Application of Neural-Fuzzy Modeling and Optimal Fuzzy Controller for Nonlinear Magnetic Bearing Systems
Shin-Shiung Yu, Shinq-Jen Wu and Tsu-Tian Lee**
Department of Electrical and Control Engineering
National Chiao-Tung University, Hsinchu, Taiwan, ROC
Department of Electrical Engineering
Da-Yeh University, Chang-Hwa, Taiwan, ROC
Abstract: - In this paper, we apply a new approach, called optimal fuzzy control based on linear TS type fuzzy model, to deal with nonlinear magnetic bearing systems. The linear TS type fuzzy model is used to represent the nonlinear plant. To obtain the linear TS fuzzy model, we use linear self-constructing neural fuzzy inference network(linear SONFIN) to model the nonlinear system. Once the TS fuzzy model of the magnetic bearing system is obtained, the optimal fuzzy controller can be applied if the system is completely controllable and observable. Simulation results show that the proposed optimal fuzzy controller can operate in widely range of shaft position.
Key-Words: - Fuzzy model, fuzzy-neural control, magnetic bearing systems, optimal fuzzy controller.
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I. Introduction
Active magnetic bearing systems have been applied in many areas. Typical application for active magnetic bearing include gas turbine engine, turbomolecular vacuum pumps, generator, and linear induction motor. The advantage of magnetic bearings over traditional mechanical bearings lies on that the former support moving machinery without physical contact. For example, they can levitate a rotating shaft and permit relative motion without friction. However, to design the control system for a magnetic bearing is difficult because it is a uncertain, highly nonlinear and unstable system. The most useful approach of dealing with nonlinear plant is to linearize it about a single nominal equilibrium point in order to use linear control technique. However, the performance of this single operating point linear controller can be quit good only near the equilibrium condition. In contrast, fuzzy control has been applied successfully to many nonlinear systems in recent years. Moreover, fuzzy control can deal with nonlinear and uncertain plant well. There has been a great deal of research investigating various fuzzy modeling and control framework.[1]-[5].
In this paper, we apply fuzzy modeling and optimal fuzzy controller to magnetic bearing system to obtain robust performance over the entire clearance. For
modeling, we use a set of linear local Takagi-Sugeno(T-S) fuzzy model[1] to represent the nonlinear magnetic bearing dynamic. The T-S fuzzy model is generated by a self-constructing neural fuzzy inference network (SONFIN) [6] such that we can dynamically adapt the system parameters and structure simultaneously. Then, we apply the optimal fuzzy controller design proposed in [8] to do the control work.
In the following section, fuzzy modeling method[6] we adopted is presented. The third section includes the optimal fuzzy controller[7], followed by two illustrative simulations of magnetic bearing models in section four. Section five provides the simulation results. Concluding remark is given in section six.
II. Fuzzy Modeling
We here adopt linear T-S fuzzy model to approximate the nonlinear magnetic bearing plant. The linear T-S fuzzy model is as follows.
Ri: If x1 is T1i,,xn is Tni,
Then
(1)
where Ri denotes the ith rule of the fuzzy model; x1,,xn are system states; T1i,,Tni are the input fuzzy terms in the ith rule; SX(t) denotes for continuous case; the state vector X(t) = [x1,,xn]T, the system output vector , and is the system input (i.e., control output); and are, respectively,, and matrices whose elements are known to be piecewise-continuous (PC) and real-valued functions defined on positive real space,.
Morever, to achieve self-constructing modeling effect, we adopt the SONFIN structure[6] to develop a SONFIN-based linear T-S fuzzy model such that we can dynamically adapt the system parameters and structure simultaneously.
Figure 1. is the self-constructing neuralnetwork structure [6]. Each of the nodes in SONFIN has some finite “fan-in” of connections represented by weight values from other nodes and “fan-out” of connections to other nodes.
net-input =
where denotes the activation function. We shall next describe the functions of the nodes in each of the six layers of the SONFIN.
Fig. 1. The structure of linear SONFIN
Layer 1:
and (2)
From the above equation, the link weight in layer one is unity.
Layer 2:
and (3)
where and are, respectively, the center (or mean) and the width (or variance) of the Gaussian membership function of the jth term of the ith input variable xi.
Layer 3:
and (4)
where n is the number of Layer-2 nodes participating in the IF part of the rule, and . The link weight in Layer 3 is then unity. The output of a Layer-3 node represents the firing strength of the corresponding fuzzy rule.
Layer 4:
and (5)
Like Layer 3, the link weight in this layer is unity, too.
Layer 5:
and (6)
Layer 6:
and (7)
Linear SONFIN realizes a fuzzy model of the following form:
Rule i : IF x1 is Ai1 andand xn is Ain,
THEN y is ai1x1++ainxn+bimum (8)
where Ai1,,Ain are fuzzy set; x1,,xn are system states; ai1x1ainxn andbim are constant; y is the system output.
III. Optimal Fuzzy Controller Design
In this section, we shall further adopt the following optimal fuzzy controller design scheme[7] to achieve optimal control of these nonlinear magnetic bearing systems.
Theorem [7]: For the fuzzy system in (1) and fuzzy controller below
If is ,,is
then u(t) = ri(t), i = 1,, (9)
Let be given constant matrices and .If is completely controllable (c.c.) and is completely observable (c.o.) for , then
1) there exists a unique symmetric positive semi-definite solution, , of the steady-state Riccati equation (S.S.R.E.)
(10)
2) the asymptotically local optimal fuzzy control law is
(11)
and their “blending” global minimizer in (12)
(12)
minimizes
(13)
3) and the optimal local feedback fuzzy subsystem
(14)
is asymptotically and exponentially stable.
Ⅳ. Magnetic Bearing Systems
Actual bearings have multiple electromagnets distributed around the rotor to control two axes of motion. To simplify, we assume that the horizontal and vertical dynamics are uncoupled. Here we only focus on the horizontal case.
In the following, two magnetic bearing systems are considered and their optimal fuzzy controller will be derived.
Case 1.
A single axis magnetic bearing is shown in Fig 2[8]. The
Fig 2. The horizontal case magnetic bearing[8]
original system in [8] is a vertical case, but here we consider the horizontal case. Denoting by x the difference between the position of the center of the ball and its nominal position, e the nominal air gap, and the forces created by the two electromagnets, and the associated currents.
In the open domain , we have
(15)
with
, (16)
The physical parameters of this simulation setup are given as follows:
m(mass of the ball): 0.2Kg
e(nominal air gap): 0.5mm
In our simulation, firstly, we shift the to and to because the input term biu of consequent part in T-S fuzzy model is linear. Thus the shifting can eliminate the nonlinearity of the magnetic bearing. The and are inputs of the T-S fuzzy model. After training, we can develop linear SONFIN based linear T-S fuzzy model:
:If is μ(-0.1725,0.3699) and is μ(-0.2240,0.4004)
then
:If is μ(-0.5483,0.2174) and is μ(0.6856,0.002)
then
:If is μ(-0.5483,0.2174) and is μ(-0.4934,0.2839)
then
:If is μ(-1.4396,0.002) and is μ(-0.6140,0.002)
then
:If is μ(-1.4396,0.002) and is μ(-0.6140,0.002)
then
where μ( m , σ ) =,
Fig 3.Simplified model of a dual-acting magnetic bearing[9]
The above linear T-S model is completely controllable and completely observable[5]. For the above T-S fuzzy model, the corresponding optimal fuzzy controller is
(17)
and the unique symmetric positive semidefinite solution,, i= 1,, 5 of the S.S.R.E in (10) is
=====
Case 2.
The second magnetic bearing system is illustrative in Fig 3[9]. The axis is known as “dual-acting” because two magnetic windings produce force in opposing directions along the motion axis. Two electromagnets are being used to move a rotor. The nonlinear dynamic equations for the dual acting magnetic system are:
(18)
where . The simulation model parameters are given as follows:
mass, m : 1kg
force constant, k : 0.0001
air gap : 0.001 m
For simulation we set
(19)
then the equation (18) becomes
(20)
and we shiftto and to . Thus, it becomes a regulator problem. After training, the linear SONFIN based linear T-S fuzzy model is as follows:
:If is μ(7256.17, 7552.82) and is μ(61.414, 12.876)
then
:If is μ(0.0012, 1.624) and is μ(6.6575, 0.002)
then
:If is μ(0.02309, 2.9807) and is μ(786.918, 0.002)
then
:If is μ(0.0011, 4353.8) and is μ(2.7081, 4353.84)
then
where μ( m , σ ) =,
The above linear T-S model is completely controllable and completely observable[5]. For the above T-S fuzzy model, the corresponding optimal fuzzy controller is in (17). And the unique symmetric positive semidefinite solution,, i= 1,, 4 of the S.S.R.E in (10) is
====
Ⅴ. Simulation results
For case 1:
The optimal controller output, i1 and i2, is shown in Fig. 4. And the corresponding optimal trajectory is shown in Fig. 5.
Fig. 4. The optimal controller output i1 and i2 for initial position x(0)=0.0004
Fig. 5. The state response of the designed optimal fuzzy controllerat for initial condition x(0)= -0.0004, 0.00009, -0.00003, 0.0004
For case 2:
The optimal controller output, i1 and i2, is shown in Fig. 6. And the corresponding optimal trajectory is shown in Fig. 7.
Fig. 6. The optimal controller output i1 and i2 for initial position x(0)=-0.0002
Fig. 7. The state response of the designed optimal fuzzy controllerat for initial condition x(0)= 0.0007, 0.0005, -0.0006, -0.0002
Ⅵ. Conclusion
A linear T-S fuzzy model-based optimal fuzzy controller to deal with nonlinear magnetic bearings is proposed in this paper. We use linear T-S fuzzy model to represent the nonlinear plant, where the nonlinear global model is approximated by a set of linear local models. For modeling, T-S fuzzy model is generated by linear SONFIN such that we can dynamically adapt the system parameters and structure simultaneously. Based on the identified T-S model, the optimal fuzzy controller design can be applied in a simple way. Simulation results have shown that the controller can achieve robust bearing performance over the entire clearance.
Acknowledgment
This work was supported by Ministry of Education of Taiwan under Grant EX-91-E-FA06-4-4.
References
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