Discrete-Time Control of Maglev System Using Switched ... - IEEE Xplore

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Discrete-Time Control of Maglev System Using Switched Fuzzy Controller Amged Sayed Abdelmageed Mahmoud, Munna Khan, Anwar Shahzad Siddique Department of Electrical Engineering, Faculty of Engineering and Technology, Jamai Millia Islamia New Delhi-110025, India [email protected], [email protected], [email protected] switched Parallel distributed compensator (switched-PDC) is feedback controllers in which the gain is switched based on the value of membership functions [5]. The quadratic Lyapunov function is applied to design the conditions for stability and performance of the system. Recently, the use of Linear Matrix Inequality (LMI) is very popular in designing conditions because The LMI-based conditions obtained in are easily calculated using convex optimization algorithm in Matlab Toolbox. By utilizing LMI technique , the performance conditions and stability criteria can be solved then the gain of feedback controller can be calculated [ 2],[4],[5] ,[6],[9].

Abstract— In this paper, the implementation of switched fuzzy controllers for Magnetic Levitation (Maglev) system which is a unstable nonlinear system is studied. The switched Parallel distributed compensator (switched-PDC) applied in the controller is a Takagi-Sugeno (T-S) fuzzy system control in a discrete form. It is a state feedback controller in which the gain switched is depending on the value of membership functions. This controller is exploited to stabilize the position of the ball in Maglev system in the existence of disturbances. So the proposed technique will guarantee the stability and performance of closed loop nonlinear system and ensure robustness to external disturbances. The simulation results for control of Maglev will compare the effect of the proposed technique with pure PDC to indicate that the switched-PDC will provide better results under external disturbances.

Magnetic Levitation (Maglev) has various uses in many fields such as Maglev trains, wind turbines, Launching Rockets, Electrodynamics' Suspension and the centrifuge of nuclear reactor [11]. These systems are highly nonlinear and unstable open loop systems, so great efforts have applied for constructing the high performance feedback controllers to stabilize and control the Maglev systems. In last decades, various studies try to manipulate the Maglev systems such as , in [8] switched controller is proposed; adaptive fuzzy neural Network has been applied by [12]; T-S fuzzy controller is presented by[7]; real-time PID control with PSO gain selections is implemented in [13]; optimal fuzzy control design using neural fuzzy inference networks is proposed in [14] .

Keywords: Takagi–Sugeno (T–S) system, Magnetic Levitation (Maglev) system , Parallel distributed compensator (PDC), switched-PDC , Linear Matrix Inequality ( LMI).

I.

INTRODUCTION

Many real systems are actually nonlinear models, where the stability analysis and control techniques for these systems are generally challenging. So in last decades, the fuzzy logic control (FLC) is considered to be effective technique to control complex nonlinear system. In this direction, FLC using Takagi-Sugeno (T-S) fuzzy system gains great attentions in the field of nonlinear control systems as a result of its simplicity, systematically and effectiveness [1]-[9]. In T-S fuzzy systems, the nonlinear model is blending of linear timeinvariant subsystems connected by weighted membership functions. This allows nonlinear system to be presented by a set of local linear models, therefore linear systems theory can be employed for systematic analysis and design closed-loop controlled systems. Based on T-S fuzzy model, the technique of PDC is used to control and stabilize complex nonlinear systems by using state feedback controller for each rule in T-S model [2],[4].

In this paper, the state feedback controller is exploited to control and stabilize discrete-time Maglev fuzzy system using switched-PDC controller. The proposed controller is TS fuzzy controller that the gain switched is depending on the value of membership functions. The switched PDC is used to control the position of iron ball in maglev in the presence of external disturbances. The switched PDC will exploit an LMI-based approach to design the controller's gain. Finally, simulation results are given to prove that the proposed technique guarantees the stability condition of the closed loop nonlinear system and ensures robustness to external disturbance. The paper organized as follows. Section II presented the model of Maglev system. Section III shows The T-S discrete fuzzy models. The Switched PDC controller which will be used is given in Section IV. the switched PDC will be analyzed and compared with other scheme technique. Section V, simulation results are presented to emphasize the efficiency of proposed

Recently, T-S fuzzy system has been extended to Switched fuzzy model which has T-S fuzzy models and switches them according to premise variables [3],[6],[8]. A switched fuzzy controller is build by extension of the PDC scheme. The This work was supported by Indian Council for Cultural Relations (ICCR) under Africa Scholarship Scheme, under the program of executive program between Arab Republic of Egypt and India, with ICCR Ref. No. ASS-248\2014-2015, Ministry of External Affairs of India.

978-1-4673-6540-6/15/$31.00 ©2015 IEEE

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methods and compare it with PDC scheme. Finally in section VI , the conclusion is given.

⎧ x(k + 1 ) = Ai x(k) + B1i w(k) + B2i u(k) Then⎨ ⎩ y(k) = Ci x(k) + D1i w(k) + D2i u(k)

Notation: The superscript “T” represents the transpose of a matrix. The notation “*” is used as an ellipsis for terms that are induced by symmetry. II.

(3)

Here i=1,2,...,r where r is the number of model rules ; Mij is the fuzzy set ; x(k) ∈ Rn is the state vector ; u(k) ∈ Rm is the control input; y(k) ∈ Rq is the output vector ; w(k) ∈ Rs is the energy-bounded disturbance.Ai, B1i,B2i,Ci,D1i and D2i are of appropriate dimensions ; δ1(k),..., δP(k) are known premise variables. The final outputs of the fuzzy systems are inferred as follows:

MAGLEV DYNAMICS

Fig.1 shows Magnetic Levitation (Maglev) system model. The control purpose is to stabilize the position of the ball by the force of electromagnet.

r

x(k + 1 ) =

∑ η (δ (k ))A x(k) + B w(k) + B i

i

1i

2 i u(k))

(4)

i =1

r

y(k) =

∑ η (δ (k ))C x(k) + D w(k) + D i

i

2 i u(k))

1i

(5)

i =1

Where δ(k) = [δ1(k) δ2(k) " δP(k)]T P

∏ M (δ (k)) ij

ηi(δ(k)) =

j

j =1 P r

∑∏ M (δ (k)) ij

Fig. 1.

For all k. The term M ij(δ j(k)) is the grade of membership of

The Maglev's dynamics equations [7] are presented as mh + f = mg + f d 2

(δ j(k))

in

M ij

brief

⎧ r ⎪ ηi (k ) = 1 ⎨ i =1 ⎪ η (k ) ≥ 0 ⎩ i

∑

expression,

we

will

∀i

(7)

The switched PDC control technique is proposed by combining PDC control law with switching controllers where the PDC controllers are switched based on the value of membership functions [3].The following definitions are necessary:

k is the electromagnet constant; i is controlled current of the electromagnet. Define the state variable as of (1) as:

x1 = h , x2 = h

δ(k) = [δ1(k),δ2(k)," ,δr (k)]T

Then the state equation 2

k ⎛ i ⎞ 1 ⎜⎜ ⎟⎟ + f d m ⎝ x1 ⎠ m

r ⎧⎪ ⎪⎫ Ω = ⎨δ(k):0 ≤ δi(k) ≤ 1,1 ≤ i ≤ r, δi(k) = 1⎬ ⎪⎩ ⎪⎭ i =1

(8.a)

r ⎧⎪ ⎫⎪ Ωl = ⎨α(k):0 ≤ δi(k) ≤ δi(k),1 ≤ i ≤ r, δi(k) = 1⎬ ⎪⎩ ⎪⎭ i =1

(8.b)

∑

x1 = x2

III.

.For

denote η i (k) = ηi (δ (k )) . We have

(1)

fd is the disturbance force; g is the acceleration gravity;

x 2 = g −

j

i =1 j =1

Maglev system model

⎛i⎞ f = k⎜ ⎟ ⎝h⎠ Where m is the mass of the iron ball ; h is the position of ball;

(6)

(2)

∑

T-S FUZZY MODEL

The T-S fuzzy model is described by fuzzy IF-THEN rules and represents nonlinear system by linear time-invariant subsystems connected by weighted membership functions [1]. So the i-th rule of the discrete fuzzy models (DFS) is of the following form:

where 1 ≤ l ≤ r. Remark 1: δi take all possible values of membership functions and Ω is the set of all the vectors δ (k)=[ δ1(k), δ2(k) ,..., δr(k)]T. Ωl is the set of all the vector δ(k) with δi(k) (i=1,2,...,r) satisfying 0≤ δi(k) ≤ δl (k) which describes the case where the ℓ-th rule plays more important or at least the same role than other rules.

DFS model rule i: If δ1(k) is Mi1 ... and δ P(k) is MiP,

2

At any moment k, we have the vector δ(k)=[ δ1(k), δ2(k),... , δr(k)]T ∈ Ω . Then there exist one ℓ, ℓ ∈ {1,...,r}, such that the vector δ(k) ∈ Ωl , which implies that the ℓ-th subsystem plays role at least as important as other subsystems. The switched PDC controllers is considered as follows: u(k) = K (δ (k )) x

T

⎡ η1 I ⎤ ⎢ # ⎥ R ijml ⎢ ⎥ ⎢⎣ ηr I ⎥⎦

[

⎡ η1 I ⎤ # ⎥⎥ ≥ 0 , ηi ∈ {0 ,1}, and ηl = 1 ⎢⎣ η r I ⎥⎦

]r × r ⎢⎢

(15)

Where

(9)

Γ ijml = Δijml + Rijml , 1 ≤ i,j,l ≤ r, m ∈ Ll

(16)

And

Where

⎧ ⎪ K1(δ (k)) = ⎪ ⎪ K(δ (k)) = ⎨ ⎪ ⎪K r(δ (k)) = ⎪⎩

r

∑

δ j K j1,

j =1

for δ(k) ∈ Ω1

#

r

∑δ K

jr ,

j

Δijml

(10) for δ(k) ∈ Ωr

j =1

⎡ Ql − Sl − Sl T ⎢ 0 = ⎢⎢ A S + B2iY jl ⎢ i l ⎢⎣Ci Sl + D2iY jl

x(k + 1 ) = ( A(δ (k )) + B (δ (k )))x(k) + B2 (δ (k ))w(k)

r

i =1

∑ δ (k ) B i

2i ,

C (δ (k )) =

i =1

(17)

i

(18)

1≤ i ≤ r

[

i

]

1≤ i ≠ j ≤ r

(19)

r

δ i (k ) D1i , D2 (δ (k )) =

i =1

∑ δ (k ) D i

Where

2i

i =1

⎡ Q − S − ST ⎢ 0 Θij = ⎢⎢ A S + B2iY j ⎢ i C ⎢⎣ i S + D2iY j

FUZZY CONTROLLER DESIGN

In this part, state feedback controller is presented using switched PDC scheme. The proposed controllers will be utilized to stabilize and control unstable system. The switched PDC will exploit an LMI-based approach to design the controller's gain to solve the problem of output feedback controller for discrete-time T-S fuzzy systems. The main result is given in the following theorems [5]:

V.

Γ iiml < 0,1 ≤ i,l ≤ r, m ∈ Ll

)

(13)

Riiml ≤ 0 , 1 ≤ i ≠ l ≤ r, m ∈ Ll

(14)

* * ⎤ ⎥ * * ⎥ −Q * ⎥ ⎥ 0 − γI ⎥⎦

SIMULATION RESULTS

With changing the amplitude of the disturbance, switched PDC controller is compared with PDC scheme. It should be noted that the proposed controller has the same response or better than PDC scheme. The Switched PDC is combining the benefit of switched gain and PDC. So it is more tolerable to disturbance than ordinary PDC controller.

(12)

1 1 Γ iiml + Γ jiml + Γ ijml < 0 ,1 ≤ i ≠ j ≤ r 2 r −1

* − γI B1i D1i

In this section, Discrete-time TS fuzzy controller is employed to maglev system to make the system stable and more robust. Switched fuzzy controller is used to control the position of iron ball in maglev exposed to disturbances. The Discrete-time Magnetic Levitation fuzzy system is presented. Then Switched Fuzzy controller is implemented and the corresponding simulations are given.

Theorem :The all subscripts of sets, to which δ(k) transit from Ωℓ for some k, are collected as new set Lℓ. Given a prescribed H∞ performance bound γ > 0, if there exist matrix Qi = QiT > 0, Rijmℓ = RTijmℓ , Sℓ, Yjℓ, 1≤i,j,m,ℓ≤r , satisfying the following LMIs:

(

− Qm 0

1 1 Θii + Θ ji + Θij < 0 , 2 r −1

∑ δ ( k )C i =1

r

IV.

Θii < 0,

r

∑ δ (k ) B

∑

1i

i =1

r

D1 (δ (k )) =

B1i D1i

If choose Q l= Qm= Q, Sl = S , Rijml = Riiml = 0, Yjl = Yj , 1≤i,j,m,l≤r in the theorem , then the followings LMIs will be satisfied:

r

δ i (k ) Ai , B1 (δ (k )) =

i

*

Renders the T-S fuzzy system (11) asymptotically stable while satisfying H∞ performance bound γ .

Where

B2 (δ (k )) =

− γI

* ⎤ ⎥ * ⎥ * ⎥ ⎥ − γI ⎥⎦

K il = Yil Sl −1

(11)

y(k) = (C (δ (k )) + D1 (δ (k )))x(k) + D2 (δ (k ))w(k)

∑

*

Then the controller(10) with

By combining (4),(5) and (10) then the closed-loop discrete fuzzy system is:

A(δ (k )) =

*

3

Consider a following discrete-time Magnetic Levitation TS fuzzy system model , which is based on Tustin's method of numerical example in [14] with sampling time T=0.2 2

x(k + 1 ) =

∑ δ (k)[A x(k) + B i

i

wi w(k) +

-3

8

Bui u(k)]

4

i =1

h (m)

∑ δ (k)[C x(k) + D u(k)] i

i

1i

i =1

Where the grade of membership functions are: ⎡ (x(k) − mi )2 ⎤ M i(x(k)) = exp ⎢ − ⎥ σ i2 ⎢⎣ ⎥⎦

i = 1,2

0

-4 -6

- 0 .006442 ⎤ ⎡ 0 .04123 ⎤ , B w1 = ⎢ ⎥ -1.064 ⎥⎦ ⎣ 0 .4123 ⎦ - 0 .005791 ⎤ ⎡ 0 .04155 ⎤ , Bw2 = ⎢ ⎥ ⎥ -1.058 ⎦ ⎣ 0 .4155 ⎦

60

80

100

State response of x1 with ρ=12

PDC Switched PDC

0.3 0.2

h dot

0.1 0 -0.1 -0.2

is

-0.3 -0.4

0

20

40

60

80

100

Time in sample

-1.9377], F12 = [454.982 7.551]

F21 = [-292.45 -4.948],

40

0.4

exposed to external disturbance w = ρ ∗ sin (10 k ) 20 (1 + 2 k ) to notice the effect of switched PDC on the output and compare the results with PDC scheme. Applying the theorem of the switched PDC technique, we get:

F11 = [-112.37

20

Fig. 2.

m1 = 0 .32 m 2 = − 0 .18 , σ 1 = 0 .21 , σ 2 = 1.76

Maglev

0

Time in sample

⎡ - 0 .010 ⎤ ⎡ - 0 .001 ⎤ Bu1 = ⎢ , Bu 2 = ⎢ ⎥ ⎥ ⎣ - 0 .095 ⎦ ⎣ - 0 .014 ⎦ - 0 .006442 ⎤ 0 ⎡ - 0 .006442 ⎡ ⎤ C1 = ⎢ ⎥ , D 1 = ⎢ 0 .004123 ⎥ 0 0 ⎣ ⎦ ⎣ ⎦ - 0 .005791 ⎤ 0 ⎡ - 0 .005791 ⎡ ⎤ C2 = ⎢ ⎥ ,D 2 = ⎢ 0 .00416 ⎥ 0 0 ⎣ ⎦ ⎣ ⎦

The

2

-2

And ⎡ -1.064 A1 = ⎢ ⎣ - 20 .64 ⎡ -1.058 A2 = ⎢ ⎣ - 20 .58

PDC Switched PDC

6

2

y(k) =

x 10

Fig. 3.

F22 = [-648.176 -10.893]

State response of x2 with ρ=12

6

γ = 0.3292 The state response of the controlled system is depicted in the Fig.2 and Fig.3. It is noticed that the switched PDC controller and PDC scheme stabilize and control the system. It is seen that the response obtained from switched PDC is better than PDC and the obtained H∞ performance via Switched PDC equal 0.3292, whilst the value using PDC equal 0.5019. The controller input u(k) is shown in Fig.4.The disturbance is shown in Fig.5. So from the above computational and from the figures it should be noticed that the switched PDC give better results than PDC scheme.

PDC Switched PDC 4

i(A)

2

0

-2

-4

-6

0

20

40

60

80

Time in sample

Fig. 4.

4

State response of control input with ρ=12

100

14

0.15

PDC Switched PDC

12

0.1

8

0

6

h (m)

Amplititud

10 0.05

-0.05

4 2

-0.1

0 -2

-0.15

-4 -0.2

0

20

40

60

80

-6

100

Time in sample

Fig. 5.

0

40

60

80

100

Time in sample

Disturbance with ρ=12

Fig. 8.

0.025

State response of x1 under large disturbance

400

PDC Switched PDC

0.02

20

PDC Switched PDC

300

0.015

0.005

100

h dot

200

h (m)

0.01

0 -0.005

0 -100

-0.01 -0.015

-200

-0.02

-300

-0.025 0

20

40

60

80

-400

100

Time in sample

Fig. 6.

Fig. 9.

40

60

80

100

State response of x2 under large disturbance

Furthermore, the effect of using of switched PDC in the disturbances is analyzed and compared with the PDC scheme result by changing the amplitude of the disturbance. Fig 6 and Fig.7 show that the switched PDC is more tolerable to disturbance than PDC.

2 PDC Switched PDC

1

Changing the disturbance to be w=1800 sin(10k)/(0.2+4k) and apply the controller to the Maglev system. The response of the system is depicted in Fig. 8 and Fig. 9. It should be noticed that the system with switched PDC is stable while the system with PDC is going to be unstable. So the switched PDC is more robust against disturbance than the pure PDC scheme.

0.5

h dot

20

Time in sample

State response of x1 with ρ=35

1.5

0

0 -0.5 -1 -1.5 -2

0

20

40

60

80

VI.

100

Time in sample

Fig. 7.

CONCLUSIONS

We have analyzed the discrete-time control problem for Maglev system using T-S fuzzy technique called Switched PDC. The switched PDC controller's gains are switched based on the value of membership functions. The proposed

State response of x2 with ρ=35

5

technique guarantees the stability and performance of closed loop nonlinear system and ensures robustness to external disturbances. The stabilization conditions and robust stability are expressed in terms of set of LMIs, which can be solved efficiently by using LMI optimization tools. Simulation results for control of Maglev system prove that the proposed method provides better results than the other controller scheme. REFERENCES [1]

K. Tanaka, H. O. W,“FUZZY CONTROL SYSTEMS DESIGN AND ANALYSIS”, John Wiley & Sons, Inc., 2001 [2] Wang, Hua O., Kazuo Tanaka, and Michael F. Griffin. "An approach to fuzzy control of nonlinear systems: stability and design issues." Fuzzy Systems, IEEE Transactions on fuzzy systems, vol. 4, no. 1, pp. 14-23 February 1996. [3] Tanaka, K.; Iwasaki, M.; Wang, H.O., "Switching control of an R/C hovercraft: stabilization and smooth switching," , IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol.31, no.6, pp.853-863, Dec 2001. [4] Tanaka, K.; Ikeda, T.; Wang, H.O., "Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: quadratic stabilizability, H∞ control theory, and linear matrix inequalities," , IEEE Transactions on Fuzzy Systems, vol.4, no.1, pp:1-13, Feb 1996. [5] Jiuxiang Dong; Guang-hong Yang, " H ∞ controller synthesis via switched pdc scheme for discrete-time t--s fuzzy systems" , IEEE Transactions on Fuzzy Systems, vol.17, no.3, pp.544-555, June 2009. [6] Weiming Xiang, Jian Xiao, Muhammad Naveed Iqbal, "H∞ control for switched fuzzy systems via dynamic output feedback: Hybrid and switched approaches," Communications in Nonlinear Science and Numerical Simulation, vol. 18, Issue 6, pp: 1499-1514, June 2013 [7] Xiaojie Su, Xiaozhan Yang, Peng Shi, Ligang Wu," Fuzzy control of nonlinear electromagnetic suspension systems" , Mechatronics, Volume 24, Issue 4, pp: 328-335,June 2014. [8] de Souza, W.A.; Teixeira, M.C.M.; Cardim, R.; Assuncao, E., "On switched regulator design of uncertain nonlinear systems using takagi– sugeno fuzzy models," IEEE Transactions on Fuzzy Systems, vol.22, no.6, pp.1720,1727, Dec. 2014. [9] Juing-Shian Chiou, Chi-Jo Wang, Chun-Ming Cheng, Chih-Chieh Wang, "Analysis and synthesis of switched nonlinear systems using the T–S fuzzy model," Applied Mathematical Modelling, Volume 34, Issue 6, pp: 1467-1481, June 2010. [10] Eduardo S. Tognetti, Ricardo C.L.F. Oliveira & Pedro L.D. Peres, "H∞ and H2 nonquadratic stabilisation of discrete-time Takagi–Sugeno systems based on multi-instant fuzzy Lyapunov functions," International Journal of Systems Science, Vol. 46, No. 1, pp: 76–87, 2015. [11] Hamid Yaghoubi, “The most important maglev

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[12] Rong-Jong Wai; Jeng-Dao Lee, "Adaptive fuzzy-neural-network control for maglev transportation system," , IEEE Transactions on Neural Networks, vol.19, no.1, pp.54-70, Jan. 2008 [13] Rong-Jong Wai; Jeng-Dao Lee; Kun-Lun Chuang, "Real-time PID control strategy for maglev transportation system via particle swarm optimization," , IEEE Transactions on Industrial Electronics, vol.58, no.2, pp.629-646, Feb. 2011. [14] Shinq-Jen Wu; Cheng-Tao Wu; Yen-Chen Chang, "Neural–fuzzy gap control for a current/voltage-controlled 1/4-vehicle maglev system," , IEEE Transactions on Intelligent Transportation Systems, vol.9, no.1, pp.122-136, March 2008

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