A Backstepping Control of LSM Drive Systems Using Adaptive Modified Recurrent Laguerre OPNNUO

Abstract

The good control performance of permanent magnet linear synchronous motor (LSM) drive systems is difficult to achieve using linear controllers because of uncertainty effects, such as fictitious forces. A backstepping control system using adaptive modified recurrent Laguerre orthogonal polynomial neural network uncertainty observer (OPNNUO) is proposed to increase the robustness of LSM drive systems. First, a field-oriented mechanism is applied to formulate a dynamic equation for an LSM drive system. Second, a backstepping approach is proposed to control the motion of the LSM drive system. With the proposed backstepping control system, the mover position of the LSM drive achieves good transient control performance and robustness. As the LSM drive system is prone to nonlinear and time-varying uncertainties, an adaptive modified recurrent Laguerre OPNNUO is proposed to estimate lumped uncertainties and thereby enhance the robustness of the LSM drive system. The on-line parameter training methodology of the modified recurrent Laguerre OPNN is based on the Lyapunov stability theorem. Furthermore, two optimal learning rates of the modified recurrent Laguerre OPNN are derived to accelerate parameter convergence. Finally, the effectiveness of the proposed control system is verified by experimental results.

I. INTRODUCTION

Permanent magnet linear synchronous motors (LSMs), which are direct-drive machines, have been widely used in industrial robots, semiconductor manufacturing systems, and machine tools [1]-[3] because of their high-performance servo-drive property.

A backstepping design involves the recursive selection of the appropriate functions of state variables as pseudo-control inputs for lower dimension subsystems of an overall system. Each backstepping stage results in a new pseudo-control design, which is expressed in terms of the pseudo control designs from the preceding design stages. The termination of the procedure results in a feedback design for true control inputs; this outcome 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 [4], [5]. Some existing methods use off-line data collected from machines under static conditions, which change during motor operation as a result of changes in motor parameters. Some methods use linear models of machines, which may not be suitable for high-performance applications with uncertainties. Neural networks (NNs) show great potential for modeling nonlinear systems, which is difficult to achieve using traditional techniques owing to the inherent parallel structure and learning ability of such systems. However, NNs feature static mapping. Moreover, the weight updates of NNs do not utilize the internal information of NNs, and function approximation is sensitive to training data. Recurrent NNs have received increasing attention because of their structural advantages in the modeling of nonlinear systems and their dynamic system control [6]-[10]. These networks are capable of effectively identifying and controlling complex process dynamics, but they entail considerable computational complexity. The recurrent Laguerre orthogonal polynomial NN [11]-[13] features dynamic mapping and demonstrates good control performance in the presence of uncertainties. Hence, the present study proposes a backstepping control system using adaptive modified recurrent Laguerre OPNNUO for LSM drive systems. The purpose of this study is to investigate and implement the proposed novel approach and thereby enhance system robustness.

This paper is organized as follows. The system structure of the LSM drive system is reviewed in Section II. A backstepping control design method using adaptive recurrent Laguerre OPNNUO is presented in Section III. The experimental results are illustrated in Section IV. The conclusions are given in Section V.

 

II. CONFIGURATION OF LSM DRIVE SYSTEM

The machine model of an LSM can be described in a synchronous rotating reference frame as follows [1]-[3]:

where

and vd , vq are the d and q axis voltages, respectively; id , iq are the d and q axis currents, respectively; Rs is the phase winding resistance; Ld , Lq are the d and q axis inductances, respectively; ωr is the angular velocity of the mover; ωe is the electrical angular velocity; λPM is the permanent magnet flux linkage; and P is the number of pole pairs. Moreover,

where vr is the linear velocity, τ is the pole pitch, ve is the electric linear velocity, and fe is the electric frequency. The developed electromagnetic power is given by [2]

Thus, the electromagnetic force is

and the mover dynamic equation is

where Fe is the electromagnetic force, M is the total mass of the moving element system, D is the viscous friction and iron-loss coefficient, and FL is the external disturbance term.

The basic control approach of an LSM servo drive is based on field orientation [2]. The flux position in the d–q coordinates can be determined with Hall sensors. In (4), (8), and (9), if id = 0, then the d-axis flux linkage λd is fixed because λPM is constant for an LSM. Moreover, the electromagnetic force Fe is proportional to , which is determined by a closed-loop control. The rotor flux is only produced in the d-axis, whereas the current vector is generated in the q-axis for field-oriented control. As the generated motor force is linearly proportional to the q-axis current while the d-axis rotor flux is constant in (4), the maximum force per ampere can be achieved. The resulting force equation is

The configuration of a field-oriented LSM servo drive system is shown in Fig. 1, which consists of an LSM, a sinusoidal pulse-width-modulation (PWM)control modulator and current control, a field-orientation mechanism, a coordinate translator, a speed control loop, a position control loop, linear scale and Hall sensors, and three sets of isulated-gate bipolar transistor (IGBT) power modules inverter. The flux position of the permanent magnet is detected by the output signals of the Hall sensors denoted as U, V, and W. Iron disks of different sizes can be mounted on the mover of the LSM to change the mass of the moving element and viscous friction. The field-oriented mechanism drive system is implemented with an field-programmable gate array (FPGA) control system, and the control law is implemented with a digital signal processor (DSP) control system.

Fig. 1.Configuration of the LSM drive system.

With the implementation of field-oriented control [1-3], the LSM drive can be simplified into a control system, the block diagram of which is shown in Fig. 2. That is,

Fig. 2.Block diagram of the backstepping control system.

where Kf is the thrust coefficient, is the command of thrust current, and s is the Laplace's operator.

The LSM used in this study features the following: 220 V, 3.5 A, 1 kW, and 213 N. For a convenient controller design, the position and speed signals in the control loop are set to 1 V = 0.075 m and 1 V = 0.075 m/s, respectively. The parameters of the system are

The "-" symbol represents the system parameter in the nominal condition.

 

III. A BACKSTEPPINGCONTROL SYSTEM DESIGN USING ADAPTIVE MODIFIED RECURRENT LAGUERRE OPNNUO

By considering an LSM servo drive system with parameter variations, external load disturbances, and friction forces, we can rewrite (10) as

where dr is the mover position of the LSM; Xp is the mover velocity of the LSM; a1 = -D/M ; b1 = Kf/M > 0 ; c1 = -1/M ; Δa and Δb denote the uncertainties introduced by system parameters M and D , respectively; and ua is the control input to the LSM drive system. By reformulating (17), the following can be derived:

where q is the lumped uncertainty defined by

The lumped uncertainty q is assessed by an adaptive uncertainty observer and is assumed to be constant during the observation. The above assumption is valid in the practical digital processing of the observer because the sampling period of the observer is short enough compared with the variation of q .

The control objective is to design a backstepping control system for the output Y of the system shown in (18) to asymptotically track the reference trajectory Yd (t) , which is dm . The proposed backstepping control system is designed to achieve the position-tracking objective. The step-by-step process is described as follows.

Step 1: For the position-tracking objective, the tracking error is defined as

and its derivative is defined as

The following stabilizing function is defined:

where k1 and k2 are positive constants and σ = ∫z1(τ)dτ is an integral action. We can ensure that the tracking error converges to zero using the integral action. Then, the first Lyapunov function L1 is chosen as

The virtual tracking error z2 = Xp - η is defined. The derivative of L1 is

Step 2: The derivative of z2 is now expressed as

To design the backstepping control system, the lumped uncertainty q is assumed to be bounded, i.e., . Then, the following Lyapunov function is defined as

Using (25) and (26), the derivative of L2 can be derived as follows:

According to (28), the backstepping control law ua can be designed as follows:

By substituting (29) into (28), (28) can be obtained as

The following term is then defined:

Then,

Given that L2 (z1(0), z2(0)) is bounded and that L2 (z1 (t), z2 (t)) is non-increasing and bounded, . Moreover, is bounded; thus, ϕ(t) is uniformly continuous [14], [15]. By using Barbalat’s lemma [14], [15], . That is, z1 and z2 converge to zero as t→∞. Moreover, , and . Therefore, the backstepping control system is asymptotically stable. The stability of the backstepping control system (Fig. 2) can be guaranteed.

Step 3:

Given that the lumped uncertainty q is unknown in practical applications, the upper bound is difficult to determine. Therefore, a modified recurrent Laguerre OPNNUO is proposed to adapt the value of the lumped uncertainty .

A three-layer modified recurrent Laguerre OPNN, which comprises an input layer (the i layer), a hidden layer (the j layer), and an output layer (the k layer), is adopted to implement the proposed control system.

are the tracking error and tracking error change, respectively. are the recurrent weight between the output layer and the input layer and the connective weight between the hidden layer and the output layer, respectively. N denotes the number of iterations. The Laguerre orthogonal polynomial [11]-[13] Gn(x) is the argument of the polynomials with -1 < x < 1; n is the order of expansion. m is the number of nodes. β is the self-connecting feedback gain of the hidden layer selected between 0 and 1. G0(x) = 1 , G1(x) = 1 - x , and G2(x) = x2 - 4x + 2. The higher-order Laguerre orthogonal polynomials may be generated by the recursive formula Gh+1(x) = [(2h + 1 - x)Gh(x) - hGh-1(x)]/(h+1). are the activation functions selected as linear functions. The recurrent modified Laguerre orthogonal polynomial NN output can be denoted as

where is the collection of adjustable parameters of the modified recurrent Laguerre orthogonal polynomial NN. , in which is determined by the selected Laguerre orthogonal polynomials and .

To develop the adaptation laws of the modified recurrent Laguerre OPNNUO u, the minimum reconstructed error ε is defined as follows:

where o* is an optimal weight vector that achieves the minimum reconstructed error. The absolute value of ε is assumed to be less than a small positive constant . Then, a Lyapunov candidate is chosen as

where δ and γ1 are positive constants and is the estimated value of the minimum reconstructed error ε. The estimation of the reconstructed error ε involves compensating for the observed error induced by the modified recurrent Laguerre OPNNUO and further guaranteeing the stable characteristics of the whole control system. The derivative of the Lyapunov function from (38) is obtained as

According to (39), a backstepping control system using adaptive modified recurrent Laguerre OPNNUO ua = ûa is proposed as follows:

By substituting (40) into (39), the following equation can be obtained:

The adaptive laws are designed as follows:

Thus, (41) can be rewritten as follows

By using Barbalat’s lemma [14], [15], ϕ(t) → 0 as t→∞. That is, z1 and z2 converge to zero as t→∞. As a result, the stability of the proposed backstepping control system using an adaptive modified recurrent Laguerre OPNNUO (Fig. 3) can be guaranteed. Then again, the guaranteed convergence of the tracking error to zero does not imply the convergence of the estimated value of the lumped uncertainty to the real values. The persistent excitation condition [14], [15] should be satisfied for the estimated value to converge to its theoretical value.

Fig. 3.Block diagram of the backstepping control system using adaptive modified recurrent Laguerre OPNNUO.

According to the Lyapunov stability theorem and gradient descent method, an on-line parameter training methodology of the modified recurrent Laguerre OPNN can be derived and trained effectively. Then, the parameter of the adaptive law shown in (42) can be computed with the gradient descent method to select the appropriate learning rate. Parameter convergence can be guaranteed, but the convergence speed is relatively slow because of the low learning rate. By contrast, parameter convergence may oscillate because of a high learning rate. In efficiently training the modified recurrent Laguerre OPNN, two optimal learning rates are derived to achieve a rapid convergence of the output tracking error. The adaptation law shown in (42) can then be rewritten as

To effectively train the parameters of the modified recurrent Laguerre OPNN, recursively obtaining a gradient vector is very important. Each component can be defined as the derivative of a cost function in the training algorithm. The gradient vector is calculated in the direction opposite to the flow of the output of each node by means of the chain rule. To describe the on-line training algorithm of the modified recurrent Laguerre OPNN, a cost function is defined as [10]

The adaptation law of the connective weight using the gradient descent method can be represented as

The above Jacobian term of the controlled system can be rewritten as . The recurrent weight from the Jacobian term of the controlled system can be updated as

where can be calculated from (33). Then, two optimal learning rates are derived to ensure the convergence of the output tracking error. The convergence analysis is provided in the following two theorems.

Theorem 1: Assume that γ1 is the learning rate of the connective weight between the hidden layer and the output layer in the modified recurrent Laguerre OPNN. Meanwhile, let Q1max be defined as Q1max ≡ maxN║Q1(N)║ , in which and ║·║ is the Euclidean norm in ℜn . If γ1 is chosen as [10,11], then

The convergence of the output tracking error is guaranteed. Furthermore, the optimal learning rate, which achieves rapid convergence, can be obtained.

Proof: Given that

Let a discrete-type Lyapunov function be selected as

The change in the Lyapunov function is obtained by

Next, the error difference can be represented by

in which Δz2(N) is the output error change and represents the change in weight. Then, (54) can be obtained by means of (45), (46), (47), and (51).

Therefore,

By substituting (53) into (57), ΔL4(N) can be rewritten as

If γ1 is chosen as 0 < γ1 < 2/{(Q1max)2[z2Ba/z2(N)]2}, then the Lyapunov stability of L4(N) > 0 and ΔL4 < 0 is guaranteed. Then, the output tracking error converges to zero as t→0 , which completes the proof of the theorem. Furthermore, the optimal learning rate, which achieves rapid convergence, corresponds to [16], [17]

i.e.,

which comes from the derivative of (58) with respect to γ1 and equals zero. The results indicate the optimal learning rate can be tuned on-line instantly.

Theorem 2: Assume that γ2 is the learning rate of the recurrent weight between the output layer and the input layer in the modified recurrent Laguerre OPNN. Meanwhile, let Q2max be defined as Q2max ≡ maxN║Q2(N)║ , where and ║·║ is the Euclidean norm in ℜn . If γ2 is chosen as [10, 11], then

The convergence of the output tracking error is thereby guaranteed. Furthermore, the optimal learning rate, which achieves rapid convergence, can be obtained as

Proof: Given that

Let a discrete-type Lyapunov function be selected as (52), and let the change in the Lyapunov function be obtained with (53). Then, the error difference can be represented by

where Δz2(N) is the output error change and represents the change in weight. Then (64), by using (46), (48), and (63), can be represented as

Therefore,

By means of (53) and (64) to (67), ΔL4(N) can be rewritten as

If γ2 is chosen as 0 < γ2 < 2/{(Q2max)2[Baz2/z2(N)]2}, then the Lyapunov stability of L4(N) > 0 and ΔL4(N) < 0 is guaranteed such that the output tracking error converges to zero as t→0. At this point, the proof of the theorem is complete. Moreover, the optimal learning rate, which achieves rapid convergence, corresponds to [16], [17]

i.e.,

which comes from the derivative of (68) with respect to γ2 and equals zero. This result shows that the optimal learning rate can be tuned on-line instantly.

In summary, the on-line tuning algorithm of the modified recurrent Laguerre OPNN is based on the adaptation laws (47) and (48) for the connective weight adjustment and recurrent weight adjustment with two optimal learning rates in (50) and (62), respectively. Moreover, the modified recurrent Legendre OPNN weight estimation errors are fundamentally bounded [18]. As long as the modified recurrent Laguerre OPNN weight estimation errors are bounded, the control signal is bounded.

 

IV. EXPERIMENTAL RESULTS

Experimental results are provided to demonstrate the control performance of the LSM drive system. A photo of the experimental setup is shown in Fig. 4. A host PC downloads the program running on DSP. The proposed controllers are implemented with the DSP control system. The current-controlled PWM VSI is implemented with the IGBT power modules with a switching frequency of 15 kHz. A DSP control board includes multi-channels of D/A and encoder interface circuits. The field-oriented mechanism drive system is implemented with the FPGA control system, and the control law is implemented with the DSP control system.

Fig. 4.Photo of the experimental setup.

The parameters of the backstepping control system are k1 = 2.2 , k2 = 1.7 , and k3 = 2.3 through some heuristic knowledge [20-22] resulting from the periodic step command from 0 mm to 84 mm at the nominal case for position tracking. In this way, good transient and steady-state control performance is achieved. The parameters of the backstepping control system using adaptive modified recurrent Laguerre OPNNUO are k1 = 2.2, k2 = 1.7, k3 = 2.3, and δ = 0.5 according to heuristic knowledge [4-5] resulting from the periodic step command from 0 mm to 84 mm at the nominal case for position tracking. In this way, good transient and steady-state control performance is achieved. First, a second-order transfer function in the following form with a rise time of 0.1 s is chosen as the reference model [19] by using the reduction of order method for the periodical step command:

The control objective is to control the mover such that it moves 84 mm periodically. Then, when the command is a sinusoidal reference trajectory, the reference model is set as a unit gain. The sampling interval of the control processing in the experiment is set at 1 ms. To show the effectiveness of the control system with a small number of neurons, the modified recurrent Laguerre OPNN is equipped with two, four, and one neuron(s) in the input layer, hidden layer, and output layer, respectively. The parameter adjustment process remains active for the duration of the experiment.

Some experimental results are provided to demonstrate the control performance of the proposed control system. Two test conditions are provided in the experiment: the nominal case and parameter variation case. The parameter variation case involves the addition of one 8.1 kg iron disk to the mass of the mover, i.e., the total mass is three times the nominal mass. The experimental results of the backstepping control system attributed to the periodic step command from 0 mm to 84 mm in the nominal case and parameter variation case are shown in Figs. 5 and 6, respectively.

Fig. 5.Experimental results of the backstepping control system attributed to the periodic step command from 0 mm to 84 mm in the nominal case. (a) Position response of the mover. (b) Response of control effort.

Fig. 6.Experimental results of the backstepping control system attributed to the periodic step command from 0 mm to 84 mm in the parameter disturbance case. (a) Position response of the mover. (b) Response of control effort.

The position responses of the mover under the nominal case and parameter variation case are shown in Figs. 5(a) and 6(a), respectively, and the associated control efforts are shown in Figs. 5(b) and 6(b), respectively. The experimental results of the backstepping control system attributed to the periodic sinusoidal command from −84 mm to 84 mm in the nominal case and parameter variation case are shown in Figs. 7 and 8, respectively. The position responses of the mover under the nominal case and parameter variation case are shown in Figs. 7(a) and 8(a), respectively, and the associated control efforts are shown in Figs. 7(b), and 8(b), respectively. Although favorable tracking responses can be obtained by the backstepping control system, the chattering in the control efforts is critical because of the large control gain.

Fig. 7.Experimental results of the backstepping control system attributed to periodic sinusoidal command from –84 mm to 84 mm at the nominal case. (a) Position response of the mover. (b) Response of control effort.

Fig. 8.Experimental results of backstepping control system attributed to the periodic sinusoidal command from −84 mm to 84 mm in the parameter disturbance case. (a) Position response of the mover. (b) Response of control effort.

The experimental results of the proposed backstepping control system using adaptive modified recurrent Laguerre OPNNUO attributed to the periodic step command from 0 mm to 84 mm in the nominal case and parameter variation case are shown in Figs. 9 and 10, respectively.

Fig. 9.Experimental results of the proposed backstepping control system using adaptive modified recurrent Laguerre OPNNUO attributed to the periodic step command from 0 mm to 84 mm in the nominal case. (a) Position response of the mover. (b) Response of control effort.

Fig. 10.Experimental results of the proposed backstepping control system using adaptive modified recurrent Laguerre OPNNUO attributed to the periodic step command from 0 mm to 84 mm in the parameter disturbance case. (a) Position response of the mover. (b) Response of control effort.

The position responses of the mover in the nominal case and parameter variation case are shown in Figs. 9(a) and 10(a), respectively, and the associated control efforts are shown in Figs. 9(b) and 10(b), respectively. The experimental results of the proposed backstepping control system using adaptive modified recurrent Laguerre OPNNUO attributed to the periodic sinusoidal command from −84 mm to 84 mm in the nominal case and parameter variation case are shown in Figs. 11 and 12, respectively. The position responses of the mover in the nominal case and parameter variation case are shown in Figs. 11(a) and 12(a), respectively, and the associated control efforts are shown in Figs. 11(b) and 12(b), respectively.

Fig. 11.Experimental results of the proposed backstepping control system using adaptive modified recurrent Laguerre OPNNUO attributed to the periodic sinusoidal command from −84 mm to 84 mm in the nominal case. (a) Position response of the mover. (b) Response of control effort.

Fig. 12.Experimental results of the proposed backstepping control system using adaptive modified recurrent Laguerre OPNNUO attributed to the periodic sinusoidal command from −84 mm to 84 mm in the parameter disturbance case. (a) Position response of the mover. (b) Response of control effort.

However, the robust control performance of the proposed backstepping control system using adaptive modified recurrent Laguerre OPNNUO under the occurrence of parameter variation at different trajectories is obvious owing to the on-line adaptive adjustment of the modified recurrent Laguerre OPNN. As indicated by the experimental results, the control performance of the proposed backstepping control system using adaptive modified recurrent Laguerre OPNNUO is better than that of the backstepping control system for the tracking of periodic steps and sinusoidal commands.

The comparison of the control performances of the backstepping control system and the proposed backstepping control system using adaptive modified recurrent Laguerre OPNNUO with two optimal learning rates is summarized in Table I with respect to the experimental results of four test cases. As shown in the table, the proposed backstepping control system using adaptive modified recurrent Laguerre OPNNUO results in smaller tracking errors in comparison with the backstepping control system. According to the tabulated measurements, the proposed backstepping control system using adaptive modified recurrent Laguerre OPNNUO indeed yields superior control performance. The comparison of the characteristic performances of the backstepping control system and the proposed backstepping control system using adaptive modified recurrent Laguerre OPNNUO is summarized in Table II with respect to the experimental results. As shown in the table, the various performances of the proposed backstepping control system using adaptive modified recurrent Laguerre OPNNUO are superior to those of the backstepping control system.

TABLE ICONTROL PERFORMANCE COMPARISON OF CONTROL SYSTEMS

TABLE IICHARACTERISTIC PERFORMANCE COMPARISON OF CONTROL SYSTEMS

 

V. CONCLUSION

A backstepping control system using adaptive modified recurrent Laguerre OPNNUO is proposed to control LSM drives for the tracking of periodic reference inputs. First, a field-oriented mechanism is applied to formulate the dynamic equation of the LSM servo drive. Then, the proposed backstepping control system using adaptive modified recurrent Laguerre OPNNUO is developed to control the LSM drive with parameter variations. With the backstepping control system, the mover position of the LSM drive achieves good transient control performance and robustness to uncertainties for the tracking of periodic reference trajectories. In increasing the robustness of the LSM drive, an adaptive modified recurrent Laguerre OPNNUO is proposed to estimate the required lumped uncertainty. The on-line parameter training methodology of the modified recurrent Laguerre OPNN is based on the Lyapunov stability theorem. Two optimal learning rates of the modified recurrent Laguerre OPNN are derived to accelerate parameter convergence. The effectiveness of the proposed control scheme is verified by experimental results.