Jérémie Kreiss

Jean-François Trégouët

Input Redundancy: Definitions, Taxonomy, Characterizations and Application to Over-Actuated Systems

Keywords: Input-redundant systems, Over-actuated systems, Linear systems

For input redundant systems, it is possible to construct distinct inputs producing identical outputs, from the same initial state. Thus, a system is input redundant if it is not left-invertible. In this paper, a rigorous framework is proposed to support this new definition. From the observation that state trajectories induced by the two inputs can be identical or not, an enriched taxonomy of input redundancy is also designed. A comprehensive set of characterizations is associated with the proposed definitions. A degree of redundancy is also conceived. This allows to carry out a thorough comparison with the state-of-the-art. Finally, the fact that over-actuated systems are input redundant is formally proved. This fact leads to a control design framework which can easily cope with over-actuated systems equipped with dynamical actuators.

Introduction

Substituting a high-capacity actuator by several ones working in concert has many technological advantages: Examples include state-of-health and/or thermal management, resilience to failure, cost reduction, etc [START_REF] Johansen | Control allocation-a survey[END_REF][START_REF] Trégouët | Reaction wheels desaturation using magnetorquers and static input allocation[END_REF][START_REF] Huang | Circuit theoretic classification of parallel connected dc-dc converters[END_REF]. Such strategy hinges on the idea of creating a so-called input redundant (IR) system.

Historically, the property of IR 1 has been characterized by the existence of a non trivial null space of the input matrices. That is

ρ := dim Ker {[ B D ]} > 0, (1)
when the considered dynamical system Σ is governed by the following equations ẋ(t) = Ax(t) + Bu(t), x(0) =: x 0 , (2a)

y(t) = Cx(t) + Du(t), (2b)
for some quadruple (A, B, C, D) of appropriate dimensions. Here, vectors u(t) ∈ R m , x(t) ∈ R n =: X and y(t) ∈ R p are the input, the state and the output at time t. Well surveyed in [START_REF] Johansen | Control allocation-a survey[END_REF], most of the existing literature on IR relies on characterization [START_REF] Johansen | Control allocation-a survey[END_REF].

However, (1) falls short in dealing with systems like the one associated with the following quadruple:

(   -1 1 0 -1   ,   1 0 0 1   , 1 0 , 0 0 ). (3)
Even if [START_REF] Johansen | Control allocation-a survey[END_REF] is not satisfied, it is clear that any of the two inputs on its own has enough authority to control the output. This suggests that this system shall be regarded as IR as well.

as an alternative characterization of IR [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF]. Indeed, it holds G = [ 1 1 ] for (3) so that this system is said weakly IR in [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF], as opposed to strongly IR when (1) holds. Focusing on right-invertible, minimal and strictly proper systems, weak IR is re-characterized in [START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF] as follows:

m ≥ rank B > p. [START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF] One can verify that this condition is satisfied for [START_REF] Huang | Circuit theoretic classification of parallel connected dc-dc converters[END_REF].

The key point is that (4) and [START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF] are not equivalent, so that definition of weak IR has implicitly changes from [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF] to [START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF]. As formally stated in the sequel, ( 5) is related to left-invertibility, whereas (4) has to do with transmission zeros.

Often misunderstood, this crucial point impacts not only how dynamical system are classified but also how the related control problem is tackled. To see this, let us illustrate the discussion on system (3). In both [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF] and [START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF], the proposed control strategy amounts to add a signal ϕ to y c delivered by a given controller

C designed beforehand, see Fig. 1. For system (3), one can define y c (t) = -x(t). One gets u = ϕ + y c
where ϕ is designed by A to optimize some criterion, e.g. u , while being as much invisible as possible from the output y. This strategy aims preserving dynamical response induced by the first controller. Following [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF], one defines ϕ(t) = 1 -1 w(t) with w sufficiently slow. Observe that ϕ belongs to Ker {G } so that internal steady state is smoothly reconfigured via w, without affecting the asymptotic value of y. However, ϕ inevitably affects the transient of y. This is in stark contrast with the control scheme proposed in [START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF] where

ϕ(t) is defined as -x2(t) w(t)
. In this case, not only w is made completely invisible from the output but also this signal can be selected arbitrarily fast while preserving internal stability. Although superior, this second strategy cannot be implemented on all systems for which the first one is applicable. As shown in this paper, this originates from the fact that (4) does not implies [START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF].

Σ C A y c + + ϕ x u y Figure 1:
The control scheme used to illustrated differences between approaches proposed in [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF] and [START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF].

Characterization of IR is therefore of utmost importance as it underpins the related control design strategy. For this reason, this paper aims setting the ground for further developments of control design for this class of systems by explicitly unveil relationships between existing characterizations of IR. This analysis is expected to clarify the implication behind the selection of a characterization, among a set of seemingly equivalent ones.

Among existing works, [START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF] is the closest to the framework that is promoted here. Yet, there is a number of issues that prevent this important study to receive the audience that it deserves. (i) Analysis is conducted under the assumption of right-invertibility, minimality and strict properness. Not only this makes the study less general, but also it does not help in fighting against widespread misconceptions which are only valid in this context. The main one is that m > p is a necessary condition for LTI system to be IR. This paper provides an explicit counter example of this fact. (ii) In the context of right-invertible, minimal and strictly proper systems, the analysis hinges on the claim that dimension of a particular vector space V , renamed N hereafter, is equal to m -p. This technical fact is of foremost importance since it can be interpreted as the number of independent input directions (after regular state-feedback) which do not affect the output. In this paper, this result is formally proved and it is shown that this dimension cannot be computed from m, p and ρ in general. (iii) The retained taxonomy is quite misleading, since a system can be strongly but not weakly IR, which challenges common sense. For instance, quadruple (0

, [1 1], 1, [0 0]) leads to m = 2 ≥ rank B = p = 1,
so that (5) does not hold, unlike condition m > rank B = p characterizing strong IR in the sense of [START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF].

For this reason, a richer taxonomy is proposed here. It distinguishes between two kinds of weak redundancy that are different in nature.

Contributions are now in order. (i) The core idea of this paper is to redefine IR as the contrary of left-invertibility, i.e. for IR system, it is possible to construct distinct inputs producing identical outputs from the same initial state. In Sec. 2, a formal and unambiguous statement is associated to this intuitive idea, which is widely applicable i.e. for a large class of system. Note that such a formulation was sometimes approached (see [5, p.4871] and [6, p.346]), but, to the best of our knowledge, no clear and rigorous statement can be found in the literature. (ii) Besides, this paper revisits and enriches the existing taxonomy of IR as well as introduces a new degree of IR by exploiting this new framework. (iii) The considered system is not necessarily right-invertible, stable, strictly proper, minimal or minimum phase. In this context, several equivalent tractable characterizations of IR, its taxonomy and its degree are derived by way of a structural

analysis adopting an open-loop view point and by exploiting existing characterizations of left-invertibility (see Sec. 3 and Sec. 4). (iv) This allows us to perform an in-depth comparison with existing definitions of IR and to unveil differences between them (see Sec. 5). (v) Closely related to IR, over-actuation refers to systems equipped of more effectors than strictly needed to meet the control objectives [START_REF] Johansen | Control allocation-a survey[END_REF]. By relying on previous characterizations, those two classes of systems are formally related in Sec. 6. This bridges the gap between two lines of research that are disconnected. As an example of the benefits of this achievement, a control design framework for over-actuated systems is offered, by exploiting results of this paper. Note that the considered actuators can be indifferently static or dynamic. This is in stark contrast with most of the existing results in the literature since the case of dynamical actuators is often considered as much more challenging than static ones [START_REF] Oppenheimer | Control allocation[END_REF]Sec. 8.5].

Context of the study and notations. From (2a), the input-to-state relationship is concisely captured via H x [x 0 ; •] which maps an input trajectory u(•) to the state trajectory x(•) produced by the system when excited by u(•) with an initial condition x(0) = x 0 ∈ R n . The corresponding input-to-output mapping

H[x 0 ; •] : u → CH x [x 0 ; u] + Du is defined by (2b).
Throughout this paper, inputs u are assumed to belong to U, the set of causal, piecewise continuous and exponentially bounded signals. This ensures that the Laplace transform of u exists.

Let x 0 ∈ R n be a given initial condition. The set of all triples (u, x, y) (resp. pairs (u, y)) compatible for

x 0 is denoted by Q(x 0 ) (resp. W(x 0 )), i.e. Q(x 0 ) := {(u, x, y) | H x [x 0 ; u] = x, H[x 0 ; u] = y}, W(x 0 ) := {(u, y) | ∃x : (u, x, y) ∈ Q(x 0 )}.
Finally, note that symbol 0 stands for anything that is not a real number and is zero (a vector, matrix, map, or subspace), according to context. The identity matrix is denoted by I. Given a set A ⊆ R n and a (not necessarily invertible or square) matrix B with n rows, the set B -1 A is defined as {u : Bu ∈ A}.

Definitions and first properties

IR and its taxonomy is defined in this section.

Input redundancy

Definition. System Σ is input redundant (IR) if there exists an output y which can be produced by (at least) two distinct inputs for some x 0 ∈ X , i.e. there exists x 0 ∈ X such that

∃(u 1 , y 1 ), (u 2 , y 2 ) ∈ W(x 0 ) : u 1 = u 2 , y 1 = y 2 . ( 6
)
Remark 1 (IR ⇒ m > p). Condition m > p is often understood as a necessary condition for IR. Let us already provide a counter-example with that respect. Consider the following square system for which

m = p = 2 holds: ẋ =      -1 0 0 0 -1 0 0 0 -1      x +      1 0 1 0 0 1      u y =   1 0 0 0 1 0   x.
Then, IR follows from the observation that distinct input trajectories u 1 : t → 0 and u 2 : t → [ 0 1 ] produce output y = 0 for x 0 = 0.

To prove that Σ is IR, it suffices to find a single output which admits at least two distinct preimages by 80 H[x 0 ; •] for some x 0 ∈ X . 2 From linearity of H, it can be easily proved that this singularity occur for all admissible outputs and for all initial states. Proposition 2.1. If system Σ is IR then it holds:

∀x 0 ∈ X , ∀(u 1 , y 1 ) ∈ W(x 0 ), ∃(u 2 , y 2 ) ∈ W(x 0 ) : u 1 = u 2 , y 1 = y 2 .
Remark 2 (Left-invertibility). System Σ is said left-invertible if identical output trajectories can only originate from identical input trajectories, i.e. for all x 0 ∈ X and for all

u 1 , u 2 ∈ U H[x 0 ; u 1 ] = H[x 0 ; u 2 ] ⇒ u 1 = u 2 . ( 7
)
Therefore, system Σ is IR if and only if Σ is not left-invertible.

Taxonomy

In this section, the state trajectory comes into play. Indeed, the ability of distinct inputs to produce not 85 only identical output but also identical state trajectory is instrumental in classifying different species of IR.

Let us now introduce state trajectories into (6):

∃x 0 ∈ X , ∃(u 1 , x 1 , y 1 ), (u 2 , x 2 , y 2 ) ∈ Q(x 0 ) : u 1 = u 2 , x 1 = x 2 , y 1 = y 2 , ( 8
)
∃x 0 ∈ X , ∃(u 1 , x 1 , y 1 ), (u 2 , x 2 , y 2 ) ∈ Q(x 0 ) : u 1 = u 2 , x 1 = x 2 , y 1 = y 2 , (9)
Observe that if Σ is IR then at least one of the above relationships holds.

Definition. System Σ is input redundant (IR) of:

• The first kind if (8) holds but [START_REF] Ntogramatzidis | Squaring down lti systems: A geometric approach[END_REF] does not, i.e., if and only if

u 1 = u 2 y 1 = y 2    ⇒ x 1 = x 2 (10)
for all x 0 ∈ X and for all (u 1 , x 1 , y 1 ), (u 2 , x 2 , y 2 ) ∈ Q(x 0 );

• The second kind if (8) does not hold but (9) does, i.e., if and only if

u 1 = u 2 y 1 = y 2    ⇒ x 1 = x 2 (11)
holds for all x 0 ∈ X and for all (u 1 , x 1 , y 1 ), (u 2 , x 2 , y 2 ) ∈ Q(x 0 );

• The third kind if ( 8) and ( 9) hold.

The previous definition induces that different kinds are mutually exclusive: No system Σ can be simultaneously of different kinds.

Characterizations

Let us now associate tractable conditions to definitions introduced in Sec. 2. To the end, define V * and R * as the weakly unobservable subspace and the controllable weakly unobservable subspace, respectively (see [START_REF] Trentelman | Control theory for linear systems[END_REF]).

Input redundancy

The following theorem, proved in Appendix A, proposes several characterizations of IR.

Theorem 3.1. Define ρ as in [START_REF] Johansen | Control allocation-a survey[END_REF] and N as follows:

R m ⊇ N := B -1 V * ∩ Ker {D} . ( 12
)
Then, the following statements are equivalent:

(i) System Σ is IR; (ii) dim (R * ) > 0 or ρ > 0;
(iii) dim(N ) > 0;

(iv) Transfer matrix G(s) of Σ is not left-invertible, i.e. there exists a non zero polynomial vector q such that G(s)q(s) = 0 for all s ∈ C;

105 (v) System matrix of Σ P (s) :=   sI -A -B C D   ( 13
)
is not left-invertible.

Remark 3. Given a matrix T (s), parametrized by s ∈ C. Its normal rank is defined as follows:

nrank T := max s∈C rank T (s).

It comes out that (iv) and (v) of Th. 3.1 are equivalent to nrank G < m and nrank P < n+m, respectively.

Taxonomy

Observe that Ker

{[ B D ]} belongs to N . The codimension of Ker {[ B D ]} in N is denoted by ν: ν := dim (N /Ker {[ B D ]}) (14)
so that dimension of N is the sum of two terms:

dim N = ρ + ν, ( 15
)
where ρ is defined by [START_REF] Johansen | Control allocation-a survey[END_REF]. From this equation and (iii) of Th. From (15), note that the following equivalence readily follows by rewriting (ii) of Th. 3.1 as ρ+dim(R * ) >

IR ρ ν 1st kind > 0 = 0 2nd kind = 0 > 0 3rd kind > 0 > 0
0: dim R * > 0 ⇔ ν > 0. ( 16
)
This relationship makes Prop. 3.2 more convenient since dim R * is typically easier to compute than ν.

Degree of redundancy and additional facts

Further discussions are provided in this section, leading to a degree of IR.

A matrix view point

Define Q = [Q a , Q b ] ∈ R n×n and H = [H a , H b , H c ] ∈ R m×m two invertible matrices satisfying Im {Q a } = R * , Im {H a } = Ker {[ B D ]} and Im {[H a , H b ]} = N .
Given any friend4 F of V * , apply feedback transformation u = F x + w to Σ and changes of coordinates ŵ = H -1 w and x = Q -1 x. Matrices of the resulting quadruple (Q -1 (A + BF )Q, Q -1 BH, (C + DF )Q, DH) with input ŵ, state x and output y has the following structure:

     A 11 A 12 0 B 12 B 13 0 A 22 0 0 B 23 0 C 2 0 0 D 3      . ( 17
)
Further, let (i) ŵ be decomposed into ŵa (t) ∈ R ρ , ŵb (t) ∈ R ν and ŵc (t) ∈ R m-ρ-ν , (ii) x be decom- Involved subsystems enjoy the following property: Pair (A 11 , B 12 ) of Σ2 is controllable and Σ3 is leftinvertible (see [9, p.170]). As a result, input ŵc is uniquely defined by pair (x 0 , y), unlike ŵa and ŵb . Hence, ŵc corresponds to part of the input w which is fixed by (x 0 , y). As a possible control purpose, ŵa,b can be used to minimized the input norm and/or to drive coordinate xa toward a relevant reference, see [START_REF] Galeani | On input allocation-based regulation for linear overactuated systems[END_REF]. Fig. 2 clarifies why N , in which ŵa (t) and ŵb (t) belongs to, is sometimes called "input unobservability subspace".

posed into xa (t) ∈ R dim

Degree of redundancy and computation of dim N

From the above discussion, the scalar dim N = ρ + ν corresponds the number of independent input directions (after regular state-feedback with any friend of V * ) which do not affect the output. Specifically, ρ of them do not affect the state whereas ν impact the state. As a result, dim N is related to the size of the set of all inputs leading to the same output. This motivates the use of ρ and ν to quantify the degree of IR.

Definition. The pair (ρ, ν), defined by ( 1) and ( 14), is called degree of redundancy of system Σ.

Next lemma, proved in Appendix C, offers an efficient way to compute dim N = ρ + ν. (18)

If Σ is right-invertible

The following corollary particularizes the previous results under the assumption of right-invertibility , which can be characterized via nrank P = n + p [8, Th. 8.13].

Corollary 1. Assume that Σ is right-invertible. It holds

dim N = m -p, ( 19
)
so that Σ is IR iff m > p holds.
Combined with Prop. 3.2 and (15), this corollary allows to conclude on the kind of IR on the basis of values of m, p and ρ solely.

Comparison with the literature

In the last decade, several definitions, characterizations and taxonomies of IR have been proposed in the literature. Let us relate the material of [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF][START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF][START_REF] Galeani | On dynamic input allocation for fat plants subject to multi-sinusoidal exogenous inputs[END_REF] with contents presented in previous sections.

Hereafter, without further details, IR (of the k-th kind) refers to definition proposed in Sec. 2 and 3. The symbols P and Z represent the sets of poles and zeros (transmission zeros) of G, respectively.

Comparison with [4]

For the first time in the literature, a taxonomy of IR has been proposed in [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF].

[4] : System Σ is said strongly IR if ρ > 0 and weakly IR if G := lim s→0 G(s) is finite and satisfies Ker {G } = {0}.

150

The following equivalence immediately derives from Th. 3.1 and Prop. 3.2:

IR 1st kind or 3rd kind ⇔ strong IR as in [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF].

Let us focus on weak redundancy in the sense of [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF]. Observe that G is finite iff 0 ∈ P holds. 5 In this case and in view of Rem. 3, IR implies rank G ≤ nrank G < m. This yields:

IR 0 ∈P

=⇒ weak (and possibly strong)

IR as in [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF].

(20)

Focusing on redundancy of the 2nd kind where ρ = 0, we also have:

IR 2nd kind 0 ∈P =⇒ weak but not strong
IR as in [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF].

Let us emphasize that the proposed definition applies even if 0 belongs to P. As an example, the quadruple (0, I 2 , [1 1], 0) is associated with an IR system which is neither strong nor weak IR as in [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF]. Indeed, in this case, matrix G(s) = (1/s)[1 1] is neither left-invertible (which proves IR) nor finite when s tends to 0, and ρ equals zero.

The next example shows that the converse implication of (20) does not hold, i.e. IR cannot be inferred 155 from weak IR as in [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF].

Example 1. Consider the following system

ẋ = -x + u, y = -x + u.
so that G(s) = s/(s + 1) holds. Therefore, the system is weakly IR in the sense of [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF], since G(0) = 0.

Yet, G(s) is invertible (and a fortiori left-invertible), so that this system is not IR: For a given initial state, identical outputs can only originate from identical inputs. 5 Clearly if 0 ∈ P, then strong IR implies weak IR since Ker

B D
⊆ G holds. The converse implication is not true in general.

10

The previous example shows that G(s) can lose rank for some specific s, even if nrank G equals m, i.e.

rank G < m does not imply nrank G < m. Those value of s satisfying rank G(s) < nrank G are the transmissions zeros. As a result, if 0 ∈ Z holds, then weak IR as in [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF] implies nrank G = rank G < m, so that:

IR 0 ∈Z
⇐= weak IR as in [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF].

To sum up, the proposed definition is more general in the sense that it can be applied regardless of P.

The proposed taxonomy is also richer as it distinguishes between two kinds of strong redundancy as in [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF] (that is IR of the 1st and 3rd kind) which are different in nature.

Comparison with [5]

The taxonomy of [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF] is revisited in [START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF] under the following assumption.

Assumption 1. Σ is minimal, strictly proper and right-invertible, so that rank C = p holds.

[5] : Under ASM 1, strong IR means m > rank B = p whereas weak IR means m ≥ rank B > p.

IR as in [START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF] is therefore implicitly characterized by m > p. It follows that this condition is equivalent to IR as proved by Cor. 1:

IR ASM 1
⇐⇒ IR as in [START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF].

Suppose that Σ is IR and that ASM 1 holds. From Cor. 1, rank [ B D ] = rank B = m -ρ > p is equivalent to ν > 0 and, in turn, to IR of the 2nd or 3rd kind:

IR 2nd or 3rd kind ASM 1
⇐⇒ weak IR as in [START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF].

On the contrary,6 rank B equals p, which is equivalent to ν = 0 in view of (19) and, therefore, to IR of the 1st kind:

IR 1st kind ASM 1
⇐⇒ strong IR as in [START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF].

Example 2. Consider the system associated with the quadruple (

-1 0 -1 -1 , [ 1 1 ] , [ 0 1
] , 0), which satisfies ASM 1. One can check that G(s) = s/(s + 1) 2 so that this system is weakly IR in the sense of [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF], but not in that of [START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF].

Summarizing, the proposed definition is more general in the sense that it applies to any proper system, i.e. ASM 1 is not required. In Cor. 1, we also demonstrate that the right-invertibility is the key to prove IR of Σ from the inequality m > p, i.e. minimality and strict properness required by ASM 1 can be dismissed.

The proposed taxonomy also refines that of [START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF] by distinguishing two kinds of weak redundancy, that is IR of the 2nd and 3rd kind. Finally, we exhibit distinctions between seemingly equivalent definitions in the literature, by providing an example which is weakly IR in the sense of [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF], but not in that of [START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF].

Remark 4. In [START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF], the discussion focuses on V := B -1 R * instead of N . If Σ is strictly proper, these subspaces are actually identical since

N = B -1 R * ∩ Ker {D} (21)
holds.

To prove (21), observe that BKer {D} and applying map B -1 yields B -1 (R * ∩ BKer {D}) ⊇ B -1 (V * ∩ BKer {D}). This is equivalent to

B -1 R * ∩ Ker {D} is trivially included in B -1 V * ∩ Ker {D} since R * ⊆ V * .
B -1 R * ∩ B -1 BKer {D} ⊇ B -1 V * ∩ B -1 BKer {D} which reduces to B -1 R * ∩ Ker {D} ⊇ B -1 V * ∩ Ker {D} since B -1
BKer {D} = Ker {D}.

Comparison with [10]

The characterization and taxonomy of IR proposed in [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF] is implicitly revisited in [START_REF] Galeani | On dynamic input allocation for fat plants subject to multi-sinusoidal exogenous inputs[END_REF] under the assumption of right-invertibility. 7Instead of referring to the transfer matrix G, [START_REF] Galeani | On dynamic input allocation for fat plants subject to multi-sinusoidal exogenous inputs[END_REF] focuses on the system matrix P , defined in (13). For some ω ∈ R ≥0 , a matrix N ω satisfying Im {N ω } = Ker {P (jω)} is called IR matrix basis at frequency ω.

Clearly, rank N ω is non zero if rank P (jω) < n + m holds. In view of Rem. 3, the following implication is valid for all ω ∈ R ≥0 :

IR =⇒ rank N ω > 0.
The converse implication requires an additional assumption:

IR jω ∈Z P ⇐= rank N ω > 0,
where Z P refers to set of invariant zeros of Σ, i.e. the values of s for which P (s) loses rank:

Z P := {s ∈ C | rank P (s) < nrank P }. (22)
Indeed, the invariant zeros on the imaginary axis might prevent IR to be inferred from the rank of N ω , as shown in the following example.

Example 3 (Ex. 1 continued). For this system, P (s) equals s+1 -1 -1 1 . It follows that nrank P = n + m = 2 and Z P = {0}, so that rank N 0 = 1 > 0 holds for this non IR system. In [10, Def. 1], a degree of redundancy is proposed. Defining µ := m -p -ρ, Σ is said ρ-strongly (equivalently, µ-weakly) IR or simply strongly (resp. weakly) IR if it is 0-weakly (resp. 0-strongly) IR. If Σ is right-invertible, Cor. 1 applies and shows that µ equals ν. Thus, strongly (resp. weakly) IR as in [START_REF] Galeani | On dynamic input allocation for fat plants subject to multi-sinusoidal exogenous inputs[END_REF] refers to 1st kind (resp. 2nd kind), whereas ρ-strongly or µ-weakly corresponds 3rd kind as soon as both ρ and µ = ν are non zero.

As compared to [START_REF] Galeani | On dynamic input allocation for fat plants subject to multi-sinusoidal exogenous inputs[END_REF], the definition of the degree of input-redundancy given in this paper can be regarded as a generalization to non right-invertible system for which dim N cannot be computed from ρ and the dimensions of the system. If Σ is right-invertible, we also formally prove that µ + ρ = ν + ρ = m -p independent input directions exist (after regular feedback) which do not affect the output, see Subsec. 3.2.

In general, the number of those inputs depends not only on the dimensions of the system but also on the normal rank of the system matrix, see Lem. 4.1.

How to control over-actuated systems ?

By relating over-actuated systems to IR, the internal structure of those systems is exploited to derive a natural control design framework.

Over-actuation implies input redundancy

System Σ is over-actuated if Σ can be decomposed into a set of actuators gathered in an IR subsystem Σ a in series with the plant Σ p , see Fig. 3. Overall control effort τ is delivered by Σ a and drives Σ p which produces y. Indeed, over-actuation is naturally translated by the existence of some trajectory τ which can be produced by way of distinct input trajectories u. This means that Σ a is input redundant.

Let (A a , B a , C a , D a ) be the quadruple associated with Σ a . Denote its controllable weakly unobservable subspace by R * a . Besides, let us simplify the analysis by assuming that output y uniquely determines τ , i.e. Σ p is left-invertible.

Ker      B D      = Ker      B a D a      , ( 23
)
R * = R * a ⊕ {0}, (24)
so that over-actuated system Σ is input redundant of the k-th kind.

Proof. See Appendix D.

Let assumptions of Prop. 6.1 hold. From Prop 3.2 and (16), it immediately follows that if Σ a is static, i.e. τ = D a u, then Σ a and, in turn Σ, are IR of the first kind. Similarly, if Σ a is purely dynamical, i.e.

D a = 0 and Ker {B a } = {0}, then Σ a and Σ are IR of the second kind.

A straightforward control scheme

Prop.

Conclusions and final remarks

The proposed definition and taxonomy of IR are formulated in terms of input/state/output signals. In addition of being more intuitive than existing characterizations (see Sec. The first matrix of this product is full rank since ρ p = 0, due to left-invertibility of Σ p . This implies (23).

By definition, R * gathers reachable states [x a , x p ] (from the origin) under the constraint y = 0, which is equivalent to τ = 0 since Σ p is left-invertible. This proves (24).

Together with Prop. 3.2 and (16), equalities (23) and ( 24) prove that Σ is input redundant of the k-th kind.

DefineG

  (s) := C(sI -A) -1 B + D as the transfer matrix of Σ. Whenever steady state arising from constant inputs is well-defined, i.e. G := lim s→0 G(s) is finite, systems like (3) motivate the introduction of inequality dim Ker {G } > 0,

Figure 2 :

 2 Figure 2: Internal structure of Σ with feedback transformation.

3

 3

  115

  R * and xb (t) ∈ R n-dim R * and (iii) Σ2 and Σ3 be associated with quadruple 120 (A 11 , [A 12 , B 13 , B 12 ], 0, 0) and (A 22 , B 23 , C 2 , D 3 ), respectively. Then, relationship between signal is depicted by Fig.2. Sparsity of matrices of (C.1) betrays the cascaded structure of this scheme. It highlights that output y is independent of both ŵa and ŵb .

Lemma 4 . 1 .

 41 It holdsdim N = n + m -nrank P.

Figure 3 :

 3 Figure 3: Over-actuated system Σ.

Figure 4 :Proposition 6 . 1 .

 461 Figure 4: Control scheme of over-actuated system Σ.

  6.1 allows to perform decomposition of Σ introduced in Subsec. 4.1 in a straightforward way. One ends up with conceptual scheme depicted on Fig 2. Since ŵa and ŵb are invisible from the output, they can be arbitrarily selected. This naturally leads to the modular controller colored in gray on Fig. 4: Control of y is handled by ŵc via C c , while ŵa and ŵb are produced via C a,b . Typically, those last two signals aimsdriving xa to a specific reference (possibly defined online) and making u = F x + H ŵ comply with input constraints, see e.g.[START_REF] Kreiss | A new framework for dealing with input constraints on parallel interconnection of buck converters[END_REF][START_REF] Trégouët | New framework for parallel interconnection of buck converters: Application to optimal current-sharing with constraints and unknown load[END_REF].

P 3 ( 1

 31 5), such a formulation is intended for direct extension, out of the class of unconstrained, proper, finite dimensional, linear, continuous time and time-invariant systems considered in this paper. Any of those adjectives leads to a possible extension of this work, where proposed definition of IR needs to be characterized in a more general context. with Since quadruple (A 22 , B 23 , C 2 , D 3 ) is left-invertible by construction, the normal rank of P 3 equals the number of its columns, that is n -dim R * + m -dim N . This proves (18). Appendix D. Proof of Prop. 6.Subscripts a and p identify variables associated with Σ a and Σ p , respectively, e.g. x a and x p refer to state vector of those subsystems.Define state x(t) of Σ as x a (t) x p (t) . Observe that

  Conversely, given any friend F of V * , it is known that R * is the smallest (A + BF )-invariant subspace containing V * ∩BKer {D} [8, Th. 7.14]. Thus, R * ⊇ V * ∩BKer {D} holds. Intersecting both side by

This means that system Σ is IR if and only if H[x 0 ; •] is not injective for some x 0 ∈ X .

Let us emphasize that dim R * and ν are by no means equal, in general.

Recall that a F is a friend of W if (A + BF )W ⊆ W holds.

From (19), it holds ¬(rank B D > p) ⇔ rank B D = p.

In[START_REF] Galeani | On dynamic input allocation for fat plants subject to multi-sinusoidal exogenous inputs[END_REF] ASM. 3], it is assumed that there exists λ ∈ C such that rank P (λ) = n + p. In this case, nrank P = n + p holds which implies right-invertibility of Σ.

Appendix A. Proof of Th. 3.1

The equivalence between (iv) and IR of Σ can be proved by straightforward adaptation of the proof of [START_REF] Trentelman | Control theory for linear systems[END_REF]Th. 8.8] to the context of this paper where U does not contain impulsive-smooth distributions. By [START_REF] Trentelman | Control theory for linear systems[END_REF]Th. 8.8], (iv) is equivalent to saying that Σ is not left-invertible (in the classical sense of [START_REF] Trentelman | Control theory for linear systems[END_REF]Def. 8.7]), so that [START_REF] Trentelman | Control theory for linear systems[END_REF]Th. 8.26] and [START_REF] Trentelman | Control theory for linear systems[END_REF]Cor. 8.10] apply and prove (i)⇔(ii) and (i)⇔(v), respectively. Regarding (iii), observe that if ρ = 0, then (i)⇔(iii) holds as proved in [START_REF] Ntogramatzidis | Squaring down lti systems: A geometric approach[END_REF]Lem. 3]. On the contrary, if ρ > 0 so that (ii) holds and Σ is input-redundant, then (iii) is valid as well since Ker {[ B D ]} ⊆ N holds.

Appendix B. Proof of Prop. 3.2

From linearity of H, Σ is IR of 1st kind (resp. 2nd kind) iff (B.1) (resp. (B.2)) holds:

where

The first line of the table follows from the fact that (B.1) is equivalent to R * = {0} (see [START_REF] Trentelman | Control theory for linear systems[END_REF]Chap. 7]), so that ν = 0 holds, by virtue of (16).

The second line of the table shows that ρ = 0 is equivalent to (B.2), if Σ is IR. Indeed, if ρ > 0 holds, then there exists a non-zero signal u ∈ Ker {[ B D ]} so that (u, 0, 0) ∈ Q(0) which contradicts (B.2). By contraposition, this proves that (B.2) implies ρ = 0. Conversely, assume that ρ = 0. By contradiction, assume that (B.2) does not hold, i.e. there exists a non-zero signal u such that (u, 0, 0) ∈ Q(0). This implies that u ∈ Ker {B} and y = Du = 0. This contradicts the equality ρ = 0, so that (B.2) must hold.

Last line of the table follows immediately from the fact that IR of the third kind is equivalent to IR of neither the first nor the second kind.

Appendix C. Proof of Lem. 4.1

Define matrices Q and H as in Subsec. 4.1. Let F be a friend of V * . Then, it holds