Intelligent Estimation of Uncertainty Bounds of An Active Magnetic Bearings Using ANFIS

Safanah M.Raafat Dept. of Mechatronics Engineering/ Intelligent

Mechatronics Systems Research Unit International Islamic University Malaysia IIUM

Kuala lumpur, Malaysia e-mail safanamr@gmail.com

Rini Akmeliawati Dept. of Mechatronics Engineering/ Intelligent

Mechatronics Systems Research Unit International Islamic University Malaysia IIUM

Kuala lumpur, Malaysia E-mail rakmelia@iium.edu.my

Abstract—Active magnetic bearings is known to be inherently unstable systems that have a widespread applications and an increased potential research area in today’s technology. So far many controllers including H∞ robust controller has been developed for the system. However, proper selection of the required weighting functions for robust and non-conservative controller synthesize is still a critical issue that needs more investigations. In this paper the selection of uncertainty weighting function for the robust controller synthesize is automated by intelligent estimation of uncertainty weighting functions using adaptive neuro fuzzy inference system (ANFIS). Then a robust H∞ controller for the magnetic bearings is designed based on the intelligent estimated uncertainty. v- gap metric is utilized to validate the estimated uncertainty bounds for improved robust stability. Comparison with another neural based estimation method of uncertainty proves the validity of the applied approach.

Keywords- -active magnetic bearing; adaptive neuro fuzzy inference systems(ANFIS); H∞ robust controller ; v- gap metric

I. INTRODUCTION Active magnetic bearing system (AMB) is an advanced

mechatronic device having wide spread applications in turbo machineries [1]. It is a collection of electromagnets producing a magnetic field to support a rotating iron shaft without any physical contact. AMBs are widely used in industries because of its advantages such as no mechanical contact and lubrication. AMBs are open loop unstable and so the stabilization of the system can only be done by feedback control. In addition, their complex structure requires powerful control system design approaches for robust stability and robust performance. Robust H∞ control approach can provide a suitable control to linear systems and high robustness to uncertainties and variations in operating conditions, as in the work of [2], where advanced robust control strategies of H∞ control of magnetic bearing are used for high speed machining applications. However, the determination of uncertainty weighting function and performance weighting function are essential requirements in the robust H∞ design procedure and usually are very hard to select. Generally, for robust H∞

controller designs, there are no ready-to-use rules available for determining these weight functions.

Some other recent research work on the design of robust controllers [3-5] implements ad hoc procedure for the selection of the weighting functions. In [1] an automatic weight selection is developed to shape the sensitivity and complementary weighting functions. An alternative intelligent approach to estimate uncertainty bound was introduced by Buckner [6]; this ‘soft computing’ approach uses an artificial neural network to bound modeling uncertainty adaptively. Based on this approach, a confidence interval neural network to estimate uncertainty bounds for robust active magnetic bearing controller is developed in [7-9].

The combination of neural networks (NNs) and fuzzy logic systems can make good use of both sensory numerical data and expert linguistic information [10]. In previous work, we propose an adaptive neuro fuzzy inference system as a tool for estimating the uncertainty bounds for robust control of servo systems[11-12]; comparison was conducted with conventional estimation method of predictive error modeling PEM and intelligent confidence interval network CIN[7],[9] for the purpose of validation; less complicated procedure and reduced number of calculations are achieved; improved uncertainty bounds are guaranteed with larger stability margins and smaller v- gap metric. Motivated by these results and the necessity for more efficient estimation of uncertainty bounds of AMB systems for robust controller design, the ANFIS approach will be extended in this paper to be applied for MIMO system of AMBs.

This paper is organized as follows: In Section II, the modeling of the AMBs system is described in brief; this is followed by the uncertainty representation in Section III. The robust controller design is explained in Section IV. We conclude with final observations in Section V.

II. MODELING OF ACTIVE MAGNETIC BEARINGS SYSTEM

A. Analytical Model The AMB system under consideration consists of a rigid

body as shown in Fig. 1. The steel rotor has a mass of 1.549 kg and a length of 0.457 m, the two steel disks can be positioned

978-1-4244-6581-1/11/$26.00 ©2011 IEEE

to modify the modal characteristics at high speeds. Two radial AMBs are located at the ends of the rotor, orthogonally aligned in the x and y directions, together with two orthogonal pairs of sensors to measure rotor displacements from the bearing line of centre. These radial AMBs comprise a four input (bearing currents) and four output (displacements) dynamic system. A linearized system dynamics obtained from a Lagrangian analysis of an AMB [9] can be expressed using a state vector composed of the rotor displacement and their time derivatives [9]:

BuAxx +=

(1) Cxy =

(2)

where,

⎥⎦

⎤⎢⎣

⎡=

M

M

zz

x , ⎥⎦

⎤⎢⎣

⎡−

= −−BBsB

IGMKM

A 11

0,

⎥⎦

⎤⎢⎣

⎡= −

iB KMB 1

0,

T

⎥⎦

⎤⎢⎣

⎡=

0I

C , MzT zv = ,

zFB MTTM 1−= , zFB GTTG 1−= ,

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

b

a

b

a

M

yyxx

z ,

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−=

cs

s

cs

s

s

kkk

kkk

000000000000

K ,

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

i

i

i

i

i

kk

kk

000000000000

K ,

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−=

ba

baF

ll

ll

001100000001

T ,

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−+

=

ba

ab

ab

baz

llll

ll

ll0000

001100

1T ,

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

y

y

Im

Im

000000000000

M ,

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡−

=

0000000

0000000

z

z

I

IΩG

with system parameters m=1.549kg, 2210392 m.Kg.II yx

−×== , 2410 m.KgI z−= ,

m.la 1530= , m.lb 1700= ,

m/N.k s310596 ×−= , A/N.ki 929=

m/N.kc31062 ×= and Ω = 627.0 rad/sec (nominal speed:

6.0 krpm). The resulting continuous-time model is unstable, with

eigenvalues [9] λ = −471 ± 3.6i, 471 ± 3.6i, −351 ± 0.006i, 351 ± 0.006i.

Figure 1. Generalized rigid rotor supported by two radial bearings [9]

III. UNCERTAINTY REPRESENTATION Generally, the nominal plant with associated uncertainty

corresponding to N different operating points can be realized as:

,)s(u)s(y)s(G k

i

kik

ii = i=1,.., 4, k=1, 2,…, N (3)

where yik(s) and ui

k(s) are the output and input of the i-th sub matrix in the k-th operating point.

Unstructured additive uncertainty description can be expressed as [13]:

)()()()(: ssWsGsG iaiNiiiai Δ+=Π ωω ∀≤Δ 1)( ji (4)

where GNi(s) is the i-th nominal model, Wai(s) is the i-th weighting representing the magnitude of the uncertainty, and Δi(s) is any stable transfer function that is less than one in magnitude for all the range of frequencies.

Wai is generated by applying "Model Error Modeling" MEM technique [14] where an error system ε is generated as

)k(v)k(u)q(G)k( iieii +=ε (5)

ui is the controlled signal, vi(k) is the measurement noise, Gei is the estimate of un-modeled dynamics relevant error. As for nominal model, it was evaluated using equations 1 and 2. The MEM technique has the ability of completely separating bias and variance errors. Moreover, the uncertainty can be estimated regardless of the order of the nominal model. However, the drawback of this technique is that it leads to conservative uncertainty sets because it is based on worst case assumptions [15]. In order to avoid overly conservative uncertainty sets, the MEM method is modified in our approach using intelligent technique of ANFIS.

A. ANFIS Structure ANFIS is an adaptive network functionally equivalent to a

first order Sugeno fuzzy inference system. ANFIS uses a hybrid-learning rule combining back-propagation, gradient descent and a least-squares algorithm to identify and optimize the Sugeno system's signals [16], [17]. The equivalent ANFIS architecture of a first-order Sugeno fuzzy model with two rules is shown in Fig. 2. The model has five layers and every node in a given layer has a similar function.

1w

2w

11 fw

22 fw

1w

2w1y

•

•

1y

1y

1x 1x

1x

Figure 2. Architecture of an ANFIS, equivalent to a first-order

Sugeno fuzzy model with two inputs and two rules. The fuzzy IF-THEN rule set, in which the outputs are

linear combinations of their inputs, is:

Rule 1: if x1 is A1 and y1 is B1 then f1=p1x1+q1y1+r1 Rule 2: if x1 is A2 and y1 is B2 then f2=p2x1+q2y1+r2 Layer 1 consists of adaptive nodes that generate

membership grades of linguistic labels based upon premise signals, using any appropriate parameterized membership function such as the generalized bell function:

j

j

b

j

j

Aj,

]a

cx[

)x(MF 2

1

11

1

1

−+

== μ j=1, 2 (6)

where output MF1,j is the output of the jth node in the first layer, x1 is the input to node j, Aj is a linguistic label ("small," "large," etc.) from fuzzy set A= (A1,A2,A3,A4) associated with the node, and aj,bj,cj is the premise parameter set used to adjust the shape of the membership function. The nodes in Layer 2 are fixed nodes designated Π, which represent the firing strength of each rule. The output of each node is the fuzzy AND (product or MIN) of all the input signals:

)y()x(wMFjj BAjj, 112 μμ ×== (7)

The outputs of Layer 3 are the normalized firing strengths. Each node is a fixed rule labeled N. The output of the jth node is the ratio of the jth rule's firing strength to the sum of all the rules firing strengths:

213 ww

wwMF j

jj, +== (8)

The adaptive nodes in Layer 4 calculate the rule outputs based upon consequent parameters using the function:

)ryqxp(wfwMF jjjjjjj, ++== 114 (9)

where iw is a normalized firing strength from Layer 3, and (pj , qi , rj) is the consequent parameter set of the node. The single node in Layer 5, labeled Σ, calculates the overall ANFIS output from the sum of the node inputs:

∑∑

∑ ==

jj

jjj

jjj

j, w

fwfwMF5 (10)

Training the ANFIS is a two-pass process over a number of epochs. During each epoch, the node outputs are calculated up to Layer 4. At Layer 5, the consequent parameters are calculated using a least-squares regression method. The output of the ANFIS is calculated and the errors are propagated back through the layers in order to determine the premise parameter (Layer 1) updates.

B. Intelligent Uncertainty Weighting Function Fig. 3 shows the applied approach to estimate the model

error frequency response |Gei(jω)|, using the generated input-output data and ANFIS of four rules. The mapping from input ui to modeling error εi is estimated using the proposed neuro fuzzy approach.

Figure 3. Intelligent estimation of the uncertainty weighting function, using ANFIS.

To develop an intelligent estimation of uncertainty bound

Gui, the hybrid learning algorithm ANFIS is used:

)),j(e),j(G(AFS)j(G kkudkeikui ωωωω 1−=),...,1( nnkk =ω

(11) where AFS is the ANFIS function, nn is the number of data samples and eud is the updating error:

)]j(G)j(G[)k(e kuikeiud ωω −= (12) eud is utilized to enhance the search for a reduced uncertainty bound through iterative minimization procedure until some stopping criteria is met [12]

δ<= |e|)e(J udud (13) where, δ is a pre-specified very small numerical value, e.g. less than 10-3.

For validation of the intelligent uncertainty weighting function Wai the v- gap metric is implemented.

The v-gap δv (GNi,Gii), between a nominal plant GNi and a perturbed plant Gii is defined as [18]:

⎩⎨⎧ =

=otherwise 1

0 if )G,G(W))e(G),e(G(Vmax)G,G( iiNio

jii

jNi

iiNiv

ωωωδ

(14) where

22 11 |)e(G||)e(G|

|)e(G)e(G|))e(G),e(G(Vj

iij

Ni

jii

jNij

iij

Ni ωω

ωωωω

++−=

(15)

and )G(~)G()GG(wno)G,G(W Niiiii*NiiiNio ηη −++= 1 ,

i=1,.,4. )e(G)e(G jk

j*k

ωω −= , )G( kη denotes the number of poles of Gk in the complement of the closed unit disc, )G(~

kη denotes the number of poles of Gk in the complement of the opened unit disc, while wno(Gk) denotes the winding number about the origin of Gk(z) as z follows the unit circle pole and zero of Gk(z). The controller Ki that stabilizes GNi also stabilizes Gii if this controller lies in the controller set:

)G,G(b|K iiNivK,Gi iNiδ> (16)

where ))e(K/),e(G(Vminb j

ij

NiK,G iNi

ωωω 1−=

(17) is a generalized stability margin of the stable loop [GNi, Ki].

The size of the set of controllers that guarantees to stabilize both GNi and Gii is related to δv(GNi, Gii).Therefore, the smaller the v-gap between the nominal plant GNi and the perturbed plant Gii, the larger is the set of controllers stabilizing GNi that is also stabilizing Gii.

For a MIMO system, it is necessary to estimate as many uncertainty weighting functions as the number of measured variables. Consequently, the resulted weighting functions will be combined in the following uncertainty weighting matrix and used to synthesize the robust controller

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

4

3

2

1

000000000000

a

a

a

a

a

WW

WW

W (18)

Fig. 4 shows the four intelligent estimated uncertainty weighting functions using ANFIS while Fig.5 shows the four intelligent estimated uncertainty weighting functions using CIN [7]. The later is developed for comparison to validate the proposed approach of ANFIS.

10-5 10-4 10-3 10-2 10-1 100-200

-190

-180

-170

-160

-150

-140

-130

-120

-110

Mag

nitu

de (d

B)

Bode Diagram

Frequency (rad/sec)

wa1wa2wa3wa4

Figure 4. The identified intelligent weighting function Wa

for AMB model, Using ANFIS

Bode Diagram

Frequency (rad/sec)10

-510

-410

-310

-210

-110

0-105

-100

-95

-90

-85

-80

-75

-70

-65

Mag

nitu

de (d

B)

wa1wa2wa3wa4

Figure 5. The identified intelligent weighting function Wa for AMB model,

using CIN

IV. ROBUST CONTROLLER SYNTHESIZE H∞ control is a robust multivariable control technique that

seeks to calculate a controller such that the effects of model uncertainty, steady state error, disturbance, and noise effects are minimized according to performance specifications. H∞ control allows for frequency dependent bounds to be placed on each of these signals during controller synthesis to specify admissible levels of these undesirable effects [13].

In order to reflect the performance objectives into optimal control setting, the configuration of Fig.6 is considered. The main idea of this setup is to shape the closed loop transfer functions S and T with weighting functions We, Wu and Wa to achieve robust stability, disturbance rejection, and noise attenuation, and to make the closed loop response close to target reference response r. The closed-loop matrix transfer function from the exogenous variables ω=[r n]T to the regulated variables z= [z1 z2]T is given by

⎥⎦

⎤⎢⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−−=⎥

⎦

⎤⎢⎣

⎡nr

GIIW

WGWWW

ez

N

u

eNee

a

00

00

(19)

H∞ strategies are applied to find a sub-optimal controller for the linear fractional transformation problem. e is the tracking error between the reference command and the displacement output, and K(s) is the designed H∞ feedback controller. H∞ multivariable controllers were designed using MATLAB’s LMI Control Toolbox (msfsyn) and the system interconnection in Fig. 6.

Figure 6 The entire-connection of the robustly -controlled system. We is shaped into low pass filter in order to penalize

steady-state error, which has low frequency properties. And to reduce the high frequency content of the control signal, Wu is selected to reduce the high-frequency content of the control signal. The additive uncertainty weighting function Wa, derived directly from the intelligently identified uncertainty bound of Fig.4 or Fig. 5 can be used for robust control synthesis of the MIMO AMB system. And since the order of the H∞ controller is directly related to the order of the uncertainty weighting function; it is desirable to substitute a low-order transfer function for the uncertainty bound.

Table I summarizes the numerical results of the intelligent estimation and robust controller design of the uncertainty bounds for the AMB system. It is obvious that the learning time required for ANFIS is almost ten times faster than the time required for CIN, and the number of iteration of ANFIS is 20 times the number of required iterations for CIN, for the same number of training data pairs. Moreover, the lower order of ANFIS- Wa results in a lower order controller matrix than that developed using CIN, obviously with similar δv value, indicating a relatively simplified controller.

TABLE I. COMPARISON BETWEEN TWO INTELLIGENTLY IDENTIFIED UNCERTAINTY WEIGHTING FUNCTIONS

Learning time (sec.)

No. of iteration

of training

No. of training

data pairs

Order of Wak

(k=1,..,4)

CIN NNs 106.2426 100 1900 4

ANFIS 16.2154 20 1900 3

v-gap metric

Best objective

LMI

K- LMI Dimension

No. of iteration

LMI CIN NNs 0.707 1.998 [4,32] 47

ANFIS 0.707 1.997 [4,28] 44

V. CONCLUSIONS In this paper, an intelligent adaptive neuro fuzzy

methodology is extended for MIMO system. The purpose is to estimate the uncertainty bound for robust controller synthesize. AMB is used to apply the proposed ANFIS estimation method. Preliminary results show that similar accuracy to CIN neural network is obtained in a shorter learning time and less number of iterations of training. Moreover the order of the estimated weighting function is reduced using the new ANFIS method. The order of the evaluated LMI robust controller is reduced as well, which is preferred for practical applications. Future work will continue on practical application of the developed method where more challenging problems are expected to be raised.

ACKNOWLEDGMENT The authors would like to thank Dr. H. Choi for providing sufficient information on the experimental AMBs system, which simplify the development of our algorithms.

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