development of a new inertial-based vibration …development of a new inertial-based vibration...

Development of a New Inertial-based VibrationAbsorber for the Active Vibration Control of

Flexible StructuresCarmine Maria Pappalardo, Domenico Guida

Abstract—In this paper, a new actively controlled inertial-based vibration absorber is proposed and used for controllingthe mechanical deformations of flexible structures. To thisend, a method for reducing the externally induced vibrationsof structural systems is developed. The method employedin this paper is based on the numerical techniques of theapplied system identification field and is grounded in theoptimal control theory. A three-story shear building systemwith a pendulum hinged on the third floor is the flexiblestructure considered as the case study of this investigation.This mechanical system is constructed using rigid and flexiblecomponents in order to reproduce a simple three-dimensionalstructure. The base of the structural system is excited by aharmonic excitation combined with a noise excitation source inorder to simulate the earthquake. By doing so, the worst casescenario in which the frequencies of the external excitationare close to the natural frequencies of the flexible structureis considered. The pendulum mounted on the third floor ofthe flexible structure serves as an actively controlled inertial-based vibration absorber. The control torque applied to thependulum is actively controlled by using a brushless motordriven by a programmable digital controller. Therefore, theinertial effects of the oscillating pendulum directly contrastthe externally induced vibrations of the three-story shearbuilding system. The feedback control torque applied on thependulum is obtained by monitoring the accelerations of thethree floors of the flexible structure and is designed employinga realistic mechanical model of the dynamical system obtainedby using a time-domain system identification approach. Thenumerical results found in this investigation by using a simplecomputer program coded in MATLAB show that the modalparameters identified for describing the dynamic behaviorof the flexible structure are consistent with those predictedemploying a simple lumped parameter model. Furthermore,in this investigation, an optimal closed-loop controller based onthe Linear-Quadratic Regulator (LQR) method and an optimalstate observer based on the Kalman filtering approach, alsoknown as Linear-Quadratic Estimator (LQE), are designedemploying the state-space model obtained from experimentaldata. Experimental results demonstrate that a considerablereduction of the structural vibrations of the three-story shearbuilding system can be obtained by means of the introductionof the feedback controller combined with the state estimator.

Index Terms—Three-story shear building system, Systemidentification, Optimal control, State estimation, Experimentalmodal parameter identification.

I. INTRODUCTION

This paper is focused on the design and implementationof a new inertial-based vibration absorber for the active

C. M. Pappalardo is with the Department of Industrial Engineering,University of Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano, Salerno,ITALY (corresponding author, email: [email protected]).

D. Guida is with with the Department of Industrial Engineering, Univer-sity of Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano, Salerno, ITALY(email: [email protected]).

vibration control of flexible structures. For this purpose, anoptimal controller is developed for reducing the mechanicalvibrations of structural systems considering the applied sys-tem identification computational techniques. In this section,background material, the formulation of the problem ofinterest for this study, the scope and the contribution ofthis investigation, and the organization of the manuscript areprovided.

A. Background Material

Applied system identification can be seen as the complexprocess of developing mathematical models of dynamicalsystems based on input-output data [1]–[3]. In general, amathematical model of a physical system can be used to pre-dict by means of numerical simulations the system dynamicbehavior in response to known external excitations [4]–[8].In the field of applied system identification, the basic laws ofmechanics are combined with statistical methods in order todevise mathematical models of dynamical systems by usingexperimental measurements. Model reduction techniques andthe methodologies for the optimal design of experimentsare also widely employed in the field of applied systemidentification [9]–[12]. In this context, numerical experimentsobtained by means of dynamic simulations can be performedby using a reliable mathematical model of a physical systemin order to reproduce the input-output relationships observedexperimentally [13]–[17].

In the field of mechanical engineering, the consistentformulation of the equations of motion that describe thenonlinear behavior of a mechanical system represents afundamental step in the solution of the optimal control,estimation, and identification problems [18], [19]. In thisrespect, the main challenge is represented by the fact that,in general, the linearized version of the equations of motionis inadequate for capturing the fully nonlinear physics ofthe problem at hand and, therefore, more advanced controlalgorithms, estimation techniques, and design methods arenecessary for solving these difficult tasks [20], [21]. Onthe other hand, in several practical applications, using thefully nonlinear dynamic equations for predicting the timeevolution of a mechanical system and calculating the controlactions for the mechanical system employing the numericalsolutions of the nonlinear set of dynamic equations canbe difficult, if not impossible, to achieve or implement inreal time [22]. One direction of research devoted to thesolution of these important issues is based on the directapplication of complex control strategies to a simplified, butstill nonlinear, version of the equations of motion of the

Engineering Letters, 26:3, EL_26_3_11

(Advance online publication: 28 August 2018)

______________________________________________________________________________________

mechanical system of interest [23]. To this end, one can usethe optimal control theory based on the Pontryagin minimumprinciple leading to the Hamilton-Jacobi-Bellman partialdifferential equation, that is the continuous-time analogue ofthe discrete deterministic dynamic programming algorithm,or to a set of nonlinear differential-algebraic equations whichform a nonlinear two-point boundary value problem whichcan be numerical solved by using the adjoint approach.Moreover, one can also use other effective analytical andcomputational methods for the design of nonlinear controlactions. For instance, the feedback linearization approach,the nonlinear H-infinity control method, the sliding modecontrol algorithm, the nonlinear control strategies based onthe control-Lyapunov function, and the extended Kalmanfilter approach represent, among the others, valid computa-tional algorithms which can be effectively employed for theconstruction of a nonlinear control and estimation scheme[24]–[27]. However, as discussed in details in this paper,the linear control method of interest for this investigation isbased on the optimal control and estimation approach since,in the case of small deformations, the structural system ofinterest can be adequately modeled as a linear dynamicalsystem.

B. Formulation of the Problem of Interest for this Study

In the fields of dynamic and control engineering, appliedsystem identification deals with the development of mechan-ical models of physical systems necessary for the design ofopen-loop (feedforward) and closed-loop (feedback) controlstrategies by using experimental measurements [28]–[32].Thus, several investigations based on system identificationtechniques focused on the design of effective control strate-gies for machines and structures can be found in the literature[33]–[37]. However, all the identification analytical methodsand numerical procedures employed in this important fieldof research have in common some fundamental principlesand mathematical tools [38], [39]. In particular, the time-domain numerical procedures of applied system investigationare of interest for this investigation because the first-orderdynamic models based on the state-space representation andthe second-order dynamic models based on the configuration-space representation can be readily used for developingoptimal controller and observer systems by using effectiveand robust control algorithms such as, for example, theLinear Quadratic Gaussian (LQG) regulation method [40]–[44]. For instance, important practical applications of thesystem identification methodologies are the refinement offinite element models of aerospace systems by using dy-namic testing, the modal parameters identification of civilstructures based on environmental and artificial vibrations,and the real-time identification of mechanical models ofsuspension mechanisms for implementing the active andsemi-active control paradigms [45]–[53]. Furthermore, thenonlinear control methods based on the techniques of ap-plied system identification are also useful for determiningstructural parameters of mechanical systems composed ofrigid and flexible bodies connected by kinematic constraintsthat are referred to as multibody systems [54]–[59].

C. Scope and Contributions of this Investigation

This paper deals with the development of a control systemcapable of reducing the vibration of flexible structures basedon a methodology for identifying first-order and second-orderdynamic models of mechanical systems using time-domaininput-output measurements. The proposed approach relieson the optimal control theory, is based on the numericaltechniques of applied system identification, and is verifiedexperimentally by using a simple test rig. The flexiblestructure considered for testing the proposed method is athree-story shear building system. A pendulum is hinged onthe third floor of the flexible structure in order to operateas an actively controlled inertial-based vibration absorber.The pendulum is driven by a brushless motor connected to aprogrammable digital controller. The floors of the three-storysystem are instrumented with piezoelectric accelerometersthat are connected to the programmable digital controller.Therefore, the torque applied on the pendulum can beactively controlled allowing for the design of a feedbackcontrol law based on a real-time estimation of the systemstate. In order to correctly design the feedback controller andthe state estimator, a preliminary mechanical model of theflexible structure is developed employing a lumped parameterapproach. After that, a refined mechanical model of thethree-story shear building system is derived from input-output experimental data by using a system identificationprocedure developed in this work. For this purpose, a methodfor constructing mechanical models from identified state-space representations is used for identifying first-order andsecond-order dynamical models of the flexible structure.Subsequently, a simple least-square method is employedin order to obtain an improved estimation of the dampingmodel of the flexible structure. The identified mechanicalmodel of the three-story shear building system is effectivelyused to design a feedback controller and a state estimatorby using the Linear Quadratic Gaussian (LQG) controlapproach. Finally, the numerical procedure developed in thepaper is experimentally tested using the three-story shearbuilding model as a demonstrative example in order to verifythe effectiveness of the identification and control approachproposed in this investigation.

D. Organization of the Manuscript

The remaining part of this paper is organized accordingto the following structure. In Section II, a description ofthe mechanical system analyzed in this paper is reported. InSection III, a simple lumped parameter model for the three-story shear building system that describes the mechanicalvibrations of the flexible structure considered as the casestudy is developed. In Section IV, the computational steps ofthe proposed system identification numerical procedure aredescribed. In Section V, the system identification numericalprocedure developed in the paper is used to determine adiscrete mechanical model of the flexible structure basedon input-output experimental measurements. In Section VI,the development and the implementation of the controlscheme for reducing the structural vibration of the three-storybuilding system are described. In Section VII, a summary ofthis investigation and the conclusions drawn in this study arereported.



______________________________________________________________________________________

II. SYSTEM DESCRIPTION

In this section, a description of the mechanical systemanalyzed in this investigation is reported. The mechanicalsystem considered as the demonstrative example of the ap-proach developed in the paper is a three-dimensional flexiblestructure and is shown in figure 1. The flexible structure

Fig. 1. Experimental test rig.

used as test rig can be modeled as a three-story shearbuilding system. The structural system is composed of sixflexible beams and three rigid connecting rods. The flexiblebeams are made of harmonic steel while the material of theconnecting rods is aluminum. The dynamic behavior of themechanical system considered as a case study is of interestfor this investigation and serves as a test rig for studyingthe mechanical vibrations of real full-scale flexible structuressubjected to the earthquake. For this purpose, the frequencyrange of interest for this study includes all the excitationfrequencies between 0 Hz and 15 Hz. In this frequencyrange, the flexible beams deform as linear elastic continuumbodies and the connecting rods behave essentially as rigidbodies. Observing the geometric configuration of the three-story system, this flexible structure is deployed in a plane.Therefore, the lateral stiffness and the torsional stiffness ofthe flexible structure are considerably larger than stiffnessalong the plane. Consequently, this three-dimensional flexiblestructure can be modeled as a planar three-story shearbuilding system with a very good approximation. The firstfloor of the flexible structure is excited by a shaker that isconnected to the structure by means of a stinger. A loadcell is collocated between the first floor and the stingerin order to measure the force transferred to the flexiblestructure by the shaker. In order to minimize the inertial

influence of the shaker on the flexible structure, the shaker issuspended through a steel cable fixed on an external supportisolated from the flexible structure of interest. The shakeris connected to a power amplifier which is controlled byan arbitrary wave function generator. In order to measurethe time response of the three-story shear building system,piezoelectric transducers that sense the system accelerationsare collocated on each floor of the flexible structure. Fur-thermore, the flexible structure is actively controlled byusing an inertial-based vibration absorber collocated on thethird floor. The actively controlled inertial-based vibrationabsorber is realized by using a physical pendulum havingan additional mass concentrated on the pendulum tip asshown in figure 2. The pendulum can oscillate along the

Fig. 2. Actively controlled inertial-based vibration absorber.

same plane in which the flexible structure is deployed inorder to contrast the mechanical vibrations of the structuralsystem by means of its inertial effects. The pendulum isactively controlled employing a control actuator driven bya brushless motor. The brushless motor is equipped withan encoder that allows for measuring the pendulum angu-lar displacement. The electromechanical actuator providesa control torque that is computed in real time by usinga digital controller that communicates with the brushlessmotor by means of a drive device. The digital controllerreads the acceleration signals coming from the piezoelectrictransducers and the force signal measured by the load cell.The digital controller can be programmed and monitoredoffline in order to calculate a feedback control law for thecontrol torque of the brushless motor based on the forceand acceleration measurements. A schematic representationof the identification and control scheme for the test rig isshown in figure 3. A detailed three-dimensional CAD modelof the flexible structure together with the actively controlledinertial-based vibration absorber is represented in figure 4.Furthermore, in order to develop an effective control lawfor the digital controller, an experimental modal analysis ofthe flexible structure was performed at first. To this end, thethree floors of the flexible structure were excited by usingan impact hammer instrumented with a load cell connectedto a spectrum analyzer. The acceleration signals measuredby the piezoelectric transducers in correspondence of theexternal excitations of the impact hammer were recorded byusing the spectrum analyzer in order to obtain the input-



______________________________________________________________________________________

Fig. 3. Identification and control scheme.

Fig. 4. Three-dimensional CAD model of the mechanical system.

output experimental data necessary for the use of systemidentification numerical procedures [60]–[62].

III. DYNAMIC MODEL

In this section, a simple lumped parameter model forthe three-story shear building system that describes themechanical vibrations of the flexible structure is developed.The analytical development of this preliminary mechanicalmodel of the flexible structure is necessary in order to guidethe subsequent identification process based on experimentalmeasurements. A schematic representation of the lumpedparameter model of the flexible structure is shown in figure5. In the analysis of this preliminary mechanical model,the flexible structure is modeled considering three lumpedmasses that represent the connecting rods and three spring

Fig. 5. Lumped parameter model.

elements that represent the flexible beams. On the other hand,the angular displacement of the pendulum is fixed in orderto model the behavior of the flexible structure in the casein which the control system is not active. Therefore, thelumped parameter model of the three-story shear buildingsystem includes only n2 = 3 degrees of freedom. Denotingthe continuous time with t, the degrees of freedom of theflexible structure are the horizontal displacements of thesystem first, second, and third floors respectively indicatedas x1(t), x2(t), and x3(t). Thus, the generalized coordinatevector that identifies the configuration of the three-storyshear building model is indicated with x(t) and is givenby x(t) =

[x1(t) x2(t) x3(t)

]T. Since in the fre-

quency range of interest and for the amplitudes of the inputsconsidered as external excitations the dynamic behavior ofthe flexible structure is linear, the equations of motion thatdescribe the lumped parameter model of the three-storyshear building system can be written in a compact matrixform as M x(t) + K x(t) = 0, where x(t) is the systemgeneralized acceleration vector, x(t) denotes the systemgeneralized coordinate vector, M is the system mass matrix,and K denotes the system stiffness matrix. In the analyticalderivation of the equations of motion of the flexible structure,the effect of the structural damping is neglected. However,a simple but realistic estimation of the viscous damping ofthe flexible structure will be subsequently obtained in thepaper by using the proposed identification procedure. Themass and stiffness matrices of the lumped parameter modelfor the flexible structure can be readily obtained using theclassical methods of analytical dynamics. In particular, themass matrix is given by M = diag (m1,1,m2,2,m3,3), while



______________________________________________________________________________________

the stiffness matrix is defined as follows:

K =

k1,1 k1,2 0k1,2 k2,2 k2,30 k2,3 k3,3

(1)

It is important to note that the mass matrix M and thestiffness matrix K of the three-story shear building modelare constant, symmetric, and positive-definite matrices. Theentries mi,j and ki,j for i = 1, 2, . . . , n2 and j = 1, 2, . . . , n2

of the system mass and stiffness matrices are given by: m1,1 = m1

m2,2 = m2

m3,3 = m3 +m4

,

k1,1 = k1 + k2, k1,2 = −k2k2,2 = k2 + k3, k2,3 = −k3k3,3 = k3

(2)where m1 = 1.281 (kg), m2 = 0.814 (kg), and m3 =1.380 (kg) are the masses of the first, second, and thirdconnecting rods, respectively, whereas m4 = 0.083 (kg)is the mass of the pendulum. The stiffness of each flexiblebeam can be readily computed assuming a set of clamped-clamped boundary conditions and a parallel configurationof the resulting lumped spring elements to yield k1 =24EJ

/L31, k2 = 24EJ

/L32, and k3 = 24EJ

/L33, where

E = 207 · 109(N/m2)

denotes the elastic modulus ofthe flexible beams, J = 2.917 · 10−12

(m4)

represents thesecond moment of area of the flexible beams, whereas L1 =23 ·10−2 (m), L2 = 28 ·10−2 (m), and L3 = 23 ·10−2 (m)are the lengths of the first, second, and third flexible beams,respectively. A modal analysis can be readily performedusing the lumped parameter model of the three-story buildingmodel in order to obtain the system natural frequencies fn,jand the corresponding modal vectors ϕj associated with thenormal mode j which can be written as ϕj = eiθjρj , wheree is the Napier’s constant, i =

√−1 is the imaginary unit, ρj

denotes the vector of relative amplitudes associated with themode j, and θj represents a diagonal matrix containing therelative phases of the components of the mode j. Thus, themodal parameters of the flexible structure obtained by usingthe preliminary three-story shear building lumped parametermodel are given by:

fn,1 = 1.962 (Hz)

ρ1 =[

1 2.509 3.085]T

θ1 = diag (0, 0, 0)

(3)

fn,2 = 5.780 (Hz)

ρ2 =[

1 0.2447 0.394]T

θ2 = diag (0, 0,−3.141)

(4)

fn,3 = 8.807 (Hz)

ρ3 =[

1 3.138 1.136]T

θ3 = diag (0,−3.141, 0)

(5)

As expected, the lumped parameter model of the flexiblestructure leads to a set of three mode shapes where the firstnormal mode has no nodes, the second normal mode has onenode, and the third normal mode has two nodes. The modalparameters obtained by using the simplified mathematicalmodel of the flexible structure will be used to guide andcorroborate the numerical results obtained employing theproposed system identification method based on experimentalinput-output data [63]–[66].

IV. SYSTEM IDENTIFICATION METHOD

In this section, the system identification method used in thepaper is described. To this end, consider a state-space modelof a mechanical system that describes the dynamic behaviorof the flexible structure of interest for this investigation. Forboth linear and nonlinear systems, a state-space model of thedynamic equations of a mechanical system can be obtainedby using a continuous-time approach or employing directly adiscrete-time representation [67]–[69]. In this investigation,a discrete-time representation is used in order to facilitate thesubsequent computer programming of the digital controller.In general, a discrete-time model of a mechanical system isdescribed by a set of dynamic and measurement equationsgiven by: z(k + 1) = Az(k)+Bu(k)

y(k) = Cz(k)+Du(k)(6)

where k represents the discrete time, z(k) is the discrete-time state vector having dimension n = 2n2, n2 is thenumber of generalized coordinates of the mechanical systemof interest, u(k) denotes the discrete-time input vector ofdimension r, y(k) is the discrete-time output vector havingdimension m, A identifies the discrete-time system statematrix of dimension n × n, B denotes the discrete-timeinput influence matrix having dimension n × r, C is theoutput influence matrix of dimension m×n, and D denotesthe direct transmission matrix having dimension m × r.These equations respectively represent the time evolutionof the system state due to its intrinsic dynamics and tothe variation in time of the physical quantities measured bythe instrumentation. In order to improve the accuracy of thedynamic analysis, one can introduce a state estimator in thestate-space model of the mechanical system described byan observed set of discrete-time dynamic and measurementequations. By doing so, one obtains: z(k + 1) = Az(k)+Bv(k)

y(k) = Cz(k)+Du(k)(7)

where z(k) is the estimated state vector of dimension n, y(k)represents the estimated measurement vector having dimen-sion m, A denotes the discrete-time observer state matrix ofdimension n×n, B identifies the discrete-time observer stateinfluence matrix having dimension n× (r +m), and v(k) isthe generalized input vector of dimension r +m. The matrixand vector quantities A, B, and v(k) are respectively definedas follows:

A = A + GC

B =[

B + GD −G]

v(k) =[

uT (k) yT (k)]T

(8)

where G represents the observer matrix having dimensionn×m. The observer matrix G is aimed at adjusting theeigenvalues of the observer state matrix A with respect tothe eigenvalues of the state matrix A. Consequently, theobserver state matrix A can be made asymptotically stableemploying an appropriate selection of the observer matrix G.Furthermore, for a general linear time-invariant dynamical



______________________________________________________________________________________

system, the input-output relationship of the discrete-time dy-namic model represented in the state-space and the recursivesequence of the discrete-time observer state-space model canbe respectively written as follows:

y(k) =k∑j=0

(Yju(k − j))

y(k) =k∑j=0

(Yjv(k − j)

) (9)

where the sequence of matrices Yk is formed by m× rrectangular matrices that are known as discrete impulseresponse functions or system Markov parameters, whereasthe sequence of matrices Yk is made of m× (r +m)rectangular matrices called observer Markov parameters. Thesequences of system and observer Markov parameters Yk

and Yk are respectively defined as follows: Y0 = D

Yk = CAk−1B,

Y0 = D

Yk = CAk−1B(10)

In order to facilitate the development of an effectivenumerical procedure for performing in the time domainthe system identification of a discrete-time dynamic modelrepresented in the state-space, one can define for a mechan-ical system the additional sequence of matrices Y0

0 = Dand Y0

k = CAk−1G, where the matrices Y0k are m×m

rectangular matrices that are referred to as observer gainMarkov parameters. By using experimental input-output data,the sequence of observer Markov parameters can be obtainedusing a simple least-square computational approach. Thisprocess leads to a sequence of matrices that represent theset of observer Markov parameters identified experimentally.Once that this process has been successfully performed,one can obtain both the set of system Markov parametersand the set of observer gain Markov parameters startingfrom the identified observer Markov parameters. For thispurpose, consider the following input-output representationof a dynamical system described by a discrete-time dynamicmodel represented in the state-space:

y(k) +p∑j=1

(Y

(2)j y(k − j)

)=

p∑j=1

(Y

(1)j u(k − j)

)+ Du(k)

(11)

where Y(1)k and Y

(2)k are rectangular matrices having re-

spectively dimensions m× r and m×m that are defined asfollows:

Y(1)k = C(A + GC)k−1(B + GD)

Y(2)k = C(A + GC)k−1G

(12)

It can easily proved that the sequence of matrices thatrepresents the observer Markov parameters can be writtenin a block matrix form as Yk =

[Y

(1)k −Y

(2)k

]. The

constant coefficients of the recursive sequence can be numer-ically calculated using a least-square approach based on inputand output experimental data. To this end, a computational

methodology based on the Observer/Kalman Filter Identifica-tion Method (OKID) can be readily used. This method allowsfor computing the sequence of Markov parameters fromexperimental data employing experimental measurements.Considering a data set having length l, the recursive rela-tionship can be mathematically expressed in a matrix form asLp = YV+

p , where Y, Lp, and Vp are rectangular matricesrespectively of dimensions m× l, m× (r + (r +m)p), and(r + (r +m)p)× l that are respectively defined as follows:

Y =[

y(0) y(1) . . . y(l − 1)]

Lp =[

Y0 Y1 . . . Yp

]

Vp =

u(0) u(1) . . . u(l − 1)

0 v(0) . . . v(l − 2)...

.... . .

...0 0 . . . v(l − p− 1)

(13)

where p is an integer number that denotes the discrete timeat which the estimated output vector y(k) approaches theactual measured output vector y(k). The sequence of first pobserver Markov parameters Yk are included in block matrixLp. Therefore, a simple least-square estimation method canbe used for obtaining an approximate solution for the ob-server Markov parameters Yk employing experimental input-output measurements as Lp = YV+

p , where V+p represents

a rectangular matrix having dimension m× (r + (r +m)p)that identifies the Moore-Penrose pseudoinverse matrix cor-responding to the matrix Vp. By using simple recursiverelationships, one can obtain the system Markov parametersand the observer gain Markov parameters using the estimatedobserver Markov parameters. Subsequently, the combinationof the identified system and observer gain Markov parame-ters, which are denoted respectively as Yk and Y0

k, can bereadily used for developing an applied system identificationalgorithm based on time domain data. For this purpose,a computational procedure based on the Eigensystem Re-alization Algorithm (ERA) can be employed for derivinga state-space set of characteristic matrices A, B, C, andD. These matrices characterize the state-space model of ageneral linear mechanical system obtained using a discrete-time representation based on the system Markov parametersidentified from experimental data. To this end, consider thefollowing block matrix that contains the system and observergain Markov parameters Pk =

[Yk Y0

k

], where Pk

is a block matrix having dimensions m× (r +m). Theidentification of the matrix Pk is necessary for constructinga generalized block Hankel matrix defined as:

N(k − 1) =

Pk Pk+1 . . . Pk+δ−1

Pk+1 Pk+2 . . . Pk+δ

......

. . ....

Pk+γ−1 Pk+γ . . . Pk+γ+δ−2

(14)

where N(k − 1) denotes a block Hankel matrix havingdimensions γm× δ(r +m) in which both the set of systemMarkov parameters and the set of observer gain Markovparameters Yk and Y0

k are included, while γ and δ aretwo integer parameters that can be assumed as γ = p andδ = l − p, where l is the length of the data set used forthe numerical implementation of the system identification



______________________________________________________________________________________

algorithm. Considering the case in which k = 0, thegeneralized Hankel matrix N(0) can be factorized using theSingular Value Decomposition (SVD) and can be written asN(0) = RQST , where the rectangular matrix Q containsthe singular values of the generalized Hankel matrix N(0),while the columns of the matrices R and S form orthonormalvectors. In particular, the block matrix Q can be expressedas Q = block

[[Qn,O

]; [O,O]

], where the submatrix

Qn is a square diagonal matrix that is given by Qn =diag (q1, q2, . . . , qn), where qi , 1 = 1, 2, . . . , n representthe nonzero singular values associated with the generalizedHankel matrix N(0). Indicating respectively with Rn andSn the rectangular matrices formed by the first n columnsof the rectangular matrices R and S, the generalized Hankelmatrix N(0) and its Moore-Penrose pseudoinverse matrixN+(0) can be respectively written as N(0) = RnQnSTnand N+(0) = SnQ−1n RT

n . The analysis of the spectrum ofthe singular values qi , 1 = 1, 2, . . . , n of the generalizedHankel matrix N(0) allows for determining the order n ofthe identified dynamical model of the mechanical systemunder consideration based on the sequences of input-outputdata. By doing so, an identified discrete-time dynamic modelrepresented in the state-space associated with the mechanicalsystem under consideration is given by the discrete-timestate-space matrices A, B, C, and D as well as by theobserver matrix G resulting from the system identificationalgorithm. Thus, one can obtain the identified discrete-timedynamic model in the state-space of the dynamical systemunder consideration employing the following equations:

A = Q−1/2n RT

n N(1)SnQ−1/2n[

B G]

= Q−1/2n STnEr+m

C = ETmRnQ

−1/2n

D = Y0 = Y0

(15)

where Em and Er+m are Boolean matrices with appropriatedimensions. The set of identified matrices A, B, C, D, andG represents a minimum order, controllable, and observablestate-space realization of a general mechanical system havinga linear mathematical structure. In particular, although thenumerical values of the entries of the identified character-istic matrices A, B, C, and D are not equal to those ofthe actual discrete-time dynamic model represented in thestate-space of the mechanical system given by A, B, C,and D, the modal parameters of the identified mechanicalmodel coincide with the actual modal parameters of thedynamical system [70], [71]. Moreover, the discrete-timedynamic model represented in the state-space of the mechan-ical system can be readily transformed in its continuous-time counterpart. Denoting with Lc the diagonal matrixcontaining the eigenvalues of the continuous-time state-spacemodel and indicating with W the matrix of eigenvectorsassociated with the system generalized coordinates, a second-order mechanical model of the dynamical system can bederived by using the identified modal parameters of the state-

space model as follows:

M =(WLcW

T)−1

K = −(WL

−1c WT

)−1R = −

(MWL

2

cWTM

)(16)

where M is the identified mass matrix, K represents the iden-tified stiffness matrix, and R denotes the identified dampingmatrix of the mechanical system under study. However, theestimate values of the entries of the system damping matrixR are considerably influenced by the noise that affectsthe input-output data set. For lightly damped mechanicalsystems, an improved estimation of the identified dampingmatrix of the mechanical system can be obtained by using theproportional damping assumption given by R = αM +βK.By doing so, one can employ a least-square approach for theestimation of the proportional damping coefficients α and βwhich characterize the the proportional damping hypothesis.To this end, the improved estimation of the system dampingmatrix R can be written as R = αM + βK, where the esti-mated proportional damping coefficients α and β can be ob-tained using the a least-square approach given by x = A+

f bξ,

where x =[α β

]T, bξ =

[ξ1 ξ2 . . . ξn2

]T, and:

Af =1

2

1

2πfn,12πfn,1

12πfn,2

2πfn,2...

...1

2πfn,n2

2πfn,n2

(17)

where A+f denotes the Moore-Penrose pseudoinverse matrix

of the coefficient matrix Af , fn,j represents the identifiednatural frequency associated with the mode j of the mechan-ical system, and ξj is the identified damping ratio relative tothe mode j of the linear dynamical system [72]–[76].

V. EXPERIMENTAL INVESTIGATION

In this section, the system identification numerical proce-dure developed in the paper is used to determine a discretemechanical model of the flexible structure based on input-output experimental measurements. The numerical resultsproposed in this section are obtained using a simple com-puter program coded in MATLAB, while the experimentalresults reported in this section are obtained by means of theexperimental apparatus described previously in the paper. Forthis purpose, an impact hammer instrumented with a load cellwas used to excite impulsively the first floor of the three-storyshear building system and the corresponding accelerations ofthe three floors of the flexible structure were recorded usingpiezoelectric transducers. The force signal recorded by theload cell of the impact hammer is shown in figure 6. Theacceleration signals of the floors of the flexible structure arerecorded by the piezoelectric transducers placed on each floorand are respectively shown in figures 7, 8, and 9. The timespan considered is Ts = 64 (s) while the sampling frequencyused for acquiring the experimental data is fs = 32 (Hz).Thus, the sampling time step is ∆t = 3.125 · 10−2 (s) and



______________________________________________________________________________________

Fig. 6. Force input measurement.

Fig. 7. Acceleration output measurement of the first floor.

the corresponding Nyquist frequency is fN = 16 (Hz). Byusing this set of parameters for acquiring the experimentalmeasurements, the frequency range of interest from 0 Hz to15 Hz is correctly captured by the experimental data. Further-more, the set of input and output data was first filtered in thetime domain and subsequently in the frequency domain. Inthe time domain, since the action of the impact hammer tookplace around t = 10 (s), the sections of the input and outsignals before t = 10 (s) were set equal to zero because theinformation contained in this interval of the measured datahad no physical meaning. In the frequency domain, a low-pass Butterworth filter having a cut-off frequency of fc =16 (Hz) was used in order to eliminate the effect of high-frequency noise. The proposed identification procedure wasapplied to the set of input-output data obtained by means ofexperimental measurements. By doing so, the system Markovparameters Yk and the observer gain Markov parameters Y0

k

were recovered from the identified sequence of the observerMarkov parameters Yk. The identified Markov parametersYk and the identified observer gain Markov parameters Y0

k

were used to construct the matrix sequence of the combinedMarkov parameters Pk. The combined Markov parameters

Fig. 8. Acceleration output measurement of the second floor.

Fig. 9. Acceleration output measurement of the third floor.

Pk were used to assemble the sequence of Hankel matricesN(k − 1) based on experimental data. In particular, a SVDof the Hankel matrix N(0) was computed and figure 10shows the magnitude of the identified singular values. Asexpected, figure 10 shows that only 6 singular values havea relatively large magnitude and, consequently, the order ofthe identified state-space model that mathematically representthe three-story shear building system is n = 6. Subsequently,the identified discrete-time state-space matrices A, B, C, D,and G representing the identified linear dynamical model ofthe flexible structure and the identified observer matrix werecomputed yielding to the following set of identified modalparameters:

fn,1 = 1.934 (Hz) , ξ1 = 0.0395

ρ1 =[

1 2.503 2.851]T

θ1 = diag (0,−0.0186,−0.1053)

(18)

fn,2 = 5.690 (Hz) , ξ2 = 0.0030

ρ2 =[

1 0.1492 0.4221]T

θ2 = diag (0,−0.0365,−3.170)

(19)



______________________________________________________________________________________

Fig. 10. Magnitude of the singular values of the identified Hankel matrix.

fn,3 = 8.793 (Hz) , ξ3 = 0.0042

ρ3 =[

1 3.443 1.476]T

θ3 = diag (0,−3.144,−0.0296)

(20)

where fn,j is the identified natural frequency of the modej, ξj denotes the identified damping ratio correspondingto the mode j, ϕj = eiθj ρj represents the identifiedmodal vector of the normal mode j, ρj is the identifiedvector of relative modal amplitudes of the mode j, and θjrepresents the relative phase matrix of the mode j. Figures11, 12, and 13 respectively show the first, second, and thirdmode shapes identified by using the computational proceduredeveloped in the paper. It is important to note that the

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

Degrees of freedom

Firs

t mod

e m

agni

tude

s

Phase = 0

Phase = −0.0186

Phase = −0.1053

Natural frequency = 1.934 (Hz), damping ratio = 3.95 (%)

Fig. 11. Identified first mode shape.

modal parameters of the identified continuous-time state-space model are consistent with those obtained by using thepreliminary lumped parameter model. Since the identifiedmodal damping is small, the identified mode shapes areapproximately in phase or out of phase. Therefore, thehypothesis of proportional damping can be assumed and theproposed method for identifying the proportional damping

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Degrees of freedom

Sec

ond

mod

e m

agni

tude

s

Phase = 0

Phase = −0.0365

Phase = −3.17


Fig. 12. Identified second mode shape.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

Degrees of freedom

Thi

rd m

ode

mag

nitu

des

Phase = 0

Phase = −3.144

Phase = −0.0296


Fig. 13. Identified third mode shape.

coefficients can be applied to yield:

α = 0.9751, β= −2.8815 · 10−4 (21)

where α and β denote the identified proportional dampingcoefficients. Furthermore, a second-order physical model canbe constructed from the identified mathematical sate-spacerepresentation by using the algorithm discussed in the paperto yield:

M =

0.3871 −0.0156 0.0011−0.0156 0.2621 −0.03230.0011 −0.0323 0.3790

(22)

K =

534.061 −221.931 25.696−221.931 614.564 −432.792

25.696 −432.792 422.005

(23)

R =

9.960 −4.823 1.018−4.823 12.480 −8.3661.018 −8.366 7.017

(24)

where M is the identified mass matrix, K denotes theidentified stiffness matrix, and R represents the identifieddamping matrix. An improved estimation of the identifieddamping matrix R can be obtained by using the proportional



______________________________________________________________________________________

damping assumption and employing the identified propor-tional damping coefficients α and β as well as the identifiedmass and stiffness matrices M and K as follows:

R = αM+βK =

0.22356 0.0487 −0.00630.0487 0.0785 0.0932−0.0063 0.0932 0.2480

(25)

The identified second-order mechanical model of the flex-ible structure will be used in the paper for designing aneffective control strategy for reducing the system vibrationsinduced by external excitations [77]–[80].

VI. CONTROL DEVELOPMENT

In this section, the development and the implementation ofthe control scheme for reducing the structural vibration of thethree-story building system are described. For this purpose,the Linear Quadratic Gaussian (LQG) control approach isemployed by using the identified second-order mechanicalmodel of the flexible structure. Thus, the design of the controlsystem based on the Linear Quadratic Gaussian (LQG) regu-lation methodology is performed by combining the identifiedsecond-order model of the vibrating structure with a linearlumped parameter model of the actuated pendulum thatworks as an actively controlled inertial-based vibration ab-sorber. Therefore, the resulting mechanical model used to de-sign the control system includes n2 = 4 degrees of freedomcontained in a new generalized coordinate vector x(t) givenby x(t) =

[x1(t) x2(t) x3(t) ϕ(t)

]T, where x1(t),

x2(t), and x3(t) represent the linear displacements of thesystem first, second, and third floors, respectively, while ϕ(t)identifies the angular displacement of the actively controlledpendulum that serves as an inertial-based vibration absorber.By using the analytical methods of classical mechanics,the mechanical model of the three-story shear building sys-tem with the pendulum hinged on the third floor can bemathematically expressed as Mx(t) + Rx(t) + Kx(t) =B2,eue(t)+B2,cuc(t), where x(t) is the system generalizedacceleration vector, x(t) represents the system generalizedvelocity vector, x(t) denotes the system generalized coor-dinate vector, M is the system mass matrix, R representsthe system damping matrix, K denotes the system stiffnessmatrix, ue(t) is the vector of uncontrollable inputs, uc(t) isthe vector of controllable inputs, B2,e is a Boolean matrixthat identifies the locations of the uncontrollable inputs, andB2,c is a Boolean matrix that identifies the locations of thecontrollable inputs. The system mass, damping, and stiffnessmatrices of the flexible structure combined with the activelycontrolled inertial-based vibration absorber are respectivelydefined as follows:

M = BTx MBx + BT

4 M4B4

K = BTx KBx + BT

4 K4B4

R = BTx RBx + BT

4 R4B4

(26)

where M, R, and K denote respectively the identifiedmass, damping, and stiffness matrices, while M4, R4, andK4 represent respectively the mass, damping, and stiffnessmatrices associated with the actively controlled pendulum

given by:

M4 =

[m4 m4L4

m4L4 m4L24 + Izz,4

](27)

K4 = m4gL4, R4 = 0 (28)

where g = 9.81 (m/s2) represents the gravity acceleration,

L4 = 8.25 · 10−2 (m) is half the length of the pendu-lum, m4 = 0.083 (kg) denotes the pendulum mass, andIzz,4 = 8.32 · 10−4 (kg · m2) identifies the mass momentof inertia of the pendulum referred to its center of mass.The matrices Bx and B4, on the other hand, are appropriateBoolean matrices that serve for assembling the mechanicalmodel of the flexible structure with the actively controlledpendulum. For the three-story shear building system, thevector of uncontrollable inputs ue(t) contains an externalexcitation force applied on the first floor of the three-storyshear building system, whereas the vector of controllableinputs uc(t) includes a control torque applied on the pen-dulum that works as a vibration absorber. The Booleanmatrices B2,e and B2,c respectively identify the locationsof the input actions ue(t) and uc(t). In the experimentaltesting, the uncontrollable input vector ue(t) is equal to anexternal force Fe(t) defined by the superposition of threeharmonic components having considerable amplitudes andcharacterized by a set of excitation frequencies close tothe first three natural frequencies of the flexible structure.Furthermore, a noise excitation source is considered inorder to simulate the earthquake in the worst case scenarioin which external excitation frequencies are close to thesystem natural frequencies. Therefore, the external forceFe(t) considered as an uncontrollable input is defined asFe(t) = F0,1 sin(2πf1t)+F0,2 sin(2πf2t)+F0,3 sin(2πf3t),where F0,1 = 0.1 (N), F0,2 = 0.1 (N), and F0,3 = 0.1 (N)are the amplitudes of the first, second, and third harmonicforces, respectively, while f1 = 1.9 (Hz), f2 = 5.7 (Hz),and f3 = 8.8 (Hz) represent the frequencies of the first,second, and third harmonic forces, respectively. The externalexcitation force can be measured using a load cell collocatedbetween the shaker and the stinger connected with the firstfloor of the flexible structure. On the other hand, the measur-able output variables of the three-story shear building systemare the accelerations of the three floors and the angulardisplacement of the actively controlled pendulum. Theseoutput variables are included in a output vector y(t) givenby y(t) =

[x1(t) x2(t) x3(t) ϕ(t)

]T. Consequently,

the set of measurement equations can be readily writtenas y(t) = Cdx(t) + Cvx(t) + Cax(t), where Cd is thegeneralized displacement influence matrix on the measuredoutput, Cv is the generalized velocity influence matrix onthe measured output, and Ca is the generalized accelerationinfluence matrix on the measured output. Considering atime span equal to Ts = 64 (s) and a time step equalto ∆t = 3.125 · 10−2 (s), an optimal state estimator andan optimal controller can be obtained by using the LQGapproach leading to the following set of discrete-time state-space equations:

z(k + 1) = Az(k) + Beue(k) + BcF∞z(k)+K∞ (y(k)− y(k))

y(k) = Cz(k) + Deue(k) + DcF∞z(k)

(29)



______________________________________________________________________________________

where z(k) is the estimated discrete-time state vector andy(k) denotes the estimated discrete-time measurement vec-tor. The discrete-time dynamic model represented in thestate-space given by the set of matrices A, Be, Bc, C,De, and Dc can be easily derived from the mechanicalmodel of the flexible structure combined with the activelycontrolled inertial-based vibration absorber. Moreover, thematrices K∞ and F∞ represent respectively a discrete-time infinite-horizon Kalman filter gain and a discrete-timeinfinite-horizon optimal feedback gain which can be readilycomputed by using the LQG method. In order to analyze thedynamic behavior of the three-story shear building systemwith and without the action of the control system, the controlactuator is deactivated in the time range between t = 0 (s)and t = 32 (s), while the control actuator is activatedin the time range between t = 32 (s) and t = 64 (s).Figures 14, 15, and 16 show, respectively, the estimateddisplacements of the first, second, and third floors of theflexible structure resulting from the implementation of thecontrol system. Figure 17 shows the control torque used

Fig. 14. Estimated displacement of the first floor.

Fig. 15. Estimated displacement of the second floor.

as a controllable input signal for the mechanical system

Fig. 16. Estimated displacement of the third floor.

that forms the activelly controlled inertial-based vibrationabsorber. It is apparent from the estimated displacements

Fig. 17. Control torque

shown in figures 14, 15, and 16 that the action of the controlsystem allows for obtaining a large amplitude reduction ofthe system mechanical vibrations. In fact, the reduction ofthe system vibrations can be estimated quantitatively bycomparing the maximum amplitude values of the systemdisplacements with and without the action of the vibrationabsorber as follows:

(xu1,max − xc1,max

)/xu1,max = 74.82 (%)(

xu2,max − xc2,max

)/xu2,max = 63.91 (%)(

xu3,max − xc3,max

)/xu3,max = 58.40 (%)

(30)

where xu1,max, xu2,max, and xu3,max denote the maximum ab-solute values of the estimated displacements when there is nocontrol action, whereas xc1,max, xc2,max, and xc3,max representthe maximum absolute values of the estimated displacementswhen the actively controlled inertial-based vibration absorberacts on the three-story shear building. It can be noticed



______________________________________________________________________________________

that the numerical results found by using the state estimatorand employed for assessing the amplitude reductions of themechanical vibrations of the flexible structure are consistentwith the system actual dynamic behavior observed in theexperimental testing [81], [82].

VII. SUMMARY AND CONCLUSIONS

The principal topics of interest for the research of theauthors are nonlinear control, multibody dynamics, andsystem identification [83]–[86]. Thus, the research effortsof the authors are devoted to the development of newoptimal control actions for dynamical systems, new methodsfor obtaining accurate analytical modelling of rigid-flexiblemultibody mechanical systems, and new numerical param-eter identification approaches based on experimental data[87]–[90]. This investigation deals with the development ofa new actively controlled inertial-based vibration absorbersuitable for controlling the mechanical vibrations of flexiblestructures. To this end, a system identification method basedon the time domain is developed in the paper. For thispurpose, the set of the system Markov parameters is usedin the paper for obtaining from experimental data the modalparameters of structural systems and, at the same time, inorder to derive reliable first-order state-space and second-order mechanical models driven from input-output measure-ments. Furthermore, a least square estimation method forimproving the estimation of the system structural dampingis also proposed in this work. Subsequently, a methodologyfor reducing the system mechanical vibrations induced byexternal excitation sources by using an actively controlledinertial-based vibration absorber is developed in the paper.The proposed inertial-based vibration absorber is realizedby using an actively controlled pendulum system mountedon the third floor of the flexible structure. This device isprogrammed with both an optimal feedback controller andan optimal state observer. The optimal closed-loop controllerand the optimal state estimator were designed by using theLinear Quadratic Gaussian (LQG) algorithm. In order toachieve this goal, the dynamic parameters obtained from theapplication of the system identification numerical proceduredeveloped in this work were used. Moreover, the proposedapproach is corroborated by means of dynamic simulationscarried out by using a computer code developed in MATLABand is experimentally verified employing a simple test rig.In particular, the analytical derivation and the experimentaltesting of the active control strategy developed in this workfor reducing the vibrations of flexible structures are carriedout in two steps. In the first step of the analysis, theflexible structure is modeled as a three-story shear buildingsystem mathematically represented by using a three-degree-of-freedom lumped parameter model obtained by means ofthe analytical methods of classical mechanics. Once thatan accurate mechanical model was obtained in the firststep of the present analysis, an optimal feedback controllerand an optimal state estimator were designed in the secondpart of this work considering the identified state-space andconfiguration-space dynamic models obtained by using thesystem identification approach elaborated in the paper. Nu-merical results and experimental measurements showed thatthe action of the actively controlled inertial-based vibrationabsorber leads to a considerable reduction of the vibrations

of the flexible structure considered as the case study. Thenumerical and experimental results obtained in this paper alsodemonstrated that the use of applied system identificationnumerical techniques can considerably facilitate the develop-ment of effective control systems for reducing the externallyinduced vibrations of machines, mechanisms, and structures.Future research will be devoted to the experimental testing ofthe proposed methodology in the case of the active vibrationcontrol problem of complex three-dimensional mechanicalsystems.

REFERENCES

[1] L. Ljung, “Perspectives on System Identification,” Annual Reviews inControl, vol. 34, no. 1, pp. 1-12, 2010.

[2] O. Nelles, Nonlinear System Identification: From Classical Approachesto Neural Networks and Fuzzy Models, Springer Science and BusinessMedia, 2013.

[3] T. Sands, “Nonlinear-Adaptive Mathematical System Identification,”Computation, vol. 5, no. 4, 47, 2017.

[4] B. Peeters and G. De Roeck, “Stochastic System Identification forOperational Modal Analysis: a Review,” Journal of Dynamic Systems,Measurement, and Control, vol. 123, no. 4, pp. 1-9, 2001.

[5] E. Reynders, “System Identification Methods for (Operational) ModalAnalysis: Review and Comparison,” Archives of Computational Meth-ods in Engineering, vol. 19, no. 1, pp. 51-124, 2012.

[6] Y. Gao, F. Villecco, M. Li, and W. Song, “Multi-Scale PermutationEntropy based on Improved LMD and HMM for Rolling BearingDiagnosis,” Entropy, vol. 19, no. 4, 176, 2017.

[7] F. Villecco and A. Pellegrino, “Evaluation of Uncertainties in theDesign Process of Complex Mechanical Systems,” Entropy, vol. 19,no. 9, 475, 2017.

[8] F. Villecco and A. Pellegrino, “Entropic Measure of Epistemic Uncer-tainties in Multibody System Models by Axiomatic Design,” Entropy,vol. 19, no .7, 291, 2017.

[9] R. Castro-Garcia, K. Tiels, O. M. Agudelo, and J. A. Suykens, “Ham-merstein System Identification through Best Linear ApproximationInversion and Regularisation,” International Journal of Control, pp.1-17, 2017.

[10] F. P. Carli, T. Chen, and L. Ljung, “Maximum Entropy Kernels forSystem Identification,” IEEE Transactions on Automatic Control, vol.62, no. 3, pp. 1471-1477, 2017.

[11] T. A. Catanach and J. L. Beck, “Bayesian System Identification usingAuxiliary Stochastic Dynamical Systems,” International Journal ofNon-Linear Mechanics, vol. 94, pp. 72-83, 2017.

[12] D. Belmonte, M. D. L. Dalla Vedova, and P. Maggiore, “Aircraft FlapControl System: Proposal of a Simulink Test Bench for EvaluatingInnovative Asymmetry Monitoring and Control Techniques,” Inter-national Journal of Mathematical Models and Methods in AppliedSciences, vol. 10, pp. 51-61, 2016.

[13] X. Hong, R. J. Mitchell, S. Chen, C. J. Harris, K. Li, and G. W. Irwin,“Model Selection Approaches for Non-linear System Identification: aReview,” International Journal of Systems Science, vol. 39, no. 10, pp.925-946, 2008.

[14] H. Unbehauen and G. P. Rao, “A Review of Identification inContinuous-time Systems,” Annual Reviews in Control, vol. 22, pp.145-171, 1998.

[15] R. De Farias Campos, E. Couto, J. De Oliveira, and A. Nied, “On-Line Parameter Identification of an Induction Motor With Closed-LoopSpeed Control Using the Least Square Method,” Journal of DynamicSystems, Measurement, and Control, vol. 139, no. 7, 071010, 2017.

[16] R. Bighamian, H. R. Mirdamadi, and J. O. Hahn, “Damage Iden-tification in Collocated Structural Systems Using Structural MarkovParameters,” Journal of Dynamic Systems, Measurement, and Control,vol. 137, no. 4, 041001, 2015.

[17] M. D. L. Dalla Vedova and P. C. Berri, “Interactions Problems inHydraulic Systems: Proposal of a New Nonlinear Speed Control LawMethod,” International Journal of Mechanics and Control, vol. 18, no.1, pp. 39-44, 2017.

[18] I. Palomba, D. Richiedei, and A. Trevisani, “Kinematic State Esti-mation for Rigid-Link Multibody Systems by means of NonlinearConstraint Equations,” Multibody System Dynamics, vol. 40, no. 1,pp. 1-22, 2017.

[19] I. Palomba, D. Richiedei, and A. Trevisani, “Two-Stage Approachto State and Force Estimation in Rigid-Link Multibody Systems,”Multibody System Dynamics, vol. 39, no. 1-2, pp. 115-134, 2017.

[20] T. Sands, “Nonlinear-Adaptive Mathematical System Identification,”Computation, vol. 5, no. 4, 47, 2017.



______________________________________________________________________________________

[21] M. Cooper, P. Heidlauf, and T. Sands, “Controlling Chaos - ForcedVan Der Pol Equation,” Mathematics, vol. 5, no. 4, 70, 2017.

[22] F. L. Lewis, D. Vrabie, and V. L. Syrmos, Optimal Control, John Wileyand Sons, 2012.

[23] H. K. Khalil, Nonlinear Control, Prentice Hall, New Jersey, 2014.[24] C. Hermosilla, R. Vinter, and H. Zidani, “HamiltonJacobiBellman

Equations for Optimal Control Processes with Convex State Con-straints,” Systems and Control Letters, vol. 109, pp. 30-36, 2017.

[25] F. Y. Wang, H. Zhang, and D. Liu, “Adaptive Dynamic Programming:An Introduction,” IEEE computational intelligence magazine, vol. 4,no. 2, pp. 39-47, 2009.

[26] D. V. Rao and N. K. Sinha, “A Sliding Mode Controller for AircraftSimulated Entry into Spin,” Aerospace Science and Technology, vol.28, no. 1, pp. 154-163, 2013.

[27] X. Lu, Z. Liu, J. Zhang, H. Wang, Y. Song, and F. Duan, “Prior-Information-Based Finite-Frequency H-Infinity Control for ActiveDouble Pantograph in High-Speed Railway,” IEEE Transactions onVehicular Technology, vol. 66, no. 10, 7922629, pp. 8723-8733, 2017.

[28] D. Tomas, J. A. Lozano-Galant, G. Ramos, and J. Turmo, “StructuralSystem Identification of Thin Web Bridges by Observability Tech-niques Considering Shear Deformation,” Thin-Walled Structures, vol.123, pp. 282-293, 2018.

[29] J. N. Juang and M. Q. Phan, Identification and Control of MechanicalSystems, Cambridge University Press, 2001.

[30] T. Katayama, Subspace Methods for System Identification, SpringerScience and Business Media, 2006.

[31] M. Sharifzadeh, A. Farnam, A. Senatore, F. Timpone, and A. Akbari,“Delay-Dependent Criteria for Robust Dynamic Stability Control ofArticulated Vehicles,” in International Conference on Robotics in Alpe-Adria Danube Region, Springer, Cham, pp. 424-432, 2017.

[32] M. Sharifzadeh, F. Timpone, A. Farnam, A. Senatore, and A. Ak-bari, “Tyre-road adherence conditions estimation for intelligent vehi-cle safety applications,” in Advances in Italian Mechanism Science,Springer, Cham., pp. 389-398, 2017.

[33] G. Mercere, I. Markovsky, and J. A. Ramos, “Innovation-basedSubspace Identification in Open-and Closed-loop,” in IEEE 55thConference on Decision and Control (CDC), pp. 2951-2956, 2016.

[34] M. D. L. Dalla Vedova, P. Maggiore, L. Pace, and S. Romeo, “Proposalof a Model Based Fault Identification Neural Technique for More-Electric Aircraft Flight Control EM Actuators,” WSEAS TransactionsOn Systems, vol. 15, pp. 19-27, 2016.

[35] A. Alenany, G. Mercere, and J. A. Ramos, “Subspace Identification of2-D CRSD Roesser Models With Deterministic-Stochastic Inputs: AState Computation Approach,” IEEE Transactions on Control SystemsTechnology, vol. 25, no. 3, pp. 1108-1115, 2017.

[36] J. A. Ramos and G. Mercere, “Subspace Algorithms for IdentifyingSeparable-in-denominator 2D Systems with Deterministic-stochasticInputs,” International Journal of Control, vol. 89, no. 12, pp. 2584-2610, 2016.

[37] A. Ruggiero, M. C. De Simone, D. Russo, and D. Guida, “Soundpressure measurement of orchestral instruments in the concert hall ofa public school,” International Journal of Circuits, Systems and SignalProcessing, vol. 10, pp. 75-812, 2016.

[38] E. Ikonen and K. Najim, Advanced Process Identification and Control,CRC Press, 2001.

[39] T. Liu and F. Gao, Industrial Process Identification and Control De-sign: Step-test and Relay-experiment-based Methods, Springer Scienceand Business, 2011.

[40] R. F. Stengel, Optimal Control and Estimation, Courier Corporation,2012.

[41] S. M. Da Conceicao, C. H. Vasques, G. L. C. M. De Abreu, J. V.Lopes, and M. J. Brennan, “Experimental Identification and Control ofa Cantilever Beam Using ERA/OKID with a LQR Controller,” Journalof Control, Automation and Electrical Systems, vol. 25, no. 2, pp. 161-173, 2014.

[42] M. C. De Simone, S. Russo, Z. B. Rivera, and D. Guida, “MultibodyModel of a UAV in Presence of Wind Fields,” in ICCAIRO 2017 -International Conference on Control, Artificial Intelligence, Roboticsand Optimization, Prague, Czech Republic, 20-22 May, 2017.

[43] A. Concilio, M. C. De Simone, Z. B. Rivera, and D. Guida, “ANew Semi-Active Suspension System for Racing Vehicles,” FMETransactions, vol. 45, no. 4, pp. 578-584, 2017.

[44] A. Quatrano, M. C. De Simone, Z. B. Rivera, and D. Guida, “De-velopment and Implementation of a Control System for a RetrofittedCNC Machine by using Arduino,” FME Transactions, vol. 45, no. 4,pp. 565-571, 2017.

[45] S. Strano and M. Terzo, “Actuator Dynamics Compensation for Real-time Hybrid Simulation: An Adaptive Approach by means of aNonlinear Estimator,” Nonlinear Dynamics, vol. 85, no. 4, pp. 2353-2368, 2016.

[46] S. Strano and M. Terzo, “Accurate State Estimation for a HydraulicActuator via a SDRE Nonlinear Filter,” Mechanical Systems andSignal Processing, vol. 75, pp. 576-588, 2016.

[47] R. Russo, S. Strano, and M. Terzo, “Enhancement of Vehicle Dynamicsvia an Innovative Magnetorheological Fluid Limited Slip Differential,”Mechanical Systems and Signal Processing, vol. 70, pp. 1193-1208,2016.

[48] S. Strano and M. Terzo, “A SDRE-based Tracking Control for aHydraulic Actuation System,” Mechanical Systems and Signal Pro-cessing, vol. 60, pp. 715-726, 2015.

[49] S. Strano and M. Terzo, “A Multipurpose Seismic Test Rig Control viaa Sliding Mode Approach,” Structural Control and Health Monitoring,vol. 21, no. 8, pp. 1193-1207, 2014.

[50] S. Pagano, R. Russo, S. Strano, and M. Terzo, “Non-linear Modellingand Optimal Control of a Hydraulically Actuated Seismic Isolator TestRig,” Mechanical Systems and Signal Processing, vol. 35, no. 1, pp.255-278, 2013.

[51] A. Ruggiero, S. Affatato, M. Merola, and M. C. De Simone, “FEMAnalysis of Metal on UHMWPE Total Hip Prosthesis During NormalWalking Cycle,” in Proceedings of the XXIII Conference of The ItalianAssociation of Theoretical and Applied Mechanics (AIMETA 2017),Salerno, Italy, 4-7 September, 2017.

[52] M. C. De Simone and D. Guida, “On the Development of a Low CostDevice for Retrofitting Tracked Vehicles for Autonomous Navigation,”in Proceedings of the XXIII Conference of The Italian Association ofTheoretical and Applied Mechanics (AIMETA 2017), Salerno, Italy,4-7 September, 2017.

[53] M. C. De Simone, Z. B. Rivera, D. Guida, “Finite Element Analysison Squeal-noise in Railway Applications,” FME Transactions, vol. 46,no. 1, pp. 93-100, 2017.

[54] T. Lauss, S. Oberpeilsteiner, W. Steiner, and K. Nachbagauer, “Thediscrete adjoint method for parameter identification in multibodysystem dynamics,” Multibody System Dynamics, vol. 42, no. 4, pp.397-410, 2017.

[55] C. M. Pappalardo, Z. Zhang, and A. A. Shabana, “Use of IndependentVolume Parameters in the Development of New Large DisplacementANCF Triangular Plate/Shell Elements,” Nonlinear Dynamics, vol. 91,no. 4, pp. 2171-2202, 2018.

[56] Z. Tian, S. Li, Y. Wang, and B. Gu, “Priority Scheduling of NetworkedControl System Based on Fuzzy Controller with Self-tuning ScaleFactor,” IAENG International Journal of Computer Science, vol. 44,no. 3, pp. 308-315, 2017.

[57] S. El Kafhali, and M. Hanini, “Stochastic Modeling and Analysis ofFeedback Control on the QoS VoIP Traffic in a single cell IEEE 802.16e Networks,” IAENG International Journal of Computer Science, vol.44, no. 1, pp. 19-28, 2017.

[58] T. Taniguchi and M. Sugeno, “Trajectory Tracking Controls for Non-holonomic Systems Using Dynamic Feedback Linearization Basedon Piecewise Multi-Linear Models,” IAENG International Journal ofApplied Mathematics, vol. 47, no. 3, pp. 339-351, 2017.

[59] X. Zhang and D. Yuan, “A Niche Ant Colony Algorithm for ParameterIdentification of Space Fractional Order Diffusion Equation,” IAENGInternational Journal of Applied Mathematics, vol. 47, no. 2, pp. 197-208, 2017.

[60] C. M. Pappalardo, “A Natural Absolute Coordinate Formulation forthe Kinematic and Dynamic Analysis of Rigid Multibody Systems,”Nonlinear Dynamics, vol. 81, no. 4, pp. 1841-1869, 2015.

[61] C. M. Pappalardo, M. Wallin, and A. A. Shabana, “A NewANCF/CRBF Fully Parametrized Plate Finite Element,” ASME Journalof Computational and Nonlinear Dynamics, vol. 12, no. 3, pp. 1-13,2017.

[62] C. M. Pappalardo, Z. Yu, X. Zhang, and A. A. Shabana, “RationalANCF Thin Plate Finite Element,” ASME Journal of Computationaland Nonlinear Dynamics, vol. 11, no. 5, pp. 1-15, 2016.

[63] D. Guida, F. Nilvetti, and C. M. Pappalardo, “Parameter Identificationof a Two Degrees of Freedom Mechanical System,” InternationalJournal of Mechanics, vol. 3, no. 2, pp. 23-30, 2009.

[64] C. M. Pappalardo and D. Guida, “On the Lagrange Multipliers ofthe Intrinsic Constraint Equations of Rigid Multibody MechanicalSystems,” Archive of Applied Mechanics, vol. 88, no. 3, pp. 419-451,2018.

[65] D. Guida and C. M. Pappalardo, “Sommerfeld and Mass ParameterIdentification of Lubricated Journal Bearing,” WSEAS Transactions onApplied and Theoretical Mechanics, vol. 4, no. 4, pp. 205-214, 2009.

[66] C. M. Pappalardo and D. Guida, “On the use of Two-dimensional EulerParameters for the Dynamic Simulation of Planar Rigid MultibodySystems,” Archive of Applied Mechanics, vol. 87, no. 10, pp. 1647-1665, 2017.

[67] M. A. Jami’in and E. Julianto, “Hierarchical Algorithms of Quasi-Linear ARX Neural Networks for Identification of Nonlinear Systems,”Engineering Letters, vol. 25, no. 3, pp. 321-328, 2017.



______________________________________________________________________________________

[68] J. Zhang, L. Yu, and L. Ding, “Velocity Feedback Control of SwingPhase for 2-DoF Robotic Leg Driven by Electro-hydraulic ServoSystem,” Engineering Letters, vol. 24, no. 4, pp. 378-383, 2016.

[69] Q. W. Guo, W. D. Song, M. Gao, and D. Fang, “Advanced GuidanceLaw Design for Trajectory-Corrected Rockets with Canards underSingle Channel Control,” Engineering Letters, vol. 24, no. 4, pp. 469-477, 2016.

[70] H. Liu and Y. Li, “Safety Evaluation of a Long-Span Steel Trestlewith an Extended Service Term Age in a Coastal Port Based onIdentification of Modal Parameters,” Engineering Letters, vol. 24, no.1, pp. 84-92, 2016.

[71] I. I. Lazaro, A. Alvarez, and J. Anzurez, “The Identification ProblemApplied to Periodic Systems Using Hartley Series,” EngineeringLetters, vol. 21, no. 1, pp. 36-43, 2013.

[72] Pappalardo, C. M., and Guida, D., 2018, “System IdentificationAlgorithm for computing the Modal Parameters of Linear MechanicalSystems”, Machines, 6(2), 12.

[73] C. M. Pappalardo, M. D. Patel, B. Tinsley, and A. A. Shabana,“Contact Force Control in Multibody Pantograph/Catenary Systems,”Proceedings of the Institution of Mechanical Engineers, Part K:Journal of Multibody Dynamics, vol. 230, no. 4, pp. 307-328, 2016.

[74] D. Guida and C. M. Pappalardo, “Control Design of an ActiveSuspension System for a Quarter-Car Model with Hysteresis,” Journalof Vibration Engineering and Technologies, vol. 3, no. 3, pp. 277-299,2015.

[75] C. M. Pappalardo, T. Wang, and A. A. Shabana, “Development ofANCF Tetrahedral Finite Elements for the Nonlinear Dynamics ofFlexible Structures,” Nonlinear Dynamics, vol. 89, no. 4, pp. 2905-2932, 2017.

[76] D. Guida, F. Nilvetti, and C. M. Pappalardo, “Instability Induced byDry Friction,” International Journal of Mechanics, vol. 3, no. 3, pp.44-51, 2009.

[77] C. M. Pappalardo, T. Wang, and A. A. Shabana, “On the Formulationof the Planar ANCF Triangular Finite Elements,” Nonlinear Dynamics,vol. 89, no. 2, pp. 1019-1045, 2017.

[78] D. Guida, F. Nilvetti, and C. M. Pappalardo, “Dry Friction Influenceon Cart Pendulum Dynamics,” International Journal of Mechanics,vol. 3, no. 2, pp. 31-38, 2009.

[79] C. M. Pappalardo and D. Guida, “Use of the Adjoint Method for Con-trolling the Mechanical Vibrations of Nonlinear Systems,” Machines,vol. 6, no. 2, 19, 2018.

[80] C. M. Pappalardo and D. Guida, “On the Computational Methods forsolving the Differential-Algebraic Equations of Motion of MultibodySystems,” Machines, vol. 6, no. 2, 20, 2018.

[81] C. M. Pappalardo and D. Guida, “System Identification and Exper-imental Modal Analysis of a Frame Structure,” Engineering Letters,vol. 26, no. 1, pp. 56-68, 2018.

[82] M. C. De Simone and D. Guida, “Dry Friction Influence on Struc-ture Dynamics,” in COMPDYN 2015 - 5th ECCOMAS ThematicConference on Computational Methods in Structural Dynamics andEarthquake Engineering, pp. 4483-4491, 2015.

[83] D. Guida and C. M. Pappalardo, “A New Control Algorithm for ActiveSuspension Systems Featuring Hysteresis,” FME Transactions, vol. 41,no. 4, pp. 285-290, 2013.

[84] D. Guida and C. M. Pappalardo, “Forward and Inverse Dynamics ofNonholonomic Mechanical Systems,” Meccanica, vol. 49, no. 7, pp.1547-1559, 2014.

[85] C. M. Pappalardo and D. Guida, “Control of Nonlinear Vibrationsusing the Adjoint Method,” Meccanica, vol. 52, no. 11-12, pp. 2503-2526, 2017.

[86] C. M. Pappalardo and D. Guida, “Adjoint-based Optimization Proce-dure for Active Vibration Control of Nonlinear Mechanical Systems,”ASME Journal of Dynamic Systems, Measurement, and Control, vol.139, no. 8, 081010, 2017.

[87] M. C. De Simone, Z. B. Rivera, and D. Guida, “Obstacle AvoidanceSystem for Unmanned Ground Vehicles by using Ultrasonic Sensors,”Machines, vol. 6, no. 2, 18, 2018.

[88] C. M. Pappalardo, “Modelling Rigid Multibody Systems using Nat-ural Absolute Coordinates,” Journal of Mechanical Engineering andIndustrial Design, vol. 3, no. 1, pp. 24-38, 2014.

[89] M. C. De Simone and D. Guida, “Modal Coupling in Presence of DryFriction,” Machines, vol. 6, no. 1, 8, 2018.

[90] M. C. De Simone and D. Guida, “Identification and control of aUnmanned Ground Vehicle by using Arduino,” UPB Scientific Bulletin,Series D: Mechanical Engineering, vol. 80, no. 1, pp. 141-154, 2018.



______________________________________________________________________________________

development of a new inertial-based vibration …development of a new inertial-based vibration...

Documents