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A Computational Approach to Existence Verification and Construction of Robust QFT Controllers P.S.V. Nataraj and Sachin Tharewal Systems & Control Engineering, IIT Bombay, India Outline Contribution Introduction to QFT Problem definition Proposed Algorithm Example Conclusions

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a computational approach to existence verification and construction of robust qft controllers
A Computational Approach to Existence Verification and Construction of Robust QFT Controllers

P.S.V. Nataraj and Sachin Tharewal

Systems & Control Engineering,

IIT Bombay, India

outline
Outline
  • Contribution
  • Introduction to QFT
  • Problem definition
  • Proposed Algorithm
  • Example
  • Conclusions
contribution
Contribution
  • Existence verification
  • Constructive approach –
    • Gauranteed existence verification
    • Give all solutions, if any solution exist.
introduction

R(s)

Y(s)

F(s)

K(s)

G(s)

-1

Introduction

2-DOF Structure for QFT formulation

introduction5
Introduction …

QFTObjective :

Synthesize K(s) and F(s) for the following specifications:

  • Robust Stability margin
  • Tracking performance
  • Disturbance Attenuation
introduction6
Introduction …

QFT Procedure :

  • Generate the plant template at the given design frequencies .
  • Generate the bounds in terms of nominal plant, at each design frequency, on the Nichols chart.
introduction7
Introduction …

QFT Procedure …

  • Synthesize a controller K(s) such that
    • The open loop response satisfies the given performance bounds,
    • And gives a nominal closed loop stable system.
  • Synthesize a prefilter F(s) which satisfies the closed loop specifications.
problem definition
Problem Definition
  • Given an uncertain plant and time domain or frequency domain specification, find out if a controller of specified (transfer function) structure exist.
feasibility test

.

Feasible

Ambiguous

.

Infeasible

Feasibility Test

.

Feasible

magnitude

Ambiguous

.

Infeasible

phase

proposed algorithm
Proposed Algorithm

Inputs:

  • Numerical bound set,
  • The discrete design frequency set,
  • Inclusion function for magnitude and angle,
  • The initial search space box z0.

Output:

Set of Controller parameters, OR

The message “No solution Exist”.

proposed algorithm11
Proposed Algorithm…

BEGIN Algorithm

  • Check the feasibility of initial search box z0.
    • Feasible  Complete z0is feasible solution.
    • Infeasible  No solution exist in z0
    • Ambiguous  Further processing required : Initialize stack list Lstack and solution list Lsol.
proposed algorithm12
Proposed Algorithm …
  • Choose the first box from the stack list Lstackas the current box z, and delete its entry form the stack list.
  • Split z along the maximum width direction into two subboxes.
proposed algorithm13
Proposed Algorithm …
  • Find the feasibility of each new subbox:
    • Feasible  Add to solution list Lsol.
    • Infeasible  Discard.
    • Ambiguous  Further processing required : Add to stack list Lstack.
proposed algorithm14
Proposed Algorithm
  • IF Lstack = {empty} THEN (terminate)
    • Lsol = {empty}  “No solution exist”, Exit.
    • Lsol {empty}  “Solution set = ” Lsol, Exit.
  • Go to step 2 (iterate).

End Algorithm

example
Example
  • Uncertain plant :
  • Uncertainty :
  • Robust stability spec:
  • Tracking spec:
example17
Example …
  • For first order controller:
    • Parameter vector
    • Initial search box
  • For second order controller:
    • Parameter vector
    • Initial search box
example18
Example …
  • For third order controller:
    • Parameter vector
    • Initial search box
  • For fourth order controller:
    • Parameter vector
    • Initial search box
results
Results
  • For the aforementioned structures and the initial search domains, the proposed algorithm terminated with the message: “No feasible solution exists in the given initial search domain”.
  • This finding is in agreement with the analytically found ‘non-existence’ of Horowitz.
conclusions
Conclusions
  • An algorithm has been proposed to computationally verify the existence (or non-existence) of a QFT controller solution.
  • Proposed algorithm has been tested successfully on a QFT benchmark example.
  • The proposed algorithm is based on interval analysis, and hence provides the most reliable technique for existence verification.