Design and Implementation of Adaptive PID Controller for Active

INTERNATIONAL CONFERENCE ON ADVANCES IN TRIBOLOGY [ ICAT14 ]

Design and Implementation of Adaptive PID Controller for Active Magnetic Bearings Kumar Akash1, Lijesh K.P.2, Vedant Chittlangia3, Harish Hirani4 1

Kumar Akash., Department Mechanical Engineering, Indian Institute of Technology, New Delhi, India -110016 Email: [email protected] 2 Lijesh K.P. Department Mechanical Engineering, Indian Institute of Technology, New Delhi, India -110016 Email: [email protected] 3 Vedant Chittlangia. Department Mechanical Engineering, Indian Institute of Technology, New Delhi, India -110016 Email: [email protected] 4 Harish Hirani., Department Mechanical Engineering, Indian Institute of Technology, New Delhi, India -110016 Email: [email protected]

Abstract: The Active Magnetic Bearings (AMBs) need a closed loop feedback controller for its stable operation. In the present work a method has been proposed to design the preliminary controller for an unknown AMB system with knowledge of the maximum value of current that can be supplied to electromagnet and sampling rate of data acquisition system. This initial controller is tuned using an adaptive technique called RAPID. The developed controllers were implemented on the active magnetic bearing setup (MBC500). The performance evaluation of initial controller (PID), RAPID and onboard controllers were carried out based on two objectives: minimum initial overshoot and settling time. Keywords: Active Magnetic Bearing, PID, Controller tuning, Adaptive Controller

1 INTRODUCTION The applications of Active Magnetic Bearings (AMBs) have been increasing due to its maintenance free operation that provides longer machine life compared to the conventional bearings [1]. Moreover, AMB systems do not need lubrication systems, so can be used in harsh environmental conditions, including extremely low temperatures, zero-gravity, and corrosive environments. Due to these reasons, AMB have been used in high speed compressors [2], turbomolecular vacuum pump [3], artificial heart blood pump [4], Flywheel energy storage systems [5], etc.

Fig.1 AMB setup

An AMB consisting of a pair of diametrically opposite poles of electromagnet, each carrying current ib+ic and ib-ic respectively with ib as the bias current and ic being the control current, has been shown in figure 1. In the figure, rotor has been shown in horizontal direction, but analysis presented in the current paper shall be applicable for vertical shaft also. Position sensors measure the position of the rotor (x, y) and send the signal to the controller. The controller decides the value of the control current ic according to the control algorithm for bringing the rotor at the set point. If the rotor moves to one side, say top for horizontal alignment of rotor, the current in the top electromagnet has to be reduced and the current in the bottom electromagnet has to be increased. As the system works in the differential mode, the controller has to give a negative value of ic resulting reduction in top electromagnet current (ib+ic), and higher current in bottom electromagnet (ib-ic). Similarly, if the rotor moves towards bottom, a positive value of ic has to be given by the controller. Therefore the controller gives the value of the control current ic according to the position sensor value x. The AMB systems are inherently unstable, multivariable, nonlinear dynamic systems [6] due to which controlling an AMB system is very difficult. The main reason for the AMB instability is its negative stiffness. The positive stiffness and negative stiffness have been illustrated in figure 2.


In the case of negative stiffness (illustrated in figure 2a) on reducing displacement (x) the force increases and vice versa. In the case of a positive stiffness case, such as spring mass system, the force increases with increase in the distance (x) value as shown in figure 2(b). Due to this reason the control algorithm for AMB must be robust to control the position of the rotor in every situation. In other words for an AMB system, feedback stabilization is essentially required.

(a) AMB System (b) Spring Mass System Fig.2 AMB and spring mass system The controller can be broadly classified into two types [7]: current control and voltage control. Voltage control requires more complex control algorithms than current control [7]. Due to the simplicity of the current control, the focus of the present research work will be on current control. To design a controller, numerous tools [8] can be used. However, most of the established tools require precise knowledge of the AMBs, sensors and rotor [9], which is often not available with a new designer. To deal with this, often robust adaptive control systems at the cost of performance are recommended [10]. Alternatively, information related to AMB, sensors, etc. can be derived experimentally by system identification methods [11]. But the system identification of an unstable system needs the system to be identified in closed loop with a feedback controller which makes the system stable [11]. In the present research, a method to establish initial controller (i.e. PID, Proportional-IntegralDerivative) with only the knowledge of the maximum bearing current and the sampling rate (i.e. controller time interval) has been proposed. Initial tuning of controller can be performed on-site, even without knowing the AMB characteristics such as the bearing’s negative stiffness (kx), force-percurrent coefficient (ki), sensor gain, offset, rotor geometry, mass and moments of inertia are all unknown. Later, a Result Adaptive PID control algorithm can be used to improve the performance of the designed controller. To evaluate the proposed

control scheme, same has been implemented on the existing magnetic bearing setup. To control the axes of magnetic bearing, a step by step method can be followed. First establish control action in one direction, and then in other direction. For horizontal rotor alignment, initially the vertical axis is controlled (with horizontal electromagnets turned off) and then with this vertical controller in action, the horizontal axis controller is designed. For vertical rotor alignment, similarly, any one axis controller is designed followed by the other. The experimental results have been detailed. 2 INITIAL TUNING OF CONTROLLER For initial controlling of AMB, it is assumed that rotor in x-direction is levitated using two opposing electromagnets operating with currents ib+ic and ib-ic. With this assumption, the linearized model for the plant can be illustrated as shown in figure 3 and can be represented as [10] mx = ic ki + xk x + Fd (1) Where; m is mass, x is displacement and Fd is disturbance force.

Fig. 3 Plant Model To design PID controller, value of the maximum current is required. Consider bias current in each (i.e. top & bottom) electromagnet is ib which is 50% of the maximum current. Let icmax be 50% of the maximum current. To stabilize the rotor, only proportional differential (PD) feedback, which overcomes the negative stiffness, is sufficient. An advantage of this method is that the input of the amplifier and the output of the sensor need not be calibrated. Until the relation between the input signal and the current supplied by the amplifier as well as the output signal and the position sensed by the sensor is linear, directly the values of the signals can be used for the calculations without any conversion. To get the value of KP and KD, following procedure can be opted: 1. The current is increased in one (say, top) electromagnet to imax and keeping the other (say, bottom) electromagnet turned off (by increasing ic from zero to icmax) such the rotor hits the electromagnet/back-up bearing. The value of displacement signal, measured using proximity sensor is recorded as x1 . 2. In the next step, the current in other side (say, bottom) electromagnet is increased to imax and


keeping the other (say, top) electromagnet turned off (by decreasing ic from icmax to –icmax) such the rotor hits the other electromagnet. The displacement value x2 is measured. Find the midpoint using x 0 = x1 + (x 2 − x1 ) . 2

3. Then, the current is slowly driven from -icmax to icmax and the position signal is monitored. As soon as the rotor crosses the midpoint (x0) of the system, the lift-off current (ic2) is recorded. 4. The previous step is reversed to measure the lift-off current from the other backup bearing. Furthermore the velocity measurement v[k ] = ( x[k ] − x[k − 1]) × f sp is derived from the position signal and the maximum value is stored as vmax. 5. To estimate the values of KP for static case, equation (1) can be modified, assuming that ..

at the time when the body lifts off and with x still being zero, the body is no longer leaning on the backup bearings and the following force balance is given at the startup from both backup rails, in following two equations:

ic1ki + ( x1 − x0 )k x + Fd = 0 ic 2 ki + ( x2 − x0 )k x + Fd = 0

2(a)

2(b)

Subtracting the above two equation gives

(ic1 - ic 2 )ki + ( x1 - x2 )k x = 0

(2c)

From the above equations we get:

kx i −i = c 2 c1 ki ( x1 − x2 )

(3)

Minimum proportional gain KP required to overcome the negative stiffness is given by kx/kp. Considering a factor of safety of ‘m’ for a stable controller,

KP = m

kx ki

(4)

The higher the KP value is, a fixed PID control will correct the real output value faster with higher precision. But the gain maximum value is restricted due to the control initial peak overshoot becomes too high with very high KP values. Thus the value of m has to be defined with the consideration of the above factors and a feasible range is:

centered. In order to eliminate this steady-state error, the integral action of the controller has to be appointed. The parameter KI is chosen such that the integration is very slow and does not significantly influence the dynamic behavior of the suspension. Therefore,

KI =

KP 10 f sp

(6)

Hence, the initial controller (PID) is designed for the given Active Magnetic Bearing setup as shown in figure 4.

Fig. 4 PID controller 3 ADVANCED PID: RESULT ADAPTIVE PID (RAPID) CONTROLLER The adaptive PID are required for two reasons: (i) The initial controller designed for a system either can provide fast response or fast settling. These two objectives are conflicting objectives and require different set of parameter for the PID controller. The initial PID is designed to achieve one of the parameter and the next parameter is tuned using adaptive techniques (ii) In a nonlinear plant, the set point changes with time to time. So for such system the controller has to be changed with time. Due to these reasons adaptive PID has to be adopted. There are a number of adaptation techniques to adjust the parameters of a controller like Gain Scheduling, Direct Adaptive Control, and Indirect Adaptive Control etc. The result adaptive PID controller adapts the parameters of an initially tuned PID controller depending on the value of the error signal for improving the position control of an AMB as shown in figure 5. In this method the coefficient of proportionate (Kp) and integral term (Ki) are adjusted in such way that it leads to faster settling and less overshoot in the system.

1 < m ≤ 1.5

To estimate the values of derivative gain, i.e. velocity feedback gain KD, it is chosen such that maximum control action is contrasting the motion when the body moves at maximum noted speed. This is so that it can provide an opposing acceleration for this velocity. Therefore, i K D = max (5) v max Thus, basic PD is designed by this method bringing the system to stability. But due to a constant disturbing force Fd, there could be off-

Fig. 5 Adaptive PID controller 3.1 KP Adaption: Faster Settling We require a smaller KP for large errors and the KP can be much larger for small error values for much stable controller. Such behavior is obtained using a continuous adaption function of the instantaneous error value such as:


f P (e) = 1 +

c1 − 1 (c2 e )2 + 1

(7)

This adaptation function is multiplied with the KP value of the PID controller. Here e is the error value. The two parameters c1 and c2 have to be adjusted for optimizing the controller. The parameter c1 (with c1>1) determines the factor by which the value of KP increases as e approaches zero value (i.e. the system approaches settling zone), such that the system comes in equilibrium in shortest possible time as observed in figure 6(a). While c2 (with c2>1) determines the steepness of the function, thus determining how much small window of error values should be considered as the settling zone of the system for the adaption to happen. The higher the value of c2, steeper will be the function and thus a narrower window of error values will be considered as the settling zone which is illustrated in figure 6(b).

Integral term (KI) is required in PID to eliminate the steady state error (i.e. the difference between the reference value and actual value at steady state). But this makes the control slower by increasing the settling time. Also, it increases the overshoot of the system as the value of KI increases. This can be avoided by incorporating the integral term only when the errors are small to bring the system to a state with no steady state error. Such behavior is obtained using a continuous adaption function of the instantaneous error value. This adaptation function is multiplied with the KI value of the PID controller. This gives a zero value of KI initially when the errors are large and the KI value increases as the value of error decreases and system approaches stability which is illustrated in figure 7. Since integration starts only after the settling phase, therefore the overshoot due to the integral term is also avoided. An appropriate function can be:

f I (e) =

3

c1=2 c2=200 c1=3 c2=200

2.8 2.6

1

(ce)2 + 1

(8)

1

2.4

c=100 c=250

f(e)

2.2

0.8

2 1.8

0.6 f(e)

1.6 1.4

0.4

1.2 1 -0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.06

0.04

0.08

0.1

Error Value (e)

0.2

(a) Variation of parameter c1 3

0 -0.1

c1=3 c2=200 c1=3 c2=500

f(e)

2

1.5

-0.05

0

0.05

0.1

Error Value (e)

(b) Variation of parameter c2 Fig. 6 Response of plant for different parameter of Kd These two parameters have to be adjusted in accordance with the order of the output sensor signals and the performance required. For large error values, i.e. when in the period of initial overshoot, the value of the function becomes 1 and maintaining the minimum overshoot and later as the error decreases and the system approaches settling zone, the value of function increases up to c1 and thus decreasing the settling time. 3.2 KI Adaption: reducing peak overshoot

0 Error Value (e)

0.05

0.1

Fig. 7 Response of plant for different parameter of Kd

2.5

1 -0.1

-0.05

The parameter c have to be adjusted for optimizing the controller. The parameter c determines the steepness of the function, thus determining how much small window of error values should be considered as the settling zone of the system for the adaption to happen. For large error values, i.e. in the initial stage with overshoot, the value of the function becomes 0 and avoiding the overshoot due to integral. The value becomes 1 for small error values, i.e. during settling zone, thus eliminating the steady state error. 4 TUNING PARAMETERS OF RESULT ADAPTIVE PID (RAPID) CONTROLLER There are three parameters which need to be tuned for a Result Adaptive PID (RAPID) Controller namely c1, c2 and c. The following range of c1 was determined experimentally for good performance of controller:

1.7 ≤ c1 ≤ 2.5


Fig. 8 RAPID controller The working principle of the RAPID is shown in figure 8 in the form of block diagram.. The parameters c2 decides the window of the settling zone. Larger their value, smaller the settling zone defined for the adaptation. Using equation (7), considering 33% of the maximum value on both the sides of the mean of the position signal (xmax) from the sensor as the settling zone and assuming that the adaptation fP(e) has increased 10% of the maximum increase required (i.e. f P (e) = 1 + 0.1(c1 − 1) ) at the beginning of the settling zone, then the value of c2 is given by: 9 (9) c2 = xmax The parameters c decides the window of the settling zone for the KI adaptation. Larger their value, smaller the settling zone defined for the adaptation. Using equation (8), considering 33% of the maximum value on both the sides of the mean of the position signal (xmax) from the sensor as the settling zone and assuming that the adaptation fP(e) has increased 10% (i.e. f I (e) = 0.1 ) at the beginning of the settling zone, then the value of c is given by

c =

9 xmax

(10)

Fig. 9 Magnetic bearing setup (MBC500) MBC 500 is an experimental setup on which various controllers can be tested. Speed of rotor in MBC 500 is controlled using pressurized air and speed can reach the maximum of 10000 rpm. The rotor is supported on two active magnetic bearings. The bearing coils have 0.5A bias current upon which control current is superimposed. Nominal air gap is 0.4 mm. Hall sensors are placed just outside of the electromagnet at each end of the rotor to measure the rotor displacement. On the front panel of MBC500 there are four switches to connect and disconnect controllers, required to handle four degree of freedoms. Any external controller can be tested by replacing the existing controller of MBC500. One of the loop switch is turned off and the digital controller is connected for that axis control. Then the step mentioned in Section 2 is followed to tune the initial PID controller. The required parameters for tuning are tabulated in table 1: Table 1 Parameters for tuning Parameter Value Vin -3V to 3V fsp 100kHz

Hence, all the parameters of the controller are tuned and the controller is designed to bring the system to stability. Input (V)

2 1 0 -1 -2 -3

0

2

4

6

8

10

Time (10us)

12 5

x 10

(a) Voltage response during tuning 4

Position Output (V)

5 EXPERIMENTAL SETUP AND RESULTS The described method was implemented to design a digital controller for an existing Active Magnetic Bearing Setup MBC500 which already have a built in on-board controllers which can be disengaged by turning off the loop switches. CompactRIO 9022 System from National Instruments along with NI 9201 Analog Voltage Input Module and NI 9263 Analog Voltage Output Module was used as digital controller hardware to acquire the sensor position signals process it using the RAPID algorithm and to give signals to the amplifier for current control in the AMB. The testing of the developed controller was carried out on active magnetic bearing setup (MBC500) procured from Launchpoint (USA).

3

3 2 1 0 -1 -2 -3

0

2

4

6

Time (10us)

8

10

12 5

x 10

(b) Position response during tuning Fig. 10 voltage and position response during tuning


A PID controller with the derived values of the parameters was implemented as the initial controller over the setup which brought the system to stability. Then the parameters c1, c2 and c is calculated for the system for implementing the Result Adaptive PID controller. Table 3 value of parameter c1, c2 and c Parameter Value c1 2.5 c2 3 c 3 The RAPID controller was implemented making the controller better in terms of settling time and overshoot. The responses of the system with various controllers are given in figure 11. 2

Position (Volts)

1 0 -1 -2 -3 -4 0

0.5

1

1.5

Time (10 us)

2 x 10

(a) Existing board t a on Co t o econtroller 1

Position (Volts)

0.5 0 -0.5 -1 -1.5 -2 -2.5 -3 0

0.5

1

Time (10us)

1.5

5

1 0.5

Position (Volts)

The parameters derived from the experiment performed are as follows: Table 2 Parameters of experimental setup Parameter Value x1 3.13628V x2 -3.13483V x0 0.00072V ic1 1.543V ic2 -1.357V vmax 22285.3V/sec KP 6.336E-1 KD 1.346E-4 sec KI 6.336E-7 sec-1

0 -0.5 -1 -1.5 -2 -2.5 -3 0

0.5

1

Time (10us)

1.5

2 x 10

5

(c) RAPID Controller Fig. 11 Response of different controller The values of overshoot and settling time for different controllers are tabulated in table 4. Table 4 Values of overshoot and settling time for different controllers Controller % Initial Settling Time Overshoot On-board 46.60% 270 ms Initial PID 29.98% 1102 ms RAPID 29.81% 170 ms From the table it can inferred that the initial PID controller had lower over shoot compared to the existing on board controllers but the settling time is very high for the initial PID. In the case of RAPID controller both the objectives; overshoot and settling time is lesser than on board controller. Hence proving the designed RAPID controller is proving to be better controller than the onboard controller and standalone PID. 6 CONCLUSIONS Detailed procedures have been presented to find the constants (KP proportional, KI integral and KD deferential) for the initial PID controller for active magnetic bearing. This controller was just making the system stable. An adaptive PID controller (RAPID) was developed to make the designed initial controller more robust. The performance of the controllers was compared using magnetic bearing setup MBC500. The objectives considered for the evaluation were: minimum initial overshoot and settling time. The initial PID was giving lesser overshoot with higher settling time compared to onboard controller while RAPID provided a lesser initial overshoot and settling time compared to other two controllers.

2 x 10

(b) Initial tuned PID Controller

5

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