control a simple electromagnetic levitation system ...

Proceedings of The 2nd International Conference on Green Technology and Sustainable Development, 2014

CONTROL A SIMPLE ELECTROMAGNETIC LEVITATION SYSTEM USING LQR Nguyen Tan Viet Tuyen1,a, Dang Thai Son1,b, Vu Quang Huy1,c, Tran Ngoc Doan1,d 1

Ho Chi Minh City University of Technology and Education

a

[email protected]; [email protected]; c [email protected]; [email protected]

ABSTRACT This paper proposed a method using LQR Observer for controlling Magnetic Levitation system. Firstly, the mathematical model of a Magnetic Levitation system and the related equations was established. Secondly, the LQR observer was applied to control Magnetic Levitation system. Next, the simulation results are performed using MATLAB/Simulink environment. Finally, the results of experiment prove that LQR method gives high performance, such as rise time, settling time, overshooting than PID and Pole Placement controller. Keywords:Linear Quadratic Regulator, Magnetic Levitation, Observer.

of this method is not require mathematical models and have the ability to approximate nonlinear systems. The drawback of Neural Network [7] method is computationally complex, when the sample size is large, the learning time can be very long. AnkurGoeland A Swarup [8] proposed Sliding Mode Control for a Magnetic Levitation System. The significant features of SildingMode Control are disturbance rejection, insensitivity to parameter variations, simple implementation by means of conventional power converter. It also includes some limit such as heavy computational requirements and difficult design and tuning procedure.

1. INTRODUCTION In recent year, Green technology has become a focal point for many pioneers that want a better planet to live on. It is applied in many areas such as, Biofiltration,Wave energy, Wind turbine, Hydroelectricity, Solar power[1-3]. Hamid [4] presented the method to apply Magnetic Levitation in the of field green technology such as, pumps, generator, motor, and compressor use Maglev to support moving machinery without physical contact, which is important in machines where lubricants can be a source of contamination. Therefore, the low- friction feature could play increasingly important roles in industrial applications. Although Magnetic Levitation is the classical experimental setup, it is still a challenging for control stability.

Several methods are applied to control Magnetic Levitation system. They have got particular advantages and disadvantages. PID controllers believable for simple linear systems. While, Neural Network method could be used forundefined parameters. This method also require a specific time for learning and adaption. The computation for LQR controller is not complex as Neural Network. In this situation, which a few variablescan not be measure specifically from Magnetic Levitation system. LQR observer could be perform to control that system robustly

Many scientists researched and suggested approach to control Magnetic Levitation system such as MáriaHypiusová and his colleagues [5] designed PID controller for unstable SISO systems. This controller approach ensures about stability and performance. However, it is not suitable for non-linear system.RBF Neural Network Controller was proposed to control Magnetic Suspension System have threedegree-of-freedom [6]. A major advantage 223


The LQR could be performed to estimate a few of variables which can’t bemeasured. The strong features of the LQR control as compared to pole placement is that instead of having to specify where n eigenvalues should be placed a set of performance weighting are specified that could have more intuitive appeal. As the result, this paper proposed LQR controller to improve performance and quality response of system.

Where: f (i, d ) = k

(2)

The state variables are defined as:

[ x1

x2

 x3 ] =  d 

d

 i 

(3)

In the equation, x1 is represented for the gap which is measure from the levitating magnet to the bottom of the coil, x2 is velocity and x3 stand for the current through the electromagnet.

2. MATHEMATICAL MODEL 2.1 Electromagnetic Levitation Model

As the result (3), the state equation of system can be write are as under:

The Electromagnetic levitation system is a highly nonlinear with an open-loop is unstable. It is a challenging to control robustly.In this section, the mathematical model of Electromagnetic Levitation [9]are established as below

    x2   x1    x  =  − k .x3 + g    2   m.x 3 1   x3   1   −R  L x3 + L u 

(4)

z = x1

(5)

y =β

1 + γx3 + α + ψ x12

(6)

Where:z is the gap between the electromagnet and the levitating magnet. The variable y asthe measured output. α , β and γ are constants which depends on the Hall sensor used as well as the geometry of the system, n is the noise process.

Figure 1. Electromagnetic levitation system model

The model of electromagnetic levitation is shown in Figure 1, R is the resistance of the cold, L is the inductance of the cold, v is the voltage across the electromagnet, i is the current through the electromagnet, m is the mass of the levitating magnet, g is the acceleration due to gravity, d is the gap between levitating magnet measured from the bottom of the coil, f is the force on the levitating magnet generated by the electromagnet and e is the voltage across the Hall effect sensor

The equilibrium point of the system is:  k .ue 1/3  ( )  x1e   g .m.R    x  = 0    2e     x3e  ue    R 

(7)

The Jacobian linearization of the system about the equilibrium point is: δ x = Aδx + B1δω + B2 δu

The system follows from Newton’s Second Law is:

m d = mg − f (i, d )

i2 d2

δz = C1δx + D11δω + D12 δu δy = C 2δx + D21δω + D22 δu

(1) 224

(8)


  0  4/3 1/3  3 g .(m.R) A= (k .u e )1/3   0 

1 0 0

 0   − g .R  ue   −R  L 

Suppose that we have sensors to measure the entire state, as the result, the signal control can be obtained as: (9)

u = − Kx To drive the state to zero.

The gain vector K is established to minimize this cost function. The balance between control energy and the state deviations specified via Q and R matrices:

  0 0     B1 = 0 B2 =  0  1 0   L

(10)

 −2 β .g .m.R  0 γ C1 = [1 0 0] C2 =  k .ue  

(11)

D11 = [ 0 ] D12 = [ 0 ] D21 = [1] D22 = [ 0]

(12)

 1  q2  1   0 Q=   0    0 

3.LQR CONTROL 3.1 LQR Controller

J = ∫  x (t ) + Qx(t ) + u (t ) Ru (t ) dt

(13)

(18)

(19)

AT P + PA − PBR−1BT P + Q = 0

(20)

As the result, the optimal control is:

(

−1

)

u (t ) = − ( R + B T PB ) BT PA x( k )

n

(14)

(21)

The feedback gain can be written as:

y ∈ Rn

K = R −1BT P

We aim to drive the initial state x0 to the smallest position value as soon as possible in an interval time [t 0 , t f ] without speeding to much control effort.

(22)

3.2LQR Control and ObserverState To address the situation where not all state variables are measured, a state estimator should be design. A schematic of state feedback control with a full-state estimator is shown in figure 2. To find an observer gain with desirable characteristics which is used LQR control, the variable L could be calculated as below [11]:

In this situation, the cost function is calculated from (13) are as under: tf

J = ∫  xT (t )Qx(t ) + u T (t ) Ru (t )  dt

0 0

(17)

In order to obtain the feedback gain, K make the optimal controller. It could be calculated via Algebraic RiccatiEquation (ARE):

Consider a state-space system:

y = Cx

.

u (k ) = − K ( k ) x (k )

This optimal control which based on the minimization of equation error could give the absolute result

x = Ax + Bu x ∈ R , u ∈ R

0

0

           

The control law of feedback is:

0

n

.

  1 0   u2   1   0   .  R=   0   .    1 1    2 qn2   un

Q = Q' > 0 R = R ' > 0

∞

T

0 0

With the cost matrices (17) satisfy:

Linear Quadratic Regulator (LQR) [10] is a method of optimal control which are found by solve a mathematical equation to produces the cost function which is used to define the total of deviations of key measurements from desired value T

(16)

(15)

t0

225


L = ( R −1BT P)T

The parameters of system are shown as:

(23)

The output of the observer is the state estimate:

R = 2.41Ω, L = 15.03×10−3 H, m = 3.02 ×10−3 kg

The state estimate error could be calculate are as under:

β = 2.92 ×10−4 Vm2 , γ = 0.48V/A, ue = 0.66V

g = 9.81m/s2 , k = 17.31×10−9 kg m5 /s2 /A, x1e = 2 ×10−2 m, x2e = 0m/s,

x&% = x& − xˆ&

= Ax + Bu − ( Axˆ + Bu + L ( y − yˆ ) )

&ˆ t ) = Axˆ + Bu + L( y − yˆ ) x(

ze = 2.00 ×10−2 m, ye = 3.34V, we = 0V (24)

x3e = 0.27A,α = 2.48V (26) The PID controller designed and simulated based on the following control parameters (26) as below:

(25)

K P = 10; K I = 4; K D = 0.125

(27)

In LQR controller design, the best LQR parameters are as under:

 280 0 0  Q =  0 0 0  R = 0.625  0 0 70 

(28)

L = [ −4.1797 −67.5319 741.4103] (29)

The simulation was carried out to analyze the output performance when using PID or LQR control. It also verified ied the LQR observer‘s ability to estimate the state of Magnetic Levitation.

Figure 2. Observer with perfect model

4. EXPERIMENT ANALYSIS

RESULT RESULTS

AND

This section present the simulation result and the analyses which is experienced from PID control and LQR observer control.The control. simple Magnetic netic Levitation is shown under

The simulation resultswere swere showed in Figure 4 and Figure 5 proved that the LQR observer control could be estimated the output signalss which couldn’t be measure directly from sensors.Both oth of these controllers are suitable to utilize to control Electromagnetic Levitation system due to both can give zero steady-state state error and no overshoots at the transient ansient response.

Figure 4. The step response for PID and LQR control

Figure 3. The simple Magnetic Levitation system

226


5. CONCLUSION This paper developed a method to control a simple Magnetic Levitation, successfully. The mathematical models of system were established, LQR observer was applied to design the control. The result experiences shows that LQR observer controller can be applied forElectromagnetic Levitation system and the output response of the magnetic levitation system can obtain high performance with high accuracy and stability.

Figure 5. The input signal for PID and LQR control

However, the LQR observer control has significant feature in fast response, and setting time is smaller than PID controller.

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[6]

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[8]

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