Chapter 6
Feedback Linearization
- Central idea:
To algebraically transform a nonlinear system
dynamics into a (fully or partly) linear one
 linear control techniques can be applied.
-Feedback linearization techniques:
Ways of transforming original system models
into equivalent models of a simpler form.
6.1 Intuitive Concepts
Feedback linearization
amounts to canceling the
nonlinearities in a nonlinear
system so that the closed loop
dynamics is a linear form
6.1.1 Feedback Linearization and
the Canonical Form
Example 6.1:Controlling the fluid level in a tank
Dynamic model of the tank is
d
dt
h
[  A ( h ) dh ]  u ( t )  a 2 gh
0
A(h): cross section of the tank
a: cross section of the outlet pipe.
Figure 6.1：Fluid level control in a tank
h0
hd
: initial level
: desired level
The dynamics (6.1) can be rewritten as
A ( h ) h  u  a 2 gh
(6.2)
If u ( t )  a 2 gh  A ( h ) v
with v being an "equivalent input" to be
specified, the resulting dynamics is linear
h v
Choosing
v   h
with
Then
h
(6.3)
: level error, and  > 0
h  h ( t )  hd
h h  0
(6.4)

h ( t )  0 as t  
Thus, the nonlinear control law
u ( t )  a 2 g h  A ( h ) h
(6.5)
-If the desired level is a known time-varying
function hd ( t ) , set
v  hd ( t )   h

h (t )  0
as t  
Applying Feedback
linearization to companion
form or controllable canonical
form system
Companion Form, or Controllable Canonical Form
System
A companion form system:
x
(n)
 f ( x )  b ( x )u
u : scalar control input
x : scalar output of interest
( n 1) T
x  [ x , x , ..., x
] : state vector
f(x) and b(x) : nonlinear function of states
Using
1
u 
[v  f ]
b

x
(n)
=v
(6.7)
Then, the control law
v   k 0 x  k1 x  ...  k n 1 x
( n  1)
with k i  s n  k n 1 s n 1  ...  k 0  0 has all its roots
strictly in the left-half complex plane exponentially
stable dynamics
x
i.e.
x (t )  0
(n)
 k n 1 x
as
( n  1)
t
 ...  k 0 x  0
Tracking Desired Output xd(t)
the control law
v  xd
(n)
 k 0 e  k 2 e  ...  k n  1 e
( n 1)
(where e(t) = x(t) -xd(t))
 exponentially convergent tracking.
(6.8)
One interesting application of
the above control design idea
Example 6.2：Feedback linearization of a
two-link robot
- Tracking control problems arise when a robot
hand is required to move along a specified path,
e. g., to draw circles.
- Control objective : To make the joint positions
q1 and q2 follow desired position histories qd1(t)
and qd2(t) specified by the motion planning
system.
Figure 6.2:
A two-link
robot- Each joint
equipped with a
motor to provide
input torque, an
encoder to measure
joint position, and a
tachometer to
measure joint
velocity
Using Lagrangian equations in classical dynamics, the
dynamics of the robot :
 H 11

 H 21
q  [ q1
τ   1
 hq1  hq 2   q1   g 1    1 
     
0
  q 2   g 2   2 
H 12   q1    hq 2
 
H 22   q 2   hq1
q2 ]
2
T
T
: two joint angles
: joint inputs
H 11  m 1l c1  I 1  m 2 [ l1  l c 2  2 l1l c 2 cos q 2 ]  I 2
2
2
2
H 22  m 2 l c 2  I 2
2
H 12  H 21  m 2 l1l c 2 cos q 2  m 2 l c 2  I 2
2
h  m 2 l1l c 2 sin q 2
g 1  m1l c1 g cos q1  m 2 g [ l c 2 cos( q1  q 2 )  l1 cos q1 ]
g 2  m 2 l c 2 g cos( q1  q 2 )
- Compact expression
H  q  q  C  q, q  q  g  q   
- Multiplying both sides by
H
1
 Companion form
Tracking control
(6-10)
Using the control law (computed torque)
  1   H 11 H 12   v 1    h q 2  h q 1  h q 2   q 1   g 1 

 
    H






0
 2   21 H 22   v 2   h q 1
  q 2   g 2 
where
2~
~

vq
 d  2  q   q
with v = [v1 v2 ]T being the equivalent input,
~ qq
q
being the tracking error and
d
~
q
 a positive number. The tracking error then
satisfies the equation
~
~   2 q
~ 0
  2  q
q
and converges to zero exponentially.
6.1.2 Input-State Linearization
x1   2 x1  ax 2  sin x1
x 2   x 2 cos x1  u cos(2 x1 )
(6.11a)
(6.11b)
-Linear control design can stabilize the system in a
small region around (0, 0)
- What controller can stabilize it in a larger region.
- Note that the nonlinearity in the first equation cannot
be canceled by the control input u.
- Consider the new set of state variables:
z1  x1
(6.12a)
z 2  ax 2  sin x1
z1   2 z1  z 2

z 2   2 z1 cos z1  cos z1 sin z1  au cos(2 z1 )
With equilibrium point at (0,0).
(6.13a)
(6.13b)
Applying
1
u 
a cos( 2 z 1 )
( v  cos z 1 sin z 1  2 z 1 cos z 1 )
v : an equivalent input to be designed

z 1   2 z 1  z 2
z 2  v
- Linear and controllable
(6.15a)
(6.15b)
(6.14)
Using state feedback control law can place the poles
anywhere with proper choices of feedback gains.
v   k 1 z1  k 2 z 2
For example,
v  2 z2
(6.16)

in the stable closed-loop dynamics
z1   2 z1  z 2
1   2  -2

z2  2 z2
The control law
u 
1
cos(2 x1 )
(  2 ax 2  2 sin x1  cos x1 sin x1  2 x1 cos x1 )
(6.17 )
The original state x is given from z by
x1  z 1
(6.18a)
(6.18b)
x 2  ( z 2  sin z 1 ) / a
Since both z1 and z2 converge to zero, the
original state x converges to zero.
Figure 6.3: Input-State Linearization
• The inner loop: achieves the linearization of the
input-state relation.
• The outer loop: achieves the stabilization of the
closed-loop dynamics.
Remarks
• The result is not global.
The control law is not well defined when x1 =
(

4

k
2
), k = 1, 2, …
• Different from a Jacobian linearization for
small range operation.
• The new state components (z1, z2) must be
available. If they cannot be measured directly,
the original state x must be measured and used
to compute them from (6.12)
• Uncertainty in the model, e. g., uncertainty on
the parameter a, will cause error in the
computation of z and u.
6.1.3 Input-Output Linearization
Consider the system
x  f ( x, u )
(6.19a)
(6.19b)
y  h(x)
Objective : make y(t) track a desired trajectory
yd(t) while keeping the whole state bounded
yd(t) : and its nth order derivative are known
and bounded.
Q: how to design the input u ?
Input-Output Linearization Approach
Consider the third-order system
x 1  sin x 2  ( x 2  1 ) x 3
(6.20a)
5
x 2  x 1  x 3
(6.20b)
x 3  x  u
(6.20c)
y  x1
(6.20d)
2
1
- Differentiate y twice
y  ( x 2  1) u  f1 (x )
(6.21)
f 1 ( x )  ( x1  x 3 )( x 3  cos x 2 )  ( x 2  1) x1
5
2
(6.22)
- Choosing
u 
1
x2  1
( v  f1 )
v : a new input to be determined.

yv
(6.23)
- Letting e = y(t) - yd (t) and choosing
v  y d  k1 e  k 2 e
(6.24)
with k1 and k 2 being positive constants

e  k 2 e  k1 e  0
(6.25)
If e (0)  e (0)  0, then e ( t )  0,  t  0,
otherwise, e(t) converges to zero exponentially.
Remarks
• The control law is defined everywhere,
except at the singularity points such that
x2 = －1.
• Full state measurement is necessary in
implementing the control law.
• If we need to differentiate the output of a
system r times to generate an explicit
relationship between the output y and input u,
the system is said to have relative degree r.
Internal Dynamics
A part of the system dynamics (described by
one state component) has been rendered
"unobservable" in the input-output
linearization, because it cannot be seen from
the external input-output relationship (6.21).
- For the above example, the internal state can
be chosen to be x3, and the internal dynamics
is represented by
x 3  x 
2
1
1
x2  1
( y d ( t )  k 1 e  k 2 e  f 1 )
(6.26)
- If this internal dynamics is stable (by which
we actually mean that the states remain
bounded during tracking, i. e., stability in the
BIBO sense), our tracking control design
problem has indeed been solved.
-Otherwise, the above tracking controller is
practically meaningless, because the instability
of the internal dynamics would imply
undesirable phenomena such as the burning-up
of fuses or the violent vibration of mechanical
members.
- The effectiveness of the above control design,
based on the reduced-order model (6.21), hinges
upon the stability of the internal dynamics.
Example 6.3: Internal dynamics
Consider the nonlinear system
 x 1   x  u 

x   
 2   u 
3
2
=> x 1  x  u
y  x1
3
2
(6.27a)
x 2  u
 y  x 1  x  u
3
2
(6.27b)
- Control objective: make y track yd (t)
- Choosing the control law
3
u   x 2  e (t )  y d (t )
substitute into (6.27b)
(6.28)

ee0
(6.29)
The dynamics of x 2 cannot be observed from (6.27b)
Applying u to the second dynamic equation :
x 2  u   x 2  e (t )  y d (t )
3
x2  x  yd  e
3
2
y d (t )  e  D
If
where D > 0, then
(perhaps after a transient)
(6.30)
x2  D
1/ 3
x 2 in the internal dynamic is bounded  (6.28)
is satisfactory
The Internal Dynamics of Linear Systems
- In general, it is very difficult to directly determine
the stability of the internal dynamics because it is
nonlinear, non-autonomous, and coupled to the
"external" closed-loop dynamics.
- Q : How to seek simpler ways of determining the
stability of the internal dynamics.
Example 6.4: Internal dynamics in two
linear systems
Consider the simple controllable and
observable linear system
 x 1   x 2 
x    u
 2  
u


y  x1
(6.31a)
(6.31b)
- Objective : make y(t) track yd (t).
- Differentiating y(t)
y  x 2  u
- Choosing
u   x2  y d  ( y  y d )
(6.32)

e  e  0
(where e = y -yd ) and the internal dynamics
x 2  x 2  y d  e (t )
- e ( t )  0 and y d is bounded  x 2 and u are
bounded  (6.32) is a satisfactory tracking
controller for system (6.31).
- Consider a slightly different system:
 x 1   x 2  u 
x     u 

 2  
(6.33a)
y  x1
(6.33b)
- Applying same control law  same tracking
error dynamics, but different internal
dynamics
x 2  x 2  e ( t )  y d

x2 and u go to  as t  .
- Transfer functions for system (6.31),
W1 ( s ) 
s 1
s
2
for system (6.33),
W2 (s) 
s 1
s
2
- Same poles but different zeros.
- Left half-plane zero at -1  stable
- Right half-plane zero at 1  unstable
- For all linear system, the internal dynamic
is stable if the plant zeros are in the left-half
plane.
To keep notations simple, let us consider a
third-order linear system in state space form
T
z  Az  b u
yc z
(6.34)
and having one zero (i.e. two more poles than
zeros).
- The system's input-output linearization can
be facilitated if we first transform it into
companion form.
- To do this, we note from linear control that
the input/output behavior of this system can
be expressed in the form
b0  b1 s
1
y  c ( sI  A ) bu 
T
a 0  a1 s  a 2 s  s
2
Thus, if we define
x1 
1
a 0  a1 s  a 2 s  s
x 2  x1
x3  x 2
2
3
u
3
u
(6.35)
the system can be equivalently represented in
the companion form
 x1   0
d   
x2 
0
dt   
 x 3    a 0
y  b0
b1
1
0
 a1
 x1 
0  x 2 
 
 x 3 
0   x 1  0 
1   x 2   0  u
   
 a 2   x 3   1 
(6.36a)
(6.36b)
-Performing input-output linearization:
y  b0 x 2  b1 x 3
y  b0 x 2  b1 x 3
 b0 x 3  b1 (  a 0 x 1  a 1 x 2  a 2 x 3  u )
(6.37)
-Input u appears in the second differentiation
-The required number of differentiations (the
relative degree) = the excess of poles over
zeros
-Since the input-output relation of y to u is
independent of the choice of state variables,
it would also take two differentiations for u
to appear if we used the original state-space
equations (6.34)).
Thus, the control law


b0
u   a 0 x1  a 1 x 2  a 2 x 3 
x3 
b1



1
b1
(6.38)
  k 1 e  k 2 e  yd 
where e = y - yd , yields an exponentially stable
tracking error with k  0 and k  0 :
1
2
e  k 2 e  k 1 e  0
which is a second order dynamics
-The internal dynamics can be described by
only one state equation. Since we can show x1,
y, y are related to x1, x2, and x3 through a
one-to-one transformation (and thus can serve
as states for the system).
-The internal dynamics can be found from
(6.36a) and (6.36b): 1
x 1  x 2 
that is,
x 1 
b0
b1
b1
x1 
( y  b0 x 1 )
1
b1
y
(6.39)
-Since y is bounded (y = e + yd), we see that
the stability of the internal dynamics depends
on the location of the pole of the internal
dynamics, which is the zero -b0 / b1 of the
original transfer function in (6.35).
-If the system’s zero is in the left-half plane,
which implies that the internal dynamics (6.39)
is stable, independently of the initial
conditions and of the magnitudes of the
desired yd , …, yd (r) (where r is the relative
degree).
Motivation for Zero Dynamics
- Previously discussed transfer functions, on
which linear system zeros are based, cannot be
defined for nonlinear systems.
- Zeros are intrinsic properties of a linear plant,
while for nonlinear systems the stability of the
internal dynamics may depend on the specific
control input.
Zero-dynamics for a nonlinear system
- Zero-dynamics: the internal dynamics when
output is kept at zero by the input.
- Consider the system
x1  x  u
3
2
x2  u
y  x1
the zero-dynamics is

x2  x2  0
3
(6.45)
The zero-dynamics (6.45) is asymptotically
stable (by using the Lyapunov function V =
x22 ).
V  2 x2 x2   2 x  0
4
2
-Similarly, for the linear system (6.34), the
zero-dynamics is (from (6.39))
x 1   b0 / b1  x 1  0
-Thus, in this linear system, the poles of the
zero-dynamics are exactly the zeros of the
system.
-This result is general for linear systems, and
therefore, in linear systems, having all zeros in
the left-half complex plane guarantees the
global asymptotic stability of the zerodynamics.
-For linear systems, the stability of the zerodynamics implies the global stability of the
internal dynamics: The left-hand side of (6.39)
completely determines its stability
characteristics, given that the right-hand side
tends to zero or is bounded.
-For nonlinear systems, it can be shown for
local asymptotic stability of the zero-dynamics
is enough to guarantee the local asymptotical
stability of the internal dynamics. Though we
will not prove it here.
To summarize, control design based on inputoutput linearization can be made in three steps:
A. differentiate the output y until the input u
appears
B. choose u to cancel the nonlinearities and
guarantee tracking convergence
C. study the stability of the internal dynamics