The paper deals with the quantiﬁcation of uncertainties in an elevator system, subjected to lateral vibration induced by the car guide rail irregularities, with and without a feedback control system. A mathematical model for the elevator system is derived,and a stochastic model for the rail proﬁle irregularities is constructed. This model of uncertainties describes the rail proﬁle as a random ﬁeld, which is represented by means of Karhunen-Lo `eve expansion. Polynomial chaos is employed to compute the propagation of uncertainties through the stochastic model, in order to evaluate the disturbance rejection properties and robustness of the closed-loop system. c/circlecopyrt2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017. Keywords: elevator systems; nonlinear dynamics; uncertainty quantiﬁcation; polynomial chaos. 1. Introduction Elevator systems are very important devices for the modern society, being essential in many places, from oﬃce buildings to industrial transportation of material and goods. A typical elevator consists of vertical cables and a car, that slides on lateral rails, which induce lateral vibrations due to the irregular shape of its proﬁle. The understandingof the impact of this vibration on a elevator dynamics, at its control, are essential for a safe operation [1]. In this sense, this paper deals with the quantiﬁcation of uncertainties in an elevator system, subjected to lateral vibrations induced by rail’s irregularities. In particular, it is interested in study the eﬀect of the uncertainties with andwithout a feedback control system for the horizontal vibration. The rest of this paper is organized as follows. Section 2 presents the modeling and a control strategy for the system dynamics. Numerical results are presented and discussed in section 3. Finally, the main conclusions and paths forfuture works are indicated in section 4. E-mail address: diego@lac.usp.br 1877-7058 c/circlecopyrt2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017.Available online at www.sciencedirect.com Procedia Engineering 00 (2017) 000–000 www.elsevier.com/locate/procedia X International Conference on Structural Dynamics, EURODYN 2017 On dynamic analysis and control of an elevator system using polynomial chaos and Karhunen-Lo `eve approaches Diego Col ´ona,, Americo Cunha Jrb, Stefan Kaczmarczykc, Jos´e M. Balthazard aUniversity of S˜ ao Paulo - USP , Escola Polit´ ecnica da USP , Laborat´ orio de Automa¸ c˜ ao e Controle (LAC -PTC), Brazil bRio de Janeito State University - UERJ, NUMERICO - Nucleus of Modeling and Experimentation with Computers, Brazil cThe University of Northampton, Faculty of Arts, Science and Technology, UK dTechnological Institute of Aeronautics - ITA, Graduate Program in Aeronautics and Mechanical Engineering, Brazil Abstract The paper deals with the quantiﬁcation of uncertainties in an elevator system, subjected to lateral vibration induced by the car guide rail irregularities, with and without a feedback control system. A mathematical model for the elevator system is derived,and a stochastic model for the rail proﬁle irregularities is constructed. This model of uncertainties describes the rail proﬁle as a random ﬁeld, which is represented by means of Karhunen-Lo `eve expansion. Polynomial chaos is employed to compute the propagation of uncertainties through the stochastic model, in order to evaluate the disturbance rejection properties and robustness of the closed-loop system. c/circlecopyrt2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017. Keywords: elevator systems; nonlinear dynamics; uncertainty quantiﬁcation; polynomial chaos. 1. Introduction Elevator systems are very important devices for the modern society, being essential in many places, from oﬃce buildings to industrial transportation of material and goods. A typical elevator consists of vertical cables and a car,that slides on lateral rails, which induce lateral vibrations due to the irregular shape of its proﬁle. The understandingof the impact of this vibration on a elevator dynamics, at its control, are essential for a safe operation [1]. In this sense, this paper deals with the quantiﬁcation of uncertainties in an elevator system, subjected to lateral vibrations induced by rail’s irregularities. In particular, it is interested in study the eﬀect of the uncertainties with andwithout a feedback control system for the horizontal vibration. The rest of this paper is organized as follows. Section 2 presents the modeling and a control strategy for the system dynamics. Numerical results are presented and discussed in section 3. Finally, the main conclusions and paths for future works are indicated in section 4. E-mail address: diego@lac.usp.br 1877-7058 c/circlecopyrt2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017.Available online at www.sciencedirect.com Procedia Engineering 00 (2017) 000–000 www.elsevier.com/locate/procedia X International Conference on Structural Dynamics, EURODYN 2017 On dynamic analysis and control of an elevator system using polynomial chaos and Karhunen-Lo `eve approaches Diego Col ´ona,, Americo Cunha Jrb, Stefan Kaczmarczykc, Jos´e M. Balthazard aUniversity of S˜ ao Paulo - USP , Escola Polit´ ecnica da USP , Laborat´ orio de Automa¸ c˜ ao e Controle (LAC -PTC), Brazil bRio de Janeito State University - UERJ, NUMERICO - Nucleus of Modeling and Experimentation with Computers, Brazil cThe University of Northampton, Faculty of Arts, Science and Technology, UK dTechnological Institute of Aeronautics - ITA, Graduate Program in Aeronautics and Mechanical Engineering, Brazil Abstract The paper deals with the quantiﬁcation of uncertainties in an elevator system, subjected to lateral vibration induced by the car guide rail irregularities, with and without a feedback control system. A mathematical model for the elevator system is derived,and a stochastic model for the rail proﬁle irregularities is constructed. This model of uncertainties describes the rail proﬁle as a random ﬁeld, which is represented by means of Karhunen-Lo `eve expansion. Polynomial chaos is employed to compute the propagation of uncertainties through the stochastic model, in order to evaluate the disturbance rejection properties and robustness of the closed-loop system. c/circlecopyrt2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017. Keywords: elevator systems; nonlinear dynamics; uncertainty quantiﬁcation; polynomial chaos. 1. Introduction Elevator systems are very important devices for the modern society, being essential in many places, from oﬃce buildings to industrial transportation of material and goods. A typical elevator consists of vertical cables and a car, that slides on lateral rails, which induce lateral vibrations due to the irregular shape of its proﬁle. The understandingof the impact of this vibration on a elevator dynamics, at its control, are essential for a safe operation [1]. In this sense, this paper deals with the quantiﬁcation of uncertainties in an elevator system, subjected to lateral vibrations induced by rail’s irregularities. In particular, it is interested in study the eﬀect of the uncertainties with andwithout a feedback control system for the horizontal vibration. The rest of this paper is organized as follows. Section 2 presents the modeling and a control strategy for the system dynamics. Numerical results are presented and discussed in section 3. Finally, the main conclusions and paths forfuture works are indicated in section 4. E-mail address: diego@lac.usp.br 1877-7058 c/circlecopyrt2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017. 1630 Diego Colón et al. / Procedia Engineering 199 (2017) 1629 –1634 2 Col´ on et al. /Procedia Engineering 00 (2017) 000–000 Fig. 1. Sketch of an idealized mechanical system which represents the elevator system. 2. Modeling elevator system dynamics 2.1. Deterministic mechanical-mathematical model A simpliﬁed schematics of the decoupled elevator system, which is the object of study of this work, is shown in Figure 1. This idealized mechanical system, subjected to lateral and vertical motions, is composed by a car attached to a vertically moving rope (cable), and is excited by an external load due to the irregularities in rail’s proﬁle. The rope’s linear speciﬁc mass is m, and it has the same linear velocity Vas the car. The rail’s proﬁle s(l) is a function of l, the vertical distance travelled by mass M, given by l=Vt. The coupling with the car is done through a linear spring, with constant k. The rope is under tension T(x,t) and its horizontal motion is described by the ﬁeld variable w(x,t), where xis the vertical coordinate. The rope length (from the traction sheave to the load) is represented byL(t). If we deﬁne the slow time factor /epsilon1=V/ω0L0, in the practical cases one has /epsilon1 << 1 and the slow time is deﬁned as τ=/epsilon1t. It is possible then separate the slow dynamic related to the motion of the car (and variation of the rope’s length) and the fast dynamic of the horizontal vibration. The time derivative of Lis related to the slow time derivative by ˙L=/epsilon1dL/dτ=/epsilon1L/prime. The dynamical behavior of this mechanical system evolves according to m(w tt+2/epsilon1L/primewxt+/epsilon12(L/prime)2wxx+/epsilon12L/prime/primewx)+γ(wt+/epsilon1L/primewx)−(Tw x)x=0, (1) which is supplemented by the boundary conditions w(0, t)=0 and M(wtt+2/epsilon1L/primewxt+/epsilon12(L/prime)2wxx+/epsilon12L/prime/primewx)x=L+k(w−s)x=L+(Tw x)x=L=0. (2) This mathematical model emulates the behavior of a distributed parameter system with a discrete mass (car) in one end, but with the length slowly variable. For details the reader is addressed to [1]. We search for solutions of the form w(x,t)=N/summationdisplay n=1Ψn(x,τ)qn(t), (3) where Ψn(x,τ)=sin(β n(τ)x), and the wave-number βn(τ), is a slowly variable in time, that satisfy the equation /parenleftbigg k−M mT(τ)β2 n/parenrightbigg sin(β nL(τ))+T(τ)β n(cos βnL(τ))=0 (4) Diego Colón et al. / Procedia Engineering 199 (2017) 1629 –1634 1631Col´ on et al. /Procedia Engineering 00 (2017) 000–000 3 forT(τ)=[M+(1/2)mL (τ)]g, a time-varying mean tension. After a substitution, we obtain a set of ordinary diﬀerential equation for the time-varying coeﬃcients of the form ¨qr(t)+2ςrωr(τ)˙qr(t)+ω2 r(τ)qr(t)=1 mr(τ)Ψr(L(τ))ks(t), r=1,2···,N, (5) where ςris the damping coeﬃcient for the rnatural mode, and mr(τ)=m/integraldisplayL 0Ψ2 r(x,τ)dx+MΨ2r(L). (6) Note that wave-numbers βnare related to the natural frequencies of the several modes by ωn(τ)=¯cβn(τ), where ¯c=√T(τ)/m represents the time-dependent velocity of the lateral wave. The set of equations in Eq.(5) are decoupled in the sense that the vibration modes are independent of one another. The equations are linear, but (slowly) time-varying. 2.2. Strategy of vibration control In order to damp the car’s horizontal vibration, a position sensor is used in order to measure the horizontal position y(t). Also, an actuator that applies an horizontal force f(y(t)) in the car must be used. This force is calculated in real time, according to y(t), as prescribed by a feedback control law. The mathematical model of the system, that is presented Eq.(5), is them modiﬁed by the inclusion of the force term f(y(t)) in the second equation, which turns it in a closed-loop system. We choose to implement a state feedback linear control law, in which f(y(t)) is calculated as a linear combination of the states in real time. Those states, that are not measured, are estimated by a state observer, based in the measurements of the position sensor y(t) [2]. 2.3. Karhunen-Lo` eve representation of random ﬁeld The function s(l) has a random nature, in reason of the rail’s proﬁle irregularities. In this way, it is modeled as a second-order, mean-square continuous, stationary random process {S(l),l∈[lmin,lmax]⊂R}, with zero-mean and autocovariance function given by Ks(l2−l1)=σ2 sexp(−|l2−l1|/lcorr), (7) where σ2 sandlcorrdenotes, respectively, the variance and the correlation length for the process S(l). In order to represent S(l), a truncated Karhunen-Lo `eeve expansion is employed, so that S(l)≈Nkl/summationdisplay n=1/radicalbig λnϕn(l)Yn, (8) where the integer Nklis the number of terms used in the expansion, the pairs (λn,ϕn)are solution of the Fredholm integral equation associated to Ks, and {Yn}N n=1is a family of, zero-mean, mutually uncorrelated random variables. In this case, there is an analytical solution to the eigenvalue problem, that can be found in [3]. 2.4. Polynomial chaos expansion Plugging S(l), deﬁned in section 2.3, into (5) we obtain a stochastic model of uncertainties for the mechanical system, deﬁned by a random system of diﬀerential equations of the form ˙X(t,Y)=F(X(t,Y),Y,t), (9) where tis the time variable, X(t,Y) is a stochastic vector of state, and Y=(Y1,···,YNkl)Tis a vector which collects the ﬁnite variance random variables which parametrize the random ﬁeld S(l). The polynomial chaos expansion of this stochastic vector of state is given by X(t,Y)=∞/summationdisplay i=0xi(t)φi(Y), (10) 1632 Diego Colón et al. / Procedia Engineering 199 (2017) 1629 –1634 4 Col´ on et al. /Procedia Engineering 00 (2017) 000–000 where {φi(Y)}∞ i=0is an orthogonal basis for the Hilbert space of the ﬁnite variance random variables. The coeﬃcients {xi(t)}∞ i=0are deterministic functions of time that must be found [4,5]. If the random variables are Gaussian, this basis is formed by Hermite polynomials, but it is also possible to do this expansion for other probability distributions and corresponding orthogonal polynomials, known as Wiener-Askey scheme [5]. There are two methods to ﬁnd the temporal coeﬃcients: 1) intrusive method; 2) non-intrusive method. The intrusive method is useful in some situations,as can be found in [5–9], on the other hand, it can represent a very heavy burden to the computational system. In the non-intrusive method, the expansion coeﬃcients are calculated by the following formula x i(tk)=E[X(tk,Y)φi(Y)] E[φi(Y)φi(Y)], (11) where the denominator is calculated by an explicit formula (the basis polynomials in the Hilbert space are known), and the numerator is calculated by solving an integral in the space RNklof the Nklrandom variables. A grid in this space is established and the original stochastic system is simulated for each point in this grid to obtain X(tk,Y). This must be done to each time value tin which we are interested [10]. If a grid is chosen such that there are nsample points, the total number of samples is nNkl, which grows very rapidly with Nkl. In order to avoid this problem, we use the Sparse Grid Integration, which consists in selecting convenient grids to RNklin which the number of point does not grow exponentially with the dimension, but in a much slower rate (depending on the smoothness of the integrand) [11]. The non-intrusive polynomial chaos approach has other advantages over the intrusive, as presented in [10,12,13]. 3. Numerical results In this section, the results of numerical simulation tests carried out to predict the dynamic behaviour of the elevator system with, and without control system are presented. The results demonstrate the eﬃcacy of the proposed control strategy. For random ﬁeld representation, it is adopted Nkl=2, which means that there will be two random parameters Y1andY2, that will be Gaussian and independent. The polynomial chaos expansion will use bi-dimensional Hermite polynomials, with maximum degree ﬁve. Only the ﬁrst vibration mode will be considered for control, and the plant state variables will be x1=q1andy1=˙q1. The damping ratio is will be considered ς1=0.4. The stiﬀness coeﬃcient for the car-rail interface is k=2083 N/m, and σs=7.07×10−4mm and lcorr=100 m. We suppose that that rope’s speciﬁc mass is m=0.65 kg/m, the total car’s weight (car +load) is M=3250 kg, the well height is 70 m, the car height is 3 .2 m and the total travel height is 60 m, which is also the maximum length of the rope L0. The vertical velocity of the elevator is V=6m/s and it is ascending in the simulations. This means that the rope has the slowly variable length L(τ)=(V//epsilon1)τ+L0, and /epsilon1=(V/ω0L0) is the slow time constant factor. The open loop equation is then /bracketleftBigg˙x1 ˙y1/bracketrightBigg =/bracketleftBigg01 −ω2 1−2ςω 1/bracketrightBigg/bracketleftBiggx1 y1/bracketrightBigg +/bracketleftBigg0 Ψ1(L(τ)) m1(τ)k/bracketrightBigg S(l;Y1,Y2)+/bracketleftBigg0 1/bracketrightBigg f (12) Figure 2 shows important parameters for the problem when the elevator is ascending with vertical velocity V=6 m/s. We can see that the velocity of the wave propagation does not change too much, but the wake number β1 approach inﬁnity, as well as the fundamental frequency, when L(τ) approaches zero. The amplitude of the forcing term in Eq.(12), that isΨr(L(τ)) mr(τ)k, also goes to inﬁnity, which means that the forcing term has growing inﬂuence as the elevator approaches the uppermost part. In Figure 3 it is presented the covariance surface for the case without the vibration control, and in Figure 4 it is presented the covariance surface for the case with vibration control. One can see that the magnitudes are reduced bymore than one order of magnitude, which means that the vibration control designed is eﬀective. The designed control law imposed as closed-loop poles −3,−4, which resulted in the state feedback gains k 1= 11.1978 and k2=6.2835. The state observer poles was imposed to be −6,−8, which is the double, in order to guarantee that the observer is more rapid than the plant dynamics. The resulting observer gains are h1=13.2835 and h2=37.6800. Diego Colón et al. / Procedia Engineering 199 (2017) 1629 –1634 1633Col´ on et al. /Procedia Engineering 00 (2017) 000–000 5 0 0.5 1 1.50102030405060L(τ) τ(s)0 0.5 1 1.5221.5221.6221.7221.8221.9222222.1222.2c(τ) τ(s)0 0.5 1 1.544.555.566.57x 10−3β1(τ) τ(s) 0 0.5 1 1.50.80.911.11.21.31.41.51.6ω1(τ) τ(s)0 0.5 1 1.5050100150200m1(τ) τ(s)0 0.5 1 1.5246810121416kΨ1(L)/m1(τ) τ(s) Fig. 2. Time-varying parameters for velocity V=6m/s. 00.20.40.60.811.21.4 00.511.5012345x 10−5 l1l2covariância Fig. 3. Covariance without vibration control. 4. Concluding remarks In this paper a model of an elevator system with a stochastic excitation originating from the guide rail proﬁle is presented. The excitation is represented as a random ﬁeld and with a truncated Karhunen-Lo `eve expansion being employed. Stochastic numerical simulations were conducted, with and without vibration control, by using polynomial chaos expansion, in its intrusive form, to compute the propagation of uncertainties. The covariance surfaces were obtained for both situations, and it was conﬁrmed that the control law designed is eﬀective in reducing the stochastic vibrations. For future works, it is intended to include an improved model for the vibration source and use diﬀerentcontrol techniques to compare its eﬃcacy. 1634 Diego Colón et al. / Procedia Engineering 199 (2017) 1629 –1634 6 Col´ on et al. /Procedia Engineering 00 (2017) 000–000 00.20.40.60.811.21.4 00.511.500.20.40.60.81x 10−6 l1l2covariance Fig. 4. Covariance with vibration control. Acknowledgments The authors are indebted to Brazilian agencies CNPq, CAPES, FAPESP, and FAPERJ for the ﬁnancial support given to this research.