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ACTIVE VIBRATION CONTROL OF A TWO-FLOORS BUILDING MODELBASED ON H2 AND H METHODOLOGIES USING LINEAR MATRIX

INEQUALITIES (LMIs).

Rodrigo Borges Santos, Douglas Domingues Bueno, Clayton Rodrigo Marquiand Vicente Lopes Jr.

Mechanical Engineering Department – UNESP – Ilha SolteiraAv. Brasil Centro, 56, Ilha Solteira, São Paulo, 15385-000, Brazil

e-mail: [email protected]

ABSTRACT

Experimental verification of structural vibrations control strategies is essential for eventual full-scale

implementations. However, few researchers have facilities readily available to them that are capable of even

small-scale structural control experiments. Appropriately constructed bench-scale models can be used to study

important aspects of full-scale structural vibrations control implementations, including: control-structure

interaction, actuator and sensor dynamics, states feedback design, control spillover, etc. In this sense, the

purpose of this article is the project of controllers based on H 2 and H methodologies using linear matrix

inequalities (LMIs) for vibration attenuation in a flexible structure. The considered structure is manufactured by

Quanser Consulting Inc., and represents a building controlled by an active mass driver (AMD). The structure

consists of a steel frame with a controllable mass located at the top, which can be configured to have either 1 or 2

floors. In this paper, a 2-floor configuration is employed. The actual experiment considered herein is an effective

way to deal with challenges associated with control design.

NOMENCLATURE

M f1 First Floor Massc

x (t)& Cart Linear Velocity Relative to the Second Floor

M f2 Second Floor Massf1

x (t)& First Floor Linear Velocity Relative to the Ground

M c Total Mass of the Cart Systemf2

x (t)& Second Floor Linear Velocity Relative to the FirstFloor

Kf1 First Floor Linear Stiffness Constantf1

x (t)&& First Floor Linear Acceleration Relative to theGround

Kf2

Second Floor Linear Stiffness Constantf2x (t)&&

Second Floor Linear Acceleration Relative to theFirst Floor

Rm Cart Motor Armature Resistancef1

X (t)&& First Floor Absolute Acceleration

Kt Cart Motor Torque Constantf2

X (t)&& Second Floor Absolute Acceleration

Km Cart Back ElectroMotive Force Constantg

X (t)&& Ground Absolute Acceleration

Kg Cart Planetary Gearbox Gear Ratio A Dynamic Matrix

mailto:[email protected]

mailto:[email protected]



rmp Cart Motor Pinion Radius B1 Matrix of Disturbance Input

Vm(t) Cart Motor Armature Voltage B2 Matrix of Control Input

Fc(t) Cart Driving Force Produced by theMotor (i.e Control Force)

C Output Matrix

w(t) Vector of Disturbance Input D Feed-through Matrix

cx (t) Cart Linear Position Relative to the

Second Floor

x(t) State Vector

f1x (t) First Floor Linear Deflection Relative to

the Groundy(t) Output Vector

f2x (t) Second Floor Linear Deflection Relative

to the First Flooru(t) Vector of Control Input

INTRODUCTION

Experimental investigations are essential to obtain a fundamental understanding of many phenomena.

However, when physical systems are scaled down to a size appropriate for laboratory study, salient features of

their behavior may be lost. This is particularly true for large civil engineering structures (Battaini, et al., 1998). In

the area of control of civil structures, it is well-recognized that experimental verification of control strategies isnecessary to focus research efforts in the most promising directions (Housner, et al. 1994a,b). However, few

researchers have experimental facilities readily available to them that are capable of even small-scale structural

control experiments. Consequently, the majority of control studies to date have been analytical in nature, a

substantial number may have employed models that lacked important features of the physical problem. One such

phenomena that has been neglected for many years is control-structure interaction (CSI). Through a series of

analytical and experimental studies, Dyke, et al. (1995) recognized that understanding CSI was key to developing

acceleration feedback control strategies and showed that accounting for CSI is fundamental to achieving high

performance controllers.

A variety of techniques have been intensively studied to reduce the structural vibration in order to achieve

high-speed and high-precision motions (Singer and Seering, 1989). Recent development in microelectronics and

smart structure techniques have provided new solutions for vibration control (Clark et al.,1998). “Smart structures”

adopt microprocessors and distributed transducers to dynamically modify structural dynamics to actively suppress

vibration in time-varying working environments. Use of smart structure techniques can lead to lightweight

structures, leading to higher performance.

There are many robust control techniques in literature that outline these problems. In this research work we

have chosen a recent technique involving linear matrix inequalities (LMI) due its advantages when compared to

conventional techniques (Grigoriadis and Skelton, 1996). LMI contributed to overcome many difficulties in control

design. In the last decade, LMI has been used to solve many problems that until then was unfeasible through

others methodologies, due mainly to the emerging of powerful algorithms to solve convex optimization problem,

as for instance, the interior point method (Boyd et al., 1994 and Gahinet et al., 1995).

The purpose of this paper is design and implement a control state feedback strategy based on H2 and H

controllers using LMI technique for control vibration in a bench-scale structure. Moreover, design full-order state

observer solved for LMI. The model structure employed in the experiment represents a building controlled by an



active mass driver (AMD) and consists of a steel frame with a controllable mass located at the top, as illustrated in

Figure 1.

EXPERIMENT SETUP

The equipment used for the active control experiment consisted of (see Figs.1 and 2):

Structure: The structural specimen, manufacutured by Quanser Consulting Inc., is a model of a flexiblebuilding, which can be configured to have either 1 or 2 floors. Herein, a 2-floor configuration of the model is

employed. The interstorey height is 490 mm, with each column being steel with a section of 1.75 x 108 mm. The

total mass of the structure is 3.5 kg, where the first floor mass is 1.16 kg, the second floor mass is 1.38 kg, and

the mass of each column is 0.24 kg. The structure has the natural frequencies of 1.7 Hz and 5.1 Hz which have

corresponding damping ratios of 0.013 and 0.062 respectively.

Figure 1: Bench-scale building model Figure 2: Experimental set-up

Active Mass Driver (AMD): The AMD provides the control force to the structure. As shown in Fig. 3, it

consists of a moving cart with a DC motor that drives the cart along a geared rack. The maximum stroke is ± 95

mm and the total moving mass is 0.65 Kg.



Figure 3: Active mass driver (AMD).

Sensors: Each floor of the building structure is equipped witch a capacitive DC accelerometer with full scale

range of ± 5 g and sensitivity of 9.81 m/s2 /V (i.e. 1g/V). It consists of a single-chip accelerometer with signal

conditioning. The cart position is directly measured using an optical encoder whose shaft meshes with the track

via an additional pinion. It offers a high resolution of 4096 counts per revolution and sensibility of 2.275E-5

m/count.Digital Controller: Digital control is achieved by use of the MultiQ - PCI board with the WinCon real time

controller. The controller is developed using SIMULINK (1997) and executed in real time using WinCon. This

board has a 14-bit analog/digital (A/D) and 13-bit digital/analog (D/A) converters with four and sixteen outputs and

inputs analog channels, respectively. The SIMULINK code is automatically converted to C code and interfaced

through the Wincon software to run the control algorithm on the CPU of the PC.

Computer: The computer used is a Pentium IV 2.8 GHz configured with 512 MB of RAM and 40 MB of Hard

Disc.

STATE-SPACE MODEL

For small floor deflection angles, both floors are modelled as standard linear spring-mass systems, as

represented in Figure 2. Both floors’ linear stiffness constants, K f1 and Kf2, for small angular structure oscillations,

are of 500 N/m. In the presented modelling approach, the structure viscous damping coefficients, Bf1 and Bf2, are

neglected. The Lagrange's method is used to obtain the dynamic model of the system. In this approach, the

inputs to the system are considered to be the vector of disturbance, w(t), and the driving force of the linear

motorized cart ,Fc(t).

In order to design and implement a state-feedback controller for our system, a state-space representation of

that system needs to be derived. Moreover, it is reminded that state-space matrices, by definition, represent a set

of linear differential equations that describe the system's dynamics. The following relationships represent the state

space model:



1 2

dx(t) = Ax(t)+B w(t)+B u(t)

dtand y(t)=Cx(t)+Du(t) (1)

where x(t) is the system's state vector. In practice, x(t) is often chosen to include the generalized coordinates as

well as their first-order time derivatives. In our case, x(t) is defined such that its transpose is as follows:

[ ]T

c f1 f2 c f1 f2x (t)= x (t) x (t) x (t) x (t) x (t) x (t)& & & (2)

Furthermore, it is reminded that the system's measured output vector is:

[T

c f1 f2y (t)= x (t) x (t) x (t)&& && (3)

Also in Equation (1), the input u(t) is set in a first time to be Fc(t). Thus, we have:

cu(t)=F (t) (4)

where Fc(t) can be expressed by:

2

g t m c

g t m

c 2m mpm mp

dK K K x (t)

K K V (t)dtF (t)=- +R rR r

(5)

According to the system’s state space representation defined by Equations (1), (2) and (3) and according to

the equation of motion expressed in Reference [2] and taking into account Equation (5) in order to convert force

(Fc(t)) to voltage input (Vm(t)) and using the model parameter values provided in Reference [1] the state space

matrices A, B1, B2, C and D result to be as follows:



A=

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

0 0 278.43 -18.69 0 0

0 -431.03 431.03 0 0 00 431.03 -766.49 5.98 0 0

1B =

0

0

0

0

10

2B =

0

0

0

3.00

0-0.96

(6)

C=

1 0 0 0 0 0

0 -431.03 431.03 0 0 0

0 431.03 -766.49 5.98 0 0

D=

0

0

-0.96

(7)

H∞ CONTROLLER BY LMI APPROACH

Considering the linear time-invariant system in state-space form the object of the regulator design is to find a

linear control law of the form

u(t) = -Gx(t) (8)

Objective of control design is to obtain G that minimizes the H norm between the input and output. The H∞

norm of a SISO system is the peak of the transfer function magnitude, in terms of its singular values. Controller

design only is possible if the pair (A, B2) will be controllable. The H∞ optimal control problem can be formulated

from a convex parameterization of Riccati’s equation or through the definition of a new parametric space (Silva etal, 2006).

The H convex problem can be solved using the following formulation

T T T T T T

2 2 1

T

1

min

> 0

W > 0

-AW + WA - B Z - Z B B WC + Z D

B I 0

0CW + DZ 0 I

,W,Z

(9)

Solving Eq. (9), is possible to obtain∞

=δ Hmin , so, H∞ optimal gain of feedback is given by



-1G = ZW (10)

where Z and W are solution of LMI. Using the Eq. (9) is possible to define the sub-optimal problem substituting the

LMI d>0 for d>d’ or d-d’>0.

H2 CONTROLLER BY LMI APPROACH

The optimization problem of the H2 norm can be formulated from the definition of a new parametric space,

that it supplies as solution the optimal H2 gain such that minimizes the H2 norm of the system.

(

T T

T T

2 2 1

T

1

minTr X

W > 0

W WC + Z D 0

CW + DZ X

-AW - WA - B Z - Z B B 0

B I

X,Z,W

(11)

The solution of the equation above supplies ( )2

2HminTr =X . The optimal H2 gain of feedback is given by

Eq. (10).

EXPERIMENTAL APPLICATION

Using LMI Toolbox of the Matlab® 6.5, H∞ and H2 controllers gains were obtained solving LMIs (9) and (11),

respectively, and Eq. (10). Sub-optimal H∞ and H2 controllers have been solved using δ>δ’, with δ’=100.

Figures 4 and 5 show the displacements of first and second floors of the structure shown in figure 1,

respectively, for the case of H∞ control. The disturbance force was applied in the first floor with a sinusoidal signal

with amplitude of 0.3 V and frequency of 1.6 Hz (approximately the first natural frequency of vibration). Figure 6

shows the control force applied by the actuator.



Figure 4: Displacement of first floor without and with H∞ control. Excitation in the first mode.

Figure 5: Displacement of second floor without and with H∞ control. Excitation in the first mode.



Figure 6: Control force applied in the structure [V] (H∞ control).

Figures 7 and 8 show the displacements of first and second floors, respectively, due to a sinusoidal

disturbance applied in the first floor with amplitude of 1.2 V and frequency of 4.8 Hz (approximately the second

natural frequency of vibration). Figure 9 shows the control force applied by the actuator.

Figure 7: Displacement of first floor without and with H∞ control. Excitation in the second mode.



Figure 8: Displacement of second floor without and with H∞ control. Excitation in the second mode.

Figure 9: Control force applied in the structure [V] (H∞ control).

The design of the H2 controller was not implemented experimentally because the displacement of the car

exceeded the limit of +/- 95 mm allowed, as it can be seen in figure 11. Besides, the force that is necessary by

actuator is greater than of the H∞ control, Fig. 10.



Figure 10: Force of control - first mode (H∞ e H2 control)

Figure 11: Displacement of the active mass driver (cart) - first mode (H∞ e H2 control)

CONCLUSION

The study of algorithms for active vibrations control in flexible structures became an area of enormous

interest, mainly due to the countless demands of an optimal performance of mechanical systems as aircraft,

aerospace and automotive structures. Smart structures, formed by a structure base, coupled with actuators and

sensor are capable to guarantee the conditions demanded through the application of several types of controllers.

This paper is addressed to explore the abilities of LMI approach in the design of controllers. The major

advantage of LMI design is to enable specifications such as stability degree requirements, decay rate, input force



limitation in the actuators and output peak bounder. It is also possible to assume that the model parameters

involve uncertainties. The LMI is a very useful tool for problems with constraints, where the parameters vary in a

range of values. Once formulated in terms of LMI a problem can be solved efficiently by convex optimization

algorithms.

This paper showed that H controller, solved through LMI, can reduce structural vibration appropriately. In

this specific structure, it wasn’t possible to implement the H2 controller in real-time because the displacement of

the car (actuator) was bigger than the maximum stroke of the structure.

ACKNOWLEDGEMENTS

The authors acknowledge the support of the Research Foundation of the State of São Paulo (FAPESP-Brazil).

REFERENCES

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[11] Silva, S., Lopes Jr, V. and Brennan M.J. Design of a Control System using Linear Matrix Inequalities for the

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