Lyapunov-Based Control of the Sway Dynamics for Elevator Ropes

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1 MITSUBISHI ELECTRIC RESEARCH LABORATORIES Lyapunov-Based Contro of the Sway Dynamics for Eevator Ropes Benosman, M.; Fukui, D. TR24-67 June 24 Abstract In this work we study the probem of rope sway dynamics contro for eevator systems. We formuate this probem as a biinear contro probem and propose noninear controers based on Lyapunov theory, to stabiize the rope sway dynamics. We study the stabiity of the proposed controers, and test their performances on a numerica exampe. American Contro Conference (ACC), 24 This work may not be copied or reproduced in whoe or in part for any commercia purpose. Permission to copy in whoe or in part without payment of fee is granted for nonprofit educationa and research purposes provided that a such whoe or partia copies incude the foowing: a notice that such copying is by permission of Mitsubishi Eectric Research Laboratories, Inc.; an acknowedgment of the authors and individua contributions to the work; and a appicabe portions of the copyright notice. Copying, reproduction, or repubishing for any other purpose sha require a icense with payment of fee to Mitsubishi Eectric Research Laboratories, Inc. A rights reserved. Copyright c Mitsubishi Eectric Research Laboratories, Inc., 24 2 Broadway, Cambridge, Massachusetts 239

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3 Lyapunov-Based Contro of the Sway Dynamics for Eevator Ropes Mouhacine Benosman and Daiki Fukui Abstract In this work we study the probem of rope sway dynamics contro for eevator systems. We formuate this probem as a biinear contro probem and propose noninear controers based on Lyapunov theory, to stabiize the rope sway dynamics. We study the stabiity of the proposed controers, and test their performances on a numerica exampe. I. INTRODUCTION The growing demand for high-rise buidings motivates the probem of rope sway contro, which is very important in order to maintain a high safety eve of the eevator system. Indeed, even sight externa disturbances on the buiding, e.g. wind gust or earthquake, at such dimensions of structures can ead to important rope sway within the eevator shaft. Considering the ength of the ropes and their heavy weight, it is cear that the rope sway can damage the equipments that are instaed in the eevator shaft and can aso cause damages to the eevator shaft structure itsef, without mentioning the potentia danger caused for the eevator passengers. For these reasons, it is very important to be abe to contro the rope dynamics within the eevator shaft. Furthermore, due to cost constraints, it is preferabe to be abe to do so, with minimum actuation capabiities. Many papers have been dedicated to the probem of modeing and contro of ong eevator ropes [], [2], [3], [4], [5], [6]. In [6], a simpe mode of a cabe attached to an actuator at its free end is used to investigate the stiffening effect of the contro force on the cabe. An energy anaysis is used to tune an open-oop sinusoida force appied to the cabe. In [4], a scaed mode for high-rise, high-speed eevators is deveoped. The mode is used to anayze the infuence of the car motion profies on the atera vibrations of the eevator cabes. An active stiffness contro of the transverse vibrations of eevator ropes is presented in []. The Mouhacine Benosman (m benosman@ieee.org) is with Mitsubishi Eectric Research Laboratories, 2 Broadway Street, Cambridge, MA 239, USA. Daiki Fukui (Fukui.Daiki@ah.MitsubishiEectric.co.p) is with Advanced Technoogy R&D Center Mitsubishi Eectric Corp, 8- - Tsukaguchi Honmachi, Amagasaki City, Hyogo, JAPAN author propose a noninear moda feedback to drive an actuator puing on one end of the rope. The contro performance is investigated by numerica tests. In [5], the authors proposed a nove idea to dissipate the transversa energy of an eevator rope. The authors used a passive damper attached between the car and the rope. Numerica anaysis of the transverse motion average energy was conducted to find the optima vaue of the damper coefficient. In this work we propose to investigate the probem of eevators rope sway mitigation as a noninear contro probem. Foowing [], we use an active actuator to pu on one-side of the ropes. We show that in this case the mode of the eevator rope together with its actuator writes as a biinear mode (in the contro theory sense), and we use this biinear mode to deveop noninear Lyapunov-based feedback controers to stabiize the rope sway dynamics. We study the stabiity of the cosed-oop dynamics, and show the performances of these controers on a numerica exampe. The paper is organized as foows: We start the paper with some preiminaries in Section II. In Section III, we reca the mode of the system and underine its state-contro biinear form. Next, in Section IV, we present the main resuts of this work, namey, the noninear Lyapunov-based controers, together with their stabiity anaysis. Section V is dedicated to some numerica resuts. Finay, we concude the paper with a brief summary of the resuts in Section VI. II. NOTATIONS AND PRELIMINARIES Throughout the paper, R, R + denotes the set of rea, and the set of nonnegative rea numbers, respectivey. For x R N we define x = x T x, we denote by A i, i =,..., n, =,..., m the eements of the matrix A, and denote by sgn(.) the signum function. III. ELEVATOR ROPE MODELLING In this section we first introduce the infinite dimension mode, i.e partia differentia equation (PDE), of a moving hoist cabe, with non-homogenous boundary

4 Fig.. Schematic representation of an eevator shaft showing the different variabes used in the mode conditions. Secondy, to be abe to reduce the PDE mode to an ODE mode using a Gaerkin reduction method, we introduce a change of variabes and re-write the first PDE mode in a new coordinates, where the new PDE mode has zero boundary conditions. Let us first enumerate the assumptions under which our mode is vaid. - The eevator ropes are modeed within the framework of string theory. - The eevator car is modeed as a point mass. - The vibration in the second atera direction is not incuded. - The suspension of the car against its guide rais is assumed to be rigid. Under the previous assumption, foowing [3], [], the genera PDE mode of an eevator rope, depicted on Figure, is given by ρ( 2 t 2 + v 2 (t) 2 y 2 + 2v(t) y t + a y T(y, t) u(y,t) y y) u(y, t) ( + c p t + v(t) y )u(y, t) = () where u(y, t) is the atera dispacement of the rope. ρ is the mass of the rope per unit ength. T is the tension in the rope, which varies depending on which rope in the eevator system we are modeing, i.e. main rope, compensation rope, etc. c p is the damping coefficient of the rope per unit ength. v = (t) t is the eevator rope veocity, where : R R is a function (at east C 2 ) modeing the time-varying rope ength. a = 2 (t) t 2 is the eevator rope acceeration. The PDE () is associated with the foowing two boundary conditions: u(, t) = f (t) u((t), t) = f 2 (t) (2) where f (t) is the time varying disturbance acting on the rope at the eve of the machine room, due to externa disturbances, e.g. wind gust. f 2 (t) is the time varying disturbance acting at the eve of the car, due to externa disturbances. In this work we assume that the two boundary disturbances acting on the rope are reated via the reation: f 2 (t) = f (t)sin ( π(h ) ), H R (3) 2H where H is the height of the buiding. This expression is an approximation of the propagation of the boundary disturbance f aong the buiding structure, based on the ength, it eads to f 2 = f for a ength (which is expected), and a decreasing force aong the buiding unti is vanishes at = H, f 2 = (which makes sense, since the effect of any disturbance f, for exampe wind gusts, is expected to vanish at the bottom of the buiding). As we mentioned earier the tension of the rope T(y) depends on the type of the rope that we are deaing with. In the seque, we concentrate on the main rope of the eevator, the remaining ropes are modeed using the same steps by simpy changing the rope tension expression. For the case of the main rope, the tension is given by T(y, t) = (m e +ρ((t) y))(g a(t))+.5m cs g+u(t) (4) where g is the standard gravity constant, m e, M cs are the mass of the car and the compensating sheave, respectivey, and U(t) is the contro tension appied by an actuator attached to the compensation sheave (the same actuator pacement has been considered in []). Next, we reduce the PDE mode () to a more tractabe mode for contro, using a proection Gaerkin method or assumed mode approach, e.g. [7], [8]. To be abe to appy the assumed mode approach, et us first appy the foowing one-to-one change of coordinates to the equation () u(y, t) = w(y, t) + (t) y f (t) + y (t) (t) f 2(t) (5) One can easiy see that this change of coordinates impies trivia boundary conditions w(, t) = w((t), t) = (6) This change of coordinates is needed to write to origina PDE mode as an equivaent PDE with homogenous boundary conditions. To the best of our knowedge, it has not been proposed in previous work on eevator ropes modeing, and is newy introduced in this paper.

5 After some agebraic and integra manipuations, the PDE mode () writes in the new coordinates as ρ 2 w t + 2v(t)ρ 2 w 2 y t + ( ρv 2 T(y, t) ) 2 w y + G(t) w 2 y +c p w t = y ( ρs (t) c p s 2 (t)) ρf (2) + s 4 (t) (7) where G(t) = ρa(t) T y +c pv(t), and the s i variabes are defined as s (t) = (2) 2 2 f 3 (t) + 2 f 2 + (3 f (2) 2 f 2 2 (2) +2 2 f f 2) f(2) 4 s 2 (t) = f f 2 f + 2 f 2 2 s 3 (t) = f2 f s 4 (t) = 2v(t)ρs 2 (t) G(t)s 3 (t) c p f (t) associated with the two-point boundary conditions (8) w(, t) =, w((t), t) =. (9) Now instead of deaing with the PDE () with non-zero boundary conditions, we can use the equivaent mode, given by equation (7) associated with trivia boundary conditions (9). Foowing the assumed-modes technique, the soution of the equation (7), (9) writes as w(y, t) = =N = q (t)φ (y, t), N N () where N is the number of bases (modes), incuded in the discretization, φ, =,..., N are the discretization bases and q, =,.., N are the discretization coordinates. In order to simpify the anaytic manipuation of the equations, the base functions are chosen to satisfy the foowing normaization constraints (t) φ 2 (y, t)dy =, (t) φ i (y, t)φ (y, t)dy =, i () To further simpify the base functions, we define the normaized variabe, e.g. [5], [3] ξ(t) = y(t) (t) and the normaized base functions (2) φ (y, t) = ψ (ξ) (t), =,..., N (3) In these new coordinates the normaization constraints () write as ψ 2 (ξ)dξ =, ψ i (ξ)ψ (ξ)dξ =, i (4) After cassica (e.g. refer to [3]) discretization of the PDE-based mode (7), (9), we can write the reduced ODE-mode based on N-modes as M q + C q + (K + βu)q = F(t), q R N, F R N where M i=ρδ i (5) C i=ρ (2 ( ξ)ψi(ξ)ψ (ξ)dξ δi)+cpδi K i= ρ 2 2 δ 4 i ρ 2 2 ( ξ)2 ψ i (ξ)ψ (ξ)dξ +ρ (g a(t)) ( ξ)ψ i (ξ)ψ (ξ)dξ+me 2 (g a(t)) ψ i (ξ)ψ (ξ)dξ +ρ( 2 2 )(.5δ i ( ξ)ψi(ξ)ψ (ξ)dξ) c p ( ψi(ξ)ψ (ξ)ξdξ+.5δi)+.5mcsg 2 ψ i (ξ)ψ (ξ)dξ +U(t) 2 ψ i (ξ)ψ (ξ)dξ β ii= 2 ψ 2 i dξ= 2 β ii β i= β i=, i F i(t)= (ρs (t)+c ps 2(t)) ψi(ξ)ξdξ + (s 4(t) ρf (2) (t)) δ i = {, i, i = ψi(ξ)dξ where s i, i =, 2, 3, 4 are given in (8). (6) Remark : - In the previous deveopments we have negected the bending stiffness of the rope, introducing it back does not change the obtained modes. Indeed, if we consider that the rope materia and shape impies a bending stiffness coefficient EI (where E is the materia Young moduus and I is moment of inertia of the cross section of the rope), the mode equation (5) remains vaid (e.g. [3]) with the addition of the foowing term in the stiffness matrix K K i = EI 4 ψ (2) i (ξ)ψ (2) (ξ)dξ (7) we see that these new terms are inversey proportiona to 4, which makes them negigibe for ong ropes, furthermore, their addition does not change the structure of the mode and thus does not ater the resuts of this paper. 2- The mode (5), (6) has been obtained for the genera case of time-varying rope ength (t), however, in this paper we ony consider the case of stationary ropes = cte, which is directy deduced from (5), (6), by setting = =, t. If we use the cassica definition of the state vector z = (q, q) T, then it is easy to see that the obtained ODE mode is a biinear mode in the state z and the contro vector U.

6 IV. MAIN RESULT: LYAPUNOV-BASED CONTROLLERS In this section we present Lyapunov-based feedback controers designed to stabiize the rope sway dynamics. Theorem : Consider the rope dynamics (5), (6), with non-zero initia conditions, with no externa disturbances, i.e. f (t) = f 2 (t) =, t, and with constant ength, then the feedback contro { q T βq umax U nom (z) = +( q T βq) 2, if qt βq >, if q T βq (8) where z = (q T, q T ) T, renders the cosed-oop equiibrium point (, ) gobay asymptoticay stabe, with U nom u max, furthermore U nom decreases as function of q T βq. Proof: We define the contro Lyapunov function as V (z) = 2 qt (t)m q(t) + 2 qt (t)kq(t) (9) where x = (q T, q T ) T. First we compute the derivative of the Lyapunov function aong the dynamics (5), without disturbances, i.e. F(t) =, t V (z) = q T ( C q Kq βuq) + q T K q = q T C q q T βqu (2) To ensure the negative definiteness of V (x) we define the first controer (8). Using the continuity of (8) at q T βq = and LaSae theorem, e.g. [9], we can concude that the states of the cosed-oop dynamics converge to the set S = {z = (q T, q T ) T R 2N, s.t. q = }. Next, we anayze the cosed-oop dynamics: Since the boundedness of V impies boundedness of q, q and by equation (5), boundedness of q. Boundedness of q, q impies the uniform continuity of q, q, which again by (5), impies the uniform continuity of q. Next, since q, and using Barbaat s Lemma, e.g. [9], we concude that q, and by invertibiity of the stiffness matrix K + βu we concude that q. Finay, the fact that V is a radiay unbounded function, ensures that the equiibrium point (q, q) = (, ) is gobay asymptoticay stabe. Furthermore the fact that U nom u max, and the decrease of U nom as function of q T βq is deduced from equation (8). Remark 2: By examining the Lyapunov derivative (2), we can see that instead of the C controer (8), we coud use a smooth controer of the form q T βq U nom (z) = u max + ( q T βq) 2 However, the advantage of the switching controer (8) is the fact that it necessitates ess contro energy, since when the condition q T βq is satisfied, it does not appy any extra contro and ony uses the system s natura damping. The nomina controer U nom given by (8) does not take into account the disturbance F(t) expicity. We present next a controer that takes into account F(t) in the design of the contro aw. However, since in practica appications we sedom have access to direct measurements of the disturbance signa F(t), we use the so caed Lyapunov reconstruction technique, e.g. [], to augment the nomina controer U nom with an additiona feedback term which is based ony on an upper bound of the disturbance signa F(t) (i.e. does not need the exact measurements of F(t)) and which ensures the stabiization of the sway to a sma ampitude, which can be tuned by the choice of the feedback gains. First et us state the foowing assumption. Assumption : The time varying disturbance functions f, f 2 are such that, the function F(t) is bounded, i.e. F max, s.t. F(t) F max, t. Theorem 2: Consider the rope dynamics (5), (6), under non-zero externa disturbances, i.e. f (t), f 2 (t), and with constant ength, then under Assumption, the feedback contro U(z) = U nom (z) + ksgn( q T βq)(fmax + ǫ) q, k >, ǫ > (2) where z = (q T, q T ) T, ensures that the soutions of (5), (6) and (2) converges to the invariant set S = {(q T, q T ) T R 2N, s.t. k 2 q T βq }. Proof: Using the same Lyapunov function (9), and writing its derivative aong (5) V (z) = q T Cq q T βqu(x) + q T F(t) (22) if we denote U(z) = U nom (z)+v(z), where v(z) = ksgn( q T βq)(fmax + ǫ) q, k >, ǫ >, we obtain V (z) = q T Cq q T βqu nom (x) q T βqv(x) + q T F(t) (23)

7 and by definition of U nom (z) we know that q T Cq q T βqu nom (z) < thus, using Assumption, we can write V (z) q T βqv(z) + q T F(t) k 2 q T βq (Fmax + ǫ) q + q F(t) k 2 q T βq (Fmax + ǫ) q + q F max k 2 q T βq q ǫ + q Fmax ( k 2 q T βq ) + q F max ( k 2 q T βq ) which proves the decrease of V (x) unti reaching the invariant set S = {(q T, q T ) T R 2N, s.t. k 2 q T βq } Remark 3: The controers (8), (2) are state feedbacks based on q, q, these states can be easiy computed from the sway measurements at N given positions y(),..., y(n), via equation (). The sway w(y, t) can be measured by aser dispacement sensors paced at the positions y(i), i =, 2,...N, aong the rope, e.g.[], subsequenty q can be computed by simpe agebraic inversion of (), and q can be obtained by direct numerica differentiation of q. Remark 4: The controer (2) has a discontinuity due to the sgn function. For practica impementation, we known reguarization can be used to smoothen the controer, for exampe a sat function can be used instead of the sgn function, and simiar stabiity resuts can be concuded (e.g. refer to []). Due to space imitation, we do not present the reguarized version of the controer here, and defer it to a onger ourna version of this work. V. NUMERICAL EXAMPLE In this section we present some numerica resuts obtained on the exampe presented in []. The case of an eevator system with the mechanica parameters summarized on Tabe I has been considered for the tests presented hereafter. We write the controers based on the mode (5), (6) with one mode, but we test them a mode with three modes (the fact is that one mode is enough since when comparing the soution of the PDE (7) to the discrete mode (5) the higher modes shown to be negigibe, and a discrete mode with one mode showed a very good match with the PDE mode, but to make the simuation tests more reaistic we chose to Parameters Definitions Vaues n Number of ropes 8[ ] m e Mass of the car 35[kg] ρ Main rope inear mass density 2.[kg/m] Rope maximum ength 39[m] H Buiding height 42.8[m] c p Damping coefficient.35[n.sec/m] TABLE I NUMERICAL VALUES OF THE MECHANICAL PARAMETERS test the controers on a three modes mode 2 ). First, to vaidate Theorem, we present the resuts obtained by appying the controer (8), to the mode (5), (6), with non-zero initia conditions q() = 2, q() =, and zero externa disturbances, i.e. f (t) = f 2 (t) =, t. In these first tests, to show the effect of the controer (8) aone, without the hep of the system s natura damping, we fix the damping coefficient to zero, i.e. c p =. Figures 2, 3 3 show the rope sway obtained at haf rope-ength y = 95 m without contro. It reaches a maximum vaue of about.45 m. We show next on Figures 4, 5 the rope sway obtained at the same rope ength but this time with the controer (8), with u max = 5 N. We see the expected effect of the controer on the sway, which is reduced by haf in about 6 sec and vanishes asymptoticay. The corresponding contro force is depicted on Figures 6, 7. We see that, as expected from the theoretica anaysis of Theorem, the contro force remains bounded by u max and decreases with the decrease of the sway. Next, we consider the mode (5), (6) with nozero disturbance signas: f (t) =.2sin(2π..8t), and f 2 being deduced from f via equation (3). We underine that we have purposey seected the disturbance frequency to be equa to the first resonance frequency of the rope, to simuate the worst-case scenario. We first show on Figures 8, 9 the sway signa in the uncontroed case. Let us consider now the controer (2) introduced in Theorem 2. We appy (2), with the parameters u max = 5 N, F max =.6, ǫ =., k = 6. The effect of the contro on the rope sway ampitude is depicted on Figures,. The rope sway is effectivey reduced, and enters 2 We aso want to inform the reviewers that these controers have been actuay vaidated on a fu-size test-bed in Japan, unfortunatey, due to IP reasons we were not aowed to report the experimenta data here (This footnote is added for the reviewers but wi be deeted in the fina version of the paper). 3 The figures zoom is incuded for the reader to have a better idea about the signas shape.

8 Contro [N] Fig. 2. Rope sway at y = 95 m- No contro with zero disturbance Fig. 6. Output of controer (8) Contro [N] Fig. 3. Zoom of the rope sway at y = 95 m- No contro with zero disturbance Fig. 7. Zoom of the output of controer (8) the invariant set defined by the upper bound condition k 2 q βq, as indicated in Theorem 2. In fact, by checking the q q pot presented on Figures 2, 3, we see that the Lagrangian variabes q, q converge to the point satisfying k 2 q βq max =.97, which is in concordance with the invariant set convergence resut of Theorem 2. The contro force is depicted on Figures 4, 5, which shows a high ampitude, due to the seected high gain vaue for k. Fig. 4. Rope sway at y = 95 m- with controer (8) VI. CONCLUSION Fig. 5. (8) Zoom of the rope sway at y = 95 m- with controer In this paper we have studied the probem of active contro of eevator rope sway dynamics occurring due to externa force disturbances acting on the eevator system. We have considered the case of constant rope ength and have proposed noninear controers based on Lyapunov theory. We have presented the stabiity anaysis of these controers and shown their efficiency on a numerica exampe. The stabiization probems reated to time-varying rope engths, i.e. moving car, wi be presented in a onger ourna version of this work.

9 Fig. 8. Rope sway at y = 95 m- No contro with non-zero disturbance Fig.. Rope sway at y = 95 m- controer (2) Fig. 9. Rope sway at y = 95 m- No contro with non-zero disturbance Fig.. Rope sway at y = 95 m- controer (2) REFERENCES [] S. Kaczmarczyk, Noninear sway and active stiffness contro of ong moving ropes in high-rise vertica transporation systems, in The th Internationa Conference on Vibration Probems, S. P. in Physics, Ed., 2, pp [2] S. Kaczmarczyk, R. Iwankiewicz, and Y. Terumichi, The dynamic behavior of a non-stationary eevator compensating rope system under harmonic and stochastic excitations, in Journa of Physics: Conference Series 8. IOP Pubishing, 29, pp. 8. [3] W.D.Zhu and G. Xu, Vibration of eevator cabes with sma bending stiffness, Journa of Sound and Vibration, vo. 263, pp , 23. [4] W.D.Zhu and L. Teppo, Design and anaysis of a scaed mode of a high-rise high-speed eevator, Journa of Sound and Vibration, vo. 264, pp , 23. [5] W.D.Zhu and Y. Chen, Theoretica and experimenta investigation of eevator cabe dynamics and contro, Journa of Sound and Vibration, vo. 28, pp , 26. [6] Y. Fuino, P. Warnitchai, and B. Pacheco, Active stiffness contro of cabe vibration, ASME Journa of Appied Mechanics, vo. 6, pp , 993. [7] L. Meirovitch, Anaytica methods in vibrations, Fred Landis ed., ser. Appied Mechanics. Coier Macmian pubishers, 967. [8] P. Homes, J. LUmey, and G. Berkooz, Turbuence, coherent q q[sec ] q q[sec ] Fig. 2. q q signa- controer (2) Fig. 3. q q signa- controer (2)

10 x Contro [N] x 4 Fig. 4. Output of controer (2) Contro [N] Fig. 5. Output of controer (2) structures, dynamica systems and symmetry. UK: Cambridge University Press, 996. [9] H. Khai, Noninear systems, 2nd ed. New York Macmian, 996. [] M. Benosman and K.-Y. Lum, Passive actuators faut toerant contro for affine noninear systems, IEEE, Transactions on Contro Systems Technoogy, vo. 8, 2. [] M. Otsuki, K. Yoshida, K. Nagata, S. Fuimoto, and T. Nakagawa, Experimenta study on vibration contro for ropesway of eevator of high-rise buiding, in American Contro Conference, 22, pp

Semi-Active Control of the Sway Dynamics for Elevator Ropes

Semi-Active Control of the Sway Dynamics for Elevator Ropes MITSUBISHI ELECTRIC RESEARCH LABORATORIES http://www.mer.com Semi-Active Contro of the Sway Dynamics for Eevator Ropes Benosman, M. TR15-35 January 15 Abstract In this work we study the probem of rope