Prevention of controller windup A framework for linear ...mexgh/gherrmann_files/... · Prevention of controller windup / Engineering Colloquium – University of Bristol - Thursday

Dr Guido HerrmannDepartment of Mechanical Engineering, UOB

Prevention of controller windup –

A framework for linear, nonlinearand adaptive control schemes

ThursdayThursday,, 26 April 26 April 20072007

Engineering Engineering Colloquium Colloquium -- University of BristolUniversity of Bristol

Prevention of controller windup / Engineering Colloquium – University of Bristol - Thursday 26 April 2007 -2-

Outline of the presentation

I. Motivation for Anti-Windup (AW) Compensation

II. AW-compensation for control loops with linear plants• A sampled-data framework

• Robustness

III. AW-compensation in application to hard-disk drive servo systems• Hard Disk Drives and Aeroplanes ?

• A novel Dual-Stage Servo

IV. AW-compensation in non-linear dynamic inversion control

V. AW-compensation for adaptive neural network controllers

VI. Summary


K Gpr u y

Nominal control loop:Designed with some known design technique, e.g.

PlantController

Control Systems with Constrained Actuator Signals

• μ-synthesis/analysis, H∞-control

• adaptive neural network approach

• non-linear dynamic inversion control

• sliding mode control


K Gpr u yum

Practical Constraint:

• control signal limitations

Performance Loss or even Instabilitye.g. sluggish response due to windup of internal states of controller or even limit cycles



u umK Gpr y

Work on Constrained Control

Actuator limitations are significant when controlling

High precision micro/nano positioning systems,e.g. hard disk drives

Aircrafts,Actuator rate & amplitude limits

Saab Gripen JAS39

Chemical Processes



K Gp+

-

r y

++ Θ

Θ1

Θ2

• acts once saturation occurs

• retains nominal controller action

+ -

u um

Advantages:• Nominal controller design can be retained (straightforward commissioning, intuitive for control engineer)

• straightforward implementation since off-line designed (any sampling frequencies are permissible, in contrast to model-predictive control)

A solution: Anti-Windup-Compensation


AW-compensation for control loops with linear plants

-A sampled-data framework

Work with:

Matthew C. Turner, Ian Postlethwaite

Further Reading:• Herrmann, G, Turner, M. C. & Postlethwaite, I. . 'Some new results on anti-windup-conditioning using the Weston-Postlethwaite approach', 43rd IEEE Conference on Decision and Control, Bahamas, (pp. 5047-5052), 2004.• Herrmann, G, Turner, M. C. & Postlethwaite, I.. 'Discrete-time and sampled data anti-windup synthesis: stability and performance', The International Journal of Systems Science (Special Issue), 37(2), (pp. 91-113), 2006.• Turner, M. C., Herrmann, G. & Postlethwaite, I.. ‘Incorporating robustness requirements into anti-windup design’, IEEE Transactions on Automatic Control, to be published, 2007.

For complete list: https://www.bris.ac.uk/iris/publications/


Kr(k)

y(t)

z(t)

Gpy(t)

sampler Sτ

zero-order hold

Hτ

Sampled-data AW-Compensation


K(z)+

r(k)

+

+ -

u um


The AW-compensator has a special structure which allows to re-draw this scheme into an equivalent configuration.

⎥⎦

⎤⎢⎣

⎡)(G)(G

p/y

p/z

ss

z(t)

y(t)

zero-order hold

Hτ

samplerSτ

)(kυ

-

+Gp/y(z)M(z)

M(z)-I

Gp/y(z)=Sτ Gp/y(s)Hτ


K(z)

Gp/y(z)M(z)

+-

+ -

+

+

M(z)-I

r

y(t)

z(t)⎥⎦

⎤⎢⎣

⎡)(G)(G

p/y

p/z

ssHτ

Sτ


rK(z)

M(z)

+ -

M(z)-I

+

-

The required structure for the AW-compensator allows to equivalently re-draw this scheme:

z

⎥⎦

⎤⎢⎣

⎡)(G)(G

p/y

p/z

ss

Gp/z(s)

Hτ

Hτ

Sτ


K(z)

M(z)

+ -

M(z)-I

+

-

z(t)⎥⎦

⎤⎢⎣

⎡)(G)(G

p/y

p/z

ss

Gp/z(s)

Hτ

Hτ

Sτ

r(k)

ulin

ud

u~

zd(t)

The aim is to retain nominal linear performance:Minimization of the L2-gain of T: ulin (k) → zd (t)



M(z)

M(z)-I

+

-Gp/z(s)Hτ

ulin

ud

u~zd(t)

The AW-stability problem is reduced to a discrete stability problem The performance problem is

the mixed problem of optimizing the L2 gain for the operator

dlin zu a :with discrete input signal and continuous output signal

zu



Lifting and Sampled-data AW-Compensation

Note that for every sampling interval [kτ,(k+1)τ), the output trajectory zd(t) is uniquelydetermined by the input value uz(k) and the state xp(kτ) at the time instant kτ.

uz(k) zd(t)

This is an important feature which justifies the usage of sampled-data lifting techniques which allows to represent zd(t) which allows to represent in a discrete-time space.

uz(k)

xp(kτ)

Gp/z(s)Hτ


Lifting and Sampled-data AW-Compensation

∑

∑

∑

∫∞

=

∞

=≠∞

=

∞

≠ ==

0

2

0

2

0

0

2

0

2

0

)(

)(~

sup)(

)(sup

klin

kd

u

klin

d

u

ku

kz

ku

dttz

linlinγ

Using these linear lifting ideas, there exists a discrete linear system )(~

/ zG zp

An upper bound on γ can now be computed using a multi-variable version of the circle criterion and g can be minimized using LMI-techniques and using the coprime factorization )()()(~ 1/ zMzNzG zp

−=

M(z)

M(z)-I

+- Gp/z(s)Hτ

ulinu~

zd(t)zu

M(z)

M(z)-I

+-

p/z(z)ulin

u~ zu )(~ tzdG~


AW-compensation in application

to hard-disk drive servo systems

Work with:

Branislav Hredzak, Matthew C. Turner, Ian Postlethwaite, Guoxiao Guo

Further Reading:

• Herrmann, G, Turner, M. C., Postlethwaite, I. & Guo, G.. 'Practical implementation of a novel anti-windup scheme in a HDD-dual-stage servo-system', IEEE /ASME Transactions on Mechatronics, 9(3), (pp. 580-592), 2004.• Hredzak, B. , Herrmann, G & Guo, G.. 'A Proximate-Time-Optimal-Control Design and Its Application to a Hard Disk Drive Dual-Stage Actuator System', IEEE Transactions on Magnetics, 42(6), (pp. 1708-1715), 2006.• Herrmann, G, Hredzak, B., Turner, M. C., Postlethwaite, I. & Guo, G.. 'Improvement of a novel dual-stage large-span track-seeking and track-following method using anti-windup compensation', American Control Conference, Minneapolis, USA, 2006.



1500 mi/s

2 mm above the ground

Hard Disk Drives and Aeroplanes ?


1500 mi/s

2 mm above the ground

Above a ‘highway’

with 3 cm wide lanes

Sorry no GPS!Informfrom the lane markers below which have about a100 m separation

ation is picked


Mechanical resonance frhigher than the Nyquistsampling frequency)

equencies are frequency (or even



Let us decrease the whole problem by a factor of 105.


Hard Disk Drive (HDD) Servo-Control

Seeking

Settling

Track-following


• The dual-stage actuator allows the controller bandwidth to be increased from 1.4kHz for the single stage to more than 2kHz.

• This improves position accuracy and disturbance rejection in a large frequency region.

• Secondary actuators are an attractive alternative to the single actuator to achieve higher track densities, measured in Track-Per-Inch (TPI)

SecondaryPiezoelectric (PZT) Micro-Actuator±0.5μm

VCM-actuator (driven by theVoice Coil Motor)

AW-Compensation and HDD Servo-Control

Seagate Cheetah 10K.7SCSI-Drive

High-tech at Low Cost !!


Dual-Stage Servo-Control

Track-Seeking Track-Following Track-Settling• Solved for dual-stagee.g.

Schroeck, S. J., Messner, W. C., & McNab, R. J. IEEE T. Mechatronics, (2001).

Herrmann, G. and Guo, G., CEP (2003).

Mori et al., 1991; Koganezawa et al., 1999;

Oboe & Murari, 1999; Semba et al., 1999

• Solved now

• Path Planning Approaches:Kobayashi & Horowitz (2001), Numasato & Tomizuka(2001), Li & Tomizuka IFAC(2005).

No direct account of actuator limitation

• Short Seeking Methods accounting for actuator limitationGuo, Wu & Chong (2002), Herrmann et al (IEEE/ASME T. Mechatronics 2004).

• Seagate Cheetah 10K.7 ?

• Short and Large Span Seeking MethodsHredzak, Herrmann & Guo (IEEE T Magnetics 2006),Herrmann, Hredzak, Turner, Postl. & Guo (ACC2006), Herrmann et al. (Intern J Adapt C & Sign Pr, acc.)


Revision of the Decoupled Dual-Stage Servo

PVCMCVCM+

+PPZTCPZT

++

-+yr

e

yT

yV

yP

PVCMCVCM+

+

PPZTCPZT+

+

-

+yr

yT

yV

e

yP-

• The decoupling does notdepend on the character of the control scheme!

• The controller can be non-linear!

(yr-yV)


Options for implementation – Original Approach

PVCMCVCM++

PPZTCPZT(z)

++

-+yr

e

yT

yV

yP

py

We have to satisfy the Circle criterion for stability ! This is not necessarily always possible !

An alternative is to introduce anti-windup (AW) techniques to deal with the saturation !

S1 S2

S3

The saturation S1 is tuned so that the limits of S2 are never reached.

Non-linear Proximate-Time Optimal Controller

py


PVCMCVCM++

PPZTCPZT(z)

++

-+yr

e

yT

yV

yP

py

++

Θ2

Θ1+ -

-+

The additional AW-compensator is only active once saturation is reached.

It retains stability and performance in case saturation is reached.

Control scheme with AW-compensation


EXPERIMENTAL RESULTS


4 μm step (Original control)

Ch3: LDV-measurement (2 μm/V), Ch2: PZT control signal, Ch1: VCM driver

input.

4 μm step (dynamic AW)Ch3: LDV-measurement (2 μm/V), Ch2:

PZT control signal, Ch1: VCM driver input.

EXPERIMENTAL RESULTSTime Responses

We investigated seeks up to the LDV-sensor limit of 200 μm step

ts ts


AW-compensation in

non-linear dynamic inversion control

Work with:

Prathyush Menon, Matthew C. Turner, Ian Postlethwaite, Declan Bates

Further Reading:

• Herrmann, G, Turner, M. C. , Menon, P., Bates, D. & Postlethwaite, I. . 'Anti-windup synthesis for nonlineardynamic inversion controllers', 5th IFAC Symposium on Robust Control Design, Toulouse, France, 2006.• Menon, P. , Herrmann, G, Turner, M. C., Postlethwaite, I. & Bates, D.. 'General Anti-windup synthesis for input constrained nonlinear systems controlled using nonlinear dynamic inversion', 45th IEEE Conference on Decision and Control, San Diego, California, 2006.



A Nonlinear dynamic inversion controller with outer loop

dBuxBGxBfAxx dm +++= )()(&1)( −xG

x

Plant

NDI-Controller with tracking

mu

NDI-controllers are significant in

• Flight Control (Fighter aircraft, Missile Control, Helicopter Control, etc.)• Control of Robots

d

C

pdD

x

)(xf−

r)(sH



Control Signal Saturation in Closed Loop Systems can cause

• performance degradation• instability Resolution via AW-compensation

dBuxBGxBfAxx dm +++= )()(&1)( −xG)(xf−

x

xNDI-Controller with tracking

mu

Plant

)(sHr

d

C

pdD




)(xf−

x

mu

)(sHr

dC

pdD



Assumption 1:

)(xBfAxx +=& is exponentially stable.

Assumption 3:The closed loop is well-posed and the origin is exponentially stable.

Assumption 2:

f(x) and G(x) are Lipschitz


)(xf−

x

mu

)(sHr

dC

pdD


K(s)+

r

+

+ -

u um

Linear Control Loop with AW-Compensation

-

+Gp/y(s)M(s)

M(s)-I

⎥⎦

⎤⎢⎣

⎡)(G)(G

p/y

p/z

ss

z

y


Coprime Factorization Approach

)()()(G 1p/y sMsNs−=

Choose M(s) to be a Coprime Factor:

⎥⎦

⎤⎢⎣

⎡0

~)(G/

p/yyp

pp

CBA

s

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ +

⎥⎦

⎤⎢⎣

⎡−

00~

)()()(G

/p/y

FC

BFBA

IsMsMs

yp

ppp

F -parameter matrix

K(s) +-

+ -

+

+

r

y(t)

z(t)⎥⎦

⎤⎢⎣

⎡)(G)(G

p/y

p/z

ss

AWpAWppAW uBxFBAx ++= )(&

AWFx

AWp xC

AWu



K(s)+

-

+ -

+

+

r

y(t)

z(t)

⎥⎦

⎤⎢⎣

⎡)(G)(G

p/y

p/z

ss

AWpAWpAWpAW uBFxBxAx ++=&

AWFx

AWpxCAWu



+

K(s)+

-

-+

+

r

y(t)

z(t)

⎥⎦

⎤⎢⎣

⎡)(G)(G

p/y

p/z

ss

AWpAWpAWpAW uBFxBxAx ++=&

AWFx

AWpxCAWu



-

K(s)+

+

-+

+

r

y(t)

z(t)

⎥⎦

⎤⎢⎣

⎡)(G)(G

p/y

p/z

ss

AWpAWpAWpAW uBFxBxAx −+=&

AWFxAWu

AWpxC


-

AW-compensation for the nonlinear dynamic inversion

K(s) ++

+ -

+

r

y(t)

z(t)⎥⎦

⎤⎢⎣

⎡)(G)(G

p/y

p/z

ss

AWpAWpAWpAW uBFxBxAx −+=&

AWFx

AWyp xC /

AWu

• h(xAW) is a parameter

How do we parameterize h(xAW) ?

The coprime factorizationapproach is to be extended to

the non-linear problem


)(xf−

x

)(sHr

d C

pdD

])()[()( AWAWAWAWAW uxhxGxBfAxx −++=&+

+

+ -

+ mu

• Note the non-linear input gain G(x) depends on x

+)( AWxh

C

)( AWxf−-

-

e.g.: h(xAW)=0 gives thealways stable Internal Model Control (IMC)-AW


K(s)

Gp/y(s)M(s)

+-

+ -

+

+

M(s)-I

r

y(t)

z(t)⎥⎦

⎤⎢⎣

⎡)(G)(G

p/y

p/z

ss

Equivalent Representations of AW – Linear Case

rK(s)

M(s)

+ -

M(s)-I

+

-

The required structure for the AW-compensator allows to equivalently re-draw this scheme:

z

⎥⎦

⎤⎢⎣

⎡)(G)(G

p/y

p/z

ss

Gp/z(s)


K(s)

M(s)

+ -

M(s)-I

+

-

z(t)⎥⎦

⎤⎢⎣

⎡)(G)(G

p/y

p/z

ss

r(k)

ulin

ud

u~

zd(t)

Design Approach of AW - Linear Case

Gp/z(s)

zdDesign approach: Minimization of the L2-gain of T: ulin→

Optimization viaLMI-methods


Equivalent Representation of Optimization for Linear Systems

K(s)

M(s)

+ -

M(s)-I

+

-

z(t)⎥⎦

⎤⎢⎣

⎡)(G)(G

p/y

p/z

ss

r(t)

ulin

udu~

zd(t)Gp/z(s)

K(s)

Gp/y(s)M(s)

+-

+ -

+

+

M(s)-I

r

y(t)

z(t)⎥⎦

⎤⎢⎣

⎡)(G)(G

p/y

p/z

ss+ -

+

w

is the equal to

the L2-gain of T: w→z

G. Herrmann, M. C. Turner & I. Postlethwaite. Some new results on anti-windup-conditioning using the Weston-Postlethwaite approach. In: Proc. Of the 42nd IEEE Conference on Decision and Control, Bahamas, 2004.

For linear nominal systems The L2-gain of T: ulin→zd


AW-Compensator Parametrization – Nonlinear Case

xCz

z

Minimization of the L2-gain of T: w→z


)(xf−

x

)(sHr

dC

pdD

])()[()( AWAWAWAWAW uxhxGxBfAxx −++=&

C

)( AWxf−

)( AWxh

+

+

-

-

+ -

+

+

w

+ -


[ ]

021)(

0)()(21)()()(

2

<

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−∗∗

−−−∗

⎥⎦

⎤⎢⎣

⎡−

∂∂

+++∂∂

I

WIW

WxhxBGxVxCCxxhxBGxBfAx

xV T

AWAW

AWzT

zT

AWAWAWAWAW

γ

ε

AW-Compensator Parametrization – Nonlinear Case

The L2-gain of T: w→z is smaller than γ>0 if the following matrix inequality is satisfied:

for suitable diagonal, W>0, and, ε>0.


Numerical Algorithm

,)(2

21

22212

112112

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

qAW

AW

AW

qqT

qT

q

qT

q

qAW

AW

AW

AW

x

xx

P

PPP

PPPPPP

x

xx

xVM

4444 34444 21L

MMM

L

L

Mand

The positive definite matrix P is now the parameter matrix for suitably chosen degree q.

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

inAW

iAW

iAW

iAW

x

xx

x

,

1,

1,

M

The MI is to be solved for the special choice of

AW

AWTTAW x

xVBxGxh ∂∂−= )()()(

021)(

0)()(21)()()(

2

<

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−∗∗

−−−∗

⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

+⎥⎦

⎤⎢⎣

⎡

∂∂

−+∂∂

I

WIW

WxBGxVxBG

xVxCCx

xVBxGxBGxBfAx

xV

AWAWAWz

Tz

TAW

T

AW

TTAWAW

AW

γ

ε

so that


Numerical AlgorithmThe non-convex matrix inequality is solved using Genetic Algorithms (GA), i.e.

[ ]

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡ +∂∂

−

−<

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−∗∗

−−−∗

⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

+⎥⎦

⎤⎢⎣

⎡

∂∂

−+∂∂

II

xBfAxxV

I

WIW

WxBGxVxBG

xVxCCx

xVBxGxBGxBfAx

xV

AWAWAW

AWAWAWz

Tz

TAW

T

AW

TTAWAW

AW

0000

00)(

21)(

0)()(21)()()(

2

δ

γ

ε

for given γ>0 and fixed ε>0 find P, W so that for δ>0

in a sufficiently large bounded set of x and xAW around the origin.Optimization Algorithm: Fix γ>0

Step 1: 1) Fix 0),( ≠AWxx2) Find from n matrices P the one with largest δ>03) Apply GA-operators

Step 2: 1) Fix P2) Find from n pairs 0),( ≠AWxx for smallest δ3) Apply GA-operators

0),( ≠AWxxUse

and decreaseγ>0if appropriate


Numerical Example

Nominal Control With Constraints ,50 1 ≤≤ u 50 2 ≤≤ u


Recent Future Work

Courtesy: Dr Mark Lowenberg, University of Bristol

A hawk modelwith

Dr. M. Lowenberg, Dr. P. Menon, Dr. M. Turner, Prof. Ian Postlethwaite, Dr D. Bates


AW-compensation for

adaptive neural network controllersWork with:

Matthew C. Turner, Ian Postlethwaite

Further Reading:

• Herrmann, G, Turner, M. C. & Postlethwaite, I.. 'Performance oriented anti-windup for a class of linear control systems with augmented neural network controller', IEEE Transactions on Neural Networks, 18(2), (pp. 449-465), 2007.



Unknown Nonlinearity

++

Motivation

LinearPlant

LinearController +

NNcompen-

sation

-

Adap-tation

NN-Control- Examples :S. S. Ge, T. H. Lee, and C. J. Harris, Adaptive Neural Network Control of Robotic Manipulators. World Scientific, Singapore, 1998.Y. Kim and F.L. Lewis, High-Level Feedback Control with Neural Networks," World Scientific, Singapore, 1998.

??Cu u

Linear control performance in combination with NN-control – Examples of practical validation:G. Herrmann, S. S. Ge, and G. Guo, “Practical implementation of a neural network controller in a hard disk drive,” IEEE Transactions on Control Systems Technology, 2005.——, “A neural network controller augmented to a high performance linear controller and its

application to a HDD-track following servo system,” IFAC 2005 (under journal review).

Anti-Windup (AW)(AW) Control - a possible approach to overcome controller saturationG. Grimm, J. Hatfield, I. Postlethwaite, A. R. Teel, M. C. Turner, and L. Zaccarian, “Antiwindup for stable linear systems with input saturation: An LMI based synthesis,” IEEE Trans. on Autom. Control, vol. 48, no. 9, pp. 1509–1525, 2003.Alternative for NN:W. Gao; R.R. Selmic, "Neural network control of a class of nonlinear systems with actuator saturation Neural Networks", IEEE Trans. on Neural Networks, Vol. 17, No. 1, 2006.


+

NNcompen-

sation

-


++

Controller conditioning

LinearPlant

LinearController

Adap-tation

Cu u

Non-linear

Algorithm

Linear AW-comp.

+ -


AW-Compensator Design Target

+

NNcompen-

sation

-


++LinearPlant

LinearController

Adap-tation

Linear AW-comp. + -

Cu u

Non-linear

Algorithm

++ -

wzdy

Linear AW-comp.

Design target for linear Design target for linear AWAW--compensator:compensator: 0, ;)()(

2

0

22

0

2 ≥+≤ ∫∫∞∞

γββγ dsswdsszMinimize γ for

This L2-gain optimization target ensures recovery of the nominal controller performance.


AW-Compensator Design Target

The conditioned linear control uL term operating in connection with the constrained NN-controller uNL, will track asymptotically any permissible steady state. The NN-weight estimates will remain bounded.

Design target for overall AWDesign target for overall AW--compensator:compensator:

+

NNcompen-

sation

-


++LinearPlant

LinearController

Adap-tation

Linear AW-comp. + -

Cu u

Non-linear

Algorithm

++ -d

y


A Simulation Example

Simulation for a direct drive DC-torque motor

Hsieh & Pan (2000)

Hsieh & Pan (2000) :

6-th order model to include issues of static friction, i.e. the pre-sliding behaviour:

650;=k0.175;=C

;015.54=m

1

s

-4⋅

The nominal model used for linear controller design

;1010

2

11

2

1 umx

x

mC

mk

xx

s⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−=⎥⎦

⎤⎢⎣

⎡&

&Other parameters:

80000;=50000;=k

2.5;=454.5;=4;=n

2

β

αλ

Assume both angle position x1 and angle velocity x2 are measurable



Control signalPosition signal

0 0 .02 0.04 0.06

-2

0

2

tim e

u

U n c on stra in ed R es po nse

0 0 .02 0.04 0.06-3

-2

-1

0

1

2

3

u

tim e

C o n s train ed R esp o n se

0 0.01 0.02 0.03 0.04 0.05 0.06-2

-1

0 x 10-4

x 1

time



0 0.02 0.04 0.06-1

-0.8

-0.6

-0.4

-0.2

0x 10-3

x 1

time

Control signalPosition signal

0 0 .0 2 0 .04 0 .0 6-15

-10

-5

0

5

10

15

tim e

u

U n c o n str a in e d R e s p o n s e

0 0 .0 2 0 .04 0 .0 6-15

-10

-5

0

5

10

15

u

t im e

C o n s train ed R e sp o n s e


SummaryOne single AW-concept suited to a linear control, a non-linear NDI-scheme and an

adaptive control context

o Linear / Nonlinear Coprime Factorization approach

o L2-gain Optimization approach

• Non-linear Operator for linear AW-compensation is mirrored to the NDI

AW-compensation problem

Design Optimization using Linear Matrix Inequalities for linear control systems

Future work: Further theoretical work and applications to practical systems

Design Optimization using Genetic Algorithms for NDI-control

Practically validated in HDD-servo systems but also in aero-space systems (DERA)

o Use of a copy of a plant model augmented with a filter for parameterization

o Straightforward Interpretation

MotivationController conditioningAW-Compensator Design TargetAW-Compensator Design TargetA Simulation ExampleA Simulation ExampleA Simulation Example

Documents

Prevention of controller windup A framework for linear ...mexgh/gherrmann_files/... · Prevention of controller windup / Engineering Colloquium – University of Bristol - Thursday