Control of a Ma Non-Lin

Muhammad Ahsan Department of Electrical an Power En

Pakistan Navy Engineering College, NationaScience and Technology, Karachi, P

ahsan_kh05@hotmail.com

Abstract— In this paper we present differenttools applied on a Magnetic Levitation syscontrol under the presence of parametric perturbations. The system in view is inherentand requires the application of Robust Constabilize the system at desired heights orapplications. Several non-linear State Feeddesign methods were applied on the system ana robust Output Feedback control was deveGain Observer to recover the performance ofcontroller. The simulations presented showeffectiveness of different design techniques.

Keywords—Non-Linear , magnetic leFeedback, High Gain Observer

I. INTRODUCTION The problem under study is a Magnetic

as described in [1]. In the system a ball of msuspended by means of an electromagnet controlled by feedback from the, opticallposition [2 pp. 192-200]. The system has theof systems constructed to levitate mass, usaccelerometers, and fast trains. The equationball with a specified vertical distance takes the viscous friction, acceleration due to gravitby electromagnet and its current, inductancstored in the electromagnet and the magnetic

Due to a large number of interlinked elgood chance of system destabilization duncertainties and small external disturbancerequires a robust control design input toinvolved parameters, either measureable or nstate within a specified domain to guarantee s

We present the performance analysis of dcontrol methods applied on the system, theiconvergence behaviors [3]. The numerous minclude: Stabilization using Linearization,Linearization [4], Tracking a time varying refeedback Linearization, Robust Stabilizatiusing Discrete and Continuous Sliding Mode10-20% parametric variation. As a speciaFeedback Controller based on Khalil Obserbecause of its novel feature of performancAttraction Recovery of the system as in State[6], [7].

agnetic Levitation Systnear Robust Design To

ngineering al University of

Pakistan

Nouman MasoDepartment of El

College of E&ME, NationTechnology, Ra

t non-linear design stem for elevation

uncertainties and tly highly unstable

ntrol techniques to r use in tracking dback and Robust nd as a special case eloped using High

f the state feedback w the results and

evitation, Robust,

Levitation system magnetic material is

whose current is ly measured, ball e basic ingredients sed in gyroscopes, ns of motion of the

into consideration ty, force generated ce of coil, energy flux linkage.

lements there is a due to parametric es. So the systems o regulate all the not, for any initial stable behavior.

different non-linear ir comparison and

methods considered , State Feedback ference using state ion and Tracking e Control [5] under al case an Output rver was designed ce and Region of

e Feedback Control

The simulations presented the analysis of different desicomparing the results as per one

II. EQUATI

The equation of motion of th

Where m is the mass of

(downward) position of the bpoint (y = 0 when the ball is nfriction coefficient, g is the accits electric current as shown ielectromagnet depends on the modeled as

where , and a are po

Figure 1. Magneti

The model represents the caits maximum value when thedecrease to a constant value asdistance i.e. y = . The energydefined as:

,

the force is given by

When the electric circuit of source with voltage v, Kirchhoffollowing relationship:

tem Using ools

ood, Fawad Wali lectrical Engineering nal University of Science and awalpindi, Pakistan

with each section complement ign methods adopted and help e’s application.

IONS OF MOTION he ball is:

(1)

the ball, y 0 is the vertical ball measured from a reference next to the coil), k is the viscous celeration due to gravity, and I is in Fig.1. The inductance of the position of the ball and can be

(2)

ositive constants.

c Suspension System

ase that the inductance will have e ball is next to the coil and s the ball is removed to a large y stored in the electromagnet is

(3)

(4) the coil is driven by a voltage

ff’s voltage law gives the

978-1-4673-5885-9/13/$31.00 © 2013 IEEE

(5)

Where R is the series resistance of the circuit and ; is the magnetic flux linkage.

III. STATE EQUATION

For the system we define the stae varaibles taken are displacement, velocity and current defined as follows:

(position) (velocitty) (6)

(current) and u = v as the control input. The magnetic suspension system is modeled as

1 22

0 32 2 2

1

0 2 33 3 2

1 1

2 ( )

1( ) ( )

x x

L axkx g xm m a x

L ax xx Rx u

L x a x

=

= − −+

= − + ++

(7) The niminal values foer the system are as follows: m = 0.1kg, k = 0.01N/m/sec, g = 9.81m/sec2, a = 0.05m,

= 0.01H, =0.02H, and R=1

IV. CONTROL TECHNIQUES IMPLEMENTATION

A. Stabilization via Linearization For the Control methods implementation we decided to

stabilize the system at y = 0.05m. To do this task we first found the Steady State Current

value i.e.

; Then we shifted the origin of the system to

x-- = x - xss i.e. x1 - 0.05 ; x2 ; x3 – 6.26 and linearized the original system at origin.

= A =

= B = (8)

= C =

Using the feedback control law

in (7), where ‘K’ is such that A – BK is Hurwitz. Using Pole Placement method to find the gains K = [k1 k2 k3] so that the Eigen values of the system are at [-20 -30 -40]:

K = = (9)

The system was then constructed using state equations in Simulink. The system is stable and settles at the desired equilibrium point i.e.

X = =

Figure 2. Results of Stabilization using Linearization

B. Stabilization via State Feedack Linearization For the application of state feedback Linearization it was

required to make a Diffeomorphism for the system to introduce a change of variables

z = T(x) through the map T such that T must be invertible; i.e. it must have an inverse map T-1(.) such that x = T-1(z) for all z � T(D), where D is the domain of T. A continuously differentiable map with a continuously differentiable inverse is called a diffeomorphism [1].

According to [1] a system is Feedback Linearizable if and only if there is a domain Do � D such that 1. The matrix has

rank 3 for all 2. The distribution is invloutive

in

is invloutive because is a null vector and distribution has rank 2.

The system has relative degree = n =3, so it is full state feedback linearizable. For our system T(x) becomes:

0 1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06

time

Sta

te x

1

0 1 2 3 4 5 6 7 8 9 10

0

0.1

0.2

0.3

0.4

0.5

time

Sta

te x

2

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2

3

4

5

6

7

time

Sta

te x

3

T(x) = (10)

For the given system the elements of T(x) are: h(x) =

=

= =

=

(11)Using this Diffeomorphism we changed the system into ‘z’

coordinates and the specific form of the input vector ‘U’ becomes:

= = =

(12)

For this specific form of the system we can easily define the input U as:

(13)Where and :

(x) = (14)

(15)With these parameters known, (13) becomes:

U =

(16)

Where ‘k’ is such that Ac – BcK is Hurwitz, with

for poles at [-20 -30 -40].

Input ‘U’ is designed such that it cancels the non-linear terms and stabilizes the system at desired position. The system now stabilizes at the desired equilibrium point [0.05 0 6.26] as shown in the Simulink results below:

Figure 3. Stabilization using State Feedback Linearization

C. Tracking Using FeedBack Linearization After the development of Diffeomorphism for the system it

was time to test it for tracking a time varying signal r(t). For this purpose we took a time varying Sinusoid with DC Offset reference defined as:

For this purpose we transform the system into error coordinates as described in [1].

R = , e = = Z – R , =Z (17)

Where R is the vector containing reference and its derivatives which are available online, e is the vector of errors of states. The purpose of this method is to reduce the rate of change of errors asymptotically to zero i.e. 0 as t .

= y(t) – r(t) 0

(18)

U = (19)

This input is applied to the system described in (12), where the gains K = [k1 k2 k3] are same as in (9). The results of the tracking are shown below:

0 2 4 6 8 10 12 14 16 18 200

0.01

0.02

0.03

0.04

0.05

0.06

time

Sta

te x

1

0 2 4 6 8 10 12 14 16 18 20-0.005

0

0.005

0.01

0.015

0.02

0.025

time

Sta

te x

20 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

6

7

time

Sta

te x

3

Figure 4. Tracking Results

D. Robust Stabilization Using Sliding Mode Control The dynamic model of the system in z coordinates is given

by (12). As in theory of SMC we define a sliding Surface i.e. (20)

(21)

(22)

(23)

The Control Law becomes:

(24)

System Parameters for simulation are:

Figure 5. Satbilization using SMC

Figure 6. Applied Control Input U

So the system reaches the Surface S = 0 in finite time and enters in the sliding phase controlled by ‘ ’. By increasing the value of prameter ‘ ’ the allowed bound near the surface S = 0 increases i.e. the constraints are not very strict any more as we are now allowing the state to settle within a small area around the surface S = 0 bounded by ‘ ’ in the settling phase rather

than origin itself and system behves like ultimately bounded as shown in fig.7.

Figure 7. Results for

Now if we perturb the system parameters by 10% to check the system robustness, i.e. m = 0.11, k = 0.011, g = 9.81, a = 0.055, = 0.011, =0.012, and R=1.1. The system asymptotically settles at the desired value.

Figure 8. Satbilization SMC with 10% Perturbation

E. Linear Observer Design In this part a linear observer was designed for the system

using Linearization described in (8) using measured output only i.e. y = x1 only and estimating states x2 and x3.

= A =

= C =

The Observability Matrix was made and it is full rank.

Observability Matrix = (25)

It has rank = 3. Then we found the Observer Gains i.e.

H = =

such that the matrix A – HC is Hurwitz for eigen values at [-30 -40 -50]. For A – BK the same gains are used as in (9). For a system = Ax + Bu; y = Cx where (A,C) is observable, the observer is given by: (26)

The design of this observer is based on two steps. First we make a State Feedback Controller that to globally stabilize the origin of the non-linear system and then an Output feedback Controller is obtained by replacing the state ‘x’ by its estimate ‘ ’ provided by the Observer. The results are shown below:

0 5 10 15 20 250

0.02

0.04

0.06

0.08

time

Ou

tpu

t

Output

0 5 10 15 20 25 300

0.01

0.02

0.03

0.04

0.05

0.06

0.07

time

Ou

tpu

t

OutputReference

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-2

-1.5

-1

-0.5

0

0.5

time

Co

ntr

ol I

np

ut

U

0 5 10 15 20 25 300

0.01

0.02

0.03

0.04

0.05

0.06

time

Ou

tpu

t

OutputReference

0 5 10 15 20 25 30 35 40 45 500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

time

Ou

tpu

t

Figure 9. Linear Observer Results

V. HIGH GAIN/ KHALIL OBSERVER DESIGN Considering the original non-linear system of (7)

1 22

0 32 2 2

1

0 2 33 3 2

1 1

2 ( )

1( ) ( )

x x

L axkx g xm m a x

L ax xx Rx u

L x a x

=

= − −+

= − + ++

where = . In part (A) we designed a locally Lipchitz stabilizing state feedback controller (27) that stabilizes the origin = 0 of the closed loop system.

To implement this feedback controller using only the measurements of output y, we use the observer design method described by [1].

The Observer becomes: (28)

As described by [1] the gains are chosen as:

= (29)

for

Using this Output Feed Back Observer we recover the performance of State Feed Back controller with its asymptotic convergence and Region of Attraction Properties. For different values of we get very close to the performance of State Feed Back Controller as is reduced to zero.

The system exhibits an overshoot in the states evolution process before converging to the original results and the estimated input also shows a large overshoot for a short time in the beginning before it goes to zero, as an inherent property of the High Gain Observer [1].

This overshoot phenomenon is called peaking which can make the system unstable and may result in existence of finite escape time. To overcome this non-linearity we saturate the input as if putting the physical constraints on the system, i.e. (30)

Using this sat we can limit the input between a specified maximum and minimum value and still recover the performance of State Feedback controller.

Figure 10. High Gain Observer Results

0 5 10 15 20 25 300

0.01

0.02

0.03

0.04

0.05

0.06

0.07

time

Sta

te x

1^

0 5 10 15 20 25 30-12

-10

-8

-6

-4

-2

0

2

4 x 10-3

time

Sta

te x

2^

0 5 10 15 20 25 306.245

6.25

6.255

6.26

6.265

6.27

6.275

6.28

6.285

time

Sta

te x

3^

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.01

0.02

0.03

0.04

0.05

0.06

0.07

time

Sta

te x

1

SFBOFB e = 0.1OFB e = 0.01OFB e = 0.005

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

time

Sta

te x

2

SFBOFB e = 0.1OFB e = 0.01OFB e = 0.005

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

20

40

60

80

100

time

Sta

te x

3

SFBOFb e = 0.1OFB e = 0.01OFB e = 0.005

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-120

-100

-80

-60

-40

-20

0

20

time

Inp

ut

U

SFBOFB e = 0.1OFB e = 0.01OFB e = 0.005

Now upon Saturating the Input U between [-2.5 Vss] to avoid the peaking and check the state results.

Figure 11. Khalil Observer input with Sat

Figure 12. Khalil Observer Results with Input Sat

VI. CONCLUSION This paper we illustrated different non-linear design tools

applied on a Magnetic Suspension System for position control under the presence of parametric uncertainties and external disturbances. The system is highly unstable and requires the application of Robust Control techniques to stabilize the system at desired displacement. Numerous non-linear State Feedback and Robust control design methods were applied on the system and at the end a robust Output Feedback control was developed using High Gain Observer to recover the performance of the state feedback controller. The simulations presented show the results and effectiveness of different design techniques.

ACKNOWLEDGMENTS The authors would like to thank the faculty of Electrical

Engineering for providing the research opportunity, Dr. Attaullah Memon for his guidance and Sir Hassan. K. Khalil for his outstanding contributions in nonlinear systems and especially High Gain Observers.

REFERENCES [1] Hassan. K. Khalil, Non-Linear Systems, 3rd ed, Prentince Hall, upper

Saddle River, New Jersy, NJ 07458 [2] H. H. Woodson and J. R. Melcher, Electromechanical Dynamics, Part I:

Discrete Systems, John wiley , New York, 1968 [3] M. Vidyasagar, Non-Linear Systems asnalysis Prentince Hall,

Englewood Cliffs, NI, 2nd ed, 1993 [4] John A. Henley, Deign And Implementation of a Feedback Linearizing

Controller and Kalman Filter for a magnetic levitation system, MS Thesis, University of Texas at Arlington, 2007

[5] V. Utkin, J. Guldner, “Sliding Mode Control in Electromechanical Systems”, Taylor and Francis, London 1999

[6] A. N. Atassi, H. K. Khalil, “A separation Pinciple for the stabilization of a class of nonlinear systems,” IEEE Trans. Automat. Contr. 44:1672-1687,1999

[7] F. Esfandiari, H. K. Khalil, “Output Feedback Linearization of Fully LInearizable Sytems”, Int. J. Contr., 56:1007-1037, 1992

The results clearly show the effectiveness of this method as the states converge to the results of State Feedback Controller with improved Transient Performance and input converges to zero, even under small parametric uncertainties.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-4

-2

0

2

4

6

8

time•

Inp

ut

U w

ith

Sat

SFBOFB e = 0.1OFB e = 0.01OFB e = 0.005

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.01

0.02

0.03

0.04

0.05

0.06

0.07

time

Sta

te x

1 u

nd

er U

Sat

SFBOFB e = 0.1OFB e = 0.01OFB e = 0.005

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.2

0

0.2

0.4

0.6

0.8

time

Sta

te x

2 u

nd

er U

Sat

SFBOFB e = 0.1OFB e = 0.01OFB e = 0.005

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-4

-2

0

2

4

6

8

time

Sta

te x

3 u

nd

er U

Sat

SFBOFB e = 0.1OFB e = 0.01OFb e = 0.005