A Normal Form for Generalized State Space Descriptions

A Normal Form for Generalized State Space Descriptions

Hakan Fortell

Department of Electrical Engineering Linkoping University

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

April 4, 1995

Abstract

In this report we will give a denition of a normal form for systems on generalized state space form. We will investigate some of the properties of this normal form and we will show how the usual normal form follows as a special case when we are working with ane polynomial systems.

The construction of the normal form is based on a generalization of the Lie derivative. The calculation of this new Lie derivative is done with Grobner bases.

Keywords: Normal Forms, Grobner Bases, Dierential Algebra

1 Introduction

The normal form of ane state space systems as described in, e.g., 8] is some- thing that has found several important applications. Two examples are exact linearization and sliding mode control. One question that one may ask oneself is then whether one can nd a normal form for generalized state space descrip- tions 4]. A canonical form that can be used for such system descriptions is the generalized controller canonical form which is based on the theorem of the dif- ferentially primitive element 9]. One problem with this canonical form is that, however, it may be dicult to compute. There are nevertheless results which utilizes the canonical form, for instance 12] where it is shown how it can be used for designing sliding mode control laws. The idea of this paper is to try to generalize the concept of a normal form to the generalized state space case us- ing one of the most important tools from commutative algebra, namely Grobner bases. To justify the name normal form we will also study how the usual normal form follows as a special case and how some of the related dierential geometric results translates to commutative algebra. The paper is organized as follows.

The next section contains some of the most basic concepts from commutative

and dierential algebra. Section 3 then shows how a generalized normal form

can be computed and in section 4 this is applied to sliding mode control. Finally

in section 5 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

The theory in this section is collected from 1] and 5] where all proofs that are left out and also further references to commutative algebra can be found. We begin by making a few necessary de nitions.

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

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

R .

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

R .

4. There are 0  1

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

R . 5. Given a

R there is b

R such that a + b = 0.

We also need the following.

Denition 2.2

A eld k is a commutative ring such that given a

k there is c

k such that a c = 1.

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

:::x

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

:::x

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

Denition 2.3

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

I .

2. If ab

I then a + b

I . 3. If a

I and b

R then b a

I .

If we now set f

:::f

to be polynomials in a ring k  x

:::x

] and introduce the notation

h

f

:::f

=

(

n X

i⁼¹

p

f

: p

k  x

:::x

]  i = 1 ::: n 

)

(1) we can note that

f

:::f

is an ideal and it is called the ideal generated by f

:::f

.

A property that will be needed in the next section is to have some general-

ization of linear dependence of polynomials. To nd such a generalization we

rst need some more de nitions.

Denition 2.4

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 2.5

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

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

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

Denition 2.6

Let f

:::f

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

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

k  x

:::x

] such that P ( f

:::f

) = 0. Otherwise the f

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 de ne a basis for these extension elds.

Denition 2.7

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 de nition is then the theorem below.

Theorem 2.1

If k is a eld and the variables x

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

:::x

) =k = n .

Proof.

We only have to note that the x

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

k ( x

:::x

) =k .

The following corollary will also be useful.

Corollary 2.1 The elements y

:::y

k ( x

:::x

) are algebraically de- pendent if N > n .

Proof.

Otherwise y

:::y

would be a transcendence basis for k ( x

:::x

) =k .

2

Another property which will be utilized is given by the following theorem. The proof can be found in 9].

Theorem 2.2

Let x

:::x

y

:::y

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

is algebraic over the eld k ( x

:::x

). Then the y

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

:::f

k ( x

:::x

).

Then if



N > n , f

:::f

are algebraically dependent.



N = n , f

:::f

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



@f

@x



ij

(2)

is identically singular.



N < n , f

:::f

are algebraically independent i the Jacobian matrix (2) has rank N .

Another useful theorem concerning algebraic dependence is given by the follow- ing theorem which can be found in 11].

Theorem 2.4

Let K=k be a eld extension and let x

:::x

y

:::y

z

K . Now suppose that each of the y

are algebraically dependent on x

:::x

over k and that z is algebraically dependent on y

:::y

over k . Then is z algebraically dependent on x

:::x

over k .

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 equation. To be able to choose which variables to eliminate we must introduce an ordering of variables.

Denition 2.8

Let x

k  x

:::x

] with  =



:::

denote x

x

. A term ordering < is an ordering on

such that for all  

0 <  and  <

 + < + (3) There are of course several ways of de ning these term orderings but we will only consider one of them.

Denition 2.9

The lexicographic term ordering is dened by

 <

j : 

<



i < j : 

<

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

In particular will one of them be ordered rst.

Denition 2.10

If f =

c

x

k  x

:::x

] then the degree of f is dened as

deg f = max

: c

= 0

(5)

The leading term of f is

LT f = c

x ^deg

(6)

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

and the leading coecient is LC f = c

.

This de nition can now be extended to cover ideals.

Denition 2.11

Let

be an ideal in k  x

:::x

]. Then

LM

=

LM f : f

deg

=

deg f :

(7)

Using this last de nition we can now go on to de ne Grobner bases.

Denition 2.12

A set G

for an ideal

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

=

LM G

.

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

=

(8)

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

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 2.13

Let fg

k  x

:::x

]. 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

:::x

] 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 2.5

Let F =

f

:::f

k  x

:::x

] and p

k  x

:::x

]. Then it is possible to nd polynomials g

:::g

r

k  x

:::x

] such that

q = p

i⁼¹

g

f

(9)

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

.

The de nition below is related to this theorem.

Denition 2.14

If all q which satises the conditions of Theorem 2.5 are equal then p reduces

to q modulo F which we write p

q .

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

Denition 2.15

Let f

f

k  x

:::x

]. Then the S-polynomial of f

and f

is

S ( f

f

) = h

f

h

f

(10) where

h

= LC f

^lcm(LM f

 LM f

)

LM f

 h

= LC f

^lcm(LM f

 LM f

)

LM f

⁽¹¹⁾

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 2.6

The set G =

g

:::g

is a Grobner base for

G

i

S ( g

g

)

0 (12)

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 2.7

Let

be an ideal in k  x

:::x

] and partition x

:::x

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

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

G is a Grobner base for k  A ]

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

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

+ y + z = 1 x + y

+ z = 1

x + y + z

= 1 (13)

Calculating a Grobner base for the polynomials



x

+ y + z

1  x + y

+ z

1  x + y + z

1

(14) with the term ordering x > y > z gives the result



x + y + z

1  y

+ z

y

z

 2 yz

+ z

z

 z

4 z

+ 4 z

z

(15) From this it is clear that

z

4 z

+ 4 z

z

x

+ y + z

1  x + y

+ z

1  x + y + z

1

(16) and we have eliminated x and y from the set of equations (13).

This example indicates that Grobner bases can be interpreted as a generalization of Gaussian elimination. The algorithm for calculating Grobner bases has been implemented in every major computer algebra language such as

and

.

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 10, 9].

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

Denition 2.16

A dierential eld k is a eld satisfying the conditions

8

a

k @a @t ^{= _} a

k (17)

8

ab

k @@t ⁽ a + b ) = _ a + _ b (18)

8

ab

k @@t ⁽ ab ) = _ ab + a b ^_ (19) Here and in the sequel will the \dot" notation be used instead of @=@t to de- note dierentiation with respect to time. We also make a few more de nitions regarding dierential elds which are the dierential variants of some of the commutative de nitions in the previous section.

Denition 2.17

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



.

We now consider a de nition of dynamics based on dierential algebra.

Denition 2.18

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

u

where u =

u

:::u

can be viewed as inputs and the outputs y is chosen as

some nite set y =

y

:::y

in D .

Using this de nition a more general form of state space description than (22) can be made.

Theorem 2.8

Let D=k

u

be a dynamic system according to denition 2.18. Then there exists a generalized state x =

x

:::x

for some nite integer n and polynomials A

( ) and B

( ) such that

A



x _

xu u:::u _

= 0  j = 1 :::n (20) B



y

xu u:::u _

= 0  i = 1 :::p (21) A theorem which is important when considering dierentially algebraic exten- sions is given below.

Theorem 2.9

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

L such that L = K

. The element  is called a dierentially primitive element.

3 A Generalized Normal Form

Here we will study how the normal form of ane state space descriptions can generalized to the more general dynamics of De nition 2.18.

3.1 The normal form of ane state space systems

The theory in this section is collected from 8]. Consider the SISO nonlinear system which is described by the following state equations

x _ = f ( x ) + g ( x ) u

y = h ( x ) (22)

where x

. The system (22) is said to have relative degree r

n if ( i ) L

L

h ( x )

0  k = 0 :::r

2

( ii ) L

L

h ( x )

0 (23)

at some point x

. Here L

and L

denote the Lie derivatives in the directions f ( x ) and g ( x ) respectively. Using the notation

dh ( x ) =



@h ( x )

@x

 @h ⁽ x )

@x

::: @h ⁽ x )

@x



(24) the following lemma can be shown to hold.

Lemma 3.1 The row vectors

dh ( x

) dL

h ( x

) :::dL

h ( x

) (25)

are linearly independent.

We can now perform the change of variables locally around x

where 

( x ) = h ( x )

( x ) = L

h ( x ) 

( x ) = ... L

h ( x )

(26)

If r < n then 

( x )  i = r +1 :::n can be found so that the change of variables becomes invertible. In particular these functions can be chosen so that

L

( x ) = 0  i = r + 1 :::n (27) We see that nding 

( x )  i = r + 1 :::n such that (27) is satis ed involves solving n

r partial dierential equations which is of course dicult in general.

However, if we do not bother with the condition (27) the invertible transforma- tion is most often very easy to nd. With this transformation the new state equations becomes

z _

= z

z _

= z

_ ...

z

= z

z _

= L

h (

( z )) + L

L

h (

( z )) u z _

= q

( z ) + p

( z ) u

_ ...

z

= q

( z ) + p

( z ) u y = z

(28)

This is known as the normal form of the system. We note that with the condition (27) we get p

( x )

0  i = r + 1 :::n .

3.2 A generalization of the normal form

We consider a dynamics D=k

u

which has dierential transcendence degree 1 and trdeg D=k

u

= n . In generalized state space form we thus get

A

x _

xu u:::u _

= 0 A

x _

xu u:::u _

= 0 ...

B

yxu u:::u _

= 0

(29)

where we we for simplicity say that the system has only one output. Since we want to be able to have a unique description of the output we assume that the

condition @B ( yx )

@y

^{= 0 (30)}

is ful lled. Our goal is then to try to nd a generalization of the normal form

(28). The rst thing we must do is then to generalize the Lie derivative and the

relative degree. The rst thing we note is that if the output polynomial explicitly depends on u and its derivatives we have a case which do not correspond to the ane case. The natural thing is then to say that such a system has relative degree zero and that (29) is already in normal form. If this is not the case we must study the following polynomial

dB ( yx )

dt ⁼

n

X

i⁼¹

@B ( yx )

@x

x _

+ @B ( yx )

@y y ^_ (31)

The correspondence to the Lie derivative (23) would then be if all _ x

and the y could be eliminated from (31). Obviously this can be done with Grobner bases as given in the de nition below.

Denition 3.1

Suppose that we have a system (29) where the output polynomial does not depend explicitly on u and that G

is a Grobner base for the polynomials

f

A

( ) ::: A

( )  B ( yx )  dB ( yx ) =dt

(32) under the term ordering

x

< < x

< y < u < _ u < < u _

< y < x _

< < x _

(33) where 

= max



:::

. The generalized Lie derivative of y w.r.t.

A =

A

( ) ::: A

( )

(34) is then dened as

L

B ( yx ) = G

k  x

:::x

u u:::u _

 y _ ] (35)

with @L

B ( yx )

@ y _

^{= 0 (36)}

The iterated generalized Lie derivative L

B ( yx ) with i > 1 is dened as L

B ( yx ) = G

k  x

:::x

u u:::u _

y

] (37)

with @L

B ( yx )

@y

^{= 0 (38)}

Here G

is the Grobner base of



A

( ) ::: A

( )  B ( yx )  L

B ( yx ) ::: L

B ( yx )  dL

B ( yx ) =dt

w.r.t. the term ordering (39)

x

< < x

< y

< u < u < < u _

< < y < y < _ x _

< < x _

(40)

Here and in the sequel will k denote the eld of real numbers. With this de ni-

tion we can de ne a relative degree for systems of the form (29).

Denition 3.2

A system (29) has generalized relative degree r

1 if L

B ( yx )  i = 1 :::r consist of only one polynomial for each i and it holds that

@L

B ( yx )

@u

⁰ 

j

0  i = 1 :::r

1 (41)

and @L

B ( yx )

@u

^{= 0}  for some j

0 (42) In order to justify the names generalized Lie derivative and generalized relative degree we must now show that the usual de nitions of these concepts follow as special cases.

Theorem 3.1

Suppose that we have a system (22) with f ( x ), g ( x ) and h ( x ) polynomials in the ring k  x

:::x

]. Then this system has relative degree r i it has generalized relative degree r . The Lie derivatives L

h ( x )  i = 0 :::r

1 are given as the solutions to

L

( y

h ( x )) = 0  i = 0 :::r

1 (43) w.r.t. y

 i = 1 :::r

1. Furthermore, L

h ( x )+ L

L

h ( x ) u is the solution

to L

( y

h ( x )) = 0 (44)

w.r.t. y

.

Proof.

For suciency we rst assume the relative degree of (22) to be 1. To calculate L

( y

h ( x )) we must nd a Grobner base for the polynomials

(

x _

f

( x )

g

( x ) u::: x _

f

( x )

g

( x ) u y

h ( x )  y _

i⁼¹

@h ( x )

@x

x _

)

under the term ordering (45)

x

< < x

< u < y < y < _ x _

< < x _

(46) It is clear that the polynomials (45) are not a Grobner base for the ideal that they generate since

S

x _

f

( x )

g

( x ) u y _

i⁼¹

@h ( x )

@x

x _

!

=

= _ y

i⁼²

@h ( x )

@x

x _

@h ( x )

@x

⁽ f

( x ) + g

( x ) u ) (47) Using Theorem 2.5 with p equal to (47) and the set F being equal to (45) we get

q = _ y

i⁼¹

@h ( x )

@x

( f

( x ) + g

( x ) u ) (48)

which is reduced w.r.t. (45). Since q

= 0 this is a contradiction to Theorem 2.6.

However, we note that the following equality holds

x _

f

( x )

g

( x ) u::: x _

f

( x )

g

( x ) u

y

h ( x )  y _

i⁼¹

@h ( x )

@x

x _

=

=

x _

f

( x )

g

( x ) u::: x _

f

( x )

g

( x ) u

y

h ( x )  y _

i⁼¹

@h ( x )

@x

( f

( x ) + g

( x ) u )



(49)

It is now easy to see that the generating set of the latter ideal, let us call it G , is in fact a Grobner base since

S

x _

f

( x )

g

( x ) u y _

i⁼¹

@h ( x )

@x

( f

( x ) + g

( x ) u )

!

=

= _ y (

f

( x )

g

( x ) u )

x _

;

n

X

i⁼¹

@h ( x )

@x

( f

( x ) + g

( x ) u )

!

;!G

0 (50) and the same is easily seen for all possible combinations of elements in G . We get

L

( y

h ( x )) = _ y

i⁼¹

@h ( x )

@x

( f

( x ) + g

( x ) u ) (51) and setting (51) to zero and solving for _ y gives L

h ( x ) + L

h ( x ) u . If now the relative degree is higher than 1 we know that L

h ( x )

0 so L

( y

h ( x )) will not depend on u . We now assume, in an inductive manner, assume that we have calculated L

( y

h ( x )) for some i < r . We know that it does not depend on u so we calculate L

( y

h ( x )) by nding a Grobner base for



x _

f

( x )

g

( x ) u::: x _

f

( x )

g

( x ) u

y

h ( x )  y _

L

h ( x ) ::: y

L

h ( x )  y

i⁼¹

@L

h ( x )

@x

x ^_



We immediately get that (52)

L

( y

h ( x )) = y

L

h ( x )

L

L

h ( x ) u (53) If i + 1 = r we are nished and otherwise we continue by calculating L

( y

h ( x )). The suciency has thus been proved.

As for the necessity we assume that (22) has no relative degree. Following the reasoning above we immediately see that then will the system have no

generalized relative degree either.

The question is now how these Lie derivatives can be used for nding a normal form for our system (29). Given a relative degree r

1 we have a set of

polynomials

p ( yx )  p ( _ yx ) ::: p

( y

x )

(54)

and in correspondence with the normal form (28) we want to perform a trans- formation such that

z

= y z

= _ y::: z

= y

(55) The y

 i = 0 :::r

1 can now be shown to have the following property.

Theorem 3.2

Suppose that the dynamic system (29) has generalized relative degree r . Then the output derivatives y

 i = 0 :::r

1 are algebraically independent over k

u

.

Proof.

Since (29) has relative degree r we have calculated the following set of polynomials

p

( yx ) = B ( yx ) p

( _ yx ) = L

B ( yx ) p

( y

x ) = ... L

B ( yx )

(56) Now suppose that there exists an algebraic relation between the y

 i = 0 :::r

1, i.e., P ~ ( y y:::y _

) = 0 (57) where ~ P ( ) is a polynomial in y

with coecients from k

u

. Multiplying out fractions in (57) we get a polynomial P ( ) with coecients from k such that

P ( y y:::y _

u u:::u _

) = 0 (58) We now have to consider two dierent cases.

1. Suppose that

@P ( )

@u

^{= 0}  for some j

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

1 or less in y . However, the de nition of generalized relative degree shows that we must dierentiate y r times in order to get an explicit dependence on the input. In other words the minimum order of y in an input output relation is r . Thus we get a contradiction.

2. Suppose that

@P ( )

@u

^{= 0} 

j

0 (60) which says that the output satis es some dierential equation

P ( y y:::y _

) = 0  0

i

r

1 (61) Taking r

i time derivatives of this polynomial we get

P ~ ( y y:::y _

y

) (62)

The conclusion to be drawn from this is that y

is algebraically dependent

of ( y y:::y _

) w.r.t. k but since each of the y

 i = 0 :::r

1 are

algebraically dependent of x

:::x

w.r.t. k we note that y

also must be algebraically dependent of x

:::x

w.r.t. k according to Theorem 2.4. Then there exists a polynomial  P ( ) with coecients from k such

that P  ( xy

) = 0 (63)

However, this contradicts the de nition of generalized relative degree.

The two cases above together show that no polynomial (58) can exist and thus that the y

 i = 1 :::r

1 are algebraically independent over k

u

.

Corollary 3.1 Suppose that the dynamic system (29) has generalized relative degree r . Then r

n .

Proof.

If r > n we would have a transcendence basis

y y::: y _

for D=k

u

which contradicts the fact that trdeg D=k

u

= n .

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.

Since trdeg D=k

u

= n we know that there exists z

D=k

u

 i = r + 1 :::n such that with z

= y

 i = 1 :::r the set

z

::: z

becomes a transcendence basis for D=k

u

. The question is how the z

:::z

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

Lemma 3.2 Suppose that the dynamic system (29) has generalized relative de- gree r

n . Then there exists n

r dierent integers i

:::i

between 1 and n such that with

z

=



y

 j = 1 :::r

x

 j = r + 1 :::n ⁽⁶⁴⁾ the set

z

::: z

becomes a transcendence basis for D=k

u

.

Proof.

It is clear that with the z

given as (64) z

D=k

u

 i = 1 :::n . The task is then to prove the existence of the n

r x

such that

z

::: z

becomes a transcendence basis for D=k

u

.

To show this we rst de ne the set

X

=

i

1 :::n

p

( y

x ) depends on x

for some j

1 :::r

Denoting  r to be the number of elements in X

then  r

r . To prove this (65)

we assume the opposite, namely  r < r . Since we already know that each

z

 i = 1 :::r is algebraic over k

u

( x

:::x

) it follows from Theorem 2.2

that z

:::z

must be algebraically dependent over k

u

which by using The-

orem 3.2 gives a contradiction.

Now suppose that  r = r < n . It is obvious that choosing the indices i

:::i

such that i

X

for each j gives us a transcendence basis for D=k

u

.

If  r > r we must nd  r

r x

which are algebraically independent of k

u

( z

:::z

). It is clear that at least one such x

must exist since if we assume the opposite we again get a contradiction using Theorem 2.2. We thus get an index i

and we set z

= x

. If  r = r + 1 we are nished with this step. Otherwise we conclude that by using Theorem 2.2 there must be yet another x

X

which is algebraically independent w.r.t. k

u

( z

:::z

).

We repeat this process until we have

z

:::z

which are algebraically inde- pendent over k

u

.

If  r < n we nd the nal z

which gives us a transcendence basis according

to the method above.

Remark 3.2 The proof of Lemma 3.2 gives an idea of what class of dynamical systems (29) have no relative degree. Suppose that we have a dynamical SISO system

A

(_ x

x

x

) = 0 A

(_ x

x

x

) = 0 A

x _

xu u:::u _

= 0 A

x _

xu u:::u _

= 0 ...

B ( yx

x

) = 0

(66)

It is clear that this system has no generalized relative degree since

L

B

( yx

x

)

k  y

x

x

]

i

0 (67) so we get no relative degree r

n . If n is large we might want to be able to spot this fact without having to calculate n generalized Lie derivatives. Using the set X

as dened in (65) we can note that as soon as the number of ele- ments in X

is less than the number of Lie derivatives that has been calculated (including L

B ( yx ) = B ( yx ) ) then by Theorem 2.2 the corresponding y

are algebraically dependent over k . This obviously gives us a condition to check the existence of a generalized relative degree.

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

which are algebraically independent over the eld k

u

( z

:::z

)  j = r::: r 

1. These x

can be found with Grobner bases in the following manner. Calculate a Grobner base, G , for (54) with the term ordering

z

< < z

< < z

< x  (68) where j

r::: r 

1

and  x is some ordering of the x

 i

X

. If we set x

to have the lowest order in  x we consider

G

k

u

 x

z

:::z

] (69)

If (69) explicitly depends on x

it is algebraic over k

u

( z

:::z

) and is thus not

of interest to us. Otherwise x

is obviously independent w.r.t. k

u

( z

:::z

)

and we set z

= x

. This procedure can then be repeated until j =  r .

The main tool for performing the transformation the system (29) will of course also be Grobner bases. First we consider the time derivatives of the generalized state variables z

:::z

. From the de nition it is easy to see that we get

z _

= z

 i = 1 :::r

1 (70) To get the equation for _ z

we rst take the time derivative of p

( z

x ) to get

dp

( z

x )

dt ⁼

n

X

i⁼¹

@p

( z

x )

@x

x _

+ @p

( z

x )

@z

z _

(71)

Calculating the Grobner base, G, for the polynomials

(

p

( z

x ) ::: p

( z

x ) 

i⁼¹

@p

( z

x )

@x

x ^_

+ @p

( z

x )

@z

z ^_

)

(72) with the term ordering

u < u < < u _

< z

< < z

< z _

< x

< < x

< x _

< < x _

and studying (73)

G

k  _ z

z

:::z

u u:::u _

] (74) we get the generalized state space equation

q



z _

zu u:::u _

= 0 (75) for z

. For the remainder of the transformation we know that by construction

z

= x

 for some j

1 :::n

(76) for all i = r + 1 :::n . Thus nding the generalized state space equation for one such z

amounts to calculating a Grobner base, G, for

n

p

( z

x ) ::: p

( z

x )  A

z _

xu u:::u _

(77) with the term ordering (73) where _ z

is replaced by _ z

. We get the generalized state space equation from

G

k _ z

z

:::z

u u:::u _

] (78) which gives

q



z _

zu u:::u _

= 0 (79) This is repeated for every i

r + 1 :::n

. The new generalized state space description then becomes

z _

= z

z _

= z

_ ...

z

= z

q

z _

zu u:::u _

= 0 q

z _

zu u:::u _

= 0 q

z _

zu u:::u _

= 0 ...

y = z

(80)

Due to the fact that if we would have performed the calculations above for an ane system (22) we would get (28) we can call this a generalized normal form for (29). We give an example of the calculations leading to this normal form.

Example 3.1

Consider the system

3 x

x _

+ 7 = 0 x

x _

u _ + 3 x

u = 0 x

x _

+ 2 x

= 0 y

+ x

= 0

(81)

The rst Lie derivative becomes

L

( y

+ x

) = 36 x

x

y _

+ 49 (82) so if a relative degree exists it is greater than one. Repeating the procedure gives

L

( y

+ x

) = 25412184 x

ux

u _

13716864 x

u y 

x

ux _

+ 28005264 x

u

+ 5764801 x

u _

+ 6223392 x

u _

y 

x

x

+ 1679616 x

u _

x

y 

x

which shows that the system has relative degree r = 2. We now compute the (83) transformation

z

+ x

= 0 36 x

x

z

+ 49 = 0

z

x

= 0 (84)

to achieve the normal form

z _

= z

2401_ u

z

z

4802_ u

z _

z

z

z

10584_ uuz

z

z

10584 u z _

uz _

z

z

+ 2401_ u

z _

z

z

+ 11664 u

z

2 z

= 0 49_ z

144 z

z

= 0 y = z

(85)

2

3.3 A comparison with the Generalized Controller Canon- ical Form

It is shown in 2] how Theorem 2.9 can be used for nding a canonical form for generalized state space representations (29).

Theorem 3.3

Suppose that  is a primitive element of the dynamics D=k

u

. Then x

=

:::x

= 

qualies a generalized state vector and the corresponding state

space description becomes

x _

= x

x _

= x

_ ...

x

= x

C

x _

x

:::x

u u:::u _

= 0

(86)

This is known as the generalized controller canonical form.

Let us compare (86) to the generalized normal form (80). Although no output is de ned one might suspect that one gets the generalized controller canonical form when the generalized relative degree is n . The theorem below con rms this suspicion.

Theorem 3.4

Suppose that the dynamics (29) has generalized relative degree n . Then is the output y of (29) a di erentially primitive element of the dynamics given by (29).

Proof.

Our dynamic system is given by the eld extension D=k

u

. A generalized state space representation of this dynamics is given by (29) and this system has generalized relative degree n . From Theorem 3.2 we know that the y

 i = 0 :::n

1 are algebraically independent over k

u

and from 2] we can deduce that it is a transcendence basis for D=k

u

. Then it is clear that y must be a dierentially primitive element of the eld extension.

We give a simple example of this.

Example 3.2

Consider the trivial system

x _

= x

x

x _

= u

y = x

⁽⁸⁷⁾

It is easy to see that the system has (generalized) relative degree 2 and per- forming some simple calculations we also get that x

is a dierentially primitive

element.

The opposite situation, i.e., that if y is a dierentially primitive element then has the system (29) generalized relative degree n , is not true in general. This is shown by the following example.

Example 3.3

Here we consider the system below which is a trivial modi cation of the system in the previous example

x _

= x

x

+ u x _

= u

y = x

⁽⁸⁸⁾

Again x

can be shown to be a dierentially primitive element but it is easy to

see that the (generalized) relative degree is equal to 1.