Basic Concepts of Nonlinear Control Theory

Basic Concepts of Nonlinear Control Theory 9

CHAPTER 1 Basic Concepts of Nonlinear Control Theory Each scientific branch has its own specific basic concepts, which show extraordinary significance because they are the basic elements and components of this theory and the essential part of the theoretical framework. In modern linear control system theory, the basic concepts include dynamic control systems, inputs and outputs, feedback, state variables and state vectors, state space and state equations, dynamic responses and state trajectory, stability, reachability, controllability and observability, performance index, optimal,control, and the basic concepts of linear algebra. Besides to above, there are some specific concepts and definitions of nonlinear control theory. First, the concepts of nonlinear coordinate transformation, nonlinear mapping in state space and diffeomorphism are illustrated in comparison with those in linear systems so as to get a deeper understanding. The next concepts is the affine nonlinear control system, which is the most common and important type of nonlinear control system in applications. Follow vector fields in state space, Lie derivative and Lie brackets. With the vector field and Lie bracket it is possible to discuss the concept of involutivity which is very important property of vector field sets and will be used in the condition of exact linearization of nonlinear control systems. To discuss the issue of exact linearization it will illustrate the concept of relative degree of a control system first and then discuss the normal form of linearized nonlinear control systems. 1.1. Nonlinear Coordinate Transformation and Diffeomorphism Nonlinear coordinate transformation can be described in the form as

( )XZ Φ= (1.1) where Z and X are vectors of equal dimensions, Φ is a nonlinear vector function which can be expanded as

( )( )

( )nnn

n

n

xxxz

xxxzxxxz

,,,

,,,,,,

21

2122

2111

L

M

L

L

ϕ

ϕϕ

=

==

(1.2)

The first condition it assumes for the nonlinear coordinate transformation in (1.2) is that its inverse transformation exists and is single-valued, i.e.

( )ZX 1−Φ= (1.3)

The second condition is that both ( )XΦ and ( )Z1−Φ are smooth vector functions, i.e. the

function of each component of both ( )XΦ and ( )Z1−Φ has continuous partial derivatives of any order. In short, the first condition is invertible and the second is differentiable. If these

Sliding-Mode Robotic Manipulators and Mobile Robots Control

10

two conditions are satisfied the expression ( )XZ Φ= should be a valid coordinate transformation and this expression ( )XΦ is called a diffeomorphism between two coordinate space. From a geometric viewpoint the coordinate transformations ( )XZ Φ= and

( )ZX 1−Φ= can be regarded as a mapping between two spaces with same dimension X and Z. However, the necessary and sufficient conditions mentioned above that a diffeomorphism must be satisfied may hold only for a neighborhood of a specific point oX rather than all points in the space (any nX ℜ∈ ). One calls it a local diffeomorphism so long as it is not defined on the whole space no matter how large it is. How to test whether a nonlinear mapping ( )XΦ is a local diffeomorphism at oX ?. The answer is given by the next proposition Proposition 1.1: Suppose that ( )XΦ is a smooth function defined on a certain subset U of

the space nℜ . If the Jacobian matrix at oXX =

oXXn

nn

n

oXX

dx

ddxd

dxd

dxd

dXd

=

=

=Φ

ϕϕ

ϕϕ

L

LOL

L

1

1

1

1

(1.4)

Is non-singular, ( )XΦ is then a local diffeomorphism in a open subset oU including oX . 1.2 Affine Nonlinear Control Systems In the engineering world, many nonlinear systems such as power systems, robot control systems, helicopter control systems and chemical control systems have the following form of state equations

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) mnmnnnnnn

mnmnn

mnmnn

uxxxguxxxgxxxfx

uxxxguxxxgxxxfxuxxxguxxxgxxxfx

LLLL&

M

LLLL&

LLLL&

,,,,,,

,,,,,,,,,,,,

21121121

212121122122

211121112111

+++=

+++=+++=

(1.5)

And output equations ( )

( )nmm

n

xxxhy

xxxhy

L

M

L

,,

,,

21

2111

=

= (1.6)

which can written in compact form as

( ) ( )( ) ( )( ) ( )

( ) ( )( )tXhtY

tutXgtXftXm

iii

=

∑+==1

& (1.7)

where nX ℜ∈ is the state vector, ( )miui ,,1L= control variables, ( )Xh m-dimensional output function vector; ( )Xf and ( ) ( )miXgi ,,1L= are n-dimensional function vectors. A


nonlinear control system like (1.7), posseing the feature that it is nonlinear to state vector ( )tX but linear to control variables ( )miui ,,1L= is called affine nonlinear system.

1.3 Vector Fields In (1.7) ( )Xf is a n-dimensional vector function, i.e.

( )( )

( )

=

nn

n

xxf

xxfXf

,,

,,

1

11

L

M

L

(1.8)

Each component of ( )Xf is smooth of variable [ ]TnxxX ,,1 L= . Then it knows that each specific point in the state space corresponds to a certain smooth vector at this point

( ) ( ) ( )[ ]Ton

oo XfXfXf L1= (1.9) Hence, ( )Xf is called a vector field of the state space. 1.4 Lie Derivative and Lie Bracket Given a differentiable scalar function of X ( )nxx ,,1 Lλ (1.10)

and a vector field (1.8). The derivative of scalar function ( )Xλ along the vector field ( )Xf is the scalar product between ( )Xλ∇ and ( )Xf , where ( )Xλ∇ is the gradient of the function ( )Xλ

( ) ( ) ( )

∂

∂∂

∂=∇

11,,

xX

xXX λλλ L (1.11)

( ) ( )XfX ,λ∇ (1.12) This formula defines a new scalar function which is called the Lie derivative of ( )Xλ along ( )Xf and denoted ( )XL f λ .

Definition 1.2: Given a differentiable scalar function ( )Xλ of [ ]TnxxX ,,1 L= and a vector

field ( ) ( ) ( )[ ]Tn XfXfXf L1= , the new scalar function denoted ( )XL f λ , is obtained by the following operation

( ) ( ) ( ) ( ) ( )XfxXXf

XXXL i

n

i if ∑

∂∂

=∂

∂=

=1

λλλ (1.13)

and called the Lie derivative of function ( )Xλ along vector field ( )Xf . From Definition 1.2, it knows that Lie derivative is a scalar function, so it is possible to repeat use of this operation to obtain lie derivative ( )XL f λ along another vector field ( )Xg , i.e.

( )( )( )

( )XgX

XLXLL f

fg ∂

∂=

λλ (1.14)


12

Certainly, it can also derive the kth order Lie derivative of ( )Xλ along ( )Xf recursively as

( ) ( )( )( )

( )

( )( )( )

( )XfX

XLXL

XfX

XLXLXLL

kfk

f

ffff

∂

∂=

∂

∂==

− λλ

λλλ

1

2

M (1.15)

The kth-order Lie derivative of ( )Xλ along ( )Xf , ( )XLkf λ is still a scalar function and thus

can be used to get Lie derivative along another vector field ( )Xg , i.e.

( )( )( )

( )XgX

XLXLL

kfk

fg ∂

∂=

λλ (1.16)

Assume two vectors fields of the same dimension are given as

( )( )

( )

=

nn

n

xxf

xxfXf

,,

,,

1

11

L

M

L

, ( )( )

( )

=

nn

n

xxg

xxgXg

,,

,,

1

11

L

M

L

(1.17)

The derivative of one vector field along another is a vector field and is defined as follows. Let ( ) ( )[ ]XgXf , denote the derivative of ( )Xg along ( )Xf , then

[ ]( )

∂∂

∂∂

∂∂

∂∂

−

∂∂

∂∂

∂∂

∂∂

=

n

n

nn

n

n

n

nn

n

g

g

xf

xf

xf

xf

f

f

xg

xg

xg

xg

Xgf M

L

M

L

M

L

M

L1

1

1

1

1

1

1

1

1

1

,,

,,

,,

,,

, (1.18)

Expression (1.18) shows a new vector field, namely Lie bracket, for it accustomed notation [ ]gf , , which can also be denoted by gad f . Definition 1.3: Suppose two vector fields ( ) [ ]TnffXf ,,1 L= and ( ) [ ]TnggXg ,,1 L= . The following operation denoted by ( ) ( )[ ]XgXf , or gad f

[ ] gXff

Xggadgf f ∂

∂−

∂∂

==, (1.19)

Obtains a new vector field which defines the Lie bracket of ( )Xg along ( )Xf . 1.5 Relative Degree of a Control System Suppose a single-input single-output nonlinear control system ( ) ( )( ) ( )( ) ( )( ) ( )( )tXhtY

tutXgtXftX=

+=& (1.20)

where nX ℜ∈ , ℜ∈u , ℜ∈y , ( )Xf and ( )Xg are vector fields. If

(i) The Lie derivative of the function ( )XhLkf along g equals zero in a neighborhood Ω of

oXX = , i.e.


( ) Ω∈∀−<= XrkXhLL kfg ,1,0 (1.21)

(ii) The Lie derivative of the function ( )XhLrf

1− along vector field ( )Xg is not equal to zero

in Ω , i.e., ( ) 01 ≠− XhLL r

fg (1.22)

Then this system is said to have relative degree r in Ω . Example 1.1: Given a nonlinear system with state equations:

uxcxxc

xX

+

−−=

10

)1( 122211

2&

where 1c and 2c are constants and the output equation is 1)( xXhy == . Calculate the relative degree. According to the definition of relative degree, to ascertain the relative degree r, it first calculates the Lie derivative of the function zro-order lie derivative of ( )Xh along ( )Xf along ( )Xg and the result is:

[ ] 010

1010

)()()()(21

0 =

=

∂∂

∂∂

=∂

∂==

xh

xhXg

XXhXhLXhLL gfg

Then it computes )(XhLL fg . Firs it calculates

[ ] 2122

211

2

)1(01)()()( x

xcxxc

xXf

XXhXhL f =

−−=

∂∂

=

Thus the following result is induced

[ ] 110

10)())((

)( =

=

∂

∂= Xg

XXhL

XhLL ffg

Hence the given system has the relative degree 2=r . Assume a system

)()()(

XhyuXgXfX

=+=&

(1.23)

where nRX ∈ , has relative degree nr = . The following expressions are true according to the definition

0)()()()( 220 ===== − XhLLXhLLXhLLXhLL nfgfgfgfg L (1.24)

0)(1 ≠− XhLL nfg (1.25)

One constructs a mapping from X to Z space. If it chooses: ),,( 11 nxxhz L= (1.26)

Then

XXXhz &&

∂∂

=)(

1 (1.27)

Substituting (1.23) into (1.27) for X& , it obtains:

uXgXXhXf

XXhz )()()()(

1 ∂∂

+∂

∂=& (1.28)

This formula can be rewritten according to the definition of Lie derivative


14

uXhLLXhLz fgf )()( 01 +=& (1.29)

From the formula (1.24), one knows that in (1.29) 0)(0 =XhLL fg . Therefore,

)(1 XhLz f=& (1.30) If setting

21 )( zXhLz f ==& (1.31)

Then it express uXhLLXhLz fgf )()(2

2 +=& (1.32) Since nr = , from (1.24) it expresses

0)( =XhLL fg , so )(22 XhLz f=& once again it sets

32

2 )( zXhLz f ==& (1.33) By analogy, it can certainly obtain:

1)( +== iifi zXhLz& (1.34)

until

nnfn zXhLz == −

− )(11& (1.35)

Since nr = , things will change if is follows (1.35), i.e. uXhLLXhLz n

fgnfn )()( 1−+=& (1.36)

From the formula (1.25) it knows 0)(1 ≠− XhLL n

fg (1.37)

Therefore, (1.36) can be also rewritten as uXbXazn )()( +=& (1.38)

where )()( XhLXa n

f= , 0)()( 1 ≠= − XhLLXb nfg (1.39)

1.6. Coordinate Transformation to a Nonlinear Control System of Relative Degree Equal with System Order Integrating the content from (1.20) through (1.39) yields an X-to-Z coordinate transformation and a Z-coordinate system. The chosen coordinates are:

)(

)(

)(

1

2

1

XhLz

XhLz

Xhz

nfn

f

−=

=

=

LL (1.40)

which can also written as )(XZ Φ= , where )(XΦ is required to be a local diffeomorphism.

This fact can be seen by showing )(,),(),( 1 XhdLXhdLXdh nff−L are linearly independent.

Thus the new dynamic system described by Z-coordinates is:


)()()( 11

32

21

ZXuXbXaz

zz

zzzz

n

nn−

−

Φ=+=

=

==

&

&

LL

&

&

(1.41)

and the output equation is 1)( zXhy == (1.42)

If an SISO affine nonlinear system has relative degree nr = where n denotes the number of the system’s order, then a coordinate mapping shown in the expressions (1.40) can transfer the original system into that as in (1.41), in which the first ( )1−n equations are linearized and do not include the control variable u apparently and only the last equation involving u is nonlinear. The above fact is very important for exact linearization of affine nonlinear system and will be further discussed in subsequent subchapter. 1.7. Relative Degree Less than System Order. Linearized Normal Form The instance that an affine nonlinear system has relative degree nr = has been discussed in the above subchapter where the original system can be transformed into the form (1.41) by the local change of coordinates shown in formula (1.40). In general, however, the relative degree r may not just equal n , but nr < . For the sake of completeness, let us consider a nonlinear system

)()()(

XhyuXgXfX

=+=&

(1.43)

where nX ℜ∈ and the relative degree r is less than n . The mapping )(XZ Φ= will be chosen by the following steps. First one chooses the first r components of the transformation as

)()(

)()()()(

1

22

11

XhLXz

XhLXzXhXz

fr

rr

f

−==

====

ϕ

ϕϕ

LL (1.44)

Then the remaining rn − components of the transformation are chosen as:

)(

)(11

Xz

Xz

nn

rr

ϕ

ϕ

=

= ++LL (1.45)

such that nirXL ig ≤≤+= 10)(ϕ (1.45)

holds. After the transformation as shown in formula (1.44) the first r equations of the original nonlinear system are transformed into

riXhLz ifi <= )(& (1.46)

In view of the definition of the relative degree of the system it can write


16

riXhLL ifg <=− 0)(1 (1.47)

Considering the mapping in formula (1.44) it knows that

1)( +== iifi zXhLz& (1.48)

and the rth equation must be uXhLLXhLz r

fgrfr )()( 1−+=& (1.49)

From the definition of relative degree, one knows that 0)(1 ≠− XhLL rfg . As a result in

equation (1.49) if one set

))(()()(

))(()()(

11)(

1

1)(

1

1

ZhLLXhLLZb

ZhLXhLZa

rfgZX

rfg

rfZX

rf

−−Φ=

−

−Φ=

Φ==

Φ==

−

−

(1.50)

then (1.49) can be written as uZbZazr )()( +=& (1.51)

Generally speaking, ( )Za and ( )Zb above are nonlinear functions of Z . Now let us consider the remaining rn − dynamic equations. It is clear from (1.45) that

)()()()()( 1111 ZXuXg

XXXf

XXz rr

r−++

+ Φ=∂

∂+

∂∂

=ϕϕ

& (1.52)

or uZLZLz rgrfr ))(())(( 1

11

11−

+−

++ Φ+Φ= ϕϕ& (1.53)

Similarly niruZLZLz irgirfir <+Φ+Φ= −

+−

++ ))(())(( 11 ϕϕ& (1.54)

and uZLZLz ngnfn ))(())(( 11 −− Φ+Φ= ϕϕ& (1.55)

Since it has known form (1.45) that nr ϕϕ ,,1 L+ satisfy nirXL ig ≤≤+= 10)(ϕ (1.56)

From (1.45) through (1.55) it obtains

))((

))((

1

111

ZLz

ZLz

nfn

rfr

−

−++

Φ=

Φ=

ϕ

ϕ

&

LL

&

(1.57)

To get the normalized form, it sets the above formulae that

))(()(

))(()(

1

111

ZLZq

ZLZq

nfn

rfr

−

−++

Φ=

Φ=

ϕ

ϕ

LL (1.58)

thus the dynamic equation from ( )thr 1+ through thn have the following forms

)(

)(11

Zqz

Zqz

nn

rr

=

= ++

&

LL

&

(1.59)


The above arguments give the following Proposition 1.2: Consider the system

)()()(

XhyuXgXfX

=+=&

(1.60)

where nX ℜ∈ and the relative degree r is less than n . If the mapping )(XZ Φ= is chosen as

)(

)(

)()(

)()()()(

11

1

22

11

Xz

Xz

XhLXz

XhLXzXhXz

nn

rr

fr

frr

f

ϕ

ϕ

ϕ

ϕϕ

=

=

==

====

++

−

LL

LL

(1.61)

in which nr ϕϕ ,,1 L+ satisfy nirXL ig ≤≤+= 10)(ϕ (1.62)

and the Jacobian matrix at oXX = :

oXXXXJ =Φ ∂

Φ∂=

)( (1.63)

is nonsingular, setting

))(()(

))(()(

11

1

ZhLLZb

ZhLZa

rfg

rf

−−

−

Φ=

Φ= (1.64)

and

))(()(

))(()(

1

111

ZLZq

ZLZq

nfn

rfr

−

−++

Φ=

Φ=

ϕ

ϕ

LL (1.65)

then the original nonlinear system can be transformed into following form

)(

)()()(

11

32

21

Zqz

ZqzuZbZaz

zzzz

nn

rr

r

=

=+=

==

++

&

LL

&

&

LL

&

&

(1.66)

The model of the system in (1.66) is called a normal form. Example 1.2: (Robot with flexible joint) The dynamic equations of a single link robot arm with a revolute elastic joint rotating in a vertical plane are given by


18

( )( )

==−−+

=+−++

1

2122

1211111 0sin

qyuqqkqFqJ

qMglqqkqFqJ

mm &&&

&&&

(1.67)

in which 1q and 2q are the link displacement and the rotor displacement, respectively. The

link inertia 1J , the motor inertia mJ , the elastic constant k , the link mass M , the gravity

constant g , the center of mass l and the viscous friction coefficients 1F , mF are positive constant parameters. The control u is the torque delivered by the motor. The control problem are: 1) Assuming that all state ( )2211 ,,, qqqq && is measured, u is to be designed so that 1q tracks a desired reference ( )tqr1 in two situations: (i) The parameters are assumed to be known; (ii) The parameters are unknown; 2) Assuming that only 1q is measured, u is to be designed so that 1q tracks a desired reference ( )tqr1 in two situations: (i) The parameters are assumed to be known; (ii) The parameters are unknown.

Fig. 1.1: Rigid link-flexible elastic joint robotic manipulator. Choosing as state variables: 11 qx = , 12 qx &= , 23 qx = , 24 qx &= the dynamical state model of robotic manipulator can be expressed like a SISO affine nonlinear system

( ) ( ) ( ) ( )( )

( ) ( )( ) ( )

( ) ( )uxgxfu

JxxJkxJFx

xxJkxJMglxJFx

xxxx

mmmm

+=

+

−+−

−−−−=

/1000

//

/sin//

314

4

31111211

2

4

3

2

1

&

&

&

&

(1.68)

Example 1.3: Defining 1xh = , the relative degree r equals the system order 4=n and the linearizing diffeomorphism is given by


( ) [ ] [ ]TTfff zzzzhLhLhLhXZ 4321

32 ,,,,,, ==Φ= (1.69)

with

( ) ( ) ( )

( ) ( ) ( )

+

−+++

+−

−−−−=Φ=

4311221

3112

2

1

sincos

sin

xJkxx

Jkx

JMglx

JF

JF

xJkx

JMgl

xxJkx

JMglx

JF

xx

xz

llll

l

l

l

ll

lll

l

(1.70) In new coordinates the system becomes

)()()( 14

43

32

21

ZXuXXz

zzzzzz

−Φ=+=

===

βα&

&

&

&

(1.71)

with

( ) ( ) ( )

( ) ( ) ( )

( )

−−+−

−++

+−+

++==

43142

31122

2

1

221214

sincos

cossin

xJF

xxJk

Jkx

J

kF

xxKxJ

MglxJF

Jk

J

Fx

JMgl

xJ

kFx

J

MglFx

JMglhLX

m

m

mll

l

lll

l

ll

l

l

l

l

l

l

lfα

(1.72)

( )ml

fg JJkhLLX == 3β (1.73)

Example 1.4: Defining 3xh = , the relative degree of the system is 2=r .

( ) ( )

( )

=

+−+−=

=−−−=

=

3

312414

43

311112

21

1

sin

xy

uJ

xxKxBx

xxxxKxMx

xx

m&

&

&

&

(1.74)

where 1x stands for the arm angle and 3x is the motor shaft angle. The input u is the voltage

of the d.c. motor that drive the arm. The parameters are defined as 1

1 JMglM = ,

11 J

kK = ,

mJkK =2 ,

m

mJF

B =1 . The inertia momentum of the link arm is considered unitary. The


20

measured output to be controlled is the position of the motor shaft. The trajectory to be tracked is defined as ( ) ( )t3cos7.0tyr −= ; 2*(1+cos(2*t)). The successive Lie derivatives are

( ) 3xxh = , ( ) 4xxhL f = , ( ) ( )312412 xxKxBxhL f −+−= ,

( ) ( ) ( ) 2231214221

3 xKxxKBxKBxhL f +−−−=

( ) ( ) ( ) ( )( ) 22131122121214

3121

4 sin2 xKBxxKKBKxKMxBKBxhL f −−−−+−−=

( ) 0=xhLg , ( )m

fg JxhLL 1= , ( )

mfg J

BxhLL 12 −= , ( )m

fg JKBxhLL 2

213 −

=

The matching condition is in this case mm JJ ˆ= and 11 BB = . The state transformation is now

( ) ( )( ) ( )

+−−−−+−

=Φ=

2231214221

31241

4

3

xKxxKBxKBxxKxB

xx

xz (1.75)

The inverse transformation can be easily found in this case

( )

++

++

=Φ= −

2

1

2

22314

2

12213

1

zzK

zKzBzK

zKzBz

zx (1.76)

The transformed state equations are

( )

−+

++−−+−−=

−=

+=

=

uJ

KBK

zKzBzKMzBzKKzKBz

uJB

zz

uJ

zz

zz

m

m

m

221

2

1221321413212114

143

32

21

sin

1

&

&

&

&

(1.77)

The following parameter uncertainties are considered (note that the matching conditions are not fulfilled): 004.0=mJ , 26=lM , 230=lK , 17832 =K , 75.1=lB , 0044.0ˆ =mJ ,

32ˆ =lM , 317ˆ =lK , 2140ˆ 2 =K , 60.1ˆ =lB . 1.8. Relative Degrees for MIMO Nonlinear Control Systems Consider the MIMO nonlinear system (1.5) and (1.6) with m inputs and m outputs.


Definition 1.4: For MIMO system (1.5) and (1.6), if the following conditions hold in a neighborhood of oX , namely for 1−< ii rk ,

( ) mjmiXhLL ikfgi

j,,2,1,,2,10 LL === (1.78)

and the mxm matrix

( )

( ) ( )( ) ( )

( ) ( )

=

−−

−−

−−

XhLLXhLL

XhLLXhLL

XhLLXhLL

XB

irfgi

rfg

irfgi

rfg

irfgi

rfg

mm

m

m

m

11

11

11

1

221

111

L

MMM

L

L

(1.79)

Is nonsingular near oX , then mrrrr L,, 21= is the relative degree set of the system and each sub-relative degree ir corresponds to output ( ) ( )( )tXhty ii = . 1.9. Linearization Normal Form MIMO Nonlinear Control Systems In order to make the discussion brief and without loss of generality, let us deal with systems with two inputs and two outputs. That is to say, the nonlinear system to be discussed is in the form

( ) ( ) ( )( ) ( ) ( ) 221212112122

221212112111,,,,,,

,,,,,,uxxxguxxxgxxxfx

uxxxguxxxgxxxfx

nnn

nnnLLL&

LLL&

++=++=

(1.80)

and output equations ( )( )n

nxxxhy

xxxhyL

L

,,,,

2122

2111==

(1.81)

Assume that its relative degrees satisfy 21 rrr += , where n is the dimension of the state vector X . Under this condition, the coordinates mapping ( )XZ Φ= should be chosen as

)()(

)()()()(

)()(

)()()()(

21

222

211

11

122

111

22

11

XhLXz

XhLXzXhXz

XhLXz

XhLXzXhXz

fr

frr

f

fr

frr

f

−

−

==

====

==

====

ψ

ψψ

ϕ

ϕϕ

LL

LL

(1.82)

From the above mapping, it can obtain

( )( )

( ) ( ) ( ) 21111

221111

1

21

)()()()(

uXhLuXhLXhL

uXguXgXfXXhX

XX

ggf ++=

++∂

∂=

∂∂

= &&ϕ

ϕ (1.83)

Furthermore from the definition 1.4, if 01 >r then ( ) ( ) 011 21

== XhLXhL gg (1.84)


22

and (1.83) can be written as ( ) 211 ϕϕ == XhL f& (1.85)

One can derive similar results for iϕ& until

11 1 rr ϕϕ =−& (1.86)

As stated by the definition of relative degree, ( ) 1111

1uXhLL r

fg− and ( ) 11

112

uXhLL rfg− are at

least not all equal zeros. It can express ( ) ( ) ( ) 22

111

111

12

11

11

uXhLLuXhLLuXhL rfg

rfg

rfr

−−= ++ϕ& (1.87)

Similar to the above it can get

( ) ( ) ( ) 221

111

11

1

21

22

21

22

22

uXhLLuXhLLuXhL rfg

rfg

rfr

rr−−

−

++=

=

=

ψ

ψψ

ψψ

&

&

M

&

(1.88)

Combining from (1.85) to (1.88), it knows when relative degrees satisfy nrrr =+= 21 , under the coordinate transformation shown in (1.82), the system (1.80) can be transformed into the following normal form

( ) ( ) ( )

( ) ( ) ( ) 221

111

11

1

21

221

111

11

1

21

22

21

22

22

12

11

11

11

uXhLLuXhLLuXhL

uXhLLuXhLLuXhL

rfg

rfg

rfr

rr

rfg

rfg

rfr

rr

−−

−

−−

−

++=

=

=

++=

=

=

ψ

ψψ

ψψ

ϕ

ϕϕ

ϕϕ

&

&

M

&

&

&

M

&

(1.89)

and the output (1.83) ( )( )n

nxxxy

xxxyL

L

,,,,

2112

2111ψϕ

==

(1.90)

The relationship (1.89) is called first type normal form of MIMO affine nonlinear system. It correspond to the condition nrrrr m =+++= L21 . Let us discuss the condition that the sum of system relative degrees miri ,,1L= is less than system order n. If for the system (1.81), its relative degrees satisfy nrrr <+= 21 , then after choosing a coordinate transformation as in (1.83), the last rn − coordinates could always be found as

( )

( )X

X

rnrn −− =

=

ηη

ηηM

11 (1.91)

such that the Jacobian matrix of the vector function ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]Trnrr XXXXXXX −=Φ ηηψψϕϕ ,,;,,;,, 111 21

LLL (1.92)

is nonsingular at oXX = . Thus, it has chosen a set of qualified local coordinates mapping.


Under this condition, the following ( )rn − equations should be added to the system described in new coordinates

( ) ( ) ( )

( ) ( ) ( ) 21

211111

21

21

uXLuXLXL

uXLuXLXL

rngrngrnfrn

ggf

−−−− ++=

++=

ηηηη

ηηηη

&

M

&

(1.93)

Therefore, the system transformation is

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) 21

211111

221

111

11

1

21

221

111

11

1

21

21

21

22

21

22

22

12

11

11

11

uXLuXLXL

uXLuXLXL

uXhLLuXhLLuXhL

uXhLLuXhLLuXhL

rngrngrnfrn

ggf

rfg

rfg

rfr

rr

rfg

rfg

rfr

rr

−−−−

−−

−

−−

−

++=

++=

++=

=

=

++=

=

=

ηηηη

ηηηη

ψ

ψψ

ψψ

ϕ

ϕϕ

ϕϕ

&

M

&

&

&

M

&

&

&

M

&

(1.94)

together with output (1.90). The system (1.94) is called the second type of normal form. Proposition 1.3: Suppose that there exists a system as shown in (1.80) and (1.81), with the sum of relative degrees nrrr =+= 21 . Choose a coordinate transformation ( )XZ Φ= as follows

)()(

)()(

)()(

)()(

)()()()(

21

222

211

11

122

111

22

1

1

11

XhLXz

XhLXz

XhXz

XhLXz

XhLXzXhXz

fr

frr

fr

r

fr

frr

f

−

+

+

−

==

==

==

==

====

ψ

ψ

ψ

ϕ

ϕϕ

LL

LL

(1.95)

Then the system can be transformed into the first type normal form in the new coordinates Z


24

( ) ( ) ( )

( ) ( ) ( ) 2221212

1

21

2121111

1

21

2

22

1

11

uZbuZbZa

uZbuZbZa

r

rr

r

rr

++=

=

=

++=

=

=

−

−

ψ

ψψ

ψψ

ϕ

ϕϕ

ϕϕ

&

&

M

&

&

&

M

&

(1.96)

where ( ) ( ) ( ) ( )[ ]Trr XXXXZ

21,,,, 11 ψψϕϕ LL= (1.97)

( ) ( )( )

( )( )

( )ZXrf

rf

XhL

XhL

ZaZa

ZA1

2

1

2

1

2

1

−Φ=

=

= (1.98)

( ) ( ) ( )( ) ( )

( ) ( )( ) ( )

( )ZXrfg

rfg

rfg

rfg

XhLLXhLL

XhLLXhLL

ZbZbZbZb

ZB1

22

11

12

21

11

11

11

11

2121

1211

−Φ=

−−

−−

=

= (1.99)

Proposition 1.4: Suppose there exists a system as in (1.80) with the sum of relative degrees

nrrr <+= 21 . Apaart from the chosen coordinate transformation as in (1.95) the last ( )rn − coordinates can always be found as

( )

( )Xz

Xz

rnrnn

r

−−

+

==

==

ηη

ηηM

111 (1.100)

Such that the Jacobian matrix of the vector related to the coordinate mapping (1.92) is nonsingular at oX , thereby the original system can be transformed into the second type normal form, which is formed by replenishing the form shown by (1.96)with the following ( )rn − equations

( ) ( ) ( ) 2211 uZpuZpZq ++=η& (1.101) which may also be written as

( ) ( ) ( )[ ]

++=

2

121 u

uZpZpZqη& (1.102)

where ( ) ( ) ( )[ ]Trn XXX −=Φ ηη ,,1 L (1.103)

( )( )

( )

( )( )

( )ZXrnf

f

rnXL

XL

Zq

ZqZq

1

11

−Φ=−−

=

=

η

ηM (1.104)


( )( )

( )( )

( )

( )( )ZX

rng

g

rn XL

XL

Zp

ZpZp

11

1 1

1

11

1

−Φ=−−

=

=

η

η

MM (1.105)

( )( )

( )( )

( )

( )( )ZX

rng

g

rn XL

XL

Zp

ZpZp

12

2 1

2

21

2

−Φ=−−

=

=

η

η

MM (1.106)

In addition, for the above propositions, their output equations are the same as (1.90). Let one comes to the third type normal form. It can be verified that, if the vector field set

( ) ( ) XgXg m,,1 L shown in (1.80) is involutive , then rn − coordinates mappings ( ) ( )XX rn−ηη ,,1 L can surely be found such that

( )

( ) rniXL

XL

ig

ig

m−==

=

,,10

01

L

M

η

η

(1.107)

Thus, the system (1.94) can be transformed into

( ) ( ) ( )

( ) ( ) ( )( )

( )XL

XL

uXhLLuXhLLuXhL

uXhLLuXhLLuXhL

rnfrn

f

rfg

rfg

rfr

rr

rfg

rfg

rfr

rr

−−

−−

−

−−

−

=

=

++=

=

=

++=

=

=

ηη

ηη

ψ

ψψ

ψψ

ϕ

ϕϕ

ϕϕ

&

M

&

&

&

M

&

&

&

M

&

11

221

111

11

1

21

221

111

11

1

21

22

21

22

22

12

11

11

11

(1.108)

with the output equations (1.90). In (1.108) ( ) ( )XX rn−ηη ,,1 L are the solutions of the partialdifferential equation set (4.14). What is expressed in (1.108) can be called the third type normal form. It should be pointed out that the third type normal form is simpler than the second one in (1.94) in the last ( )rn − equations. However, this simplicity is at the cost of solving the set of partial differential equations (1.107). Proposition 1.5: Suppose there exists a system as in (1.80) with the sum of relative degrees

nrrr <+= 21 . If its vector field set ( ) ( ) XgXg m,,1 L is involutive , the last ( )rn − coordinates ( ) ( )XX rn−ηη ,,1 L can be chosen to satisfy (1.107) such that the original system can be transformed into the third normal form, which is formed by replenishing the form shown in (1.96)with the following ( )rn − equations

( )Zq=η& (1.109)


26

1.10. References [1]. Andreson B., Bitmead R., Jonhson C., Kosut R., Kokotovic P., Kosut R., Mareels I.,

Praly L. and Rietle B., Stability of Adaptive Systems: Passivity and Averaging Analysis. Cambridge, Ma: M.I.T. Press, 1986

[2]. Filipescu A., Dugard L. and Stamatescu S., “Robots control based on parameter identification and adaptive gain smooth sliding observer-controller,” in CD Preprints 16th IFAC World Congress, Prague, July, 2005.

[3]. Filippov, A.F., Application of the theory of differential equations with discontinuous right-hand sides to non-linear problems of automatic control, Proceedings of 1st IFAC Congress II, Butterworths, London, 1961

[4]. Khalil H. K ., Nonlinear systems. Prentice-Hall, Englewood Cliffs, NJ, 1996. [5]. Kokotovic P.V. and Sussman J.H . A Positive Real Condition for Global Stabilization

of Nonlinear Systems., Syst. Control Lett., vol. 13, pp. 125-133, 1989 [6]. Itkis Yu., U., Control Systems of Variable Structure, Wiley, New York, 1976. [7]. Isidori A., Nonlinear Control Systems, 2nd ed., Springer-Verlag, Berlin, 1989. [8]. Marino R. and Tomei P., Non-linear control design. Prentice-Hall, Englewood Cliffs,

NJ, 1995. [9]. Neimark, Yu.I., Note on Filippov’s A. paper, Proceedings of 1st IFAC Congress II,

Butterworths, London, 1961 [10]. Pomet B.J. and Praly L., Adaptive Nonlinear Regulation: Estimation from the

Lyapunov Equations, IEEE Trans. Automat. Contr., vol.37, pp. 729-740, July, 1992 [11]. Popov M.V., Hyperstability of Control Systems. New-York: Springer-Verlag, 1973 [12]. Sabery A. , Kokotovic P.V. and Sussman J.H . Global Stabilization of Partially Linear

Composite Systems, SIAM J. Control Optim., vol. 28, no. 6, pp. 1491-1503, 1990 [13]. Samson C., Control of Chained Systems: Application to Path Following and Time-

Varying Point Stabilization of Mobile Robots., IEEE Trans. Automat. Contr., vol. 40, pp. 64-77, Jan. 1995

[14]. Varadarajan V. S., Lie Groups, Lie Algebra, and their Representations, Springer-Verlag, New York, 1984.

[15]. Sontag D.E. and Wang Y. New Caracterization of Input to State Stability, IEEE Trans. Automat. Contr., vol. 41, pp. 1283-1294, Sept. 1996

[16]. Xu J. X. and Hashimoto H., “Parameter identification methodology based on variable structure control,” Int. J. Control, 57(5); 1207-1220, 1993.

[17]. Zinober A.S., Deterministic Non-Linear Control, Ed., Peter Peregrinus Limited, UK, 1990.

[18]. Zinober A.S., Variable Structure and Lyapunov Control, Ed.,Springer Verlag, London, 1993.

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Basic Concepts of Nonlinear Control Theory