2.1.1 Sensor

In this subsection the position sensor characteristics is identified. If you click the Sensor button the following window opens (see Fig. 4)

Fig. 4. Sensor signal in [V] vs. the sphere distance from the electromagnet in [mm]

The following procedure is required to identify the characteristics.

1.

Screw in the screw bolt into the seat.

1. Screw in the black sphere and lock it by the butterfly nut. Notice, that the sphere is fixed to the frame!
2. Turn round the screw so the sphere be in touch with the bottom of the electromagnet.

4.

Switch on the power supply and the light source.

3. Start the measuring and registration procedure. It consists of the following steps:
4. Push the Measure button – the voltage from the position sensor is stored and displayed as Measured value [V]. One can correct this value by measuring it again.

° Push the Add button – the measured value is added to the list. A rotation number value is automatically enlarged by one (see Fig. 5).

Fig. 5. Characteristics of the sphere position sensor

° Manually make one full rotation of the screw. ° Repeat three last steps so many times as none change in the voltage vs. position characteristics is observed.

7. Push the Export Data button – the data are written to the disc (see Fig. 6). Data are stored in the ML_Sensor.mat file as the SensorData structure with the following signals: Distance_mm, Distance_m andSensor_V.

In the Simulink real-time models the above characteristics is used as a Look-Up-Table model. The block named Position scaling is located inside the device driver block of MLS (see Fig. 7). Notice, that the characteristics shows meters vs. Volts. In Fig. 6 there were shown Volts vs. meters. It is obvious that we require the inverse characteristics because we need to define the output as the position in meters.

Fig. 6. The sensor characteristics after being measured and exported to the disc

Fig. 7. The Simulink Look Up Table model representing the position sensor characteristics

If we click this block the window shown in Fig. 8 opens. Any time you like to modify the sensor characteristics you can introduce new data related to the voltage measured by the sensor. The voltage corresponds to the distance of the sphere set by a user while the identification procedure is performed. The sensor characteristics is loaded from the ML_sensor.dat file which has been created during the identification procedure. If the curve of the Position scaling block is not visible please load the file with data.

The sensor characteristics can be approximated by a polynomial of a given order. For example, we can use a fifth order polynomial.

P(x) = p₅ x⁵ + 2 + p₀

p₅ =−25697073504.59, p₄ = 1245050011.25, p₃ =−18773635.92, p₂ = 79330.24, p₁ =−150.21 and p₀ = 5.015.

Fig. 8. Look-Up Table to be fulfilled with vectors of input and output values

The approximated polynomial (the red line) is shown in Fig. 9. The polynomial approximation will be not used in this manual due to the fact that the entire model is built in Simulink. Therefore we recommend to model the characteristics as a Look-Up Table block (see Fig. 7 and Fig. 8).

Fig. 9. The sensor characteristics approximated by the fifth order polynomial

2.1.2 Actuator static mode

In this subsection we examine static features of the actuator i.e. the electromagnet. Notice, that the sphere is not present!

Click the Actuator static mode button and the window shown in Fig. 10 opens.

Now, we can perform button by button the operations depicted in Fig. 10. We begin from the Build model for data acquisition button. The window of the real-time task shown in Fig. 11 opens and the RTW build command is executed (the executable code is created).

Click the Set control gain button. It results in activation of the model window and the following message is displayed (see Fig. 12):

In Fig. 11 one can notice the Control signal block. In fact the control signal increases linearly. We can modify the slope of this signal changing the Control Gain value.

Click the Data acquisition button. Within 10 seconds data are acquired and stored in the workspace.

Click the Data analysis button. The collected values of the coil current are displayed in Fig. 13.

The characteristics is linear except a small interval at the beginning. We can locate the cursor at the point where a new line slope starts (see the red line in the picture). We can move the cursor in two ways: by writing down a value into the edition window or by drugging the slider. In this way the current characteristics is prepared to be analyzed in the next step. The line is divided into two intervals: the first – from the beginning of measurements to the cursor and the second – from the cursor to the end of measurements.

After setting the cursor position, consequently, click the Analyze button. The following message (see Fig. 14) appears. We obtain the dead zone values corresponding to the control and current. The constants a and b of the linear part are the parameters of the line equation: i(u) = au + b .

These parameters, namely: = 0.00498 , = 0.03884 , k = 2.5165 and c_i = 0.0243

^uMIN ^x3MIN i are going to be used in the simulation model in section 2.3.1 (see the differential equations parameters).

To obtain a family of static characteristics for linear controls with different slopes we repeat the following experiment. We apply a PWM voltage signal in the time interval from 0 to 10 s. The PWM duty cycles for the subsequent ten experiments are varying linearly in the ranges: [0, 0.1], [0, 0.2], ..., [0, 1.0] (see Fig. 16 ).

012 34 567 8910 Time [s]

Consequently, we obtain diagrams of the currents corresponding to ten experiment (see Fig. 16). Each characteristics is approximated by a polynomial of the first order. Finally the entire current vs. PWM duty cycle relation is depicted (black points) in Fig. 17. The red line represents the linear approximation of measurements. We obtain the following numerical values of linear characteristics:

i(u) = k_iu + c ; a = 2.60798876298869, b = −0.01077522109792.

The constant c is obtained for u = 0. The family of linear characteristics is used to obtain the coefficients k_i vs. control u.

Fig. 16. Family of the output (current) characteristics

Fig. 17. Current vs. PWM duty cycle

2.1.3 Minimal control

In this subsection we examine the minimal control to cause a forced motion of the sphere from the supporting structure (tablet) toward the electromagnet against the gravity force. Notice, that in this experiment the sphere is not levitating! It is kept nearby the electromagnet by the supporting structure.

Click the Minimal control button and the window shown in Fig. 18 opens.

Now, we proceed button by button the operations depicted in Fig. 18 similarly to the procedure described in the previous subsection. We begin from the Build model for data acquisition button. The window of the real-time task shown in Fig. 19 opens.

Click the Set control gain button. It results in activation of the model window and the following message is displayed (see Fig. 20).

It means that we can set a duty cycle of the control PWM signal. The sphere is located on the support and the experiment starts. Click the Data acquisition button. A forced motion of the ball toward the electromagnet begins.

Click the Data analysis button. The collected values of the ball position are displayed in Fig. 21.

The sphere motion is visible. We can locate the cursor at the point slightly before a position jump occurs (takes place) (see the red line in the picture). We can move the cursor in two ways: by writing down a value into the edition window or by drugging the slider. In this way the acquired data are prepared to be analyzed in the next step.

After setting the cursor position, consequently, click the Analyze button. The following message (see Fig. 22) appears. This information means that the sphere located 15.82 mm from the electromagnet begins to move toward it when the PWM control over-crosses the 0.49485 duty cycle value.

2.1.4 Actuator dynamic mode

In this subsection we examine dynamic features of the actuator i.e. the electromagnet. It means that the moving sphere generates an electromotive force (EMF). EMF diminishes the current in the electromagnet coil. Click the Actuator static mode button and the window shown in Fig. 23 opens.

A user should perform three experiments: without the sphere (Without ball), with the sphere on the supported structure (Ball on the tablet) and with the sphere fixed to the rigid screw (Ball fixed).

We begin from the Build model for data acquisition button. The window of the real-time task shown in Fig. 24 opens. We have to set the control gain. If we are going to modify the control magnitude then we set the default gain to 1 and the subsequent duty cycles to: 0.25, 0.5, 0.75 and 1. Click the Data acquisition button and save data under a given file name.

Click the Data analysis button. It calls the ml_find_curr_dyn.m file. The following window opens (see Fig. 25). The parameters optimization procedure starts. The optimization routine is based on themlm_current.mdl model.

When ml_find_curr_dyn.m runs the optimization function fminsearch is executed. Fminsearch uses the ml_opt_current.m file.

The ki and fi parameters are iteratively changed during the optimization procedure. The current curve is fitted four times. This is due to the control signal form.

Fig. 25. Current curve – the fitting result of the optimization procedure

Finally the information about the mean values is displayed (see Fig. 26). The advanced user can use the functions code to perform a detailed analysis.

Current scalling

[V] to [A]

Current [A]

2.5 2

1.5

1

0.5

0

2 10

a₂ = 0.0168 a₁ = 1.0451 a₀ =−0.0317

2.3.1 Open Loop

° Simulink model

Next, you can click the first Open Loop button. The following window opens (see Fig. 32). Notice that the scope block writes data to the MLSimData variable defined as a structure with time. The structure consists of the following signals: Position [m], Velocity [m/s], Current [A], Control [PWM duty 0÷1].

If you click the Magnetic Levitation model block the following mask opens (see Fig. 33).

In Fig. 32 enter into the File option and choose Look under mask. The interior of the Magnetic Levitation model block shown in Fig. 34 opens.

Notice two integrator blocks in Fig. 34. In fact we deal with third order dynamical system. The third integrator related to the coil current is visible in Fig. 35.

The Simulink model is also equipped with the animation block. When a simulation starts the following window opens (see Fig. 36). The animation screen is updated in every sample time. All state variables: the ball position and velocity, and also the coil current are animated.

° Mathematical model

The Simulink model is consistent with the following nonlinear mathematical model Model nieliniowy

where:

The parameters of the above equations are given in the table below

Parameters	Values	Units
m	0.0571 (big ball)	[kg]
g	9.81	[m/s2]
emF	function of x1 and x3	[N]
emP1F	1.7521⋅10-2	[H]
emP2F	5.8231⋅10-3	[m]
)(x1fi	function of x1	[1/s]
iP1f	1.4142⋅10-4	[m·s]
iP2f	4.5626⋅10-3	[m]
ci	0.0243	[A]
ik	2.5165	[A]
x3MIN	0.03884	[A]
uMIN	0.00498

The electromagnetic force vs. position diagram is shown in Fig. 37 and the electromagnetic force vs. coil current diagram is shown respectively in Fig. 38.

Fig. 37. Electromagnetic force vs. position. The gravity force of the big ball (dashed horizontal line) is crossing the curve at the 0.009 m distance from the electromagnet

Fig. 38. Electromagnetic force vs. coil current. The gravity force of the big ball (dashed horizontal line) is crossing the curve at the 0.9345 A coil current

The electromagnetic force depends on two variables: the ball distance from the electromagnet and the current in the electromagnetic coil. This is clearly presented in Fig.

37 and Fig. 38. We can show these dependencies in three dimensional space (see Fig. 39). The ball is stabilized at [x , x , x ]= col(9⋅10⁻³, 0, 9.345⋅10⁻¹). It means that the ball

123 velocity remains equal to zero. The ball is levitating kept at the 9 mm distance from the bottom of the electromagnet. The 0.9345 A current flowing through the magnetic coil is the appropriate value to balance the gravity force of the ball.

° Linear continuous model

ML is a highly nonlinear model. It can be approximated in an equilibrium point by a linear model. The linear model can be described by three linear differential equations of the first order in the form:

The elements of the A matrix are expressed by the nonlinear model parameters in the following way:

The C vector elements correspond to an applied controller. For example, The PID controller shown in the next subsection requires C in the form:

2.3.2 PID

If you click the PID button the following windows open (see Fig. 41). The interior of the Magnetic Levitation model block has been shown in Fig. 34. The PID controller is built in the form:

PK	IK	DK
130	500	6

The simulated stabilization results are shown below.

0.15

Coil current [A]

Ball position [m]

0.1

0.05

0 -0.05

Time [s] T ime [s]

0.8

0.6

0.4

0.2 1

0 Time [s] Time [s] Fig. 42. PID simulation – the desired position is a constant.

-3 -3

x10 x10

2

1

0

-1

-2

0.4

0.35

Coil current [A]

Coil current [A]

0.3

Time [s] Fig. 43. PID simulation – the desired position is in a sine wave form.

0.15

0.1

0.05

0

-0.05

0.8

0.6

0.4

0.2

0 Time [s] Fig. 44. PID simulation – the desired position is in a square wave form.

2.3.3 LQ

If you click the LQ button the following windows open (see Fig. 45).

The continuous LQ regulator is depicted on the basis of ml_model4lq.mdl (see Fig. 46). The user can use two files

• ML_calc_steady_state.m
• ML_calc_lq.m

The first one calculates the equilibrium point of the system. The second one calculates the LQ controller parameters using linmod and lqr. linmod obtains linear models from systems of ordinary differential equations. In theML_calc_lq.m file we encounter the following command:

[A, B, C, D] = linmod( ml_model4lq );

The state-space linear model of the system of ordinary differential equations described in the block diagram model4lq is returned in the form of the A, B, C, D matrices. The state variables and inputs are set to the defaults specified in the block diagram. Having obtained the linear model calculated at x10, x20, x30 equilibrium point for the assumed value u0 of

the control we are ready to calculate the K gains of the LQ controller. We only need to assume the Q and R matrices. From the ML_calc_lq.m file we have

Q=eye(3,3); Q(1,1)= 300; Q(2,2)= 0.001; Q(3,3) = 10; R=10.5;

The following assumptions corresponding to the Q and R weighting matrices have to be satisfied:

The following command from the ML_calc_lq.m file:

[K,S,E] = lqr(A,B,Q,R) calculates the optimal gain matrix K such that the state-feedback law u= −Kx minimizes the cost function — (x′Qx + u′Ru )dt subject to the state dynamics ^dx = Ax + Bu .

Now, the gain vector K can be used as the optimal feedback (see the Simulink diagram in Fig. 45). We start the LQ simulation for a constant desired value and for the desired position assumed in a sine wave form. We obtain the results shown in Fig. 47 and Fig. 48.

9.5	0.03
9Ball position [m]	0.01 0.02 Ball velocity [m/s]
8.5	0
	-0.01

0 0.5 1 1.5 00.5 1 1.5 Time [s] Time [s]

Coil current [A]

0.3

0.2

0.1 0.9

0

00.5 1 1.5 2 Time [s] Time [s]

Fig. 47. LQ simulation – the desired position is a constant

0.04

0.02

0

Coil current [A] Coil current [A]

-0.02

0.8

0.6

0.4

0.2

0 01234 01234 Time [s] Time [s]

Fig. 48. LQ simulation – the desired position is in a sine wave form

0.15

0.1

0.05

0

-0.05 Time [s] Time [s]

0.8

0.6

0.4

0.2

0 Time [s] Time [s] Fig. 49. LQ simulation – the desired position is in a square wave form

MLS User’s Manual -33

Similarly, we perform the LQ simulation for the desired position assumed in a square wave form. The simulation results are consequently shown in Fig. 49.

Remember that the obtained results are correct as long as the control and _Û state variables do not saturate. Otherwise, the control algorithm has nothing to do with the LQ policy.

2.3.4 LQ tracking

If you click the LQ tracking button the following windows open (see Fig. 50). We do remember that the LQ control policy has been calculated for a given equilibrium point. To improve the LQ control action we introduce the LQ tracking policy. For each new value of the ball position the ball velocity, coil current values and the control are recalculated on the basis of nonlinear dynamical equations of ML (the ML_GetStState s-function is used). In fact we should introduce a no-stationary LQ – it means solve the Riccati equation for every new equilibrium point to obtain a new value of the gain vector K.

Now, the gain vector K can be used as the optimal feedback (see the Simulink diagram in Fig. 50). We start the LQ tracking simulation for a constant desired value and for the desired position assumed in sine and square wave forms. We obtain the results shown in Fig. 51, Fig. 52 and Fig. 53.

0.03

0.02

0.01

0 -0.01

Coil current [A]

Coil current [A] Ball position [m]

00.5 1 1.5 2 Time [s] Time [s]

0.3

0.2

0.1 0.9

0

0 0.5 1 1.5 2 00.5 1 1.5 2 Time [s] Time [s]

Fig. 51. LQ tracking simulation – the desired position is a constant

0.03

0.02

0.01

0

-0.01 Time [s] Time [s]

0.8

0.4

0.2

0

Fig. 52. LQ tracking simulation – the desired position is in a sine wave form.

0.15

0.1

Coil current [A]

0.05

0

-0.05 Time [s] Time [s]

0.8

0.6

0.4

0.2

0 Time [s] Time [s] Fig. 53. LQ tracking simulation – the desired position is in a square wave form.

2.4.1 PID

Double click the PID button. The real-time PID controller opens (see Fig. 55). The results of the real-time experiment are shown in: Fig. 56, Fig. 57 and Fig. 58.

0.1

0.05

0

-0.05

Coil current [A] Coil current [A]

Ball position [m] Ball position [m]

-0.1 Time [s] Time [s]

1

0.5 1.2

0 01234 01234 Time [s] Time [s]

Fig. 56. PID real-time experiment. The desired position as a constant.

-0.1 Time [s] Time [s]

01234 Time [s] Time [s]

Fig. 57. PID real-time experiment. The desired position in a sine wave form.

0.3

0.2

Coil current [A]

0.1

0

-0.1 Time [s] Time [s]

1.4

1

0.8

0.6

Time [s] Time [s] Fig. 58. PID real-time experiment. The desired position in a square wave form.

2.4.2 LQ

Double click the LQ button. The real-time LQ controller opens (see Fig. 59). The results of the real-time experiment are shown in: Fig. 60, Fig. 61 and Fig. 62.

0.1

0.05

0

-0.05

Coil current [A]

Ball position [m] ^{Coil current [A]} Ball position [m]

-0.1 Time [s] Time [s]

0.8

0.6

0.4

0.2 1.2

0 01234 01234 Time [s] Time [s]

Fig. 60. LQ real-time experiment. The desired position as a constant.

-0.1 Time [s] Time [s] 7

01234 Time [s] Time [s]

Fig. 61. LQ real-time experiment. The desired position in a sine wave form.

0.2

0.1

0

-0.1

-0.2

1

0 Time [s] Fig. 62. LQ real-time experiment. The desired position in a square wave form.

2.4.3 LQ tracking

Double click the LQ tracking button. The real-time LQ tracking controller opens (see Fig. 63). The results of the real-time experiment are shown in Fig. 64, Fig. 65 and Fig. 66.

0.01

0

-0.01

-0.02

-0.03

1

0.8

0.6

0.4

0.2 1.2

Time [s] Time [s] Fig. 64. LQ tracking real-time experiment. The desired position as a constant.

0.02

0.01

0

Ball position [m]

-0.01

0.8

0.6

0.4

0.2

0 Time [s] Fig. 65. LQ tracking real-time experiment. The desired position in a sine wave form.

0.2

0.1

Coil current [A]

0

-0.1

-0.2 Time [s] Time [s]

0 Time [s] Time [s] Fig. 66. LQ tracking real-time experiment. The desired position in a square wave form.

ml = MagLev;

Display their properties by typing the command:

ml

BA = get( ml, ‘BaseAddress’ );