A computational approach to existence verification and construction of robust qft controllers
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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 Construction of Robust QFT Controllers

  • Contribution

  • Introduction to QFT

  • Problem definition

  • Proposed Algorithm

  • Example

  • Conclusions


Contribution
Contribution Construction of Robust QFT Controllers

  • Existence verification

  • Constructive approach –

    • Gauranteed existence verification

    • Give all solutions, if any solution exist.


Introduction

R(s) Construction of Robust QFT Controllers

Y(s)

F(s)

K(s)

G(s)

-1

Introduction

2-DOF Structure for QFT formulation


Introduction1
Introduction … Construction of Robust QFT Controllers

QFTObjective :

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

  • Robust Stability margin

  • Tracking performance

  • Disturbance Attenuation


Introduction2
Introduction … Construction of Robust QFT Controllers

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.


Introduction3
Introduction … Construction of Robust QFT Controllers

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 Construction of Robust QFT Controllers

  • Given an uncertain plant and time domain or frequency domain specification, find out if a controller of specified (transfer function) structure exist.


Feasibility test

. Construction of Robust QFT Controllers

Feasible

Ambiguous

.

Infeasible

Feasibility Test

.

Feasible

magnitude

Ambiguous

.

Infeasible

phase


Proposed algorithm
Proposed Algorithm Construction of Robust QFT Controllers

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 algorithm1
Proposed Algorithm… Construction of Robust QFT Controllers

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 algorithm2
Proposed Algorithm … Construction of Robust QFT Controllers

  • 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 algorithm3
Proposed Algorithm … Construction of Robust QFT Controllers

  • 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 algorithm4
Proposed Algorithm Construction of Robust QFT Controllers

  • 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 Construction of Robust QFT Controllers

  • Uncertain plant :

  • Uncertainty :

  • Robust stability spec:

  • Tracking spec:


Qft bounds
QFT Bounds Construction of Robust QFT Controllers


Example1
Example … Construction of Robust QFT Controllers

  • For first order controller:

    • Parameter vector

    • Initial search box

  • For second order controller:

    • Parameter vector

    • Initial search box


Example2
Example … Construction of Robust QFT Controllers

  • For third order controller:

    • Parameter vector

    • Initial search box

  • For fourth order controller:

    • Parameter vector

    • Initial search box


Results
Results Construction of Robust QFT Controllers

  • 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 Construction of Robust QFT Controllers

  • 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.


Thank you
Thank you ! Construction of Robust QFT Controllers


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