[IEEE 2009 Fifth International Conference on Natural Computation - Tianjian, China...

Extended Luenberger Observer Based on Dynamic Neural Network for Inertia Identification in PMSM Servo System

Xianqing Cao1, Meng Bi2

1College of Information Engineering, Shenyang Institute of Chemical Technology, Shenyang 2Network Management Center, Shenyang University of Technology, Shenyang, China

[email protected]

Abstract

A new scheme to estimate the moment of inertia in the motor drive system in very low speed is proposed. The simple speed estimation scheme, which is used in most servo systems for low-speed operation, is sensitivity to variations in machine parameters especially the moment of inertia. To estimate the motor inertia value, an extended Luenberger observer (ELO) is applied. The observer gain matrix can be adjusted on-line based on dynamic neural network. The effectiveness of the proposed ELO is verified by simulation results. 1. Introduction

In most servo systems, motor speed is obtained by counting the number of pulses generated by incremental encoders. Since this method gives the average speed, not instantaneous speed, it causes a detection lag. Under low-speed conditions, in particular, when the interval between pulses is as large as or larger than the sampling time, the detection lag has much influence on the speed controller. Especially at low speed, instantaneous speed estimation is adopted to give a better result [1]. The proposed instantaneous speed estimation derived from mechanical modeling is sensitive to the variation of machine parameters, especially the moment of inertia. Information on the moment of inertia is essential in the instantaneous speed observer. In servo systems, as the moment of inertia is changed by variations in the load and the speed, a robust estimation strategy against the variation should be designed for high-precision servo response.

To estimate the moment of inertia and the load torque at the same time, adaptive control theories such as the least square method are applied [2]. However, the problem with this method is that the speed command must be changed all the time, so problems

arise when the moment of inertia and the load torque change simultaneously. Several methods to estimate the moment of inertia separately, for example, RELS (recursive extended least square) and the Kalman filter, have been proposed [3]. Since the gain matrix has to be tuned every sampling period to meet some objective criteria, these methods are computationally intensive.

The potential of powerful mapping and representational capabilities of artificial neural network architectures have long been recognized in the neural network community [4]. The introduction of back propagation algorithm by Rumelhart et.al enabled real-time applications of neural networks as adaptive systems [5]. In [6], Puskorius and Feldkamp investigate the application of recurrent multiplayer perceptrons (MLP) to the control of nonlinear dynamical systems and propose an alternative training algorithm to update the dynamic weights of the network based on parameter-based extended Kalman filter (EKF) estimates. Their simulation results with a number of nonlinear systems favor the use of EKF-based training algorithm over the conventional back propagation. Zhu et.al focus on the application of dynamic recurrent neural networks (DRNN) as observers for nonlinear systems [7]. They consider a class of single-input-single-output (SISO) nonlinear time-varying systems in their work, where they prove the boundedness of the observer error and the DRNN weights during adaptation using Lyapunov stability theory and the well-known universal approximation theorem for neural networks. With an alternative approach, Wang and Wu exploit the multiplayer recurrent neural networks as matrix equation solvers and utilize this scheme to synthesize linear state observers in real-time by solving the Sylvester’s equation for pole placement [8].

In this paper, a new estimation strategy, which estimates the moment of inertia by using ELO (extended Luenberger observer) is proposed. In the proposed estimator, the moment of inertia is identified

2009 Fifth International Conference on Natural Computation

978-0-7695-3736-8/09 $25.00 © 2009 IEEE

DOI 10.1109/ICNC.2009.357

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978-0-7695-3736-8/09 $25.00 © 2009 IEEE

DOI 10.1109/ICNC.2009.357

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978-0-7695-3736-8/09 $25.00 © 2009 IEEE

DOI 10.1109/ICNC.2009.357

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978-0-7695-3736-8/09 $25.00 © 2009 IEEE

DOI 10.1109/ICNC.2009.357

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978-0-7695-3736-8/09 $25.00 © 2009 IEEE

DOI 10.1109/ICNC.2009.357

48


978-0-7695-3736-8/09 $25.00 © 2009 IEEE

DOI 10.1109/ICNC.2009.357

48


978-0-7695-3736-8/09 $25.00 © 2009 IEEE

DOI 10.1109/ICNC.2009.357

48


978-0-7695-3736-8/09 $25.00 © 2009 IEEE

DOI 10.1109/ICNC.2009.357

48


978-0-7695-3736-8/09 $25.00 © 2009 IEEE

DOI 10.1109/ICNC.2009.357

48


978-0-7695-3736-8/09 $25.00 © 2009 IEEE

DOI 10.1109/ICNC.2009.357

48


978-0-7695-3736-8/09 $25.00 © 2009 IEEE

DOI 10.1109/ICNC.2009.357

48

based on the disturbance torque, which is estimated by a full order observer at the previous sampling period. In the first step, the moment of inertia is assumed to be a constant and the disturbance load torque and speed are estimated. In the second step, the moment of inertia is identified by use of the disturbance load torque estimated in the first step. Figure 1 shows the overall block diagram of the proposed observer using the ELO. Although there is a solid theory behind the linear Luenberger observer and there are rigorous analytical methods of selecting the observer gain matrix, such results are not available for the extended version, yet. However, as we will demonstrate in this paper, dynamic neural network is used to overcome this difficulty by letting L adapt on-line while the system is running. Simulation is carried out to illustrate the performance of the proposed estimator at very low speed.

Motor and

encoder

Extended Luenberger observer

eT

L

Kmθ+

−

LT

mω

mθ

mJ

mθ mω LTFull order observer 3

Figure 1 Overall block diagram

2. Modeling of mechanical system and full

order observer

The mechanical system of the servo motor system can be represented as shown in figure 2.

mm

1BsJ + s

1eTLT

mω mθ

Figure 2 Dynamic model of mechanical system

It is assumed that the time constant of the

disturbance load torque variation is much larger than that of a servo controller. Thus the variation of the disturbance load torque can be considered as zero[9]. The state equation of the mechanical system can be written as

CxBuAxx

=+=

y (1)

where

−−=

000

10010

mm

m

JJA B

=

0

10

mJB [ ]001=C

[ ]TLmm Tωθ=x , mθ=y , eT=u , mθ is angular

position, mω is angular speed, mJ is moment of inertia,

mB is friction coefficient, eT is motor torque, LT is disturbance torque.

Considering (1), the full order state estimator for this system can be arranged as

xCKBuxAx

ˆˆ)ˆ(ˆˆ

=−++=

yyy (2)

where, [ ]T

Lmmˆˆˆˆ Tωθ=x , m

ˆˆ θ=y , K is the gain matrix.

The characteristic equation of (2) is given in (3) and the observer gain can be determined using the pole placement method

( )[ ]

0

s-det

m

3

m

m1m2

2

m

mm13

=−++

++=

Jk

sJ

BkJkJ

BJks-s KCAI

(3)

Suppose that the design specifications for this system require that the three roots of the characteristic equation be placed at 11 β=s 22 β=s and 33 β=s . Denoting the desired characteristic equation as )(scα , we have

( ) ( )( )( )( )

( ) 0-

321133221

2321

3

321

=+++++−=

−−−=

ββββββββββββ

βββα

sss

ssssc

(4)

The design is completed by choosing the gains 1k , 2k and 3k such that the coefficients of (3) are equal to those of (4)

( )

( )

( )

m3213

2

m

m

m

m321

1332212

m

m3211

Jk

JB

JB

kJB

k

βββ

βββ

ββββββ

βββ

=

++++

++=

−++−=

(5)

Suppose that 321 ββββ === and 0m ≈B , (5) can

be rearranged as

m3

3

22

1

33

Jkkk

βββ

==

−= (6)

4343494949494949494949

3. Inertia identification using ELO 3.1 Extended luenberger observer

The moment of inertia changes slowly, in comparison with the sampling period of the servo controller. Thus the variation of the moment of inertia can be considered as zero. When the motor speed and the moment of inertia are taken as the state variables of the mechanical system, the resultant system equation can be given as

( )Hx

uxx==

ytf ,,

(7)

where, [ ]Tmm Jω=x , [ ]01=H , mω=y , eT=u ,

( )T

em

Lm

mm

m 01ˆ1 +−−= TJ

TJJ

Bf ωx

Considering (7), the full order state estimator for this system can be arranged as

( )xH

uxxˆˆ

,,ˆˆ==

ytf (8)

where, mω=y , [ ]T

mmˆˆˆ Jω=x ,

( )T

m

L

m

m

m

m 0ˆ1ˆ

ˆ1ˆ

ˆˆ +−−= T

JT

JJB

f ωx

Obviously, (8) is nonlinear. Although there is a solid theory behind the linear Luenberger observer in (2) and there are rigorous analytical methods of selecting the observer gain matrix K, such results are not available for the extended version in (8), yet. However, as we will demonstrate in the following sections, there is a way to overcome this difficulty by letting L adapt on-line while the system is running.

3.2 Additive model for recurrent neural networks

The most widely used dynamic neural network is the additive model by Grossberg [10]. The state dynamics of the additive model is described by

)())(()()( Is tttt IWxWxx ++⋅−= σ (9)

Usually, the weights matrix multiplying the input vector I(t) is chosen to be identity and the passive decay matrix, is a diagonal positive definite matrix and the interactions between states is provided through Ws and the nonlinearity of the neurons, )(⋅σ , but these are not necessities. The biological motivation for the additive model provided by Sejnowski [11] served to the increased popularity of this structure in numerous applications. The static MLP is just a special case of

the additive model obtained by setting the time-derivative of the states to zero, thus imposing the static constraint on the states and restricting the weight matrix Ws to be strictly lower diagonal. In this case, the feedforward MLP is expressed as

IWxWWIWxWx I1

s0I1

s1 )( )( −−− +=+= τσσ (10)

where some of the states may be designated as the outputs of the MLP. A special case of interest is when the nonlinearity of the neurons in (9) is chosen to be a linear function. For the choice σ (a) = a, (9) reduces to a linear dynamic neural network whose dynamics are of the form

)()()()( Is ttt IWxWx +⋅−= (11)

Considering the connection between the additive model in (9) and the extended Luenberger observer (ELO) in (8), with proper adaptation, it is possible for the linear dynamical neural network to approximate the stable extended Luenberger observer (ELO), since with sufficient number of neurons and proper choice of weight matrices the additive model can approximate any function with an arbitrarily small error. In fact, [12] demonstrates how approximately a neural network can learn the unknown system dynamics.

In cases where a complete model of the dynamical system is not available, such approaches can be taken to obtain approximators of system models and substituted in the observer structure in proper places. From this point on, however, we will assume that either the full system dynamic equations are known or a neural network has been trained to sufficient accuracy as described for this purpose. This is because; the main focus of this study is to determine the capabilities of the adaptive observer structure, not to investigate the function approximation capabilities of the mentioned additive model. 3.3 Stochastic gradient adaptation for extended luenberger observer

In this section, we will consider the discrete-time equivalent of the ELO for reasons of analytical simplicity in computing and evaluating the gradient for off-line adaptation. In this context, the back propagation refers to the back propagation-in-time of the partial derivatives with respect to the observer gains. According to (7) and (8), the system and the observer we are considering are given by the following equations.

kk

kkk

yf

Hxkuxx

==+ ),,(1 (12)

4444505050505050505050

kk

kkkkk

yyykf

xHLuxx

ˆˆ)ˆ(),,ˆ(ˆ

1

=−+=+ (13)

Suppose we want to train for L such that the mean-square error (MSE) along a given training trajectory { } 1, −

=

N

oiii yu is minimized. We would like to remark at this point that MSE is not the sole possibility as the performance criterion. When the instantaneous squared error is used as a stochastic approximation to MSE, the computed the gradient of this stochastic cost function with respect to the weights, one gets Widrow’s stochastic gradient for MSE[13]. Note that in computation of the stochastic gradient with respect to the weights of a recursive system Widrow suggests the designer approaches the problem with care and caution. Although the cost function depends only on the instantaneous value of the error, due to the recursion the error still exhibits a back propagation property and it is easy to oversee this. In the case of the ELO, at time step k, one may use the actual gradient expression computed with full consideration of the recursive structure of the topology or use an approximate version of the gradient, which is very accurate if the learning rate is chosen to be a small value. The former allows the use of larger learning rates, whereas the latter may go unstable for those same values of learning rates. The stochastic cost function and its approximate gradient (without consideration of the recursive nature of the system) are simply computed using (14) and the current value of the observer gains.

[ ][ ]

LLL

LxHxLux

Lx

LxHx

L

∂∂⋅−=

⋅−+∂

∂⋅−−=

∂∂

∂∂

⋅−−=∂∂

−=

+

−−

−−−

J

yy

kf

yyJ

yyJ

kk

nxnkk

kkkx

k

kkk

kk

ˆ1)ˆ(

ˆ)1,,ˆ(

ˆ

ˆ)ˆ(2

ˆ)ˆ(ˆ

1

11

111

2

η

(14)

where, η is the learning rate, fx(.) represent the Jacobians of the corresponding functions with respect to the state vector, and nn×]1[ represents an all-ones square matrix of the size of the state vector.

It is a well-known fact that Widrow’s stochastic gradient algorithm makes the weights converge to the optimal MSE solution in the mean. Furthermore, the LMS algorithm is a well understood and proven algorithm that is useful in real-time adaptation problems (yet, there are stochastic versions of the

information theoretic adaptation criteria also, and in studies they are shown to exhibit all the advantageous properties of LMS and more [14]).

4. Simulations

In this section, computer simulations of the PMSM

servo system using the proposed ELO based on dynamic neural network are performed. The parameters of the PMSM used in simulation research are as follows

Table 1 Parameters of PMSM

Rated power PN(kW) 5 Rated voltage UN(V) 300 Magnetic pole pairs pn 4 Rated speed(r/min) 3000 Inertia (kg/m2) 0.01215 viscous damping coefficient (Nm.s) 0.001 Stator resistance Rs( ) 0.15 Rotor flux linkage f (Wb) 0.3775

In simulation, the sampling period of the system is

100 s. The speed control system is in nominal condition, the load torque with 4 Nm and step rotor speed commands (1 rpm) are given at the beginning of the simulation. At 0.2s, the moment of inertia is ten times of nominal value. The identification curve of inertia is shown in figure 3. At 0.8s, the load torque is changed to 10Nm. Figure 4 shows the load torque observe curve. It can be seen that the proposed ELO can observe the moment of inertia and load torque well and truly from figure 3 and figure 4. Figure 5 shows step rotor speed response curve of PMSM servo system. The PMSM servo system based on proposed ELO shows good results for low-speed operation against the parameter variation and the load disturbance.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Time [s]

Iner

tia [k

gm2 ]

Figure 3 Identification result of inertia

4545515151515151515151

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

2

4

6

8

10

12

Time [s]

Load

torq

ue [N

m]

Figure 4 Identification result of load torque

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.2

0.4

0.6

0.8

1

1.2

1.4

Time [s]

Spe

ed [r

pm]

Figure 5 Angular velocity step response curve of PMSM servo system

5. Conclusions

A new estimation algorithm, which observes the

moment of inertia using ELO, is proposed. Dynamic neural network is used to adjust the observer gain matrix on-line while the system is running. In the proposed algorithm, the moment of inertia is estimated by use of the disturbance torque, which was estimated by a full order observer at the previous sampling period. With this identified inertia value, the parameter of the full order observer is appropriately updated. The speed control performance in a low-speed region is improved with the inertia identification method using ELO. The theoretical basis of the proposed ELO was

derived, and the effectiveness of the proposed ELO was confirmed by the simulation results.

6. References [1] Hori, Y., Umeno, T., Uchida, T., and Konno, Y.: “An instantaneous speed observer for high performance control of DC servomotor using DSP and low precision shaft encoder”. EPE’91, 1991, Vol. 3, pp. 647–652 [2] Awaya, I., Yoshiki,Miyake, I., and Ito, M.: “New motion control with inertia identification function using disturbance observer”. IECON’92, pp. 77–81 [3] Hong, S.-J., Kim, H.-W., and Sul, S.-K.: “A novel inertia identification method for speed control of electric machine”. IECON’96, pp. 1234–1239 [4] K. Hornik, M. Stinchcombe, H. White, “MLP’s are universal approximators,” Neural Networks, vol. 2, pp.359-366. [5] D. Rumelhart, G. Hinton, R. Williams, “Learning internal representations by error backpropagation,” Nature, vol 323, pp.533-536. [6] G.V. Puskorius, L.A. Feldkamp, “Neurocontrol of Nonlinear Dynamical Systems with Kalman Filter Trained Recurrent Networks,” IEEE Trans. Neural Networks, vol. 5, no. 2, pp.279-297. [7] R. Zhu, T. Chai, C. Shao, “Robust Nonlinear Adaptive Observer Design Using Dynamic Recurrent Neural Networks,” Proc. Amer. Cont. Conf., , New Mexico, June 1997,pp. 1096-1100. [8] J. Wang, G. Wu, “Real-Time Synthesis of Linear State Observers Using a Multilayer Recurrent Neural Network,” Proc. IEEE Int. Conf. Industrial Tech. , 1994, pp. 278-282. [9] K.B. Lee, J.Y. Yoo, J.H. Song and I. Choy. Improvement of low speed operation of electric machine with an inertia identification using ROELO, IEE Proc.-Electr. Power Appl., Vol. 151, No. 1, January 2004,pp:116-120 [10] J.C. Principe, N. Euliano, C. Lefebvre, Neural and Adaptive Systems, John Wiley and Sons Inc, NY, 1999. [11] T.J. Sejnowski, “Skeleton Filters in the Brain,” in Parallel Model of Associative Memory, Hinton and Anderson (eds), LEA Inc., 1981,pp. 189-212. [12] Deniz Erdogmus, A. Umut Genç, José C. Príncipe. A neural network perspective to extended luenberger observers Measurement and Control, v 35, n 1, Feb. 2002, pp 10-16 [13] B. Widrow, S.D.Stearns, Adaptive Signal Processing, Prentice Hall, NJ, 1985. [14] D. Erdogmus, J.C. Principe, “An On-Line Adaptation Algorithm For Adaptive System Training with Minimum Error Entropy: Stochastic Information Gradient,” submitted to Independent Component Analysis 2001.

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