pole placement and active vibration control in...
TRANSCRIPT
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
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