On the Generalized Normal Form for MIMO-Systems

On the Generalized Normal Form for MIMO-Systems

Hakan Fortell

Department of Electrical Engineering Linkoping University

S-581 83 Linkoping, Sweden Phone: +46 13 284028 E-mail: hafor@isy.liu.se

August 3, 1995

Abstract

In this report will we investigate how a generalized normal form can be calculated for MIMO-systems. The method given is constructive and it will be shown how the calculation of the normal form of ane polynomial state space systems follow as a special case. Also exact input-output linearization and noninteracting control using the generalized normal form will be discussed.

Keywords: Normal Form, Grobner Bases, Dierential Algebra, Exact Linearization

1 Introduction

In the report 8] it was shown how a normal form can be calculated for dynamic SISO-systems as it is dened in, e.g., 3, 2]. This normal form was shown to be a generalization of the normal form as it is dened in 10]. The purpose of this report is to generalize this normal form to MIMO-systems and to study how it can be used for control purposes. In particular will exact linearization and noninteracting control be studied. In the next section will we give an overview of the concepts from algebra that we need. Then in section 3 it will be shown how the generalized normal form can be calculated. In section 4 we discuss exact linearization and noninteracting control and section 5 contains an electrical circuit example. Finally in section 6 we give some concluding remarks.

2 Basic Algebraic Concepts

In this section we present some of the most basic concepts from commutative and

dierential algebra that will be used in later sections. The purpose is basically

to display the notation.

2.1 Commutative Algebra

Most of the theory in this section is collected from 1] and 6] where also further references to commutative algebra can be found. We begin by making a few necessary denitions.

Denition 2.1

A commutative ring consists of a set R and two binary operations \ " and \+"

dened on R and such that:

1. ( a + b ) + c = a + ( b + c ) and ( a b ) c = a ( b c ) for all abc

²

R . 2. a + b = b + a and a b = b a for all ab

²

R .

3. a ( b + c ) = a b + a c for all abc

²

R .

4. There are 0  1

²

R such that a + 0 = a 1 = a for all a

²

R . 5. Given a

²

R there is b

²

R such that a + b = 0.

Rings can contain dierent types of elements.

Denition 2.2

A zero divisor in a ring R is a nonzero element x

²

R such that

9

y

²

R

^nf

0

^g

with xy = 0 (1) We also need the following.

Denition 2.3

A eld k is a commutative ring such that given a

²

k there is c

²

k such that a c = 1.

The commutative ring that we will mostly work with here is k  x

¹

:::x

_n

] which denotes the set of all polynomials in the variables x

¹

:::x

_n

with coecients taken from the eld k . Usually we will assume that k is the real numbers.

Denition 2.4

A subset I of a commutative ring R is an ideal if it satises 1. 0

²

I .

2. If ab

²

I then a + b

²

I . 3. If a

²

I and b

²

R then b a

²

I .

If we now set f

¹

:::f

_n^

to be polynomials in a ring k  x

¹

:::x

_n

] and introduce the notation

h

f

¹

:::f

n^i

=

(

n X

i⁼¹

p

i

f

i

: p

i²

k  x

¹

:::x

n

]  i = 1 ::: n 

)

(2)

we can note that

^h

f

¹

:::f

n^i

is an ideal and it is called the ideal generated by

f

¹

:::f

^n

. One can divide ideals into classes with dierent properties. We shall

mention one such property here.

Denition 2.5

An ideal I is a prime ideal of the commutative ring R prime if for all xy

²

R it holds that

xy

²

I

⁾

x

²

I or y

²

I (3)

A property that will be needed in the next section is to have some generalization of linear dependence of polynomials. To nd such a generalization we rst need some more denitions.

Denition 2.6

If the eld k is a subset of the eld K then K is said to be an extension eld of k . This is denoted by K=k .

Denition 2.7

Let K be an extension eld of k and suppose that 

^

K . The smallest eld containing k

^

 is then called the eld obtained by adjoining  to k and is denoted k (  ). Furthermore,  is said to be algebraic over k if there exists a nonzero polynomial f

²

k  x ] such that f (  ) = 0. If  is not algebraic it is said to be transcendental over k .

A denition can also be made in a similar manner for several elements in an extension eld.

Denition 2.8

Let f

¹

:::f

_m

be elements in an extension eld K of k . The f

_i

are said to be algebraically dependent over k if there exists a nonzero P

²

k  x

¹

:::x

_m

] such that P ( f

¹

:::f

m

) = 0. Otherwise the f

i

are said to be algebraically independent.

Since algebraic dependence can be viewed as a generalization of linear depen- dence in linear algebra we can go on to dene a basis for these extension elds.

Denition 2.9

Let K be an extension eld of k and let U be a subset of K such that all elements of U are algebraically independent. The maximum number of elements that such a U can contain is called the transcendence degree of K=k and is denoted by trdeg K=k . The elements of such a U is called the transcendence basis for K=k . An important consequence of this denition is then the theorem below.

Theorem 2.1

If k is a eld and the variables x

i

 i = 1 :::n are algebraically independent over k then trdeg k ( x

¹

:::x

n

) =k = n .

Proof.

We only have to note that the x

i

 i = 1 :::n is a transcendence basis for

k ( x

¹

:::x

n

) =k .

²

The following corollary will also be useful.

Corollary 2.1 The elements y

¹

:::y

N ²

k ( x

¹

:::x

n

) are algebraically de-

pendent if N > n .

Proof.

Otherwise y

¹

:::y

N

would be a transcendence basis for k ( x

¹

:::x

n

) =k .

2

Another property which will be utilized is given by the following theorem. The proof can be found in 12].

Theorem 2.2

Let x

¹

:::x

n

y

¹

:::y

m

be elements of a eld extension of k and let n < m . Now suppose that each y

i

is algebraic over the eld k ( x

¹

:::x

n

). Then the y

i

are algebraic over k .

We also give a theorem which in some sense can be interpreted as connecting algebraic and linear dependence.

Theorem 2.3

Suppose k is a eld of characteristic zero and that f

¹

:::f

N ²

k ( x

¹

:::x

n

).

Then if



N > n , f

¹

:::f

N

are algebraically dependent.



N = n , f

¹

:::f

N

are algebraically dependent i the Jacobian matrix J ( f ) =



@f

i

@x

j



ij

(4)

is identically singular.



N < n , f

¹

:::f

N

are algebraically independent i the Jacobian matrix (4) has rank N .

Another useful theorem concerning algebraic dependence is given by the follow- ing theorem which can be found in 15].

Theorem 2.4

Let K=k be a eld extension and let x

¹

:::x

n

y

¹

:::y

m

z

²

K . Now suppose that each of the y

i

are algebraically dependent on x

¹

:::x

n

over k and that z is algebraically dependent on y

¹

:::y

m

over k . Then is z algebraically dependent on x

¹

:::x

n

over k .

A property that we will use is the dimension of an ideal. First of all we give the denition.

Denition 2.10

If I is a prime ideal in a commutative ring R then is the dimension of I equal to the maximum length r of all strictly ascending chains of prime ideals in R given as

I = I

^0

I

^{1 }

I

r

(5)

Other equivalent denitions of dimension can also be given. Intuitively one can

think of the dimension of an ideal as the dimension of its solution space. This

means that if a set of polynomial equations has a nite number of solutions the

corresponding ideal has dimension 0. A theorem which can be used for practical

computations of the dimension is given below.

Theorem 2.5

The dimension r of a prime ideal I

^

k  x

¹

:::x

_n

] is the maximal number such that there exists i

¹

:::i

r

with

I

^\

k  x

¹

:::x

_n

] = 0 (6) We will in the next section see how the intersection (6) can be calculated. With such a tool the calculation of dimension becomes quite simple.

Ideals can also be used for dening equivalence relations on rings.

Denition 2.11

Let

^I

be an ideal in the ring R . The quotient ring R=

^I

is dened as the ring R with the equivalence relation such that for xy

²

R it holds that x

y

^,

x

^;

y

^2I

.

Denition 2.12

An integral domain is a ring without zero divisors.

There is in fact an important connection between integral domains and prime ideals.

Theorem 2.6

The quotient ring k  x

¹

:::x

n

] =

^I

is an integral domain i

^I

is a prime ideal in k  x

¹

:::x

n

].

2.2 Grobner bases

The reason why we introduce the concept of Grobner bases is that we wish to have some way of eliminating variables from a set of polynomial equations. To be able to choose which variables to eliminate we must introduce an ordering of variables.

Denition 2.13

Let x

^2

k  x

¹

:::x

n

] with  =

^f



¹

:::

n^g2Nn

denote x

^11

x

^_nⁿ

. A term ordering < is an ordering on

^Nn

such that for all  

^2Nn

0 <  and  <

⁾

 + < + (7) There are of course several ways of dening these term orderings but we will only consider one of them.

Denition 2.14

The lexicographic term ordering is dened by

 <

^{() 9}

j : 

j

<

j



⁸

i < j : 

i

<

i

(8) With a term ordering will now the dierent terms in a polynomial be ordered.

In particular will one of them be ordered rst.

Denition 2.15

If f =

^P

c



x

^2

k  x

¹

:::x

n

] then the degree of f is dened as

deg f = max

^f

: c

 ⁶

= 0

^g

(9)

The leading term of f is

LT f = c

^degf

x

^degf

(10)

and correspondingly the leading monomial of f is LM f = x

^degf

and the leading coecient is LC f = c

^degf

.

This denition can now be extended to cover ideals.

Denition 2.16

Let

^I

be an ideal in k  x

¹

:::x

n

]. Then

LM

^I

=

^h

LM f : f

^2Ii

deg

^I

=

^f

deg f : f

^2Ig

⁽¹¹⁾

Using this last denition we can now go on to dene Grobner bases.

Denition 2.17

A set G

^{ I}

for an ideal

^I

is a Grobner base with respect to a given term ordering i LM

^I

=

^h

LM G

ⁱ

.

At a rst glance it not at all clear how the concept of Grobner bases can be useful but it can be shown to have several nice properties. One of these properties is that

h

G

ⁱ

=

^I

(12)

which says that the set of solutions to the generating set of

^I

is the same for the equations described by the Grobner base. To be able to describe the algorithm we need to introduce a few more concepts.

Denition 2.18

Let fg

²

k  x

¹

:::x

n

]. f is said to be reduced w.r.t. g if there is no term in f that is divisible by the leading term of g . A subset of k  x

¹

:::x

n

] in which all elements are reduced w.r.t. each other is said to be auto-reduced

If some polynomials are not reduced we can make them reduced.

Theorem 2.7

Let F =

^f

f

¹

:::f

m^g

k  x

¹

:::x

n

] and p

²

k  x

¹

:::x

n

]. Then it is possible to nd polynomials g

¹

:::g

m

r

²

k  x

¹

:::x

n

] such that

q = p

^;Xm

i⁼¹

g

i

f

i

(13)

and q is reduced w.r.t. all the f

i

.

The denition below is related to this theorem.

Denition 2.19

If all q which satises the conditions of Theorem 2.7 are equal then p reduces to q modulo F which we write p

^;!F

q .

The algorithm for calculating Grobner bases uses the so called S-polynomials.

Denition 2.20

Let f

¹

f

²²

k  x

¹

:::x

_n

]. Then the S-polynomial of f

¹

and f

²

is

S ( f

¹

f

²

) = h

¹

f

^1;

h

²

f

²

(14) where

h

¹

= LC f

²

^lcm(LM f

¹

 LM f

²

)

LM f

¹

 h

²

= LC f

¹

^lcm(LM f

¹

 LM f

²

)

LM f

²

⁽¹⁵⁾

and lcm(  ) denotes the least common monomial.

The Grobner base algorithm can now be given as follows given some set of polynomials F .

Algorithm 2.1

1. Make F auto-reduced.

2. Add to F the S-polynomial of two elements in F .

3. If all S-polynomials reduce to zero modulo F then F is a Grobner base.

Otherwise return to step 1.

The following theorem ensures that the result of this algorithm is a Grobner base.

Theorem 2.8

The set G =

^f

g

¹

:::g

m^g

is a Grobner base for

^h

G

ⁱ

i

S ( g

i

g

j

)

^;!G

0 (16) A property of Grobner bases that we have mentioned earlier is that we can eliminate variables from a set of polynomial equations. How this elimination is performed is given in the following theorem.

Theorem 2.9

Let

^I

be an ideal in k  x

¹

:::x

n

] and partition x

¹

:::x

n

into two disjoint sets A and B . Then if G is a Grobner base for

^I

with a lexicographic term ordering where A < B then k  A ]

^\

G is a Grobner base for k  A ]

^\I

w.r.t. the term ordering given by < on k  A ].

What this theorem says is that with a proper choice of term ordering we can

nd out if there are elements in

^I

which are polynomial in the variables A only.

If this is the case we also get a generating set for the corresponding ideal. We give a simple example.

Example 2.1

Consider the following set if polynomial equations x

²

+ y + z = 1 x + y

²

+ z = 1

x + y + z

²

= 1 (17)

Calculating a Grobner base for the polynomials



x

²

+ y + z

^;

1  x + y

²

+ z

^;

1  x + y + z

^2;

1

^

(18)

with the term ordering x > y > z gives the result



x + y + z

^2;

1  y

²

+ z

^;

y

^;

z

²

 2 yz

²

+ z

^4;

z

²

 z

^6;

4 z

⁴

+ 4 z

^3;

z

^2

(19) From this it is clear that

z

^6;

4 z

⁴

+ 4 z

^3;

z

^22h

x

²

+ y + z

^;

1  x + y

²

+ z

^;

1  x + y + z

^2;

1

ⁱ

(20) and we have eliminated x and y from the set of equations (17).

²

This example indicates that Grobner bases can be interpreted as a generalization of Gaussian elimination. A theorem which has an obvious interpretation in terms of equation solving is given below.

Theorem 2.10

Suppose that the ideal I

^

k  x

¹

:::x

n

] is prime and has dimension 0. Then will the Grobner base, G , of I with lexicographic term ordering x

¹

> > x

n

consist of n elements with the structure

G =

^f

g

¹

( x

¹

:::x

n

) g

²

( x

²

:::x

n

) :::g

n

( x

n

)

^g

(21) where g

_i

( )

²

k  x

¹

:::x

_n

] for i = 1 :::n .

The proof can be found in 9]. It is easily seen that the ideal in Example 2.1 does not fulll the conditions of Theorem 2.10. However, it is shown in 9] that when the conditions of Theorem 2.10 are fullled will the structure of G \generically"

be G =

^f

x

^1;

p

¹

( x

n

) ::: x

n^;1;

p

n^;1

( x

n

)  p

n

( x

n

)

^g

(22) When this is the case the ideal is said to be in generic position.

The algorithm for calculating Grobner bases has been implemented in every major computer algebra language such as

Maple

and

Mathematica

.

2.3 Dierential algebra

The theory presented in this section is gathered from 4] and the proofs can be found either there or in the books 13, 12].

The main dierence between dierential and commutative algebra is that we add dierentiation w.r.t. time, d=dt , to the set of allowed operations.

Denition 2.21

A dierential eld k is a eld satisfying the conditions

8

a

²

k da dt ^{= _} a

²

k (23)

8

ab

²

k ddt ⁽ a + b ) = _ a + _ b (24)

8

ab

²

k ddt ⁽ ab ) = _ ab + a b ^_ (25)

Here and in the sequel will the \dot" notation be used instead of d=dt to de-

note dierentiation with respect to time. We also make a few more denitions

regarding dierential elds which are the dierential variants of some of the

commutative denitions in the previous section.

Denition 2.22

If the di erential eld k is a subset of the di erential eld K then K is said to be a dierential extension eld of k . This is denoted by K=k . If K is a di erential extension eld of k and 

^

K then the smallest di erential eld containing k

^

 is called the di erential eld obtained by adjoining  to k and is denoted k

^h



ⁱ

.

We now consider a denition of dynamics based on dierential algebra.

Denition 2.23

A dynamic system is a nitely generated di erential algebraic extension D=k

^h

u

ⁱ

where u =

^f

u

¹

:::u

_m^g

can be viewed as inputs and the outputs y is chosen as some nite set y =

^f

y

¹

:::y

_p^g

in D .

Using this denition a more general form of state space description than (28) can be made.

Theorem 2.11

Let D=k

^h

u

ⁱ

be a dynamic system according to denition 2.23. Then there exists a generalized state x =

^f

x

¹

:::x

n^g

for some nite integer n and polynomials A

j

( ) and B

i

( ) such that

A

j



x _

j

xu u:::u _

^(j)

= 0  j = 1 :::n (26) B

_i^

y

_i

xu u:::u _

^(i)

= 0  i = 1 :::p (27) A theorem which is important when considering dierentially algebraic exten- sions is given below.

Theorem 2.12

Let L=K be a nitely generated di erential algebraic extension such that K contains non constant elements. Then there exists an element 

²

L such that L = K

^h



ⁱ

. The element  is called a dierentially primitive element.

3 The Generalized Normal Form for MIMO Sys- tems

We will in this section show how a normal form can be calculated for MIMO- systems. The results will be based on the normal form for SISO-systems as described in 8].

3.1 The Normal Form of Ane State Space Systems

Here will the normal form for ane state space descriptions be given. The presentation follows that of 10]. We consider a system with m inputs and m outputs

x _ = f ( x ) +

^Xm

i⁼¹

g

i

( x ) u

i

y

¹

= h

¹

( x ) y

m

= ... h

m

( x )

(28)

where x

^2Rn

. For each output we dene a constant r

i

as

L

gj

L

_kf

h

i

( x

⁰

)

^

0  j = 1 :::m  k = 0 :::r

i^;

2

L

gj

L

^r_f^i;1

h

i

( x

⁰

)

^6

0  for some j

^2f

1 :::m

^g

⁽²⁹⁾ for some x

⁰

belonging to the state space. The system (28) is then said to have vector relative degree

^f

r

¹

:::r

m^g

at a point x

⁰

if the matrix

A ( x ) =

0

B

B

B

B

B

B

@

L

g¹

L

^r_f^1;1

h

¹

( x ) L

gm

L

^r_f^1;1

h

¹

( x ) L

_g¹

L

^r_f^2;1

h

²

( x ) L

_gm

L

^r_f^2;1

h

²

( x )

... ... ...

L

g¹

L

^r_f^m;1

h

m

( x ) L

gm

L

^r_f^m;1

h

m

( x )

1

C

C

C

C

C

C

A

(30)

is nonsingular at x

⁰

. Using the notation dh ( x ) =



@h ( x )

@x

¹

 @h ⁽ x )

@x

²

::: @h ⁽ x )

@x

n



(31) the following lemma is shown to hold in 10].

Lemma 3.1 The row vectors

dh

¹

( x

⁰

) dL

f

h

¹

( x

⁰

) :::dL

^r_f^1;1

h

¹

( x

⁰

) dh

²

( x

⁰

) dL

f

h

²

( x

⁰

) :::dL

^r_f^2;1

h

²

( x

⁰

) dh

m

( x

⁰

) dL

f

h

m

( x

⁰

) ... :::dL

^r_f^m;1

h

m

( x

⁰

)

(32)

are linearly independent.

It can also be shown that r = r

¹

+ + r

m^

n so that we can put



_ij

( x ) = L

^j_f^;1

h

i

( x )  j = 1 :::r

i

 i = 1 :::m (33) When r < n we can nd functions so that a locally invertible variable trans- formation can be found. One restriction here is, however, that these function can only be chosen so that L

gj



i

( x ) = 0

⁸

i = 1 :::m j = r + 1 :::n i

f

g

¹

( x ) :::g

m

( x )

^g

is involutive. Dividing the state space into

ⁱ

=

0

B

B

B

B

B

B

@

¹ⁱ

²ⁱ

_iri

...

^;1

1

C

C

C

C

C

C

A

=

0

B

B

B

B

B

B

@



ⁱ¹

( x ) 

ⁱ²

( x ) 

_iri^;1

... ( x )

1

C

C

C

C

C

C

A

i = 1 :::m  =

0

B

B

B

B

B

B

@

¹

²

...

^m

1

C

C

C

C

C

C

A

 =

0

B

B

B

B

B

B

@



¹



²

 ...

^m

1

C

C

C

C

C

C

A

=

0

B

B

B

B

B

B

@



¹_r¹

( x ) 

²_r²

( x ) 

_mrm

... ( x )

1

C

C

C

C

C

C

A

 =

0

B

B

B

B

B

B

@



¹



²



n

...

^;r

1

C

C

C

C

C

C

A

=

0

B

B

B

B

B

B

@



r⁺¹

( x ) 

r⁺²

( x ) 

n

... ( x )

1

C

C

C

C

C

C

A

(34)

the resulting normal form becomes _ = M



+ M





 _ = b (  ) + A (

^;1

(  )) u

 _ = q (  ) + p (  ) u y = H



+ H





(35)

where b ( ) is a vector with the elements

b

i

(  ) = L

^r_fⁱ

h

i

(

^;1

(  ))  i = 1 :::m (36) and q

i

( ), p

ij

( ) are some nonlinear functions. The constant matrices in (35) are

M

_

=

0

B

B

B

B

B

B

@

M

_¹

0 0

0 M

_²

0

... ... ... ...

0 0 M

_^m

1

C

C

C

C

C

C

A

with M

_i

=

0

B

B

B

B

B

B

@

0 1 0

... ... ... ...

0 0 1

0 0 0

1

C

C

C

C

C

C

A

2Rri^ri

, M



= (0 ::: 0  1)

^T

and H



, H



chosen in the obvious manner. (37)

3.2 A generalization of the normal form

Consider a dynamics D=k

^h

u

ⁱ

which has dierential transcendence degree m and trdeg D=k

^h

u

ⁱ

= n . Assuming that a choice of p outputs has been made we get a generalized state space representation

A

^1

x _

¹

xu

¹

 u _

¹

:::u

⁽m^1m)



= 0 A

n

...



x _

n

xu

¹

 u _

¹

:::u

⁽m^nm)



= 0 B

^1

y

¹

xu

¹

 u _

¹

:::u

⁽m^1m)



= 0 B

p

...



y

p

xu

¹

 u _

¹

:::u

⁽m^pm)



= 0

(38)

Since we want to be able to have a unique description of the outputs we assume that the condition

@B

i

( )

@y

i ⁶

= 0 (39)

is fullled for all i

^{2 f}

1 :::p

^g

and that the y

i

are algebraically independent

over k ( x

¹

:::x

n

u

¹

 u _

¹

:::u

⁽_m^max)

) (40)

The basic tool for calculating the normal form will as in 8] be the generalized Lie

derivative. Using this Lie derivative we calculate constants r

j

 j = 1 :::p in

the following manner. We suppose that L

ⁱ_A^j

B

j

( ) consist of only one polynomial for all i

^2f

0 :::r

j^g

and j

^2f

1 :::p

^g

where for each j the r

j

satises

@L

ⁱ_A^j

B

_j

( )

@u

⁽_l^{) }

⁰ 

⁸



^

0  i

j

= 1 :::r

j^;

1  l = 1 :::m (41) and @L

^r_A^j

B

j

( )

@u

⁽_l^{) 6}

^{= 0}  for some 

^

0 and l

^2f

1 :::m

^g

(42) The following denition is now made.

Denition 3.1

Suppose that the constants r

¹

:::r

_p

with each r

_i ^

0 has been calculated for the system (38) as above. Then is (38) said to have generalized vector relative degree if

h

L

^r_A¹

B

¹

( ) :::L

^r_A^p

B

p

( )

^i\

k  x

¹

:::x

n

y

^(1r1)

:::y

_p^(rp)

] = 0 (43) In order to justify the name generalized vector relative degree we now show that the usual denition of vector relative degree follow as a special case.

Theorem 3.1

Suppose that we have a system (28) with f ( x ), g

i

( x ) and h

i

( x ) polynomials in the ring k  x

¹

:::x

n

] for all i

^2f

1 :::m

^g

. Then this system has vector relative degree ( r

¹

:::r

m

) i it has generalized vector relative degree ( r

¹

:::r

m

). The Lie derivatives L

ⁱ_f^j

h

j

( x )  i

j

= 0 :::r

j ^;

1  j = 1 :::m are given as the solutions to

L

ⁱ_A^j

( y

j^;

h

j

( x )) = 0 (44) w.r.t. y

_j^(ij)

. Furthermore, L

^r_f^j

h

j

( x ) +

^Pm_l⁼¹

L

gl

L

^r_f^j;1

h

j

( x ) u

l

is the solution to

L

^r_A^j

( y

j^;

h

j

( x )) = 0 (45) w.r.t. y

^(rj)

for each j = 1 :::m .

Proof.

The calculation of the Lie derivatives (44) and (45) together with the fact that the r

i

are equal follows immediately from Theorem 3.1 in 8]. The only thing we have to prove is then that the condition (43) is equivalent to the non- singularity of the matrix A ( x ) as dened in (30).

First assume that A ( x ) is singular. From the denition of vector relative degree we know that we can write

0

B

B

@

y

^(1r1)

y

⁽m^r

...

^m)

1

C

C

A

=

0

B

B

@

L

⁽_f^r1)

h

¹

( x ) L

⁽_f^rm)

h ...

m

( x )

1

C

C

A

+ A ( x )

0

B

@

u

¹

u ...

_m

1

C

A

(46)

Since A ( x ) is singular we can nd a row vector

q ( x ) = ( q ( x ) :::q

m

( x )) (47)

consisting of polynomials from k  x

¹

:::x

n

] such that q ( x ) A ( x ) u = 0. Studying (46) we immediately see that the condition (43) is not fullled.

Now assume that A ( x ) is nonsingular and set

 v

¹

=

0

B

B

@

y

^1(r1)

y

m^(r

...

^m)

1

C

C

A

 v 

²

=

0

B

B

@

L

⁽_f^r1)

h

¹

( x ) L

⁽_f^rm)

... h

m

( x )

1

C

C

A

(48)

Using that

A

^;1

( x ) = 1

det A ( x )Adj A ( x ) = 1

det A ( x ) A  ( x ) (49) where Adj A ( x ) =  A ( x ) denotes the adjoint matrix of A ( x ) see, e.g., 11]. The equality below can then be shown to hold.

h

y

^(r1);

L

^r_f¹

h

¹

( x )

^;

A

¹

( x ) u:::y

^(r1);

L

^r_f^m

h

m

( x )

^;

A

m

( x ) u

ⁱ

=

h

A 

¹

( x ) v

^1;

A ^

¹

( x ) v

^2;

det A ( x ) u

¹

::: A ^

m

( x ) v

^1;

A ^

m

( x ) v

^2;

det A ( x ) u

mⁱ

It is now easy to see that subject to the term ordering (50)

u > x > y

^1(r1)

> > y

_m^(rm)

(51) are in fact the generators of the latter ideal in (50) a Grobner base. Thus the condition (43) must be fullled since every generator in the Grobner base

contains some u

i

. This concludes the proof.

²

We now give a theorem which is needed if we wish to calculate a normal form of (38). It is a generalization of Theorem 3.2 in 8].

Theorem 3.2

Suppose that the dynamic system (38) has generalized vector relative degree ( r

¹

:::r

p

) and dene the set P

r

to be

P

r

=

^f

j

^2f

1 :::p

^{g j}

r

j^

1

^g

(52) Then are the elements in the set

Y

=

ⁿ

y

⁽_j^{ij) j}

i

_j

= 0 :::r

_j^;

1  j

²

P

_r^o

(53) algebraically independent over k

^h

u

ⁱ

.

Proof.

Suppose that there exists an algebraic relation between the y

^(ij)

 i

j

= 0 :::r

_j^;

1  j

²

P

_r

, i.e.,

P ~

^

:::y

⁽_j^ij)

:::

^

= 0 (54) where ~ P ( ) is a polynomial in y

^(ij)

with coecients from k

^h

u

ⁱ

. Multiplying out fractions in (54) we get a polynomial P ( ) with coecients from k such that

P

^

:::y

⁽_j^ij)

:::u

¹

 u _

¹

:::u

⁽_m^m)

= 0 (55)

We now have to consider two dierent cases.

1. Suppose that

@P ( )

@u

⁽_i^{j) 6}

^{= 0}  for some i

^2f

1 :::m

^g

and j

^

0 (56) and choose l such that

@P ( )

@y

⁽_l^{jl) 6}

^{= 0}  for some l

²

P

r

and 0

^

j

l^

r

l^;

1 (57) Using the Lie derivatives that has been calculated to eliminate all y

⁽_j^ij)

 i

_j

= 1 :::r

j^;

1  j

²

P

r^nf

l

^g

from (55) we get

P

_l^

y

_l

:::y

⁽_l^rl;1)

xu

¹

 u _

¹

:::u

⁽_m^m)

(58) ,i.e., that we have an input output relation which is of order r

l^;

1 or less in y

l

. However, the denition of r

l

shows that we must dierentiate y

l

r

l

times in order to get an explicit dependence on the input. In other words the minimum order of y

l

in an input output relation is r

l

. Thus we get a contradiction.

2. Suppose that

@P ( )

@u

⁽_i^{j) }

⁰  for all i

^2f

1 :::m

^g

and j

^

0 (59) which says that the output satises some dierential equation

P

^

:::y

⁽_j^ij)

:::

^

= 0  0

^

i

j^

r

j^;

1  j

²

P

r

(60) Taking min

j

( r

j^;

i

j

) time derivatives of this polynomial we get

P ~

^

:::y

_j^(ij)

:::

^

= 0  0

^

i

j ^

r

j

 j

²

P

r

(61) Eliminating the y

_j^(ij)

 i

j

= 1 :::r

i^;

1  j

²

P

r

from (61) we get a relation P 

^

x:::y

_j^(rj)

:::

^

= 0  j

²

P

_r

(62) but this contradicts the denition of generalized vector relative degree.

The two cases above together show that no polynomial (55) can exist and thus that the output derivatives (53) are algebraically independent over k

^h

u

ⁱ

.

²

Corollary 3.1 Suppose that the dynamic system (38) has generalized vector relative degree ( r

¹

:::r

p

). Then  r = r

¹

+ ::: + r

p ^

n .

Proof.

If  r > n would the elements in the set

^Y

be a transcendence basis for D=k

^h

u

ⁱ

which contradicts the fact that trdeg D=k

^h

u

ⁱ

= n .

²

Remark 3.1 This theorem has a nice interpretation in the ane polynomial case namely that the proof of Lemma 3.1 follows directly from Theorem 3.2 and Theorem 2.3.

We now split up the elements of

^Y

as follows

ⁱ

=

0

B

B

B

B

B

@

¹ⁱ

²ⁱ

_iri

...

^;1

1

C

C

C

C

C

A

=

0

B

B

B

B

@

y

i

y _

i

y

⁽_i^ri

...

^;2)

1

C

C

C

C

A

8

i

²

P

r

such that r

i^

2

=

0

B

B

B

B

B

@

¹

²

...

^p

1

C

C

C

C

C

A

(63)

and

 =

0

B

B

B

B

B

@



¹



²

 ...

p

1

C

C

C

C

C

A

=

0

B

B

B

B

B

B

@

y

^(1r1;1)

y

^(2r2;1)

...

y

⁽p^rp;1)

1

C

C

C

C

C

C

A

8

i

²

P

r

(64)

where we have set

ⁱ

=

^

if r

i

< 2 and 

j

=

^

if r

i

= 0. Since trdeg D=k

^h

u

ⁱ

= n we know that there exists 

i ²

D=k

^h

u

ⁱ

 i = 1 :::n

^;

r  such that the set

f

  

^g

becomes a transcendence basis for D=k

^h

u

ⁱ

. The question is how the 

¹

:::

n^;r

can be chosen. The lemma below gives a partial answer to this question.

Lemma 3.2 Suppose that the dynamic system (38) has generalized relative de- gree ( r

¹

:::r

p

) with  r = r

¹

+ ::: + r

p

< n and that and  are given by equations (63) and (64) respectively. Then there exists n

^;

r  di erent integers l

¹

:::l

n^;r

between 1 and n such that with



j

= x

lj

 j = 1 :::n

^;

 r (65) the set

^f

  

^g

becomes a transcendence basis for D=k

^h

u

ⁱ

.

Proof.

It is clear that with the 

i

given as (65) 

i²

D=k

^h

u

ⁱ

 i = 1 :::n

^;

 r . The task is then to prove the existence of the n

^;

 r x

i

such that

^f

  

^g

becomes a transcendence basis for D=k

^h

u

ⁱ

.

To show this we rst dene the sets

X

ij

=

ⁿ

l

^2f

1 :::n

^g

L

ⁱ_A^j

B

j

( y

j

x ) depends on x

l

o

(66)

for all i

j

= 0 :::r

j^;

1 and j

²

P

r

. Taking the union of these sets we get X

^r

=

ij

X

ij

(67)

Denoting ~ r to be the number of elements in X

^r

then ~ r

^

r  . To prove this we assume the opposite, namely ~ r < r  . Since we already know that each of the elements in

^Y

are algebraic over k

^h

u

ⁱ

( x

¹

:::x

r^~

) it follows from Theorem 2.2 that the elements of

^Y

must be algebraically dependent over k

^h

u

ⁱ

which by using Theorem 3.2 gives a contradiction.

If ~ r < n then it is obvious that all x

i

such that i

⁶²

X

r^

are algebraically independent of the elements in

^Y

so that they can be included in a transcen- dence basis for D=k

^h

u

ⁱ

. However, if ~ r > r  there still remains ~ r

^;

r  elements in the transcendence basis. To nd these last x

_i

we use the fact that since in any collection of  r + 1 x

_i

with i

²

X

^_r

there is at least one element which is algebraically independent to the elements in

^Y

since the opposite situation is a contradiction to Theorem 2.2. The method below can then be used to nd the x

i

.

1. Set ~ X

r^

X

^r

to be any collection of  r +1 indices from X

^r

and set  X

r^

=

^

. 2. Find one x

l

with l

²

X ^~

^r

which is algebraically independent to

^Y

w.r.t.

k

^h

u

ⁱ

and set  X

^r

:=  X

^r^f

l

^g

. 3. Take an element j

²

X

^rⁿ



X ~

^r^

X 

r^



if such an element exists and set X ~

^r

:=

^

X ~

^r^nf

l

^gf

j

^g

. Otherwise end.

4. Go to step 2.

It is clear that this process will end precisely when the number of elements in  X

^r

is ~ r

^;

 r . A set of x

ij

which together with

^Y

will give a transcendence basis for k

^h

u

ⁱ

are then given by the indices

X

=

^f

l

¹

:::l

n^;r^g

= (

^f

1 :::n

^gn

X

r^

)

^

X 

r^

(68)

2

Remark 3.2 Lemma 3.2 only shows one way in which the 

i

can be chosen. It is clear that other choices can be made with possibly some nicer properties than the choice indicated by the lemma above.

A constructive note that must be made now is that in Lemma 3.2 it is not shown how to choose the x

i

which are algebraically independent over the eld k

^h

u

ⁱ

(  ). These x

i

can be found with Grobner bases in the following manner.

Consider the ideal

^I

generated by L

ⁱ_A^j

B

_j

( y

_j

x ) for i = 0 :::r

_j^;

1 and j

²

P

_r

. Calculate a Grobner base, G , for this ideal with the term ordering



_r^

< < 

¹

< x  (69) where  x is some ordering of the x

i

 i

²

X

r

. If we set x

l

to have the lowest order in  x we consider

G

^\

k

^h

u

ⁱ

 x

_l

  ] (70) If (70) explicitly depends on x

l

it is algebraic over k

^h

u

ⁱ

(

^Y

) and is thus not of interest to us and we repeat the process with another ordering  x . Otherwise x

l

is obviously independent w.r.t. k u (  ) and we stop.

The transformation that we now shall perform is given by ,  and  as given in equations (63), (64) and Lemma 3.2. What we seek is then a new generalized state space description with generalized state variables ,  and  . We can write the implicit transformation as

p

ij

(

_ij

x ) = L

ⁱ_A^j

B

j

( y

j

x )

^j_y⁽jij^;1)=_ij

= 0  i

j

= 1 :::r

j^;

2  j

²

P

r

 r

j^

2

q

i

( 

i

x ) = L

^r_A^i;1

B

i

( y

i

x )

^j_y⁽iri^;1)

= 

i

= 0  i

²

P

r



_i

= x

_j

 j

^2X

 i = 1 :::n

^;

r 

(71)

with

^X

given by equation (68).

We now study the time derivatives of our new transcendence basis. For the situation is simple since we get that

_ = M



+ M



 (72)

where M



is a diagonal matrix

M



=

0

B

B

B

B

B

B

@

M

_¹

0 0

0 M

_²

0

... ... ... ...

0 0 M

_^p

1

C

C

C

C

C

C

A

(73)

with

M

_i

=

0

B

B

B

B

B

B

@

0 1 0

... ... ... ...

0 0 1

0 0 0

1

C

C

C

C

C

C

A

2Rri^ri

(74) if r

i^

2 and M

_i

=

^

otherwise. In a similar manner is M



dened as

M



=

0

B

B

B

B

B

B

@

M

_¹

0 0

0 M

_²

0

... ... ... ...

0 0 M

_^p

1

C

C

C

C

C

C

A

(75)

with M

_i

= (0  0 ::: 1)

^{T 2Rri}

if i

²

P

r

and M

_i

=

^

otherwise.

As for the time derivatives of the elements in  we rst consider the polyno- mials

q 

i



 _

i

xu

^()

= L

^r_Aⁱ

B

i

( y

i

x )

^j_y⁽ri⁾⁼^_i

 i

²

P

r

(76) and we must now eliminate the x in order to get the new state space description.

The tool for performing the elimination will of course be Grobner bases. We