2012 IEEE International Conference on Control System, Computing and Engineering, 23 - 25 Nov. 2012, Penang, Malaysia
A Hybrid Neuro-Fuzzy - P.I. Speed Controller for B.L.D.C. Enriched with an Integral Steady State Error Eliminator Mahdi Mansouri
S. Hr. Aghay Kaboli
Jalil Ahmadian
Jeyraj Selvaraj
University of Malaya Kuala Lumpur, Malaysia
[email protected]
University of Malaya Kuala Lumpur, Malaysia
[email protected]
University of Malaya Kuala Lumpur, Malaysia
[email protected]
University of Malaya Kuala Lumpur, Malaysia
[email protected]
aforementioned system has become the most desirable especially in the area of controlling nonlinear systems [8]. Neuro-Fuzzy controllers are conventionally improved with a P.I. [9], P.D. [10] or an adaptive controller [11] how a paralleled S.S.E.E. can be accompanied with to improve the system performance and efficiency. In this hybrid controlling system, a P.I. (Proportional Integral) N.F.C. is the main follower controller paralleled with an Integral S.S.E.E., which are used to track the exact control law. The in-use N.F.C. training algorithm in a direct adaptive control scheme is back-propagation algorithm and the trained N.F.C. is used considering the “Fuzzy Set Theory Based Control on a Phase Controlled Converter D.C. Machine Drive” [12]; although a smooth and simple activation mechanism is applied for the Integral S.S.E.E. to modify the law of control adaptively [1]. The performance of the represented speed control is evaluated under varying parameters and loads to show the efficiency and performance of the applied controlling system.
Abstract—This paper is subject to present a hybrid NeuroFuzzy (N.F.) - P.I. fed Controller for controlling the speed of B.L.D.C. (Brush Less D.C.) motors to evolve the drives controlling performance at both transient and steady state conditions by considering a paralleled robust integral S.S.E.E. (Steady State Error Eliminator) to enrich the whole controlling process. In the presented hybrid system, P.I. N.F.C. is the main controller loop while the paralleled integral S.S.E.E. controller reimburses the steady state errors. The presented B.L.D.C. drive contains the capabilities of quick tracking, small steady state error and high stability despite of all load and parameter variations. MATLAB simulation results depict the impressiveness of the presented controlling system. Index Terms—Nonlinear Control, Hybrid Neuro-Fuzzy, B.L.D.C. Speed Control, Integral Steady State Error Elimination.
I. INTRODUCTION A B.L.D.C. machine is a synchronous machine accompanying with a P.M. (Permanent Magnet) in the rotor circuit. The windings of the armature are mounted on the stator switched electronically respecting to the rotor position. The B.L.D.C. motors are widely applied in robotic servo implementation, machine tools and dynamic actuators, based on their preferred electrical and mechanical features, efficiency increasing feature and the ability of lowering the inertia momentum [3]-[5] and [13]. The high degree of precision is not something imperious for the most electrical drives, although, a favorable controlling performance has to be prepared in high performance drive application even when the motor and load parameters are changing during the functioning. The constant gain controllers are the most conventional system implemented in high performance variable speed drives. Their prominent drawback is the poor performance when the load is nonlinear, parameters are changing and there are uncertainties. Consequently, the strategy of control in high performance electrical drives has to be robust and adaptive. When there is a demand for high performance implementation of drive, a favorable controlling performance has to be served while the motor and load parameters are changing during operating. It can be resulted that there is a vivid interest in developing adaptive controlling systems how different schemes of adaptive control for B.L.D.C. motors are proposed based on nonlinear models [6], and [7]. A.N.F.I.S. (Adaptive Neuro-Fuzzy Inference System) is a given name for neural fuzzy network-based systems. The
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II. A HYBRID N.F.- P.I. SPEED CONTROLLER FOR B.L.D.C. DRIVES The B.L.D.C. machine can be modelled as V=E+(R+jωl)I, how V and E are sinusoidal at the frequency of ω, R is the phase resistance and ωL is phase inductance (Fig. 1). Due to E = jωλm when λm stands for the linkage flux of windings of the stator per phase respecting to the permanent magnet. Still by assuming and implementing a position feedback that keeps V and E (and henceforward I) in the same phase, the equation of the voltage can be simplified in the form of algebraic as: (1) When substituting relations of ~ and ~ , we result in:
Fig. 1. Steady state per phase equivalent circuit of B.L.D.C. motors
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2012 IEEE International Conference on Control System, Computing and Engineering, 23 - 25 Nov. 2012, Penang, Malaysia
Fig. 2. Block diagram of the B.L.D.C. drive
(4) The above equation explicates that controlling the voltage is the way of system stabilizing while there is a varying torque demanded with constant rotational speed. The basis of the presented controlling system has been based on these inferences [14]. Block diagram of B.L.D.C. drive system is shown in Fig. 2. As shown in the figure, there are two control loops in the system. The inner controlling loop provides bridge pulse synchronization with the forces of electromotive. The outer loop consists of two paralleled controlling loops also, an adaptive Neuro-Fuzzy P.I.D. that regulates the speed of the motor by changing the voltage of the D.C. bus and a S.S.E.E. which is taken into the controlling loop after the system goes into stability and removes any steady state error effect.
Fig. 3. The hybrid neuro-fuzzy controller for the B.L.D.C. drive
(2) , and the main equation which declares the relation between input D.C. voltage and the speed of motor is: (3)
A. The Hybrid N.F. Controller Applying the feedback voltage controlling technique and assuming ideal voltage control, the controlling block diagram of B.L.D.C. drive can be simplified as shown in Fig. 3, where” speed reference” indicates the demanded operational speed and “Motor Rotational Speed” stands for motor momentum velocity. Applied Neuro-Fuzzy in this paper constructed on the Mamdani fuzzy model which is adopted for controlling the speed of B.L.D.C. motors P.I. type N.F.C. has favorable characteristics of transient response; although a steady state error occurs that is easily removed by the integral steady state error eliminator.
In the above equation, V is sinusoidal at a frequency of ω, P is the number of poles, λm stands for the linkage flux of windings of the stator per phase respecting to the permanent magnet, R is the phase resistance, m stands for the number of phases and, Tem shows the electro-magnetic torque [2]. By assuming a constant imposed torque, the aforementioned equation indicates that changing V directly influences the motor rotation speed. Controlling the voltage of the D.C. bus influences the amplitude of the drive generated semisinusoidal waveform, and consequently by referring from the Equation 3, varies the motor rotation speed. Just by transferring the parameters of the Equation 3, another equation can be inferred as:
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2012 IEEE International Conference on Control System, Computing and Engineering, 23 - 25 Nov. 2012, Penang, Malaysia
(the links weights) in the Equation 7, represent the r The system output is using action of the output of the kth rule. the central defuzzification for Mamdani M fuzzy model given as: ∑
The N.F.C. can regulate the rules of fuzzy control by o the outputs. The function manipulating the weights of membership parameters and thhe weights of outputs of N.F.C. are modified applying the back-propagation b algorithm to lessening E, the performance inndex, . (9) . , where the error of the speeed tracking is indicates the reference speedd and ω stands for the actual rotational speed. The parameteer in the Equation 7 can be manipulated as: (10)
Fig. 4. The Second Layer Degree of Membershiip Calculation
It should be considered that in similaar hybrid systems which are based on P.D., there usually are significant s amount of steady state error when the load is appliedd since there are no integral mechanisms. The appeared steady state s error in those systems can be reduced just by long period time of training for the N.F.C.; although, this usually results inn high gains how causes noise in the controlling systems. The N.F.C.-P.D. C.-P.I that contains drawbacks may be conquered with a N.F.C an integral compensator [1].
,where the parameter of η represent the rate of learning. The performance index gradiennt is deducted as below: .
∑
, it should be noticed that
III. NEURAL-FUZZY ARCHITECTURE
(11)
has to be calculated
using the dynamics of the mootor. By applying Equation 5 to Equation 8, the membership fuunctions parameters performance index gradient is derived as:
The adopted Neuro-Fuzzy model basiss in this paper is Mamdani Fuzzy model. The aforementionedd model is a fourlayer neuro-fuzzy system applied to control the t desired system. In Mamdani N.F.C. a common rule set withh two fuzzy if-then is as following: If x1 is A1 and x2 is B1 thenn ω1 If x1 is A2 and x2 is B2 thenn ω2 The input nodes transfer the input signalss to the next layer; it means , where stands for the ith input to the node of the first layer. The x1 signal indicatess the error of speed and x2 signal shows the error changing rate, x2 =∆e. The degree of membership function is calculated in the G activation second layer for the input values. The Gaussian function is implemented to depict the functioon of membership. It is assumed that the weight between the meembership function and the input is unity. The output if this layerr is shown in Fig. 4 and Equation 7. ,
(8)
∆
. .
∑
∆ ∆ ∆
∑
(12) (13)
IV. SIMULATTION RESULTS The whole system is esttablished and simulated under MATLAB R2012a. Simulatingg under such powerful software provides a high degree of accuuracy and assures the result will be as close as possible to thhe real world practices. After simulating the system and viiewing the result, it has been proved that the system yields a prominent priority over the one of the most conventional comppetitors, the pure P.I. controlling technique. Initially at the time of 0.00 second, both systems have a
(5)
Fuzzy rule base is included in the third laayer of N.F.C. how Π is representing the nodes in this layer that declare d the rules of fuzzy. In this model, each node takes two inpuuts, one input from the membership value standing for the speeed error, and the other input is from the changing rate in the sppeed error. For the kth rule node, we have: ∏ , (6) th In the above equation, stands for the j input to the rule layer node, how w3jk supposed to be unity. The output layer functions as defuzzifier. All incoming signnals from the rule layer in this node are gathered by the singlee node to grip the final results ∑ (7)
Fig. 5. Speed Control Comparison Diagram, D Points (a) and (b) are impact of integral steady sttate error eliminator
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2012 IEEE International Conference on Control System, Computing and Engineering, 23 - 25 Nov. 2012, Penang, Malaysia
rotational velocity of 0 R.P.M. noting the desired set point is 3000 R.P.M. By passing the time, both systems try to reach the demanded speed as faster as possible. Figure 5 explicates that increasing rate of Hybrid N.F.C. P.I. tuned is much faster than pure P.I. controller and A.N.F.I.S. takes less than 0.1 Sec. to draw its system into the stable steady state condition, while the conventional P.I. technique takes the similar system into the steady state condition hardly around 0.15 Sec. However the Hybrid controlling method rapidly reaches into the steady state condition, a rather prominent steady state error occurs in the hybrid N.F.-P.I. After the Hybrid N.F.-P.I. reaches to the steady state condition at the time of 0.1 Sec., the paralleled S.S.E.E. is activated automatically, calculates the error between set-point and real value then adds it to the input to level the speed (note the point (a) in Fig. 5). A load of 10 Nm applied at 0.3 Sec on both two controlling systems. Figure 5 clearly shows the conventional P.I. lag more than 300 R.P.M. (around 10% deviation from the desired speed) under such sudden heavy load besides depicting the hybrid N.F.C.-P.I. controlling system just lags less than 50 R.P.M. (1.7% deviation from the desired speed), proves the robustness and flexibility of hybrid controlling technique to tolerate and compensate such abrupt disturbances within very short period of time. Also here after passing 0.1 Sec. (the time hybrid system needs to draw its system into steady state condition) from load applying, around Second 0.42, the paralleled S.S.E.E. is again activated automatically and levels the speed (note the point (b) in Fig. 5). Points (a) and (b) are distinguishingly marked as the impact points of paralleled integral S.S.E.E. (Fig. 5).
B. Simulation Test Parameters Speed Set Point: 3000 RPM, Applied Torque on Motor: 10 Nm, Time of Torque Applying: 0.3 Sec. REFERENCES [1] M. Gökbulut, B. Dandil, and C. Bal, “A Hybrid Neuro-Fuzzy Controller for Brushless DC Motors Artificial Intelligence and Neural Networks.” vol. 3949, F. Savaci, Ed., ed: Springer Berlin / Heidelberg, 2006, pp. 125-132. [2] C.-L. Xia, “Mathematical Model and Characteristics Analaysis of the BLDC Motor,” in Permanent Magnet Brushless DC Motor Drives and Controls, 1st ed: John Wiley & Sons Singapore Pte. Ltd., 2012, pp. 25-40. [3] M. A. Rahman and P. Zhou, “Analysis of brushless permanent magnet synchronous motors,” Industrial Electronics, IEEE Transactions on, vol. 43, pp. 256-267, 1996 [4] I. Boldea and S. A. Nasar, Vector Control of AC Drives: Taylor & Francis, 1992. [5] M. Gokbulut, “Adaptive control of brushless DC motors using neural networks,” PhD, Erciyes University Kayseri, 1998. [6] M. A. El-Sharkawi, “Development and Implementation of High Performance Variable Structure Tracking Control for Brushless Motors,” Power Engineering Review, IEEE, vol. 11, p. 40, 1991. [7] A. A. El-Samahy, M. A. El-Sharkawi, and S. M. Sharaf, “Adaptive multi-layer self-tuning high performance tracking control for DC brushless motor,” Energy Conversion, IEEE Transactions on, vol. 9, pp. 311-316, 1994. [8] D. Feipeng and S. Wenzhong, “Fuzzy neural networks for direct adaptive control,” Industrial Electronics, IEEE Transactions on, vol. 50, pp. 507-513, 2003. [9] L. Faa-Jeng, W. Rong-Jong, and C. Hong-Pong, “A PM synchronous servo motor drive with an on-line trained fuzzy neural network controller,” Energy Conversion, IEEE Transactions on, vol. 13, pp. 319-325, 1998. [10] E. Meng Joo and G. Yang, “Robust adaptive control of robot manipulators using generalized fuzzy neural networks,” Industrial Electronics, IEEE Transactions on, vol. 50, pp. 620628, 2003. [11] L. Faa-Jeng, W. Rong-Jong, and T. Mao-Sheng, “Adaptive fuzzy-neural-network control for induction spindle motor drive,” in Power Electronics and Motion Control Conference, 2000. Proceedings. IPEMC 2000. The Third International, 2000, pp. 990-995 vol.2. [12] G. C. D. Sousa and B. K. Bose, “A fuzzy set theory based control of a phase-controlled converter DC machine drive,” Industry Applications, IEEE Transactions on, vol. 30, pp. 34-44, 1994. [13] C. C. Hwang, P. L. Li, C. T. Liu, and C. Chen, “Design and analysis of a brushless DC motor for applications in robotics,” Electric Power Applications, IET, vol. 6, pp. 385-389, 2012. [14] M. Mansouri, S. Hr. Aghay Kaboli, J. Ahmadian, and J. Selvaraj, “A Hybrid Neuro-Fuzzy - P.I. Speed Controller for P.M.D.C. Enriched with an Integral Steady State Error Eliminator,” in Asia-Oceania Top University League on Engineering, 2012. AOTULE’12. International Conference On, University of Malaya, Kuala Lumpur, 2012, in press.
V. CONCLUSION In this paper MATLAB simulation results show that the proposed hybrid N.F.C.-P.I. is free of previous versions of N.F.C.-P.D. and conventional P.I. controlling techniques drawbacks. The mentioned hybrid system consists of a main P.I. type N.F.C. which is paralleled with an Integral S.S.E.E. for compensating the steady state errors that activated in the stabilized region to enhance the system controlling performance at transient and steady state conditions. Respecting to the simulation, the desirable characteristics of control are gripped by using an alternative approach for N.F.C. The advantages and effectiveness of the proposed controlling system under various loads and parameters is hitherto depicted. A sorted list of publications in the references part has been also given to provide a quick source to the students, researchers, users, designers, and manufacturers. VI. APPENDIX A. B.L.D.C. Machine Simulation Parameters Torque Constant: 1.4 Nm/Apeak, Number of Phases: 3, Back E.M.F. Waveform: Trapezoidal, Rotor Type: Round, Stator Phase Resistance Rs: 2.875 ohm, Stator Phase Inductance Ls: 8.5e-03 H, Flux Linkage Established by Magnets: 0.175 V.S., Voltage Constant: 146.6077 Vpeak L-L/KRPM, Back E.M.F. Flat Area: 120 degrees, Inertia: 0.8e-03 J, Viscous Damping: 1e-03 F, Pole Pairs: 4.
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