# Experimental Active Vibration Control in  Truss Structures Considering Uncertainties in System Parameters

Authors: Douglas Domingues Bueno; Clayton Rodrigo Marqui; Rodrigo Borges Santos; Camilo Mesquita Neto; Vicente Lopes
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

This paper deals with the study of algorithms for robust active vibration control in flexible structures considering uncertainties in system parameters. It became an area of enormous interest, mainly due to the countless demands of optimal performance in mechanical systems as aircraft, aerospace, and automotive structures. An important and difficult problem for designing active vibration control is to get a representative dynamic model. Generally, this model can be obtained using finite element method (FEM) or an identification method using experimental data. Actuators and sensors may affect the dynamics properties of the structure, for instance, electromechanical coupling of piezoelectric material must be considered in FEM formulation for flexible and lightly damping structure. The nonlinearities and uncertainties involved in these structures make it a difficult task, mainly for complex structures as spatial truss structures. On the other hand, by using an identification method, it is possible to obtain the dynamic model represented through a state space realization considering this coupling. This paper proposes an experimental methodology for vibration control in a 3D truss structure using PZT wafer stacks and a robust control algorithm solved by linear matrix inequalities.


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## Body

## 1. Introduction

Lightweight space  structures are
the future of space vehicles and satellite technology. Possessing ideal space
launching characteristics, such as minimal storage volume and minimal mass,
these lightweight structures will propel the space industry into the next generation.
Space satellites must be expertly controlled from a vibration standpoint
because signal transmission to and from the earth mandates tight tolerances.
Vibration control is critical to mission success as well as satellite longevity [1].Large and light space structures
are basically flexible due to their low stiffness and damping. These
characteristics may cause problems since flexible structures present many
vibration modes within or beyond the bandwidth of the controller. When only a
few modes are dealt in the controller, spillover may occur because uncontrolled
higher modes or unmodeled modes may become excited. The effects of spillover can occur where
structure, sensors, or actuators are poorly modeled,  and the numbers of sensors and
actuators are low.In order to achieve better dynamic
properties, great attention has been paid to the control of structural
vibration using intelligent structures. So, the application of active vibration
control in flexible structures has been increasingly used as a solution for space
structures to achieve the degree of vibration suppression required for
precision pointing accuracy and to guarantee the stability.In truss structures, active members
are integrated because of their multiple functions. They serve as structural
elements, and as load device. As actuators, the active member exerts internal
forces, and as sensor elements, it allows measurement of the elastic strains.
The piezoelectric stack actuators are remarkable because they are light weight,
high force, and low power consumption [2].Several researchers have proposed
the use of piezoelectric material for active vibration control. In truss
structures, the control force can be accomplished by piezoelectric active
members, known as “PZT wafer stacks,” that are mechanically linked in series producing
an axial force in the bar that are positioned.State space realization is used in
modern control formulation to obtain the dynamic model, but in many cases this
model has significant uncertainties in relation to the real system. These
uncertainties can be caused, for instance, by parameter variation during the
operation, or by dynamic uncertainties (nonlinearities, higher modes, noises,
etc.). So,  for an efficient experimental control design, it is important to qualify and
quantify the uncertainties.In this context, this paper
proposes a methodology for robust control design considering uncertainties in
the dynamic model, represented by state space realization. It is designed as an active controller
to attenuate vibrations in a truss structures. The active members are composed
by PZT wafer stacks actuators, and the control design is based on linear
quadratic regulator solved through linear matrix inequalities (LMIs). LMI presents
advantages when compared to conventional techniques, and it has contributed to
overcome many difficulties in control design [3].
In the last decade, LMI has been used to solve many problems that until then were
unfeasible through other
methodologies, mainly due to the emergence of powerful algorithms to solve
convex optimization problem, for instance, the interior point method (see Boyd et al. [4] and Gahinet et al.  [5]). Sarracini and Serpa [6] apply Hinfinite
control approach solved through LMI for model reduction. Silva et al. [7]  present a
consistent formulation for control design based on LMI approach. Chen and Zhu [8]
present a formulation based on H2,  H∞, and
mixed H2/H∞ control strategies  for a flexible
rotor system under seismic excitation by means of linear matrix inequality (LMI) to attenuate the transient vibration
of the flexible rotor system under a nonstationary seismic excitation and to
improve robust performance of the flexible rotor system.In the present work, the numerical
method of subspace and the Kalman estimator were used to identify the dynamic
model with experimental data and to estimate the state vector, respectively.
Experimental results, obtained through dSPACE control board and the Simulink/Matlab,
are shown in order to validate the proposed approach.


## 2. Dynamic Modeling of Robust Control

Modern
linear control formulation is based on the state space realization. Using this
formulation, the design of a vibration control for multi-input multioutput
(MIMO) system is similar of simple-input simple-output (SISO) system approach. This
realization is appropriate for experimental applications because there are many
numerical methods to solve it in real time, as for instance, the numerical
method of Runge-Kutta. A linear differential inclusion (LDI) system, in modal state space form,
considering the matrices with appropriate dimensions and assumed to be known,
is given by(2.1)x˙(t)=A(t)x(t)+Bw(t)w(t)+Bu(t)u(t)+Einvin(t),y(t)=C(t)x(t)+Du(t)+Eoutvout(t),[A(t)Bw(t)Bu(t)C(t)]∈Ω,  where Ω is a polytope that is described by  a
list of vertexes in a convex space, A(t)
is the dynamic matrix, Bw(t)
is the matrix of disturbance, Bu(t) is the matrix of control input, C(t) is the output matrix, w(t) is the vector of disturbance
input, u(t) is the vector of control
input, y(t) is the output vector, and vin(t) and vout(t) are stationary zero-mean
Gaussian white process and measurement noises vectors with unit intensity,
respectively. In this paper, some variables, as the matrices in (2.1), are
represented as time function to emphasize the uncertainties in the system parameters.
The vectors Ein and  Eout are the process and
measurement noise vectors, respectively.The
state vectorx(t) of the modal
coordinates system consists of n independent components, xi(t), that represent a
state of each mode, where n   is the number of modes. The xi(t)
(ith state component), related to (2.3),  is defined as [9] (2.2)xi(t)={qmi(t)qmoi(t)},whereqmoi(t)=ζi(t)qmi(t)+q˙mi(t)ωi(t),  where qmi and  qmoi are named modal displacement and velocity for ith  vibration mode, respectively. Using
modal coordinates, these parameters have no physical interpretation. Also, ζi and   ωi are damping factor and natural frequency of the ith mode. These parameters are
represented as time function to emphasize the uncertainties.The
modal state space realization is characterized by the block-diagonal dynamic
matrix and the related input and output matrices [9]: (2.3)A(t)=diag(Ami(t)),Bw(t)=[Bwm1(t)Bwm2(t)⋮Bwmn(t)],Bu(t)=[Bum1(t)Bum2(t)⋮Bumn(t)],C(t)=[Cm1(t)Cm2(t)⋯Cmn(t)], where i=1,2,…,n; Ami, Bwmi, Bumi,
and Cmi are 2×2,  2×k,  2×s,
and r×2 blocks, respectively; k is the number of disturbances; s is the number
of control inputs; and r is the number of outputs. These blocks can be obtained
in several different forms and also it is possible to convert it in another
realization through a linear transformation. One possible form to block Ami(t) is (2.4)Ami(t)=[−ζi(t)ωi(t)ωi(t)−ωi(t)(ζi2(t)−1)−ζi(t)ωi(t)].The dynamic
model of the truss structure was initially identified using experimental data through
subspace identification method.nI>1 identification tests were considered to characterize
the uncertainties in the system parameters. The  members that include PZT stack
actuator are nominated by active members. It is considered a variation in the properties
of these members caused by the insertion of these actuators. These
uncertainties are described by a polytopic linear differential inclusion (PLDI): (2.5)Ω=[Ac(t)Bwc(t)Buc(t)Cc(t)],Co{S1,…,Sv},whereSi=[AciBwciBuciCci],i=1,…,v,  where the subscript c is
relative to controlled modes; Ω is a polytope described by a list
of vertexes in a convex space Co [4], and v is the number of vertexes of the
polytopic system. Usually, in practical situations, it is very difficult to
define the polytopic
vertexes, but these vertexes are not variant in time. So,  it is possible to project
the vibration control using an invariant model.A
reduced-order model is obtained by truncating the states. Letx(t) and the state (A(t), Bw(t), Bu(t), C(t)) be partitioned considering the
canonical modal decomposition. From the Jordan
canonical form, the following
can be obtained: (2.6){x˙c(t)x˙r(t)}=[Ac(t)00Ar(t)]{xc(t)xr(t)}+[Bwc(t)Bwr(t)]w(t)+[Buc(t)Bur(t)]u(t),y(t)=[Cc(t)Cr(t)]{xc(t)xr(t)},  where Ac(t) is given by (2.4) and
the subscript r  is
relative to the residual modes. Generally, in practical applications, Ein and Eout are not identified, but there are, always, some
process and measurement noises.


## 3. Control Methodology

In
this section, a robustness analysis is conduced for understanding the LQR-LMI
controller performance. Controller
design can be done through rigorous mathematical optimization techniques. One
of these, which was originated in the sixties [10], is called modern
optimal control theory that is a time-domain technique.Control
systems robustness is defined as the ability for maintaining satisfactory
stability and performance features in the presence of parameters variations and
uncertainties [11]. Traditional LQR solved by Riccati’s
equation can be obtained in text books [12]. In the following, one
presents the procedure for the LQR-LMI approach.Firstly,
mathematical definitions of some terms are given, and LQR control problem is
defined. Then, the LQR problem is represented as an equivalent eigenvalue
problem (EVP) in terms of LMI using theH2  representation of the LQR
problem.


### 3.1. Basic Definitions of LMIs and EVPs

A linear matrix inequality has the
form(3.1)F(z)=F0+∑i=1mziFi>0, where z  is a real vector and F0 and Fi are real symmetric
matrices. Inequality (3.1) is shorthand for saying that F(z) is positive
definite. A vector z that satisfies
inequality (3.1) is known as a feasible solution of the LMI.Inequality (3.1) is a convex
constraint on z. This property is important because powerful numerical
techniques are available for the solution of problems involving convex LMIs [4]. On the
other hand, no efficient algorithm is available for the solution of nonconvex
problems. Hence, nonconvex inequalities which may arise from a control problem
should be converted to convex LMIs to be solvable numerically. One useful
example for such manipulations is the LMI representation of the following
nonconvex inequalities: (3.2)Q(z)−S(z)R(z)−1ST(z)>0,R(z)>0,  where Q,  R, and S  are affine functions of z,
and Q and R are symmetric matrix. Inequality (3.2) is equivalent to (3.3)[Q(z)S(z)ST(z)R(z)]>0. This transformation can be achieved easily premultiplying
inequality (3.3) by (3.4)[I−SR−10I]>0 and postmultiplying it by its transpose.One of the concepts related with
LMIs and control problems is the eigenvalue problem. An eigenvalue problem may
have several representations, one of which is given by(3.5)mincTzsubjecttoA(z)>0,  where A(z) is a symmetric and affine function
of  z. An LQR problem may be
transformed into this form of the EVP to be represented in terms of LMIs.


### 3.2. LQR Control Problem

There are various representations
of LQR problem in the literature. Here, definitions are given in a way that
aids the derivation of the specific problem defined above; hence they may not
be the most general forms of LQR problem. The LQR problem is to find the
control gainK that satisfies the
optimization (3.6)minKE[yTQ˜y+uTR˜u+yTN˜u+uTN˜Ty],subjectto(2.1)andu(t)=−αKx(t),  where Q˜≥0 and R˜>0 are
symmetric weighting matrices, N˜ is the weighting
matrix between input and output vectors, and α is a scalar amplifier.
Substituting the output equation in (2.1) into the optimization problem (3.6), and
assuming Ein and Eout as zero vectors; one
obtains another form of the LQR problem (3.7)minKE[xTQx+uTRu+xTNu+uTNTx],subjecttox˙(t)=A(t)x(t)+Bw(t)w(t)+Bu(t)u(t),u(t)=−αKx(t),  where (3.8)Q=CTQ˜C,N=CTQ˜D+CTN˜,R=R˜+DTQ˜D+DTN˜+N˜D.


### 3.3. EVP Representation of the LQR Problem

Lyapunov’s stability criteria can
be used to state that a system given by (2.1) with control force u(t)=−αKx(t)=−Gx(t),
where G=αK, is stable if there exists a matrix S=ST>0 that satisfies (3.9)(A−BuG)S+S(A−BuG)T+BwBwT≤0, where S that satisfies inequality (3.9) is the optimal state covariance matrix P. So, combining (3.7) and (3.9) and optimizing S, the H2 optimization
problem may be stated as [13] (3.10)min(K,S)Tr(R1/2GSGTR1/2)+Tr(QS)−Tr(GSN)−Tr(NTSGT),subjecttoAS−BuGS+SAT−SGTBuT+BwBwT≤0,S=ST>0,  where Tr() is the trace of the matrix. This is
not an EVP problem since it is not convex because of the terms involving GS. To obtain a convex version of the
problem, two new variables are introduced X=R1/2GSGTR1/2 and Y=GS. Substituting these variables
into (3.10) and using the transformation given by (3.2) and (3.3), the EVP
representation of LQR problem is obtained as (3.11)min(X,S,Y)Tr(X)+Tr(QS)−Tr(YN)−Tr(NTYT),subjecttoAS−BuY+SAT−YTBuT+BwBwT≤0,[XR1/2YYTR1/2S]>0. The optimal control gain is then computed from G=YS−1, so K=αYS−1. For well-posed
problem with no additional constraints, the S that optimizes (3.11) is identical to the optimal state covariance
matrix P. In this paper, LQR in LMI
version was implemented using the LMI Toolbox of Matlab.


### 3.4. Kalman States Estimator

The Kalman estimator was named after Rudolf E. Kalman, though Thorvald
Nicolai Thiele and Peter Swerling actually developed a similar algorithm
earlier. Stanley
F. Schmidt is generally credited with developing the first implementation of a
Kalman estimator. It was during a visit of Kalman to the NASA Ames Research Center
that he saw the applicability of his ideas to the problem of trajectory
estimation for the Apollo program, leading to its incorporation in the Apollo
navigation computer. The estimator was developed in papers by Swerling [14],
Kalman [15], and Kalman and Bucy [16]. In control theory, the Kalman estimator
is most commonly referred to as Kalman filter or, mainly, as linear quadratic
estimation (LQE). In this paper, the Kalman estimator gain was obtained using
the software Matlab through command “lqe.”


### 3.5. Dynamic and Modal Uncertainties Representation

This
paper presents a methodology to design a robust control considering dynamic or
modal uncertainties in the state space model. The uncertainty ranges in the parameters
were quantified through experimental identification considering different
excitations. The mathematical model in state space realization was obtained
using the numerical method of subspace identification (N4SID). An expressive
part of identification methods concerns with computing polynomial models,
which, typically, give rise to numerically ill-conditioned mathematical
problems, especially for multi-input multioutput systems [17]. N4SID
algorithms are then viewed as optimal alternatives. This approach is
advantageous, especially for high-order multivariable systems, where the
parameterization is not trivial. The parameterization is needed to start up
classical identification algorithms, which means that a priori knowledge of the
order and of the observability or controllability indices is required [18].Usingnt(nt>1)  data acquisition tests, it is possible to
realize nt model identification (through N4SID algorithm)  and,
consequently, nt dynamic models. Each one can be used as a polytopic
vertex,  Si (see (2.5)). In this way, it is possible to define the
polype Ω  to describe the convex
space Co. Considering this convex
space to solve the controller, it is possible to obtain a robust gain, and so
to get a controller with the ability for maintaining satisfactory stability and
performance features in the presence of parameters uncertainties and variations.


## 3.1. Basic Definitions of LMIs and EVPs

A linear matrix inequality has the
form(3.1)F(z)=F0+∑i=1mziFi>0, where z  is a real vector and F0 and Fi are real symmetric
matrices. Inequality (3.1) is shorthand for saying that F(z) is positive
definite. A vector z that satisfies
inequality (3.1) is known as a feasible solution of the LMI.Inequality (3.1) is a convex
constraint on z. This property is important because powerful numerical
techniques are available for the solution of problems involving convex LMIs [4]. On the
other hand, no efficient algorithm is available for the solution of nonconvex
problems. Hence, nonconvex inequalities which may arise from a control problem
should be converted to convex LMIs to be solvable numerically. One useful
example for such manipulations is the LMI representation of the following
nonconvex inequalities: (3.2)Q(z)−S(z)R(z)−1ST(z)>0,R(z)>0,  where Q,  R, and S  are affine functions of z,
and Q and R are symmetric matrix. Inequality (3.2) is equivalent to (3.3)[Q(z)S(z)ST(z)R(z)]>0. This transformation can be achieved easily premultiplying
inequality (3.3) by (3.4)[I−SR−10I]>0 and postmultiplying it by its transpose.One of the concepts related with
LMIs and control problems is the eigenvalue problem. An eigenvalue problem may
have several representations, one of which is given by(3.5)mincTzsubjecttoA(z)>0,  where A(z) is a symmetric and affine function
of  z. An LQR problem may be
transformed into this form of the EVP to be represented in terms of LMIs.


## 3.2. LQR Control Problem

There are various representations
of LQR problem in the literature. Here, definitions are given in a way that
aids the derivation of the specific problem defined above; hence they may not
be the most general forms of LQR problem. The LQR problem is to find the
control gainK that satisfies the
optimization (3.6)minKE[yTQ˜y+uTR˜u+yTN˜u+uTN˜Ty],subjectto(2.1)andu(t)=−αKx(t),  where Q˜≥0 and R˜>0 are
symmetric weighting matrices, N˜ is the weighting
matrix between input and output vectors, and α is a scalar amplifier.
Substituting the output equation in (2.1) into the optimization problem (3.6), and
assuming Ein and Eout as zero vectors; one
obtains another form of the LQR problem (3.7)minKE[xTQx+uTRu+xTNu+uTNTx],subjecttox˙(t)=A(t)x(t)+Bw(t)w(t)+Bu(t)u(t),u(t)=−αKx(t),  where (3.8)Q=CTQ˜C,N=CTQ˜D+CTN˜,R=R˜+DTQ˜D+DTN˜+N˜D.


## 3.3. EVP Representation of the LQR Problem

Lyapunov’s stability criteria can
be used to state that a system given by (2.1) with control force u(t)=−αKx(t)=−Gx(t),
where G=αK, is stable if there exists a matrix S=ST>0 that satisfies (3.9)(A−BuG)S+S(A−BuG)T+BwBwT≤0, where S that satisfies inequality (3.9) is the optimal state covariance matrix P. So, combining (3.7) and (3.9) and optimizing S, the H2 optimization
problem may be stated as [13] (3.10)min(K,S)Tr(R1/2GSGTR1/2)+Tr(QS)−Tr(GSN)−Tr(NTSGT),subjecttoAS−BuGS+SAT−SGTBuT+BwBwT≤0,S=ST>0,  where Tr() is the trace of the matrix. This is
not an EVP problem since it is not convex because of the terms involving GS. To obtain a convex version of the
problem, two new variables are introduced X=R1/2GSGTR1/2 and Y=GS. Substituting these variables
into (3.10) and using the transformation given by (3.2) and (3.3), the EVP
representation of LQR problem is obtained as (3.11)min(X,S,Y)Tr(X)+Tr(QS)−Tr(YN)−Tr(NTYT),subjecttoAS−BuY+SAT−YTBuT+BwBwT≤0,[XR1/2YYTR1/2S]>0. The optimal control gain is then computed from G=YS−1, so K=αYS−1. For well-posed
problem with no additional constraints, the S that optimizes (3.11) is identical to the optimal state covariance
matrix P. In this paper, LQR in LMI
version was implemented using the LMI Toolbox of Matlab.


## 3.4. Kalman States Estimator

The Kalman estimator was named after Rudolf E. Kalman, though Thorvald
Nicolai Thiele and Peter Swerling actually developed a similar algorithm
earlier. Stanley
F. Schmidt is generally credited with developing the first implementation of a
Kalman estimator. It was during a visit of Kalman to the NASA Ames Research Center
that he saw the applicability of his ideas to the problem of trajectory
estimation for the Apollo program, leading to its incorporation in the Apollo
navigation computer. The estimator was developed in papers by Swerling [14],
Kalman [15], and Kalman and Bucy [16]. In control theory, the Kalman estimator
is most commonly referred to as Kalman filter or, mainly, as linear quadratic
estimation (LQE). In this paper, the Kalman estimator gain was obtained using
the software Matlab through command “lqe.”


## 3.5. Dynamic and Modal Uncertainties Representation

This
paper presents a methodology to design a robust control considering dynamic or
modal uncertainties in the state space model. The uncertainty ranges in the parameters
were quantified through experimental identification considering different
excitations. The mathematical model in state space realization was obtained
using the numerical method of subspace identification (N4SID). An expressive
part of identification methods concerns with computing polynomial models,
which, typically, give rise to numerically ill-conditioned mathematical
problems, especially for multi-input multioutput systems [17]. N4SID
algorithms are then viewed as optimal alternatives. This approach is
advantageous, especially for high-order multivariable systems, where the
parameterization is not trivial. The parameterization is needed to start up
classical identification algorithms, which means that a priori knowledge of the
order and of the observability or controllability indices is required [18].Usingnt(nt>1)  data acquisition tests, it is possible to
realize nt model identification (through N4SID algorithm)  and,
consequently, nt dynamic models. Each one can be used as a polytopic
vertex,  Si (see (2.5)). In this way, it is possible to define the
polype Ω  to describe the convex
space Co. Considering this convex
space to solve the controller, it is possible to obtain a robust gain, and so
to get a controller with the ability for maintaining satisfactory stability and
performance features in the presence of parameters uncertainties and variations.


## 4. Experimental Control Design

The proposed  methodology was
experimentally applied in a 3D truss structure, as shown in Figure1. The
physics and geometric properties of the truss are given in Table 1. The properties
of the PZT wafer stacks elements are shown in Table 2, and the output signals
were obtained with an accelerometer, model 352C22 PCB Piezotronics. Figures 2(a)
and 2(b) show the accelerometer in joint 4, and the detail of the PZT wafer actuator,
respectively. In this application, the PZT 1 was used to apply the disturbance
input, w(t), and PZT 2 was used as
control force input, u(t).Table 1
Physics and geometric properties of the truss
structures.
Length of structural element (mm)30Diameter of structural element (mm)5Young’s Modulus (GPa)210Density (Kg/m3)7800Table 2
Physics and geometric properties of PZT wafer stack
actuators, based on material designation APA60 M (Amplified Piezo Actuators,
CEDRAT).
PropertyUnitAPA60 MDisplacement(μm)80Blocked Force(N)110Stiffness(N/μm)1,38Resonance Frequency (Free-Free)(Hz)10400Response Time (Free-Free)(ms)0,05Resonance Frequency (Blocked-Free)(Hz)2800Response Time (Blocked-Free)(ms)0,18Voltage Range(V)−20⋯150Capacitance(μF)1,55Resolution(nm)0,8Thermo-Mechanical Behaviour(μm/K)1,02Height H (in actuation direction)(mm)13,0Length(mm)26,9Width (ind. Edges, wires)(mm)11,5Mass(g)20,0Standard Mechanical Interface [TH]2 flat surface 5*10 mm2 with M2.5
threaded holeStandard Electrical Interface2 PTFE insulated AWG30 wires 100 mm long with ø 1
banana plug3D truss structures and PZT wafer stack actuators.
(a)(b)(c)Accelerometer in joint 4 and details of the PZT wafer stack actuator.
(a)(b)The
connections of the piezoelectric actuators in the structural elements were made
through adapters, shown in Figure3(a). Figure 2(b) shows details of this
connection. These adapters were made using aluminum rod in order to connect the
structural part and the PZT wafer actuator. This kind of actuator amplifies the
displacement in the axial direction of the structural member, and it is named active
member. The joint connections were made of copper, with 24 mm of diameter in the
geometric format of eight sides, as shown in Figure 3(b).(a) Adapters to connect PZT stacks; (b) structural nodes.
(a)(b)The
dynamic  model represented through state space realization was identified using
the N4SID algorithm consideringnt=6. Therefore, the dynamic
uncertainties were considered through identification of six models, then, the convex space was obtained with six vertexes (S1,S2,…,S6). The order of
the model (dimension of state vector) was chosen as 2, so the first mode was
identified. Using the first identified mode, the Kalman estimator gain was
computed by the “lqe” command of the Matlab software. It was computed  as L=[463,998265,8124]T.
Considering the weighting matrices Q and R as 5*I and 1*I, respectively,
where I is the identity matrix, the
controller gain was obtained as G=[−1,54262,5514]. The scalar amplifier α was chosen as 80 to
the first mode. The controller was designed to the first mode, however,  at the
practice test, it was verified that the second mode also had a significant
attenuation in the vibration amplitude.To
verify the results of the active vibration control, two cases were considered. Figure4 shows the configuration of the experimental setup used. In the first case,
the disturbance input was a sine signal with frequency of 16 Hz (approximately
the first natural frequency). Figure 5 shows the output signal with and without
control obtained in joint 4. Figure 6 shows the experimental output measured
using the accelerometer in joint 4 and the estimated output through Kalman estimator
algorithm. These results were obtained using the dSPACE 1103 control board and
the Simulink/Matlab.Figure 4
Disposition of the experimental setup.Figure 5
Output signal measured in the joint 4 using accelerometer-controlled and uncontrolled systems.Figure 6
Estimated output signal in joint 4 through Kalman Estimator and experimental output.In
the second case, a disturbance
input was considered as a sine signal with frequency of 26 Hz
(approximately the second natural frequency). Figure7 shows the response in
time domain for the uncontrolled and controlled systems. The controller was applied approximately after
4.5 seconds. Figures 8 and 9 show the control force in the PZT stack actuator 2
and the output estimated through Kalman estimator, respectively. Figure 10
shows the frequency response function (FRF) of the uncontrolled and controlled
truss structures.
It was attenuated approximately by 6 dB and 9 dB to the first and second modes,
respectively.Figure 7
Experimental output signal of uncontrolled and controlled truss structure in joint 4.Figure 8
Control force applied by PZT wafer stack 2 for the second case of disturbance.Figure 9
Output signal estimated through Kalman Estimator algorithm for the second case of disturbance (sine signal with 26 Hz).Figure 10
Frequency response function of uncontrolled and controlled system.


## 5. Final Remarks

Over the last two decades, the use
of piezoceramics as actuators and sensors has increased considerably,  since
they provide an effective means of high-quality actuation and sensing mechanism.
Piezoceramics have been considered as an alternative due features as low-cost, light
weight, and easy-to-implement for active control of structural vibration.In this paper, the subspace identification
method was used to obtain the parameters of the system and to
characterize the uncertainty ranges present in the model. In the experimental
application, the uncertainties were defined in a polytopic with six vertexes.
The system identification technique was used to identify the model in the state
space realization that was converted to modal coordinates. The LQR controller
solved through LMI was experimentally implemented and applied in a 3D truss
structure that contains nonlinearities and uncertainties. The disturbance was
applied through PZT wafer stack. LMI techniques that  are classified by some authors
as postmodern control present many advantages, mainly due to the facilities of solving
numerical problems for complex structure, where the analytical solution should
be difficult to implement. Uncertainties in the dynamic matrix were considered
in order to design a robust active vibration control. However, any other
uncertain parameter should be added, for instance, damping coefficients. In
this case, it is only needed to consider new vertexes in the box with all
uncertain parameters and write the respective LMIs. The proposed approach
showed that an efficient robust controller design can be obtained for complex
structures with nonlinearities and uncertainties.



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*Source: 754951-2008-12-02.xml*