Applied Nonlinear Control - Chapter 6: Feedback Linearization - Nguyễn Tân Tiến

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Chapter 6 Feedback linearization 
27 
Example 6.2: Feedback linearization of a two-link robot 
Consider the two-link robot as in the Fig. 6.2 
lc1
l1
l2
lc2
I2, m2
I1, m1
q2,τ2
q1,τ1
Fig. 6.2 A two-link robot 
The dynamics of a two-link robot 


=

+



 −−−=




2
1
2
1
2
1
1
212
2
1
2221
1211
0 τ
τ
g
g
q
q
qh
qhqhqh
q
q
HH
HH
&
&
&
&&&
&&
&&
 (6.9) 
where, 
[ ]Tqqq 21= : joint angles 
[ ]T21 τττ = : joint inputs (torques) 
2221
2
2
2
121
2
1111 )cos2( IqllllmIlmH ccc +++++= 
2
2
2222122112 cos IlmqclmHH c ++== 
2
2
2222 IlmH c += 
2212 sin qllmh c= 
]cos)cos([cos 1121221111 qlqqlgmqglmg cc +++= 
)cos( 21222 qqglmg c += 
Control objective: to make the joint position 1q and 2q 
follows desired histories )(1 tqd and )(2 tqd 
To achieve tracking control tasks, one can use the follow 
control law 


+



 −−−=



=


2
1
2
1
1
212
2
1
2221
1211
2
1
0 g
g
q
q
qh
qhqhqh
v
v
HH
HH
&
&
&
&&&
τ
τ
 (6.10) 
where, 
qqqv d
~~2 2λλ −−= &&&& 
[ ]Tvvv 21= : the equivalent input 
dqqq −=~ : position tracking error 
λ : a positive number 
The tracking error satisfies the equation 0~~2~ 2 =++ qqq λλ &&& 
and therefore converges to zeros exponentially. 
⊗ When the nonlinear dynamics is not in a controllability 
canonical form, one may have to use algebraic transforms to 
first put the dynamics into the controllability canonical form 
before using the above feedback linearization design. 
6.1.2 Input-State Linearization 
Consider the problem of design the control input u for a 
single-input nonlinear system of the form 
)( ,uxfx =& 
The technique of input-state linearization solves this problem 
into two steps: 
- Find a state transformation )(xzz = and an input trans-
formation )( vx,uu = , so that the nonlinear system 
dynamics is transformed into an equivalent linear time-
invariant dynamics, in the familiar form vbzAz +=& . 
- Use standard linear technique to design v . 
Example: Consider a simple second order system 
1211 sin2 xxaxx ++−=& (6.11a) 
)2cos(cos 1122 xuxxx +−=& (6.11b) 
Even though linear control design can stabilize the system in a 
small region around the equilibrium point (0,0), it is not 
obvious at all what controller can stabilize it in a large region. 
A specific difficulty is the nonlinearity in the first equation, 
which cannot be directly cancelled by the control input u. 
Consider the following state transformation 
11 xz = (6.12a) 
122 sin xxaz += (6.12b) 
which transforms (6.11) into 
211 2 zzz +−=& (6.13b) 
)2cos(sincoscos2 111112 zuazzzzz ++−=& (6.13b) 
The new state equations also have an equilibrium point at (0,0). 
Now the nolinearities can be canceled by the control law of 
the form 
)cos2sincos(
)2cos(
1
1111
1
zzzzv
za
u +−= (6.14) 
where v is equivalent input to be designed (equivalent in the 
sense that determining v amounts to determining u, and vise 
versa), leading to a linear input-state relation 
11 2 zz −=& (6.15a) 
vz =2& (6.15b) 
Thus, 
the problem of
stabilizing the new
dynamics (6.15)
using the new
input v
the problem of
stabilizing the original
nonlinear dynamics
(6.11) using the original
control input u
input
transformation
(6.14)
state
transformation
(6.12)
Applied Nonlinear Control Nguyen Tan Tien - 2002.5 
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Chapter 6 Feedback linearization 
28 
Now, consider the new dynamics (6.15). It is linear and 
controllable. Using the well known linear state feedback 
control law ,2211 zkzkv −−= one could chose 0,2 21 == kk 
or 
22 zv −= (6.16) 
resulting in the stable closed-loop dynamics 211 2 zzz +−=& 
and 22 2 zz −=& . In term of the original state, this control law 
corresponds to the original input 
)cos2sincossin22(
)2cos(
1
111112
1
xxxxxxa
xa
u +−−−= 
 (6.17) 
The original state x is given from z by 
11 zx = (6.18a) 
azzx /)sin( 122 −= (6.18b) 
The closed-loop system under the above control law is 
represented in the block diagram in Fig. 6.3. 
v=-kTz u=u (x,v) = f(x,u)x&
z=z (x)
x0
pole-placement loop
linearization loop
z
Fig. 6.3 Input-State Linearization 
⊗ To generalize the above method, there are two equations: 
- What classes of nonlinear systems can be transformed 
into linear systems ? 
- How to find the proper transformations for those which 
can ? 
6.1.3 Input-Ouput Linearization 
Consider a tracking control problem with the following system 
)( ,uxfx =& (6.19a) 
)(xhy = (6.19b) 
Control objective: to make the output )(ty track a desired 
trajectory )(tyd while keeping the whole state bounded. 
)(tyd and its time derivatives are assumed to be known and 
bounded. 
Consider the third-order system 
3221 )1(sin xxxx ++=& (6.20a) 
3
5
12 xxx +=& (6.20b) 
uxx += 213& (6.20c) 
1xy = (6.20d) 
To generate a direct relationship between the output and input, 
let us differentiate the output 3221 )1(sin xxxxy ++== && . 
Since y& is still not directly relate to the input ,u let us 
differentiate again. We now obtain 
)()1( 12 xfuxy ++=&& (6.21) 
2
12233
5
11 )1()cos)(()( xxxxxxf ++++=x (6.22) 
Clearly, (6.21) represents an explicit relationship between y 
and u . If we choose the control input to be in the form 
)(
1
1
1
2
fv
x
u −+= (6.23) 
where v is the new input to be determined, the nonlinearity in 
(6.21) is canceled, and we apply a simple linear double-
integrator relationship between the output and the new input v, 
vy =&& . The design of tracking controller for this double-
integrator relation is simple using linear technique. For 
instance, letting )()( tytye d−= be the tracking error, and 
choosing the new input v such as 
ekekyv d &&& 21 −−= (6.24) 
where 21, kk are positive constant. The tracking error of the 
closed-loop system is given by 
012 =++ ekeke &&& (6.25) 
which represents an exponentially stable error dynamics. 
Therefore, if initially 0)0()0( == ee & , then 0,0)( ≥∀≡ tte , 
i.e., perfect tracking is achieved; otherwise, )(te converge to 
zero exponentially. 
⊗ Note that: 
- The control law is defined anywhere, except at the 
singularity point such that 12 −=x . 
- Full state measurement is necessary in implementing the 
control law. 
- The above controller does not guarantee the stability of 
internal dynamics. 
Example 6.3: Internal dynamics 
Consider the nonlinear control system 



 +=


u
ux
x
x 32
2
1
&
&
 (6.27a) 
1xy = (6.27b) 
Control objective: to make y track to )(tyd 
uxxy +== 321&& ⇒ )()(32 tytexu d&+−−= (6.28) 
yields exponential convergence of e to zero. 
0=+ ee& (6.29) 
Apply the same control law to the second dynamic equation, 
leading to the internal dynamics 
eyxx d −=+ && 322 (6.30) 
which is non-autonomous and nonlinear. However, in view of 
the facts that e is guaranteed to be bound by (6.29) and dy& is 
Applied Nonlinear Control Nguyen Tan Tien - 2002.5 
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Chapter 6 Feedback linearization 
29 
assumed to be bounded, we have Detyd ≤−)(& , where D is 
positive constant. Thus we can conclude from (6.30) that 
3/1
2 Dx ≤ , since 02 , and 02 >x& when 
3/1
2 Dx −< . Therefore, (6.28) does represent a satisfactory 
tracking control law for the system (6.27), given any trajectory 
)(tyd whose derivative )(tyd& is bounded. 
⊗ Note: if the second state equation in (6.27a) is replaced by 
ux −=2& , the resulting internal dynamics is unstable. 
▲ The internal dynamics of linear systems ⇒ refer the test book 
▲ The zero-dynamics 
Definition: The zeros-dynamics is defined to be the internal 
dynamics of the systems when the system output is kept at 
zero by the input. 
For instance, for the system (6.27) 



 +=


u
ux
x
x 32
2
1
&
&
 (6.27a) 
1xy = (6.27b) 
the out put 01 ≡= xy 01 ≡=→ xy && 32xu −≡→ , hence the 
zero-dynamics is 
0322 =+ xx& (6.45) 
This zero-dynamics is easily seen to be asymptotically stable 
by using Lyapunov function 22xV = . 
⊗ The reason for defining and studying the zero-dynamics is 
that we want to find a simpler way of determining the stability 
of the internal dynamics. 
- In linear systems, the stability of the zero-dynamics 
implies the global stability of the internal dynamics. 
- In nonlinear systems, if the zero-dynamics is globally 
exponentially stable only local stability is guaranteed for 
the internal dynamics. 
⊗ To summarize, control design based on input-output 
linearization can be made in three 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. 
6.2 Mathematical Tools 
6.3 Input-State Linearization of SISO Systems 
6.4 Input-Output Linearization of SISO System