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CHAPTER 3

LQR BASED PID TUNING FOR TRAJECTORY

TRACKING OF MAGNETIC LEVITATION SYSTEM

3.1 INTRODUCTION

Magnetic levitation systems have received wide attention recently

because of their practical importance in many engineering systems such as

high-speed maglev passenger trains, frictionless bearings, levitation of wind

tunnel models, vibration isolation of sensitive machinery, levitation of molten

metal in induction furnaces, and levitation of metal slabs during

manufacturing (Almuthari & Zribi 2004). Magnetic levitation (maglev)

technology reduces the physical contact between moving and stationary parts

and in turn eliminates the friction problem. However, Maglev systems are

inherently nonlinear, unstable and are described by highly nonlinear

differential equations which present additional difficulties in controlling these

systems (Hajjaji & Ouladsine 2001). So the design of feedback controller for

regulating the position of the levitated object is always a challenging task. In

recent years, many methods have been reported in the literature for

controlling magnetic levitation systems. The feedback linearization technique

has been used to design control laws for magnetic levitation systems (Barie &

Chaisson 1996). The input-output, input-state, and exact linearization

techniques have been used to develop nonlinear controllers (Charara et al

1996). The current control scheme was employed for the control purpose

instead of the conventional voltage control by Oliveira et al (1999). Control

48

laws based on the gain scheduling approach (Kimand & Kim 1994), linear

controller design (Rifai & Youcef 1998), and neural network techniques

(Lairi & Bloch 1999) have also been used to control magnetic levitation

systems.

Classical optimal control theory has evolved over decades to

formulate the well known LQR, which minimizes the excursion in state

trajectories of a system while maintaining minimum controller effort. This

typical behavior of LQR has motivated control designers to use it for the

tuning of PID controllers (Das et al 2013). PID controllers are most common

in process industries due to its simplicity, ease of implementation and

design problem reduces to the ARE which is solved to calculate the state

feedback gains for a chosen set of weighting matrices. These weighting

matrices regulate the penalties on the deviation in the trajectories of the state

variables (x) and control signal (u). Indeed, with an arbitrary choice of

weighting matrices, the classical state-feedback optimal regulators seldom

show good set-point tracking performance due to the absence of integral term

unlike the PID controllers. Thus, combining the tuning philosophy of PID

controllers with the concept of LQR allows the designer to enjoy both optimal

set-point tracking and optimal cost of control within the same design

framework. The objective of this work is to present a PID controller tuning

approach via LQR based on dominant pole placement method. PID tuning

based on LQR method is tested on a benchmark magnetic levitation system

for trajectory tracking application, and the performance of the controller in

tracking the reference trajectory is assessed using various test signals.

3.2 MAGNETIC LEVITATION SYSTEM MODEL

Magnetic levitation systems are electromechanical devices which

suspend ferromagnetic materials using electromagnetism. The schematic

49

diagram of the magnetic levitation system is shown in Figure 3.1. A steel ball

in a magnetic levitation system is levitated in mid air by controlling the

current applied to the coils. The maglev system consists of an electromagnet,

a steel ball, a ball post, and a ball position sensor.

Figure 3.1 Magnetic levitation system diagram

Figure 3.2 Block diagram of closed loop magnetic levitation system

Figure 3.2 illustrates the overall block diagram of magnetic

levitation system. The entire system is encased in a rectangular enclosure

which contains three distinct sections. The upper section contains an

electromagnet, made of a solenoid coil with a steel core. The middle section

consists of a chamber where the ball suspension takes place. One of the

electro magnet poles faces the top of a black post upon which a one inch steel

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ball rests. A photo sensitive sensor embedded in the post measures the ball

elevation from the post. The last section of maglev system houses the signal

conditioning circuitry needed for light intensity position sensor. The entire

system is decomposed into two subsystems, namely, mechanical subsystem

and electrical subsystem. The various parameters of magnetic levitation

system are given in Table 3.1. The coil current is adjusted to control the ball

position in the mechanical system, where as the coil voltage is varied to

control the coil current in an electrical system. Thus, the voltage applied to

the electromagnet indirectly controls the ball position. In the following

section, the nonlinear mathematical model of the maglev system is obtained

and linearized around the operating region in order to design a stabilizing

controller.

Table 3.1 Magnetic levitation system parameters

Symbol Description Value

Lc Coil inductance 412.5mH Rc Coil resistance Nc Number of turns in the coil wire 2450 lc Coil length 0.0825m rc Coil steel core radius 0.008m Rs Current sense resistance Km Electromagnet force constant 6.5308E-005 rb Steel ball radius 1.27E-002 m

Mb Steel ball mass 0.068kg Kb Ball position sensor sensitivity 2.83E-003 m/V g Gravitational constant 9.81 m/s2

51

3.3 MATHEMATICAL MODEL OF MAGLEV SYSTEM

Figure 3.3 Schematic of the maglev plant

the electrical circuit shown in Figure 3.3

( ) dV R R I L Ic cc c s dt (3.1)

The transfer function of the circuit can be obtained by applying

Laplace transform to Equation (3.1)

( )( ) 1( ) cI s Kc cG sc sV sc (3.2)

where

1Kc R Rc s and

Lcc R Rc s

52

3.3.1 Equation of Motion of a Ball

The force applied on the ball due to gravity can be expressed by

F M gg b (3.3)

Force generated by the electromagnet is given by

212 2

K Im cFc xb (3.4)

The total force experienced by the ball is the sum of Fg and Fc

212 2

K Im cF F M gc g bxb (3.5)

can be obtained as

2 212 22

d x K Ib m c gM xdt b b (3.6)

At equilibrium point, all the time derivative terms are set to zero.

21 02 2K Im c gM xb b (3.7)

From Equation (3.7), the coil current at equilibrium position, 0cI ,

can be expressed as a function of 0xb and Km .

53

2

0 0M gbI xc bKm (3.8)

The electromagnet force constant, Km , as a function of the nominal

pair 0 0( , )b cx I , can be obtained from Equation (3.7).

22 020

M gxb bKmIc (3.9)

The nominal coil current 0cI for the electromagnet ball pair can be

obtained at the static equilibrium point of the system. The static equilibrium at

a nominal operating point 0 0( , )b cx I is characterized by the ball being

suspended in air at a stationary point 0xb due to a constant attractive force

created by 0cI .

3.3.2 Linearization of EOM

In order to design a linear controller, the system must be linearized around equilibrium point where the suspension of ball takes place. The nonlinear system equations are linearized around the operating point using

0 0( , )b cx I to

Equation (3.6),

2 2 2 111 0 0 0 12 2 3 22

0 0 0

d x K I K I xm m b K I Ib c c m c cgM x M x M xdt b b bb b b (3.10)

Substituting Equation (3.9) into Equation (3.10)

22 21 1 1

2 0 0

d x gx gIb b cx Ib cdt (3.11)

54

Applying Laplace transform to Equation (3.11)

2( ) 2 2

Kb bG sb s b (3.12)

where

00

xbKb Ic and 2

0g

b xb

The open loop transfer function of a maglev system is a type zero,

second order system. The two open loop poles of the system are located at

s b . Location of pole on the right half of the s-plane proves that the open

loop system is unstable in nature. Thus, the feedback controller is necessary

for stabilizing the system. In this work, a PID controller combined with the

feed forward component is designed to not only levitate the ball but also to

make the ball follow the reference trajectory.

3.4 PID PLUS FEED FORWARD CONTROLLER DESIGN

The objective of the control strategy is to regulate and track the ball

position in mid-air. The control scheme should regulate the ball in such a way

that in response to a desired 1mm square wave position set point, the ball

position response should satisfy the following performance requirements.

1. Percentage overshoot 15%

2. Settling time 1s

The proposed control scheme, as shown in Figure 3.4, consists of a

PID controller with a feed forward component. The gain of the feed forward

controller is determined based on the equilibrium point at which the ball

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suspends in mid air. Three separate gains are used in PID controller design,

which introduces two zeros and a pole at origin so that the entire system

becomes a Type 1 system, allowing for zero steady state error. The objective

of the feed forward control action is to compensate for the gravitational bias.

When the PID controller compensates for dynamic disturbances around the

linear operating point 00( , )cbx I , the feed forward control action eliminates the

changes in the force created due to gravitational bias.

Figure 3.4 PID plus Feed forward control loop for ball position control

The dynamics of the electromagnet is taken into account while

obtaining the following open loop transfer function ( )mG s , which relates the

output variable (ball position) and the input variable (coil current).

02

0

2( )( ) ( ) 2des

b cc

b

gIx sG sm I s gsx (3.13)

The current feed forward action is represented by

_ _I K xc ff ff b des (3.14)

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and

1 _I I Ic c c ff (3.15)

At equilibrium point, 0x xb b and 0I Ic c . Thus, the feed forward

gain is given by

0

0

IcK ff xb (3.16)

The closed loop transfer function of the system is

2 (( ) )( )( )

( ) 2 2 223 2( )00 0 00

g K K s Kx s p iffbG sc x s gK gK gKg pbdes d iI s s sc I x I Ic c cb (3.17)

The normalized characteristic equation of electromechanical system is

2 2 223 2 00 0 00

gK gK gKg pd is s sI x I Ic c cb (3.18)

3.5 LQR BASED PID TUNING

Figure 3.5 LQR based PID tuning of second order magnetic levitation

system

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LQR design technique is well known in modern optimal control theory and has been widely used in many applications (Lewis & Syrmos 1995). It has a very nice stability and robustness properties. If the process is of single-input and single-output, then system controlled by LQR has a minimum phase margin of 60o and an infinite gain margin, and this attractive property appeals to the practicing engineers. Thus, the LQR theory has received considerable attention since 1950s. Optimal control theory has been extended to PID tuning in the recent years. Saha et al (2012) have proposed a methodology for tuning PID controller via LQR for both over-damped and critically damped second order systems. They also suggested that one of the real poles can be cancelled out by placing one controller zero at the same position on the negative real axis in complex s-plane. So, the second order system which is to be controlled with a PID controller can be reduced to a first order system to be controlled by a PI controller. However, this approach is not valid for lightly damped processes because a single complex pole of the system cannot be removed with a single complex zero of controller, as they are in conjugate pairs. To address this issue, Das et al (2013) have given a formulation for tuning the PID controller gains via LQR approach with guaranteed pole placement. In this work, the idea has been extended to a magnetic levitation system which has two control schemes such as PID controller and feed forward controller. In this approach, the error, error rate and integral of error are chosen as state variables to obtain the optimal gains of the PID controller.

Let the state variables be

1( ) ( )x t e t dt 2 ( ) ( )x t e t 3( ) ( )dx t e tdt (3.19)

From Figure 3.5,

2 2( ) ( )( ) ( )2 ( )o o o

n n

Y s K E sU s U ss s (3.20)

58

In the state feedback regulator design, the external set point does not affect the controller design, so the reference input is zero (r(t)=0) in Figure 3.5. When there is no change in the set point, the relation y(t)=-e(t) is valid for standard regulator problem. Thus, Equation (3.20) becomes,

2 22 ( ) ( ) ( )o o on ns s E s KU s (3.21)

Applying inverse Laplace transform,

22 ( )o o on ne e e Ku (3.22)

Thus the state space representation of the above system is of the form

1 1

2 22

3 3

0 1 0 00 0 1 0

0 ( ) ( 2 )o o on n

x xx x ux x K

(3.23)

From Equation (3.23), the system matrices are

2

0 1 00 0 1

0 ( ) ( 2 )o o on n

A

00BK (3.24)

The following quadratic cost function which weighs both control input and state vector is minimized to obtain the optimal gains.

0

( ) ( ) ( ) ( )T TJ x t Qx t u t Ru t dt (3.25)

The minimization of above cost function gives the optimal control input as

1( ) ( ) ( )Tu t R B Px t Fx t (3.26)

59

where 1 TF R B P and P is the symmetric positive definite solution of the

following Continuous Algebraic Riccatti Equation (CARE).

1 0T TA P PA PBR B P Q (3.27)

The weighting matrix Q is a symmetric positive definite and the

weighting factor R is a positive constant. In general, the weighting matrix Q is

varied, keeping R fixed, to obtain optimal control signal from the linear

quadratic regulator. The corresponding state feedback gain matrix is

11 12 131 1 1

12 22 23 13 23 33

13 23 33

0 0Tpi d

P P PF R B P R K P P P R K P P P K K K

P P P (3.28)

The corresponding expression for the control signal is

1

2

3

( )( ) ( ) ( ) ( ) ( ) ( )

( )p pi id d

x tdu t Fx t K K K x t K e t dt K e t K e tdt

x t (3.29)

The third row of symmetric positive definite matrix P can be obtained in terms of PID controller gains as

13 1iK

PR K 23 1

pKP

R K 33 1dK

PR K (3.30)

The closed loop system matrix for the system given in Equation

(3.24) with state feedback gain matrix represented using Equation (3.28) is

1 2 2 1 2 1 213 23 33

0 1 00 0 1

( ) (( ) ) ( 2 )c

o o on n

A

R K P R K P R K P (3.31)

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The corresponding characteristic polynomial for the closed loop system is

( ) 0cs sI A3 2 1 2 2 1 2 1 2

33 23 13(2 ) (( ) ) 0o o on ns s R K P s R K P R K P (3.32)

The characteristic polynomial of closed loop system in terms of the

desired damping ratio and natural frequency is given by,

3 2 2 2 2 3( (2 ) (( ) 2 ( ) ( ) ) ( ) ) 0c c c c c c cn n nns s m s m m (3.33)

On equating the coefficients of Equation (3.32) with Equation (3.33),

1 233

2 1 2 2 2 223

1 2 2 313

(2 ) (2 )

( ) ) ( ) 2 ( ) ( )

( ) ( )

o o c cn n

o c c cn n n

c cn

R K P m

R K P m

R K P m (3.34)

The elements of the third row of matrix P are determined by

knowing the open loop process characteristics ( , , )o on K and desired closed

loop dynamics ( , , )c cn K .

2 3

13 1 2

2 2 2 2

23 1 2

33 1 2

( ) ( )

( ) 2 ( ) ( ) ( )

(2 ) 2

c cn

c c c on n n

c c o on n

mP

R Km

PR K

mP

R K (3.35)

by solving the ARE given in Equation (3.27). With the known third row

elements of P matrix the other elements of P and Q matrices can be obtained

as follows.

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2 1 211 13 13 23

1 212 13 13 23

1 2 222 23 23 33 33 13

( )

2

2 ( )

ono o

no o o

n n

P P R K P P

P P R K P P

P P R K P P P P (3.36)

1 2 21 13

1 2 2 22 23 12 23

1 2 23 33 23 33

2( ( ) )

2( 2 )

ono o

n

Q R K P

Q R K P P P

Q R K P P P (3.37)

3.5.1 Design Steps

Step 1: Specify both the open loop characteristics ( , , )o on K and the

desired closed loop system dynamics ( , , )c cn K .

Step 2: Choose the weighting factor R in LQR and determine the weighting

matrix Q using Equation (3.37).

Step 3: Obtain the transformation matrix P, solution of ARE, using

Equation (3.35) and Equation (3.36).

Step 4: Calculate the system matrices A and B as specified in

Equation (3.24)

Step 5: Determine the solution of state feedback control using Equation

(3.28) and obtain the PID controller gains.

3.6 SIMULATION RESULTS

Numerical simulations are carried out to evaluate the performance

of the proposed LQR based PID tuning for magnetic levitation system using

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MATLAB. The simulink block diagram of the overall closed loop system is

shown in Figure 3.6. The open loop parameters of magnetic levitation system

are found to be K=7, 1.8on , and the desired parameters of the closed loop

system which satisfy the controller specification given in section 3.4 are

0.8c , 1.4cn , and m=9. Based on the closed loop requirements of the

system given in terms damping ratio and natural frequency, the gains of the

PID controller are determined using the LQR weighting matrices approach as

presented in the above section. The corresponding system matrices are found

to be

0 1 00 0 10 3.24 0

A

007

B

The resultant state feedback gain matrix obtained by solving the ARE is

65.8 28.14 5.03 pi dF K K K

At the beginning of the simulation, the magnetic ball started in a

certain arbitrary position and was controlled to track the desired trajectory.

Figure 3.7 shows the simulation result of a set point change of the position

from 6mm to 8mm, and it is observed that the ball position response has a

settling time of 0.8s and an overshoot of 7%, which suggest that the design

requirement of the system are met with the above controller gains.

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Figure 3.6 Simulink block diagram of PID plus feed forward controller

Figure 3.7 Simulated ball position response

3.7 EXPERIMENTAL RESULTS

The experimental set up, as shown in Figure 3.8, consists of a

control algorithm is realized in the PC using the real time algorithm, QUARC,

which is similar to C like language. The sampling interval is chosen to be

0.002s, and in order to attenuate the high frequency noise current which

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increaes the oscillation in steel ball beyond the 6mm range, a low pass filter

of cut off frequency 80Hz is added to the ball position sensor output.

Furthermore, by differentiating the ball position, the ball vertical velocity is

estimated.

Figure 3.8 Snap shot of magnetic levitation experimental set up

3.7.1 Trajectory Tracking

Three signals namely square, sine and saw-tooth signals with a

frequency of 1 Hz is chosen as reference signals to test the trajectory tracking

performance of the controller. Figure 3.9, 3.10, and 3.11 show the results for

trajectory following. Figure 3.12 shows the response of coil current, which

tracks the specified command to make the ball follow the reference trajectory.

Figure 3.13 depicts the coil input voltage V(t), and it can be observed that the

amount of voltage supplied to the coil is comparatively high at the beginning

because of the unstable nature of the system, and it is worth to note that the

coil voltage is restricted to 8V during the stabilization phase. Figure 3.14

shows the tracking error, which is the difference between actual trajectory and

reference trajectory. The performance indices namely IAE, ITAE, ISE and

ITSE are considered to evaluate the performance of the controller, and they

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are given in Table 3.2. Reduced tracking error suggests that the position of the

magnetically levitated ball quite accurately tracks the desired trajectory.

Figure 3.9 Square wave trajectory

Figure 3.10 Sine wave trajectory

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Figure 3.11 Saw tooth trajectory

Figure 3.12 Coil current response

67

Figure 3.13 Coil voltage response

Figure 3.14 Trajectory tracking error

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Table 3.2 Performance indices of LQR based PID tuning method

IAE ITAE ISE ITSE

0.0041 0.1252 0.000017 0.000015

3.8 CONCLUSION

A combined design approach of PID tuning with LQR has been

applied for a trajectory tracking application of magnetic levitation system.

Conventional PID controller design has been formulated as a LQR problem

for the control of second order oscillatory system. For levitating the ball, the

conventional PID controller is combined with the feed forward controller in

order to nullify the effect of gravitational bias existing in the magnetic

levitation system. The non linear mathematical model of the plant from

fundamental physical laws has been obtained and the non linear equation has

been linearized around the equilibrium point using Taylor's series. Combining

the tuning philosophy of PID controllers with the concept of LQR theory, the

simple mathematical gain formulae has been formulated to obtain the

satisfactory response. The experimental results demonstrated the effectiveness

of the proposed approach not only in stabilizing the ball but also in tracking

the various reference trajectories given as an input.