2.N 1.I FEEDBACKLINEARIZATIONBASEDNONLINEARCONTROLOFMAGNETICLEVITATIONSYSTEM

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STUDIA Z AUTOMATYKI I INFORMATYKI VOL. 41 – 2016

Jakub Jaroszy´nski, Joanna Zietkiewicz^∗

FEEDBACK LINEARIZATION BASED NONLINEAR CONTROL OF MAGNETIC LEVITATION SYSTEM

Keywords:Feedback linearization, linear quadratic control, magnetic levitation

1. I

NTRODUCTION

Nowadays magnetic levitation systems are in great interest because of the ability of re- duction of friction force, e.g., in vehicles like trains and in other applicattions [2]. Feedback linearization method is an effective tool for approaching nonlinear systems as it provides linear exact model of a nonlinear model, mapping its dynamics. It is decribed in [1] and it is widely used, example of using cen be seen e.g. in [4]. Despite problems with applicability for some systems (e.g. not continuous) it can be used for many nonlinear models. Even expected problems with robustness (as the model is dependent on nonlinear model and its parameters) is not very troublesome in practical applications [5].

In this paper, feedback linearization with linear quadratic control method was used for levitation system with one input. Simulations shows that this approach provides satisfying results.

2. N

ONLINEAR MODEL OF MAGNETIC LEVITATION SYSTEM

The system consists of an electromagnet and a ferromagnetic ball as can be seen in Figure 1. The balance between the magnetic force and the gravity force allow to sustain the ball in the given position. The dynamics of the system is given by the equations (time variable, t, has been omitted for brevity)

˙x1 = x2, (1)

˙x2 = ga−aL1

2mx²₃e^−ax¹, (2)

˙x3 = u − Rx3

L0+ L1e⁻^ax¹, (3)

and the parameters used in this paper are as follows: m= 0.1 kg, R = 2 Ω, ga= 9.81 m/s², L0= 0.007 H, L1= 0.003 H, a = 10. The input of this system is the voltage u [V] used for the electromagnet, whereas the output of the system is the distance between the electromagnet and the center of the ball

∗Poznan University of Technology, Faculty of Electrical Engineering, Institute of Control and Information Engi- neering, Piotrowo 3A, 60-965 Poznan, e-mail: Joanna.Zietkiewicz@put.poznan.pl

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x1

i electro- u magnet

ball light

source

light sensor

Fig. 1. Levitation system

3. F

EEDBACK LINEARIZATION

Feedback linearization (FBL) is a method that provides exact linear model calculated from nonlinear one. This linear model is obtained through calculations of nonlinear feedback (using new input v, u = ϕ(z, v)) and a map between state vectors of nonlinear and linear system (diffeomorphism z = ψ(x)), see Figure 2. To use feedback linearization, nonlinear model should be presented in the form

˙x = f (x) + g(x)u, (4)

y = h(x), (5)

therefore it has to be affine (linear from the input). Additional condition for FBL is that the system has smooth vector functions f(x), g(x), h(x). For the levitation system we have

ϕ(x, v)

˙x = f (x)+g(x)u ψ(x)

h(x)

c^Tz v

u x z

y

Fig. 2. Feedback linearization scheme

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FEEDBACK LINEARIZATION BASED NONLINEAR CONTROL. . . 37

f(x) =







x2

ga−^aL_2m¹x²₃e⁻^ax¹

−Rx3

L0+L1e^−ax1







, g(x) =





 0 0

1 L0+L1e^−ax1







, y= x1 (6)

The functions f(x), g(x), h(x) are indeed smooth, and therefore we can apply feedback linearization. There are two kinds of feedback linearization: input-output and input-state linearization. For this system input-output feedback linearization can be used. This also happens to be input-state linearization for this example. Input-output linearization can be performed by succeeding differentiation of the output y. It is convenient to use Lie derivatives, e.g., Lfh is the derivative of h(x) of x along the vector function f (x):

Lfh= dh(x) dx f(x).

This way we can apply time derivative using derivatives with the argument x and internal derivative with the argument t. The derivative^dh_dx is a gradient

dh

dx = ∂h(x)

∂x1

,∂h(x)

∂x2

, ...,∂h(x)

∂xn

, (7)

Higher orders of the Lie derivatives are described as L²_fh= Lfh

dx f(x).

For the system of magnetic levitation the output

y= x1= z1, (8)

the first derivative

˙y = Lfh+ Lghu= [1, 0, 0] f (x) + [1, 0, 0] g(x)u, (9) here Lgh= [1, 0, 0] g(x) = 0; therefore,

˙y = Lfh= x2= z2, (10)

and the second derivative:

¨

y = L²_fh+ LgLfhu= [0, 1, 0] f (x) + [0, 1, 0] g(x)u, (11)

¨

y = ga−aL1

2mx²₃e⁻^ax¹= z3. (12)

The second derivative also does not depend of u; therefore, LgLfh= 0, and we can assign z3to this derivative. The third derivative of y should contain the input u, and the obtained function is the new input v of the linear system

y⁽³⁾= L³fh+ LgL²_fhu= a²L1

2m x²₃e⁻^ax¹,0, −aL1

m x3e⁻^ax¹

(f (x) + g(x)u) (13)

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v=a²L1

2m x2x²₃e^−ax¹+ aL1R

mL(x1)x²₃e^−ax¹− aL1

mL(x1)x3e^−ax¹u, (14) where L(x1) = L0+ L1e^−ax¹. Feedback linearization performed this way results in obtain- ing linear model:

˙z1 = z2, (15)

˙z2 = z3, (16)

˙z3 = v, (17)

y = z1, (18)

where the states x and z are connected through the diffeomorphism z= ψ(x):

z1 = x1, (19)

z2 = x2, (20)

z3 = g − aL1

2mx²₃e^−ax¹, (21)

and the connection between the inputs (u and v) in function of x or z can be calculated from (14), (using diffeomorhism (21) if variables z are needed); e.g., the function u= ϕ(z, v) is described by equation

u= s

2m(g − z3) aL1e⁻^az¹

R+a

2L(z1)z2+ L(z1) 2(z3− g)v

, (22)

where L(z1) = L0+ L1e^−az¹.

For the linear model linear control can be applied, e.g., linear quadratic control.

4. L

INEAR QUADRATIC CONTROL

For the system of levitation linear quadratic control is applied. It assures minimum of the cost function:

J= Z ^∞

0

(z^TQz+ r²v+ 2z^Tnv)dt (23)

where Q >0, n and r includes weights for appropriate signals. Here z and v are not physical values therefore it is difficult to chose weights in the elements Q, r and n. The entries besides the main diagonal are needed in Q and the vector n (that usually is omitted) is also needed.

The solution of minimization is the input

v= −k^Tz (24)

where

k^T = r⁻¹b^TS (25)

and S is the solution of Riccatii equation

A^TS+ SASbr⁻¹b^TS+ Q = 0. (26)

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Calculation of S and k is performed numerically.

One of the way to choose the weights is to calculate them from the assumed objective function for the variables of nonlinear system (x and u). The procedure is described in [3].

Assuming that the objective function for nonlinear system is Jx=

Z ^∞

0

(x²₁+ x²₂+ x²₃+ u²)dt, (27) the weights calculated for the system (23) are the following

Q =





8595 33.87 −87.61 33.87 1.17 −0.35

−87.61 −0.35 0.89



, (28)

r = 1.7 ∗ 10⁻⁶, (29)

n =





−0.1075

−0.0005

−0.0011



, (30)

then

k^T = 7.15, 1.46, 0.075 ∗ 10⁴ (31) The weights presented above are used for some of the simulations, but also weights are ad- justed to provide better results.

5. R

ESULTS

In the first simulation weights only on the main diagonal entries of Q was used, all equaled one, r= 1, and the vector n was not used. The result with mentioned weights can be see in Figure 3. Weights are also written down in the captions of every simulation figure. We can see that this solution is not sufficient. Here the weights do not influence directly signals x and v, but through the functions u= ϕ(z, v) and z = ψ(x). In order to obtain better performance, we tried to modify the weights for x1, but with consideration of the rest of signals, especially with the care that the voltage u and the current i= x3were not too hight or not too low and that they do not jumps with large distance. Using the weights only on the main diagonal of Q without vector n we obtained the solution for Q=





100000 0 0

0 1 0

0 0 1



, r= 0.001, n =



 0 0 0



; the result can be seen in Figure 4 Here the output (x1) has good performance, but the other values can be too high or too low. Therefore, we applied higher weight on u, r=0.1, see Figure 5 We see that x1has longer rising time but x3and u are closer to their reference points.

Afterwards we tried to modify the rest of the weights. In Figure 6 we used weights calculated in the section 4. The solution has no oscilations and has well performance. We could adjust weights in better way, but the problem occurs that here many parameters are to adjust in Q, r, n.

For the next simulation we have tried to add more pressure on x1, by increasing Q(1, 1):

Q=





1000000 0 10

0 −8 0

11 0 −100



, r = 1, (here, as in the latter examples, we do not use n), see Figure 7.

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t

0 5 10

x 1[m]

0 0.005 0.01 0.015

t

0 5 10

x 2[m/s]

×10^-3

-2 0 2 4

t

0 5 10

x 3[A]

8 8.2 8.4 8.6

t

0 5 10

u[V]

16 16.5 17 17.5

Fig. 3. Simulations with Q =



 1 0 0 0 1 0 0 0 1



, r = 1, n =



 0 0 0



.

Afterwards, r = 10 was used, to add the higher weight on u, the results are presented in Figure 8. It can be seen that we obtained smoother signals. Considering the tested values of weights, both from calculations and from tries, we have decided to use the weights: Q=





100000 33.87 −20 33.87 1.2 −0.35

−20 0 0.1



, r= 0.0001, n =





−0.1075

−0.0005

−0.0011



. The obtained results can be seen in Figure 9. Here we obtained the best rising time and the stabilization time, keeping the input on the value not lower than14.

6. C

ONCLUSIONS

Feedback linearization and linear quadratic control was applied to the model of magnetic levitation system. Simulation research allowed to choose the best weights in objective function in linear quadratic problem. Solution assures stable and fast change of the ball position to the reference point, at the same time maintaining the voltage and the current in electromagnet close to the equilibrium point.

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STUDIA Z AUTOMATYKI I INFORMATYKI VOL. 41 – 2016

t

0 5 10

x 1[m]

×10^-3

-5 0 5 10 15

t

0 5 10

x 2[m/s]

-0.05 0 0.05 0.1

t

0 5 10

x 3[A]

7.5 8 8.5 9

t

0 5 10

u[V]

15 16 17 18





100000 0 0

0 1 0

0 0 1



, r = 0.001, n =



 0 0 0



.

R

EFERENCES

[1] J.-J. E. Slotine and W. Li. Applied Nonlinear Control. Prentice Hall Inc, Englewood Cliffs, New Yersey, 1991.

[2] H. Yaghoubi. The most important maglev applications. Journal of Engineering, 2013:1–19, 2013.

http://dx.doi.org/10.1155/2013/537986.

[3] J. Zietkiewicz. Linear quadratic control with feedback linearized models. Studies in Automation and Information Technology, 40:37–49, 2015. http://sait.cie.put.poznan.pl/40/SAIT_40_02.pdf.

[4] J. Zietkiewicz. Non-minimum phase properties and feedback linearization control of nonlinear chemical reaction. In Proc. of 20th International Conference on Methods and Models in Automation and Robotics (MMAR), pages 489–494, Miedzyzdroje, 2015. DOI:10.1109/MMAR.2015.7283924.

[5] J. Zietkiewicz, A. Owczarkowski, and D. Horla. Performance of feedback linearization based con- trol of bicycle robot in consideration of model inaccuracy. In Challenges in Automation, Robotics and Measurement Techniques, pages 399–410, Warsaw, 2016. DOI: 10.1007/978-3-319-29357- 8_36.

ABSTRACT

Control of unstable nonlinear system is challenging. When the method relying on objective function is used, this function may be non-convex. Feedback linearization derives exact linear model that imitates

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t

0 5 10

x 1[m]

×10^-3

-5 0 5 10 15

t

0 5 10

x 2[m/s]

-0.02 0 0.02 0.04

t

0 5 10

x 3[A]

8 8.2 8.4 8.6 8.8

t

0 5 10

u[V]

16 16.5 17 17.5





100000 0 0

0 1 0

0 0 1



, r = 0.1, n =



 0 0 0



.

the dynamics of nonlinear model. In this paper linear quadratic control is applied for feedback linearized system. The method is simulated on magnetic levitation system. Simulation shows that the control assures good performance.

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t

0 5 10

x 1[m]

×10^-3

-5 0 5 10 15

t

0 5 10

x 2[m/s]

-0.02 0 0.02 0.04

t

0 5 10

x 3[A]

7.5 8 8.5 9

t

0 5 10

u[V]

14 16 18 20





8595 33.87 −87.61 33.87 1.17 −0.35

−87.61 −0.35 0.89



, r = 1.7∗ 10⁻⁶, n =





−0.1075

−0.0005

−0.0011



.

NIELINIOWE STEROWANIE PREDYKCYJNE OPARTE NA LINEARYZACJI PRZEZ SPRZ ˛E ˙ZENIE ZWROTNE DLA UKŁADU LEWITACJI MAGNETYCZNEJ

STRESZCZENIE

Sterowanie nieliniowymi, niestabilnymi obiektami jest wymagaj ˛ace je´sli metoda opiera si˛e na funkcji celu, funkcja ta mo˙ze by´c niewypukła. Linearyzacja przez sprz˛e˙zenie zwrotne zapewnia dokładny lin- iowy model odwzorowuj ˛acy dynamik˛e modelu nieliniowego. W artykule do zlinearyzowanego systemu zastosowane jest sterowanie liniowo-kwadratowe. Metoda jest przetestowana symulacyjnie na układzie magnetycznej lewitacji. Otrzymane symulacje pokazuj ˛a dobr ˛a jako´s´c takiego rozwi ˛azania.

Received: 2016-10-06 Accepted: 2016-11-19

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t

0 5 10

x 1[m]

×10^-3

-5 0 5 10 15

t

0 5 10

x 2[m/s]

-0.04 -0.02 0 0.02 0.04

t

0 5 10

x 3[A]

8 8.2 8.4 8.6 8.8

t

0 5 10

u[V]

16 16.5 17 17.5 18





1000000 0 10

0 −8 0

11 0 −100



, r = 1, n =



 0 0 0



.

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t

0 5 10

x 1[m]

×10^-3

-5 0 5 10 15

t

0 5 10

x 2[m/s]

-0.02 0 0.02 0.04

t

0 5 10

x 3[A]

8 8.2 8.4 8.6 8.8

t

0 5 10

u[V]

16 16.5 17 17.5





1000000 0 10

0 −8 0

11 0 −100



, r = 10, n =



 0 0 0



.

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t

0 5 10

x 1[m]

×10^-3

-5 0 5 10 15

t

0 5 10

x 2[m/s]

-0.1 -0.05 0 0.05 0.1

t

0 5 10

x 3[A]

7 8 9 10

t

0 5 10

u[V]

14 16 18 20





100000 33.87 −20 33.87 1.2 −0.35

−20 0 0.1



, r = 0.0001, n =





−0.1075

−0.0005

−0.0011



.