On Pointwise Feedback Invariants of Linear Parameter-varying Systems

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.


Introduction
Let's consider the family of linear control systems Σ(λ) = ẋ = A(λ)x(t) + B(λ)u(t) y(t) = C(λ)x(t) (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 1 C(Λ,R) and 0 C(Λ,R) are respectively the constant functions λ → 1 and λ → 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.
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.

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

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On Pointwise Feedback Invariants of Linear Parameter-varying Systems 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 λ m ∈ Λ such that f (λ m ) = 0 for every f ∈ m, and conversely, given a point λ ∈ Λ, the set m λ = {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/m λ0 obtained by extension of scalars Now we recall the definition of pointwise feedback equivalence of linear systems as introduced in [8].
Pointwise feedback equivalence is weaker than feedback equivalence. That is to say, Theorem 2.3. If systems Σ = (A, B) and Σ = (A , B ) are feedback equivalent via the action of element (P, Q, K) ∈ F nm (C (Λ, R)) then Σ(λ) and Σ (λ) are feedback equivalent via (P (λ), Q(λ), K(λ)) ∈ F nm (R) Proof. Suppose that (A, B) is feedback equivalent to (A , B ) via (P, Q, K); that is to say A P = P (A + BK), and B = P BQ it follows that above equalities hold on every evaluation; that is 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)) = 0 and det(Q(λ)) = 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 N Σ i generated by columns of the n × im ma- 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: (ii) The canonical homomorphism (iv) If Σ is a reachable system of simple input ndimensional then the modules N Σ i 1≤i≤n and M Σ i 1≤i≤n are free.
(v) If Σ is a Brunovsky system then the modules N Σ i 1≤i≤n and M Σ i 1≤i≤n are free.
Now we deal with the case of fields K: We prove that above invariant K-vector spaces N Σ i 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 κ 1 ≥ κ 2 ≥ · · · ≥ κ s uniquely determined by Σ with n = κ 1 + κ 2 + · · · + κ s , such that Σ is feedback equivalent to the system The integers κ = {κ 1 , κ 2 , . . . , κ s } are called the Kronecker indices of Σ. They are a complete set of invariants for Σ by the action of the feedback group.
Note that if m = 1 and system is controllable it is easy to see that there is just one nonzero index and κ 1 = 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: Proof. See [10].
According the above result rank R M Σ i 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 is the list of invariants we need: 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: Proof. Two reachable linear systems Σ and Σ over C(Λ, R) are pointwise feedback equivalence if and only if (by definition) linear systems Σ(λ) and Σ (λ) are feedback equivalent over R. By Brunovsky's Theorem this is equiva- Hence data (i) is a complete set of pointwise feedback equivalence. Statements (ii) and (iii) are proved in the same way.

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 = (a ij ) an n × m matrix with entries in R and let be i a nonnegative integer. The i−th determinantal ideal of M, denoted by U i (M ) , is the ideal of R generated by all the i × i minors of M. By construction we have Definition 3.1. Let be n ∈ N and κ = {κ 1 , κ 2 , . . . , κ s } with κ 1 ≥ κ 2 ≥ · · · ≥ κ s a partition of n. We call dual partition of κ to the partition η = {n 1 , n 2 , . . . , n p } with n 1 ≥ n 2 ≥ · · · ≥ n p of n where n i , is the number of κ j which are more or equal than i.
where the matrix Then the following properties can be easily verified. i) Let be n 1 the number of κ j greater than or equal to 1, then the definition n i and by the properties i) and ii) it is followed rank(B) = s = n 1 , Conversely, by the equalities rank B Σκ i = n 1 + n 2 + . . . + n i rank B Σκ i−1 = n 1 + n 2 + . . . + n i−1 and the property iii) before it is followed In consequence, with n 1 ≥ n 2 ≥ · · · ≥ n p By the definition of n i (that is 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 κ i and so the complete set of invariants for the Brunovsky form Σ κ , (for which the partition η = {n 1, n 2, . . . , n p } is its dual associated partition).
The following properties of Brunovsky canonical forms over any field K are easily derived: is the Brunovsky form over the field K = R, whose associated partition is κ = {κ 1 , κ 2 , . . . , κ s } , by Theorem 3.4 it is and therefore Σ(λ 0 ) is feedback equivalent to Σ κ .

The pointwise feedback relation
then Σ(λ) is equivalent feedback to Σ (λ) for all λ. The converse is not true in general, and this motivates the study of following relationship. We say Σ and Σ are pointwise feedback equivalent if the systems Σ(λ) = (A(λ), B(λ)) 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).
where f 1 , f 2 , . . . , f k are the minors of order n of the matrix and in consequence rank B Σ(λ) = n, or equivalently Σ(λ) is reachable for all λ in Λ.
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 The following result (see [8]) gives a set of invariants for the pointwise feedback relation. (i) Systems Σ and Σ are pointwise feedback equivalents; that is to say, Σ(λ) and Σ (λ) are feedback equivalents for λ ∈ Λ The inverse contention is also proved because feedback equivalence is a equivalence relation, and symmetric property solves the case.
are isomorphic for all i and all λ ∈ Λ and consequently Σ(λ) and Σ (λ) are feedback equivalent for all λ ∈ Λ or equivalently systems Σ and Σ are 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 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 (ii) Let us consider the elements a ∈ a and b ∈ b (for example, built as in the previous item) such that The result is followed considered the elements b = b ∈ b y a = a b ∈ a. Hence Recall that Theorem 4.2 states that the n 2 sets given by 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 verifies: U j B Σ i = R, for all j ≤ i. In terms of sets of zeroes of reachability maps, one has the following result: Proof. First note that since system is reachable then  Proof. With the notation of Theorem 3.4 we have rank B Σκ i+1 = n 1 + n 2 + . . . + n i + n i+1 < j and as n i+1 ≥ 1, it is followed that Theorem 4.7. Let be Σ = (A, B) a reachable linear dynamical system of type (n, m) with coefficients in the ring Proof. It is immediate from the previous Lemma.
But this is the same that and taking into account only the column system vectors linearly independent e 1 , e 2 , A κ e 1 , A κ e 2 , . . . , but, by hypothesis and it is contrary to the course. Therefore, it should be and also 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.

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.

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.