On Pointwise Feedback Invariants of Linear Parameter varying Systems

(1)

On Pointwise Feedback Invariants of Linear

Parameter-varying Systems

R. Marta Garc´ıa Fern´andez

∗

,

Miguel V. Carriegos

Departamento de Matemáticas, Universidad de León, 24071 León, Spain

Abstract

Linear systems with constant real coefficients are completely described in terms of feedback actions. In this paper the problem is studied in the framework of linear systems where coefficients depending continuously on a set of parameters. Some invariants are given as well as criteria to find a complete classification in low dimension.

Keywords

Feedback Classification, Systems over Commutative Rings, Controllability

1 Introduction

Let’s consider the family of linear control systems

Σ(λ) =

˙

x=A(λ)x(t) +B(λ)u(t)

y(t) =C(λ)x(t) (1)

wherex(t) is the n-dimensional vector of states, u(t)is them-dimenional vector of external inputs andy(t)is the

p-dimensional vector of outputs. MatricesA(λ),B(λ)and

C(λ)depend continuously on some parametersλliving in a compact topological spaceΛ. We’ll denote byC(Λ,_R)the ring of continuous real functions defined onΛwith point-wise sum and product. This is a commutative ring where

1C(Λ,R)and0C(Λ,R)are respectively the constant functions

λ7→1andλ7→0.

It is well known that if the matrices have constant coeffi-cients then there is a canonical form forΣ, the Brunovsky’s Canonical Form [5].

Our main goal in this paper is to stablish the so-called pointwise feedback equivalence, see [8], for systems over

C(Λ,R). This equivalence is just given by all evaluations of parameters of the given linear systems. Therefore point-wise feedback equivalence is studied at every point just like classical feedback equivalence for linear systems with con-stant coefficients.

We also are interested in stablish global invariants of linear systems for such pointwise feedback equivalence.

Here some sets of zeroes of functions will arise. A com-plete set of invariants will be given. Despite this classifica-tion result, we are not given canonical forms for pointwise feedback action. This is left as a future work.

The paper is organized as follows. Section 2 reviews main results involving feedback equivalence of linear sy-stems which are traslated to the study of pointwise feed-back equivalence. Section 3 reviews determinantal ranks of matrices in order to get main pointwise feedback inva-riants which are introduced in section 4. This section 4 is also devoted to prove that the set of zeroes of determinantal ideals of reachability maps are invariant for the pointwise feedback equivalence. Low dimensional casesn ≤ 5 are completely described and characterized in section 5. Fi-nally, we list our conclusions.

2 Feedback actions and feedback

classification

LetR be a commutative ring with unit1 6= 0. A m -input,n-dimensional linear system overRis just a pair of matricesΣ = (A, B) ∈ Rn×n _×_Rn×m_{representing the}

right-hand-side (dynamic) equation

x+(t) =Ax(t) +Bu(t),

wherex+(t)represents time-derivative in continuous time framework or time-shift for discrete systems.

Linear systems Σ0 = (A0, B0) and Σ are equivalent (feedback) whenΣcan be transformed to Σ0 by one ele-ment of the feedback groupFnm(R)and we will note this

byΣ ∼Σ.We recall that feedback groupFnm(R)is the

generated group by the following three types of transfor-mations:

(1) A −→ A0 = P AP−1,B −→ B0 = P Bfor some invertible matrix P. The transformation is a conse-quence of a change of base inRn,the state module. (2) A−→A,B −→B0 =BQfor some invertible

(2)

(3) A −→ A0 ₌_A₊_BK _,_B _−→ _B _{for some}_m_×_n

matrixKwhich is called a feedback matrix.

Note 2.1. The feedback classification problem is wild in the sense of Representation Theory (see [4]). Hence it is an open problem in the general case and it is unlikely to be solvable. However in some cases it is possible to find solutions: WhenR = _Kis a field the problem is known as classical case and a classical result of Brunovsky [5, 13, 19] characterizes the class of equivalence ofΣby the action of the feedback group, see below.

Throughout this paper we focus on commutative ring

R = C(Λ,_R)of real valued continuous functions defi-ned on compact topological spaceΛ. Note that sinceΛis compact then maximal idealsmofRare in one to one cor-respondence with points inΛ, that is, given a maximalmof

Rthere is a unique pointλm ∈Λsuch thatf(λm) = 0for

everyf ∈m, and conversely, given a pointλ∈Λ, the set

mλ ={f : f(λ) = 0}is a maximal ofR(the reader can

see [1] for details).

Let be Σ = (A, B) a linear dynamical system over

R=C(Λ,R).Then linear system overR∼=R/mλ0

obtai-ned by extension of scalarsC(Λ,R) −→ C(Λ,R)/mλ0

sendingf 7→f(λ0)is just obtained by the evaluation at the pointλ0∈Λ; that is,Σ(λ0) = (A(λ0), B(λ0)).

Now we recall the definition of pointwise feedback equi-valence of linear systems as introduced in [8].

Definition 2.2. Linear systems Σ and Σ0 are pointwise feedback equivalent if systemsΣ(λ) = (A(λ), B(λ))and

Σ0(λ) = (A0(λ), B0(λ))over Rare feedback equivalent

for allλ∈Λ.

Pointwise feedback equivalence is weaker than feedback equivalence. That is to say,

Theorem 2.3. If systems Σ = (A, B) and Σ0 = (A0, B0)are feedback equivalent via the action of element

(P, Q, K) ∈ Fnm(C(Λ,R)) then Σ(λ) and Σ0(λ) are

feedback equivalent via(P(λ), Q(λ), K(λ))∈Fnm(R)

Proof. Suppose that (A, B) is feedback equivalent to

(A0, B0)via(P, Q, K); that is to say

A0P =P(A+BK), andB0=P BQ

it follows that above equalities hold on every evaluation; that is

A0(λ)P(λ) =P(λ)(A(λ) +B(λ)K(λ)), B0(λ) =P(λ)B(λ)Q(λ) (2)

Now the proof is complete once it is assured that P(λ)

and Q(λ) are invertible. But this is trivial because, for instance, det(P) anddet(Q) are units inC(Λ,R)and a fortioridet(P(λ))6= 0anddet(Q(λ))6= 0.

We are interested in give a complete (and minimal if pos-sible) set of invariants for pointwise feedback relation. First we recall classical invariants and canonical form for con-stant linear systems over a field.

Definition 2.4. Let beΣ = (A, B)a linear dynamical sy-stem of size(m, n)over commutative ringR. Consider the

R-module NΣ

i generated by columns of the n×im

ma-trixBΣ

i = (B|AB|. . .|Ai−1B). We’ll denote byMiΣthe

quotient moduleMΣ

i =Rn/NiΣ.

Above modules are feedback invariant associated to a gi-ven linear (over any commutative ring), and they form a minimal complete set in the case of controllable systems over fields. In fact, the following results are well known. First, modules are shown to be feedback invariants: Theorem 2.5. (cf. [10]) Let beΣ = (A, B)a linear dyna-mical system of size(m, n)over a ringR.Then

(i) (0)⊆NΣ

0 ⊆N1Σ⊆. . .⊆NnΣ.

(ii) The canonical homomorphism

ϕi:NiΣ/N

Σ

i−1→N Σ

i+1/N Σ

i

x+N_iΣ₋₁→Ax+N_iΣ

is surjective for1≤i≤n−1.

(iii) IfΣis feedback equivalent toΣ0thenN_iΣandM_iΣare isomorphic toNΣ0

i andMΣ 0

i respectly, for1≤i≤n.

(iv) If Σ is a reachable system of simple input n -dimensional then the modules NΣ

i 1≤i≤n and

MΣ

i ₁_≤_i_≤_nare free.

(v) If Σ is a Brunovsky system then the modules

NΣ

i 1≤i≤nand

MΣ

i 1≤i≤nare free.

Now we deal with the case of fieldsK: We prove that above invariantK-vector spacesNiΣform a complete and

minimal set of feedback invariants.

Theorem 2.6. (cf. [5]) Let beΣ = (A, B)a reachable linear dynamical system of size(m, n)over a fieldK. Then

there exist positive integersκ1 ≥κ2 ≥ · · · ≥κsuniquely

determined byΣwithn=κ1+κ2+· · ·+κs, such thatΣ

is feedback equivalent to the systemΣκ= (Aκ, Bκ)where

Aκis the block matrix

Aκ=



   

Aκ1 0 · · · 0 0 Aκ2 · · · 0

..

. ... . .. ...

0 0 · · · Aκs 

   

,

and blockAκiis theκi×κimatrix

Aκi= 

       

0 1 0 0 · · · 0 0 0 1 0 · · · 0 0 0 0 1 · · · 0

..

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

0 0 0 0 · · · 1

0 0 0 0 0 0



(3)

and

Bκ=

s

z }| {

m−s

z }| {



              

0 0 · · · 0

..

.

0

· · ·

0

1 0 · · · 0

0 0 · · · 0

0

... · · ·

0

1 0 · · · 0

..

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

0 0 · · · 0

0 0

· · · ...

1 0 · · · 0 

              

 



κ1

 



κ2

 



κs

The integersκ = {κ1, κ2, . . . , κs} are called the

Kro-necker indices ofΣ.They are a complete set of invariants forΣby the action of the feedback group.

Proof. See [5] or [10].

Above linear systemΣκ = (Aκ, Bκ)is called

Bruno-vsky’s Canonical Form associated to indicesκ.

Note that ifm= 1and system is controllable it is easy to see that there is just one nonzero index andκ1 =nhence

s= 1in above matrices and consequently Brunovsky’s Ca-nonical Form in this case is just the CaCa-nonical Controller Form

Brunovsky’s Canonical form gives rise a complete and minimal set of feedback invariants for controllable linear systems over a field. This is the case ofR = _R, which is the base for our study. To be precise, the following result summarizes the list of invariants.

Theorem 2.7. (cf.[10]) Let beΣ = (A, B)a reachable linear dynamical system of size(m, n) overR. Then the

feedback equivalence class ofΣis characterized for each one of the following sets:

(i) The Kronecker’s indices{κi}1≤i≤s

(ii) dim(N_iΣ) ₁_≤_i_≤_n

(iii) dim N_iΣ/N_iΣ₋₁ ₁_≤_i_≤_n

Proof. See [10].

According the above resultrankRMiΣ 1≤i≤nis a

com-plete set of invariants for the class of feedback of a Bruno-vsky form. If K is a field and Σ = (A, B)a reachable dynamical systemrank MΣ

i

= n−rankB˜Σ

i

and in consequencenrankB˜Σ_i o

1≤i≤n is the list of invariants

we need:

Theorem 2.8. A complete set of pointwise feedback invari-ants for the reachable linear systemΣ(λ) = (A(λ), B(λ))

is given by the one (and hence all) follo–wing data:

(i) dim_R(N_iΣ(λ)),1≤i≤n,λ∈Λ

(ii) ndimN_iΣ(λ)/N_iΣ(₋₁λ)o,1≤i≤n,λ∈Λ

(iii) nrank_RB˜_iΣ(λ)o,1≤i≤n,λ∈Λ

Proof. Two reachable linear systems Σ and Σ0 over

C(Λ,_R)are pointwise feedback equivalence if and only if (by definition) linear systemsΣ(λ)andΣ0(λ)are feedback equivalent over_R. By Brunovsky’s Theorem this is equiva-lent todim_R(N_iΣ(λ)) = dim_R(N_iΣ0(λ)), for all1≤i≤n,

λ∈Λ. Hence data(i)is a complete set of pointwise feed-back equivalence. Statements(ii)and(iii)are proved in the same way.

3 Determinantal ranks

Note that pointwise feedback invariants found in above section involves a potentially infinite data (for instance, when compact topological space Λ is infinite). Thus we need to refine the result in order to find a minimal complete set of pointwise feedback invariants. This section is devo-ted to briefly review determinantal rank of a matrix and to compute the determinantal ranks of Brunovsky’s canonical forms in order to find our invariants in terms of determinan-tal ranks in the sequel.

Let beM = (aij)ann×mmatrix with entries inRand

let beia nonnegative integer. Thei−th determinantal ideal ofM,denoted byUi(M),is the ideal ofRgenerated by

all thei×iminors ofM.By construction we have

R=U0(M)⊇ U1(M)⊇. . .⊇ Ui(M)⊇. . .

andUi(M) = 0fori >min{m, n}.The rank ofM,

de-noted byrankR(M), is the largestisuch thatUi(M)6= 0.

ThenΣis reachable if and only ifUn BnΣ

=R.

Now we give a technical result of characterization of Brunovsky’s canonical formsΣκ = (Aκ, Bκ) depending

on the sequence of determinantal ideals. Fist, we give with some definitions and notations.

Definition 3.1. Let ben ∈ Nandκ = {κ1, κ2, . . . , κs}

withκ1 ≥κ2 ≥ · · · ≥ κsa partition ofn. We call dual

partition ofκto the partitionη = {n1, n2, . . . , np} with

n1 ≥ n2 ≥ · · · ≥ npofnwhereni,is the number ofκj

which are more or equal thani.

Note 3.2. The applicationκ→ηis biyective on the set of partitions ofn. See [2].

Note 3.3. Note thatn1=s.

Theorem 3.4. Let beκ={κ1, κ2, . . . , κs}a partition ofn

andη ={n1, n2, . . . , np}be its associated dual partition.

The following conditions are equivalent.

(i) κare the Kronecker indices of systemΣ = (A, B)

(ii) For1≤i≤p−1we have

Un1+n2+...+ni

˜

BΣκ i

= K

Un1+n2+...+ni+1

˜

BΣκ i

= (0)

and fori=p

Un1+n2+...+np

˜

BΣκ p

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Proof. By Brunovsky Theorem, 2.6,

A= 

   

E1

E2 . ._.

Es



   

,

where the matrix

Ei =



      

0 1 0 · · · 0

0 0 1 . .. 0

..

. ... ... . .. 0 0 0 0 · · · 1 0 0 0 . . . 0



      

,

is of dimensionκi×κiand

B= 

   

e1 0 · · · 0 0 · · · 0

0 e2 · · · 0 0 · · · 0 ..

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

0 0 0 es 0 · · · 0



   

,

whereei = (0, . . . ,0,1)tis of dimensionκi×1. Then the

following properties can be easily verified. i)

AhB= 

   

E₁he1 0 · · · 0 0 · · · 0

0 E₂he2 · · · 0 0 · · · 0 ..

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

0 0 · · · Eshes 0 · · · 0



   

ii) Ifh > κiisEihei= 0

iii) Im(Ah_B₎_∩_Im₍_Ah0_B_{) =}_{₀_}_if_h₆₌_h0

Let ben1the number ofκjgreater than or equal to1, then

the definitionni and by the properties i) and ii) it is

follo-wed

rank(B) =s=n1,

rank(AB) =n2,

rank(A2_B_{) =}_n 3,

rank(A3_B_{) =}_n 4, ..

.

rank(Anp−1_B_{) =}_n p,

and by property iii)

rankB˜Σκ i

=n1+n2+. . .+ni

for1≤i≤p, or equivalently

Un1+n2+...+ni

˜

BΣκ i

=_K

Un1+n2+...+ni+1

˜

BΣκ i

= (0)

for1≤i≤p.

Conversely, by the equalities

rankB˜Σκ i

=n1+n2+. . .+ni rankB˜Σκ

i−1

=n1+n2+. . .+ni−1

and the property iii) before it is followed

rank(Ai−1B) =ni

In consequence,

rank(B) =n1=s,

rank(AB) =n2,

rank(A2_B_{) =}_n 3, ..

.

rank(Ap−1B) =np,

with

n1≥n2≥ · · · ≥np

By the definition ofni(that is

n1is equal to the number ofκjgreater than or equal to 1,

n2is equal to the number ofκjgreater than or equal to 2, ..

.

npis equal to the number ofκjgreater than or equal to p)

it is obtained κi and so the complete set of invariants

for the Brunovsky formΣκ, (for which the partition η =

{n1,n2,. . . , np}is its dual associated partition).

The following properties of Brunovsky canonical forms over any field_Kare easily derived:

Corollary 3.5. A Brunovsky canonical form Σκ = (Aκ, Bκ)over the fieldKverifies

Ui

˜

BΣκ i

=_K 1≤i≤n.

Corollary 3.6. A Brunovsky canonical form Σκ = (Aκ, Bκ)over the fieldKverifies

rankB˜Σκ i

=n1+n2+. . .+ni ≥i for1≤ i≤p

whereη ={n1, . . . , np}is the dual partition ofκ. In

par-ticular

rankB˜Σκ p

.=n1+n2+. . .+np =n.

Corollary 3.7. The following conditions hold for_K-vector spacesNΣκ

i generated by columns of then×immatrix. ˜

BΣκ

i = (Bκ|AκBκ|. . .|Aiκ−1Bκ)

associated to Brunovsky canonical formΣκ

(i) dimNΣκ i

≤dimNΣκ i+1

for alli= 1,2, . . .

(ii) IfdimNΣκ t

= dimNΣκ t+1

then

dimNΣκ t

=· · ·= dim NΣκ n

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Now, we are ready to give a characterization of pointwise feedback equivalence in terms of dimensions of pointwise invariants.

Theorem 3.8. Let be Σ = (A, B) and Σ0 = (A0, B0)

two reachable linear dynamical systems of size(m, n)over

R = C(Λ,_R).Forλ0 ∈ Λ,the following conditions are

equivalent.

(i) Σ(λ0) ∼ Σκ whereΣκ = (Aκ, Bκ)is the

Bruno-vsky’s linear form associated to the Kronecker’s indi-cesκ={κ1, κ2, . . . , κs}

(ii) Let beη ={n1, n2, . . . , np}the dual partition ofκ.

Then

dim_RNΣ(λ0) i

=n1+n2+. . .+ni; 1≤i≤p

.

Proof. (i)⇒(ii) As (A(λ0), B(λ0)) ∼ (Aκ, Bκ) and Σκ= (Aκ, Bκ)is the Brunovsky form over the fieldK=

R, whose associated partition isκ={κ1, κ2, . . . , κs},by

Theorem 3.4 it is 





Un1+n2+...+ni

˜

BΣκ i

=_R

Un1+n2+...+ni+1

˜

BΣκ i

= (0) for 1≤i≤p

or equivalently 





Un1+n2+...+ni

˜

BΣ(λ0) i

=R

Un1+n2+...+ni+1

˜

BΣ(λ0) i

= (0) for 1≤i≤p,

then

dimNΣ(λ0) i

= rankB˜Σ(λ0) i

=n1+n2+. . .+ni

for1≤i≤p.

(ii)⇒(i) Conversely, let beλ0∈Λsuch that

dimNΣ(λ0) i

=n1+n2+. . .+ni for 1≤i≤p,

then

rankB˜Σ(λ0) i

=n1+n2+. . .+ni= rank

˜

BΣκ i

for1≤i≤pAlso

Uj

˜

BΣ(λ0) i

=Uj

˜

BΣκ i

for 1≤i≤n, 1≤j≤n,

and thereforeΣ(λ0)is feedback equivalent toΣκ.

4 The pointwise feedback relation

Note that ifA = (fij)is a matrix overR = C(Λ,R) thenA(λ)is the matrix(fij(λ)). Let beΣ = (A, B)and Σ0 = (A0, B0)two systemsn-dimensional withminputs over R = C(Λ,_R) , if Σis equivalent feedback to Σ0,

thenΣ(λ)is equivalent feedback to Σ0(λ)for allλ. The converse is not true in general, and this motivates the study of following relationship. We sayΣandΣ0 are pointwise feedback equivalent if the systemsΣ(λ) = (A(λ), B(λ))

andΣ0(λ) = (A0(λ), B0(λ))over_Rare feedback equiva-lents for allλ∈ Λ.Since reachability is a property that it is preserved by feedback, let us see this concept in the ring

R=C(Λ,R).

Theorem 4.1. Let beΛ a compact topological space and

Σ = (A, B)be a reachable linear dynamical system of size

(m, n)overR=C(Λ,_R).Then the following conditions are equivalent.

(i) Σis reachable overC(Λ,R)

(ii) Σ(λ)is reachable overR, for allλ∈Λ.

Proof. (i)⇒(ii) IfΣis reachable inC(Λ,_R)then

Un

˜

BΣ= (f1, f2, . . . , fk) = (1) =C(Λ,R),

wheref1, f2, . . . , fkare the minors of ordernof the matrix ˜

BΣ.

SinceB˜Σ₍_λ_{) = ˜}_BΣ(λ)_{, it follows}

Un

˜

BΣ(λ)=Un

˜

BΣ⊗C(Λ,_R)/mλ=C(Λ,R)/mλ,

and in consequence rankB˜Σ(λ) ₌ _n, _{or equivalently}

Σ(λ)is reachable for allλinΛ.

(ii)⇒(i) Conversely, assume

Un

˜

BΣ=Un B|AB|. . .|An−1B

= (f1, f2, . . . , fk),

wheref1,f2, . . . , fkare all minors of ordernof the matrix ˜

BΣ_._As_Σ(_λ_{) = (}_A₍_λ₎_{, B}₍_λ₎₎_{is reachable for all}_λ_in_Λ_, we haverankB˜Σ(λ)=n.

Then the set of zeroes (see below)Z(f1, . . . , fk) = ∅

is empty. Thus ideal generated by{f1, f2, . . . , fk} is the

whole ring. A fortiori f2

1 +f22 +. . .+fk2 is an unit in

C(Λ,R).ThenUn

˜

BΣ ₌ _R _and_Σ_{is a reachable} sy-stem.

In order to introduce the sets of invariants for pointwise feedback relationship we need to remark usual notation of ideal of zeroes of a function. Let be a an ideal ofR =

C(Λ,_R).We’ll denote byZ(a)the set

Z(a) ={λ∈Λ/ f(λ) = 0 for all f ∈a}

The following result (see [8]) gives a set of invariants for the pointwise feedback relation.

Theorem 4.2. Let beΣ = (A, B)andΣ0 = (A0, B0)two reachable linear systems of type(n, m)over the ringR=

(6)

(i) SystemsΣandΣ0_{are pointwise feedback equivalents;} that is to say,Σ(λ) andΣ0(λ) are feedback equiva-lents forλ∈Λ

(ii) For all 1 ≤ i, j ≤ n one has ZUj

˜

BΣ

i

=

ZUj

˜

BΣ0

i

1≤i≤n, 1≤j≤n

Proof. (i) ⇒ (ii). If λ ∈ ZUj

˜

BΣ

i

then

Uj

˜

B_iΣ(λ) = (0)and hencedimN_iΣ(λ) < j. Since

Σ(λ)andΣ0(λ)are feedback equivalent overRit follows that it is also satisfied dimN_iΣ0(λ) < j which yields

λ∈ZUj

˜

BΣ_i0. The inverse contention is also proved because feedback equivalence is a equivalence relation, and symmetric property solves the case.

(i)⇐(ii)Let beλ∈Λ. Observe that(ii)yields that

dimN_iΣ(λ)= minnj :λ∈ZUj

˜

B_iΣo=

= minnj:λ∈ZUj

˜

BiΣ0

o

= dimN_iΣ(λ)

ThereforeRvector spacesNiΣ(λ)andN

Σ0(λ)

i are

isomor-phic for all iand all λ ∈ Λand consequently Σ(λ) and

Σ0(λ)are feedback equivalent for allλ∈Λor equivalently systemsΣandΣ0are point wise feedback equivalents.

At this point let us see how the generation of these zeroes sets can be done by only one element.

Theorem 4.3. Sets of zeroes of finitely generated ideals of

C(Λ,_R) can be obtained as the set of zeroes of a single function. That is to say, one has the following properties:

(i) Let beaa finitely generated ideal ofC(Λ,R). Then

there existsa∈asuch thatZ(a) =Z(a)

(ii) Ifa⊆bare finitely generated ideals ofC(Λ,_R)then we can choosea∈aandb∈bwitha=λbsuch that

Z(a) =Z(a)⊇Z(b) =Z(b)

Proof. (i) If a is generated by f1, f2, . . . , fk with fi ∈

C(Λ,R),it is enough to consider

a=f₁2+f₂2+. . .+f_k2.

(ii) Let us consider the elementsa0 ∈aandb0 ∈ b(for example, built as in the previous item) such that

Z(a) =Z(a0)⊇Z(b0) =Z(b).

The result is followed considered the elementsb =b0 ∈b

ya=a0b0∈a. Hence

Z(a) =Z(a0b0) =Z(a0)∪Z(b0) =Z(a)∪Z(b) =Z(a).

Theorem 4.4. Let beaan ideal ofC(Λ,_R)generated by

f1, f2, . . . , fk. The following conditions are equivalent.

(i) a6=C(Λ,_R)

(ii) Z(f1)∩Z(f2)∩. . .∩Z(fk)6=∅

(iii) f12+f22+. . .+fk2is not an unit ofC(Λ,R)

Proof. It is sufficient to note that

Z(a) =Z(f₁2+f₂2+. . .+f_k2) =

Z(f1)∩Z(f2)∩. . .∩Z(fk)6=∅.

Recall that Theorem 4.2 states that then2sets given by n

ZUj

˜

BiΣ

o

for1≤i≤n,1≤j≤n,are a complete system invariants for pointwise feedback relation. We conclude this section by studying that set of invariants. Main properties are given in the next results.

First of all, note that a reachable linear system overR

verifies: Uj

˜

B_iΣ =R, for allj ≤i. In terms of sets of zeroes of reachability maps, one has the following result: Theorem 4.5. Let beΣ = (A, B)a reachable linear dy-namical system of type(n, m)overR=C(Λ,_R)then one hasZUj

˜

B_iΣ=∅for allj≤i.

Proof. First note that since system is reachable then

Uj

˜

BΣ

i

=R.

Now, by contradiction, suppose thatλ∈ Z(UB˜_iΣit follows thatf(λ) = 0for allf ∈ UB˜Σ

i

and therefore

1∈ U/ B˜_iΣ, which is a contradiction.

Lemma 4.6. Let beΣκ= (Aκ, Bκ)a Brunovsky form of

type(n, m)over the fieldK.IfrankB˜Σκ i+1

< j,then

rankB˜Σκ i

< j−1

.

Proof. With the notation of Theorem 3.4 we have

rankB˜Σκ i+1

=n1+n2+. . .+ni+ni+1< j

and asni+1≥1,it is followed that

rankB˜Σκ i

=n1+n2+. . .+ni=

rankB˜Σκ i+1

−ni+1< j−ni+1≤j−1

.

Theorem 4.7. Let beΣ = (A, B)a reachable linear dy-namical system of type(n, m)with coefficients in the ring

R=C(Λ,R) then

ZUj

˜

B_iΣ⊇ZUj+1

˜

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Proof. It is immediate from the previous Lemma.

Theorem 4.8. Let beΣ = (A, B)a reachable dynamical system of type(n, m)overR=C(Λ,R), then

ZUj

˜

B_iΣ=ZUj+1

˜

B_iΣ₊₁forj≤2i,

where

˜

B_iΣ= B|AB|. . .|Ai−1B

Proof. By means of Theorem 4.7 it is enough to prove

ZUj

˜

B_iΣ⊆ZUj+1

˜

B_iΣ₊₁.

Let beλ0 ∈ Z Uj B|AB|. . .|Ai−1B

and we consi-der the system Σ(λ0), reachable over R, then Σ(λ0) is feedback equivalent to a Brunovsky formΣκ = (Aκ, Bκ)

whereκ={κ1, κ2, . . . , κs}is a partition ofn withκ1≥

κ2≥ · · · ≥κs.

As

rank B(λ0)|A(λ0)B(λ0)|. . .|Ai−1(λ0)B(λ0)

< j

it is

rank Bκ|AκBκ|. . .|Aiκ−1Bκ

< j,

or equivalent

rank Bκ|AκBκ|. . .|Aκi−1Bκ≤j−1.

Let us see

rank Bκ|AκBκ|. . .|AiκBκ

≤j.

Arguing by contradiction, assume

> j,

we’ll have

rank Ai_κBκ

>1,

thenEi

1e16= 0andE2ie26= 0and it must be

i < κ1 and i < κ2. But this is the same that

i < κ2≤κ1,

and taking into account only the column system vectors li-nearly independent

e1, e2, Aκe1, Aκe2, . . . , Aκi−1e1, Aiκ−1e2

in the matrixB˜Σ

i = Bκ|AκBκ|. . .|Aiκ−1Bκ,we have rank Bκ|AκBκ|. . .|Aiκ−1Bκ

≥2i,

but, by hypothesis

rank Bκ|AκBκ|. . .|Aiκ−1Bκ

≤j−1,

then

j−1≥2i,

so that

j≥2i+ 1,

and it is contrary to the course. Therefore, it should be

≤j,

and also

rank B(λ0)|A(λ0)B(λ0)|. . .|Ai(λ0)B(λ0)≤j,

then

λ0∈Z Uj+1 B|AB|. . .|Ai−1B

.

In the table at the Note 4.10, Theorem 4.7 test conten-tions chains parallel to the main diagonal, while Theorem 4.8 proves the equalities. In addition, these equalities can not be extended toj > 2i,as it is shown in the following result.

Theorem 4.9. Let beΣ = (A, B)a reachable linear dy-namical system of type (n, m) over R = C(Λ,_R), and let be λ0 ∈ Λ such that Σ(λ0) ∼ Σκ where κ =

{κ1, κ2, . . . , κs},withη={n1, n2, . . . , np}the dual

par-tition associated checking

2i≤n1+n2+. . .+ni< j

< j+ 1≤n1+n2+. . .+ni+ni+1,

then

λ0∈Z

Uj

˜

B_iΣ\ZUj+1

˜

B_iΣ₊₁

Proof. By Theorem 3.4, and being

n1+n2+. . .+ni< j,

we have

Uj

˜

BΣ(λ0) i

=Uj

˜

BΣκ i

= 0,

thenλ0∈Z

Uj

˜

BΣ

i

.

Also, as

j+ 1≤n1+n2+. . .+ni+ni+1

it is

Uj+1

˜

BΣ(λ0) i+1

=Uj+1

˜

BΣκ i+1

6

= 0,

andλ0∈/ Z

Uj+1

˜

B_iΣ₊₁.

Note 4.10. From Theorems 4.5, 4.7 and 4.8 con-tentions and equalities are deduced between sets

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[image:8.595.56.296.87.573.2]

Figure 1.Some relations inn

Z

U_jB|AB|. . .|Ai−1B

o

5 Low dimensional cases

(

n

≤

5)

For the casesn= 2, n= 3andn= 4it follows Corollary 5.1. The sets of zeroes

Z(U2(B))

are a complete set of invariants for the feedback pointwise class of the systemΣ = (A, B)reachable of type(2, m),

over the ringR=C(Λ,_R).

For systems of type(3, m)we can give the similar result. Corollary 5.2. The sets of zeroes

Z(U2(B))⊇Z(U3(B))

Corollary 5.3. The sets of zeroes

Z(U4(B))⊇Z(U3(B))⊇Z(U4(B|AB))⊇Z(U2(B))

After these general reductions, there are another reducti-ons depending onn. For example in the casen= 5we can give the following result.

Theorem 5.4. Let beΣ = (A, B)a reachable linear dyn-amical system of type(5, m)over the ringR=C(Λ,_R).

Then

Z(U5(B))⊇Z(U4(B))⊇Z(U5(B|AB))⊇

Z(U3(B))⊇Z(U4(B|AB))⊇Z(U2(B))

Proof. It is enough to prove

Z U5 B|AB . . .|An−1B⊇Z(U3(B))

.

Let beλ0∈Z(U3(B))thenΣ (λ0) = (A(λ0), B(λ0)) is a reachable system over the fieldK = R. By Bruno-vsky theorem,Σ (λ0)only can be equivalent feedback to a Brunovsky formΣκ= (Aκ, Bκ),whereκis one of the

following partitions forn= 5:

κ1= 3≥κ2= 2>0 or

κ1= 4≥κ2= 1>0 or

κ1= 5>0,

where U5(Bκ, AκBκ) = (0).By the feedback

equiva-lence is

U5(B(λ0), A(λ0)B(λ0)) =U5(Bκ, AκBκ) = (0),

andλ0∈Z(U5(B|AB)).

6 Conclusion

The problem of obtaining invariants in the pointwise feedback equivalence overR=C(Λ,R)has been conside-red.The next step will be to construct a canonical form for eachnand the opportunity of stratify the spaceΛsorting these invariants as a lattice. For example given the table of the Figure 1, open question is to generalize reducing results as Theorem 5.4 and can extend results to the ringCk_(Λ_,

R) whereΛis a differentiable manifold or to extend results to the ring of holomorphic functions

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7 Acknowledgements

The Instituto Nacional de Ciberseguridad (Spanish Na-tional Institute for Cybersecurity) (INCIBE) has partially supported this work.

We are also grateful to the annonymous referee for valu-able comments.

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