This lecture presents Backstepping controller design for another industrial example, a magnetic levitation train.
The control design procedure to be presented in this lecture provides some additional design tools: (i) Nonlinear Damping, and (ii) A very simple model based observer design.
1

Before the design, I would like to remind that the experimental setup is ready to test. By using the guide posted at course website, please install the software set, make the required laptop configuration, and then go to Riggs 25 (passcode is 1495#).
At workstation 3, you will find the experimental setup. Connect the ethernet cable to your laptop (host computer), launch xPC Target Explorer, upload analog_loopback.mdl file, build it, and run the model.
As you remember, this mdl file sends a sin signal to the Quanser Q4 analog output port.
If you see a sin wave at the target PC monitor, then all your installations are ok !
2

Magnetic levitation (maglev) trains present a powerful alternative to land, air, and classical rail transportation systems. Because of the friction between wheel and rail, conventional trains have speed limitations, operate at high noise levels and require frequent maintenance. Maglev trains replace wheel by electromagnets and produce the propulsion force without any contact. This motivates researchers to investigate some novel maglev topologies to increase the ride quality and to decrease the cost of the overall system.
3

• A novel topology for maglev systems was patented by Levi and Zabar. This system uses only one air-cored tubular linear induction motor to produce levitation, propulsion and guidance forces simultaneously
.
• Specifically, we design, implement, test and control the newly proposed maglev system in this study. Main aim of the study is to prove that the experimental performance of this maglev system is satisfactory to use it in a commercial application.
4

The motor has two main parts as in a classical rotary induction motor: the primary; which consists of the drive coils, and the secondary; which consists of an aluminum slit sleeve. Drive coils can be placed either on the track or onboard the vehicle. Fig. 1 shows the proposed maglev system with energization from the wayside, and Fig. 2 shows the other configuration of the proposed system with energization onboard the vehicle.
Fig. 1. Proposed maglev system with energization from the wayside.
(1-vehicle, 2-drive coils, 3-aluminum sleeve, 4support)
Fig. 2. Proposed maglev system with energization onboard the vehicle.
4-support)

Main Advantages:
• This new system produces levitation, propulsion and guidance forces simultaneously by using only one motor.
• It is not needed to control the levitation and guidance forces because the restoring force centers the moving part, as will be proved experimentally in the following.
6

Drive Coils Entire System Acceleration Sensor

5,00
4,00
3,00
2,00
1,00
0,00
0
-1,00
-2,00
0,2 0,4 0,6 0,8
Time (s)
Propulsion
Levitation
Guidance
1 1,2 1,4

State-Space Model i qs
  



R s
L s

1

 
T r


 i qs

 pL m

L L h s r
 dr
V i ds
 qr
 
 




R s
L s

1


1
T r
 qr


T r


 i ds
 p V h
 dr


 pL m

L L h s r
L m
T r i qs
 dr
 
1
T r
 dr

 p V h
 qr

L m
T r i ds
 qr
V


L m
L L T s r r

L m

L L T s r r
V
 
B

K
V
M M f

 i dr qs
  i qr ds


F
M
L

 qr dr


1
L s


1
L s
V qs
V ds where i ds and i qs are stator current components on d and q-axis, λ d s and λ q s depict rotor flux components on d and qaxis, V is linear velocity, R s resistance per phase, L s and L r is stator represent stator and rotor inductances per phase, L m is magnetizing inductance per phase, p denotes number of poles, h is pole pitch, σ depicts leakage coefficient, T r is rotor time constant, K f represents force constant, F
L depicts load force, M is the total mass of the moving part, B denotes viscous friction coefficient, and finally, voltages on d and q -axis.
V d and V q are stator

To simplify the control design procedure, system dynamics given can be written in a more compact form as x
1
  a x
1 1
 a x x
2 4 5
 a x
3 3
 a u
4 1 x
2
  a x
1 2
 a x x
2 3 5
 a x
3 4
 a u
4 2 x
3
  b x
1 3
 b x
2 1
 b x x
3 4 5 x
4
  b x
1 4
 b x
2 2
 b x x
3 3 5 x
5
  c x
1 5
 c
 x x
2 1 4
 x x
2 3
  c
3
Control problem can be defined as follow; drive the linear velocity x
5 to a desired velocity profile x
5d while the rotor flux components x
3 and x
4 are unmeasurable. It is assumed that all parameters related electrical and mechanical subsystems are known. To determine the performance of the controller to be designed, an error signal can be defined as e
 x
5
 x
5 d
.

Due to the states x
3 based observer as and x
4
(rotor flux components) are unmeasurable, let design a model x
1
  a x
1 1
 a x x
2 4 5
 a x
3 3
 a u
4 1 x
2
  a x
1 2
 a x x
2 3 5
 a x
3 4
 a u
4 2 x
3
  b x
1 3
 b x
2 1
 b x x
3 4 5 x
4
  b x
1 4
 b x
2 2
 b x x
3 3 5 x
5
  c x
1 5
 c
 x x
2 1 4
 x x
2 3
  c
3
Observer x
3
  b x
1 3
 b x
2 1
 ˆ
3 4 5
ˆ
4
  ˆ
1 4
 b x
2 2
 ˆ
3 3 5 x
ˆ
3 x
ˆ
4 estimation errors as x
3 x
3
 x
3
 x
ˆ
3 x
4
 x
4
 x
ˆ .
4 and x
4
. Define the state
This implies the state estimation error system will be x
3
  b x
1 3
 b x x
3 4 5 x
4
  b x
1 4
 b x x
3 3 5
.

To show the stability of this observer, following Lyapunov function can be used:
V obs

1
2 x
3
2 
1
2 x
4
2
V obs
 x x
3 3
 x x
4 4
  b x
2
1 3
 b x x x
3 3 4 5
 b x
2
1 4
 b x x x
3 3 4 5
  b x
2
1 3
 b x
2
1 4
State estimation errors go to zero exponentially, and thus we can use estimated values of this unmeasurable states during the control design. ■

By going back to the control design, we investigate the error system dynamics; e
 x
5
 x
5 d
  c x
 c
2
 x x
 x x
3
  
3 x
5 d
If x
3 and x
4 were available for measurement, one could choose the produced electromechanical force F e
= c
2
( x
1 x
4
x
2 x
3
) as the virtual control input and initiate directly the backstepping procedure. Since these state variables are not measured but estimated,
observer backstepping should be applied.
e
  c x
1 5
 c
 x x
ˆ
2 1 4
 x x
ˆ
2 3
  c
 x x
2 1 4
 x x
2 3
  
3 x
5 d e
  c x
1 5
 c
 x x
ˆ
2 1 4
 x x
ˆ
2 3
  c
 x x
2 1 4
 x x
2 3
  
3 x
5 d
 
1

1
 
K e x c x c d c x x e e

5 d

1 5
 
3
2
1 2

1
2  2
2
 Nonlinear
Damping
Term
The nonlinear damping term defined as N d c x x e d

2
1 2

1
2

2
2 coefficient, will be used to damp the state estimation errors
 x
3 the control input signals.
where d
1 and x
4 is the damping while designing

In this step, a new error variable (a new coordinate) is defined as z
1
 ˆ e
 
1
 c
2
 x x
ˆ
1 4
 ˆ
2 3
  
1
Then the final expression for the error system dynamics will be e
 
K e e
 c
 x x
2 1 4
 x x
2 3
  d c
2
1 2
 x
1
2  x
2
2
 e
 z
1
By following the standard backstepping procedure, let’s backstep on z
1
.
To reach the control input signals u
1 z
1 expression for z
1 and u
2
, dynamics of z
1 dynamics is obtained as x must be investigated. Note that z
1
  a u
1 4 1
  a u
2 4 2
   
3 x
4 3
  x
5 4 where
 i
, i

{1, 2,3, 4,5} are the factors of related signals which contain all measurable states and known parameters. For simplicity, their explicit expressions are written here due to their length. Then the feedback rule is
 a u
1 4 1
  a u
2 4 2
 
K z z 1 1 e
3 d
2

  
4
2
 z
5 1 where d
2 is the second damping coefficient and K z1 is a control gain. If control inputs are designed as above, the final dynamics for z
1 will be z
1
 
K z z 1 1 e x
4 3
  x
5 4
 d
2

  
4
2
 z
5 1
.

A very important subtask in meeting the control objective is to ensure a bounded rotor flux, i.e., rotor flux should be forced to track a bounded signal. Since x
3 and x
4 measurable, we replace them by their estimates and define a new error variable as are not
  x
ˆ 2
3
 x
ˆ 2
4
  d where ψ d is the desired flux. Investigating ε dynamics yields
  
2
1

2
3
 x
ˆ 3
4


2 b
 x x
ˆ
2 1 3
 ˆ
2 4
   d as virtual control input, then add and subtract a second stabilizing function α
2 hand side of above equation as shown in the following.
2

1 3
ˆ
2 4 to the right

  
2
1

ˆ 2
3
 x
ˆ 3
4


2 b
2
 x x
ˆ
1 3
 x x
ˆ
2 4
   d
 
2 z
2

2

ˆ
2 1 3
 x x
ˆ
2 4
  
2

2
 
K

 
2
1

2
3
 x
3
4

  d
  
K

  z
2
.

z
2
  a u
6 4 1
  a u
7 4 2
   
8 x
9 3
  x
10 4
 a u
6 4 1 z
2
  a u
7 4 2
 
K z z 2 2

 
K z z 2 2 x
9 3
 

8 d
3

  
9
2
10 x
10 4
 d
3

  
9
2
10
 z
2
.
 z
2
By combining two expressions for control input signals, which are
 a u
1 4 1
 a u
6 4 1
  a u
2 4 2
  a u
7 4 2
 
K z z 1 1 e
3 d
2

  
4
2
 z
5 1
 
K z z 2 2

8 d
3

  
9
2
10
 z
2 we can write the control input signal in vector matrix form as

1 a
4




1
6


2
7



1  

K z z 1 1
K z z 2 2
 e
8 d d
2
3


  
  
9

 z
5 1 z
10 2






1 a
4




1
6


2
7



1  

K z z 1 1
K z z 2 2
 e
8 d d
2
3


  
  
9

 z
5 1 z
10 2



The designed control law is well defined if the matrix
D
 
1 2
  
6 7

 is globally invertible.
Before presenting the stability analysis, let me remind the final dynamics of all error variables. We will need them to complete the stability analysis.
e
 
K e e
 
1 c
2
 x x
1 4
 x x
2 3
  d c
2
1 2
 x
1
2  x
2
2
 e z
1
 
K z z 1 1 e x
4 3
  x
5 4
 d
2

  
4
2
 z
5 1
 z
2
 
K

  z
2
 
K z z 2 2
 x
9 3
  x
10 4
 d
3

  
9
2
 z
10 2 x x
4
3
  b x
1 3
 b x x
3 4 5
  b x
1 4
 b x x
3 3 5

To show the stability of the overall system, let’s select the Lyapunov function as follows:
V

1
2
 e
2
 z
1
2
 
2
 z
2
2


2
1



1

1
 b d d d
1 1 2
1
3



 x
2
3
 x
2
4

Why do we add this term to the final Lyapunov function?
We already showed the exponential stability of the observation errors !!!
The answer is hidden in another characteristic behavior of the nonlinear systems:
------------- FINITE ESCAPE TIME --------------
Please see the following side note, which is a well-known demonstration of Finite Escape Time.

Consider the system x x x
2

 already showed the observation error goes to zero exponentially, i.e.,

( )
 
(0) exp k

R

Then the solution of the differential equation is

 1 x (0)(1
 k )

(0) (0) e t  
(0) (0) e
 kt which escape to infinity in a finite time, which is t f

1
1
 k ln





  k )



   k )
Consequently,
. This is the reason to put the observation errors into the final Lyapunov function, and also to use Nonlinear Damping terms in the design, to damp the terms related observation errors. !!!!!!
■

Back to the stability analysis:
V

1
2
 e
2  z
1
2  
2  z
2
2


2
1
1



1
1

1
2
 b d d d
1
3



 x
2
3
 x
2
4

V
 
K e e
2 
K z z 1 1
2 
K


2 
K z z 2 2
2 
3
4



1

1
 d d d
1 2
1
3



 x
3
2  x
2
4

 d
1



1
2 d
1 x
4
 c ex
2 1



2
 d
1



1
2 d
1 x
3
 c ex
2 2



2
 d
2


1
2 d
2 x
3
  z
4 1


2
 d
2


1
2 d
2 x
4
  z
5 1
 2
 d
3



1
2 d
3
V x
3
  z
9 2


 2
 
K e K z K e
2  z 1 1
2 


2
 d
3



1
2 d
3

K z z 2 2
2 x
4
  z
9 2



2

3
4



1

1
 d d d
1 2
1
3



 x
2
3
 x
2
4

GAS of the E.P. is achieved.

x
5 d
4
3.5
2


1 exp
3
2.5
2
1.5
1
0.5
0
0 2 4 6

8 10
Zaman (saniye)
12 14 16 18 20
Tracking Error
0.115
0.11
0.105
0.1
0.095
0.09
0.085
0.08
0.075
0 2 4 6 8 10
Zaman (saniye)
12 14 16 18 20

Your guns before today
Backstepping
Your guns as of today
Backstepping Nonlinear Damping
You will have lots of guns at the end of the semester to control the systems in nature.