Pole Placement and Active Vibration Control in Aeroelasticity

John E Mottershead1, Yitshak Ram2, Shakir Jiffri1, Sabastiano Fichera1 and Xiaojun Wei1

1University of Liverpool UK2Louisiana State University, Baton Rouge, USA

j.e.mottershead@liv.ac.uk

1

Adaptive Impact Absorption (AIA’15)IPPT PAN, Warsaw

15-16 October 2015

Contents

• Flutter control in aeroelastic systems

• Receptance method – LTI control based on measured vibration data

• Wind-tunnel aerofoil example

• Nonlinear aeroservoelasticity using feedback linearisation.

• Wind-tunnel example – nonlinear stiffness in plunge

• LCO suppression

• Feedback linearisation for non-smooth structural nonlinearity

• Numerical example – piecewise stiffness nonlinearity and freeplay

• MODFLEX wing

• Conclusions

2

Receptance Method

• Dynamic stiffnesses Receptances:

• No need to evaluate or to know the system matrices M, C, K.

• Any input-output transfer function may be used – dynamics of actuators,

sensors, filters etc. included in the measurement.

3

12 KCMH ssssss xfH

2s s s sM C K x f

Receptance Method Partial Pole Placement Problem

4

2

21,2, ,2

( )

( )

k k k

k k k

T T

k k

k nt

t

M C K v 0

M C K w Bu

u F G w

Open–loop and closed-loop systems:

Assigned eigenvalues are distinct from eigenvalues 1

p

k k

2

1

n

k k

While eigenvalues kk

nppk 2,...,2,1 are unchanged.

Unchanged Eigenvalues5

mbbbB 21 mfffF 21 mgggG 21

kTm

Tmkm

TTk

TTkkkk wgfbgfbgfbwKCM ...222111

2

May be re-written as,

Non-trivial solution:

0vgfbgfb kTm

Tmkm

TTk ...111 nppk 2,...,2,1

0

0

0

1

1

m

m

Tk

Tkk

Tk

Tkk

Tk

Tkk

g

g

f

f

v00v00

0v00v0

00v00v

2 T T

k k k k kM C K w B F G w

k kw v

Assigned Eigenvalues6

kTm

Tmkm

TTk

TTkkk wgfbgfbgfbHw ...222111 pk ,...,2,1

Let jkjkbHr , and k

Tj

Tjkjk

wgf, mj ,...,2,1

Then

mmk kkkkkk ,,2,2,1,1, ... rrrw

m

m

m

Tk

Tkk

Tk

Tkk

Tk

Tkk

k

k

k

,

2,

1,

1

1

g

g

f

f

w00w00

0w00w0

00w00w

where

,k jThe closed-loop mode shape is determined by the choice of

Receptance Method7

0 0 0 0

0 0 0 0

0 0 0 0

T T

k k k

T T

k k kk

T T

k k k

v v

v vQ

v v

0 0 0 0

0 0 0 0

0 0 0 0

T T

k k k

T T

k k kk

T T

k k k

w w

w wP

w w

The control gains are then given by the solution of,

1 1

1 1

2

p m

p

n m

1

p

P f α

P f α

Q g 0

Q g 0

Y.M. Ram and J.E. Mottershead, Multiple-input active vibration control by partial poleplacement using the method of receptances, Mechanical Systems and Signal Processing, 40, 2013, 727-735.

General Procedure

• Measure the open loop input-output FRF over a desired frequency range.

• Fit MIMO rational fraction polynomials to the measure FRF and obtain the input-output transfer functions.

• Select force distribution vectors bk(s).

• Apply the Receptance Method to obtain unknown gains, gk, fk.

• Implementation of the controller using dSPACE in real time.

8

0

0

α

α

g

g

f

f

Q

Q

P

P

p

1

m

m

n

p

p

1

1

2

1

1

Torsion Bar

Torsional StiffnessVertical Stiffness

Flap

9

Flutter SuppressionWind-Tunnel Aerofoil Rig

V-stack piezo actuator

Aerofoil

Curve Fitting10

Open loop FRFs include not only the dynamics of the aerofoil system but also

the power amplifier, the actuator, the sensors and the effects of A/D and D/A

conversion, numerical differentiation (Simulink/dSPACE) of displacements and

high- and low-pass Butterworth filters with cut-off frequencies of 1Hz and 35 Hz.

11

Frequency/Damping Trends – Root Locus

Poles assigned: 1.5 38i, 0.7 23ipitch heave

Separation of pitch and heave frequencies

at wind speed, 7m/s.

12

Flutter Margin

Predicted flutter speed increased from 17 m/s to 20 m/s.

Separation of pitch and heave frequencies.

28.2 29.5 0.0284 0.0232T T

g f

Quadratic flutter speed prediction.

Zimmerman N.H. and Weissenburger J.T. (1964)

Vibration control – 2 DOF aeroelastic system

3

2

1Controller ON - oscillation eliminated

Controller OFF - oscillation

Tensioned-Wire Plunge NonlinearityFeedback Linearisation

14

S. Jiffri, S. Fichera, A. Da-Ronch and J.E. Mottershead, Nonlinear control for the suppression of flutter in a nonlinear aeroelastic system, AIAA Journal of Guidance, Control and Dynamics, in preparation.

1 2 2 2 2

3 4 4 4 4

5 1 1 5 6 1 2 6

7 3 1 7 8 3 2 8

9 1 9 10 2 10

11 3 11 12 4 12

Pitch

Plunge

Aerodynamic

states

x x x f g u

x x x f g u

x x x x x x

x x x x x x

x u x x u x

x x x x

x

x

States x5 and x6 are associated with pitch, x7 and x8 are associated with

plunge, x9 and x10 are associated with flap motion and x11 and x12 are

associated with gusts.

1u y x ux f x g

Pitch Linearisation

15

Input-output linearisation with pitch displacement chosen as the output y

1 1 2 1 1z y x z z y x

2 2 2 2z y x f g ux

In matrix form,

1 1

2 2

2 2

0 1 0, ,

0 0 1

z zv v f g u

z zx

Linear and nonlinear terms are located in f2(x). denotes the artificial input

– it is the term that achieves the desired linear control objective.

Relative degree r=2. When r is less than the number of states the system can

only be partly linearised.

Pole Placement16

The artificial input may be defined as,

1 1 2 2,v k z k z

Choosing2

1 2, 2

n nk k will assign the natural frequency n

and damping . Then,

1 1

2 1 2 2

0 1.

z z

z k k z

And the physical nonlinear input is given by,

2 1 1 2 2 2

2 2

1 1.u v f k z k z f

g gx x

This input cancels the system dynamics and implements the linear control

requirement.

Internal Dynamics17

A linear coordinate transformation is carried out to obtain the system in Normal

Form – such that the input u does not appear explicitly,

,

4 910,:

, 1 1:12,j j

jz Tx T

T g 0

remaining terms equal to 0.

The zero dynamics is then obtained by setting the controlled coordinates to

zero, i.e. z1=z2=0,

Stability of the zero dynamics must be examined to ensure the stability of the

nonlinear controller.

4

2

2

2

3 4 4 2 4 5 1 5 6 2 6

17 3 1 7 8 3 2 8 9 2 1 9

110 2 2 10 11 3 11 12 4 12

, , , ,

, , ,

, , ,

g

g

g

g

z z z f f z z z z

z z z z z z z f z

z f z z z z z

z z

z

z

Tuned Numerical Model18

Linear frequency-domain tests:

0 5 10 15 202

3

4

5

air speed (m/s)

d (

Hz)

0 5 10 15 200

0.05

0.1

0.15

0.2

air speed (m/s)

pitch

plunge

LFS

flap

flap

shaker

shaker

Nonlinear time-domain tests:

-1

0

1

pitch (

deg)

-20

0

20

pitch v

elo

city (

deg/s

)

0 0.2 0.4 0.6 0.8 1-0.01

0

0.01

time (s)

plu

nge (

m)

0 0.2 0.4 0.6 0.8 1

-0.2

-0.1

0

0.1

0.2

time (s)

plu

nge v

elo

city (

m/s

)

-1

0

1

pitch (

deg)

0 0.2 0.4 0.6 0.8 1-0.01

0

0.01

plu

nge (

m)

time (s)

Numerical Experimental Phase portrait

-1 0 1

-40

-20

0

20

40

pitch (deg)

pitch v

elo

city (

deg/s

)

experimental numerical

-10 0 10

-200

0

200

plunge (mm)

plu

nge v

elo

city (

mm

/s)

Embedding the Numerical Model in the Aeroelastic Control Loop

19

(1)dSPACE Inputs

(2)Compute structural

states

(5)Compute artificial inputs

(4)Select x5-x12

(6)Compute physical (nonlinear) input(3)

Numerical aeroelastic

model (7)dSPACEoutput

x1-x4

3 laserdisplacement

sensors

Piezo-stackactuator

x

u

Test ResultsAssigned Damping at =0.3, U=15m/s

20

-1

0

1

pitch (

deg)

measured

controller ON

-0.01

0

0.01

plu

nge (

m)

measured

controller ON

2 3 4 5 6 7 8

-1

0

1

pitch (

deg)

time (s)

predicted

controller ON

2 3 4 5 6 7 8-0.01

0

0.01

plu

nge (

m)

time (s)

predicted

controller ON

Closed-loop response

2 3 4 5 6 7 8-2

-1

0

1

2

flap r

ota

tion (

deg)

time (s)

measured

controller ON

2 3 4 5 6 7 8-2

-1

0

1

2

flap r

ota

tion (

deg)

time (s)

simulated

controller ON

Flap motion

Feedback LinearisationAerofoil with Non-smooth Pitch Nonlinearity

21

S. Jiffri, P. Paoletti and J.E. Mottershead, Feedback linearization in systems with non-

smooth nonlinearities, AIAA Journal of Guidance, Control and Dynamics, in press.

Edwards, J. W., Ashley, H., and Breakwell, J. V. "Unsteady aerodynamic modeling for

arbitrary motions," AIAA Journal, Vol. 17, No. 4, 1979, pp. 365-374.

Aerodynamic model with 8 states, 6 structural (pitch, plunge, flap) and 2

aerodynamic states (Edwards, 1979).

Output y: plunge displacement . Input u: flap command com.

Pitch Nonlinearity22

,

, 1

,

nl

K g

K g g

K g

f

g

g defines the initial (lower) stiffness region on

either side of =0

Inner region g : Net stiffness=(1- )KOuter region >g : Net stiffness K

=1 produces freeplay. 1 produces piecewise

nonlinearity

K : chosen as the pitch stiffness of the desired

linear system.

Piecewise nonlinearity

Freeplay

Feedback LinearisationNon-smooth Nonlinear System Parameters

23

1 2

, where ,a nl

p a

u

q 0

x f x g x f x Ψq Φq Λq Ωf g x Π

E q E q F q 0

No requirement to differentiate the non-smooth nonlinearity.

The transformation matrix T is invertible.

The usual smoothness requirement on the nonlinearity can be removed.

2 2

1 1 1 1 1 2 2

1 1

2 2 2 2

pl n

n n n n n n

n n ker n n

n

T

0 I 0 0T

0 Π

0 I

Input u eliminated in

the internal dynamics

Transformation for the linearised part

Stability of the Plunge Zero Dynamics24

3

3:8

zz Az b

z z

The zero dynamics are found to take the form:

The equilibrium points are

found when:

1;T

eq

eq a

eq a

g

g

A be z 0

Az b 0

Az b 0

3 3

3 3

3

,

,

,

z z g

z g z g

g z g

Assuming non-singular and and solving above for each of

the 3 conditions,

1 3

T ze z

A

1

TA be

3

1

3

1

3

,

,

,

eq

z g

g z g

g z g

0

z A b

A b

Stability of the Plunge Zero Dynamics25

There are 2 non-zero equilibria.

Lyapunov function based around the particular equilibrium point:

12

ˆ ˆ ˆ, , 0,T T

eqVz z z z Pz P P P

The derivative of V for each of the three regions may be expressed as

(Shevitz and Paden, 1994):

1 3

1

3

1

3

ˆ ˆ ,0

ˆ ˆ , for

ˆ ˆ ,

T T

eqeq

T

eq eq

T eqeq

z g

V g z g g

gg z g

z P A be z zz

z P Az Az b z A b

z A bz P Az Az b

These derivatives must be strictly negative for asymptotic stability. 3 z3 stability

regions, for each of the 3 equilibria → 9 equations to be investigated.

Shevitz, D., and Paden, B. "Lyapunov stability theory of nonsmooth systems," IEEE

Transactions on Automatic Control, Vol. 39, No. 9, 1994, pp. 1910-1914.

S. Jiffri, P. Paoletti and J.E. Mottershead, Feedback linearization in systems with non-

smooth nonlinearities, AIAA Journal of Guidance, Control and Dynamics, in press.

Uncontrolled Nonlinear Response26

Plunge Pitch Flap

(a) Freeplay

(b) Piecewise Linear Stiffness

Reduced air speed U*=2.0

0 2 4 6-0.05

0

0.05

time (s)

plu

nge,

0 2 4 6-5

0

5

time (s)pitch,

(deg)

0 2 4 6-5

0

5

time (s)

flap,

(deg)

0 2 4 6-0.05

0

0.05

time (s)

plu

nge,

0 2 4 6-5

0

5

time (s)

pitch,

(deg)

0 2 4 6-2

0

2

time (s)

flap,

(deg)

Zero-dynamics: Freeplay 27

Superimposed simulated time-domain responses with randomly generated initial conditions

Reduced air speed U*=2.0

5 6 7

-0.02

0

0.02

time (s)

z3

5 6 7-0.05

0

0.05

time (s)z

45 6 7

-0.1

0

0.1

time (s)

z5

5 6 7-4

-2

0

2

4

time (s)

z6

5 6 7-1

0

1x 10

-6

time (s)

z7

5 6 7

-2

0

2

x 10-5

time (s)

z8

The trivial equilibrium point , , is unstable. The non-trivial equilibria

are stable in each of the 3 regions.

0eqz

1

eq gz A b

Zero-dynamics: Piecewise Linear Stiffness28

Superimposed simulated time-domain responses with randomly generated initial conditions

Reduced air speed U*=2.0

12 14 16 18-1

0

1x 10

-5

time (s)

z3

12 14 16 18-5

0

5x 10

-3

time (s)z

4

12 14 16 18-0.01

0

0.01

time (s)

z5

12 14 16 18-2

0

2

time (s)

z6

12 14 16 18

-2

0

2

x 10-7

time (s)

z7

12 14 16 18-5

0

5x 10

-6

time (s)

z8

The trivial equilibrium point, zeq=0, is stable in each of the 3 regions. The non-trivial equilibria are found not to be feasible.

Time-domain Closed-loop Response with Feedback Linearisation

29

(a) Freeplay

(b) Piecewise Linear Stiffness

Assigned frequency and damping 1Hz, 0.1n

Reduced air speed U*=2.0

0 2 4 6-0.01

0

0.01

time (s)

plu

nge,

0 2 4 6-5

0

5

time (s)

pitch,

(deg)

0 2 4 6

-10

0

10

time (s)

flap,

(deg)

0 2 4 6-0.01

0

0.01

time (s)

plu

nge,

0 2 4 6-5

0

5

time (s)

pitch,

(deg)

0 2 4 6

-10

0

10

time (s)flap,

(deg)

Commanded and Actual Flap Angles30

Reduced air speed U*=2.0

(a) Freeplay

(b) Piecewise Linear Stiffness

0 0.5 1 1.5 2

-10

0

10

time (s)

and input

(deg)

commanded flap input

flap state

0 0.5 1 1.5 2

-10

0

10

time (s)

and input

(deg)

commanded flap input

flap state

31High Bandwidth Morphing Actuator (HBMA) & Modular Flexible Aeroelastic Wing (ModFlex)

Conclusions

• Linear flutter suppression demonstrated by active control using the method of receptances – based on data from a vibration test with no requirement to evaluate or to know the system matrices.

• Nonlinear aeroelastic system - plunge nonlinearity representative of future aircraft with very flexible wings.

• Feedback linearisation - demonstration of LCO suppression.

• Non-smooth nonlinearities – piecewise stiffness and freeplay.

• Feedback linearisation for non-smooth nonlinearity.

• Multiple equilibria – zero and nonzero.

• Numerical demonstration.

• MODFLEX wing in preparation – multiple control surfaces, conventional motor-driven flaps, morphing using piezo-benders .

32

Selected Papers

• Y.M. Ram and J.E. Mottershead, The receptance method in active vibration control, American Institute of Aeronautics and Astronautics Journal, 45(3), 2007, 562-567.

• M. Ghandchi Tehrani, R.N.R. Elliott and J.E. Mottershead, Partial pole placement in structures by the method of receptances: theory and experiments, Journal of Sound and Vibration, 329(24), 2010, 5017-5035.

• M. Ghandchi Tehrani, J.E. Mottershead, A.T. Shenton and Y.M. Ram, Robust pole placement in structures by the method of receptances, Mechanical Systems and Signal Processing, 25(1), 2011, 112–122.

• Y.M. Ram and J.E. Mottershead, Multiple-input active vibration control by partial pole placement using the method of receptances, Mechanical Systems and Signal Processing, 40, 2013, 727-735.

• S. Jiffri, P. Paoletti, J.E. Cooper and J.E. Mottershead, Feedback linearisation for nonlinear vibration problems, Shock and Vibration, vol. 2014, Article ID 106531, 16 pages, 2014. doi:10.1155/2014/106531.

• X. Wei and J.E. Mottershead, Block-decoupling vibration control using eigenstructure assignment, Mechanical Systems and Signal Processing, http://dx.doi.org/10.1016/j.ymssp.2015.03.028i .

• X. Wei, J.E. Mottershead and Y.M. Ram, Partial pole placement by feedback control with inaccessible degrees of freedom, Mechanical Systems and Signal Processing, in press.

• S. Jiffri, P. Paoletti and J.E. Mottershead, Feedback linearization in systems with non-smooth nonlinearities, AIAA Journal of Guidance, Control and Dynamics, in press.

• S. Jiffri, S. Fichera, A. Da-Ronch and J.E. Mottershead, Nonlinear control for the suppression of flutter in a nonlinear aeroelastic system, AIAA Journal of Guidance, Control and Dynamics, in preparation.

33