Intelligent Adaptive Backstepping Control System for ... - CiteSeerX

IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 5, MAY 2007

2009

Intelligent Adaptive Backstepping Control System for Magnetic Levitation Apparatus Faa-Jeng Lin, Li-Tao Teng, and Po-Huang Shieh Department of Electrical Engineering, National Dong Hwa University, Hualien 974, Taiwan, R.O.C.

We propose an intelligent adaptive backstepping control system using a recurrent neural network (RNN) to control the mover position of a magnetic levitation apparatus to compensate for uncertainties, including friction force. First, we derive a dynamic model of the magnetic levitation apparatus. Then, we suggest an adaptive backstepping approach to compensate disturbances, including the friction force, occurring in the motion control system. To further increase the robustness of the magnetic levitation apparatus, we propose an RNN estimator for the required lumped uncertainty in the adaptive backstepping control system. We further propose an online parameter training methodology, derived by the gradient descent method, to increase the learning capability of the RNN. The effectiveness of the proposed control scheme has been verified by experiment. With the proposed adaptive backstepping control system using RNN, the mover position of the magnetic levitation apparatus possesses the advantages of good transient control performance and robustness to uncertainties for the tracking of periodic trajectories. Index Terms—Adaptive backstepping control, lumped uncertainty, magnetic levitation, recurrent neural network.

I. INTRODUCTION AGNETIC levitation systems have been successfully implemented for many applications, such as frictionless bearings, high-speed maglev passenger trains, and planar positioning systems [1]–[5]. Due to the features of the open-loop instability and inherent nonlinearities in electromechanical dynamics of the magnetic levitation apparatuses, the development of the high-performance control design for the position control of the levitated object is very important. In general, the electromechanical dynamics of the magnetic levitation apparatuses is represented by a nonlinear model consisting of the state variables of position, velocity, and coil current signals. Therefore, applications of the feedback linearization control techniques have been presented in many studies [5]–[9]. In [5] and [6], the feedback linearization technique through transforming the system model into an equivalent model with simpler form. Although the dynamic model of the magnetic levitation apparatus was simplified, only the constant system parameters were considered in feedback linearization design. This usually leads to problems with deteriorated performance and instability since the system parameters are usually varied with thermal drift. In [7], the nonlinear dynamics was approximated by the use of the Taylor’s series expansion. However, the high-order terms in Taylor’s series expansion were neglected for simplifying the original dynamic model to a second-order differential equation. As a result, development of the high control performance based on this simplified model was very difficult to complete. Moreover, the sliding-mode controller was introduced to enhance the robustness of the feedback linearization control design [8]. However, stability and control performance under the operating condition of the presence of parametric uncertainties were not

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Digital Object Identifier 10.1109/TMAG.2006.890325 Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

analyzed explicitly. A robust feedback linearization controller for an electromagnetic suspension system was presented in [9]. Although the stability of the control system was guaranteed theoretically, it seems that the relatively large overshoots and oscillation in transients existed in their experimental results. This deteriorated the transient performance and resulted in impracticality of applications of the levitation systems. Therefore, some adaptation laws should be applied to solve the mentioned difficulty. In the past decade, the amount of research about adaptive backstepping control has increased [10]–[13]. The adaptive backstepping is a systematic and recursive design methodology for nonlinear feedback control. In many cases, the feedback linearization method using a geometric approach [14] is only valid in some local region and with a disturbance-free setting. The adaptive backstepping design alleviates some of these limitations [10]. Moreover, while the feedback linearization method requires a precise model and often cancels some useful nonlinearities [14], the adaptive backstepping design offers a choice of design tools for accommodation of uncertainties and nonlinearities and can avoid wasteful cancellations. Furthermore, the adaptive backstepping control approach is capable of keeping almost all the robustness properties [10]–[13]. The idea of backstepping design is to select recursively some appropriate functions of state variables as pseudo-control inputs for lower dimension subsystems of the overall system. Each backstepping stage results in a new pseudo-control design, expressed in terms of the pseudo-control designs from preceding design stages. When the procedure terminates, a feedback design for the true control input results which achieves the original design objective by virtue of a final Lyapunov function, which is formed by summing up the Lyapunov functions associated with each individual design stage [10]. There has been considerable interest in the past years in exploring the applications of fuzzy neural networks (FNNs) or RNNs to deal with nonlinearities and uncertainties of the control systems [15]–[21]. It is well known that an FNN is capable

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of approximating any continuous functions closely. However, the FNN is a static mapping. Without the aid of staggered delays, the FNN is unable to represent a dynamic mapping. Although much research has used the FNN with staggered delays to deal with dynamical problems, the FNN requires a large number of neurons to represent dynamical responses in the time domain [15], [16]. Moreover, the weight updates of the FNN do not utilize the internal information of the neural network and the function approximation is sensitive to the training data. On the other hand, RNNs [17]–[21] have superior capabilities, such as dynamics and the ability to store information for later use. Since the recurrent neuron has an internal feedback loop, it captures the dynamic response of a system without external feedback through delays. Of particular interest is their ability to deal with time-varying input or output through their own natural temporal operation [20]. Thus, the RNN is a dynamic mapping and demonstrates good control performance in the presence of uncertainties. In this study, first, the controlled magnetic levitation apparatus is represented by a nonlinear dynamic model, in which both the system uncertainties and external disturbance force are considered. Then, an adaptive backstepping approach is proposed to control the position of the levitated object to track periodic reference trajectories with the compensation of disturbances including the friction force. Moreover, an adaptive backstepping control system using an RNN uncertainty estimator is proposed to increase the robustness of the magnetic levitation apparatus, where an RNN uncertainty estimator is proposed to estimate the required lumped uncertainty in the adaptive backstepping controller. Furthermore, the adaptation law of the RNN is derived using the Lyapunov stability theorem. Finally, some experimental results illustrate the validity of the proposed adaptive backstepping controller without RNN and adaptive backstepping control system using RNN for the magnetic levitation apparatus are compared and discussed. II. DYNAMIC ANALYSIS OF MAGNETIC LEVITATION APPARATUS The magnetic levitation apparatus is shown in Fig. 1, [22]. The plant consists of a drive coil that produces a magnetic field in the response to a dc current. The drive coil is a multiple-layer pancake coil, which is air-coil type with low inductance and high field constant. One magnetic levitated object travels along a precision ground glass guide rod. By energizing the drive coil, the magnet is levitated through a repulsive magnetic force. The magnet is of an ultra-high BH product rare earth (Nd–Fe–B) type and is designed to provide large levitated displacement. One laser-based sensor is implemented to measure the magnet position. The magnet is acted on by force from the coil, the gravity, the friction, and the external disturbance as follows:

Fig. 1. Magnetic levitation apparatus.

where and are constants depending on the geometry of permanent magnet and is a parameter describing the decrease in magnetic field with increasing axial distance; is the control variable. This equation has been approximated by constant parameters in the region of interest. However, because of the inherent inhomogeneity of magnetic fields, these constants can vary when the excursion from the calculated system is large. III. PROPOSED CONTROL SYSTEM Substituting (2) into (1), then (3) Rewrite the second-order nonlinear, single-input–single-output (SISO) magnetic levitation apparatus described by (3) and assume the parameters of the system are well known as follows: (4) where are the nominal values, and is the control variable . Equation (4) is the nominal condition of the magnetic levitation apparatus. Therefore, it is selected to be the nominal model. If the uncertainties occurred, i.e., the parameters of the system are deviated from the nominal values, the dynamic equation of the coupling system can be modified as [22], [25]:

(5)

(1) where is the distance between the coil and the magnet; is the mass of the magnet; is the friction constant; is the gravity constant; is the magnetic force; and is the external force disturbance. The magnetic force terms are modeled as follows [23], [24]: (2)

where

and denote the time-varying uncertainties; is called the lumped uncertainty and is defined as (6)

The proposed hybrid control system is shown in Fig. 2, where and are the magnet position and reference magnet position, respectively.

LIN et al.: INTELLIGENT ADAPTIVE BACKSTEPPING CONTROL SYSTEM FOR MAGNETIC LEVITATION APPARATUS

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Fig. 2. Control block diagram of adaptive backstepping control system using RNN.

Fig. 3. Network structure of RNN.

A. Adaptive Backstepping Controller The control issue is to find a control law such that the magnet position can track the desired trajectory in the presence of uncertainties. For the position-tracking objective, define the tracking error as

the related terms of Lyapunov function

and are added into (12) to obtain a new

(14)

(7) and its first derivative is (8) Then, a stability function is defined as

where is the estimated error defined as and , the disturbance is assumed to be a constant during the estimation. The above assumption is valid in practical digital processing of the observer since the sampling period of the estimator is short enough compared to the variation of . is a positive constant. Then, the first derivative of is taken

(9) where as

is a positive constant. The Lyapunov function is chosen

(10) Moreover, define

as (11)

Taking the first derivative of (11) into it, one can obtain

and substituting (8), (9), and

(12) Furthermore, taking the first derivative of and substituting (5) and (9) into it, the following equation can be derived:

(13) In order to design a controller including the tracking performance and the ability of rejecting external force disturbance,

(15)

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Fig. 4. Experimental setup of magnetic levitation system.

According to (15), the backstepping control input tained as

can be ob-

is also bounded. Thus, is uniformly conIn addition, tinuous. Using Barbalat’s lemma [26], the following result can be obtained: (22)

(16) is well known and with the assumption that positive constant, the adaptation law can be expressed as

is a

(17) Substitute (16) and (17) into (15), then

That is

and

will converge to zero at . Additionally, and . Therefore, the adaptive backstepping control system is asymptotically stable, even if parametric uncertainty, external force disturbance, and friction force exist. Unfortunately, the lumped uncertainty is unknown in actual applications. Therefore, an RNN uncertainty estimator is proposed to adapt the value of the lumped uncertainty, which is denoted , online.

B. RNN Uncertainty Estimator (18)

A three-layer RNN is shown in Fig. 3, which comprised an input layer, a hidden layer, and an output layer. The RNN maps according to the following equation:

Define the following term: (23) (19) Then (20) Since is bounded and is nonincreasing and bounded, the following result can be concluded: (21)

where represents the th input to the node of input layer; denotes the output of RNN; is the number of iterations; represents the connective weight berepresents the tween the hidden layer and the output layer; connective weight between the input layer and the hidden layer; is the internal feedback gain; and is the delay operator. In this study, and . The firing weight can be represented as (24)


Fig. 5. Experimental results of adaptive backstepping controller without RNN due to periodic sinusoidal command at Case 1. (a) Reference trajectory and tracking response of magnet. (b) Control variable. (c) Tracking error.

Fig. 6. Experimental results of adaptive backstepping control system using RNN due to periodic sinusoidal command at Case 1. (a) Reference trajectory and tracking response of magnet. (b) Control variable. (c) Tracking error.

selected sigmoid function and can be rewritten as follows:

where

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. Moreover,

(25) (29) and (26) Define vectors and hidden layer in RNN as

collecting all parameters related to

(27)

where the tracking error vector is the input of the RNN; are the collections of the adjustable parameters of the RNN. To develop the adaptation laws of RNN uncertainty estimator, the minimum reconstructed error is defined as

(30)

in which and are initialized to be zero and adjusted during online operation. Then the output of RNN can be represented in a vector form

is an optimal weight vector to achieve the minimum where reconstructed error, and the absolute value of is assumed to (i.e., ). Then be less than a small positive constant, a Lyapunov function is chosen as

(28)

(31)

, in which is initialwhere ized to be zero and adjusted during online operation; , in which is determined by the

where and are positive constants; is the estimated value of the minimum reconstructed error . The estimation of the reconstructed error is to compensate the estimated

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Fig. 7. Experimental results of adaptive backstepping controller without RNN due to periodic sinusoidal command at Case 2. (a) Reference trajectory and tracking response of magnet. (b) Control variable. (c) Tracking error.

error induced by the RNN uncertainty estimator and to further guarantee the stability of the whole control system. The reconstructed error is assumed to be a constant during the adaptation. Take the first derivative of the Lyapunov function and using (15), then

Fig. 8. Experimental results of adaptive backstepping control system using RNN due to periodic sinusoidal command at Case 2. (a) Reference trajectory and tracking response of magnet. (b) Control variable. (c) Tracking error.

Substituting (33) into (32), then

(34) (32) The adaptation laws for

and

are designed as follows:

According to (32), an adaptive backstepping control law using RNN is proposed as follows:

(35) (36) Thus, (34) can be rewritten as follows:

(33)

(37)


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Fig. 9. Experimental results of adaptive backstepping controller without RNN due to periodic trapezoidal command at Case 1 (a) Reference trajectory and tracking response of magnet. (b) Control variable. (c) Tracking error.

Fig. 10. Experimental results of adaptive backstepping control system using RNN due to periodic trapezoidal command at Case 1. (a) Reference trajectory and tracking response of magnet. (b) Control variable. (c) Tracking error.

By using Barbalat’s lemma [26], it can be shown that tends to zero at . Therefore, and will converge to zero at . As a result, the stability of the proposed adaptive backstepping control system using RNN, which is shown in Fig. 2, can be guaranteed.

The update laws of and can be obtained by the gradient descent search algorithm, i.e., (40) (41)

C. Online Parameter Learning In order to train the RNN effectively, an online parameter training methodology is proposed by using the gradient descent method. Not only the connecting weights between layers are adjusted online, but also the recurrent weights. This training scheme will increase the learning capability of the RNN. The update law shown in (35) can be rewritten as (38) Thus, the Jacobian of the controlled system is . The approximated error term needs to be calculated and propagated by the following equation: (39)

The derivations of the adaptation laws (40) and (41) can overcome the inappropriate selection of connecting weights between input layer and hidden layer and the recurrent weights. It will not affect the stability property.

IV. EXPERIMENTAL RESULTS A curve-fitting technique, which is based on the experimental data due to the varying of control variable to produce force that equals to magnet weight, is applied here to find the model of the magnetic levitation apparatus. In this study, the ratio mm mm , where is maximum displacement of the center of the permanent magnet from the coil; is the mean radius of the coil. The resulting parameters of the magnetic levitation apparatus are kg

(42)

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Fig. 11. Experimental results of adaptive backstepping controller without RNN due to periodic trapezoidal command at Case 2. (a) Reference trajectory and tracking response of magnet. (b) Control variable. (c) Tracking error.

Fig. 12. Experimental results of adaptive backstepping control system using RNN due to periodic trapezoidal command at Case 2. (a) Reference trajectory and tracking response of magnet. (b) Control variable. (c) Tracking error.

Furthermore, various learning rates and control gains of the control system are chosen as follows:

first. Then, the ISR with 2.65 ms sampling rate is used for the encoder interface and execution of the control algorithm. The ISR first reads the movable magnet position from the encoder of the DSP card. Then it calculates the control variable according to the proposed control algorithm, and then sends the calculated control variable to the driver of the magnetic levitation apparatus via DAC. The control objective is to control the movable magnet to move periodically. The initial position of is 20 mm for a sinusoidal command and the controlled stroke of the magnet, , is set to be 10 mm. The initial position of is also 20 mm for a periodic trapezoidal command and the controlled stroke of the magnet, , is set to be 5 mm. Two test conditions are provided, which are the nominal case (Case 1) and the parameter variation case (Case 2). In the experimentation, the parameter variation case is created by adding one acrylate plastic disk with the equal weight (0.121 kg) to the mass of the magnet. The experimental results of the adaptive backstepping controller without RNN and adaptive backstepping control system using RNN due to periodic sinusoidal, trapezoidal, and composite periodic commands are compared. First, the experimental results of the tracking responses, control variables, and tracking errors at Case 1 using the adaptive backstepping controller without RNN, where is set to zero, and the adaptive backstepping controller with RNN due to periodic sinusoidal command are shown in Figs. 5 and 6, respectively. Moreover, the experimental results of the tracking responses,

(43) All the learning rates and control gains in the adaptive backstepping controller are chosen to achieve the best transient control performance in the experimentation, considering the requirement of asymptotical stability. In addition, to show the effectiveness of the control system with small number of neurons, the RNN has two, nine, and one neurons at the input, hidden, and output layers, respectively. The experimental setup is shown in Fig. 4. A personal computer (PC) is the control core of the magnetic levitation system. Moreover, the PC includes a DSP card with multi-channels of ADCs, DACs, encoder, and PIO. The electromechanical plant consists of the magnetic levitation apparatus including its driver and sensor. The design features high BH product rare earth permanent magnet and high flux drive coil with maximum of 3300 ampere-turns magnetomotive force to provide more than 0.04 m of controlled levitation range. Laser sensor provides wireless position-feedback. The proposed hybrid control system is realized in the PC using a real-time algorithm, which is a “C-like” language. The methodology proposed for the implementation of the real-time hybrid control system is composed of main program and one interrupt service routine (ISR) and is executed in the DSP. In the main program, parameters initialization is set


Fig. 13. Experimental results of adaptive backstepping control system using RNN due to composite command at Case 1. (a) Reference trajectory and tracking response of magnet. (b) Control variable. (c) Tracking error.

control variables, and tracking errors at Case 2 using the adaptive backstepping controller without RNN and the adaptive backstepping controller with RNN due to periodic sinusoidal command are shown in Figs. 7 and 8, respectively. The experimental results of the tracking responses, control variables, and tracking errors at Case 1 using the adaptive backstepping controller without RNN and the adaptive backstepping controller with RNN due to periodic trapezoidal command are shown in Figs. 9 and 10, respectively. Moreover, the experimental results of the tracking responses, control variables, and tracking errors at Case 2 using the adaptive backstepping controller without RNN and the adaptive backstepping controller with RNN due to periodic trapezoidal command are shown in Figs. 11 and 12, respectively. From the experimental results, the tracking errors converge to zero quickly, and the robust control characteristics of the adaptive backstepping control system using RNN under the occurrence of uncertainties for both the test conditions can be clearly observed. To further test the tracking control performance of the proposed control system, a composite periodic command (mm), where Hz, Hz, and Hz, is adopted in this study for the two test conditions. The period of the composite command is 5 s. The experimental results of the tracking responses, control variables, and tracking errors at Case 1 and Case 2 using the adaptive backstepping controller with RNN due to composite periodic command are shown in Figs. 13 and 14, respectively. From the experimental results shown

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Fig. 14. Experimental results of adaptive backstepping control system using RNN due to composite command at Case 2. (a) Reference trajectory and tracking response of magnet. (b) Control variable. (c) Tracking error.

in Figs. 13 and 14, the tracking performance of the adaptive backstepping control system using RNN is also very good for the tracking of a composite periodic command. Furthermore, the existence of nonzero value of the control variable in steady state is necessary as a holding force for magnetic levitation and in the confrontation of uncertainties even if the actual response tracks the desired trajectory precisely in the experimentation. V. CONCLUSION This study successfully demonstrates the application of an adaptive backstepping control system using RNN to control the position of a magnetically levitated object. First, the mathematical model of the magnetic levitation apparatus was introduced. Then, the theoretical bases of the proposed adaptive backstepping controller and RNN uncertainty estimator were described in detail. Moreover, experimentation was carried out to test the effectiveness of the proposed adaptive backstepping control system using RNN for periodic sinusoidal and trapezoidal reference trajectories. The advantages of the proposed control scheme can be summarized as follows: high performance of the transient response to position tracking of the levitated object is guaranteed, and robustness to system uncertainties and external disturbance is obtained. ACKNOWLEDGMENT This work was supported by the National Science Council of Taiwan, R.O.C., under Grant NSC 95-2221-E-259-040.

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Manuscript received July 25, 2005; revised December 26, 2006. Corresponding author: F.-J. Lin (e-mail: [email protected]).

Faa-Jeng Lin (M’93–SM’99) received the B.S. and M.S. degrees in electrical engineering from the National Cheng Kung University, Tainan, Taiwan, R.O.C., in 1983 and 1985, respectively, and the Ph.D. degree in electrical engineering from the National Tsing Hua University, Hsinchu, Taiwan, in 1993. From 1993 to 2001, he was an Associate Professor and then a Professor in the Department of Electrical Engineering, Chung Yuan Christian University, Chung Li, Taiwan. From 2001 to 2003, he was Chairperson and a Professor in the Department of Electrical Engineering, National Dong Hwa University, Hualien, Taiwan. From 2003 to 2005, he was Dean of Research and Development, National Dong Hwa University, Hualien, Taiwan. He is currently Dean of Academic Affairs at the same university. His research interests include ac and ultrasonic motor drives, DSP-based computer control systems, fuzzy and neural network control theories, nonlinear control theories, power electronics, and micro-mechatronics. Prof. Lin received the Outstanding Research Professor Award from the Chung Yuan Christian University, Taiwan, in 2000; the Excellent Young Electrical Engineer Award from the Chinese Electrical Engineering Association, Taiwan, in 2000; the Crompton Premium Best Paper Award from the Institution of Electrical Engineers (IEE), United Kingdom, in 2002; the Outstanding Research Award from the National Science Council, Taiwan, in 2004; the Outstanding Research Professor Award from the National Dong Hwa University, Taiwan, in 2004. Moreover, he was the recipient of the Outstanding Professor of Electrical Engineering Award in 2005 from the Chinese Electrical Engineering Association, Taiwan.

Li-Tao Teng was born in Taipei, Taiwan, R.O.C., in 1982. He received the B.S. degree in electrical engineering from the National Dong Hwa University, Hualien, Taiwan, in 2004, where he is currently working toward the Ph.D. degree. His research interests include nonlinear control theories, artificial intelligence control theories, magnetic levitation systems, and wind-driven induction generator systems.

Po-Huang Shieh was born in Chang-hua, Taiwan, R.O.C., in 1979. He received the B.S. degree in electrical engineering from the Da Yeh University, Chang-hua, Taiwan, in 2002 and the M.S. degree from the Department of Electrical Engineering, National Dong Hwa University, Hualien, Taiwan, in 2004. He is currently working toward the Ph.D. degree in the Department of Electrical Engineering, National Dong Hwa University, Hualien, Taiwan. His research interests include ac motor drives, artificial intelligence control theories, and CNC machines.