A PID Sliding Mode Control for Ropeless Elevator Maglev Guiding System

Energy and Power Engineering, 2012, 4, 158-164 http://dx.doi.org/10.4236/epe.2012.43022 Published Online May 2012 (http://www.SciRP.org/journal/epe)

Hasan Alipour, Mohammad Bagher Bana Sharifian, Hadi Afsharirad Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran Email: [email protected]

Received March 5, 2012; revised April 12, 2012; accepted April 19, 2012

ABSTRACT In this paper, three different controllers are proposed and simulated for maglev guiding systems to have convenient and smooth elevator motion. The proposed controllers are PID, sliding mode, and PID sliding mode controllers. The advantages and disadvantages of the proposed controllers are discussed. Although, PID controller is fast, its response affected considerably by external disturbances. Unlike PID, the sliding mode controller is so robust, but its transient is unsuitable based on application conditions. However, an acceptable controller for ropeless elevator guiding system should guaranty the passengers safety and convenient. Consequently, the response of the system should be fast, robust, and without considerable overshoots and oscillations. These required advantages are compromised in the proposed parallel PID sliding mode controller. The affectivity of the introduced controllers for maglev guiding system is investigated through conducted simulations in MATLAB/Simulink environment. The obtained results illustrate that PID sliding mode controller is a so fast and robust controller for a ropeless elevator maglev guiding system.

Keywords: Guiding System; Linear Ropeless Elevator; Non-Linear Control; PID Sliding Mode Controller

1. Introduction the guiding system manipulating [3-6]. Electromagnetic actuators are used in a wide ﬁeld of Land in the world’s largest cities is extremely expensive, applications: [7] is proposed an actuator for an axial blood which drives the expansion of rentable spaces into higher pump, an electromagnetic vibratory actuator for material and higher buildings and underground areas. However, conveying is shown by [8], or a spherical actuator based the larger and taller the buildings, the more elevators are on magnetic-dipole-moment principle introduced by [9]. required to keep acceptable waiting time for dispatching. Suspension technologies have brought interesting in- Ultra high buildings pose new problems in constructing novations. The plan-form of non-contact guiding device of high speed elevator systems, i.e. vertical oscillations, for a linear elevator is shown in Figure 1. horizontal swaying, car noise, cable length limitation, There are a few studies in the literature about magnetic and low efficiency [1]. Therefore, conventional elevators elevator guiding systems. A fuzzy, fuzzy PI and PID con- with counterweight are improper for skyscrapers. How- trollers based on feedback linearization are proposed in ever, ropeless elevators with linear motor propulsion and [2,10,11], respectively. In these studies, just one of the electromagnetic guiding system can be a proper solution for this problem. In the conventional elevators, mechanical guiding systems such as slide-ways or rollers are used [2]. However, compared with electromagnetic non-conducted solutions, the conventional lead frame has many disadvantages such as: low efficiency, more deterioration and required fre- quently lubrication and regular maintenance, more car swaying and audible noise. Particularly, it is important to make the air gap of the linear motor constant, which af- fects the magnitude of the propulsion force. Therefore, magnetic levitation technologies with electromagnetic actuators are applied to elevators’ guide shoes to restrain the car disturbance and vibration problems. Furthermore, Figure 1. Linear elevator with non-contact guiding system the air gap of elevator linear motors can be adjusted by [2].

Copyright © 2012 SciRes. EPE H. ALIPOUR ET AL. 159 counterpart electromagnets is controlled, and the other one In the single side controlling method, one of the coun- produces constant pulling force. However, it is unrealistic terpart electromagnets is energized permanently by a con- and cannot be applicable. A method is proposed in [12] for stant current or replaced by a PM. Therefore, the current of controlling the ropeless elevator air gaps by manipulating the other electromagnet is controlled for placing the ele- driven double side linear motor normal forces. This con- vator car in the proper position [10]. This method does troller can handle the car air gap just in one direction. not seem so attractive, because of considerable perma- However, composition of this system with electromag- nent currents in the electromagnets and consequently, re- netic guiding system can be promising. A guiding system lative high-power losses. with ﬁve controlled spatial degrees of freedom with im- In the other method, both of counterpart electromag- plementation of an additional torsion controller is aug- nets can be energized by a variable current. In this paper, mented in [4] and [13]. A solution for the design of both a new controller based on PID and sliding mode control- the electromagnets and the feedback control of an eleva- lers is proposed for double side elevator guiding system. tor guiding system is presented in [6]. Also, an elevator According to Newton’s second law, dynamic equation test bench for the evaluation of the simulation results is of a single side guiding system is as Equation (1). introduced in this paper. mt  F Fi,  Ft (1) In this paper, a PID sliding mode controller is proposed   kD    for ropeless elevator suspension system. The proposed where δ,  , F(i,δ), Fk(t), and FD(t) are the air gap length, method has robustness and short transient time with low second differential of the air gap, electromagnetic force, overshoot. Moreover, it is very simple and does not have electromagnetic force of the opposite actuator with con- complex mathematics. Therefore, the proposed method stant current or PM, and external disturbance force, re- can be easily used for controlling the elevator suspended spectively. guiding system in the real applications. Since the magnetic resistances of the electromagnets This paper is organized as follows. Mathematics of the and guiding rail are negligible, the reluctance of the mag- system is defined in Section 2. A brief review about slid- netic circuit composes of the electromagnets and guiding ing mode control is discussed in Section 3. In Sections 4 rail is focused on the air gap between them. Therefore, and 5, A PID controller method is introduced and simu- the effective air-gap reluctance is as follows: lated, respectively. A new application of sliding mode  control (SMC) in elevator guiding system is introduced R  (2) in Section 6 and simulation results are illustrated in Sec- 0 A tion 7. A parallel PID SMC method and its simulation where μ and A are permeability of the vacuum and cross results are presented in Section 8. Finally, Section 9 con- 0 section of the air gap, respectively. Consequently, the cludes the paper. pulling force is calculated as Equation (3).

22 2. System Definition 0 ANi t Fi,  2 (3) A set of hybrid electromagnetic actuators is shown in Fig- 4 t ure 2. In this figure, the electromagnets are equipped by where N and i(t) are the number of winding turns and the permanent magnets (PMs) which provide a magnetic off- current of single guiding equipment, respectively. set. These electromagnets can be energized by two differ- Voltage of single guiding equipment is: ent approaches. In this paper, these methods are named as single side and double side controlling methods.  AN 2 ddit  ANit2   t ut rit  0 0 (4) 2d tt2 2 t d t Consequently, mathematical model of single electromagnetic equipment system is obtained as follows:  mt   FkD  Fi ,   Ft     ki2 t    k'  Fi,, Fk     22t  (5) 42ref  t    2 2  0 AN ddit 0 ANit  t ut rit  2  2d tt2 t d t 2 where, k = (μ0AN )/4, δref is reference value for δ, and k' Figure 2. A set of hybrid electromagnetic guiding system [4]. is a constant coefficient which depends on the current of

Copyright © 2012 SciRes. EPE 160 H. ALIPOUR ET AL. the counterpart electromagnet or characteristics of the Consequently, controlling a system with n-degrees is PM. converted to control of sliding surface. The SMC effect Equation (6) illustrates that the control system can be consists of two parts: one part is equivalent control effect; more complicated by voltage controlling. Therefore, in the other one is switching control effect. Certainties of this paper, current signal is chosen as control input. the system are controlled by equivalent control effect. On In the double side controlling method, the counterparts the other hand, uncertainties of the system are controlled of the guiding system can be controlled simultaneously. by switching control effect. Thus, this system has strong Therefore, dynamic model of the guiding system is cor- robustness [15]. rected as Equation (6). Equivalent effect of the SCM can be attained by deriv- ing from sliding surface. Differential of s(t) should be   mFiF11,, 2i  FtD  2 maintained on zero by proper input signal selection. Fur- thermore, a switching term for uncertainty controlling is ki2  t  added to the system input signal, as: Fi,  1  11 2 (6) 2 t uukˆ sgn s (12) ref   ki2  t where, u is the control input of the system, uˆ is a con- Fi,  2  22  2 t trol signal for certainties of the system, k is switching coefficient, and sgn(s) is sign function that is defined as: sgnsifs   1 0 3. Sliding Mode Control Definition (13) SMC is a nonlinear control method which can control a sgnsifs  1 0 system even when there is not accurate mathematical Therefore, if Equations (10) and (12) are satisfied, s(t) model. Obviously, controlling a system with one degree is stabled and converged. is simpler than a system with n-degrees. Hence, SMC method tries to control an n-degrees system by manipu- 4. A PID Controller for Ropeless Elevator lating its one-degree representative system. This agent Guiding System system is named sliding surface [14]. Assume that dynamic of a nonlinear system can be In this paper, currents (i1, i2) are applied instead of apply- formed as Equation (7). ing a voltage signal as a reference control value to the guiding system. Moreover, if we assume that two coun- xn fx bxu  (7) terpart electromagnets cannot be simultaneously energized, the problem becomes simpler. In other words, the where x, and u are state variable and control input vectors, output of a PID controller, which is used to control the respectively. amount of air gaps in the reference value (δ ), is applied A sliding surface should be defined for controlling this ref to the system according to Equation (14). The guiding system. Sliding surface can be defined in various forms system with PID controller is shown in Figure 3. Since such as Equation (8). the current and guiding force is directly controlled, the n1 d system has suitable stability, in this case. sxt,  x (8) dt if I Output of PID 0: i 0, i I . 12 (14) is an error vector such as Equation (9), and λ is a constant if I Output of PID0: i12 I , i 0. parameter which confines the steady state's errors.

x xxref (9) 5. Simulation Results for Elevator Guiding System with PID Controller Error value converges to zero, If s(x,t) is stabled in zero. This convergence is guaranteed if Equation (10) is The efficiency of the introduced PID controller is inves- satisfied. tigated through conducted simulations in MATLAB/Simu- link environment. The parameters of the elevator guid- 1d s2  s (10) ance system are illustrated in Table 1. 2dt An external pulsed disturbance force is applied to the Reaching time on the zero-surface is regulated by η system as Figure 4. Moreover, the considered values for such as Equation (11). PID parameters (KP, KI, and KD) are 5000, 2000, and 20, respectively. In these conditions, the resulting air gap st 0   variation of the guiding system is shown in Figure 5. treach  (11)  Obviously, the position of the elevator car is disturbed

Figure 3. Double elevator guiding system with PID controller.

Table 1. Parameters of the ropeless elevator guidance system. considerably by influence of the external disturbance Parameter Notation Value force. However, using fuzzy PID controllers or a PID Mass of the elevator cars M 500 kg controller with lookup table and various parameters can lead to a more robust system [16]. The input currents of Number of the winding turns N 400 turns the counterpart electromagnets are shown in Figure 6. Cross section of the air gap A 0.65 × 0.65 m2 Initial air gap δ0 0 mm 6. A Sliding Mode Controller for Ropeless Reference air gap δref 5 mm Elevator Guiding System External disturbance force FD 50 N Accuracy of a ropeless elevator guiding system has a 60 major role in safety and convenience of the passengers,

50 and reduction of horizontal swaying and car noise. There- fore, a more robust controller than the simple PID is 40 necessary. SMC is a very robust controller and can have

30 suitable compatibility with nonlinear nature of elevator guiding system dynamics. 20 Assuming that the currents of counterpart electromag- Disturbance Force (N) Force Disturbance nets are not independent, and they cannot be energized 10 simultaneously, Equation (6) is rewritten as Equation (15). 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Therefore, the system equation can be considered as Equa- Time (sec) tion (7).

Figure 4. External disturbance force (FD(t)) for PID controller. 25 i1 -3 x 10 i2 7 20

6 15

5 10 4 Current (A) Current 5 3 Delta (mm) Delta 0 2 -5 1

0 -10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (sec) Time (sec) Figure 5. Simulation result for air gap of the PID controlled Figure 6. Input currents of the electromagnet of the PID system. controlled system.

having a realistic system, the input current is limited to 0 ku t ku t   12  to 12 amperes. Furthermore, an external disturbance force m 22FD t 2 t  t is applied Figure 7. The simulated air gap and input cur- ref   rents of the guiding electromagnets are shown in Figures 2   (15) ut11 i t utˆ  step 8 and 9, respectively. The results illustrate that the pro- posed method is more robust against disturbances; how- ever, it has unsuitable and long initial transients. ut it2 utˆ step  22    60 where, step(x) is equal to 1 if x has positive value, other- wise it will be zero. 50 For this problem, an integrated function of error is chosen as a sliding surface as Equation (16). 40 2 d ttderror 2 30 s errord2tt    error error d ddtt00 (16) 20 Disturbance Force (N) where, δerror is difference between δref and δ(t), supposing 10 that δref has a constant value. 0  t (17) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 error ref Time (sec) The value of sliding surface in the stable condition must Figure 7. External disturbance force for SMC controlled be constant. Therefore, the differential of s(t) should be system. equal to zero. Consequently, considering on Equations (16) and (17), ds/dt can be written as Equation (18). 0.01

ds dd2 error 20error 2 (18) 0.008 ddttdt 2 error 2 2 Finally, substituting d δ(t)/dt from Equation (15) in 0.006 Equation (18) results Equation (19).

Delta (mm) Delta 0.004 ut ut dsk12  22FD dtm2 t  0.002 ref  (19) d error 2 0  2error 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 dt Time (sec) In this method, input signal has two parts. One part Figure 8. Simulation result for air gap of the SCM con- controls the certainty of the system, and the other one is a trolled system. switching signal that is adjusted to uncertainty of the 14 system. Solving Equation (19) by considering Equation i1 i2 (15), first term of the control signal is achieved as Equa- 12 tion (20). Therefore, the control signal of the system is expressed as Equation (12). 10

8 2 derror error 2  uˆ  dt (20) 6  k steperror steperror 4 Input Current (A) Current Input 2 FD m 2 t 2  t ref  2

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 7. Simulation Results for Sliding Mode Time (sec) Controller Figure 9. Input currents of the electromagnet of the SCM Parameters of the system are assumed as Table 1. For controlled system.

x 10-3 8. A PID SCM Controller and Simulation 5.0001 Results An acceptable controller should have suitable transient 5 and robustness. Therefore, a combination of PID and sliding mode controllers with limited input current, which may have both advantages, is more acceptable. 5

In this paper, a parallel PID SMC controller is pro- (mm) Delta posed. In other words, the transient part of the system is 4.9999 controlled just by PID. Then, after the rise time of the system response, the controller is switched to SMC. The simulation results for the proposed controller are 4.9999 0.18 0.2 0.22 0.24 0.26 0.28 0.3 shown in Figures 10 and 11. In this test, a 50N-pulsed Time (sec) external disturbance is applied to the system between 0.2 Figure 12. Zoomed air gap of the PID SCM controlled sys- and 0.25 seconds. The response of the external distur- tem. bance is zoomed in Figure 12. Consequently, results illustrate that the proposed controller is a robust control- very simple and applicable. The PID controller has ac- ler with a suitable transient. ceptable inertial transient, but it is not a robust controller. The proposed sliding mode controller is so robust. How- 9. Conclusions ever, its response has long and unsuitable transients. In this paper, three new control methods are proposed for Since guiding system of a linear ropeless elevator has ropeless elevator guiding system. These methods are an important role in passengers’ safety and convenient, the controller should be robust with an acceptable tran- -3 x 10 sient. Hence, the PID SMC controller is proposed. This 6 controller has proper balances between these properties. 5 The performances of PID SMC controller has been veri- fied by simulation results. 4

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