ELECTRICAL ENGINEERING
Robust
H
1
control of Magnetic Levitation system
based on parallel distributed compensator
Munna Khan
a, Anwar Shahzad Siddiqui
a,
Amged Sayed Abdelmageed Mahmoud
a,b,*a
Department of Electrical Engineering, Faculty of Engineering & Technology, Jamia Millia Islamia, New Delhi 110025, India b
Department of Industrial Electronics & Control Engineering, Faculty of Electronic Engineering, Menoufia University, Menof 32952, Egypt
Received 12 January 2016; revised 4 June 2016; accepted 12 June 2016
KEYWORDS
Takagi–Sugeno fuzzy (TSF) system;
Magnetic Levitation (Maglev) system; Parallel Distributed Com-pensator (PDC); Linear Matrix Inequality (LMI);
RobustH1controller
Abstract This paper concerns with designing of a robustH1 controller for Magnetic Levitation
system exposed to external disturbances. First, the Maglev is modeled in a discrete form of Tak-agi–Sugeno fuzzy (TSF) system. Then the nonlinear fuzzy controller is synthesized based on such a technique called Parallel Distributed Compensation (PDC) schemes and sufficient conditions are developed to control and guarantee the robust stabilization of a complex nonlinear system. So the proposed algorithm is used to enhance the performance and to ensure the stability of the system. Furthermore, the design conditions and criteria for quadratic stability of the TSF system are formulated as a Linear Matrix Inequality (LMI) to robustly stabilize the position of the iron ball in a Maglev system in the existence of disturbances. The proposed technique will guarantee H1performance to be less than one and attenuate the effect of exogenous disturbances. Moreover,
the comparison results for the control of Maglev will show the capability of the proposed approach.
Ó 2016 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
Earlier studies intensively explore methods for robustly stabi-lizing real systems which are actually nonlinear model.
There-fore, the control techniques for nonlinear systems are generally challenging. Fuzzy logic control (FLC) is considered as a pop-ular technique to stabilize and control complex real system due to its simplicity, systematically and effectively. In this direc-tion, FLC using Takagi–Sugeno fuzzy (TSF) system gains con-siderable attention and numerous research activities have been carried out for quadratic stability and controller synthesis of the TSF system in the field of nonlinear control systems for last three decades[1–13]. In TSF systems, the nonlinear model is blending of linear time-invariant subsystems connected by weighted membership functions [1]. This allows a nonlinear complex system to be presented by a blending of linear models; hence, linear control techniques can be employed for systems
* Corresponding author at: Department of Electrical Engineering, Faculty of Engineering & Technology, Jamia Millia Islamia, India. E-mail addresses:[email protected](M. Khan),[email protected]
(A.S. Siddiqui),[email protected](A.S.A. Mahmoud). Peer review under responsibility of Ain Shams University.
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analysis and design closed-loop controlled systems. Based on TSF model, a simple feedback control is synthesized by a tech-nique called parallel distributed compensation (PDC) to con-trol and stabilize complex nonlinear systems. The design concept of PDC is to develop a control system by using a feed-back controller for each rule in TSF model[2–4]. Hence, the designed fuzzy controller reflects the structure of the related TSF model.
The aim of this paper is to propose a robust controller to highly nonlinear Magnetic Levitation system using TSF model control in the discrete form. By using the PDC concept, aH1 controller will be applied to Maglev exposed to external distur-bances. Thus, the proposed controller is TSF control system that the stability analysis is achieved and then LMI conditions are presented to calculate the gain of the controller. The nov-elties here is that a proposed controller will robustly stabilize the Maglev system for both current-controlled and voltage-controlled schemes exposed to external disturbances using straightforward algorithms. Finally, simulation results are given to prove that the proposed technique ensures the stabil-ity conditions and guarantees robustness against external dis-turbance for a complex nonlinear system. The paper is organized as follows. First, the related work is presented in Section2. Then some preliminaries and problem formulation are given in Section 3. Section 4 shows the fuzzy controller design. In Section5, the simulations are given to emphasize the efficiency of the proposed method and compare it with other PDC schemes for voltage and current controlled Maglev system. Finally, the paper is concluded in Section6.
2. Related work
Recently, PDC scheme has been intensively used in solving many academic and industrial problems because the concept of PDC is simple and easy to develop. Such as, the TSF model of an inverted pendulum is presented and the PDC scheme is developed to stabilize it in[3]. In[5]switched PDC is synthe-sized for R/C Hovercraft. Baranyi et al.[8]propose a PDC con-troller for stabilization of a real 3-DOF RC helicopter to get a good speed response. A practical approach of a Takagi–Sugeno fuzzy (TSF) modeling and control for F16 aircraft is presented in[9]. The design and implementation of PDC controller for temperature control are introduced in[10]. In[11], the fuzzy controller is presented utilizing the PDC technique and it is syn-thesized in the experimental tank level system. PDC-fuzzy based controller is proposed for chaotic systems in[12].
The literature reveals that there have been significant prob-lems in controllability, stability analysis and robustness of nonlinear systems. Therefore, a lot of efforts have been made in developing stabilization conditions and designing PDC con-trollers for guaranteeing not only the stability, but also the robustness performances of closed-loop nonlinear systems. Thus a significant research works have been exploited in the field ofH1 robust control system for fuzzy system via TSF
model [3,6,7]. The use of Lyapunov function in terms of
LMI technique is popular in designing stability conditions for the TSF model with parallel distributed compensation (PDC)[4,13]. By utilization of a convex program, the perfor-mance conditions and stability criteria can be checked and then the parameter of a fuzzy model control system can be
obtained by solving the LMIs. Although LMI-based
approaches have been successful, the obtained conditions lead to conservatism and the local stability problem. Thus, consid-erable works have been investigated to develop effective meth-ods so as to reduce the conservatism that comes from the usage of Lyapunov function, for example, piecewise Lyapunov func-tion [14–16], Fuzzy Lyapunov functions [17–20], switched
fuzzy system [5–7,21] and nonquadratic
membership-dependent Lyapunov function[22].
Nowadays Magnetic Levitation (Maglev) has various usages in many fields such as Maglev trains, wind turbines, Launching Rockets, Electrodynamics’ Suspension and the cen-trifuge of nuclear reactor[23]. These systems are highly nonlin-ear and unstable open loop systems because of magnetic force. Therefore, designing a robust controller to stabilize the Mag-netic Levitation is a challenging task. Thus, great efforts have been applied for constructing the high-performance feedback controllers to stabilize and control the Maglev systems. In the last decades, various studies try to manipulate the Maglev systems such as[24]a multilevel fuzzy logic control strategy is proposed, and in[25]switched controller is applied; adaptive fuzzy neural Network has been utilized by [26]; real-time PID control with PSO gain selections is implemented in[27]; optimal fuzzy control design using neural fuzzy inference net-works is proposed in[28]. Nevertheless, a lots of the existing techniques that manage the nonlinearity of the Maglev system usually need complicated algorithms. Therefore, TSF model control has been applied to deal with the nonlinearity of Maglev as it is simpler and more effective technique [13]. In
[13], a fuzzy model control is employed using the non-PDC to regulate the Maglev system for voltage-controller scheme. But, this controller has some shortcomings. First the con-trolled model needs more time to converge to the equilibrium point. And there are a lot of LMI variables which take much time to calculate and evaluate the gain. In this paper, we will develop a robustH1fuzzy controller based on PDC to control and stabilize discrete time Maglev system. The proposed con-troller is a state feedback concon-troller and LMI conditions are presented to calculate the gain of the controller. The resulting algorithms are much simpler with fewer variables involved which result in more efficient computationally. The LMI con-ditions are included by slack matrix variables to minimize the conservatism arising from using the Lyapunov function and to improve the performance as compared to corollary[6].
Therefore, the goal of the proposed PDC is to robustly con-trol the position of the iron ball in Maglev in the presence of
disturbances for both current-controlled and
voltage-controlled schemes. The convex optimization algorithms were
solved using a MATLAB toolbox YALMIP[29]with the
sol-ver SeDuMi[30]. MATLAB environment has some successful
applications to simulate systems and validate the algorithms
[5–9,26–32]. The main advantage of using YALMIP is much faster solver than LMI toolbox and it is so simple to use.
Notation: The superscript ‘‘T” represents the transpose of a matrix. The notation ‘‘*” in symmetric matrix is used as a transposed element in the symmetric position.
3. Preliminaries and problem formulation
The TSF model is represented by fuzzy IF-THEN rules and described a nonlinear system by linear time-invariant subsys-tems connected by nonlinear membership functions[1]. In this
paper, we studied discrete form of TSF system with distur-bances[2]. So the l-th rule of the discrete fuzzy system models (DFS) is given as follows: IF d1ðnÞ is M1l and and dpðnÞ is Mpl Then xðn þ 1Þ ¼ AlxðnÞ þ BwlwðnÞ þ BuluðnÞ yðnÞ ¼ ClxðnÞ þ DwlwðnÞ þ DuluðnÞ ð1Þ where l = 1, 2,. . .,q where q is the number of inference rules; Mlj are the fuzzy sets (j = 1, 2,. . .,p) where p is number of fuzzy sets in each rule; x(n)e Rkis the state vector; u(n)e Rm is the control signal; y(n)e Rris the output vector; w(n)e Rs is the energy-bounded disturbance; (Al, Bwl, Bul, Cl, Dwl, Dul) is the l-th local model of TSF(1)with appropriate dimensions and d1(n),. . .,dP(n) are known premise variables.
The global model of a discrete-time TSF system is inferred by using product inference engine and a weighted average method defuzzifier, as follows:
xðn þ 1Þ ¼X q l¼1 glðdðnÞÞðAlxðnÞ þ BwlwðnÞ þ BuluðnÞÞ ð2Þ yðnÞ ¼X q l¼1 glðdðnÞÞðClxðnÞ þ DwlwðnÞ þ DuluðnÞÞ ð3Þ where dðnÞ ¼ ½d1ðnÞ d2ðnÞ . . . dPðnÞ T and glðdðnÞÞ ¼ QP j¼1MljðdjðnÞÞ Pq l¼1 QP j¼1MljðdjðnÞÞ ð4Þ For all n. The term MljðdjðnÞÞ is the grade of membership of fuzzy setsðdjðnÞÞ at M
j
l. For brief expression, we will denote glðnÞ ¼ glðdðnÞÞ.We have Xq l¼1 glðnÞ ¼ 1 glðnÞ P 0 8 > > > < > > > : 8l ð5Þ
For the above TSF model, Fuzzy controller can be con-structed using the Parallel Distributed Compensation (PDC) concept. The PDC control technique is synthesized by design-ing a linear compensator to control each of fuzzy rules. So the fuzzy controller uses the same fuzzy sets of TSF model. Control rule l: IF d1ðnÞ is M1l and . . . and dpðnÞ is Mpl; then uðnÞ ¼ Fl xðnÞ; l ¼ 1; 2; . . . ; q
where Fl ðl ¼ 1; 2; . . . ; qÞ is the gain of feedback controller. Hence, the PDC controller can be obtained by
uðnÞ ¼ X
q
l¼1
glðnÞ FlxðnÞ; ð6Þ
By substituting(6)in(3) and (4), we have
xðn þ 1Þ ¼ ½AðgðnÞÞ BuðgðnÞÞFðgðnÞÞxðnÞ þ BwðgðnÞÞwðnÞ yðnÞ ¼ ½CðgðnÞÞ DuðgðnÞÞFðgðnÞÞxðnÞ þ DwðgðnÞÞwðnÞ ð7Þ Where AðgðnÞÞ ¼X q l¼1 glðnÞAl; BuðgðnÞÞ ¼ Xq l¼1 glðnÞBul; BwðgðnÞÞ ¼ Xq l¼a glðnÞBwl; CðgðnÞÞ ¼X q l¼1 glðnÞCl; DuðgðnÞÞ ¼ Xq l¼1 glðnÞDul; DwðgðnÞÞ ¼ Xq l¼b glðnÞDwl; 8 > > > > < > > > > : 4. PDC controller design
In this part, state feedback controller is presented using PDC technique. The proposed controllers will be used to control and stabilize unstable system. The proposed PDC will exploit an LMI-based approach to calculate the controller’s gain to solve the problem of output feedback controller for discrete-time TSF systems.
Lemma [6]. Stability of closed loop system(7)is fulfilled with H1performance level c > 0 if for 16 l , j 6 q, if Pll< 0 and Q= QT, if following two LMIs conditions hold:
Pll< 0; 1 6 l 6 q; ð8Þ 1 q 1 Pllþ 1 2ðPjlþ PljÞ < 0; 1 6 l – j 6 q ð9Þ where Plj¼ Q E ET 0 cI AlEþ B2lYj B1l Q ClEþ D2lYj D1l 0 cI 2 6 6 6 4 3 7 7 7 5; 1 6 l j 6 q;
Remark 1. The main problem in this PDC algorithm is the conservatism. So to reduce it, the slack matrix is introduced to the LMIs conditions. The proposed algorithm will improve the performance and give better result as it will be justified in the simulation results. So based on the lemma, LMI conditions
for designing H1 controller are given in the following
theorem:
Theorem. For a given H1 performance level c > 0, if there exists a symmetric matrix Q = QT> 0, Wlj= W
T
lj, E > 0, Yj, 16 l,j 6 q, satisfying the following LMIs:
WllP 0; ð10Þ eWljP 0; 1 6 l; j 6 q; e 2 f1; 1g ð11Þ Cll< 0; 1 6 l 6 q; ð12Þ 1 q 1Cllþ 1 2 Cjlþ Clj < 0; 1 6 l – j 6 q ð13Þ where Clj¼ Q E ET 0 cI AlEþ B2lYj B1l Q ClEþ D2lYj D1l 0 cI 2 6 6 6 4 3 7 7 7 5þ Wlj; 1 6 l; j 6 q; ð14Þ Then the controller(6)with
Kl¼ YlQ1 ð15Þ
Provides the TSF system is asymptotically stable with guar-anteedH1performance level c.
Proof. Since Pll should be negative definite from Lemma [6] and condition(9)should also be negative definite to guarantee the stability of the system, so with add positive slack matrices Wlj, Wllto both conditions without change in the negative def-initeness, therefore the theorem will be ensured the condition of stability will remain with less conservative than Lemma
[6]. Notice that if we choose Wlj= Wll= 0, 16 l, j 6 q in the theorem, then the LMIs in theorem will be the same PDC scheme as in corollary in[6].h
Remark 2. The theorem is presented in LMI terms is straight-forward to obtain the gains of state feedback controller. So it
can be simply solved using MATLAB toolbox YALMIP[29]
with the efficient solver SeDuMi [30]. And the time needed to compute the solution is a little in spite of the existence of the slack Matrix.
5. Simulation results
In this section, the proposed scheme is employed in a Maglev system to ensure that the system is quadratically stable and more robust. The discrete-time TSF system of voltage-controlled and current-voltage-controlled Magnetic Levitation is pre-sented. Next, fuzzy controller is introduced and the
corre-sponding simulations are presented. Fig. 1 shows Magnetic
Levitation (Maglev) system model. The proposed technique is used to control the position of iron ball for both current-controlled and voltage-current-controlled Maglev systems exposed to exogenous disturbances.
Where m is the mass of the iron ball; h is the gap between the ball and the magnet; g is the acceleration gravity; f is a force of electromagnetism; i is current of the electromagnet; vis the electromagnetic voltage on the coil.
5.1. Voltage-controlled Maglev system
After applying the Kirchhoff’s voltage law and the Newton’s second law, the nonlinear equations of motion of Maglev is given as[25] md2h dt2þ f ¼ Mg þ fd vþ2bi h2 dhdt¼ iR þ 2b h di dt f¼ b i h 2 8 > > < > > : ð16Þ
where b is electromagnetic constant (b = 0.25 l0N2A); l0is the permeability of air; N is the number of coil’s turns; A is cross-sectional area of magnetic; fd is the exogenous distur-bance force.
In this scheme the voltage on the coil is considered to be the control input. The three state variables are assumed to be the position of ball h, the speed of the ball _hand the cur-rent i and the control variable is the voltage v. Since the gap is required to be kept fixed at h0= 4 mm, so the state variables will be
x1ðtÞ ¼ h 0:004; x2ðtÞ ¼ _h; x3ðtÞ ¼ i 0:004b; Then the state equation is given as[13]:
_x1¼ x2 _x2¼ b M x3 x1 2 þ g þ 1 Mfd _x3¼ x2x3 x1 x1x3 2b Rþ x1 2bu u¼ v y¼ x1 ð17Þ
The discrete-time TSF system model of Magnetic Levita-tion with sampling time T = 0.5 ms:
xðn þ 1Þ ¼X 4 l¼1 glðnÞ½AlxðnÞ þ BwlwðnÞ þ BuluðnÞ yðnÞ ¼X 4 l¼1 glðnÞClxðnÞ
The parameters of local system are
A1¼ 1 0:0005 0 1:764 1 0:0027 0:3267 0:3704 0:9996 2 6 4 3 7 5; A2¼ 1 0:0005 0 1:764 1 0:0019 0:3267 0:1704 0:9996 2 6 4 3 7 5; A3¼ 1 0:0005 0 3:8111 1 0:0076 0:3267 0:6173 0:9997 2 6 4 3 7 5; A4¼ 1 0:0005 0 3:8111 1 0:0053 0:3267 0:2840 0:9997 2 6 4 3 7 5; Bw1¼ Bw2¼ Bw3¼ Bw4¼ 0 0:00000333 0 2 6 4 3 7 5 Bu1¼ Bu2¼ 0 0 0:0003886 2 6 4 3 7 5 ; Bu3¼ Bu4¼ 0 0 0:0002331 2 6 4 3 7 5 C1¼ C2¼ C3¼ C4¼ 1 0 0½
And where w(n) is the discrete form of fd(t). The premise variables are d1= x1(n) and d2= x3(n). And so the member-ship functions are described as follows:
M1 1ðd1ðnÞÞ ¼ M12ðd1ðnÞÞ ¼ d1ðnÞ þ 0:001 0:002 ; M1 3ðd1ðnÞÞ ¼ M14ðd1ðnÞÞ ¼ 0:001 d1ðnÞ 0:002 ; M2 1ðd2ðnÞÞ ¼ M23ðd2ðnÞÞ ¼ d2ðnÞ þ 1 2 ; M2 2ðd2ðnÞÞ ¼ M24ðd2ðnÞÞ ¼ 1 d2ðnÞ 2
The grade of membership functions is glðnÞ ¼ Q2 j¼1MljðdjðnÞÞ P4 l¼1 Q2 j¼1MljðdjðnÞÞ
Consequently, by using a modeling language YALMIP with SeDuMi solver, we solve the LMI of theorem and the controller gain matrices for PDC(6)are obtained as follows:
F1¼ 105 ½6:4070 0:0289 0:0060; F2¼ 106 ½1:4964 0:0733 0:0050; F3¼ 106 ½4:9009 0:2377 0:0217; F4¼ 106 ½3:3630 0:1871 0:0136
The state with zero initial condition and the exogenous dis-turbance w(n) = 103(0.2 + 5n)1sin(10n) are chosen. So the simulation results of the state response of the closed loop for
each of the proposed theorem and[13]are depicted inFig. 2. The results of using the proposed theorem and non-PDC[13]
with initial value of gap x1(0) = 0.0005 m and x1(0)
= 0.001 m exposed to external disturbance are given inFigs. 3 and 4 respectively. The proposed algorithm gives less
maxi-mum overshoot than that in [13]. The simulation results
address that the proposed theorem minimizes the effect of external disturbances and has outperformance over non-PDC. In[13], a fuzzy model control by non-PDC method to reg-ulate the Maglev system for voltage-controller scheme and this controller has shortcomings. A considerable number of LMI variables need much time to solve the problem and evaluate the gain which results in a weak solution. And the response just takes a longer time to converge to its equilibrium. While the proposed algorithms are much simpler with much fewer
variables involved so more efficient computationally. Also, it
is obvious from Fig. 2 for an energy bounded disturbance,
the proposed algorithm provides better performance in terms of less overshoot that’s because introducing the slack matrices
(10) in the new algorithm gives less conservative results and produces higher controller’s signal (high voltage) without los-ing the stability behavior which makes the system converges to its equilibrium faster and leads to better performance. Even if the initial value is changed, the system still has a stable behav-ior with good performance and there is no overshoot in the gap and the variation of the gap is smoother as shown inFigs. 3 and 4, that is because the new algorithms are more relaxed conditions. Thereby it is concluded that the proposed PDC system is an effective and a proper technique. Moreover, when the controller’s gain for the proposed algorithm is higher, the voltage will be a little bit more which make the current in the case of proposed algorithm not exceed 1A. Therefore, the pro-posed algorithm is more practical and more suitable than that of[13]and can be synthesized to control the Maglev exposed to external disturbances.
5.2. Current-controlled Maglev system
In this case the current in the coil will be the control input. The Maglev’s dynamic equations[25]are presented as
m €hþ f ¼ Mg þ fd
f¼ b i
h
2 ð18Þ
Define the state variable as follows: x1¼ h x2¼ _h
Then the state equation
Figure 1 Maglev system model.
_x1¼ x2 _x2¼ g Mk u x1 2 þ 1 Mfd u¼ i 8 > > < > > : ð19Þ
Consider a following discrete-time Magnetic Levitation TSF system model, which is based on Tustin’s method of numerical example in[28]with sampling time T = 0.2 s
xðn þ 1Þ ¼X2 l¼1 dlðnÞ A½ lxðnÞ þ BwlwðnÞ þ BuluðnÞ yðnÞ ¼X 2 l¼1 dlðnÞ C½ lxðnÞ þ DluðnÞ
Where the grades of membership functions are as follows: MlðxðnÞÞ ¼ exp ðxðnÞ mlÞ2 r2 l " # l¼ 1; 2
Figure 3 State response for initial gap = 0. 5 mm.
A1¼ 1:064 0:006442 20:64 1:064 ; Bw1¼ 0:04123 0:4123 A2¼ 1:058 0:005791 20:58 1:058 ; Bw2¼ 0:04155 0:4155 Bu1¼ 0:010 0:095 ; Bu2¼ 0:001 0:014 C1¼ 0:006442 0:006442 0 0 ; D1¼ 0 0:004123 C2¼ 0:005791 0:005791 0 0 ; D2¼ 0 0:00416 m1¼ 0:32 m2¼ 0:18; r1¼ 0:21; r2¼ 1:76
Applying the theorem of the proposed PDC technique, we get the following:
F1¼ 103:921 1:8200½ ; F1¼ 323:0482 5:5537½ c¼ 0:2117
The state trajectories of the closed loop controlled system are depicted inFigs. 5 and 6with initial value x0= [10–30]T. Fig. 7shows the controller input u(n) (the current of electro-magnetic) and it indicates that the control signal is higher in the case of proposed algorithms that result in rapid changing of the position of the iron ball as referred inFig. 6. The pro-posed controller stabilizes and compensates the overall system.
Figure 5 State response of x1.
Based on the comparison between the proposed method and corollary in[6], it should be noted that the main problem of the algorithm in corollary[6] is the conservatism. To reduce it, the slack matrix is introduced to the LMIs conditions. Thus, this matrix will relax the conservatism and improve the perfor-mance without increasing the complexity of the LMI. Then, the efficiency of the controller will be increased and the robust-ness will be ensured using the proposed PDC. So the iron ball will converge to its equilibrium state quickly. Accordingly, it is noted that the response obtained from proposed PDC is better
than PDC and the obtainedH1 performance equals 0.2117,
while the value using PDC equals 0.5019 which indicates that the proposed algorithm gives less conservatism than that of lemma. As inFig. 8, the controlled system response of the
pro-posed algorithm decays quickly and approaches the origin if change in the initial position to 10 mm and exposure to distur-bance equal to w = e(n/2)sin(10n) that’s because of the high controller input signal and the stabilization condition in the theorem.
So based on the lemma, LMI conditions for designingH1 controller are given in the following theorem: the proposed robustH1state feedback control method gives less overshoot and less settling time for the states of the controlled nonlinear system as compared to the method in[6]because slack matrix in proposed algorithm reduces the conservatives of stability condition and then the performance of the system gets better. Furthermore, the Maglev is exposed to changing in amplitude of exogenous disturbance w = q sin(10n) exp(n/2) and
ini-Figure 7 State response of control input u.
tial position of iron ball equals to 1 mm to notice the effect of disturbances on the controlled system and compare the results with other PDC scheme. Where q is the gain of the distur-bance. So the performance of the proposed will be addressed by changing the amplitude of the disturbance. With increasing the amplitude of the disturbance, the proposed PDC controller is compared with PDC[6]. Therefore,Figs. 9 and 10show that the proposed PDC is more tolerable to disturbance than other PDC scheme. Note that despite high disturbance amplitude, the system is quadratically stable.
6. Conclusions
This paper synthesizes robustH1controller for a Maglev sys-tem using the TSF model control technique for both current-controlled and voltage-current-controlled schemes exposed to external disturbances. The TSF controller is applied based on the PDC scheme in a discrete form and the sufficient conditions and cri-teria by adding matrix variable have been derived in set of Lin-ear Matrix Inequality (LMI) to ensure the robustness of the nonlinear system. The results show that the proposed
Figure 9 State response of x1with q = 2.
approach stabilizes the complex nonlinear system and guaran-teesH1performance of Maglev system for both schemes. The comparison results for the control of Magnetic Levitation sys-tem prove that the proposed PDC has a superior performance in all respects and outperforms the other techniques.
Acknowledgments
This work was supported by Indian Council for Cultural Rela-tions (ICCR) under Africa Scholarship Scheme, under the pro-gram of executive propro-gram between Arab Republic of Egypt (A.R.E) and India, with ICCR Ref. No. ASS-248n2014– 2015, Ministry of External Affairs of India.
References
[1] Takagi T, Sugeno M. Fuzzy identification of systems and its application to modeling and control. IEEE Trans Syst Man, Cybern 1985;15(1):116–32.http://dx.doi.org/10.1109/TSMC.1985. 6313399.
[2] Wang HO, Tanaka K, Griffin MF. An approach to fuzzy control of nonlinear systems: stability and design issues. IEEE Trans Fuzzy Syst 1996;4(1):14–23.http://dx.doi.org/10.1109/91.481841. [3] Tanaka K, Ikeda T, Wang HO. Robust stabilization of a class of
uncertain nonlinear systems via fuzzy control: quadratic stabiliz-ability, H1 control theory, and linear matrix inequalities. IEEE Trans Fuzzy Syst 1996;4(1):1–13. http://dx.doi.org/10.1109/ 91.481840.
[4]Kazuo T, Wang HO. Fuzzy control systems design and analysis: a linear matrix inequality approach. John Wiley & Sons Inc; 2001. [5] Tanaka K, Iwasaki M, Wang HO. Switching control of an R/C hovercraft: stabilization and smooth switching. IEEE Trans Syst Man Cybern Part B Cybern 2001;31(6):853–63.http://dx.doi.org/ 10.1109/3477.969489.
[6] Dong J, Yang G. H1 controller synthesis via switched PDC scheme for discrete-time T–S fuzzy systems. IEEE Trans Fuzzy Syst 2009;17(3):544–55. http://dx.doi.org/10.1109/TFUZZ.2008. 924328.
[7] Wang T, Tong S. H1 control design for discrete-time switched fuzzy systems. Neurocomputing 2015;151:782–9. http://dx.doi. org/10.1016/j.neucom.2014.10.021.
[8]Baranyi P, Korondi P, Tanaka K. Parallel distributed compen-sation based stabilization of a 3-DOF RC helicopter: a tensor product transformation based approach. J Adv Comput Intell Inform 2009;13(1):25–34.
[9]Tanaka M, Yamaguchi K, Ogura D, Chen Y, Tanaka K. Nonlinear control of F16 aircraft via multiple nonlinear model generation for any trimmed equilibriums. Int J Fuzzy Syst 2014;16 (2):140–52.
[10] Yordanova S, Sivchev Y. Design and tuning of parallel distributed compensation-based fuzzy logic controller for temperature. J Autom Control 2014;2(3):79–85. http://dx.doi.org/10.12691/au-tomation-2-3-3.
[11] Sadeghi MS, Farughian A. Parallel distributed compensator design for process control based on fuzzy Takagi–Sugeno model. Appl Soft Comput 2014;21:280–5. http://dx.doi.org/10.1109/ PEAM.2011.6135042.
[12] Hsiao FH. Robust H1 fuzzy control of dithered chaotic systems. Neurocomputing 2013;99:509–20. http://dx.doi.org/10.1016/j. neucom.2012.08.003.
[13] Su X, Yang X, Shi P, Wu L. Fuzzy control of nonlinear electromagnetic suspension systems. Mechatronics 2014;24: 328–35.http://dx.doi.org/10.1016/j.mechatronics.2013.08.002. [14] Wang L, Feng G. Piecewise H infinity: controller design of
discrete time fuzzy systems. IEEE Trans Syst Man Cybern B
Cybern 2004;34(1):682–6. http://dx.doi.org/10.1109/TSMCB. 2003.809229.
[15] Chen YJ, Ohtake H, Tanaka K, Wang WJ, Wang H. Relaxed stabilization criterion for T–S fuzzy systems by minimum-type piecewise lyapunov function based switching fuzzy controller. IEEE Trans Fuzzy Syst 2012;20(6):1166–73. http://dx.doi.org/ 10.1109/TFUZZ.2012.2196049.
[16] Campos VCS, Souza FO, Torres LaB, Palhares RM. New stability conditions based on piecewise fuzzy lyapunov functions and tensor product transformations. IEEE Trans Fuzzy Syst 2013;21 (4):748–60.http://dx.doi.org/10.1109/TFUZZ.2012.2230178. [17] Rhee B-J, Won S. A new fuzzy Lyapunov function approach for a
Takagi–Sugeno fuzzy control system design. Fuzzy Sets Syst 2006;157(9):1211–28.http://dx.doi.org/10.1016/j.fss.2005.12.020. [18] Lee DH, Park JB, Joo YH. A fuzzy Lyapunov function approach
to estimating the domain of attraction for continuous-time Takagi–Sugeno fuzzy systems. Inf Sci (Ny) 2012;185(1):230–48.
http://dx.doi.org/10.1016/j.ins.2011.06.008.
[19] Faria Fa, Silva GN, Oliveira Va. Reducing the conservatism of LMI-based stabilisation conditions for TS fuzzy systems using fuzzy Lyapunov functions. Int J Syst Sci 2013;44(10):1956–69.
http://dx.doi.org/10.1080/00207721.2012.670307.
[20] Lee DH, Joo YH, Tak MH. Local stability analysis of contin-uous-time Takagi–Sugeno fuzzy systems: a fuzzy Lyapunov function approach. Inf Sci (Ny) 2014;257:163–75.http://dx.doi. org/10.1016/j.ins.2013.08.036.
[21] Xiang W, Xiao J, Han L. Switching PDC control for discrete-time T–S fuzzy system: a membership function ranking approach. J Frank Inst 2014;351(7):3536–58. http://dx.doi.org/10.1016/j. jfranklin.2013.04.010.
[22] Zhao XD, Zhang LX, Shi P, Karimi HR. Novel stability criteria for T–S fuzzy systems. IEEE Trans Fuzzy Syst 2014;22(2):313–23.
http://dx.doi.org/10.1109/tfuzz.2013.2254491.
[23] Yaghoubi H. The most important maglev applications. J Eng 2013;2013:1–19.http://dx.doi.org/10.1155/2013/537986.
[24] Moinuddin, Siddiqui AS, Sharma AK, Gupta JP. Control-A multi-level fuzzy rule based control strategy for Maglev system. EPE J 2000;10(1):26–31.
[25] Xu J, Zhou Y. A nonlinear control method for the electromag-netic suspension system of the maglev train. J Mod Transp 2011;19(3):176–80. http://dx.doi.org/10.3969/j.issn.2095-087X.2011.03.005.
[26] Wai RJ, Lee JD. Dynamic control of maglev transportation system via adaptive fuzzy-neural-network. IEEE Trans Neural Networks 2008;19(1):54–70. http://dx.doi.org/10.1109/ IJCNN.2007.4371019.
[27] Wai RJ, Lee JD, Chuang KL. Real-time PID control strategy for maglev transportation system via particle swarm optimization. IEEE Trans Ind Electron 2011;58(2):629–46. http://dx.doi.org/ 10.1109/TIE.2010.2046004.
[28] Wu S, Wu C, Chang Y. Neural – fuzzy gap control for a current/voltage-controlled 1/4-vehicle MagLev system. IEEE Trans Intell Transp Syst 2008;9(1):122–36.
[29] Lofberg J. YALMIP: a toolbox for modeling and optimization in MATLAB. In: Proc IEEE int symp comput aided control syst des. p. 284–9.http://dx.doi.org/10.1109/CACSD.2004.1393890. [30] Sturm JF. Using SeDuMi 1.02, a Matlab toolbox for optimization
over symmetric cones. Optim Methods Softw 1999;11(1– 4):625–53.http://dx.doi.org/10.1080/10556789908805766. [31] Valipour M. Optimization of neural networks for precipitation
analysis in a humid region to detect drought and wet year alarms. Meteorol Appl 2016;23:91–100. http://dx.doi.org/10.1002/ met.1533.
[32] Valipour M, Banihabib ME, Behbahani SMR. Comparison of the ARMA, ARIMA, and the autoregressive artificial neural network models in forecasting the monthly inflow of Dez dam reservoir. J Hydrol 2013;476:433–41.http://dx.doi.org/10.1016/j.jhydrol.2012. 11.017.
Dr. Munna Khan is working as a Professor at Jamia Millia Islamia, New Delhi, India. He completed 4 research projects and published 120 papers in International/ National journals and conferences. He is recipient of various National and International awards including DE Gold Medal from IIT Delhi, Young Investigator award, Space Medicine Branch of NASA, USA for paper ranked to one of top 10% performer, Career award for young tea-cher from AICTE. He elected as an Associate Fellow of AsMA, USA and ISAM, IAF, India. His research interest in Bio-Implants, vBio-MEMS, Biomaterials and Biomechanics, Biomedical Engineering and Aerospace Medicine.
Prof. Anwar Shahzad Siddiqui obtained his B.Sc. Engg. (Electrical Engineering) and M.Sc. Engg. (Power Systems and Electrical Drives) degrees from AMU, Aligarh with Honors in 1992 and 1994 respectively. He earned his Ph.D. degree from Jamia Millia Islamia New Delhi. He has been teaching and guiding research for about two decades at AMU, Aligarh; JMI, New Delhi and BITS Pilani – Dubai campus. His research interests include Power System operation, control and management and Applications of Artificial Intelligence Techniques in Power System. He has more than 80 research papers published in refereed international and national journals and conferences of repute.
Amged Sayed Abdelmageed Mahmoud is an assistant Lecture in the Department of Industrial Electronics and Control Engineer-ing, Faculty of Electronic Engineering, Menoufia University, Menof, Egypt. He obtained the B.Sc. and M.Sc. degrees in automatic control engineering from Faculty of Electronic Engineering, Menoufia University, Menof, Egypt. He is currently pursuing his Ph.D. degree in Electrical Engineering at Faculty of Engineering & Technology, Jamai Millia Islamia, India. His current research interest includes robust control and fuzzy logic control.