Semi-Active Control of the Sway Dynamics for Elevator Ropes

Transcription

1 MITSUBISHI ELECTRIC RESEARCH LABORATORIES 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 sway dynamics contro for eevator systems. We choose to actuate the system with a semi-active damper mounted on the top of the eevator car. We propose noninear controers based on Lyapunov theory, to actuate the semi-active damper and stabiize the rope sway dynamics. We study the stabiity of the proposed controers, and test their performances on a numerica exampe. arxiv 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., 15 1 Broadway, Cambridge, Massachusetts 139

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3 Semi-Active Contro of the Sway Dynamics for Eevator Ropes Mouhacine Benosman Abstract In this work we study the probem of rope sway dynamics contro for eevator systems. We choose to actuate the system with a semi-active damper mounted on the top of the eevator car. We propose noninear controers based on Lyapunov theory, to actuate the semi-active damper and stabiize the rope sway dynamics. We study the stabiity of the proposed controers, and test their performances on a numerica exampe. I. INTRODUCTION Modern eevators instaed in high-rise buidings are required to trave fast and ensure comfort and safety for the passengers. Unfortunatey, the dimensions of such high-rise buidings make them more susceptibe to the impact of bad weather conditions. Indeed, when an externa disturbances, ike wind gust or earthquake, hits a buiding it can ead to arge rope sway ampitude within the eevator shaft. Large ampitudes of rope sway might ead to important damages to the equipments that are instaed in the eevator shaft and to the eevator shaft structure itsef, without mentioning the potentia danger caused for the eevator passengers. It is important then to be abe to contro the eevator system and damp out these ropes osciations. However, due to cost constraints, it is preferabe to be abe to do so, with minimum actuation capabiities. Many investigations have been dedicated to the probem of modeing and contro of eevator ropes [1], [], [3], [4], [5], [6], [7]. In [4], a scaed mode for high-rise, high-speed eevators was deveoped. The mode was used to anayze the infuence of the car motion profies on the atera vibrations of the eevator cabes. The author in [1] proposed a noninear moda feedback to drive an actuator puing on one end of the rope. The contro performance was investigated by numerica tests. In [6], a simpe mode of a cabe attached to an actuator at its free end was used to investigate the stiffening effect of the contro force on the cabe. An off-ine energy anaysis was used to tune an open-oop sinusoida Mouhacine Benosman m benosman@ieee.org) is with Mitsubishi Eectric Research Laboratories, 1 Broadway Street, Cambridge, MA 139, USA. force appied to the cabe. 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 [7], [8], [9], the present author studied the probem of an eevator system using a force actuator puing on the ropes to add contro tension to the ropes. The force actuator was controed based on Lyapunov theory. Athough, the sway damping performances obtained in [7], [8], [9] were satisfactory, we decided to investigate other forms of actuation and contro. Indeed, in [7], [8], [9] we proposed a force contro agorithm to moduate the ropes tensions, using an externa force actuator, which was introduced at the bottom of the eevator shaft to pu on the ropes. However, retrofitting existing eevators with such a cumbersome actuator, can prove to be chaenging and expensive. Instead, we propose to investigate here another actuation method, namey, using semi-active dampers mounted between the eevator car and the ropes. These actuators are ess cumbersome than the typica force actuators and thus can be easier/cheaper to insta. This type of actuation has aready been proposed in [5]. The difference between the present work and [5], is that instead of using a static damper with constant damping coefficient tuned off-ine, we use a semi-active damper, and use noninear contro theory to design a feedback controer to compute onine the desired time-varying damping coefficient that reduces the rope sway. We study the stabiity of the cosed-oop dynamics, and show the performances of these controers on a numerica exampe. One more noticeabe difference with the work in [7], [8], [9], is that due to the semi-active actuation, the mode of the actuated system is quite different from the one presented in [7], [8], [9]. Indeed, in [7], [8], [9] the mode of the actuated system exhibited terms where the contro variabe was mutipied by the sway position variabe ony. In this work, the actuated mode, exhibits terms where the contro variabe, i.e. the semi-active damper coefficient, is mutipied both

4 by the sway position and the sway veocity variabes, furthermore, the contro variabe term appears in the right hand side of the dynamica equations of the system, i.e. as part of an externa disturbance on the system refer to Section II). These differences in the mode, make the controer design and anaysis more chaenging than in [7], [8], [9]. The paper is organized as foows: In Section II, we reca the mode of the system when actuated with a semi-active damper mounted between the ropes and the eevator car. Next, in Section III, we present the main resuts of this work, namey, the noninear Lyapunov-based semi-active damper controers, together with their stabiity anaysis. Section IV is dedicated to some numerica resuts. Finay, we concude the paper with a brief summary of the resuts in Section V. 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, and we denote by A ij, i = 1,..., n, j = 1,..., m the eements of the matrix A. II. ELEVATOR ROPE MODELLING In this section we first reca the infinite dimension mode, i.e partia differentia equation PDE), of a moving hoist cabe, with non-homogenous boundary 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 rewrite 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: 1) The eevator ropes are modeed within the framework of string theory, ) The eevator car is modeed as a point mass, 3) The vibration in the second atera direction is not incuded, 4) The suspension of the car against its guide rais is assumed to be rigid, 5) The mass of the semi-active damper is considered to be negigibe compared to the eevator car mass. Under the previous assumption, foowing [3], [1] and [5], the genera PDE mode of an eevator rope, depicted on Figure 1, is given by ρ t + v t) y + vt) y t + a ) y uy, t) uy,t) y T y, t) y + c p t + vt) ) y uy, t) +k dp t + vt) ) y uy, t)δy + dp ) = 1) Fig. 1. Schematic representation of an eevator shaft showing the different variabes used in the mode where uy, 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 ) modeing the time-varying rope ength. a = t) t is the eevator rope acceeration. The ast term in the righthand-side of 1) has been added to mode the semiactive inear damper effect on the rope at the contact point δy + dp ), where δ is the Dirac impuse function, k dp is the damping coefficient the contro variabe) and dp is the distance between the car-top and the point of attach of the damper to the controed rope, e.g. [5]. The PDE 1) is associated with the foowing two boundary conditions: u, t) = f 1 t) ut), t) = f t) ) where f 1 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 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 t) = f 1 t)sin πh ) ), H R 3) H where H is the height of the buiding. This expression is an approximation of the propagation of the boundary

5 disturbance f 1 aong the buiding structure, based on the ength, it eads to f = f 1 for a ength which is expected), and a decreasing force aong the buiding unti is vanishes at = H, f = which makes sense, since the effect of any disturbance f 1, 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 + ρt) y))g at)) +.5M cs g 4) where g is the standard gravity constant, m e, M cs are the mass of the car and the compensating sheave, respectivey. Next, we reduce the PDE mode 1) to a more tractabe mode for contro, using a projection Gaerkin method or assumed mode approach, e.g. [1], [11]. To be abe to appy the assumed mode approach, et us first appy the foowing one-to-one change of coordinates to the equation 1) uy, t) = wy, t) + t) y f 1 t) + y t) t) f t) 5) One can easiy see that this change of coordinates impies trivia boundary conditions w, t) = wt), t) = 6) After some agebraic and integra manipuations, the PDE mode 1) writes in the new coordinates as ρ w t + vt)ρ w y t + ρv T y, t) ) w y +Gt) + vk dp δx + dp )) w y +c p + k dp δx + dp )) w t = y ρs 1 t) c p s t)) ρf ) w t y 1 + s 4 t) f 1 t) + y f t) ) k dp δx + dp ) f vk f 1 dp δx + dp ) 7) where Gt) = ρat) T y +c pvt), and the s i variabes are defined as s 1 t) = ) 3 f f ) f ) + f f ) f ) 4 1 s t) = f f 1 1 f + f f 1 t) + s 3 t) = f f1 s 4 t) = vt)ρs t) Gt)s 3 t) c p f 1 t) 8) associated with the two-point boundary conditions w, t) =, wt), t) =. 9) Now instead of deaing with the PDE 1) 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 wy, t) = j=n j=1 q j t)φ j y, t), N N 1) where N is the number of bases modes), incuded in the discretization, φ j, j = 1,..., N are the discretization bases and q j, j = 1,.., 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 jy, t)dy = 1, t) φ i y, t)φ j y, t)dy =, i j 11) To further simpify the base functions, we define the normaized variabe, e.g. [5], [3] ξt) = yt) t), and the normaized base functions φ j y, t) = ψjξ), j = 1,..., N. t) In these new coordinates the normaization constraints write as 1 ψ j ξ)dξ = 1, 1 ψ iξ)ψ j ξ)dξ =, i j. After discretization of the PDE-based mode 7), 8) and 9) e.g. refer to [3]), we can write the reduced ODE-mode based on N-modes as M q + C + CU) q + K + KU)q = F t) + F t)u, q R N where U=k dp 1) M ij=ρδ ij C ij=ρ ξ)ψiξ)ψ j )+c ξ)dξ δij pδ ij C ij= 1 ψ i dp )ψ j dp ) K ij= 1 4 ρ δ ij ρ 1 1 ξ) ψ i ξ)ψ j ξ)dξ +ρ 1 g at)) 1 1 ξ)ψ i ξ)ψ j ξ)dξ+me g at)) 1 ψ i ξ)ψ j ξ)dξ +ρ 1 ).5δ 1 ij 1 ξ)ψiξ)ψ j ) ξ)dξ c p 1 1 ψiξ)ψ j ξ)ξdξ+.5δij)+.5mcsg 1 ψ i ξ)ψ j ξ)dξ k ij= ψ j dp +ψ j dp )ψ i dp ) )ψ i dp ) ) dp ).5ψ i dp )ψ j dp ) F it)= ρs 1 1t)+c ps t)) ψiξ)ξdξ + ) s 4t) ρf ) 1 t) 1 ψiξ)dξ f F it)= 1 ψ i dp )+ dp ψ i dp ) f 1 f )+ 1 f f 1)) {, i j δ ij = 1, i = j 13) where i, j {1,..., N}, and s k, k = 1,, 3, 4 are given in 8). Remark 1: The mode 1), 13) has been obtained for the genera case of time-varying rope ength t), however, in this

6 paper we ony consider the case of stationary ropes = cte, which is directy deduced from 1), 13), by setting = =, t. The case where the car is static and the contro system has to reject the ropes osciations is of interest in practica setting. Indeed, besides the case of commercia buidings at night where the eevators are not in use), in many situations where the buiding is swaying due to externa strong weather conditions, the eevators are stopped, for the security of the passengers. The contro system is then used to damp out the ropes sway, to avoid the ropes from damaging the eevator system, and be sti functiona after the externa disturbances have passed. 1 III. MAIN RESULT: LYAPUNOV-BASED SEMI-ACTIVE DAMPER CONTROL The first controer deas with the case where the buiding, hosting a stationary eevator stopped at a given foor), sustains a brief impuse-ike) externa disturbance. For exampe, an earthquake impuse with a sufficient force to make the top of the buiding osciate, or a strong wind gust that happens over a short period of time, exciting the buiding structure and impying residua vibrations of the buiding even after the wind gust interruption. In these cases, the eevator ropes wi vibrate, starting from a non-zero initia conditions, due to the impuse-ike externa disturbances i.e., happening over a short time interva), and this case correspond to the mode 1), 13) with non-zero initia conditions and zero externa disturbances. We can now state the foowing theorem. Theorem 1: Consider the rope dynamics 1), 13), with non-zero initia conditions, with no externa disturbances, i.e., f 1 t) = f t) =, t, then the feedback contro q T C q Uz) = u max 1 + q T C 14) q) where z = q T, q T ) T, impies that qt), for t, furthermore U u max, t, and U deceases with the decrease of q T C q. Proof: We define the contro Lyapunov function as V z) = 1 qt t)m qt) + 1 qt t)kt)qt) 15) where z = q T, q T ) T. First we compute the derivative of the Lyapunov function 1 Of course, there are aso practica cases were the cars are in motion and an externa disturbance occurs. These cases correspond to a time-varying rope ength, which we have aso studied, however, due to space imitation, we coud not incude a the resuts in this paper. The case of time-varying rope ength wi be presented in another report. aong the dynamics 1), without disturbances, i.e., F t) =, F t) = t V z) = q T C q CU q Kq) + q T K q = q T C q q T C qu 16) Next, using U defined in 14), we have V z) u max q T C q) 1+ q T C q) 17) Using LaSae theorem, e.g. [1], and the fact that C is symmetric positive definite we can concude that the states of the cosed-oop dynamics converge to the set S = {z = q T, q T ) T R N, s.t. q = }. Next, we anayze the cosed-oop dynamics: Since the boundedness of V impies boundedness of q, q and by equation 1), boundedness of q. Boundedness of q, q impies the uniform continuity of q, q, which again by 1), impies the uniform continuity of q. Next, since q, and using Barbaat s Lemma, e.g. [1], 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 u max, and the decrease of U as function of q T C q is deduced from q equation 14), since T C q 1 and 1 1+ q T C q) q T C q) Remark : It is cear from equation 16) that the trivia choice of a constant positive damping contro U, wi aso impy a convergence of the sway dynamics to zero. However, the controer 14) has the advantage to require ess contro energy comparativey to a constant damping, since by construction of the contro aw 14), the stabiizing damping force decreases together with the decrease of the sway. The controer U given by 14) does not take into account the disturbance F t) expicity. Next, we present a controer which deas with the case of a static eevator in a buiding under sustained externa disturbances, e.g. sustained wind forces on commercia buidings at night where the cars are static. In this case F t), F t) over a non-zero time interva, and satisfy the foowing assumption. Assumption 1: The time varying disturbance functions f 1, f are such that, the functions F t), F t) are bounded, i.e. F max, Fmax ), s.t. F t) F max, F t) F max, t. Theorem : Consider the rope dynamics 1), 13), under non-zero externa disturbances, i.e., f 1 t), f t) satisfying Assumption 1, with the feedback contro q Uz)=u T C q maxp 1+ q T C q) + q T v 1max C Fmax q v 1max + Fmax q q umaxp ) 1+ q T C q) Fmax q v 1max + Fmax q umaxp ) + q T v C q max q Fmax+v max Fmax q ) 1+ q T C q) q Fmax+v max Fmax q ) 18) where u maxp, v 1max, v max >, are chosen s.t. u maxp + v 1max + v max ) u max and z = q T, q T ) T. Then, if we

7 define the two invariant sets:, and S 1 = {q T, q T ) T R N, s.t. q T C q) 1+ q T C q) F max q v 1max+ F max q u maxp ) 1 v 1max } S = {q T, q T ) T R N, s.t. q T C q) 1+ q T C q) q F max+v max Fmax q ) 1 v max }, the controer 18) ensures that the state vector z converges to the the invariant set S 1 or S, and that U u max, t. Proof: Let us consider again the Lyapunov function 15). Its derivative aong the dynamics 1), with non-zero disturbance, i.e. F t), F, writes as V x) = q T C q CU q Kq + F U) +q T K q + q T F t) = q T C q q T CU q + q T F U + q T F t) q T CU q + q T F U + q T F t) 19) Now, we use the concept of Lyapunov reconstruction, e.g. [13], we write the contro U = u nomp + v 1 + v ) where, u nomp is the nomina controer, given by 14) with u max = u maxp, designed for the case where F t) = F t) =, t. The remaining terms v 1, v are added to compensate for the effect of the disturbances F and F, respectivey. We design v 1 and v in two steps: - First step: We assume that F, F t) =, t, and design v 1 to compensate for the effect of F. In this case U = u nomp + v 1. In this case, the Lyapunov function derivative is bounded as as foows V x) q T Cunomp + v 1 ) q + q T F unomp + v 1 ) q T Cv1 q + q T F unomp + v 1 ) q T C qv1 + q T F unomp + q T F v1 which under Assumption 1, gives q T C qv1 + q F max u maxp + q F max v 1max Now, if we define to simpify the notations), the term T 1 = + q F max u maxp + q F max v 1max and if we choose the controer v 1 = v 1maxT 1 q T C q 1 + T 1 qt C q) This eads to the foowing Lyapunov function derivative upper bound T 1 1 v 1max q T C q) 1+T 1 qt C q) ) = B1 - Second step: We assume that F t), F t), and design v to compensate for F. In this case we write the tota contro as U = u nomp + v 1 + v. Now the upper-bound of the Lyapunov function derivative aong the tota contro) writes as B 1 q T C qv + q T F v + q T F B 1 q T C qv + q F max v max + q F max Next, if we define the term T = q F max v max + q F max and if we choose the controer v = v maxt q T C q 1 + T qt C q) This eads to the foowing Lyapunov function derivative upper-bound B 1 + T 1 v max q T C q) ) 1+T qt C q) B 1 + B where B = T 1 v max q T C q) 1+T qt C q) ). By choosing v 1max, v max high enough the two terms B 1, B wi be made negative. We can then anayze two cases: 1- First, the trajectories keep decreasing unti they reach the invariant set S 1 = {q T, q T ) T R N, s.t. B 1 } Then, the trajectories can either keep decreasing if B > B 1 ) unti they enter the invariant set S = {q T, q T ) T R N, s.t. B } or they get stuck at S 1. - Second, the trajectories decrease unti they reach the invariant set S first, and stay there, or keep decreasing unti they reach the invariant set S 1. In both cases, the trajectories wi end up in either S 1 or S. Finay, the fact that U u max is obtained by construction of the three terms, since from 14), we can write u nomp u maxp and by construction v 1 v 1max, v v max, which eads to U u maxp + v 1max + v max u max. Remark 3: We want to underine here the fact that, contrary to the previous case of impuse disturbances, in this case of sustainabe externa disturbances, the use of a simpe passive damper, i.e. U = cte, might actuay destabiize the system. Indeed, by examining equation 19), we can see that if the U is constant the positive term + q T F U, which is due the externa disturbance, coud overtake the damping negative term q T CU q, eading to instabiity of the system. Remark 4: The controers 14), 18) are state feedbacks based on q, q, these states can be easiy computed from the sway measurements at N given positions y1),..., yn), via equation 1). The sway wy, t) can be measured by aser dispacement sensors paced at the positions yi), i = 1,,...N, aong the rope, e.g.[14], subsequenty q can be computed by simpe agebraic inversion of 1), and q can be obtained by direct numerica differentiation of q.

8 Parameters Definitions Vaues n Number of ropes 8[ ] m e Mass of the car 35[kg] ρ Main rope inear mass density.11[kg/m] Rope maximum ength 39[m] H Buiding height 4.8[m] c p Damping coefficient.315[n.sec/m] TABLE I NUMERICAL VALUES OF THE MECHANICAL PARAMETERS IV. NUMERICAL EXAMPLE In this section we present some numerica resuts obtained on the exampe presented in [1]. 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 1), 13) with one mode, but we test them on a mode with two modes. The fact is that one mode is enough since when comparing the soution of the PDE 7) to the discrete mode 1) 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 test the controers on a two modes mode. Furthermore, to make the simuation tests more chaenging we added a random white noise to the states fed back to the controer equivaent to about ±1 cm of error on the rope sway measurement from which the states are computed, see Remark 1), and we fitered the contro signa with a first order fiter with a cut frequency of 1 hz and a deay term of 5 samping times, to simuate actuator dynamics and deays due to signa transmission and computation time. First, to vaidate Theorem 1, we present the resuts obtained by appying the controer 14), to the mode 1), 13), with non-zero initia conditions q) =, q) = 5, and zero externa disturbances, i.e. f 1 t) = f t) =, t. In these first tests, to show the effect of the controer 14) aone, without the hep of the system s natura damping, we fix the damping coefficient to zero, i.e. c p =. We appy the controer 14), with u max = 1 9 Nsec/m. Figures, 3 show the rope sway obtained at haf rope-ength y = 195 m with and without contro. Without contro the rope sway reaches a maximum vaue of about 1.5 m. With contro we see ceary the expected damping effect of the controer, which reduces the sway ampitude by haf. The corresponding contro force is depicted on Figures 4, 5. We see that, as expected from the theoretica anaysis of Theorem 1, the contro force remains bounded by a maximum vaue of 4kN sec/m, which is easiy reaizabe by existing semi-active damper, e.g. magnotorheoogica damper. Furthermore, as proven in The figures zoom is incuded for the reader to have a better idea about the signas shape. Sway at y=195 [m] Fig.. Rope sway at y = 195 m: No contro thin ine)- With controer 14) bod ine) Sway at y=195 [m] Fig. 3. Zoom of rope sway at y = 195 m: No contro thin ine)- With controer 14) bod ine) Theorem 1, the contro ampitude decreases with the decrease of the sway. Next, we consider the mode 1), 13) with no-zero disturbance signas: f 1 t) =.sinπ..8t), and f being deduced from f 1 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. In this case we appy the controer 18), with the parameters u maxp = 1 9 Nsec/m, v 1max = v max = 1 5 Nsec/m, F max = F max = 1. We show on Figures 6, 7 the sway signa in the uncontroed and the controed case. We see that the sway steady state maximum ampitude is reduced from 8.4 m in the uncontroed case to.4 m with contro. The noisy due to the simuated measurements noise) bounded and continuous contro signas are reported on Figures 8, 9.

9 4 x 14 1 Contro [N sec/m] Sway at y=195 [m] Contro [N sec/m] Fig. 4. Output of controer 14) Fig. 6. Rope sway at y = 195 m: No contro thin ine)- With controer 18) bod ine) Sway at y=195 [m] Fig. 5. Output of controer 14)- Zoom Fig. 7. Zoom of rope sway at y = 195 m: No contro thin ine)- With controer 18) bod ine) V. CONCLUSION In this paper we have studied the probem of semi active contro of eevator rope sway dynamics occurring due to externa force disturbances acting on the eevator system. We have considered the case of a static car, i.e. constant rope ength and have proposed two noninear controers based on Lyapunov theory. We have presented the stabiity anaysis of these controers and shown their efficiency on a numerica exampe. The semi-active stabiization probems reated to time-varying rope engths, i.e. moving car, wi be presented in a future report. REFERENCES [1] S. Kaczmarczyk, Noninear sway and active stiffness contro of ong moving ropes in high-rise vertica transporation systems, in The 1th Internationa Conference on Vibration Probems, S. P. in Physics, Ed., 11, pp [] 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 181. IOP Pubishing, 9, pp [3] W.D.Zhu and G. Xu, Vibration of eevator cabes with sma bending stiffness, Journa of Sound and Vibration, vo. 63, pp , 3. [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. 64, pp , 3. [5] W.D.Zhu and Y. Chen, Theoretica and experimenta investigation of eevator cabe dynamics and contro, Journa of Sound and Vibration, vo. 18, pp , 6. [6] Y. Fujino, P. Warnitchai, and B. Pacheco, Active stiffness contro of cabe vibration, ASME Journa of Appied Mechanics, vo. 6, pp , [7] M. Benosman, Lyapunov-based contro of the sway dynamics for eevator ropes, IEEE, Transactions on Contro Systems

10 Conference,, pp x 15 Contro [N sec/m] Contro [N sec/m] x Fig. 8. Output of controer 18) Fig. 9. Output of controer 18)- Zoom Technoogy, 14, to appear onine version avaiabe). [8], Lyapunov-based contro of the sway dynamics for eevator ropes with time-varying engths, in IFAC 17th Word Congress, 14, pp [9], Lyapunov-based contro of the sway dynamics for eevator ropes, in American Contro Conference, 14, pp [1] L. Meirovitch, Anaytica methods in vibrations, Fred Landis ed., ser. Appied Mechanics. Coier Macmian pubishers, [11] P. Homes, J. LUmey, and G. Berkooz, Turbuence, coherent structures, dynamica systems and symmetry. UK: Cambridge University Press, [1] H. Khai, Noninear systems, nd ed. New York Macmian, [13] M. Benosman and K.-Y. Lum, Passive actuators faut toerant contro for affine noninear systems, IEEE, Transactions on Contro Systems Technoogy, vo. 18, 1. [14] M. Otsuki, K. Yoshida, K. Nagata, S. Fujimoto, and T. Nakagawa, Experimenta study on vibration contro for ropesway of eevator of high-rise buiding, in American Contro