me677c9 feedbacklinearization t

Feedback Linearization Lecture Notes by B.Yao

FEEDBACK LINEARIZATION Problem Formulation

Consider the SISO nonlinear plant modeled by ( , ) ( )

nx f x u x Ry h x

(F1)

where f and h are nonlinear functions. Control of a nonlinear system modeled by (F1) is in general difficult. One idea is to use a combination of certain nonlinear coordinate transformation:

12 2( ) or ( ),z g x x g z (F2)

and certain feedback linearization control law:

1( , )u g x v (F3) so that the transformed system from the new synthesis input v to the output y becomes an LTI system. Linear control techniques can then be applied to obtain a control input v to achieve the desired response as shown below:

Feedback controllersynthesized via techniques for LTI systems

Nonlinear Plant

An Equivalent LTI System

yCz 2g xx ,x f x uu 1 ,g x vvr

Feedback controllersynthesized via techniques for LTI systems

Nonlinear Plant


yCz 2g xx ,x f x uu 1 ,g x vvr


Linearization via Nonlinear Input Transformation

Let us consider a nonlinear system modeled by a Hammerstein model given by

1 2

2( ) ( ) ( ) ( ) ( ) ( ) ( )k

kp p py t G p u t G p u t G p u t (F4)

where ( )

ipG p is a TF which could be unknown. In general, feedback linearization techniques cannot be directly applied to (F4). However, for a special class of Hammerstein model, a nonlinear input transformation can be used to handle the nonlinearities effectively as follows. Consider the Hammerstein model where

( ) ( ), 1,2, ,ip i pG p c G p i k (F5)

where ci is a constant. From (6.1.3)

21 2

21 2

( ) ( ) ( ) ( ) ( )

( ) ( ) , ( ) ( ) ( ) ( )

kp k

kp k

y t G p c u t c u t c u t

G p f u t f u t c u t c u t c u t

(F6)


Suppose that there exists a nonlinear mapping 1( )g v such that

1( )f g v v (F7) For example,

21 1

13 3

1

21 1

( ) ( ) or ( )

( ) ( )

1 1 4 1 1 4( ) ( ) or ( ) 2 2

f u u g v v g v v

f u u g v v

v vf u u u g v g v

(F8)

Then, the nonlinear input transformation defined by

1( )u g v (F9) will result in an LTI system given by

1( ) ( ) ( ) ( )p py G p f g v G p v t (F10) which is linear from the new synthetic input v to the output y .


A control law can then be synthesized for the new input v to handle the model uncertainties in ( )pG p effectively since control of an LTI system is well documented. The resulting overall system is graphically shown below, where

1f̂ represents an approximate inverse of the nonlinear function f in implementation.

yv

)( pGp

f


( )pG p( )f

Nonlinear Plant

u1ˆ ( )f Feedback controller

synthesized via techniques for LTI systems

r yv

)( pGp )( pGp

f f


( )pG p( )f

Nonlinear Plant

u1ˆ ( )f Feedback controller

synthesized via techniques for LTI systems

r

Fig.2 Nonlinear Input Transformation


Feedback Linearization Example F1: System

1 2 1

2

1

x x f xx uy x

(F11)

Coordinate Transformation Define a new set of coordinates 1 2

T as

1 1 12

2 1 2 2 1( ) , or

( ) ( )x x

g xx f x x f x

(F12)

System in New Coordinate

1 1 2

2 2 1 1 1 2 1 2

1

( ) ( ) ( )

x

x f x x u f x u fy

(F13)


Feedback Linearization Control Law Choose the control law as

1 2 1 2 1( ) ( ) ( )u f v u v f x x f x (F14)

From (6.12), the new system is

1 2

2

1

vy

(F15)

which is a linear system. A control law can then be synthesized for v. For example, let

2 1 1 2 1 2, 0, 0.cv a a u a a (F16) The resulting system (6.1.14) would be

2 1

0 1 01

1 0

cua a

y

(F17)


which is a stable LTI system with a closed-loop TF

21 2

( ) 1( )( )c

Y sG sU s s a s a

.

Viewing (F12) and (F16), the nonlinear feedback control law u can be expressed in terms of original state x as

2 1 1 1 2 1( ) ( ) cu a x a f x x f x u (F18)


Normal Form of SISO Nonlinear Systems System

1 1( ) ( )( ) ( )

, , ( ) , ( )( )

( ) ( )

n

n n

f x g xx f x g x u

x R f x g xy h x

f x g x

(F19)

Relative Degree Roughly speaking, the relative degree r of the system (F19) is defined to be the lowest order of the derivative of the output y that involves the control input explicitly. To make this point clear, let us take the derivative of y :

( ) ( ) ( ) ( )hy x dh x f x dh x g x ux

(F20)

where 1

( ) , ,n

h h hdh xx x x

is called the differential of h. If

( ) ( ) 0dh x g x , then, y involves u explicitly and the relative degree of the system would be r=1. Otherwise,

( ) ( ) ( ) 0gL h x dh x g x and we would have

( ) ( ) ( )fy dh x f x L h x (F21)


in which the short hand notation ( )gL h x is called Lie derivative of h with respect to the vector field g or along g, and the short hand notation fL h is called the Lie derivative of h with respect to the vector filed f or along f. We can then take the derivative of y to see if y involves u explicitly

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

f f

f f

y L h x x L h x f x g x ux x

L h x f x L h x g x ux x

(F22)

Similarly, if the Lie derivative of ( )fL h x with respect to g is non-zero, i.e.

( ) ( ) ( ) 0g f fL L h x L h x g xx

(F23)

then, the relative degree of the system would be r=2. Otherwise, we can continue the process until we reach a step where ( )ry includes u explicitly. Mathematically, the above process can be summarized by the following formal definition of the relative degree r.


Definition [Page 510 of REF1]: [Relative Degree] Consider the system (F19), where : nf D R , : ng D R , and :h D R are smooth on a domain D. The system has relative degree r on D iff

1

( ) 0, 0 2

( ) 0,

ig f

rg f

L L h x i r

L L h x x D

(F24)

Coordinate Transformation

Let us assume that the system has a relative degree of r. Then, we can define r new state variables 1 2

T rr R as

1

2

1( 1) 1

( ) ( )( ) ( )

or ( ), ( )

( ) ( )

ff

rr rfr f

y h x h xL h xy L h x

x x

L h xy L h x

(F25)

Since r n , the above r new state variables cannot characterize the system completely if r n . So additional n-r independent new variables

1 2T n r

n r R have to be introduced:

( )x (F26)


where 1 2( ) ( ) ( ) Tn r(x) x x x are smooth functions of x. The

new set of state variables given by

( )( )

( )x

z T xx

(F27)

can then characterize the system completely if 1 1 1

1 2

1 2

1 1 1

1 2

1 2

n

r r r

n

n

n r n r n r

n

x x x

x x xTx

x x x

x x x

(F28)

is nonsingular on D.


Dynamics in New Coordinates From (F19), (F25) and (F26), the system dynamics in the new coordinates can be written as

1 2

2 3

1

, ,

, ,

r r

r z z

u

x x u u

u

(F29)

where

1

1

1

1

( , )

1( , )

( , )

( , )

, , ( )

, , ( )

( ), ( )

( ), ( )

rz fx T

rz g fx T

x T

u x T

x x L h x

x x L L h x

x f xx

x g xx

(F30)


As seen from (F30), if ( )x in the coordinate transformation can be chosen such that

( ) 0 ( ) 0,ig x g x ix x

(F31)

then, ( , ) 0u (F32)

and the internal dynamics will be

( , ) (F33)

which is independent of u. The transformed system thus becomes

[ ( ) ( )] ( )[ ( )], ( ) ( ) ( )

( , )

c c

c c

c

A B x u xA B x u x x x x

y C

(F34)

where

0 1 0 00 0

, , 1 0 01 0

0 0 0 1

c c cA B C

(F35)


is said to be in the normal form. For SISO nonlinear systems described by (F19), the existence of the coordination transformation (F31) to transform the system into the normal form is always guaranteed (Theorem 13.1 of REF1).

Input-Output Feedback Linearization Control Law

Let us consider the following “input-output feedback linearization” control law

1 1 ,, z

zu x v v

x

(F36)

where v is a new synthesis input. Then (F29) becomes

0 1 0 0 00 0 1 0 0

0 0 0 1 00 0 0 0 1

, , ,

,

1 0 0

uz

z

v

v

y

(F37)


Thus, if we only care about the dynamics from v to the output y, i.e., the -

dynamics, the system is an LTI system with a TF ( ) 1( ) r

Y sV s s

. A control law can

thus be easily synthesized for the new input v such that the output y tracks any smooth desired trajectory ( )dy t . For example, choose v as

( ) ( 1)

1 1 2 1

( ) ( 1)1 1

r rd r d r d r d

r rd r r

v y a y a y a y

y a e a e a e

(F38)

where 1 d de y y y is the output tracking error. Then, from (F37), we have

( ) ( 1)1 0r r

re a e a e (F39) and thus

( ) 0 as e t t (F40) as long as 1

1( ) r rrD s s a s a is Hurwitz. In summary, a control law

given by (F36) and (F38) can be used to achieve output tracking.


Internal Dynamics and Zero Dynamics

The control law (F38) guarantees that the output tracking error e(t) converges to zero exponentially, which in turn guarantees that is bounded since

( ), , , 0re e e as t and the desired trajectory ( )dy t is bounded with bounded derivatives. However, the internal dynamics described by -dynamics of (F37) may not be I/O stable, i.e.

( )1 1 2

( 1)1

,,,

,

rz d r d r du

rz r d

y a y a y

a y

(F42)

may not be BIBO stable ( as the output and , dy as inputs). Thus, an unbounded may result even if and ( )dy t are bounded. In turn, the control input calculated by (F36) and (F38) or (F41) may become infinite and cannot be realized in implementation. Therefore, to be able to implement the “feedback linearization” control law (F41), it is also necessary to verify that the internal dynamics (F42) is BIBO stable.

Zero Dynamics Verification of the BIBO stability of the internal dynamics (F42) is in general difficult. As a starting point, we should first check if certain internal dynamics such as the zero-dynamics of the system is stable or not. Zero dynamics is


defined to be the internal dynamics of the system when the output y(t) is constrained to zero all the time (zero output). To obtain zero dynamics, let ( )( ) ( ) ( ) 0r

d d dy t y t y t and 1 ( ) 0,y t

2 ( ) 0, ,y t ( 1) ( ) 0rr y t in (F42) (since the output y is constrained

to be zero all the time),

0,

0, 0,0,

uz

z

(F43)

or when the internal states are chosen according to (F31),

0,

which depends on only. Stability of the zero-dynamics (F43) can then be studied by using standard nonlinear stability theorems. Definition [Minimum Phase Nonlinear Systems]:

The system is said to be minimum phase if the zero dynamics has an asymptotically stable equilibrium point in the domain of interests.

Remark F1:

In general, the zero-dynamics (F43) being stable does not mean that the internal dynamics (F42) is BIBO stable since (F42) is nonlinear.


Internal Dynamics and Zero Dynamics of LTI Systems

Consider a SISO LTI system with a TF of

10 1

11

( )m m

mn n

n

s sG ss a s a

(F44)

The state–space realization of (6.1.39) is (observable canonical form)

1

2

01

01 0 0

1 zeros0 1 0

0

0 0 10 0 0

1 0 0

n

nm

an m

ax x u

aa

y x

,

1

1

n m

n m

n

x

xx

x

x

(F45)


It is seen that

1 1 1 2 1 1 2

1 1 2 1 1 1 2 2 1 3

12 3

2

1( ) ( 1)

1

1

1

1

1

n m n mn m n m

n m

n m n m

n m

n

y x a x x E x xy a x x a a x x a x x

xE x

x

xd dy y E xdt dt

x

xE x

x

E

1

1 0m n m

n m

xx u

x

(F46)


where iE are some constant matrices. Thus, the system has a relative degree of r=n-m, which is consistent with our usual understanding of the relative degree of a TF G(s). If we use the standard notation in the above section, then

1 1

2 1 1 2

1( 1)

1

1

1

rr n m n m

n m

n mm

n

y xy E x x

xy E x

x

xR

x

(F47)

To obtain the zero-dynamics, set 0 . From (6.1.42),

1 2 0n mx x x (F48) and the input u for the zero output can be obtained from the (n-m)-th equation of (F45) or the last equation of (F46) as


10 0

1 1 1 0 0n mu x (F49)

From (F48) and (F49), the zero-dynamics (F43) described by is given by

1 1

0

1

0

2

0

1

0

0

0 1 0 0 0 1 0 00 0 1 0 0 0 1 0

1 1 0 00 0 0 1 0 0 0 10 0 0 0 0 0 0 0

1 0 0

0 1 0

0 0 1

0 0 0

m m

m

m

u

(F50)

which has a characteristic equation of


11

0 00m m ms s

Thus, the zero-dynamics is asymptotic stable iff the numerator of the TF G(s) is Hurwitz, i.e., the system is strictly minimum phase, and is unstable if the TF G(s) is non-minimum phase. Since internal dynamics for a linear system is still linear, the asymptotic stability of zero-dynamics also implies the BIBO stability of the internal dynamics.


Input/Output Feedback Linearization

It is seen from the above development that the zero-dynamics (F43) and the internal dynamics (F42) depend on the choice of the output or the function h(x), even though we have the same dynamical system characterized by the state x; for example, for an output defined by ( )ay h x , the system may have a relative degree of ar while picking up another output ( )by h x may result in a relative degree of br . Thus, if we can pick up an output ( )y h x such that the resulting system has a relative degree of r n , then, there will be no internal dynamics or zero dynamics and the “feedback linearization” control law (F41) can be applied without worrying about the stability of the resulting internal dynamics. The technique is called feedback linearization as illustrated by the following example.

Example F2: System

31 2 1

2

1

x x x ux uy x

,

32 1( )

0x x

f x

, 1

( )1

g x

Relative Degree: 3

1 2 1 1y x x x u r


Coordinate Transformation:

1 2

( ) 0 0g xx x x

So pick 1 2x x

Then, 1 1

1 2 2

x xx x x

3 3

2 13 3

1 2 2 1

x x u ux x x x

y

Q: Can we use I/O Feedback Linearization Control Law?

32 1 , 0u x x k k

to obtain

, 0k as t


A: No, as the zero dynamics given by

is unstable, which demands infinite control input u to implement control law-- not possible in reality!

Let us try another output:

1 2newy x x

31 2 2 1

2 2 2 22 1 1 1 2 1 13 3 3 (3 1)

new

new

y x x x xy x x x x x x x u

2r no internal dynamics.

We can always apply the I/O Feedback Linearization Control law:

2 21 2 1 1 22

1

1 3 3 ( )3 1 new newd newu x x x k y y k y

x

to obtain: 2 1 0new new newy k y k y

0 asnew new newdy y y t


Nonminimum Phase Systems

Two approaches to Nonminimum Phase Systems:

1. Redefine the output y , for example, ( )new newy h x , such that newy still has a relative degree of r as y, but has “good” internal dynamics, from which one can apply the I/O feedback linearization control law.

2. Try to find a transformation that provides input/state linearization, i.e., find an output ( )y h x such that its relative degree is n as follows.

Input/State Feedback Linearization

Conditions for r=n:

1

12

( )differential of ( )1

( ( )) 0,

( ) 0

, ( ( )) , ,( ) 0

( ( )) 0

g

g f

gnn

ng f dh xh xn

g f

L h x

L L h gh h hL h x gx x x

gL L h

L L h x

(F51)


The first 1n equality conditions are equivalent to the following conditions:

( ( )) 0, 0, , 2k

fad gL h x k n (F52)

where k

fad g is the Lie bracket of f and 1kfad g . The Lie bracket of two vector

field ( ) , ( )n nf x R g x R is defined by:

[ , ] ( ) ( ) ( )fg fad g f g x f x g xx x

(F53)

Notations: 0 ( )fad g g x 1[ , ] ( ) , 1k k

f fad g f ad g x k Note that (F52) is the same as:

2( ) 0, , , nf fdh x g ad g ad g (F54)


Geometric Terminology

For vector fields 1 2( ), ( ), , ( ) on nkf x f x f x D R , let

1 2

1

( ) { ( ), ( ), , ( )}

( ) : ( ) ( ) ( ), ( ) are scalar functions

k

kn

j j jj

x span f x f x f x

f x R f x c x f x c x

be the subspace of nR spanned by the vectors 1( ), , ( )kf x f x at any fixed x D . The collection of all vector spaces ( ) forx x D is called a distribution and referred to by:

1 2{ , , , }kspan f f f The dimension of ( )x at each x D is defined by:

1dim( ( )) [ ( ), , ( )]kx rank f x f x If dim( ( ))x k for all x D , we say that is a nonsingular distribution on D, generated by 1, , .kf f


A distribution is involutive if

1 2 1 2, [ , ]g g g g If is a nonsingular distribution on D, generated by 1, , kf f , then is involutive if and only if

[ , ] , 1 ,i jf f i j k

is said to be completely integrable if for each ox D , there exists a neighborhood of N of ox and n k real-valued smooth functions 1( ),h x ,

( )n kh x such that

( )( ) 0 or 0, 1 , 1j

i j i

h xf x dh f i k j n k

x


FROBENIUS THEOREM

A nonsingular distribution is completely integrable if and only if it is involutive.

Using Frobenius Theorem, we can find the geometric condition for (F54) to be true as:

THEOREM F1 [Theorem 13.2 of REF1]

For ( ) ( )x f x g x u ,

there exists a ( )h x whose relative degree is n (or full state feedback linearizable) if and only if:

A. The matrix 1( ) [ ( ), ( ), , ]nf fx g x ad g x ad g has rank n

B. The distribution 2{ , , , }nf fspan g ad g ad g is involutive.


Example F3: System

32 1 1

( ) , ( )10

x xf x g x

Then, 2 21 113 1 1 3

[ , ]10 0 0f

x xg fad g f g f gx x

211 1 3

( )1 0

xx

has rank of 2 for any x that 113

x

{ ( )}D span g x is always involutive as , 0,g g g .

There exists a ( )h x that has a relative degree of 2.


Example F4: Field-Controlled DC Motor with negligible shaft damping System:

1 1

2 2 1 3

3 1 2

x ax ux bx k cx x

x x x

1

2 1 3

1 2

1( ) , ( ) 0

0

axf x bx k cx x g x

x x

where 1 :x field current

2 :x armature current 3 :x angular velocity Case 1: Speed Control

3 1 2y x y x x 1 2 1 2y x x x x 1 2 1 2 1 3( ) ( )ax u x x bx k cx x

2

1 2 1 2 1 1 3 2( )( ) xx

ax x bx x kx cx x x u

3

22 in : 0or D x R x


I/O Linearization Law:

1 1( )( ) ( )

u x v y vx x

Zero Dynamics: 1( ) ( ) ( ) 0 , ( )( )

y t y t y t t u xx

3 10 and 0,x x t Thus, the zero-dynamics are:

2 2x bx k

which is a stable dynamics w.r.t. the equilibrium point at 2ekxb

.

Case 2: Full-State Feedback Linearization

3 1 3

2 1 2

0 0 1, 0

0 0f

a afad g f g g cx b cx cxx

x x x


12

2 1 3 3 1

1 2 2 1

2

3

2

0 0 0 0 0, 0 0

0 0 0

( )( )

f f f

ax aad g f ad g c bx k cx x cx b cx ad g

x x x x

aa b cx

b a x k

2

23 3

2 2

1( ) 0 ( )

0 ( )f f

a ax g ad g ad g cx a b cx

x b a x k

2 3( ) ( 2 )x c k bx x

( )x has rank 3 for 2 3, 02kx xb

The distribution , fspan g ad g is involutive as


0 0 0 1 0( )

, 0 0 0 00 0 0 0

ff

ad gg ad g g c

x

Thus, the conditions of Theorem 13.2 are satisfied in the domain

32 3: , 0

2okD x R x xb

Suppose that the control goal is to stabilize the system to the equilibrium point:

0T

e okxb

where o is the desired speed. Let us find ( )h x that has a relative degree of 3, i.e.,

2( ) ( )

0 , 0 , 0f fL h L hh g g gx x x

with ( ) 0eh x so that the I/O linearization for ( )y h x can be applied to stabilize the system for zero output or at ex . For this purpose, note

2 31

0 0 ( , )h hg h h x xx x


Then, 2 3 2 3

2 1 3 1 22 3

( , ) ( , )( )fh x x h x xL h bx k c x x x x

x x

Note 2 3 2 3

3 21 2 3

( ) ( ) ( , ) ( , )0 0 0f fL h L h h x x h x xg cx xx x x x

which is satisfied if ( )h x is chosen as

2 22 3 1 2 3 2( , )h x x c x cx c

where 1 2andc c are any constant. Let 2 21 2 2 31and ( ) ( )e ec c x c x so that

( ) 0eh x . For this ( ) :h x

2 2 1 3 3 1 2(2 )( ) (2 )fL h x bx k cx x cx x x 2 22 ( )x k bx

1

22 2 1 3

1 2

( )[0 2 ( 2 ) 0]f

f

axL h

L h f k bx bx k cx xx

x x

2 2 1 32 ( 2 )( )k bx bx k cx x


2 2

3 21

( ) ( )2 ( 2 ) 0f fL h L h

g cx k bxx x

on oD .

Indeed, if we use the coordinate transformation:

2 2 2 22 3 2 3

2 2

2 2 1 3

( ) ( )( ) 2 ( )

2 ( 2 )( )

e ex cx x c xz T x x k bx

k bx bx k cx x

then, in the new coordinates:

1 2

2 33 2

3 ( )f g f

z zz zz L h L L h u


MIMO Nonlinear Systems Consider a “square” system

1

1

1

( ) ( )

( ) ( ) , ( ) ( ), , ( )

[ ( ), , ( )]

m

i ii

m

Tm

x f x g x u

f x G x u G x g x g x

y h x h x

(F55)

Let :kr relative degree of the output ( )kh x

to be 1 0( )k

i

rg f kL L h x for some i

and 0, , 0 2( )

i

jg kf kL i j rL h x

Define: 1

11

1

1 1( ( )), ( ) ,

( ( ))mm

m

r

rrf

r m

rrf m mm

r

d yL h x udt

y R x uL h x ud y

dt

(F56)


( ) ( )ry x J x u (F57) Let

1 1

1

1

1 11 1

1 1

, ,( ) ( )

( )

, ,( ) ( )

m

m m

m

r rg gf f

r rg gf m f m

L LL h L h

J x

L LL h L h

(F58)

which is called the “decoupling” matrix, also the “invertability” matrix. If ( )J x is nonsingular in an area of interest, then, we have a linear I/O relation between y and v when one uses the following I/O Feedback Linearization Law:

1( ) ( ( ))u J x v x (F59)

ry v


Internal Dynamics Let

1

m

T kk

r r n

We can transform the system into a “normal” form as follows. Define:

1 1 1( ), , 1i i ii j jz h x z z j r

1 2

1 1 2 21 1 1[ , , , , , , , , , ] T

m

rm m Tr r rz z z z z z z R

( ) ,Tn rx R state variables of internal dynamics Then,

1 2

1

1

( )( , ) ( ),

( )( ) ( ) ( , ) ( )

( , ) ( , )

i i

i k

i i

mr ri

r f i g kf ik

z zxz f x

xxz L h L uL h P z G x

xz P z u

(F60)

Note that unlike SISO case, we cannot guarantee that one can always pick up

( )x such that ( , ) 0P z .


In order to be able to apply the I/O Feedback Control law, we have to make sure that the resulting internal dynamics is stable in the sense that is bounded, which is normally difficult. So as a first cut design, let us look at the Zero Dynamics instead:

Zero Dynamics The control input for the zero dynamics is:

1( ) ( )u J x x , ( , )x x z

1( ,0) ( ,0) ( ( ,0)) ( ( ,0))P J x x (F61)

Dynamic Expansion If ( )J x is singular, we may be able to add some dynamic compensators to extend the state to provide a nonsingular J as shown in the following example. Example F5:

1 1

2 1

2

(cos )(sin )

x ux u

u

, 1

2

xx x

, 1 1

2 2

y xy

y x


Then, 1 1 1 1

2 2 1 2

(cos ) 1(sin ) 1

y x u ry x u r

cos 0

( )sin 0

J x

which is singular and we cannot use the I/O feedback linearization law.

Let us try to extend system states by letting

3 1 3 1 3,x u x u u Then,

1 3

2 3

2

3 3

(cos )(sin )

x xx x

ux u

, 1 1

2 2

y xy x

Thus, 1 1 3(cos )y x x

1 3 3(sin ) (cos )y x x 2 3 3 1(sin ) (cos ) 2u x u r


2 3 2 3 2(cos ) (sin ) 2y x u u r and

3 21

3 32

( )

(sin ) cos(cos ) sin

J x

x uyx uy

Note 2 2

3 3 3( ) (sin ) (cos )J x x x x J(x) is nonsingular for any x that 3 0x which indicates that we can apply the I/O feedback control law to linearize the system since 1 2 4 .r r n

2 1

3

( )u

J x vu

1 1

2 2

y vy v

Remark: Instead of synthesizing a control law for the physical input 1u directly, the above dynamic expansion treats its changing rate as the fictitious control input instead. The resulting control action is thus smoother.


Q: Is it good to cancel nonlinearity? Keep in mind that canceling all nonlinearities may not be a good idea. We should try to preserve the “good” nonlinearities as illustrated below. Example F6:

3 , 0, 0x ax bx u a b

For stabilization purpose, the nonlinearity 3bx is helpful and should not be cancelled.

Limitations of Feedback Linearization Aside from the need for precise knowledge of system nonlinearities and parameters, the basic philosophy of feedback linearization that is to cancel all the nonlinear terms of the system might not be a good idea in some applications at all. The motivation to do so has been pure mathematically driven; linearize the system to make it more tractable and to use the relatively well-developed linear control theory. From a performance viewpoint, a nonlinear term could be “good” or “bad” and the decision whether we should use feedback to cancel a nonlinear term is, in reality, problem dependent as in the above example.


Stabilization via Feedback Linearization

Consider the following partially feedback linearizable system:

1

( , ) , ,

( ) [ ( )]of x y

A B x u x

, (F62)

where (A, B) is controllable and ( ) 0x . Assume (0,0) 0,of i.e., the origin is an equilibrium point. We want to stabilize the system around the origin. I/O Feedback Linearization Law:

1( ) ( ) ,u x x v A Bvv K

(F63)

where K is chosen that A BK is Hurwitz. CL System:

( , )

( )of

A Bk

(F64)


Lemma F1 [Lemma 13.1 of REF1]: The origin of (F64) is locally asymptotic stable if the origin of ( ,0)of is asymptotic stable (i.e., the system is of minimum phase).

Proof of Lemma F1:

By converse Lyapunov Theorem 4.16, as the origin of zero dynamics is asymptotic stable, 1( ) 0V such that

13

( ) ( ,0)oV f

where 3 is a class function. Let 0TP P that

( ) ( )TP A BK A BK P I

Now consider

1( , ) ( ) , 0TV V c P c

which is positive definite (why?). Then,

1

21 1

( , )2

( ,0) ( , ) ( ,0)2

T To T

o o o T

V cV f P PP

V V cf f fP


Within a sufficient small neighborhood of the origin of ( , ) 0

1 ( , ) ( ,0)o oV f f c

3max

( )2 ( )

cV cP

which is negative definite for

max2 ( )c c

P

This proves that the origin of ( , ) system (F64) is locally asymptotically stable. #


Lemma F2 [Lemma 13.2 of REF1]

The origin is globally asymptotic stable if the system ( , )of is input-to-state stable.

Note:

Input-to-state stability of ( , )of does not directly follow from globally exponential stability of the origin of zero dynamics ( ,0)of , unless

( , )of is globally Lipschitz in ( , ) , which is sometimes referred to as linear growth condition.

Example F6:

2

,v v k

The origin of zero dynamics globally exponentially stable.

But since 2( , )of is not globally Lipschitz, we cannot conclude that interval dynamics is input-to-state stable. In fact, it is true that internal dynamics is not input-to-state stable.

We can obtain the region of attraction of the above system as follows. Let , Then,

22 (1 )kk


Its origin 0 is locally asymptotically stable but not globally asymptotically stable as 1 k is another equilibrium point. As 0, 0 1 k , the exact region of attraction of the origin is given by : 1 k , which expands as k increases. In fact, by choosing k large enough, we can include any compact set in the region of attraction. Thus, the feedback control law v k achieves semi-global stabilization of the original nonlinear system.

Peaking Phenomenon of Linear System and Nonlinear Growth of Internal Dynamics

If the origin of the zero-dynamics ( ,0)f is globally asymptotically stable, one might think that the original system (F62) can be globally stabilized, or at least semi-globally stabilized by designing the linear feedback control v k to assign the eigenvalues of A BK far to the left in the complex plane so that decays to zero arbitrarily fast to make the actual internal dynamics ( , )f approaching the well-behaved zero-dynamics

( ,0)f quickly. It may even appear that this strategy is the one used to achieve semi-global stabilization in Example F6. Unfortunately, the following example shows that why such strategy may fail due to the so-called “peaking phenomenon” of the linear system and the possibility of “finite-escape time” of the nonlinear internal dynamics.


Example F7: System

32

1 2

2

1 (1 )2

v

which has a globally asymptotically stable zero dynamic of 312

.

I/O Feedback Control:

2 21 22 , [ , 2 ]v k k K K k k

2

0 12

A BKk k

has two identical poles at 1,2 k

( )2

(1 )( ,0)

(1 )

kt ktA BK t

kt kt

kt e tet e

k t e kt e

which converges to zero arbitrarily fast as k . However, notice that the (2, 1) element of the state transient matrix contains a quadratic function of k


and it can be shown that the absolute value of this element, 2 ktk t e , reaches

a maximum value of k e at 1tk

. This means that while can be made to

decay to zero arbitrarily fast by choosing k large, its transient behavior may exhibit a peak of the order of k for certain initial conditions (in this case,

2 ( )t peaks for non-zero initial condition of 1(0)). Such a phenomenon is known as “the peaking phenomenon”. The interaction of the peaking with the nonlinear growth of the internal dynamics could destabilize the system as illustrated below. For example, for the initial condition of 1(0) , (0) 1,o

2 (0) 0, we have, 2

2 2,1( ) ( ,0) ktt t k t e

and the internal dynamics become

2 31 (1 )2

ktk t e

During the peaking period, the coefficient of 3 is positive, causing ( )t to grow in a rate more than exponential. Eventually, the coefficient of 3 will become negative, but that may not happen soon enough due to the finite escape time of the nonlinear internal dynamics. Indeed, the exact solution

2 22

2( )1 [ (1 ) 1] ( )

o okt

o

tt kt e t


shows that if 2 1o , the above solution will have a finite escape time if k is chosen large enough (as the denominator ( )t is positive at 0, (0) 1t , and for large enough k , ( )t become zero after some finite time t).

0 0 .2 0 .4 0 .6 0 .8 1-1 .5

-1

-0 .5

0

0 .5

1

1 .52( ) 1 [ (1 ) 1], 2, 10kto ot t kt e k

0 0 .2 0 .4 0 .6 0 .8 1-1 .5

-1

-0 .5

0

0 .5

1

1 .52( ) 1 [ (1 ) 1], 2, 10kto ot t kt e k

References [REF1] Khalil, H. K. (2002), Nonlinear Systems, Third edition, Prentice-Hall.

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