Blending of Inputs and Outputs for Modal Velocity Feedback
Manuel Pusch
1
and Daniel Ossmann
1
Abstract— Dynamical systems like mechanical structures can
be effectively damped by applying forces which oppose the
velocity measured at the very same location. To apply this
principle also to systems with multiple actuators and sensors
of different type and at different locations, a novel control
approach is presented in this paper. The control approach
aims to damp individual modes by a minimum-gain feedback
of blended measurement outputs to blended control inputs.
To that end, a numerically efficient algorithm is proposed for
computing input and output blending vectors which yield the
desired isolation of the target mode(s). The effectiveness of the
proposed approach is demonstrated by increasing the modal
damping of an aeroelastic system.
I. INTRODUCTION
Active control technologies are commonly applied in order
to increase the damping of dynamical systems like mechani-
cal structures. To that end, the underlying system needs to be
equipped with actuators and sensors, where a larger number
in general allows for a better controller performance but
complicates controller design. A well established approach
to tackle this problem is the principle of identical location
of sensors and actuators, referred to as ”collocated control”
[1] or ”direct output feedback” [2]. Therein, each pair of
actuators and sensors is commonly augmented with a single-
input single-output (SISO) controller, which is subject to be
tuned. Considering that, collocated control provides a great
robustness with respect to stability, it is recommended when-
ever possible [1]. A successful application is presented for
example in [3], where cockpit vibrations of the Rockwell B-1
Lancer are suppressed by feeding back acceleration measure-
ments to nearby control vanes. Using velocity measurements,
it can even be shown that a simple static gain feedback
allows increasing the damping of a dynamical system, which
is also known as direct velocity feedback (DVF) [4]. The
restriction to collocated actuators and sensors, however, lim-
its the applicability of these approaches for active damping.
Furthermore, the collocated control loops generally affect not
only the Eigenmodes of interest but rather all controllable
and observable Eigenmodes of the considered system. As a
remedy, [5] proposes to isolate and damp critical Eigenmodes
by blending inputs and outputs yielding a static gain feedback
controller. In the presented algorithm, denoted as ”modal
isolation and damping for adaptive aeroservoelastic suppres-
sion” (MIDAAS), the input and output blending vectors
are computed iteratively since they are interdependent of
each other. Avoiding the issue of selecting appropriate initial
values, the algorithm proposed in [6] directly yields input
1
Manuel Pusch and Daniel Ossmann are with the Department of System
Dynamics and Control at the German Aerospace Center (DLR), Germany
{manuel.pusch,daniel.ossmann}@dlr.de
and output blending vectors maximizing the controllability
and observability of the target mode(s) in terms of the H
2
norm. The H
2
-optimal blending approach has successfully
been applied to different aeroelastic systems with the goal to
damp Eigenmodes causing large structural loads, see [7], [8],
[9], [10] for more details. The basic idea is to split a multiple-
input multiple-output (MIMO) control design problem into a
blending vector design problem and a SISO controller design
problem, which offers rich insight into the actual controller
structure and allows for a dedicated controller tuning.
In this paper, a novel control approach is presented which
combines the H
2
-optimal blending vector design from [6]
and the DVF from [4]. The proposed control approach,
denoted as blending-based modal velocity feedback (MVF),
is described in Section II and aims to damp an individual
mode by a constant feedback of blended measurements to
blended control inputs. To that end, input and output blending
vectors are derived in Section III, which yield a SISO system
with a modal velocity output and a modal force input when
applied. Eventually, a numerical example is presented in
Section IV, where the proposed modal control approach is
successfully applied to a simplified flexible aircraft model.
II. MODAL CONTROL
A. Modal Decomposition
A linear time-invariant (LTI) system with n
u
inputs, n
y
outputs and n
x
states which is physically realizable can be
described as
G :
˙x
y
=
A B
C D
x
u
,
where A ∈ R
n
x
×n
x
, B ∈ R
n
x
×n
u
, C ∈ R
n
y
×n
x
, D ∈
R
n
y
×n
u
. Assuming that A is diagonalizable, the real Jordan
normal form [11] of G can be computed as
˜
G :
˙
˜x
y
=
˜
A
1
0
˜
B
1
.
.
.
.
.
.
0
˜
A
n
i
˜
B
n
i
˜
C
1
. . .
˜
C
n
i
D
˙
˜x
u
by applying the similarity transformation
x = T˜x =
T
1
. . . T
n
i
˜x.
For a real eigenvalue p
i
with a real eigenvector v
i
, the
submatrix T
i
= v
i
and
˜
A
i
= p
i
,
˜
B
i
=
˜
b
i
,
˜
C
i
= ˜c
i
with
i = 1, .., n
i
. For a conjugate complex pole pair p
i
= <(p
i
)±
=(p
i
) associated with the conjugate complex eigenvector pair
v
i
= <(v
i
) ± =(v
i
), the submatrix T
i
= [<(v
i
) =(v
i
)] and
˜
A
i
=
<(p
i
) =(p
i
)
−=(p
i
) <(p
i
)
,
˜
B
i
=
"
<(
˜
b
i
)
T
−=(
˜
b
i
)
T
#
,
˜
C
i
=
<(˜c
i
) =(˜c
i
)
.
(1)
The vectors
˜
b
i
and ˜c
i
are real for real poles and are called
pole input and output vectors, respectively. Based on that,
the output
y =
n
i
X
i=1
y
i
+ Du
can be described as a superposition of the direct feedthrough
Du and the responses of the individual modes
˜
M
i
:
˙
˜x
i
y
i
=
"
˜
A
i
˜
B
i
˜
C
i
0
#
˜x
i
u
.
Hence, a mode
˜
M
i
is a strictly proper LTI system of first
(real pole) or second (conjugate complex pole pair) order
and has n
u
inputs and n
y
outputs.
B. Oscillating Modes
In case of a conjugate complex pole pair p
i
= <(p
i
) ±
=(p
i
) with =(p
i
) 6= 0, the mode
˜
M
i
describes a harmonic
oscillator with a natural frequency ω
n
= |p
i
| and a relative
damping ζ = −<(p
i
)/ω
n
. This motivates the description of
˜
M
i
in physical coordinates where the modal deflection ξ
and its derivative, the modal velocity
˙
ξ, are the two state
variables. Given the mode
˜
M
i
in real Jordan normal form
(1), its physical realization is
M
i
:
˙
ξ
¨
ξ
y
i
=
0 1 b
T
1
−ω
2
n
−2ζω
n
b
T
2
c
1
c
2
0
ξ
˙
ξ
u
, (2)
which is obtained by the similarity transformation ˜x
i
=
˜
T
i
[ξ
˙
ξ]
T
. While M
i
features a unique system matrix
˜
T
−1
i
˜
A
i
˜
T
i
=
0 1
−ω
2
n
−2ζω
n
,
the transformation matrix
˜
T
i
and the vectors b
1
∈ R
n
u
, b
2
∈
R
n
u
, c
1
∈ R
n
y
and c
2
∈ R
n
y
are not unique. To obtain all
possible physical realizations,
˜
T
i
can be parametrized as
˜
T
i
(α, φ) = α
−<(p
i
) sin φ + =(p
i
) cos φ sin φ
−=(p
i
) sin φ − <(p
i
) cos φ cos φ
, (3)
where α ∈ R \ {0} and φ ∈ R, see [11] for more details.
Applying the similarity transformation (3) on Equation (1)
hence yields
b
T
1
b
T
2
=
b
T
1
(α, φ)
b
T
2
(α, φ)
=
˜
T
−1
i
(α, φ)
"
<(
˜
b
i
)
T
−=(
˜
b
i
)
T
#
(4)
and
c
1
c
2
=
c
1
(α, φ) c
2
(α, φ)
=
<(˜c
i
) =(˜c
i
)
˜
T
i
(α, φ).
(5)
Note that a special physical realization for single input
systems (n
u
= 1) is the controllable canonical form, which
is derived by setting α = −|
˜
b
i
| and φ = arg (−j
˜
b
i
) yielding
[b
1
b
2
] = [0 1].
C. Direct Velocity Feedback
In order to increase the damping of dynamical systems like
mechanical structures, numerous control approaches have
been proposed, see for example [1] for a comprehensive
overview. A very intuitive concept is to generate a force input
proportional to the velocity measured at the very same loca-
tion (collocation), which is known as DVF [4]. Considering
an individual (oscillating) mode M
i
, this requires a special
form of its physical realization (2) featuring c
2
= βb
2
6=
0, β ∈ R
≥0
and c
1
= b
1
= 0, which yields the differential
equation
¨
ξ + 2ζω
n
˙
ξ + ω
2
n
ξ = b
T
2
u, y
i
= c
2
˙
ξ = βb
2
˙
ξ. (6)
Closing the loop with the feedback matrix Λ ∈ R
n
u
×n
y
by
setting
u = −Λy
i
= −Λc
2
˙
ξ
i
= −Λβb
2
˙
ξ
i
changes Equation (6) to
¨
ξ +
2ζω
n
+ βb
T
2
Λb
2
˙
ξ + ω
2
n
ξ = 0. (7)
This means that the relative damping of M
i
is changed by
∆ζ = βb
T
2
Λb
2
/(2ω
n
) while its natural frequency ω
n
remains
unchanged. To ensure that damping is rather increased but
not decreased, Λ needs to be positive definite [1]. Note,
however, that this only holds in general if actuators and
sensors are collocated (i.e., c
2
=βb
2
), which certainly limits
the application of the DVF approach. Furthermore, the DVF
approach may also be applied to higher order systems [4],
where all controllable and observable modes are commonly
affected by the derived feedback controller. Hence, DVF
generally does not allow damping individual modes without
affecting others.
D. Blending-based Modal Velocity Feedback
Based on DVF, a novel control approach is presented
herein for damping individual modes using multiple actuators
and sensors of different type and at different locations. The
control approach, denoted as blending-based MVF, aims to
control individual modes by means of a constant feedback of
blended measurement outputs to blended control inputs. The
resulting tunable feedback loops are dedicated to individual
modes, whereas in the DVF approach, they are dedicated to
individual pairs of collocated sensors and actuators.
To begin with, the idea is to blend the measurement
outputs such that the resulting virtual measurement signal
v
y
= k
T
y
y represents the modal velocity
˙
ξ of the mode to be
controlled. Similarly, a virtual control input v
u
is generated
which is distributed to the actual control inputs u = k
u
v
u
in a way that enables an explicit excitation of the target
mode. In other words, by means of the input and output
blending vectors k
u
∈ R
n
u
and k
y
∈ R
n
y
, the target mode
is isolated yielding Equation (6) with a single input and
a single output, whereby it can be directly damped by a
single static feedback gain λ ∈ R. Certainly, this requires
a sufficient controllability and observability of the target
mode (given by the actual actuator and sensor configuration),
which is assumed to be the case here. The resulting feedback
Fig. 1. Closed-loop interconnection of plant G with controller K.
interconnection is depicted in Figure 1, where the static gain
feedback controller
K = λk
u
k
T
y
is encircled with a dashed line. In order to balance be-
tween common control design criteria like damping increase,
robustness margins or saturation limitations, the controller
K can easily be tuned by adjusting the feedback gain λ
accordingly. In case it is desired to damp multiple modes, it is
proposed to repeat the described control design procedure for
each mode to be controlled and superimpose the respective
control commands. Note that the modal velocity may also
be reconstructed using observer-based methods, see [12] for
more details. This, however, typically results in controllers
of an increased order since the included observers commonly
have the same order as the underlying plant models [12].
III. BLENDING VECTOR DESIGN
A. Problem Formulation
Applying the input and output blending vectors k
u
and k
y
to an individual mode M
i
defined in Equation (2) yields
k
T
y
M
i
k
u
:
˙
ξ
¨
ξ
v
y
=
0 1 b
T
1
k
u
−ω
2
n
−2ζω
n
b
T
2
k
u
k
T
y
c
1
k
T
y
c
2
0
ξ
˙
ξ
v
u
.
In order to allow for the blending-based MVF described in
Section II-D, the blending vectors k
u
and k
y
need to be
designed such that b
T
1
k
u
= k
T
y
c
1
= 0. This means that the
virtual control input v
u
becomes a pure modal force input
while the virtual measurement output v
y
represents the modal
velocity
˙
ξ and hence, DVF can be applied. Besides, it is
desired to find blending vectors which allow increasing the
relative damping by ∆ζ with a minimum feedback gain λ.
Small feedback gains in general imply small control inputs
and high robustness margins, see also [12] for more details.
Similar to the DVF loop in Equation (7), the MVF loop with
blended inputs and outputs is given as
¨
ξ +
2ζω
n
+ λk
T
y
c
2
b
T
2
k
u
˙
ξ + ω
2
n
ξ = 0,
where the change in relative damping ∆ζ =
λk
T
y
c
2
b
T
2
k
u
/(2ω
n
). For a fixed ∆ζ > 0, |λ| becomes
minimal when |k
T
y
c
2
b
T
2
k
u
| is maximal since the natural
frequency ω
n
does not change. Summing up, the blending
vector design problem for MVF can be formulated as
maximize
k
u
∈R
n
u
,k
y
∈R
n
y
k
T
y
c
2
b
T
2
k
u
subject to kk
u
k
2
= 1
kk
y
k
2
= 1
b
T
1
k
u
= 0
c
T
1
k
y
= 0,
(8)
where the length of the blending vectors is restricted to one
to avoid an undesired scaling of the optimization problem.
Furthermore, the two constraints b
T
1
k
u
= k
T
y
c
1
= 0 enforce
that the transfer function of the blended mode is given in the
form
k
T
y
M
i
k
u
= k
T
y
c
2
b
T
2
k
u
s
s + 2ζω
n
s + ω
2
n
, (9)
where s denotes the Laplace variable. From Equation (9),
it can be seen that blending vectors which maximize
|k
T
y
c
2
b
T
2
k
u
| also maximize the H
2
norm
k
T
y
M
i
k
u
H
2
,
considering the constraints from Equation (8). In this respect,
the derived blending vectors can be considered as H
2
-
optimal under the side constraints enabling MVF. In case
MVF is not enforced, corresponding H
2
-optimal blending
vectors can be computed according to [7], where therein, a
static gain feedback in general does not yield a pure damping
increase.
B. Blending Vector Computation
In the optimization problem (8), the constraints b
T
1
k
u
= 0
and c
T
1
k
y
= 0 enforce the optimal input and output blending
vectors to lie in the null space of b
T
1
and c
T
1
, respectively.
This means, the original optimization variables k
u
and k
y
can be substituted by
k
u
= N
u
ˆ
k
u
(10)
k
y
= N
y
ˆ
k
y
, (11)
where N
u
and N
y
denote an orthonormal basis of the null
space of b
T
1
and c
T
1
. Since N
u
and N
y
act as unitary linear
transformations preserving the inner product, k
u
and k
y
have the same length as
ˆ
k
u
and
ˆ
k
y
. Hence, the optimization
problem (8) can be reformulated as
maximize
ˆ
k
u
,
ˆ
k
y
ˆ
k
T
y
Q(φ)
ˆ
k
u
subject to
ˆ
k
u
2
= 1
ˆ
k
y
2
= 1,
(12)
where
Q(φ) = N
T
y
c
2
b
T
2
N
u
. (13)
Recall that according to Equations (4) and (5), the vectors
b
1
, b
2
and c
1
, c
2
are computed by means of a similarity
transformation and depend on the parameters α and φ.
Considering Equation (13), this means that each of the given
variables depends on α and φ, where α actually cancels out,
i.e., Q(α, φ) = Q(φ). From Equation (12), it can be seen that
the actual goal is to find unit vectors
ˆ
k
u
and
ˆ
k
y
which yield
a maximum absolute value when multiplied with Q(φ) from
the right- and left-hand side, respectively. This is equivalent
to finding the right and left singular vectors associated to
the largest singular value of Q(φ). Hence, an equivalent
unconstrained optimization problem can be formulated as
φ
∗
= arg max
φ∈R
kQ(φ)k
2
, (14)
where kQ(φ)k
2
is nothing but the largest singular value of
Q(φ). Note that due to the given periodicity of
˜
Q(φ), the
optimization variable φ may be restricted to an interval of
size π, for instance φ ∈ [0, π[. Solving the optimization
problem (14), the optimal blending vectors k
∗
u
= N
u
ˆ
k
∗
u
and
k
∗
y
= N
y
ˆ
k
∗
u
can be directly derived by means of a singular
value decomposition (SVD) of
Q(φ
∗
) = UΣV
T
=
h
ˆ
k
∗
y
•
i
σ
∗
0
0 •
ˆ
k
∗
u
•
T
, (15)
where the placeholder • denotes a matrix of adequate size.
In Equation (15), the rectangular diagonal matrix Σ ∈
R
(n
y
−1)×(n
u
−1)
lists the singular values of Q(φ
∗
) in de-
scending order on its diagonal. The largest singular value
is σ
∗
= kQ(φ
∗
)k
2
∈ R
≥0
, which is associated to the right
and left singular vector
ˆ
k
∗
u
and
ˆ
k
∗
y
, respectively. Note that
ˆ
k
∗
u
as well as
ˆ
k
∗
y
feature a length of one since both U ∈
R
(n
y
−1)×(n
y
−1)
and V ∈ R
(n
u
−1)×(n
u
−1)
are orthogonal
matrices.
C. Mode Decoupling
So far, input and output blending vectors are derived
which yield a minimum static feedback gain for damping
the targeted mode. For mode decoupling, however, it is
additionally desired that feeding back the blended outputs
to the blended inputs prevents an excitation of the residual
modes as good as possible. This can be achieved by enforcing
the input and output blending vectors to be orthogonal on the
respective residual modes, or more specifically on its pole
input and output vectors
˜
b
i
and ˜c
i
, see Section II for more
details. For a complex-valued pole vector, this means that
orthogonality is enforced on both the real and imaginary part.
Collecting the real and imaginary parts of the respective pole
input and output vectors as column vectors in the matrices
P
u
and P
y
, the original optimization problem (8) can be
augmented as
maximize
k
u
∈R
n
u
,k
y
∈R
n
y
k
T
y
c
2
b
T
2
k
u
subject to kk
u
k
2
= 1
kk
y
k
2
= 1
b
T
1
k
u
= 0
c
T
1
k
y
= 0
P
T
u
k
u
= 0
P
T
y
k
y
= 0,
(16)
where the constraints P
T
u
k
u
= 0 and P
T
y
k
y
= 0 enforce the
desired mode decoupling. In order to solve the optimization
problem (16), the same procedure as in Section III-B is
applied. The only difference is that the matrices N
u
and N
y
are adapted such that they represent an orthonormal basis
of the null space of [b
1
P
u
]
T
and [c
1
P
y
]
T
, respectively. If
one of the null spaces is empty, the augmented optimization
problem (16) is infeasible. This also implies that for a finite
number of inputs and outputs, the number of residual modes
which can be made uncontrollable or unobservable is limited.
Note, however, that for mode decoupling it may be sufficient
to make the respective residual modes either uncontrollable
or unobservable but not both.
Alternatively, mode decoupling may also be achieved
using dynamic filtering, where the target mode needs to be
well separated in frequency from the rest of the system. In
that case, it is proposed to first band-pass filter the measure-
ment signals to emphasize the response of the target mode.
Based on the plant augmented with the band-pass filters, the
blending vectors for MVF are then designed as described
in Section III-B. Note that the band-pass filters introduce
additional tuning parameters and result in controller which
is not static.
IV. NUMERICAL EXAMPLE
To demonstrate the effectiveness of the proposed control
approach, it is applied to a flexible aircraft with lightly
damped modes. For replicability reasons, a low-order ap-
proximation of the high-order aeroelastic model is used in
this paper. Thereby, the numerical values of the model as
well as the resulting controller can be provided herein.
A. System Description
The example given in this paper is based on an aeroelastic
model of a large transport aircraft with distributed flaps and
measurements taken from [7]. The model used represents
only the three most dominating modes in terms of wing
bending, where the corresponding properties are summarized
in Table I. The underlying state space matrices are provided
in the Appendix, with the system featuring four control
inputs and eight measurement outputs which are certainly not
collocated. The four control inputs are symmetric deflections
commands for three pairs of trailing edge flaps on the wing
and one pair of elevators. The measurement outputs are four
vertical acceleration and four rotational rate sensors placed
on the wings of the aircraft.
TABLE I
MODES M
i
IN THE FREQUENCY RANGE OF INTEREST.
i natural frequency ω
n
relative damping ζ
1 1.6 rad/s 0.42
2 10.9 rad/s 0.12
3 18.4 rad/s 0.03
B. Blending-based Modal Velocity Feedback Controller De-
sign
In order to reduce structural loads of the aircraft, the
control objective herein is to increase the damping of mode
2 and 3. To that end, the inputs and outputs of the underlying
system are blended to isolate both modes so that a minimum
gain MVF is enabled. For comparison reasons, the blending
vector design is carried out with and without explicit mode
decoupling constraints.
In a first step, a pair of blending vectors is designed
for each of the two modes without considering any mode
decoupling constraints. The underlying optimization problem
is described in Equation (8) and solved according to Equa-
tion (12). Normalizing the corresponding objective function
kQ(φ)k
2
by the constant factor 2
√
ω
n
ζ kM
i
k
H
2
yields the
blending efficiency
η(φ) =
k
y
(φ)
T
M
i
k
u
(φ)
H
2
kM
i
k
H
2
=
kQ(φ)k
2
2
√
ω
n
ζ kM
i
k
H
2
, (17)
where the vectors k
u
(φ) and k
y
(φ) are the right and left
singular vectors associated to the largest singular value of
Q(φ). The blending efficiency η ∈ [0 1] is originally
introduced in [7] as a modal controllability and observability
measure, where η = 0 indicates that the mode can not
be controlled at all. Herein, however, it is more related to
the feedback gain λ required to increase modal damping as
described in Section III-A. In Figure 2, η is plotted over φ for
mode 2 ( ) and mode 3 ( ), where it can be seen that
the respective maxima can easily be found using some global
optimization algorithm. Blending the inputs and outputs with
the obtained blending vectors results in a system with two
virtual control inputs and two virtual control outputs, which
is plotted in Figure 3. It has to be acknowledged that all three
modes can be controlled and observed by the blended inputs
and outputs instead of being dedicated to the respective target
modes as desired.
0 π/4 π/2 3π/4
π
0
0.2
0.4
optimization variable φ [rad]
efficiency η [-]
Fig. 2. Comparison of efficiency factors for mode 2 and mode 3 with
( , ) and without ( , ) mode decoupling constraints,
respectively.
Hence, in a second step, blending vectors are designed
taking into account also mode decoupling constraints. In
order to leave mode 1 unaffected, it needs to be either
uncontrollable by the blended inputs or unobservable by
the blended outputs. For the latter, the maximum achievable
blending efficiency η is considerably larger because the
number of independent measurement outputs is much larger
than the number of control inputs. Thus, mode 1 is only
made unobservable by enforcing the corresponding output
blending vectors k
y,2
and k
y,3
to be orthogonal on its pole
output vector ˜c
1
. In other words, an explicit decoupling from
mode 1 is achieved with the constraints ˜c
T
1
k
y,2
= 0 and
˜c
T
1
k
y,3
= 0. Additionally, an independent control of mode 2
0
1
2
to: v
y,1
from: v
u,1
0 10 20 30
0
1
2
frequency [rad]
to: v
y,2
from: v
u,2
10 20 30
frequency [rad]
magnitude [abs]magnitude [abs]
Fig. 3. Comparison of frequency response from blended inputs to blended
outputs with ( ) and without ( ) mode decoupling constraints
together with the closed-loop response ( ).
and mode 3 is desired. This can be achieved by enforcing
the input and output blending vectors of one mode to be
orthogonal on the pole input and output vectors of the other
mode. The corresponding constraints for input blending are
˜
b
T
3
k
u,2
= 0 and
˜
b
T
2
k
u,3
= 0, where
˜
b
2
and
˜
b
3
are the pole
input vectors of mode 2 and mode 3, respectively. Similarly,
the output blending vectors are constrained by ˜c
T
3
k
y,2
= 0
and ˜c
T
2
k
y,3
= 0 with ˜c
2
and ˜c
3
denoting the respective
pole output vectors. Summing up, the matrices of the mode
decoupling constraints in Equation (16) are given for the
output side as
P
y,2
=
<(˜c
1
) =(˜c
1
) <(˜c
3
) =(˜c
3
)
,
P
y,3
=
<(˜c
1
) =(˜c
1
) <(˜c
2
) =(˜c
2
)
,
and for the input side as
P
u,2
=
<(
˜
b
3
) =(
˜
b
3
)
,
P
u,3
=
<(
˜
b
2
) =(
˜
b
2
)
.
Solving the augmented optimization problem (16), the op-
timal blending vectors are computed according to Equa-
tion (15) as
k
y,2
=
0.477
0.031
−0.224
−0.407
−0.238
−0.221
0.671
0.018
, k
y,3
=
−0.317
0.263
0.054
0.018
0.164
0.244
−0.777
0.37
,
and
k
u,2
=
0.056
0.489
0.85
−0.189
, k
u,3
=
−0.558
−0.036
0.669
0.489
.
The successful decoupling of modes can be seen in Figure 3
( ), where the modes to be controlled are clearly empha-
sized in the respective channels while no other modes are