Automatic Loop Shaping in QFT by Using CRONE Structures

Motivation Preliminaries Proposed solution Design Example Summary

Automatic Loop Shaping in QFT by Using CRONE Structures J. Cervera and A. Baños Faculty of Computer Engineering Department of Computer and Systems Engineering University of Murcia (Spain) [email protected] - [email protected]

2nd IFAC Workshop on Fractional Differentiation and Its Applications 19-21 July 2006, Porto, Portugal J. Cervera and A. Baños

ALS in QFT by Using CRONE Structures

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Outline 1

Motivation

2

Preliminaries QFT CRONE

3

Proposed solution CRONE 2 Modified CRONE 3

4

Design Example logoumu

J. Cervera and A. Baños



The QFT Automatic Loop Shaping Problem

QFT = Quantitative Feedback Theory: robust frequency domain control design methodology Key design step: loop shaping Loop shaping = nonlinear nonconvex optimization problem.

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Previous Approaches

APPROACH Simplify the problem (linearize/convexify) Use fixed (rational) structure (with few parameters) nonlinear + nonconvex optimization algorithm

DRAWBACKS conservative not flexible - computationally demanding - global optimum not always guaranteed

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Proposed Approach

No simplification = not conservative. Evolutionary algorithms. Fixed structure: Flexible enough to approach optimum With few parameters

⇒ FRACTIONAL STRUCTURES, in particular CRONE STRUCTURES

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QFT CRONE

QFT introduction (1) Basic idea: quantitative relation uncertainty ⇔ control effort Typical configuration:

F(s)

+ _

C(s)

P(s)

For P(s), template P and nominal P0 . Design of C(s): in Nichols chart, with L0 (s) = P0 (s)C(s) logoumu




QFT CRONE

QFT introduction (2) Ω: discrete set of design frequencies. Robust stability/performance specifications ⇒ ⇒ boundaries Bω , ω ∈ Ω Basic step: loop shaping – design of L0 (jω) which satisfies boundaries is reasonably close to optimum

Optimization: minimization of high frequency gain (Khf ) Optimum characteristics: on performance boundaries tightly following UHFB. logoumu




QFT CRONE

QFT design example

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QFT CRONE

CRONE - Introduction CRONE = Contrôle Robuste d’Ordre Non Entier Based on the use of noninteger differential operator CRONE 1 & 2: real non integer differentiation: β(s) = ksn , n, k ∈ R s n 1+ band defined: β(s) = k 1+ ωsh ωl

CRONE 3: a+ib complex differentiation: D(s) = ωsu , a, b, ωu ∈ R a+ib 1+ s band defined: D(s) = C0 1+ ωsh ωl

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QFT CRONE

CRONE 1 & 2 - Purpose

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QFT CRONE

CRONE 3 - Purpose

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QFT CRONE

CRONE - Structures CRONE 1 & 2: L(s) = k

ω

l

s

nI +1

1+ 1+

s ωh s ωl

!n

1

s ωh

+1

nF

CRONE 3: L(s) = k

ω

l

s

+1

nI

C0

1+ 1+

s ωh s ωl

!a

" cos b Log C0


1+ 1+

s ωh s ωl

!#

1

s ωh

+1


nF logoumu


CRONE 2 Modified CRONE 3

ALS with CRONE 2 structure

Original semantic is lost Structure: original, slightly rewritten ω nI ω + s n 1 l h L(s) = k 0 +1 s ωl + s (s + ωh )nF nI , nF : fixed/conditioned by np0 and npe k conditioned by specified ωcg

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ALS with (modified) CRONE 3 Original semantic is lost Structure: decoupled (to obtain more flexibility)

L(s) = k

ω

l

s

+1

nI

C0

1+ 1+

s ωh s ωl

!a

"

1+

cos b Log cC0

1+

s ωh0 s ωl0

!#

1

s ωh4

+1

nF

nI , nF : fixed/conditioned by np0 and npe k conditioned by specified ωcg J. Cervera and A. Baños


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RHP zeros avoidance 1+ ωs0 h possible RHP zeros from Q(s) = cos b Log cC0 1+ s ω0 l

no RHP zeros if and only if: if b > 0,

π (2kceil − 1) − ln c 2b π ln C0 < − (2kfloor − 1) + ln c 2b ln C0

Automatic Loop Shaping in QFT by Using CRONE Structures

Automatic Loop Shaping in QFT by Using CRONE Structures

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