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Farooq Azam wrote:
>
> Hi,
>
> I am looking for references on magnetic levitation system (control of a suspended ball)
> modeling and control using Fuzzy logic or neural networks. I was only able to find
> Passino's paper in SMC Transactions on the subject so far.
>
> 
>
> FarooqeAzam
I recommend you to try:
Analysis and mBased Controller Design for an Electromagnetic Suspension
System, J.L. Lin and B.C. Tho, IEEE Trans. on Education, Vol. 41, No.
2, May 1998, pp. 116129.
Precision Motion Control of a Magnetic Suspension Actuator using a
Robust nonlinear Compensation Scheme, S. Mittal and C.H. Menq,
IEEE/ASME Trans. on Mechatronics, Vol 2, No. 4, Dec. 1997, pp. 268280.
GOLOB, Marjan. Decomposition of a fuzzy controller based on the
inference breakup method. In: R. Roy, T.Furuhashi, P. K.Chawdhry
(Eds). Advances in soft computing : engineering design and
manufacturing. London; Berlin; Heidelberg: Springer, 1998, cop. 1999,
pp. 215227. Online version is reachable on:
http://www.au.feri.unimb.si/~marjan/WSC3/IC10.htm
Best regards,
Marjan Golob
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Marjan Golob, Laboratory for Process Automation, Faculty of Electrical Engineering and Computer Science, University of Maribor, Smetanova 17, 2000 Maribor, SLOVENIA, Tel: +386 62 2207161, Fax: +386 62 211178 , Email: mgolob@unimb.si , URL: http://www.au.feri.unimb.si/~marjan/
A concept called the decomposition of multivariable control rules is presented. Fuzzy control is the application of the compositional rule of inference and it is shown how the inference of the rule base with complex rules can be reduced to the inference of a number of rule bases with simple rules. A fuzzy logic based controller is applied to a simple magnetic suspension system. The controller has proportional, integral and derivative separate parts which are tuned independently. This means that all parts have their own rule bases. By testing it was formed out that the fuzzy PID controller gives better performance over a typical operational range then a traditional linear PID controller. The magnetic suspension system and the contactless optical position measurement system have been developed and applied for the comparative analysis of the realtime conventional PID control and the fuzzy control.
2. A Theoretical Approach to Fuzzy Control Based on the Inference Breakup Method
3. Fuzzy PID Controller Design
4. The RealTime Application of the Fuzzy Control for a Magnetic Suspension system
5. Experiments and Results monstrations
Fuzzy set theory was introduced by Zadeh [1] in 1965 and has been evolved as a powerful modelling tool that can cope with the uncertainties and nonlinearity’s of modern control systems. Fuzzy controllers have become popular in recent years because they do not necessarily require a theoretical model of the plant which is to be controlled. Therefore, in order to develop a fuzzy controller, one needs to first have access to a human expert, find quantifiable means to present the expert’s experience, and determine a mapping from states of the plant to the fuzzy measures with which the expert’s knowledge is quantified. In [2] a supervisory fuzzy logic based controller is applied to the magnetic suspension system. The control architecture consists of two loops; a pole placement controller is utilised in the internal loop, and a supervisory fuzzy controller with a proportionalderivative structure is embedded in the other loop to enhance the transient response of the system. This control design is compared in simulation studies with a classical pole placement controller. Several magnetic suspension systems have been developed and applied for magnetically levitated transit systems by Japanese and American corporations during the last years [3], [4]. In most cases the control system and energy supply requirements to levitate the vehicle have a higher level of complexity. The nonlinear nature of the system dynamics coupled with nonlinear characteristics of the actuators complicate the controller design. The classical controller development approach relies on a linearization of the system dynamics and on the application of a PID controller to compensate the effects of the nonmodelled nonlinearity’s. By this approach certain system is stabilized close to its nominal operating point. Problems could be existent in the case when the set point is changeable within wide operating range. The method of robust stabilisation based on control theory was applied and tested in a laboratory prototype system. The results are presented by Kazuo et al. in [5]. A method called inference breakup is presented in this paper and used to model the multivariable fuzzy controller using the decomposition of the control rule base into a set of simple rule bases. We provide a comparative analysis of the conventional PID control and the fuzzy control for the magnetic suspension application. Our main objective is to make an initial assessment of what advantages a fuzzy control approach has over conventional control approaches. The structure of this paper is as follows. A theoretical approach to the fuzzy control based on inference breakup method is described in the next section. The development of the fuzzy controller is presented in Section 3. A realtime laboratory model with contactless position measurement system is described in the section 4. Comparative analysis of the realtime conventional PID control and the fuzzy control are discussed in Section 5. The concluding remarks are given in Section 6.
The control algorithm is represented by fuzzy rules. A multivariable fuzzy system with three inputs and one output is considered. The linguistic description of the process is given by
IF X_{1(1)} AND X_{2(1)} AND X_{3(1)} THEN Y_{(1)} AND
IF X_{1(2)} AND X_{2(2)} AND X_{3(2)} THEN Y_{(2)} AND
: (1)
IF X_{1(i)} AND X_{2(i)} AND X_{3(i)} THEN Y_{(i)} AND
:
IF X_{1(m)} AND X_{2(m)} AND X_{3(m)} THEN Y_{(m)} AND
where Xk(i) is the fuzzy set of the kinput variable defined in the universe of discourse Xk, k=1,2,3, and Y(i) is the output fuzzy set defined in the universe of discourse Y. m is the number of rules. The fuzzy relation R of the system is expressed as follows
(2)
where s the aggregation operator and is the implication operator. For each rule a fuzzy relation R(i) has to be constructed. To obtain the fuzzy controller relation R, fuzzy relations R(i) are aggregated. Dimension of relation matrix R is dim[R] = dim[X1]´dim[X2]´dim[X3]´dim[Y]. To obtain the new fuzzy output Y’, given the current fuzzy inputs X’1, X’2 and X’3 the compositional rule of inference is used:
(3)
where is the composition operator of fuzzy relations. Because of the multidimensionality of the fuzzy relation (2) the composition rule of inference (3) is difficult to perform or analytical solutions can usually not be obtained. To overcome this difficulty, it is proposed to breaks up the inference of a multidimensional rule base into the three rule bases of which the inference is easier to perform or has analytical solutions. Demirli and Turksen [6] showed that rules with two ore more independent variables in their premise can be simplified to a number of inferences of rule bases with simple rules (only one variable in their premise). Their proof is only valid for submin composition operator and the min operation for conjunction in the premise, and is based on the fact that S and R implications are non increasing with respect to their first argument. When the rule:
IF X_{1} AND X_{2} AND X_{3} THEN Y
is considered and the implication operator min is used, the rule can be broken into a set of three rules:
(4)
The proof can be generalised to cases with more then three variables in the rule premise. Normally a rule base consist of more then just one rule. The inference of a rule base with more than one rule also can be broken up into the inference of a number of rule bases with simple rules. When considering a rule base with two rules, the rule base break up results in:
(5)
The result of the inference breakup can easily be extended to a large number of rules or more complex rules. The number of parts resulting from the inference breakup can become quite large (in the case of n variables in the premise of the m rules, the number of crossproducts is n^{m}), but many simplifications are possible. First simplification of the inference of a simple rule base can be achieved when the rule base can be divided into a set of simple rule bases which do not interact. The interaction of rule bases means overlap of the rules premises. When the premises of the rules are assumed to take fuzzy sets which form a fuzzy partition (fuzzy sets are convex and normalized) with no more then two overlapping fuzzy sets, then it can be derived thet the number of necessary simple rule bases is 2N1, where N is the number of fuzzy sets defined on the domain of the output variable. At this point, a crucial simplifying argument is made. From all products over n^{m} crossproducts only three terms are selected. The inference (5) reduces to
(6)
Using the inclusion operator, the otput Y from inference (5) can be written as Y’ Y’_{s}. The inclusion sign means that the output fuzzy set Y’ is contained in the set obtained by neglecting the terms in the inference (5). In many practical cases this neglected term may not be very significant. In the Section 3 the development of the simple fuzzy PID controller based on the clasic discrete PID controller is presented. The output of the the regular fuzzy controller is a nonlinaer function of three inputs (error, changeoferror and intgraloferror). The brakup method of regular fuzzy controller inference leads to similar result described by equation (5), with intersections of three basic simple inferences for each input and intersections of many simple crossinferences. The simple fuzzy PID controller inference is obtained by neglecting all simple crossinferences and the output Y’_{s} is a intersection of outputs of three basic simple inferences. Comparing this result with the output of the discrete PID controller, where y_{PID} is the linear algebraic equation, the discrete PID controller can be seen as a subset of the simple fuzzy controller. When designing a simple fuzzy controller and applying specific choices for the shape of membership functions, logical operators and the scalong of inputs and outputs, the simple fuzzy controller can amulate a linear control. Now define the fuzzy relation R_{k} from (5) as
(7)
and the following is obtained from (6):
(8)
To understand better the linguistic description (1) and its mathematical presentation (8), a blockdiagram form is proposed in Figure 1.
Figure 1: Block diagram of multivariable fuzzy system
The multivariable structure of the fuzzy system which has n inputs contains n functional blocks and an intersection block. Advantages of the block diagram presentation of a multivariable fuzzy system are in the fact that this kind of presentation allows evaluation of the contribution of each component to the overall performance of the system. An interesting possibility concerning the inference breakup is a processor which performs composition of a fuzzy set and a twodimensional fuzzy relation. A speedup of the inference can be achieved by using optimised code to perform the composition of a fuzzy set and a fuzzy relation. Because of the inference breakup, the code optimisation can be applied to software implementation.
For a PIDlike fuzzy controller the number of rules increases as the third power of the number of membership functions. In this section we would explain how to minimise the number of rules in the case of fuzzy PID controller.
The basic idea of the discrete PIDcontroller is to choose the control law by considering error e(kT), changeoferror de(kT):= (e(kT)  e((k1)T))/T and the numerically approximated integral of error ie(kT):=ie((k1)T)+Te((k1)T)). The PID control law is
(9)
where K_{P} is a proportional constant, K_{D} is a differential constant and K_{I} is a integral constant. For a linear process the control parameters K_{P}, K_{D} and K_{I} are designed in such a way that the closedloop control is stable. The corresponding analysis can be done by means of the knowledge of process parameters taking into account special performance criteria. In the case of nonlinear processes which can be linearized around the operating point, conventional PIDcontrollers also work successfully. However, the PIDcontroller with constant parameters in the whole working area is robust but not optimal. In this case, tuning of the PIDparameters has to be performed.
Fuzzy PID controller start from the same assumptions which are decisive for the conventional PIDcontroller:
The output of the fuzzy controller u(kT) is given by
(10)
where N(x) is a nonlinear function determined by fuzzy parameters.
In the case when we assume the structure of fuzzy PID controller with three input variables and the one output variable with five base fuzzy sets on each fuzzy variable, we get one rule base with maximum 125 rules. To minimise the number of rules the simplification of the fuzzy PID structure is proposed. The basic idea is to decompose of multivariable control rule base into three sets of one dimensional rule bases for each input. The output of the decomposed fuzzy controller u(kT) is given by
(11)
where N_{P}(×), N_{D}(×) and N_{I}(×) are nonlinear functions determined by three separated rule bases. The structure and principal components of the fuzzy logic based PID controller is shown in Figure 2.
Figure 2: The structure of the fuzzy logic based PID controller
The linguistic description of the knowledge base is given by three inference rules: The output signal is sum of defuzzified proportional, differential and integral action as given in equation (12):
(12)
The functional diagram of the realtime application of the simple suspension system is shown in Figure 3.
Figure 3: Scheme of the realtime application of the simple suspension system
Our fuzzy logic controller based on the personal computer (PC) is extended by the OMRON FB30AT fuzzy board (with the fuzzy processor FP 3000). An eight channels analog to digital (A/D) converter and two channels digital to analog (D/A) converter with 12 bit resolution is realised on the plugin PC board. The first cannel of the A/D converter is used to measure input of the control system, namely: ball position which is output of the optical position measurement system. The fuzzy controller software was implemented in ANSIC. The realtime sampling frequency of 1 kHz was attained. The realisation of the model is presented on figure 4.
Figure 4: The realisation of the laboratory model.
The electromagnet used in this study has 1200 coils of copper wire with diameter 1,5 mm. The ferromagnetic core has standard E form. Nominal parameters are: the coil resistance R is 2,8 , the electromagnet inductance L is 520mH and the maximum air gap D is 25mm. Electrical and mechanical time constants and gains can be calculated from mentioned nominal parameters and obtained from the measured static characteristic Fm = f(I,X). Nominal parameters, electrical and mechanical time constants and gains calculated with Equation (8) are shown in Table 1.
Table 1: Nominal system parameters
ontactless position measurement system is realised. The transmitter D1 (LED NLPB500) with emission angle of 15 degrees is used to generate blue light. The homogenous light wave is provided by optical system of two optical lenses (L1, L2). The light detector (D2) provides an output voltage proportional to the position of the ball in the light wave. The roomlight influence is compensated by differential structure of the light detector. By use of the 12 bits A/D converter the error between the set point and the current ball position the integer values from 0 to 4095 are mapped. The same normalisation procedure is used for the derivative input and for the integral input. Since ball position measurement is made in the fixed time frame the differential control (determines the rate of error) was implemented by simply subtracting the value of the current error from the previous error in ball position. All fuzzy inputs are divided into five ranges: Negative Large (NL), Negative Small (NS), Zero (ZE), Positive Small (PS) and Positive Large (PL). Triangular membership functions with 50% overlap are applied to inputs of the fuzzy controller, as shown in Figure 5.
a. .
b.
Figure 5: Input and output membership functions
The output of the controller is the output voltage (ranges of 0 to 10 V) from the D/A converter. The final control law is a sum of proportional, differential and integral action. Singleton memberships functions are applied to fuzzy outputs, as shown in Figure 5.
The knowledge base is given by three inference rule bases: the proportional rule base, the differential rule base and the integral rule base. Five rules of the rule base for the proportional part of the fuzzy controller are described in table 2.
Table 2: The rule base of the proportional part of the fuzzy PID controller
E  NB  NS  ZE  PS  PB 
U_{E}  NB  NS  ZE  PS  PB 
The differential part and the integral part of the fuzzy logic controller are realised with the same type of rule base. The inference is performed using the minimum operator and the composition is done using the maximum operator (Mamdani type of the inference engine). Two defuzzification methods are implemented: Standard methods such as Mean of Maxim and Centre of Gravity (COG). The Center of Gravity defuzification method is used in experiments presented in the next Section.
The realtime implementation of the control for a magnetic suspension system allowed to perform several interesting experiments where the fuzzy controller proved to be very efficient. At first, the ball position control by the discrete PID controller with optimal parameters is considered. The optimisation of parameters for the setpoint h=16 mm was done. The obtain performance (see Figure 6) of PID controller at high setpoints (14  18 mm) was good and highly oscillatory at low setpoints (4  8 mm).
Figure 6: The performance of the discrete PID controller
For a more serious comparison of the conventional discrete PID controller and the fuzzy PID controller, a discrete PID controller with gain scheduling was created. The gain schadulling use the setpoint to choose suitable parameters which have been optimised at different setpoints. The responses of the adaptive controller are shown in Figure 7.
Figure 7: The performance of the discrete PID controller with gain schedding
Results shown in Figure 8 prove that the fuzzy PID controller is similar to the discrete PID controller with gain scheduling in their performances.
Figure 8: The performance of the fuzzy PID controller
Differences between updown step responses in Figures 5 to 9 are not evident. It is the reason that results have been compared by use of the performance index IAE and the performance index ISE. Results are presented in Table 3. The discrete PID controller has been optimized for the set point at 14 mm. Increased effect of the differential part of the PID controller become problematical in the case of greater changes of the set point. By changing of the set point at 6 mm parameters of the discrete PID controller become nonoptimal, which is expressed by greater overshoot and by oscillation.
Table 3: The IAE performance index and ISE performance index
Set Point

Per. Index  IAE  ISE 
PID  0,392  0,613  
6 mm  Adaptive PID  0,256  0,362 
Fuzzy PID  0,319  0,400  
PID  0,163  0,136  
14 mm  Adaptive PID  0,170  0,142 
Fuzzy PID  0,159  0,127 
Performance indexes are satisfactory small in the case of the case of the set point 14 mm and much greater in the case of the set point of 6 mm. The adaptive discrete PID controller has also at the set point of 6 mm a nonoscilatory response and a great overshoot. IAE and ISE are suitable small. A step response of the fuzzy controller is similar to step response of the adaptive discrete PID controller. It should be mentioned that results of the simple fuzzy controller have been obtained by the basic shape of membership functions. It could be expected that fine tuning of membership functions will results in better performance (smaller overshoot). Fifteen rules in three rule basis have been used in the mentioned test. Note that the rule base can easily be extended with some multivariable control rules.
An approach of a fuzzy realtime controller based on the decomposition into the multivariable rule base is presented. The multidimensional fuzzy equation has been decomposed on the set of onedimensional fuzzy equations. Calculated outputs of the decomposed fuzzy equations are a little more fuzzified then the output of the original multivariable fuzzy equation. This is due to the simplification of the multivariable structure and the lack of a good mapping property of the multivariable fuzzy equation. A loss of accuracy must be accepted.
The model of the magnetic suspension system has been realised by the simple electromagnet and the steel ball. It is evident from test results that the performance of the discrete PID controller depends heavily on the operating parameters of the system. The implementation of the gain scheduling algorithm in the PID structure causes a better performance. The same or better performance has been achieved with the fuzzy PID controller. Advantages of an simplified fuzzy PID structure are:
An interesting possibility concerning the inference breakup is a processor which performs composition of a fuzzy set and a twodimensional fuzzy relation. A speedup of the inference can be achieved by using optimised code to perform the composition of a fuzzy set and a fuzzy relation. Because of the inference breakup, the code optimisation can be applied to software implementation. Further researchs will be directed to the control of theoretic aspects of simplified fuzzy PID control in vague environments like stability analysis or robustness.
1 Zadeh L A 1965 Fuzzy sets. Information and Control 8:338353.
2 Tzes A, Chen J C, Peng P Y 1994 Supervisory fuzzy control design for a magnetic suspension system. Proc. of the 3th IEEE Conf. on Fuzzy Systems, Vol 1, pp. 138143.
3 Rule R , Gilliland R 1980 Combined magnetic levitation and propulsion: the magtransit concept. IEEE Trans. On Vehicular Technology, Vol. VT29, No. 1. pp. 4149.
4 Ion B, Trica A, Papusoiu G, Nasar S A 1988 Field tests on a maglev with passive guideway linear inductor motor transportation system. IEEE Trans. On Vehicular Technology, Vol. VT38, No. 4, pp. 230236.
5 Kazuo S, Tadashi T, Hidenori K 1991 Robust stabilisation of a magnetic levitation system. Proc. of the 30th IEEE Conf. In Decision and Control, vol 3, pp. 23682371.
6 Demirli K, Turksen I B 1992 Rule break up with compositional rule of inference. Proc. of the 1th IEEE Conf. on Fuzzy Systems, San Diego, pp. 949956.
A Slideshow presentation of a paper "Decomposition of a Fuzzy Controller Based on Inference Breakup Method":
http://www.au.feri.unimb.si/~marjan/WSC3presentation.htm
Oneline and Real Time control of a ball position in magnetic field: http://www.au.feri.unimb.si/~marjan/BallOnline.html
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