths
  vary in accordance with the prescribed bound
  accuracy.

45
Reliability of Bounds

• All computations in the proposed procedure are
  done using machine interval arithmetic.
• Machine interval arithmetic automatically
  produces bound values that are guaranteed to be
  reliable, despite all kinds of computational
  errors, such as
• round off,
• truncation, and
• approximation.

46
Unified Procedure for Class B - Inputs

• Inputs
• Design frequency range ?0,
• Prescribed adjacent bound spacing ?
• Prescribed bound accuracy ?.

47
Unified Procedure for Class B - Steps

• BEGIN Procedure
• Split bound accuracy as ? ?Q ?G
• Subdivide working phase interval till width of
  function Q? is at most ?Q, over each subdivision.
  Denote the smallest width of these subdivision as
  w.
• Subdivide design frequency range till the
  spacing criteria is satisfied at each frequency
  interval.
• IATG Step Generate template at each frequency
  interval such that the magnitude side length of
  template rectangle ? ?G, and phase side length ?
  w. Then, extract the boundary template
  rectangles.

48
Unified Procedure for Class B Steps (Contd.)

• 5. IABG Step Compose bounds at each frequency
  interval over the phase subdivisions obtained..
• 6. Plot Step plot the upper and lower controller
  magnitude bounds on y-axis versus the
  corresponding controller phase intervals on
  x-axis in the Nichols chart. Annotate also the
  corresponding frequency interval ?j in the plot.
• END Procedure

49
An Example of Class B Problem

• The TF of a system in active noise and vibration
  control is given by
• G(s)?n/s22? ?ns ?n2
• where ?n?0.75,1.25, ? ?0.02,0.06
• The robust gain-phase margin specification is
  ?m1.2dB.
• The design frequency range is ?00.5,1.5.
• Objective is to generate bounds such that
• Adjacent bound spacing is ? 7.5dB, and
• Bound accuracy is ? 2.5dB.

50
Robust Gain-Phase Margin Bounds Generated by
Unified Procedure
Bound Frequencies 0.5,.75 0.75,.88 0.88,
1 1,1.12 1.12,1.25 1.25,1.5
51
Example of class B problem (Contd.)

• Six frequency intervals are obtained in the range
  0.5,1.5.
• The frequency intervals have different widths.
• The prescribed bound spacing is achieved.
• Unified procedure does this automatically.
• The prescribed bound accuracy is achieved at all
  frequency intervals. That is, maximum error is
  2.5 dB.
• The controller phase intervals have different
  widths at a given frequency interval.
• Phase widths also vary with frequency interval

52
Conclusions

• IQFT lends rigor and reliability to Horowitzs
  basic QFT.
• Selection of design frequencies, controller
  phases and template parameter combinations, are
  treated systematically in IQFT.
• Studies are underway to develop an efficient loop
  shaping method of IQFT using interval global
  optimization.
• An IQFT method for uncertain nonlinear plants has
  just been proposed see IJRNL 2003
• Cases of multivariable is yet to be touched by
  IQFT.

53
Interval - QFT A QFT Tool for the Future ?

• - Thank You
• The Interval Analysis and Control Group,
• Systems and Control Engineering IDP
• IIT Bombay