This document describes research into controlling the motion of a tall tower with a tuned mass damper system after it has been disturbed. The researcher will create an open-loop and closed-loop controller for a linearized model of the system. An observer will be paired with the closed-loop controller and applied to the original nonlinear system. The nonlinear dynamics of the system are modeled using differential equations. Numerical analysis and simulation show that the controller and observer are effective at damping vibrations in the linearized and nonlinear systems, even with imperfect parameter estimation.

1. Control of an Active Tuned Mass-Damper System
Rian Rustvold
Abstract
We will investigate the nonlinear dynamics
of a tall tower with a mass damper system such
as Taipei 101. The goal will be to control the
motion of the tower after it has been disturbed
by articulating the spring and damper rates of
the tower’s tuned mass-damper system (TMD).
We will create an open loop and a closed loop
controller for the linearized system. An ob-
server will be paired with the closed loop con-
troller, and these will be applied together to
the original nonlinear system.
1. Introduction
Tall structures such as the Taipei 101
skyscraper in Taiwan must be able to with-
stand external forces such as winds or earth-
quakes. If an external force were to act on a
tower at a frequency near the structure’s crit-
ical frequency, the structure could be made
to resonate. Resonance can lead to failure or
occupant discomfort. That is why tall build-
ings such as Taipei 101 include tuned mass
dampers throughout their structure. The mass
damper in Tapei 101 is a 660 tonne pendu-
lum that is suspended from the 92nd floor [1].
The giant pendulum effectively stabilizes the
tower, so that some of the energy from exter-
nal forces are absorbed by the mass-damper
system, and the building as a whole vibrates
less than it would have without the damper.
This paper studies a more sophisticated
tuned mass damper (TMD) called a semi-
active tuned mass damper (SA-TMD). In an
SA-TMD it is possible to control the proper-
ties of the TMD making it more effective at
stabilizing a system.
2. Model
We will investigate a similar SA-TMD that
Lai et al examine [2]. Fig. 1 shows what
such a system would look like in practice.
We see the large pendulum suspended from
a strucutre. The pendulum is also supported
from below with spring/damper type supports.
The effective length of the pendulum can be
actuated by moving some kind of linear ac-
tuator. A more abstract representation of the
system is given in fig. 2. Here the structure
(tower) is the bottom cart. The bottom cart is
attached to a rigid support with a spring and
a damper. This is similar to how a tower is
fixed to a rigid foundation. The stiffness and
damping coefficient of the spring and damper
are due to the building’s natural stiffness and
damping properties. A second cart represents
the mass of the SA-TMD. It is connected to
the first cart through a spring and damper. The
stiffness of this second spring is controllable
by actuating the length of the pendulum. Ex-
ternal forces such as wind are modeled by ap-
plying a force to the bottom cart.
To make the analysis more interesting, we
will suppose that the spring representing the
structure’s stiffness, ks, is non-linear such that
the spring rate relationship for the force on the
structure by the spring is given as F = ksx +
ksnx3. Where ksn is the nonlinear portion of
the stiffness, and x is the displacement of the
spring.

2. The inputs to the system include the stiff-
ness kd of the TMD system, and the damping
coefficient of the TMD cd. The outputs and
states of the system are the absolute position
and velocities of the structure and TMD which
we will call:
x1 = TMD position
x2 = Structure position
x3 = TMD velocity
x4 = Structure velocity. (1)
We will not model the external forces as an in-
put or a disturbance. Instead, when we go to
simulate the system, we will assume that an
external force has already disturbed the sys-
tem. Then the goal will be to control the two
inputs to bring the system back to its equilib-
rium state.
The mass of the structure and TMD, along
with all other spring stiffness and damper co-
efficients are taken as parameters to the sys-
tem.
An actual tower such as Taipei 101 is
much more complex than the model presented
here. For example, Taipei 101 includes sev-
eral smaller TMDs throughout its structure [1].
The inherent stiffness and damping qualities
of a tower are most likely not so easily rep-
resented as a single spring and damper. Our
model also allows for spring and damper rates
that are less than 0 which is not realistic.
The nonlinear model is then stated with the
following system of differential equations:
˙x1 = x3 (2)
˙x2 = x4 (3)
md ˙x3 = −kd(x1 −x2)−cd(˙x3 − ˙x4) (4)
ms ˙x4 = −ksx2 −ksnx3
2 −csx4 +
kd(x1 −x2)+cd(x3 −x4). (5)
The system is nonlinear due to the x3
2 term in
equation (5). To obtain numerical results, we
Figure 1. Practical application of a semi-active
tuned mass damper (SA-TMD) [2].
will use the following parameter values that
Lai et al use:
ms = 52.7×106
kg
ks = 42.3×106
N/m
ksn = 42.3×106
N/m
cs = 47.2×106
Ns/m
md = 0.7×106
kg
kd0 = 0.5×106
N/m
cd0 = 0.6×106
Ns/m. (6)
Where kd0 and cd0 are the stiffness and damp-
ing coefficients of the TMD when the inputs
are zero. To make the analysis easier, we can
scale these inputs down by considering our
unit of mass to be 106 kg

3. Figure 2. Symbolic represenrtation of the semi-
active tuned mass damper (SA-TMD) [2]. For
our example we take both the TMD’s spring and
damper as controllable.
3. Analysis
3.1. Linearized System
We will begin the analysis by linearizing
the system about an equilibrium point x∗. We
will choose the state x∗ = 0 as our equilibrium
since this state represents the tower in a state
of rest without any deflection or movement of
the building or TMD. Then, after taking the
appropriate derivatives we obtain the follow-
ing linearized system:
A =




0 0 1 0
0 0 0 1
0 0 0 0
0 −ks −3ksnx2
2(x∗) 0 −cs



 (7)
B =




0 0
0 0
(x2 −x1)(x∗) (x4 −x3)(x∗)
−(x2 −x1)(x∗) −(x4 −x3)(x∗)



 (8)
C = I4×4 (9)
D = 0 (10)
x = x1 x2 x3 x4
T
(11)
u = kd cd
T
(12)
˙x = Ax+Bu (13)
y = Cx+Du. (14)
Notice that if x∗ = 0 the control matrix B will
become identically zero. This would pose a
problem later on, so to allow us to control the
system, we will let x∗ = x0 inside the matrix
B, where x0 is the initial condition of x. Inside
A, we will have x∗ = 0 as stated before.
3.2. Stability
We can address the stability of the system
by finding the eigenvalues of A:
λ1,2 = −0.448±0.776 i
λ3,4 = 0. (15)
The real parts of all eigenvalues lie in the
closed left hand side of the complex plane.
Therefore, the system is marginally stable
about the equilibrium point x∗ = 0. Addition-
ally, since the non-zero eigenvalues have non-
zero imaginary part, that indicates the system
is underdamped and will oscillate about the
fixed point if disturbed.
3.3. Controllability and Observability
To address controllability and observabil-
ity we will first discretize A and B. This
will be necessary to carry out the simulation
as well. Then we obtain:
A =




1 0 ∆t 0
0 1 0 ∆t
0 0 1 0
0 ∆t(−ks −3ksnx2
2(x∗)) 0 −∆tcs +1




(16)
B =




0 0
0 0
∆t(x2 −x1)(x∗) ∆t(x4 −x3)(x∗)
−∆t(x2 −x1)(x∗) −∆t(x4 −x3)(x∗)



.
(17)

4. Then the controllability and observability ma-
trices are formed as:
C = B AB A2B A3B (18)
O =




C
CA
CA2
CA3



. (19)
We find that the ranks of C and O are 4,
which indicates the system is fully control-
lable and fully observable. As mentioned pre-
viously, the system is controllable only if we
take x∗ = 0 inside B.
4. Computation
4.1. Open Loop Control
We can create an open loop control using
the singular value decomposition (SVD) of C .
We will take as our initital condition:
x(0) = 0.2 0.5 0.03 0.02
T
(20)
Where these represent the position and veloc-
ities of the structure and TMD in meters and
meters per second. We will aim to return the
positions and velocities of the structure and the
TMD to zero in 10 seconds ( tf = 10 s). And
we will use a time step ∆t = .005 s for a total
of N = 2001 iterations. First note that:
x(N)−AN
x(0) = C ¯u (21)
where ¯u is a tall column vector of the inputs.
Then we can use SVD to form a pseudoinverse
for C as:
C = UΣV∗
(22)
which allows us to solve for the inputs:
¯u = VΣU∗
(x(N)−AN
x(0)). (23)
When we simulate the system using these in-
puts we obtain the response shown in fig. 3.
The position of the structure and TMD go to
zero at 10 seconds. And by inspection we see
the velocities are also zero.
Figure 3. Open loop control response on the
linearized model.
4.2. Closed Loop Control
We can obtain a closed loop controller for
the system by assuming a controller of the
form u = −Kx. When this is done we obtain
the following system dynamics:
˙x = (A−BK)x. (24)
Using Octave it is possible to obtain K by
placing the eigenvalues where we desire. If
we choose the eigenvalues to be:
λ1 = −1.0
λ2 = −1.5
λ3 = −0.4478+0.776i
λ4 = −0.4478−0.776i (25)
we will obtain a system that asymptotically
approaches the equilibrium as shown in fig.
4. This is a reasonable selection of eigenval-
ues, because the resulting response does not

5. Figure 4. Closed loop control response on the
linearized model.
involve large displacements of either the TMD
or structure. The resulting response also does
not require extremely large values for the con-
trol inputs which could be difficult to achieve
in practice.
4.3. State Estimation
We can construct an estimate of the system
dynamics as:
˙ˆx = Aˆx+Bu−L(ˆy−y). (26)
This will allow us to account for error be-
tween the actual system states and the mea-
sured states. For this system the output is the
state. i.e y = x and the estimate of the out-
put ˆy = ˆx. Then for the estimation error to
go to zero we must have the eigenvalues of
(A − LC) have negative real parts. As before,
Octave can be used to place the eigenvalues.
We will choose the eigenvalues of (A − LC)
Figure 5. Closed loop response of system with
true state and estimated state.
to be:
λ1 = −1+i
λ2 = −1−i
λ3 = −0.4478+0.776i
λ4 = −0.4478−0.776i. (27)
Then for an initial state estimation with some
error, we can observe in fig. 5 the difference
between actual state and the state estimation.
The error approaches zero very quickly (in
well under one second).
5. Application to the Nonlinear System
We can apply the feedback gain K and the
observer gain L to the nonlinear model (2) –
(5). When this is done we obtain the response
shown in fig. 6. This time the result is not
as good as when the control was applied to
the linearized model. The structure’s position
(shown in blue) is brought back to the origin

6. quite well, but the damper is still oscillating at
10 seconds. The estimated states are still very
close to the true states, but there is noticeably
more error present, especially in the damper’s
position x1.
It is unlikely that the system’s parameters
(mass, spring rates, damper rates) would be
known exactly. Keeping K and L the same,
we can then change these parameters to the
following to simulate the effect of imperfect
parameter estimation:
ms = (52.7+5)×106
kg
ks = (42.3+3)×106
N/m
ksn = (42.3−2)×106
N/m
cs = (47.2−5)×106
Ns/m
md = (0.7−0.1)×106
kg
kd0 = (0.5+0.2)×106
N/m
cd0 = (0.6+0.01)×106
Ns/m. (28)
We can re-simulate the system and exam-
ine the response shown in fig. 7. This result
looks very similar to the result shown in fig.
6, the magnitudes of the responses are slightly
different, however. This implies that the con-
troller and observer can handle imperfections
in the parameters.
6. Discussion
We have taken a nonlinear dynamic system
and successfully created an observer and con-
troller to damp out vibrations to the structure.
We’ve seen that the observer and controller, al-
though created for a linearized system, work
well on the non-linear system. We have also
seen that the observer and controller tolerate
imperfection in parameter values. We haven’t
created any metrics to measure the perfor-
mance of the controllers, however. It’s easy
enough to examine the response of the sys-
tem when there is zero input (when the TMD’s
spring and damper rates are fixed). Such a re-
sponse is plotted in fig. 8, and can be com-
pared to the controlled response in fig. 6. The
uncontrolled system’s structure returns to the
origin about the same as in the controlled sys-
tem. The biggest difference is the response of
the TMD which, in the uncontrolled version,
continues to oscillate.
From here there are a number of improve-
ments that can be made. First, it would be
important to limit the control inputs to real-
istic ranges. i.e. we should only be allowed
to apply positive values of spring stiffness and
damping, and there should be a practical max-
imum to these parameters too. We’ve also ne-
glected the exciting forces (e.g. wind) in this
analysis. In actuality, we would like to be able
to stabilize the tower for a continuous random
wind force, which is the objective of Lai et
al [2]. Instead we have only considered the
case where the wind has already disturbed the
tower, and we are trying to gently bring the
system back to rest. Another important con-
sideration would be to design the controller so
that the amount of energy dissipated by the
structure is minimized. There could be other
alternative ways to optimize the performance
too, such as minimizing the acceleration or
velocity on the structure to improve occupant
comfort.

7. Figure 6. Closed loop response of the nonlin-
ear system with true state and estimated state.
Figure 7. Closed loop response of the non-
linear system with imperfect parameter esti-
mates.
Figure 8. Zero input response of the nonlinear
system with imperfect parameter estimates.
References
[1] Wikipedia Taipei 101 2016
https://en.wikipedia.org/wiki/Taipei 101 Online; ac-
cessed 9-December-2016
[2] Yong-An Lai, CS Walter Yang, Kuan-Hua Lien, Lap-
Loi Chung, and Lai-Yun Wu Suspension-type tuned mass
dampers with varying pendulum length to dissipate en-
ergy Struct. Control Health Monit. 2016; 23:12181236