On Pointwise Feedback Invariants of Linear Parameter-varying Systems

Universal Journal of Applied Mathematics 5(5): 87-95, 2017 DOI: 10.13189/ujam.2017.050501

http://www.hrpub.org

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

c Copyright 2017 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License

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  x˙ = A(λ)x(t) + B(λ)u(t) Σ(λ) = y(t) = C(λ)x(t)

Here some sets of zeroes of functions will arise. A complete set of invariants will be given. Despite this classification 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 systems which are traslated to the study of pointwise feedback equivalence. Section 3 reviews determinantal ranks of matrices in order to get main pointwise feedback invariants 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 cases n ≤ 5 are completely described and characterized in section 5. Finally, we list our conclusions.

2 (1)

where x(t) is the n-dimensional vector of states, u(t) is the m-dimenional vector of external inputs and y(t) is the p-dimensional vector of outputs. Matrices A(λ), B(λ) and C(λ) depend continuously on some parameters λ living in a compact topological space Λ. We’ll denote by C (Λ, R) the ring of continuous real functions defined on Λ with pointwise sum and product. This is a commutative ring where 1C(Λ,R) and 0C(Λ,R) are respectively the constant functions λ 7→ 1 and λ 7→ 0. It is well known that if the matrices have constant coefficients then there is a canonical form for Σ, 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 pointwise feedback equivalence is studied at every point just like classical feedback equivalence for linear systems with constant coefficients. We also are interested in stablish global invariants of linear systems for such pointwise feedback equivalence.

Feedback actions and feedback classification

Let R be a commutative ring with unit 1 6= 0. A minput, n-dimensional linear system over R is just a pair of matrices Σ = (A, B) ∈ Rn×n × Rn×m representing the right-hand-side (dynamic) equation x+ (t) = Ax(t) + Bu(t), where x+ (t) represents time-derivative in continuous time framework or time-shift for discrete systems. 0 0 0 Linear systems Σ = (A , B ) and Σ are equivalent 0 (feedback) when Σ can be transformed to Σ by one element of the feedback group Fnm (R) and we will note this by Σ ∼ Σ. We recall that feedback group Fnm (R) is the generated group by the following three types of transformations: (1) A −→ A0 = P AP −1 , B −→ B 0 = P B for some invertible matrix P. The transformation is a consequence of a change of base in Rn , the state module. (2) A −→ A , B −→ B 0 = BQ for some invertible matrix Q. The transformation is a consequence of change of base in Rm , the input module.

88

On Pointwise Feedback Invariants of Linear Parameter-varying Systems

(3) A −→ A0 = A + BK , B −→ B for some m × n matrix K which 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: When R = K is a field the problem is known as classical case and a classical result of Brunovsky [5, 13, 19] characterizes the class of equivalence of Σ by 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 defined on compact topological space Λ. Note that since Λ is compact then maximal ideals m of R are in one to one correspondence with points in Λ, that is, given a maximal m of R there is a unique point λm ∈ Λ such that f (λm ) = 0 for every f ∈ m, and conversely, given a point λ ∈ Λ, the set mλ = {f : f (λ) = 0} is a maximal of R (the reader can see [1] for details). Let be Σ = (A, B) a linear dynamical system over R = C (Λ, R) . Then linear system over R ∼ = R/mλ0 obtained by extension of scalars C (Λ, R) −→ C (Λ, R) /mλ0 sending f 7→ f (λ0 ) is just obtained by the evaluation at the point λ0 ∈ Λ; that is, Σ(λ0 ) = (A(λ0 ), B(λ0 )). Now we recall the definition of pointwise feedback equivalence of linear systems as introduced in [8]. 0

Definition 2.2. Linear systems Σ and Σ are pointwise feedback equivalent if systems Σ(λ) = (A(λ), B(λ)) and 0 0 0 Σ (λ) = (A (λ), B (λ)) over R are feedback equivalent for all λ ∈ Λ. Pointwise feedback equivalence is weaker than feedback equivalence. That is to say, 0

Theorem 2.3. If systems Σ = (A, B) and Σ = (A0 , B 0 ) 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 , B 0 ) via (P, Q, K); that is to say A0 P = P (A + BK), and B 0 = 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)), B 0 (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 ) and det(Q) are units in C(Λ, R) and a fortiori det(P (λ)) 6= 0 and det(Q(λ)) 6= 0. We are interested in give a complete (and minimal if possible) set of invariants for pointwise feedback relation. First we recall classical invariants and canonical form for constant linear systems over a field.

Definition 2.4. Let be Σ = (A, B) a linear dynamical system of size (m, n) over commutative ring R. Consider the R-module NiΣ generated by columns of the n × im matrix BiΣ = (B|AB| . . . |Ai−1 B). We’ll denote by MiΣ the quotient module MiΣ = Rn /NiΣ . Above modules are feedback invariant associated to a given 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 be Σ = (A, B) a linear dynamical system of size (m, n) over a ring R. Then (i) (0) ⊆ N0Σ ⊆ N1Σ ⊆ . . . ⊆ NnΣ . (ii) The canonical homomorphism Σ Σ ϕi : NiΣ /Ni−1 → Ni+1 /NiΣ Σ → Ax + NiΣ x + Ni−1

is surjective for 1 ≤ i ≤ n − 1. (iii) If Σ is feedback equivalent to Σ0 then NiΣ and MiΣ are 0 0 isomorphic to NiΣ and MiΣ respectly, for 1 ≤ i ≤ n. (iv) If Σ is a reachable system of simple input ndimensional then the modules NiΣ 1≤i≤n and  Σ Mi 1≤i≤n are free. (v)  If Σ is a Brunovsky  system then the modules NiΣ 1≤i≤n and MiΣ 1≤i≤n are free. Now we deal with the case of fields K: We prove that above invariant K-vector spaces NiΣ form a complete and minimal set of feedback invariants. Theorem 2.6. (cf. [5]) Let be Σ = (A, B) a reachable linear dynamical system of size (m, n) over a field K. Then there exist positive integers κ1 ≥ κ2 ≥ · · · ≥ κs uniquely determined by Σ with n = κ1 + κ2 + · · · + κs , such that Σ is feedback equivalent to the system Σκ = (Aκ , Bκ ) where Aκ is the block matrix   Aκ1 0 ··· 0  0  Aκ2 · · · 0   Aκ =  . , . . .. ..  ..  . .. 0

···

0

Aκ s

and block Aκi is the κi × κi matrix 

Aκi

0 0 0 .. .

    =    0 0

1 0 0 .. .

0 1 0 .. .

0 0 1 .. .

··· ··· ··· .. .

0 0

0 0

0 0

··· 0

0 0 0 .. .



       1  0

Universal Journal of Applied Mathematics 5(5): 87-95, 2017

and

(iii) s

z        Bκ =         

0 .. .

m−s

}|

{

z 0

0

···

0 .. .

1 .. .

0 0

0 0 ··· .

···

0

··· ···

0 0

0 0

..

···

0

1 0 .. .

}|

.. . 0 .. .

0

1

0

··· .. . ··· ···

0 0

{                

0

    

κ1

κ2

  

κs



The integers κ = {κ1 , κ2 , . . . , κs } are called the Kronecker indices of Σ. They are a complete set of invariants for Σ by the action of the feedback group. Proof. See [5] or [10]. Above linear system Σκ = (Aκ , Bκ ) is called Brunovsky’s Canonical Form associated to indices κ. Note that if m = 1 and system is controllable it is easy to see that there is just one nonzero index and κ1 = n hence s = 1 in above matrices and consequently Brunovsky’s Canonical Form in this case is just the Canonical 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 of R = 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 be Σ = (A, B) a reachable linear dynamical system of size (m, n) over R. Then the feedback equivalence class of Σ is characterized for each one of the following sets: (i) The Kronecker’s indices {κi }1≤i≤s  (ii) dim(NiΣ ) 1≤i≤n 
Σ (iii) dim NiΣ /Ni−1 1≤i≤n

n  o ˜ Σ(λ) , 1 ≤ i ≤ n, λ ∈ Λ rankR B i

Proof. Two reachable linear systems Σ and Σ0 over C(Λ, R) are pointwise feedback equivalence if and only if (by definition) linear systems Σ(λ) and Σ0 (λ) are feedback equivalent over R. By Brunovsky’s Theorem this is equivaΣ(λ) Σ0 (λ) lent to dimR (Ni ) = dimR (Ni ), for all 1 ≤ i ≤ n, λ ∈ Λ. Hence data (i) is a complete set of pointwise feedback 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 devoted 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 determinantal ranks in the sequel. Let be M = (aij ) an n × m matrix with entries in R and let be i a nonnegative integer. The i−th determinantal ideal of M, denoted by Ui (M ) , is the ideal of R generated by all the i × i minors of M. By construction we have R = U0 (M ) ⊇ U1 (M ) ⊇ . . . ⊇ Ui (M ) ⊇ . . . and Ui (M ) = 0 for i > min {m, n} . The rank of M, denoted by rankR (M ), is the largest i such that
Ui (M ) 6= 0. Then Σ is reachable if and only if Un BnΣ =R. Now we give a technical result of characterization of Brunovsky’s canonical forms Σκ = (Aκ , Bκ ) depending on the sequence of determinantal ideals. Fist, we give with some definitions and notations. Definition 3.1. Let be n ∈ N and κ = {κ1 , κ2 , . . . , κs } with κ1 ≥ κ2 ≥ · · · ≥ κs a partition of n. We call dual partition of κ to the partition η = {n1 , n2 , . . . , np } with n1 ≥ n2 ≥ · · · ≥ np of n where ni , is the number of κj which are more or equal than i. Note 3.2. The application κ → η is biyective on the set of partitions of n. See [2].

Proof. See [10].  According the above result rankR MiΣ 1≤i≤n is a complete set of invariants for the class of feedback of a Brunovsky form. If K is a field and Σ = (A, B) a reachable 
˜ Σ and in dynamical system rank MiΣ = n − rank B i n  o Σ ˜ consequence rank Bi is the list of invariants

Note 3.3. Note that n1 = s.

we need:

(ii) For 1 ≤ i ≤ p − 1 we have   ˜ Σκ = K Un1 +n2 +...+ni B i   ˜ Σκ Un1 +n2 +...+ni +1 B = (0) i

1≤i≤n

Theorem 2.8. A complete set of pointwise feedback invariants for the reachable linear system Σ(λ) = (A(λ), B(λ)) is given by the one (and hence all) follo–wing data: Σ(λ)

89

(i) dimR (Ni ), 1 ≤ i ≤ n, λ ∈ Λ n  o Σ(λ) Σ(λ) (ii) dim Ni /Ni−1 , 1 ≤ i ≤ n, λ ∈ Λ

Theorem 3.4. Let be κ = {κ1 , κ2 , . . . , κs } a partition of n and η = {n1 , n2 , . . . , np } be its associated dual partition. The following conditions are equivalent. (i) κ are the Kronecker indices of system Σ = (A, B)

and for i = p   ˜pΣκ = K Un1 +n2 +...+np B

90

On Pointwise Feedback Invariants of Linear Parameter-varying Systems

Proof. By Brunovsky Theorem, 2.6,  E1  E2  A= ..  .

Conversely, by the equalities   ˜ Σκ = n1 + n2 + . . . + ni rank B i   ˜ Σκ = n1 + n2 + . . . + ni−1 rank B i−1

   , 

and the property iii) before it is followed

Es

rank(Ai−1 B) = ni

where the matrix     Ei =    

0

1

··· .. . .. . ··· ...

0

0 .. .

0 .. .

1 .. .

0 0

0 0

0 0

0

In consequence,



 0   , 0   1  0

rank(B) = n1 = s, rank(AB) = n2 , rank(A2 B) = n3 , .. . rank(Ap−1 B) = np ,

is of dimension κi × κi and  e1 0 · · ·  0 e2 · · ·  B= . .. ..  .. . . 0

0

0

with 0 0 .. .

0 0 .. .

··· ··· .. .

0 0 .. .

es

0

···

0



n1 ≥ n2 ≥ · · · ≥ np By the definition of ni (that is

  , 

t

where ei = (0, . . . , 0, 1) is of dimension κi × 1. Then the following properties can be easily verified. i)    Ah B =  

E1h e1 0 .. .

0 E2h e2 .. .

0

0

··· ··· .. . ···

0 0 .. .

0 0 .. .

··· ···

0 0 .. .

Esh es

0

···

0

ii) If h > κi is Eih ei = 0 0

iii) Im(Ah B) ∩ Im(Ah B) = {0} if h 6= h0 Let be n1 the number of κj greater than or equal to 1, then the definition ni and by the properties i) and ii) it is followed rank(B) = s = n1 , rank(AB) = n2 , rank(A2 B) = n3 , rank(A3 B) = n4 , .. . rank(Anp −1 B) = np , and by property iii)   ˜ Σκ = n1 + n2 + . . . + ni rank B i for 1 ≤ i ≤ p, or equivalently   ˜ Σκ = K Un1 +n2 +...+ni B i  ˜ Σκ = (0) Un1 +n2 +...+ni +1 B i for 1 ≤ i ≤ p.

n1 is equal to the number of κj greater than or equal to 1, n2 is equal to the number ofκj greater than or equal to 2, .. . np is equal to the number of κj greater than or equal to p)

it is obtained κi and so the complete set of invariants for the Brunovsky form Σκ , (for which the partition η =  {n1, n2, . . . , np } is its dual associated partition).  The following properties of Brunovsky canonical forms   over any field K are easily derived:  Corollary 3.5. A Brunovsky canonical form Σκ (Aκ , Bκ ) over the field K verifies   ˜ Σκ = K Ui B 1 ≤ i ≤ n. i

=

Corollary 3.6. A Brunovsky canonical form Σκ = (Aκ , Bκ ) over the field K verifies   ˜ Σκ = n1 + n2 + . . . + ni ≥ i for 1 ≤ i ≤ p rank B i where η = {n1 , . . . , np } is the dual partition of κ. In particular   ˜pΣκ . = n1 + n2 + . . . + np = n. rank B Corollary 3.7. The following conditions hold for K-vector spaces NiΣκ generated by columns of the n × im matrix. ˜ Σκ = (Bκ |Aκ Bκ | . . . |Ai−1 B κ Bκ ) i associated to Brunovsky canonical form Σκ     Σκ (i) dim NiΣκ ≤ dim Ni+1 for all i = 1, 2, . . .     Σκ (ii) If dim NtΣκ = dim Nt+1 then  
dim NtΣκ = · · · = dim NnΣκ = n.

Universal Journal of Applied Mathematics 5(5): 87-95, 2017

Now, we are ready to give a characterization of pointwise feedback equivalence in terms of dimensions of pointwise invariants. 0

0

0

Theorem 3.8. Let be Σ = (A, B) and Σ = (A , B ) two reachable linear dynamical systems of size (m, n) over R = C (Λ, R) . For λ0 ∈ Λ, the following conditions are equivalent. (i) Σ(λ0 ) ∼ Σκ where Σκ = (Aκ , Bκ ) is the Brunovsky’s linear form associated to the Kronecker’s indices κ = {κ1 , κ2 , . . . , κs } (ii) Let be η = {n1 , n2 , . . . , np } the dual partition of κ. Then   Σ(λ ) dimR Ni 0 = 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 field K = R, whose associated partition is κ = {κ1 , κ2 , . . . , κs } , by Theorem 3.4 it is    ˜ Σκ = R  Un1 +n2 +...+ni B i   ˜ Σκ = (0) for 1 ≤ i ≤ p  Un1 +n2 +...+ni +1 B i or equivalently    ˜ Σ(λ0 ) = R  Un1 +n2 +...+ni B i  ˜ Σ(λ0 ) = (0)  Un1 +n2 +...+ni +1 B i

Σ(λ0 )

dim Ni





Theorem 4.1. Let be Λ a compact topological space and Σ = (A, B) be a reachable linear dynamical system of size (m, n) over R = C (Λ, R) . Then the following conditions are equivalent. (i) Σ is reachable over C (Λ, R) (ii) Σ(λ)is reachable over R, for all λ ∈ Λ. Proof. (i)⇒(ii) If Σ is reachable in C (Λ, R) then   ˜ Σ = (f1 , f2 , . . . , fk ) = (1) = C (Λ, R) , Un B where f1 , f2 , . . . , fk are the minors of order n of the matrix ˜ Σ. B ˜ Σ (λ) = B ˜ Σ(λ) , it follows Since B

  ˜ Σ(λ) = n, or equivalently and in consequence rank B for

1 ≤ i ≤ p,



˜ Σ(λ0 ) = n1 + n2 + . . . + ni = rank B i

for 1 ≤ i ≤ p. (ii)⇒(i) Conversely, let be λ0 ∈ Λ such that   Σ(λ ) dim Ni 0 = n1 + n2 + . . . + ni for 1 ≤ i ≤ p, then     ˜ Σ(λ0 ) = n1 + n2 + . . . + ni = rank B ˜ Σκ rank B i i for 1 ≤ i ≤ p Also     ˜ Σ(λ0 ) = Uj B ˜ Σκ for Uj B i i

0

then Σ(λ) is equivalent feedback to Σ (λ) for all λ. The converse is not true in general, and this motivates the study 0 of following relationship. We say Σ and Σ are pointwise feedback equivalent if the systems Σ(λ) = (A(λ), B(λ)) 0 0 0 and Σ (λ) = (A (λ), B (λ)) over R are feedback equivalents for all λ ∈ Λ. Since reachability is a property that it is preserved by feedback, let us see this concept in the ring R = C (Λ, R).

    ˜ Σ(λ) = Un B ˜ Σ ⊗ C (Λ, R) /mλ = C (Λ, R) /mλ , Un B

then 

Σ(λ) is reachable for all λ in Λ. (ii)⇒(i) Conversely, assume  
˜ Σ = Un B|AB| . . . |An−1 B = (f1 , f2 , . . . , fk ), Un B where f1, f2 , . . . , fk are all minors of order n of the matrix ˜ Σ . As Σ(λ) = (A(λ), B(λ)) is reachable for all λ in Λ, B   ˜ Σ(λ) = n. we have rank B Then the set of zeroes (see below) Z(f1 , . . . , fk ) = ∅ is empty. Thus ideal generated by {f1 , f2 , . . . , fk } is the 2 2 2 whole ring. A fortiori  f1 + f2 + . . . + fk is an unit in ˜ Σ = R and Σ is a reachable syC (Λ, R) . Then Un B stem.

1 ≤ i ≤ n, 1 ≤ j ≤ n,

and therefore Σ(λ0 ) is feedback equivalent to Σκ .

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 of R = C (Λ, R) . We’ll denote by Z(a) the set Z(a) = {λ ∈ Λ / f (λ) = 0

4

91

The pointwise feedback relation

Note that if A = (fij ) is a matrix over R = C (Λ, R) then A(λ) is the matrix (fij (λ)). Let be Σ = (A, B) and 0 Σ = (A0 , B 0 ) two systems n-dimensional with m inputs 0 over R = C (Λ, R) , if Σ is equivalent feedback to Σ ,

for all f ∈ a}

The following result (see [8]) gives a set of invariants for the pointwise feedback relation. 0

Theorem 4.2. Let be Σ = (A, B) and Σ = (A0 , B 0 ) two reachable linear systems of type (n, m) over the ring R = C(Λ, R) then the following conditions are equivalent.

92

On Pointwise Feedback Invariants of Linear Parameter-varying Systems

(i) Systems Σ and Σ0 are pointwise feedback equivalents; 0 that is to say, Σ(λ) and Σ (λ) are feedback equivalents for λ ∈ Λ    ˜Σ (ii) For all 1 ≤ i, j ≤ n one has Z Uj B = i   0  ˜Σ Z Uj B 1 ≤ i ≤ n, 1 ≤ j ≤ n i 



(i) a 6= C (Λ, R) (ii) Z(f1 ) ∩ Z(f2 ) ∩ . . . ∩ Z(fk ) 6= ∅ (iii) f12 + f22 + . . . + fk2 is not an unit of C (Λ, R) Proof. It is sufficient to note that Z(a) = Z(f12 + f22 + . . . + fk2 ) =



˜Σ Proof. (i) ⇒ (ii). If λ ∈ Z Uj B then i     Σ(λ) Σ(λ) ˜ Uj B = (0) and hence dim Ni < j. Since i Σ(λ) and Σ0 (λ) are feedback equivalent  over R it follows Σ0 (λ) that it is also satisfied dim Ni < j which yields   0  ˜Σ λ ∈ Z Uj B . The inverse contention is also proved i because feedback equivalence is a equivalence relation, and symmetric property solves the case. (i) ⇐ (ii) Let be λ ∈ Λ. Observe that (ii) yields that   n   o Σ(λ) ˜iΣ dim Ni = min j : λ ∈ Z Uj B = n   0 o   Σ(λ) ˜Σ = min j : λ ∈ Z Uj B = dim N i i Σ(λ)

Σ0 (λ)

Z(f1 ) ∩ Z(f2 ) ∩ . . . ∩ Z(fk ) 6= ∅.

Recall that Theorem 4.2 states that the n2 sets given by n   o ˜iΣ Z Uj B for 1 ≤ 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 over R ˜ Σ = R, for all j ≤ i. In terms of sets of verifies: Uj B i zeroes of reachability maps, one has the following result:

Therefore R vector spaces Ni and Ni are isomorphic for all i and all λ ∈ Λ and consequently Σ(λ) and Σ0 (λ) are feedback equivalent for all λ ∈ Λ or equivalently systems Σ and Σ0 are point wise feedback equivalents.

Theorem 4.5. Let be Σ = (A, B) a reachable linear dynamical of type (n, m) overR = C(Λ, R) then one  system   ˜Σ has Z Uj B = ∅ for all j ≤ i.

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

Proof. note that since system is reachable then  First  ˜ Σ = R. Uj B i   ˜ Σ it Now, by contradiction, suppose that λ ∈ Z(U B i   Σ ˜ follows that f (λ) = 0 for all f ∈ U Bi and therefore   ˜ Σ , which is a contradiction. 1∈ /U B i

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 be a a finitely generated ideal of C(Λ, R). Then there exists a ∈ a such that Z(a) = Z(a) (ii) If a ⊆ b are finitely generated ideals of C(Λ, R) then we can choose a ∈ a and b ∈ b with a = λb such that

i

Lemma 4.6. Let be Σκ = (Aκ , Bκ) a Brunovsky form of  Σκ ˜ < j, then type(n, m) over the field K. If rank B i+1





˜ Σκ < j − 1 rank B i

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 + . . . + fk2 . (ii) Let us consider the elements a0 ∈ a and b0 ∈ b (for example, built as in the previous item) such that Z(a) = Z(a0 ) ⊇ Z(b0 ) = Z(b).

Proof. With the notation of Theorem 3.4 we have   ˜ Σκ = n1 + n2 + . . . + ni + ni+1 < j rank B i+1 and as ni+1 ≥ 1, it is followed that   ˜ Σκ = n1 + n2 + . . . + ni = rank B i   ˜ Σκ − ni+1 < j − ni+1 ≤ j − 1 rank B i+1

0

The result is followed considered the elements b = b ∈ b y a = a0 b0 ∈ a. Hence 0 0

0

0

Z(a) = Z(a b ) = Z(a ) ∪ Z(b ) = Z(a) ∪ Z(b) = Z(a).

Theorem 4.4. Let be a an ideal of C (Λ, R) generated by f1 , f2 , . . . , fk . The following conditions are equivalent.

. Theorem 4.7. Let be Σ = (A, B) a reachable linear dynamical system of type (n, m) with coefficients in the ring R = C(Λ, R) then       Σ ˜iΣ ˜i+1 Z Uj B ⊇ Z Uj+1 B

Universal Journal of Applied Mathematics 5(5): 87-95, 2017

Proof. It is immediate from the previous Lemma.

93

so that j ≥ 2i + 1,

Theorem 4.8. Let be Σ = (A, B) a reachable dynamical system of type (n, m) over R = C(Λ, R), then       ˜Σ ˜Σ for j ≤ 2i, = Z Uj+1 B Z Uj B i+1 i

and it is contrary to the course. Therefore, it should be
rank Bκ |Aκ Bκ | . . . |Aiκ Bκ ≤ j,

where

and also ˜iΣ = B|AB| . . . |Ai−1 B B




Proof. By means of Theorem 4.7 it is enough to prove       Σ ˜iΣ ˜i+1 Z Uj B ⊆ Z Uj+1 B .

Let be λ0 ∈ Z Uj B|AB| . . . |Ai−1 B and we consider the system Σ(λ0 ), reachable over R, then Σ(λ0 ) is feedback equivalent to a Brunovsky form Σκ = (Aκ , Bκ ) where κ = {κ1 , κ2 , . . . , κs } is a partition of n with κ1 ≥ κ2 ≥ · · · ≥ κs . As
rank B(λ0 )|A(λ0 )B(λ0 )| . . . |Ai−1 (λ0 )B(λ0 ) < j it is rank Bκ |Aκ Bκ | . . . |Ai−1 κ Bκ < j,


or equivalent
rank Bκ |Aκ Bκ | . . . |Ai−1 κ Bκ ≤ j − 1.


rank B(λ0 )|A(λ0 )B(λ0 )| . . . |Ai (λ0 )B(λ0 ) ≤ j, then λ0 ∈ Z Uj+1 B|AB| . . . |Ai−1 B





.

In the table at the Note 4.10, Theorem 4.7 test contentions chains parallel to the main diagonal, while Theorem 4.8 proves the equalities. In addition, these equalities can not be extended to j > 2i, as it is shown in the following result. Theorem 4.9. Let be Σ = (A, B) a reachable linear dynamical 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 } , with η = {n1 , n2 , . . . , np } the dual partition associated checking 2i ≤ n1 + n2 + . . . + ni < j

Let us see

< j + 1 ≤ n1 + n2 + . . . + ni + ni+1 ,
rank Bκ |Aκ Bκ | . . . |Aiκ Bκ ≤ j.

then

Arguing by contradiction, assume

      ˜ Σ \ Z Uj+1 B ˜Σ λ0 ∈ Z Uj B i i+1


rank Bκ |Aκ Bκ | . . . |Aiκ Bκ > j,

Proof. By Theorem 3.4, and being

we’ll have
rank Aiκ Bκ > 1, then E1i e1 6= 0 and E2i e2 6= 0 and it must be i < κ1

and

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

i < κ2 .

    ˜ Σ(λ0 ) = Uj B ˜ Σκ = 0, Uj B i i

But this is the same that i < κ2 ≤ κ1 , and taking into account only the column system vectors linearly independent  i−1 e1 , e2 , Aκ e1 , Aκ e2 , . . . , Ai−1 κ e1 , Aκ e2
˜ Σ = Bκ |Aκ Bκ | . . . |Ai−1 Bκ , we have in the matrix B κ i
rank Bκ |Aκ Bκ | . . . |Ai−1 κ Bκ ≥ 2i, but, by hypothesis
rank Bκ |Aκ Bκ | . . . |Ai−1 κ Bκ ≤ j − 1, then j − 1 ≥ 2i,

   ˜Σ . then λ0 ∈ Z Uj B i Also, as j + 1 ≤ n1 + n2 + . . . + ni + ni+1 it is

    ˜ Σκ 6= 0, ˜ Σ(λ0 ) = Uj+1 B Uj+1 B i+1 i+1    ˜Σ and λ0 ∈ / Z Uj+1 B . i+1 Note 4.10. From Theorems 4.5, 4.7 and 4.8 contentions and equalities

are deduced between sets  Z Uj B|AB| . . . |Ai−1 B for 1 ≤ i ≤ n, 1 ≤ j ≤ n, that are showed in the following scheme in the Figure 1.

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On Pointwise Feedback Invariants of Linear Parameter-varying Systems

are a complete set of invariants for the feedback pointwise class of the system Σ = (A, B) reachable of type (3, m), over the ring R = C(Λ, R). Corollary 5.3. The sets of zeroes Z (U4 (B)) ⊇ Z (U3 (B)) ⊇ Z (U4 (B|AB)) ⊇ Z (U2 (B)) are a complete set of invariants for the feedback pointwise class of the system Σ = (A, B) reachable of type (4, m), over the ring R = C(Λ, R). After these general reductions, there are another reductions depending on n. For example in the case n = 5 we can give the following result. Theorem 5.4. Let be Σ = (A, B) a reachable linear dynamical system of type (5, m) over the ring R = 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−1 B





⊇ Z (U3 (B))

. Let be λ0 ∈ Z (U3 (B)) then Σ (λ0 ) = (A (λ0 ) , B (λ0 )) is a reachable system over the field K = R. By Brunovsky theorem, Σ (λ0 ) only can be equivalent feedback to a Brunovsky form Σκ = (Aκ , Bκ ), where κ is one of the following partitions for n = 5: κ1 = 3 ≥ κ2 = 2 > 0 κ1 = 4 ≥ κ2 = 1 > 0 κ1 = 5 > 0,

or or

where U5 (Bκ , Aκ Bκ ) = (0) . By the feedback equivalence is Figure 1. Some relations in

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

n   o Z Uj B|AB| . . . |Ai−1 B

and λ0 ∈ Z (U5 (B|AB)) .

5

Low dimensional cases (n ≤ 5) For the cases n = 2, n = 3 and n = 4 it follows

Corollary 5.1. The sets of zeroes Z (U2 (B)) are a complete set of invariants for the feedback pointwise class of the system Σ = (A, B) reachable of type (2, m), over the ring R = 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))

6

Conclusion

The problem of obtaining invariants in the pointwise feedback equivalence over R = C(Λ, R) has been considered.The next step will be to construct a canonical form for each n and the opportunity of stratify the space Λ sorting 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 ring C k (Λ, R) where Λ is a differentiable manifold or to extend results to the ring of holomorphic functions H(Ω) where Ω ⊆ C.

Universal Journal of Applied Mathematics 5(5): 87-95, 2017

7

Acknowledgements

The Instituto Nacional de Ciberseguridad (Spanish National Institute for Cybersecurity) (INCIBE) has partially supported this work. We are also grateful to the annonymous referee for valuable comments.

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