14th IEEE Mediterranean Electrotechnical Conference. 5-7 May 2008, Ajaccio, France. Baños, Cervera, Lanusse & Sabatier. Bode optimal loop shaping with ...
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Bode optimal loop shaping with CRONE compensators A. Baños1
J. Cervera1
P. Lanusse2
J. Sabatier2
1 Faculty
of Computer Engineering, Department of Computer and Systems Engineering, University of Murcia (Spain) – [jcervera,abanos]@um.es — 2 Université de Bordeaux, CNRS UMR 5218, Laboratoire IMS, Talence Cedex (France) – [patrick.lanussej,jocelyn.sabatier]@laps.ims-bordeaux.fr
14th IEEE Mediterranean Electrotechnical Conference 5-7 May 2008, Ajaccio, France Baños, Cervera, Lanusse & Sabatier
logoumu
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Outline 1
Introduction: Bode Optimal Loops Four Parameters Bode Optimal Loop Eight Parameters Bode Optimal Loop Goal
2
Loop Shaping with CRONE Compensators Why a CRONE compensator? Real Differentiator Term Low and High Frequency Terms Complex Differentiator Term Maximizing Loop Phase Lag
3
Desig Example logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Four Parameters Bode Optimal Loop Eight Parameters Bode Optimal Loop Goal
Problem Statement
PROBLEM: for operational bandwith 0 ≤ ω ≤ ω0 where desired |L(jω)| = M0 1 given crossover frequency ωc compute L(jω) which maximizes ω0
SOLUTION: decrease |L(jω)| as fast as possible...
logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Four Parameters Bode Optimal Loop Eight Parameters Bode Optimal Loop Goal
Problem Statement
PROBLEM: for operational bandwith 0 ≤ ω ≤ ω0 where desired |L(jω)| = M0 1 given crossover frequency ωc compute L(jω) which maximizes ω0
SOLUTION: decrease |L(jω)| as fast as possible...
logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Four Parameters Bode Optimal Loop Eight Parameters Bode Optimal Loop Goal
Problem Statement
logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Four Parameters Bode Optimal Loop Eight Parameters Bode Optimal Loop Goal
Ideal Bode Characteristic
( |L(jω)| = M0 , 0 < ω < ω0 For ideal optimal structure ∠L(jω) = −απ, ω > ω0 solution is equivalent to maximize phase lag, or minimizing stability margin, so it is necessary to trade-off between ω0 and stability margin...
logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Four Parameters Bode Optimal Loop Eight Parameters Bode Optimal Loop Goal
Ideal Bode Characteristic
( |L(jω)| = M0 , 0 < ω < ω0 For ideal optimal structure ∠L(jω) = −απ, ω > ω0 solution is equivalent to maximize phase lag, or minimizing stability margin, so it is necessary to trade-off between ω0 and stability margin...
logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Four Parameters Bode Optimal Loop Eight Parameters Bode Optimal Loop Goal
Ideal Bode Characteristic
( |L(jω)| = M0 , 0 < ω < ω0 For ideal optimal structure ∠L(jω) = −απ, ω > ω0 solution is equivalent to maximize phase lag, or minimizing stability margin, so it is necessary to trade-off between ω0 and stability margin...
logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Four Parameters Bode Optimal Loop Eight Parameters Bode Optimal Loop Goal
Four Parameters Bode Optimal Loop
In general, the problem is well defined as a function of parameters: M0 α ω0 ωc
Not all of them independent.
logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Four Parameters Bode Optimal Loop Eight Parameters Bode Optimal Loop Goal
Practical considerations about high frequency
In practice, four parameters Bode Optimal Loop has to be modified: To cope with sensor noise amplification Because it is not realistic to assume good control of |L(jω)| for high frequency.
logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Four Parameters Bode Optimal Loop Eight Parameters Bode Optimal Loop Goal
Seven Parameters Bode Optimal Loop
logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Four Parameters Bode Optimal Loop Eight Parameters Bode Optimal Loop Goal
Seven Parameters Bode Optimal Loop
logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Four Parameters Bode Optimal Loop Eight Parameters Bode Optimal Loop Goal
Eight Parameters Bode Optimal Loop
In order to add integrators to the loop, for a good steady state response ...
logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Four Parameters Bode Optimal Loop Eight Parameters Bode Optimal Loop Goal
Eight Parameters Bode Optimal Loop
logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Four Parameters Bode Optimal Loop Eight Parameters Bode Optimal Loop Goal
Eight Parameters Bode Optimal Loop
Parameters: M0 M1 α ω0 ωc ω1 n e
Not all of them are independent. logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Four Parameters Bode Optimal Loop Eight Parameters Bode Optimal Loop Goal
Our Goal
Establish relations between these eight parameters and the parameters of a proposed CRONE structure, so that a first approach to Bode optimal loop can be obtained in an easy and fast way.
logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Why a CRONE compensator? Real Differentiator Term Low and High Frequency Terms Complex Differentiator Term Maximizing Loop Phase Lag
CRONE Features for Bode Optimal Loop Shaping Easy to tune Few parameters For the generation 2/3 CRONE band defined compensator Dr =
1+ ωs
C0 1+
l s ωh
a
1+ ωs
cos −b Log C0 1+
l s ωh
,
Phase and gain slope only depend on a (real differentiation order) Gain and phase slope only depend on b (complex differentiation order)
Idea: for a Bode optimal loop shape, with constant phase at (ωl , ωh ) and constant gain at (ωl0 , ωh0 ), use real differentiator at (ωl , ωh ) (b=0) and complex differentiator (a=0) at (ωl0 , ωh0 ). Baños, Cervera, Lanusse & Sabatier
logoumu
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Why a CRONE compensator? Real Differentiator Term Low and High Frequency Terms Complex Differentiator Term Maximizing Loop Phase Lag
CRONE Features for Bode Optimal Loop Shaping Easy to tune Few parameters For the generation 2/3 CRONE band defined compensator Dr =
1+ ωs
C0 1+
l s ωh
a
1+ ωs
cos −b Log C0 1+
l s ωh
,
Phase and gain slope only depend on a (real differentiation order) Gain and phase slope only depend on b (complex differentiation order)
Idea: for a Bode optimal loop shape, with constant phase at (ωl , ωh ) and constant gain at (ωl0 , ωh0 ), use real differentiator at (ωl , ωh ) (b=0) and complex differentiator (a=0) at (ωl0 , ωh0 ). Baños, Cervera, Lanusse & Sabatier
logoumu
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Why a CRONE compensator? Real Differentiator Term Low and High Frequency Terms Complex Differentiator Term Maximizing Loop Phase Lag
CRONE Features for Bode Optimal Loop Shaping Easy to tune Few parameters For the generation 2/3 CRONE band defined compensator Dr =
1+ ωs
C0 1+
l s ωh
a
1+ ωs
cos −b Log C0 1+
l s ωh
,
Phase and gain slope only depend on a (real differentiation order) Gain and phase slope only depend on b (complex differentiation order)
Idea: for a Bode optimal loop shape, with constant phase at (ωl , ωh ) and constant gain at (ωl0 , ωh0 ), use real differentiator at (ωl , ωh ) (b=0) and complex differentiator (a=0) at (ωl0 , ωh0 ). Baños, Cervera, Lanusse & Sabatier
logoumu
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Why a CRONE compensator? Real Differentiator Term Low and High Frequency Terms Complex Differentiator Term Maximizing Loop Phase Lag
CRONE Features for Bode Optimal Loop Shaping Easy to tune Few parameters For the generation 2/3 CRONE band defined compensator Dr =
1+ ωs
C0 1+
l s ωh
a
1+ ωs
cos −b Log C0 1+
l s ωh
,
Phase and gain slope only depend on a (real differentiation order) Gain and phase slope only depend on b (complex differentiation order)
Idea: for a Bode optimal loop shape, with constant phase at (ωl , ωh ) and constant gain at (ωl0 , ωh0 ), use real differentiator at (ωl , ωh ) (b=0) and complex differentiator (a=0) at (ωl0 , ωh0 ). Baños, Cervera, Lanusse & Sabatier
logoumu
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Why a CRONE compensator? Real Differentiator Term Low and High Frequency Terms Complex Differentiator Term Maximizing Loop Phase Lag
CRONE Features for Bode Optimal Loop Shaping
Additionally, two terms to shape low and high frequencies Final structure: L(s) = k
ω
l
s
+1
nl
C0
1+ 1+
s ωl s ωh
!a
" cos −b Log
C00
1+ 1+
s ωl0 s ωh0
!#
1
s ωh
n h +1
logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Why a CRONE compensator? Real Differentiator Term Low and High Frequency Terms Complex Differentiator Term Maximizing Loop Phase Lag
Real Differentiator Term
a 1+ ωs l L2 (s) = C0 1+ s ωh
Design relations: a
π − 2θl (ωu ) = 2 −a ωh ≈ M0 M1 2ωl
(1 − α)π
logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Why a CRONE compensator? Real Differentiator Term Low and High Frequency Terms Complex Differentiator Term Maximizing Loop Phase Lag
Real Differentiator Term
a 1+ ωs l L2 (s) = C0 1+ s ωh
Design relations: a
π − 2θl (ωu ) = 2 −a ωh ≈ M0 M1 2ωl
(1 − α)π
logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Why a CRONE compensator? Real Differentiator Term Low and High Frequency Terms Complex Differentiator Term Maximizing Loop Phase Lag
Real Differentiator Term
a 1+ ωs l L2 (s) = C0 1+ s ωh
Design relations: a
π − 2θl (ωu ) = 2 −a ωh ≈ M0 M1 2ωl
(1 − α)π
logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Why a CRONE compensator? Real Differentiator Term Low and High Frequency Terms Complex Differentiator Term Maximizing Loop Phase Lag
Real Differentiator Term
a 1+ ωs l L2 (s) = C0 1+ s ωh
Design relations: a
π − 2θl (ωu ) = 2 −a ωh ≈ M0 M1 2ωl
(1 − α)π
logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Why a CRONE compensator? Real Differentiator Term Low and High Frequency Terms Complex Differentiator Term Maximizing Loop Phase Lag
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Low and High Frequency Terms
LCRONE2 (s) = k
ωl s
+1
nl
C
1+ ωs
l 0 1+ s ωh
a
1 n
s +1 ωh
h
Design relations: nl ≥ n nh ≥ ep ≥ n |LCRONE2 (jωc )| = 1 M +M |LCRONE2 (jωu )| = 0,dB 2 1,dB
logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Why a CRONE compensator? Real Differentiator Term Low and High Frequency Terms Complex Differentiator Term Maximizing Loop Phase Lag
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Low and High Frequency Terms
LCRONE2 (s) = k
ωl s
+1
nl
C
1+ ωs
l 0 1+ s ωh
a
1 n
s +1 ωh
h
Design relations: nl ≥ n nh ≥ ep ≥ n |LCRONE2 (jωc )| = 1 M +M |LCRONE2 (jωu )| = 0,dB 2 1,dB
logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Why a CRONE compensator? Real Differentiator Term Low and High Frequency Terms Complex Differentiator Term Maximizing Loop Phase Lag
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Low and High Frequency Terms
LCRONE2 (s) = k
ωl s
+1
nl
C
1+ ωs
l 0 1+ s ωh
a
1 n
s +1 ωh
h
Design relations: nl ≥ n nh ≥ ep ≥ n |LCRONE2 (jωc )| = 1 M +M |LCRONE2 (jωu )| = 0,dB 2 1,dB
logoumu
Baños, Cervera, Lanusse & Sabatier
Bode optimal loop shaping with CRONE compensators
Introduction: Bode Optimal Loops Loop Shaping with CRONE Compensators Desig Example Conclusions
Why a CRONE compensator? Real Differentiator Term Low and High Frequency Terms Complex Differentiator Term Maximizing Loop Phase Lag
Complex Differentiator Term
L3 (s) = cos −b Log
1+ ωs0 0 C0 1+ sl 0 ω
h
Complements L2 (s), to increase phase lag at [ωl0 , ωh0 ] To avoid non minimum phase: 0 ωh b log
