Feedback Linearization: SISO Systems

University of Mhamed Bouguerra Boumerdes

Institute of Electrical and Electronic Engineering

Done by:

Karim CHERIFI

Ikram ALLOUCHE

Abdesselam GUERROUDJ

Tarek BOUAMER

Abdellah Nabih ELGHRIBI

Professor

Pr. K. HARRICHE

2014/2015

Table of Contents

Contents

Introduction __________________________________________________________________________ 1

1. Feedback Linearization Definition ____________________________________________________ 2

2. Mathematical Tools _______________________________________________________________ 3

3. Feedback Linearization Approaches___________________________________________________ 7

3.1. Input/output linearization _____________________________________________________ 7

3.2. Input/state linearization: _____________________________________________________ 11

4. Control Problem _________________________________________________________________ 19

Conclusion __________________________________________________________________________ 21

References __________________________________________________________________________ 22

Pg. 01

Introduction

In nonlinear control systems design, we tend to find ways of transforming original

system models into equivalent models of a simpler form that are fully, or partly linear.

This procedure is system linearization. We are tempted to linearize nonlinear systems

because the mathematical tools needed to analyze linear systems are simple,

straightforward, and general.

One of the techniques of linearization is feedback linearization. It differs from the

conventional Jacobian method of linearization, because it is an exact state

transformation, rather than a linear approximation of the dynamics.

Pg. 02

1. Feedback Linearization Definition

Feedback linearization is one of the linearization methods; it stands for control

techniques where the input is used to linearize all, or part of the systems model.

Feedback linearization method is applied to a class of nonlinear systems of the form

(control affine)

= () + ()

= ()

Where : and :

Remark: is the number of states, and is the number of inputs. In single input

single output (SISO) systems (which is the case in our study) = 1.

We look for a control signal of the form:

= () + ()

In addition, a change of variables:

= ()

That transforms the nonlinear system into an equivalent linear system.

() must be a diffeomorphism (defined in the next section)

Mainly, two feedback linearization approaches exist. Based on input-output

linearization, or input-state linearization.

Pg. 03

2. Mathematical Tools In this section, we introduce the mathematical tools needed for feedback linearization

method. To name them:

Diffeomorphism

Lie Derivative

Lie Brackets

Relative Degree

2.1. Diffeomorphism :

Definition: Suppose continuously differentiable with respect to each of its

arguments (it is of class 1). is a diffeomorphism of onto if

a) T() = b) is one to one

c) The inverse function 1: is also continuously differentiable with

respect to each of its arguments (it is also of class 1)

Remark: is called a smooth diffeomorphism if both and 1 are smooth functions.

2.2. Lie Derivative:

Definition: Let : be a smooth scalar function and : be a smooth

vector field on , then the Lie derivative of with respect to is a scalar function

defined by = .

Pg. 04

We have, as before:

= () + ()

= ()

=()

=

()

=

()

() +

()

()

() =()

()

() =()

()

= +

2.3. Lie brackets:

Definition: Let and be two vector fields on .The lie bracket of and is a third

vector field defined by

[, ] =

It is commonly written where stands for adjoint.

Repeated Lie brackets are defined:

0 =

= [,

1] = 1,2,

Pg. 05

Some properties of Lie brackets:

Bilinearity [11 + 22, ] = 1[1, ] + 2[2, ]

[, 11 + 22] = 1[, 1] + 2[, 2]

Skew commutatively [, ] = [, ]

Jacobi identity =

2.4. Relative degree:

Definition: The nonlinear system has relative degree at the point 0if

[()] = 0 For all in the neighborhood of 0 and all < 1

[1()] 0

In other words using the properties of Lie derivatives:

=

= 2

() = + (

1)

Therefore, the derivation of the output sees the input signal . Then the value is the relative degree of the system. Remark: in linear systems, the relative degree is the difference between the number of poles and number of zeros.

Pg. 06

Pg. 07

3. Feedback Linearization Approaches As mentioned, two approaches to feedback linearization exist. One is based on

input/output linearization, and the other is based on input/state linearization.

The choice of approach depends on the relative degree of the system, as follows:

Case of = : the input/output linearization, and input/state linearization

approaches coincide. The input/output linearization approach is used.

Case of < : there are some internal dynamics, that dont appear in the input/output

linearized model of the states, so the states are not affected by the control,

then input/state linearization is necessary before / linearization.

3.1. Input/output linearization

Given the nonlinear system:

= () + ()

= ()

The idea is to introduce a new input that makes the system linear.

First, let us consider the following:

=

= 2

.

.

.

() = + (

1)

Pg. 08

If we set

() =

() = (1)

Then

() = () + () We choose the input in such a way to compensate the nonlinearities of the system.

() = () + () Which results in the following

= () + () So

=1

()[() + ]

In other words, this approach is based on defining a new state space vector, which consists of the derivatives of the output:

1 = = 0

2 = = 1 =

.

. =

(1) = 1 = 1

+1 = () = = = () + ()

Pg. 09

The state space form of the system becomes

(

12...)

=

(

0 10 0

0 0 01 0 0

.

.00

00

. .

.

. .

.

.10 )

(

12...

)

+

(

00...1)

= [1 0 . . 0]

(

12...

)

A cascade of integrators can do this. Remark: the matrix above is of the controller canonical form.

To summarize the steps: 1) Find the relative degree of the nonlinear system 2) Make state transformations

= () =

(

0

1..

1)

3)

() =

() = (1)

= () + ()

Pg. 10

4) Check stability of the internal dynamics Example: ( = )

= [0

1 + 22

1 2

] + [2

2

0]

= () = 3

Finding the relative degree:

(0) = =0

(1 ) = [1 1 0] = 0

(2) = [1 22 0] =

2(1 + 22)

Relative degree is 3 if 1 + 22 0 so = . Make the 3 states transformations

1 = 0 = 3

2 = 1 = 1 2

3 = 2 = 1 2

2

We make the new input so that the states are linearized,

() = =

3 = 22(1 + 22)

() = (1) = (

2) = 2(1 + 22)

So =1

()[() + ] =

22(1+22)

2(1+22)

1

2(1+22)

Pg. 11

Using this input, we get the system with the following equations:

(123

) = (0 1 00 0 10 0 0

)(

123

) + (001)

= (1 0 0) (

123

)

Now we can design a controller using the new input of the linearized system .

3.2. Input/state linearization:

So far we have seen that / linearization is applied only if the relative degree =

( is degree of the system) where all the internal dynamics are shown. In the case

where the relative degree < and some internal dynamics are hidden additional

variables must be introduced to complete the coordinate transformation and the

design of the controller is not possible with = () (unstable system). The idea is to

select a new output = () where the linearization through () satisfies the

condition =

= () + ()

= = ()

The goal is to find a transformation () between new state space and .

= ()

[ 11..

]

[ 12..

]

Pg. 12

1 = = () 0()

2 = 1 = =()

=

()

[ + ]

=()

+

()

= () + ()

(() = 0)

3 = 2 = =[()]

=

[()][ + ]

=[()]

+

[()]

= 2() + ()

(() = 0)

= 1 =

1() + 2()

(2() = 0)

+1 = =

() + 1()

(1() 0)

=

= () + () = ( )

For the previous analysis we have to ensure that the equalities

{

() = 0

() = 0

(2()) = 0

Pg. 13

And

(1()) 0

Using Jacobi identity:

() = (()) (())

We get

() = () = = ()2 = 0

But

()1 0

If we use

{()[, , , 2] = 0

(x)1 0

The last equation has an advantage of () being separated from and

Question: Does the partial differential equation have a solution?

Frobenius theorem:

The nonlinear system with () and () being smooth vector fields is input state

linearizable iff there exists a region such that the following conditions are satisfied:

1. In the vector field = {, , , 1} vectors are linearly independent

in .

2. The set [, , , 2] is involutive.

Pg. 14

Remark: The matrix has a full rank=> the system is fully controllable = .

Involutivity:

Definition: A set of linearly independent vector [1 ] is said to be involutive if

[ ] = ()()

=1

Remark: This is equivalent to saying that the lie brackets of vector do not generate

new vectors.

Input/state linearization procedures:

Construct the matrix

= [, , , 2]

Check the controllability of and involutivity of matrix [, , , 2]

If is controllable and involutive we look for () where () satisfies

{()[, , , 2]

()1 0

Compute the state transformation matrix ()

() =

[

(0)()..

(1)

()]

Compute the input transformation

() = ()

() = (1())

Pg. 15

=1

()[() + ]

Remark: In the input/state linearization, we cannot analyze all the internal dynamics.

The zero dynamics of the system are defined to be the internal dynamics of the

system when the output of the system () is kept zero for a given input 0 at 0

(equilibrium point).

In this case, we can conclude local stability on system around the equilibrium point.

Example: Consider a mechanism given by the dynamics that represent a single link flexible joint robot. Its equations of motion yields:

1 + (1) + (1 2) = 0

J2 (1 2) =

Because nonlinearities (due to gravitational torques) appear in the first equation, while the control input enters only in the second equation, there is no easy way to design a large range controller.

= [

1122

] , =

[

2

sin(1)

(1 3)

4

(1 3) ]

,

[ 0001

]

Check controllability

= [, ] =

=

Pg. 16

=

[

0 1 0 0

cos(x1)

0

0

0 0 0 1

0

0]

[ 0001

]

=

[

00

1

0 ]

2 = [, ] =

=

=

[

0 1 0 0

cos(x1)

0

0

0 0 0 1

0

0]

[

00

1

0 ]

=

[

0

0

2]

3 = [,

2] =

2

2 =

=

[

0 1 0 0

cos(x1)

0

0

0 0 0 1

0

0]

[

0

0

2]

=

[

0

2

0 ]

[ 2

3 ] =

[

0 00 0

0 // 0

0 1/1/ 0

0 /2

/2 0 ]

It has rank 4. Furthermore, since the above vector fields are constant, they form an involutive set. Therefore the system input-state linearizable.

Lets find the state transformation = () and the input transformation = () + () So the input state linearization is achieved.

Pg. 17

(x)g = [

x1

x2

x3

x4 ] g = 0 =>

x4= 0

(x)adfg = [

x1

x2

x3

x4 ]

[

00

1

J0 ]

= 0 =>

x3= 0

(x)adf2g = [

x1

x2

x3

x4 ]

[

0k

IJ0

k

J2]

= 0 =>

x2= 0

(x)adf3g = [

x1

x2

x3

x4 ] [

k/IJ0

k/J2

0

] 0 =>

x1 0

It is obvious that () should be a function of 1 only. Therefore, we choose a function, which is a diffeomorphism, e.g. () = 1 The other states are obtained from the following calculations:

2 = = 1 = 2

3 = 2 = 2 =

sin(1)

(1 3)

4 = 3 = 3 =

x2cos(1)

(2 4)

The input transformation is then: () =

4() = 4 =

= [

2 sin(1)

cos(1)

0

]

[

2

sin(2)

(1 3)

4

(1 3) ]

=

Pg. 18

=

2

2 sin(1) (

cos(1)

)(

sin(2) +

(1 3)) +

2

(1 3)

() = 3() = 4 =

With

=1

( )

We end up with the following set of linear equations

1 = 2 2 = 3 3 = 4 4 =

Thus completing the input/output linearization.

Pg. 19

4. Control Problem After applying feedback linearization, we get linearized system as shown in the

following figures:

=1

[ ()]

Remark: = () + () is the overall system including the integrator

given the new linear system we can use state feedback as follow

=

= +

= ( )

= ( )

Pg. 20

Pg. 21

Conclusion Feedback linearization is a very important concept in the field of nonlinear control

systems. Feedback linearization permits to transform a nonlinear system in to linear

system.

Compared to the conventional Jacobian linearization, feedback linearization is an

exact linearization of the nonlinear systems.

As seen, depending on the cases two approaches are used in feedback linearization:

Input/output linearization, and Input/State linearization.

Once the system is linearized, the well-established theory of linear systems can be

applied.

Feedback Linearization is used in numerous applications in the field of control

systems. However, even though very powerful in theory, Feedback Linearization

method has limitations in practice due to its lack of robustness and the required

exactness of the parameters of the model.

Pg. 22

References

- Applied nonlinear control E.Slotine , Li

- Nonlinear control systems Alberto Isidori

- Nonlinear systems K.Khalil

- Nonlinear systems analysis M.Vidyasagar