Automatic Loop Shaping in QFT by Using CRONE Structures

Transcription

1 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) - 2 nd IFAC Workshop on Fractional Differentiation and Its Applications July 2006, Porto, Portugal

2 Outline Motivation 1 Motivation 2 QFT CRONE 3 CRONE 2 Modified CRONE 3 4

3 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.

4 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

5 Proposed Approach No simplification = not conservative. Evolutionary algorithms. Fixed structure: Flexible enough to approach optimum With few parameters FRACTIONAL STRUCTURES, in particular CRONE STRUCTURES

6 QFT introduction (1) Motivation QFT CRONE Basic idea: quantitative relation uncertainty control effort Typical configuration: Fs () + _ Cs () Ps () For P(s), template P and nominal P 0. Design of C(s): in Nichols chart, with L 0 (s) = P 0 (s)c(s)

7 QFT introduction (2) Motivation QFT CRONE Ω: discrete set of design frequencies. Robust stability/performance specifications boundaries B ω, ω Ω Basic step: loop shaping design of L 0 (jω) which satisfies boundaries is reasonably close to optimum Optimization: minimization of high frequency gain (K hf ) Optimum characteristics: on performance boundaries tightly following UHFB.

8 QFT design example Motivation QFT CRONE

9 CRONE - Introduction QFT CRONE CRONE = Contrôle Robuste d Ordre Non Entier Based on the use of noninteger differential operator CRONE 1 & 2: real non integer differentiation: β(s) = ks n, n, k R ( ) 1+ s n ω band defined: β(s) = k h 1+ s ω l CRONE 3: ( ) a+ib s complex differentiation: D(s) = ω u, ( ) a+ib 1+ s ω band defined: D(s) = C h 0 1+ s ω l a, b, ωu R

10 CRONE 1 & 2 - Purpose QFT CRONE

11 CRONE 3 - Purpose Motivation QFT CRONE

12 CRONE - Structures Motivation QFT CRONE CRONE 1 & 2: L(s) = k ( ) ( ωl ) s + 1 ni 1 + s n ω h s ( ) ω s l + 1 nf ωh CRONE 3: L(s) = k ( ( ωl ) s + 1 ni 1 + s ω C h s ω [ ( l cos b Log ) a C s ω h 1 + s ω l )] 1 ( s ωh + 1 ) nf

13 ALS with CRONE 2 structure CRONE 2 Modified CRONE 3 Original semantic is lost Structure: original, slightly rewritten L(s) = k ( ω l s + 1 ) ni n I, n F : fixed/conditioned by n p0 and n pe k conditioned by specified ω cg ( ) ωh + s n 1 ω l + s (s + ω h ) n F

14 ALS with (modified) CRONE 3 CRONE 2 Modified CRONE 3 Original semantic is lost Structure: decoupled (to obtain more flexibility) L(s) = k ( ( ωl ) s + 1 ni 1 + s ω C h s ω l [ ( cos b Log ) a cc s ω h 1 + s ω l n I, n F : fixed/conditioned by n p0 and n pe )] 1 ( s ω h4 + 1 ) nf k conditioned by specified ω cg

15 RHP zeros avoidance CRONE 2 Modified CRONE 3 [ )] 1+ possible RHP zeros from Q(s) = cos b Log (cc s ω h 0 1+ s ω l no RHP zeros if and only if: if b > 0, ln C 0 < π 2b (2k ceil 1) ln c ln C 0 < π 2b (2k floor 1) + ln c if b < 0, ln C 0 < π 2b (2k floor 1) ln c ln C 0 < π 2b (2k ceil 1) + ln c for k ceil = ( 1 2 ln c 2b π + 1), k floor = ( 1 2 ln c 2b π + 1)

16 CRONE 2 design

17 (original) CRONE 3 design

18 (modified) CRONE 3 design

19 Comparison Motivation loop structure K hf (db) order 2 in Chen et al order 3 in Chen et al CRONE CRONE decoupled CRONE

20 Motivation A QFT ALS method, based on fractional structures, has been developed. CRONE structures use has been considered, including: decoupling of CRONE 3 structure for higher flexibility RHP zeros characterization in original/modified CRONE 3