Feedback Linearization
(ECES-817)
Presented by : Shubham Bhat
Feedback Linearization- Single Input case
Consider a system described by
x  f ( x)  ug ( x)
... 1
where f and g are smooth vector fields on some open set X  R"
containing 0, and f (0)  0.
There exists smooth functions q, s  S ( X ) with s ( x)  0 for all x
in some neighborhood of the origin, and a local diffeomorphism
T on R n with T (0)  0.
W e define
v  q( x)  s( x)u
... 2
z  T ( x)
... 3
Feedback Linearization- Single Input case
The resulting var iables z and v satisfy a linear differential equation
of the form
z  Az  bv
... 4
where the pair( A, b) is controllable..
In this case, the system is called feedback linearizable.
q ( x)
1
u  

v,
... 5
s ( x) s ( x)
q( x)
1
where 
and
are also smooth functions.
s ( x)
s ( x)
Hence if we think of v as the external input applied to the system,
then 5 represents the nonlinear feedback, and a nonlinearstate  depenedent
pre  filter, applied to the system.
Feedback Linearization- Contd.
By applying a state  transformation
z  M 1z such that the resulting system is in controllable canonical form
z  M 1 AMz  M 1bv,
where
1
0
0 
 0
0 


 
0
0
1
.
0
0
1
1



M AM 
,M b 
,
 ..



..
.
.
:


 
 a0  a1  a2 ..  an1 
1
and the ai ' s are the coefficients of the characteristic polynomial
n 1
| sI  A | s   a j s j
n
i 0
Feedback Linearization- Contd.
A further state  feedback form
v  v  [ a0 a1... an 1 ] z
results in the closed loop system
z  A z  b v ,
where
0

0

A
.


0
1
0
0
..
0
1..
..
0
0
0 

 
0
0

,b 

:
..

 
0


1 

z  M 1T ( x), v  v  az  q ( x)  a ' M 1T ( x )  s ( x )u
where
a '  [ a0 a1 ... an 1 ].
Problem Statement
Given the system as in 1, does there exists ??
(i) a smooth function q  S ( X )
(ii) a smooth function s  S ( X ) such that s( x)  0 for all x
in the neighborhood of 0.
(iii) a local diffeomorphism T : R n  R n such that T (0)  0,
satisfyingthe following conditions:
if new var iables v and z are defined ,
then z1  z2 , z2  z3 ,.... zn1  zn , zn  v
Example- Controlling a fluid level in a tank
Consider the control of the level h of fluid in a tan k to a specified
level hd . The control input is the flow u int o the tan k , and the initial
level is h0 .
The dynamic mod el of the tan k is
d h
[  A(h)dh]  u (t )  a 2 gh
dt 0
where A(h) is the cross sec tion of the tan k and a is the cross sec tion
of the outlet pipe.
If the initial level h0 is quite different from the desired level hd , the
control of h involves a nonlinear regulation problem.
Example – Contd.
The dynamics can be written as
A(h)h  u  a 2 gh
If u (t ) is chosen as
u (t )  a 2 gh  A(h)v
with v being an " equivalent input" to be specified, the resulting dynamics is linear
h  v
~
Choo sin g v as v  h
~
with h  h(t )  hd being the level error, and  being a positive cons tan t ,
the resulting closed loop dynamics is
h  h~  0
~
This implies that h (t )  0 as t  .
Example – Contd.
The actual input flow is det er min ed by the nonlinear control law
~
u (t )  a 2 gh  A(h)h
The first part on the RHS is used to provide the output flow a 2 gh
while the sec ond part is used to raise the fluid level.
If the desired level is a known time  var ying function hd (t ),
the equivalent input v can be chosen as
~

v  hd (t )  h
~
so as to still yield h (t )  0 as t  .
The idea of feedback linearizat ion can be applied to a classs
of nonlinear systems described by the controllab le companion
form.
Example – Contd.
A system is said to be in companion form if its dynamics are
x ( n )  f ( x)  b( x)u
where u is the scalar control input , x is the scalar output and
f ( x) and b( x) are nonlinear function of the states.
 x1  
x2


 

...
...
d 



xn
dt  xn1  

 

 xn   f ( x)  b( x)u 
U sin g the control input(assumin g b to be nonzero)
1
u  [v  f ]
b
multiple int egrator form
xn  v
Example – Contd.
Thus, the control law
v  ko x  k1x  ....  kn1x(n  1)
with the ki chosen so that the polynomial p n  kn1 p n1  ....  k0
has all its roots strictly in the left  half complex plane, leading to
x( n)  kn1x( n1)  ... k0 x  0
which implies that x(t )  0
Input State Linearization
x1  2 x1  ax2  sin x1
x2   x2 cos x1  u cos(2 x1 )
However , if we consider the new set of var iables
z1  x1
z2  ax2  sin x1
then, the new state equations are
z1  2 z1  z2
z2  2 z1 cos z1  cos z1 sin z1  au cos(2 z1 )
Input State Linearization-Contd.
The nonlinearities can be canceled by the control law :
1
u
(v  cos z1 sin z1  2 z1 cos z1 )
a cos(2 z1 )
where v is an equivalent input to be designed, leading
to a linear input  state relation
z1  2 z1  z2
z2  v
Thus, through the state transformation and input transformation,
the problem of stabilizing the original dynamics u sin g the original
control input u has been transformed int o the problem of stabilizing
the new dynamics u sin g the new input v.
Input State Linearization-Contd.
The linear state feedback control law
v   k1z1  k2 z2
can place the poles anywhere with proper choices of
feedback gains.
For example, we may choose v  2 z2
resulting in the stable closed  loop dynamics
z1  2 z1  z2
z2  2 z2
whose poles are placed at  2.
In terms of the original state x1 and x2 ,
1
u
(2ax2  2 sin x1  cos x1 sin x1  2 x1 cos x1
cos(2 x1 )
Input State Linearization-Contd.
The original state x is given from z by
x1  z1
( z2  sin z1)
x2 
a
0
v  K z
T
u  u( x, v)
x  f ( x, u)
Linearization Loop
pole-placement loop
z
z  z(x)
x
Input Output Linearization
Consider the system
x  f ( x, u )
y  h( x )
Our objective is to make the output y(t) track a desired trajectory
yd(t) while keeping the whole state bounded, where yd(t) and its
time derivatives up to a sufficiently high order are assumed to be
known and bounded.
e.g Consider the third  order system
x1  sin x2  ( x2  1) x3
x2  x15  x3
x3  x12  u
y  x1
Input Output Linearization-Contd.
To generate a direct relationship between the output y and the input u,
differentiate the output y
y  x1  sin x2  ( x2  1) x3
Since y is not directly related to the input u, we differentiate again.
y  ( x2  1)u  f1 ( x)
where f1 ( x) is a function of the state defined by
f1 ( x)  ( x15  x3 )( x3  cos x2 )  ( x2  1) x12
This represents an exp licit relationship between y and .
If we choose a control input in the form
1
u
(v  f1 )
x2  1
where v is a new input to be det er min ed .
Input Output Linearization-Contd.
W e obtain a simple linear double int egrator relationship
between the output and the new input v,
y  v
The design of a tracking controller for this double int egrator
is simple.
v  yd  k1e  k2e where e  y (t )  yd (t )
where k1 and k2 being positive cons tan ts.
The tracking error of the closed loop system is given by
e  k2e  k1e  0
which represents an exp onentially stable error dynamics.
Note : The control law is defined everywhere, except at the
singularity points such that x2= -1.
Internal Dynamics
•If we need to differentiate the output r times to generate an
explicit relationship between output y and input u, the system is
said to have a relative degree r.
•The system order is n. If r<= n, there is an part of the system
dynamics which has been rendered “unobservable”. This part is
called the internal dynamics, because it cannot be seen from the
external input-output relationship.
•If the internal dynamics is stable, our tracking control design has
been solved. Otherwise the tracking controller is meaningless.
•Therefore, the effectiveness of this control design, based on
reduced-order model, hinges upon the stability of the internal
dynamics.
Internal Dynamics
Consider the nonlinear system
 x1   x23  u 

   
x
 2   u 
y  x1
Assume that the control objective is to make y track yd(t).
Differentiating y leads to the first state equation.
Choosing control law
u   x2  e(t )  y d (t )
3
which yields exp onential convergence of e to zero
e  e  0
The same control input is also applied to the sec ond
dynamic equation, leading to the int ernal dynamics
Internal Dynamics- Contd
x2  x23  y d  e
which is characterstically, non  autonomousand nonlinear.
If e is bounded and y d is assumed to be bounded, we get
y d (t )  e  D
where D is a positive cons tan t .
Thus, we conclude that
x2  D1 / 3
sin ce x2  0 when x2  D1 / 3 ,
and x2  0 when x2   D1 / 3.
Therefore the above controller does provide satisfactory control
given any trajectory yd (t ) whose derivative y d (t ) is bounded.
Internal Dynamics in Linear Systems
Consider the simple controllable and observablelinear system
 x1   x2  u 
   

 x2   u 
y  x1
where y (t ) is required to track a desired output yd (t ).
W ith one differentiation of the output, we get
y  x2  u
which exp licitly contains u. Thus, the control law
u   x2  y d  e yields the tracking equation e  e  0
where (e  y  yd ) and the int ernal dynamics x2  x2  y d  e(t )
W e see that while y (t ) tends to yd (t ) ( y (t ) tends to y d (t ),
x2 remains bounded, so does u.
Internal Dynamics in Linear Systems
Consider a slightly different system :
 x1   x2  u 
   

 x2    u 
y  x1
The same control law as above yields the same
tracking error dynamics, but the int ernal dynamics is
x2  x2  e(t )  y d
This implies that x2 , and accordingly u, both go to
inf inity as t  .
Therefore this is not a suitable controller for the system.
Internal Dynamics in Linear Systems
To unders tan d this fundamental difference between the two systems
we consider the transfer functions,
p 1
W1 ( p )  2
p
p 1
W2 ( p )  2
p
Both the systems have the same poles but different zeros.
Specficall y, for the first case, there is a left half  plane zero at  1.
while for the sec ond case, there is a right half  plane zero at 1.
Thus, int ernal dynamics of first system is stable because it is a
min imum phase system.
For non  min imum phase systems, perfect tracking requires
inf inite effort.
Extension of Internal Dynamics to Zero Dynamics
•Extending the notion of zeros to nonlinear systems is not a trivial
proposition.
•For nonlinear systems, the stability of the internal dynamics
may depend on the specific control input.
•The zero-dynamics is defined to be the internal dynamics
of the system when the system output is kept at zero by the input.
•A nonlinear system whose zero dynamics is asymptotically stable is
an asymptotically minimum phase system.
•Zero-Dynamics is an intrinsic feature of a nonlinear system, which
does not depend on the choice of control law or the desired
trajectories.
Mathematical Tools
•Lie derivative and Lie bracket
•Diffeomorphism
•Frobenius Theorem
•Input-State Linearization
•Examples
•The zero dynamics with examples
•Input-Output Linearization with examples
•Opto-Mechanical System Example
Lie Derivatives
Given a scalar function h( x) and a vector field, L f h,
called the Lie derivative of h with respect to f .
Definition: Let h : R n  R be a smooth scalar function, and
f : R n  R be a smooth vector field on R n , then the Lie derivative
of h with respect to f is a scalar function defined by L f h  h f
Example:
x  f ( x)
y  h( x )
The derivatives of the output are
[ L f h]
h
y 
x  L f h ; y 
x  L f 2h
x
x
Lie Brackets
Let f and g be two vector fields on R n .The Lie Bracket of
f and g is a third vector defined by
f , g  g f  f g
The Lie Bracket [ f , g ] is commonly written as ad f g ( where ad
s tan ds for adjo int)
Re peated Lie Brackets can be defined recursively by
ad f g  g
o
ad f g  [ f , ad f
i
i 1
g]
for i  1,2,....
Example - Lie Brackets
Example
Let x  f ( x)  g ( x)u
with the two vector fields f and g defined by
 2 x1  ax2  sin x1 
0

f 
g ( x)  


cos(
2
x
)

x
cos
x
1 
1

 2

The Lie bracket can be computed as
0
0  2 x1  ax2  sin(x1 )  2  cos x1
a  0 

[ f , g]  





2
sin(
x
)
0

x
cos
x
x
sin
x

cos
x
cos(
2
x
)
1
2
1
1
1
1 


  2
a cos(2 x1 )




cos
x
cos(
2
x
)

2
sin(
2
x
)(

2
x

ax

sin
x
)
1
1
1
1
2
1 

Properties of Lie Brackets
Lie Brackets have following properties
(i ) bilinearity :
[1 f1   2 f 2 , g ]  1[ f1, g ]   2 [ f 2 , g ]
[ f ,1g1   2 g 2 ]  1[ f , g1 ]   2 [ f , g 2 ]
where f , f1, f 2 , g , g1, g 2 are smooth vector fields, and
1 and  2 are cons tan t scalars.
(ii) skew  commutativity :
[ f , g ]  [ g , f ]
(iii) Jacobi identity:
La d f g h  L f Lg h  Lg L f h
where h( x) is a smooth function of x.
Diffeomorphisms and State transformations
Definition:
A function : R n  R n , defined in a region , is called a
diffeomorphism if it is smooth, and if its inverse  1 exists
and is smooth.
Lemma :
Let  ( x) be a smooth function defined in a region  in R n . If
the Jacobian matrix  is nonsin gular at a po int x  x0 of ,
then  ( x) defines a local diffeomorphism in a subregionof .
Example
Consider the dynamic system described by
x  f ( x)  g ( x)u
y  h( x )
and let the new states be defined by
z   ( x)
Differentiation of z yields
z 


x 
( f ( x)  g ( x)u )
x
x
New state representation :
z  f * ( z )  g * ( z )u
y  h* ( z )
where x   1 z
Frobenius Theorem- Completely Integrable
Definitionof completely int egrable:
A linearly independent set of vector fields[ f1, f 2 ,... f m ] on R n
is said to be completely int egrable if , and only if , there exists
n  m scalar functions h1 ( x), h2 ( x),... hnm ( x) satisfyingthe
system of partial differential equations
hi f j  0
where 1  i  n  m, 1  j  m, and the gradientshi are linearly
independent.
Frobenius Theorem- Involutivity
Definitionof involutivity condition:
A linearly independent set of vector fields [ f1, f 2 ,... f n ] is said
to be involutiveif , and only if , there are scalar functions
 ijk : R n  R such that
m
[ fi , f j ]( x)    ijk ( x) f k ( x)
k 1
i, j
Frobenius theorem
Theorem :
Let f1, f 2 ,... f m be a set of linearly independent
vector fields.
The set is completely int egrable if , and only if ,
it is involutive.
Consider the set of partial differential equations
h
h
4 x3

0
x1 x2
 x1
h
h
h
 ( x32  3x2 )
 2 x3
0
x1
x2
x3
Frobenius theorem- example
The associated fields are [ f1, f 2 ] with
f1  [4 x3  1 0]T
f 2  [ x1 ( x32  3x2 ) 2 x3 ]T
In order to det er min e whether this set is solvable, let us check
the involutivity of the set of vector fields[ f1, f 2 ].
[ f1, f 2 ]  [12x3 3 0]T
Since [ f1, f 2 ]  3 f1  0 f 2 , this set of vector fields is involutive.
Th erefo re, th e two p a rtiald ifferen tia l eq u a tio n sa re so lvab le.
Input-State Linearization
Definition:
A sin gle input nonlinear system in the form x  f ( x)  g ( x)u
with f ( x) and g ( x) being smooth vector fields on R n , is said to
be input  state linearizable if there exists a region  in R n ,
a diffeomorhpism  :   R n , a nonlinear feedback control law
u   ( x )   ( x )v
such that the new state var iables z   ( x) and the new input v
satisfy a linear time  in var iant relation
z  Az  bv
where
0 1 0..

0 0 1..

A
.. .
.

0 0 0
The new state z
0
0 

 
.
.

b
.
.

 
0
1
is called the linearizing state and the control law
is called the linearizing law.
Conditions for Input-State Linearization
Theorem :
The nonlinearsystem with f ( x) and g ( x) being smooth vector fields,
is input  state linearized if , and only if , there exists a region  such
that the following conditionshold :
(i ) the vector fields {g , ad f g ,... ad f n1g} are linearly independent in 
(ii) the set {g , ad f g ,... ad f n1g} is involutivein 
Few Re marks :
The first conditioncan be int erpreted as controllability condition for the
nonlinear system.
The involutivity conditionis less int uitive. It is trivially satisfied for linear
systems, but not generally satisfied in the nonlinear case.
How to perform input-state Linearization
The input  state linearization of a nonlinear system can be performed
throughthe following steps :
(i ) Construct the vector fields g , ad f g ,... ad f n1g for the given system.
(ii) Check whether the controllability and involutivity conditionsare satisfied.
(iii) If both are satisfied, find the first stage z1 (the output function leading
to input  output linearization of the relative deg ree n)
z1ad f n1g  0
i  0,... n  2
z1ad f n1g  0
(iv) Compute the state transformation z ( x)  [ z1 L f z1 .... L f n1z1 ]T and the input
transformation, with
 ( x)  
 ( x) 
L f n z1
Lg L f n1z1
1
Lg L f n1z1
Example system
Consider a mechanism given by the dynamics which represents a single link flexible joint
robot.
Its equations of motion is derived as
Iq  MgLsin q1  k (q1  q2 )  0
Jq2  k (q1  q2 )  u
Because nonlinearities ( due to gravitational torques) appear in the first equation,
While the control input u enters only in the second equation, there is no easy way
to design a large range controller.
x  [q1 q1 q2 q2 ]
and corresponding vector fields f and g can be written as
MgL
k
k
sin x1  ( x1  x3 ) x4 ( x1  x3 )T
l
l
j
1 T
g  [0 0 0 ]
J
f  [ x2 
Example system- Contd.
Checking controllability and involuvity conditions.

0

0
[ g ad f g ad 2 f g ad 3 f g ]  
0

1
 J

0
0
0
k
IJ
1
J
0
0

k
J2
k 
 
IJ 
0 

k 
2 
J 
0 

It has rank 4 for k>0 and IJ> infinity. Furthermore, since the
above vector fields are constant, they form an involutive set.
Therefore the system is input-state linearizable.
Example system - Contd.
Let us find out the state-transformation z = z(x) and the input
transformation u   ( x)   ( x)v so that input-state linearization is achieved.
z1
0
x2
z1
0
x3
z1
0
x4
z1
0
x1
Thus, z1 must be a function of x1 only.
The simplest solution to the above equation is
z1  x1
The other states can be obtained from z1
z2  z1 f  x2
MgL
k
sin x1  ( x1  x3 )
I
I
MgL
k
z4  z3 f  
x2 cos x1  ( x2  x4 )
I
I
z3  z2 f  
Example system- Contd.
Accordingly, the input transformation is
u  (v  z4 f ) /(z4 g )
which can be written exp licitly as
IJ
u
(v  a( x))
k
where
MgL
MgL
k
k
k k MgL
a ( x) 
sin x1 ( x2 2 
cos x1  )  ( x1  x3 )(  
cos x1 )
I
I
I
I
I J
I
W e end up with the following set of linear equations
z1  z2
z2  z3
z3  z4
z4  v
thus completing the input  state linearization.
Example system- Contd.
Finally, note that
The above input-state linearization is actually global, because the
diffeomorphism z(x) and the input transformation are well defined everywhere.
Specifically, the inverse of the state transformation is
x1  z1
x2  z2
I
MgL
x3  z1  ( z3 
sin z1 )
k
I
I
MgL
x4  z2  ( z4 
z2 cos z1 )
k
I
which is well defined and differentiable everywhere.
Input-Output Linearization of SISO systems
Given a nonlinear sin gle  input system described by the state space
representation
x  f ( x)  g ( x)u
y  h( x )
where y is the system output.
Issues :
(i) How to generate a linear input  output relation for a nonlinear system?
(ii) W hat are the int ernal dynamics and zero  dynamics associated with
the input  output linearization?
(iii) How to design stable controllers based on input  output linearizations?
Generating a linear input-output relation
The basic approach of input  output linearization is simply to
differentiate the output function y repeatedly until the input u
appears, and then design u to cancel the nonlinearity.
However , in some cases, the systems relative deg ree is undefined.
Definition: The SISO system is said to have relative deg ree r in a
region  if , x  
Lg L f i h( x)  0
Lg L f r 1h( x)  0
0  i  r 1
Normal Forms
Let
  [ 1  2 ....  r ]T  [ y y ... y ( r 1) ]T
In a neighborhood  of a po int x0 , the normal form of the system can be
written as
1




..

  


r


a (  , )  b(  , )u 
  w(  , )
with the output defined as
y  1
The i and  i are referred to as normal coordinates or normal states
in (or at x0 )
Zero Dynamics
The int ernal dynamics associated with the input  output
linearization simply corresponds to the last (n  r ) equations
  w(  , ) of the normal form.
Generally, this dynamics depends on the output states .
However , we can define an int rinsic property of the
nonlinear system by considering the system' s int ernal
dynamics when the control input is such that the output
y is ma int ained at zero. Studying zero dynamics will
allow us to make some conclusions about the stabilityof
the int ernal dynamics.
Zero Dynamics
The constraint that the output y is identically zero implies that all of its time
derivatives are zero. Thus, the zero dynamics of a system is its dynamics when
its motion is restricted to the (n  r ) dimensional smooth surface in R n
Further, input must be such that y stays at zero.
In order for the system to operate in zero dynamics,
The original input u must be given by the state feedback
u* (t )  
L f r h( x)
Lg L f r 1h( x)
Therefore, corresponding to the zero dynamics, the system states x evolves
according to
x  f ( x)  g ( x)u* ( x)
Zero Dynamics- Contd.
Assu min g that the system' s initial state is on the
surface, i.e.  (0)  0,
the system dynamics can be written in normal form as
  0
  w(0, )
By definition, the above equation is the zero dynamics
of the nonlinear system
The control input u0 can be written as a function only
of the int ernal states
a(0, )
u0  
b(0, )
Local Asymptotic Stabilization
Theorem :
Assume that the system has relative deg ree r , and its zero  dynamics
is locally asymptotically stable.
Let us assume that v   kr 1 y ( r 1)  ....  k1 y  k0 y where ki
Choose cons tan ts ki such that the polynomial
K ( p)  p r  kr 1 p r 1  ... k1 p  k0
has all its roots strictly in the left half plane.
Then, the control law
1
r
( r 1)
  k0 y ]
u ( x) 
[

L
y

k
y

...

k
y
f
r

1
1
Lg L f r 1 y
leads to a locally asymptotically stable closed  loop system
Example System
x1  x12 x2
x2  3x2  u
The system' s linearization at x  0 ( where x  x1
x1  0
x2 T is
x2  3x2  u
and thus has an uncontrollable mod e corresponding to a pure int egrator.
Let us define the output function y  2 x1  x2
Corresponding to this output, the relative deg ree of the system is 1, because
dy
 2 x1  x2  2 x12 x2  3x2  u
dt
The associated zero  dynamics(obtained by setting y  0 ) is simply
x1  2 x13
and thus is asymptotically stable. Therefore, the control law
u  2 x12 x2  4 x2  2 x1
locally stablizesthe nonlinear system.
Global Asymptotic Stability
Zero Dynamics only guarantees local stability of a control system
based on input-output linearization.
Most practically important problems are of global stabilization
problems.
An approach to global asymptotic stabilization based on partial
feedback linearization is to simply consider the control problem as a
standard lyapunov controller problem, but simplified by the fact that
putting the systems in normal form makes part of the dynamics
linear.
The basic idea, after putting the system in normal form, is to view 
as the “input” to the internal dynamics, and  as the “output”.
Steps for Global Asymptotic Stability
•The first step is to find a “ control law” 0  0 ( ) which stabilizes
the internal dynamics.
•An associated Lyapunov function V0 demonstrating the stabilizing
property.
•To get back to the original global control problem.
•Define a Lyapunov function candidate V appropriately as a modified
version of V0
•Choose control input v so that V be a Lyapunov function for the
whole closed-loop dynamics.
Local Tracking Control
The simple pole  placement controller can be
extended to asymptotic
tracking control tasks. Let
( r 1) T

d  [ yd yd ... yd
]
and define the tracking error vector by
~ (t )   (t )   (t )
d
Tracking Control
Theorem :
Assume the system has relative deg ree r (defined and cons tan t
over the region of int erest), that  d is smooth and bounded,
and that the solution d of the equation
 d   (  d , d )
 d (0)  0
exists, is bounded, and is uniformlyasymptotically stable.
Then by u sin g the control law
1
r
(r )
~]

u
[

L


y

k


....

k

f
1
d
r 1 r
0 1
Lg L f r 11
the whole state remains bounded and the tracking error ~
converges to zero exp onentially.
Inverse Dynamics
For systems described by previous sec tion,
let us find out what the initial conditions x(0) and control input u
should be in order for the plant output to track a reference output
yr (t ) perfectly.
Let us assume that the system output y (t ) is identical to the reference
output yr (t ) i.e y (t )  yr (t ), t  0.
y k (t )  yr k (t )
k  0,1,....,. r  1
t  0
In terms of normal coordinates,
 (t )  r (t )  [ yr (t ) y r (t ) .... yr ( r 1) (t )]T
Thus, the control input u (t ) must satisfy
yr r (t )  a( r , )  b( r , )u (t )
t  0
Inverse Dynamics- Contd.
where  (t ) is solutionof the differential equation
 (t )  w[ r (t ), (t )]
Given a reference trajectory yr (t ) , we can obtain the
required control input for output y (t ) identically
equal to yr (t ).
Pr evious equations allow us to compute the input u (t )
corresponding to reference output history yr (t ).
Therefore, they are called inverse dynamics of the system.
Application of Feedback Linearization to
Opto-Mechanics
For the double slit aperture, the irradiance at any point
in space is given as:
I ( x)  A sin c 2(
kb
x
ka
x
sin (tan 1 ( ) ) ) cos2 ( sin(tan 1 ( )))
2
z
2
z
 = wavelength = 630 nm
k = wave number associated with the wavelength
a = center-to-center separation = 32 um
b = width of the slit = 18 um
z = distance of propagation =1000 um
Plant Model
U
+
-
1
( s  1)
X2
Motor Dynamics
Plant
Model
Y= X1
Plant Model
X2
1

U  X1 s 1
Y  X1  Asinc( X 2 )
X 2
 X 2  X1  U
t
Y  sinc( X 2 )
A  1 (assume)
X 2
  sinc( X 2 )  X 2  U
t
Y  sinc( X 2 )
A  1 (assume)
Input-State Linearization
X 2
  sinc( X 2 )  X 2  U
t
U sin g a transformation z1  x2
W e get z1   sin c( z1 )  z1  u
Therefore u  z1  sin c( z1 )  z1
u  v  sin c( z1 )  z1
v  z1
Select v  k1z1
u  k1z1  f ( z ) where f ( z )  sin c( z1 )  z1
By proper choice of feedback gains , we can get a stable closed
loop dynamics.
Input-State Linearization- Block diagram
0
v   K1z1
U(x,v)
Pole-Placement loop
+
-
-
z
1
( s  1)
X2
Motor Dynamics
z1  x2
Plant
Model
Plant Model
Y
Input-Output Linearization
sin(x2 )
y  sin c( x2 ) 
x2
[cos(x2 ) x2 ]x2  sin(x2 ) x2
y 
x2 2
x2 (cos(x2 )) x2  sin(x2 ) x2

x2 2
y 

x2 cos(x2 )[ sin c( x2 )  x2  u ]  sin(x2 )[ sin c( x2 )  x2  u ]
x2 2
 x2 cos(x2 ) sin c( x2 )  x2 2 cos(x2 )  x2 cos(x2 )u 
sin(x2 ) sin c( x2 )  sin(x2 ) x2  u sin(x2 )
x2
2
Input-Output Linearization
 x2 cos(x2 ) sin c( x2 )  x2 2 cos(x2 )  x2 cos(x2 )u
 sin(x2 ) sin c( x2 )  sin(x2 ) x2  u sin(x2 )
y 
[
x2 2
x2 cos(x2 )u  sin(x2 )u
x2
2
]
[ x2 cos(x2 ) sin c( x2 )  x2 2 cos(x2 )  sin(x2 ) sin c( x2 )  x2 sin(x2 )]
x2 2
 u[
x2 cos(x 2 )  sin(x2 )
x2
2
]
[sin c( x2 ){sin(x2  x2 cos(x2 )}  x2 2 cos(x2 )  x2 sin(x2 )]
x2 2
Input-Output Linearization
y  u[
cos(x2 ) sin(x2 )

]  f ( x)
2
x2
x2
where
v  f ( x)
u
cos(x2 ) sin(x2 )
[

]
2
x2
x2
y  v
Comparing with x  Ax  Bu, we get A  1, B  0
Select v  y d  k1e
[e  y  y d ]
Control law is defined everywhere except at sin gularity
cos(x2 ) sin(x2 )
po int s where [

]0
2
x2
x2
Zero Dynamics
Zero Dynamics is given by :
x2   x2  u
This zero dynamics is asymptotically stable,
hence the feedback controllerlocally stabilizesthe nonlinear
system.
There are no int ernal dynamics associated with this system
because relative deg ree r  n ( system order).
Conclusion
Control design based on input-output linearization can be made
in 3 steps:
•Differentiate the output y until the input u appears
•Choose u to cancel the nonlinearities and guarantee tracking
convergence
•Study the stability of the internal dynamics
If the relative degree associated with the input-output linearization is
the same as the order of the system, the nonlinear system is fully
linearized.
If the relative degree is smaller than the system order, then the
nonlinear system is partially linearized and stability of internal
dynamics has to be checked.
Homework Problems
Design a linear input  output controller for
x1  x2
x2   x1  ax12 x2  ( x2  1)u
y  x1
Check global stabilityof the zero dynamics for
x1  kx1  2 x2u
x2   x2  x1u
y  x2