Keywords: linear systems, non regular control, state feedbacks, diagonal decoupling, structural properties

We propose a structural solution by non-regular static state feedback to the diagonal, or row by row decoupling problem for linear systems. Without being completely general, this solution concerns the reduced Morgan's problem, that is we do not want it increases the essential orders of the open loop system. The solution that we propose relies on properties between some partial infinite structures extracted from well chosen interactors and controllability indices of a specific shifted system. To my knowledge, there is at this date only very partial solutions to this problem deemed structurally very difficult.

Solution of Static Reduced Decoupling

Problem for Linear Systems

Jean-François Lafay * In memoriam A.N. Herrera H.

INTRODUCTION

Diagonal decoupling by a static state feedback, u = F x + Gv, or Morgan's Problem, is a very difficult control problem when G is not invertible. Such non regular feedbacks correspond to a loss of inputs for the controlled system. They can transform all invariant structures underlying the regular solutions to control problems. The challenge is to find new structures that will allow us to solve Morgan's Problem, knowing that non regular static solutions can exist even when there is none with regular laws. The non-regular diagonal decoupling by dynamic state feedback is much simpler as by static state feedback. It was solved by [START_REF] Dion | The minimal delay decoupling problem: feedback implenentation with stability[END_REF] due to new invariants, the essential orders. Non regular decoupling always comes down to a problem of increasing structures. The relative simplicity of the dynamic case comes from the fact that the essential orders are the minimum infinite structure to achieve for decoupling and that it is always possible to get it if there is a solution. The reason is that, to make this increase, dynamic feedback uses dynamics (integrators) that are external to the system. This is no more the case for static feedbacks that must only use internal dynamics. We consider here the Static Reduced Morgan's Problem (SRMP), say the static decoupling without increasing the essential orders: it provides insight into the complex mechanisms of structural changes by non-regular controls and, from the practical point of vue, this restricted problem is not whitout interest because it is necessary that a dynamic solution exists, (therefore maintaining the essential orders), to hope to find a static one. There are so far only very partial results for the SRMP: when it is suffisant to increase only one element of the infinite structure [START_REF] Herrera | New results about the Morgan's problem[END_REF]), or for trivial internal structures, [START_REF] Zagalak | On a special case of the Morgan problem[END_REF]), [START_REF] Lafay | Minimal decoupling indices for linear systems In IFAC joint TDS-SSSC-FDA 2013 Conference[END_REF]). The specific difficulties of SRMP are of two kinds: firstly, although their number is the same and is given by the necessary and sufficient condition of the dynamic problem, the increases of infinite structure for solving SRMP depend on the order of the outputs of the system, while knowing that the sum of the sizes of these increases does not depend of this order. This first lock has been lift in [START_REF] Lafay | Minimal decoupling indices for linear systems In IFAC joint TDS-SSSC-FDA 2013 Conference[END_REF], where it is proved that there is a unique "minimal list of decoupling indices" such that if SRMP has a static solution, one solution exist with this minimal indices. Secondly we must take into account internal couplings which are unobservable from the outputs to be decoupled. For that, we develop a non trivial general formulation of SRMP inspired by [START_REF] Herrera | New results about the Morgan's problem[END_REF]). The general solution of SRMP relies on the controllability of a well defined shifted system .

NOTATIONS AND BACKGROUND

Notations

Let Σ(C, A, B), denoted shortly Σ, a linear system whose state is supposed to be measured or reconstructible:

Σ ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t). (1)
x ∈ X ⊂ R n , u ∈ U ⊂ R m and y ∈ Y ⊂ R p are outputs to be controlled. We suppose B monic and C epic. Σ is controllable, right invertible and without finite zeros. This last assumption implies that V * =R * , V * and R * being the supremal (A -B)invariant and the supremal controllability subspaces in the kernel of C(finite zeros concern only internal stability, which is not adressed here Definition 1. The interactor of Σ, [START_REF] Wolovich | Invariants and canonical forms under dynamic compensation[END_REF]), is the unique p × p triangular and non-singular polynomial matrix Φ[s] = [ϕ i,j [s]] such that there exists a biproper m × m (non unique) matrix B(s) satisfying :

T (s) = C(sI n -A) -1 B = Φ -1 0 B(s)
where:

• ϕ i,i = s di , i = 1, ..., p, d i being positive integers • ϕ i,j is zero or ϕ i,j /s di is divisible by s, ∀j > i.
The interactor is invariant under the action of the group (T, F, G), where T and G are changing of bases on X and U, and F is a state feedback. Without any loss of generality, we can suppose that it is row reduced (r.r.), [START_REF] Lafay | Reduced form of the interactor matrix[END_REF]). Four lists of integers characterize partially the structure of Σ, (see [START_REF] Morse | Structural invariants of linear multivariable systems[END_REF], [START_REF] Commault | New decoupling invariants: the essential orders[END_REF]):

-{c i } m : controllability indices of (A, B) , -{n i } p : I 4 Morse's list of the orders of the zeroes at infinity. If the interactor of Σ is r.r,

d i = n i , i = 1, ..., p.
-{σ j } m-p : I 2 Morse's list providing the structure of R * , -{n ie } p : the p essential orders of the outputs y i (t):

n ie = ∂ ci Φ[s], i = 1, ..., p, .
These lists are invariant under the action of the group (T, F, G, Π), where Π is a permutation of the outputs of Σ and G is invertible [START_REF] Herrera | A semi-canonical form for a class of right invertible linear systems[END_REF]).

An extended system Σ e (C e , A, B) is defined in [START_REF] Herrera | A semi-canonical form for a class of right invertible linear systems[END_REF] by adding m -p ficticious outputs, to reflects the structure of R * taking:

C e =   C (0) p× m-p j=1 σj (0) (m-p)× p i=1 n i diag [1 0 ... 0] 1×σj m-p   . (2)
Proposition 1. The m × m interactor of Σ e , called "extended interactor of Σ", has the structure:

Φ e [s] = Φ e 1 [s] (0) Φ e 2 [s] Φ e 3 [s] = ϕ e i,j [s] , where (3)
• [START_REF] Dion | The minimal delay decoupling problem: feedback implenentation with stability[END_REF]) : Proposition 2. There exists output's permutation(s) Π such that the p × p interactor Φ Π of Σ Π (ΠC, A, B) has the structure:

Φ e [s] is row reduced • Φ e 1 [s] is the p × p interactor of Σ, then ϕ e i,i = s n i , i = 1, ...p, • Φ e 3 [s] = diag {s σ1 , ..., s σm-p }, and • Polynomials ϕ e i,j [s] = h j=l α j s j of Φ e 2 [s] are zero or verify: n j + 1 ≤ l and h ≤ σ i -1 . Factorize Φ[s] as Φ[s] = W (s)diag{s nie } p . W (s) is a proper matrix, of rank at infinity k, called "proper part of Φ[s]."
Φ Π = Φ Π,1 [s] (0) Φ Π,2 [s] Φ Π,3 [s] = ϕ Πi,j [s] , where (4)
• The k × k polynomial matrix Φ Π,3 [s] is the diagonal matrix = diag{s nje } k , the k integers n je being the essential orders of the k last outputs of Σ Π , • Non zero elements of the infinite structure of the proper part of Φ Π are given by the list

{δ i } p-k = {n ie -f i } p-k , where f i = ∂ϕ Πi,i [s] for i = 1, ..., p-k. • {δ i } p-k is the list of decoupling indices of Σ π ,
• There are always, among permutations Π, some permutations Π m such that the list {δ i } p-k is the unique minimal list {∆ i } p-k of decoupling indices of Σ, • Note that in general Φ Π (and Φ Πm ) are no longer r.r.

The minimal list of decoupling indices, is defined in [START_REF] Lafay | Minimal decoupling indices for linear systems In IFAC joint TDS-SSSC-FDA 2013 Conference[END_REF]) using the notion of "minor" list: Definition 2. Let two lists of integers {δ i } k1 and {γ i } k2 such that k1 i=1 δ i = k2 i=1 γ i . Note { δi } supδi and {γ i } supγi their dual lists. {δ i } k1 "minore" the list {γ i } k2 if, for i = 1, ..,sup(sup j (δ j ), sup j (γ j )), we have:

i j=1 δj ≤ i j=1 γj . (5)
.

This list {∆ i } p-k and an algorithmic procedure to obtain it are presented in Appendix A in a better way as in Lafay [2013] .

Impact of a permutation Π on the extended interactor

We consider the extended interactors of Σ(C, A, B) and Σ Π (ΠC, A, B). We note Φ[s] = [ϕ i,j ], and Φ e [s] is structured as in (3). Property 1. Let a right invertible system Σ. Let Π any permutation matrix acting on the outputs of Σ. The extended interactor of Σ Π is given by

Φ e Π [s] = Φ e Π,1 [s] (0) Φ e Π,2 [s] Φ e Π,3 [s]
, where : (6)

• Φ e Π,1 [s] = Φ Π [s] = B 1 (s)Φ[s]Π -1 , where Φ Π [s] is the interactor of Σ Π and B 1 (s) is a biproper p × p matrix, • Φ e Π,2 [s] = B 2 (s)Φ 2e [s]Π -1
, where B 2 (s) is a proper (m -p) × p matrix, which can add polynomials to whose of Φ e 2 [s] but, in each column i, these polynomials are all of degree less than or equal to n ie .

• Φ e Π,3 [s] = Φ e 3 [s]
The proof is based on the properties of biproper transformations. It is not given here for sake of shortness. Remark 1. Let us consider equations (3) and (6). Suppose that, for i = 1, ...p -k, one keeps in the ith columns of Φ e 2 [s] and Φ e Π,2 [s] only monomials of degree higher than n ie : then, the so "truncated" matrices Φ [START_REF] Hautus | Linear feedback, an algebraic approach[END_REF]). Then they did not affect the static decouplability of Σ and we can suppose that the transfer matrix

T m (s) of Σ m is Φ m -1 (s). We note Σ m (C m , A m , B m ), where A m = A+BF m , B m = BG m and C m = Π m C, F m and G m being a regular state feedback .
In all what follows it will be assumed, unless explicitly mentioned, that Σ is already structured as Σ m , 2

(Proposition 2), with {∆ i } p-k = {δ i } p-k . To summarize, the extended interactor Σ e m is : Φ e m [s] = ϕ e mi,j [s] =   Φ e m1,1 (0)
(0) Φ e m1,2 Φ e m1,3 (0) Φ e m2,1 Φ e m2,2 Φ e m2,3   (8)
where the p × p interactor of Σ m is:

Φ m [s] = Φ e m1,1 (
0) Φ e m1,2 Φ e m1,3 (9)
,

• Φ e m1,3 = diag{n ie } k , where i = p -k + 1, ..., p , n ie are the essential orders of the k last outputs of Σ m .

• Φ e m1,2 is a p -k × k matrix with ∂ ci Φ e m1,2 = n ie , • Φ e m1,1 is a p -k × p -k matrix with: ϕ e mi,i = s fi , polynomials ϕ e
mi,j = 0 for j > i , and for i < j the non null polynomials ϕ e mi,j are such that

f j + 1 ≤ ∂ min ϕ e m i,j and ∂ max ϕ e m i,j ≤ n je -1, • the list {n ie -f i } p-k = {∆ i } p-k , where {∆ i } p-k is the unique minimal list of decoupling indices of Σ, • Φ e m2,3 is the m -p × m -p diagonal matrix s σi , • non null polynomials of Φ e m2 = Φ e m2,1 Φ e m2,2
are such that f j + 1 ≤ ∂ min ϕ e mi,j for i ≤ p. • Note that, for i ≤ p, ∂ max ϕ e m i,j can be greater than n je .

MORGAN'S PROBLEM

Formulation of the problem and consequences

The static diagonal decoupling without stability of Σ, or Static Morgan's Problem, can be stated as follows: Given Σ described by (1), does it exists a static state feedback

u = F x + Gv = F x + G 1 v 1 + ... + G p v p , v i ∈ U and G i ∈ R m×1 such
that, for any i ∈ 1, 2, ..., p, v i controls the scalar output y i without affecting the p -1 other outputs y j ? If such a feedback exists, the transfer matrix (square and invertible) this problem has a solution if and only if the ordered lists {n i } p and {n ie } p are the same. So, the interactor of Σ is diagonal [START_REF] Commault | New decoupling invariants: the essential orders[END_REF]). If {n i } p = {n ie } p , Σ will be decouplable if and only if it is possible to increase the structure at infinity so that it matches to (new) essential orders. This modification can only be achieved by a non regular feedback, that is with a loss of inputs. With such controls, the list of essential orders is not always the minimal infinite structure to reach for the static decoupling. Static Restricted Morgan's Problem, (SRMP), is the particular case of Static Morgan's Problem where the essential orders of Σ should be the infinite structure of the decoupled system. SRMP is not without interest because, if static decoupling is only possible with a infinite structure "bigger" than the original essential orders, Σ can always be decoupled without modification of these initial essential orders by a non regular dynamic state feedback. More details on that analysis can be found in [START_REF] Herrera | New results about the Morgan's problem[END_REF]. Finally, as any non regular decoupling reduces to increase infinite structure, the solution will be based on a theorem of [START_REF] Loiseau | Sur la modification de la structure à l'infini par retour d'état statique[END_REF] on modifying this structure via non-regular state feedbacks that we remind now:

T F G (s) of the closed loop system Σ(C, A + BF, BG) is T F G (s) = diag{s -r1 , ..., s -rp }, r i ∈ R. If G is regular,
Theorem 1. Let a linear system for which {n i } is the infinite structure and {σ i } the I 2 Morse's list. Let {p i } a given list of integers. Note {v i }, {α i } and {π i } the dual lists of, respectively, {n i }, {σ i } and {p i }. Let {Γ i } the list obtained by arranging the differences (π i -v i ) in a non increasing order. Then, there exists a static state feedback such that the structure at infinity of the closed loop system is the list {p i } if and only if:

v 1 -v i ≥ π 1 -π i , ∀i ≥ 1, ( 10
) j i=1 α i ≥ j i=1 Γ i , ∀j ≥ 1. ( 11
)
Before addressing the static problem, let us consider now the dynamic solution of decoupling:

The Dynamic Morgan's Problem

Proposition 3. [START_REF] Dion | The minimal delay decoupling problem: feedback implenentation with stability[END_REF]).

Π (Proposition 2). The list {δ i } p-k = {n ie -f i } p-k
is not necessarily the list of minimal decoupling indices. Dynamic decoupling amounts to annihilating the indices {δ i } p-k applying the following iterative procedure for i = 1, 2, ..., p -k:

-for i = 1, u 1 is replaced by an external chain of δ 1 integrators controlled by an entry of R * , for instance v 1 = u p+1 . This external chain is independant of the (m -p) chains of σ i integrators of R * . This substitution amounts to multiply the first row of (4) by s δ1 . So ∂ϕ Π1,1 becomes n 1e and it is possible, by a left biproper operation, to eliminate all the other polynomials of the first column of Φ Π .

-we successively made the same operation for the p -k -1 other rows of Φ Π,1 , taking at each step a different entry v i of R * . This is possible because m -p ≥ p -k.

The final interactor is the p × p matrix diag{s nie } and the system with entries {v 1 , v 2 , ..., v p-k , u p-k+1 , ..., u p } is regularly decouplable. So we need p-k chains of independant integrators coming from a dynamic extension of dimension p i=1 n ie -p i=1 n i . This dynamic procedure only require that the number of independent entries of R * is at least p-k. The global number of integrators wich are necessary to decouple does not depend of the permutation Π but, even if the list of decoupling indices depends of it. This remark point out one of the main difficulties of Morgan's problem. Another difficulty is the possible existence of couplings between R * and the rest of the system that are not taking into account by the dynamic solution. , where (12)

• Φ S,e m1 [s] = diag{s nie } p • Φ S,e m3 [s] = diag{s σ i } m-p • Φ S,e m2 [s] is deduced from the m -p × p matrix Φ e m2 [s] = Φ e
m2,1 Φ e m2,2 by eliminating, in each column i = 1, ..., p, all monomials of degree less than or equal to n ie by a left biproper transformation.

• Φ S,e m [s] has the structure of an interactor.

4.1 Shifted system Definition 3. The shifted system Σ S m (C S m , A S m , B S m ) associated with Σ m is the invertible system of which Φ S,e m [s], equation ( 12), is the extended interactor.

Let us extend the state space X of Σ m by adding an extermal dynamic state extension X a , of dimension n a = p-k i=1 ∆ i , composed of p -k independant controllable and observable chains of integrators of lenghts ∆ 1 , ..., ∆ p-k with entries w 1 (t), ..., w p-k (t) in U a and outputs z 1 (t), ..., z p-k (t) in Y a . Noting ⊕ the external direct sum of subspaces, the state, control and output spaces of Σ m completed with this dynamic extension are X

S = X ⊕ X a , U S = U ⊕ U a and Y S = Y ⊕ Y a .
The shifted system is obtained by replacing the entries u b (t) of B b by z i (t) for i = 1, ..., p -k. The state space realization (C S m , A S m , B S m ) of Σ S m is:

A S m =      A m A S 1 . . . A S p-k (0) J ∆1 (0) (0) (0) (0) . . . (0) (0) . . . (0) J ∆ p-k     
where : (13)

• for i = 1, ..., p -k, J ∆i is the upper ∆ i × ∆ i Jordan block and the n × ∆ i matrices A S i are given by

A S i = b i(n×1) (0) n×∆i-1 . • B S m = (0) (n×p-k) B s(n×k) B r (n×m-p) B a(n a ×p-k) (0) (0) , (14)
where

B a = [b a,1 , ..., b a,p-k ] , b a,i = [0 . . . 0 1] T being ∆ i × 1 vectors. • C S m = C m(p×m) (0) (p×na) . (15)
Then, as in noted in Subsection 3.2 the interactor of Σ S m is given by

Φ S m [s] = diag{s nie } p (0) p×m-p , (16)
and the corresponding extended interactor of Σ S m is:

Φ S,e m [s] = diag{s nie } p (0) p×m-p Φ S,e m,2 diag{s σi } m-p , (17)
where Φ S,e m2 [s] is deduced from Φ e m2 [s] by eliminating in each column j = 1, ...p the monomials of degree less than or equal to n je .

A convenient formulation of SRMP

SRMP will be solved if it is possible to replace the pk independant chains of external integrators solving the dynamic Morgan's problem by p -k independant chains of integrators extracted from Σ. A chain of lenght L is a set of L connected integrators, actived only at its beginning by a function f a (t) to generate a function f e (t), ie f (L) e (t) = f a (t). Let set of q chains of integrators defined by f i e (Li) (t) = f i a (t), i = 1, ..., q. For SRMP, we require that each function f i a (t) contains at least one input of the the system. These chains are independant if, ∀j = i, the term containing the inputs in f i a (t) is not a linear combination of terms containing the inputs in f j a (t), and if f i e (t) is not a linear combination of functions f j e (t). The following lemma characterize internal chains that will increase the infinite structure without changing the essential orders: it is a new formulation for SRMP, generalizing in a non trivial way Lemma 4.2 in [START_REF] Herrera | New results about the Morgan's problem[END_REF])valid when k = p -1. Lemma 1. Let a right invertible system Σ m with R * = V * , k the rank of the proper part of its interactor and ∆ i (p-k) its decouplability indices (say the minimal decoupling indices of Σ). Then, SRMP has a solution if and only if it is possible to extract from R * p -k independant chains of integrators of lengths ∆ 1 , ..., ∆ p-k described by f i e (t) (∆i) = f i a (t), i = 1, ..., p -k such that: (a) The output f i e (t) of each chain of integrators is only function of the vector state space x(t) and these p -k functions are independant, (b) For i = 1, ..., p -k, the entry f i a (t) of each chain of integrators do not contain derivatives of inputs u j (t), j = p -k + 1, ..., p. (c) For i = 1, ..., p -k, the entry f i a (t) of each chain of integrators do not contain derivatives of inputs u j (t) of order greather than or equal to ∆ j , for all j = 1, ..., p -k . Proof 1. of Lemma 1 IF. Let a state feedback (F, G) which decouples Σ m (C m , A m , B m ). According to [START_REF] Dion | The minimal delay decoupling problem: feedback implenentation with stability[END_REF] this feedback is equivalent to the precompensator

C(s) = (I m -F (sI n -A m ) -1 B m ) -1 G = W 1,m (s) X(s)
, where X(s) is a proper m -p × p matrix and, noting d = s -1 , the p × p proper part W 1,m (s) of the interactor of Σ m can be written as:

W 1,m (s) =            d ∆1 φ2,1 . . . (0) d ∆ p-k ( φi,j ) . . . 1 . . . (0) . . . φp,1 φp,p-k 0 • • • 1            . ( 18
) φi,j [d] are polynomials in d = s -1 such that ∂ φi,j [d] < ∆ j = n je -f j for i ≤ p -k.
Matrix G of the feedback (F, G) is obtained (Herrera H [1992]) by: G

= lim s→∞ W 1,m (s) = (0) p-k×p-k (0) (g i,j ) k×p-k I k where g i,j ∈ R .
So, the control law u(t) = F x(t) + Gv(t) is given by: 4

u i (t) =    F i x(t), i = 1, ..., p -k (a) F i x(t) + p-k j=1 g i,j v j (t) + v i (t), i = p -k + 1, .., p (b) (19) As Σ m,(F,G) (C m , A m + B m F, B m G
) is assumed to be decoupled without changing the essential orders, we have:

y (nie) i (t) = u (∆i) i (t) = v i (t), i = 1, ...p -k u (0) i (t) = v i (t), i = p -k + 1, ..., p. (20)
Therefore, equations ( 20) and (19a) describe p -k chains of lenghts ∆ 1 , ..., ∆ p-k generating independant functions F i x(t) of the state x(t). Moreover, from (19b), these chains are not activated by derivatives of the entries u p-k+1 (t), ..., u p (t) and for i and j = 1, ..., p -k, the ith chain is not actived by derivatives of v j (t), i = j. But, by equation (19b), if the role of the chains is only to make Σ m regularly decouplable, the ith chain can be eventually activated by derivatives of entries u i (t) of order less than ∆ i for i = 1, ..., p-k, because for each chain must generate a function of the state x(t), say:

f i a (t) = f i e (∆i) (t) and f i a (t) = F i x(t)
According [START_REF] Loiseau | Sur la modification de la structure à l'infini par retour d'état statique[END_REF], since any increasing of infinite structure can only be done using entries of R * , entries v 1 (t), ..., v p-k (t) should contain independant linear combinations of entries of R * . (Only if ): Assume that these p -k independant chains can be constructed. For i = 1, ..., p -k, each chain is characterized by its lenght ∆ i , its entry f i a (t) and its output f i e (t) linked by f i e (∆i) (t) = f i a (t). As f i e (t) is uniquely a function of the state space of Σ m , we can write f i e (t) = F i x(t), where F i ∈ R 1×n , and as the chains are

independant, rank F =    F 1 . . . F p-k    = p -k.
Now, consider the entries of the chains. As these chains are extracted fom R * , each function f i a (t) contains at least one linear combination of the entries u p+1 (t), ..., u m (t) and these linear combinations are independant: this implies that dim B ∩ R * ≥ p -k. This is the necessary and sufficient condition of dynamic decoupling for Σ. For static decoupling, we must add the condition:

p-k i=1 ∆ i ≤ m-p i=1 σ i , where {σ i } m-p is the Morse's list I 2 .
By assumption, no derivative of entries u p-k+1 (t), ..., u p (t) appears in functions f i a (t). However there may be terms which depend on the state x(t) and/or on entries u p-k+1 (t), ..., u p (t), and/or, for i = 1, ..., p -k, on derivatives of entries u i (t) of order less than ∆ i , (because each function f i a (t) = f i e (∆i) (t) does not include derivative of u i (t) of order greather than or equal to ∆ i from the fact that f i e = F i x(t). So, for i = 1, ..., p -k, the general form of functions f i a (t) is:

f i a (t) = f i a1 (u p+1 (t), ..., u m (t)) + f i a2 (x(t), u p-k+1 (t), ..., u p (t)) + p-k j=1 f i a3,j (u (
f i a (s) = f i a1 (u p+1 (s), ..., u m (s)) + f i a2 (x(s), u p-k+1 (s), ..., u p (s)) + p-k j=1 Ψ i,j [s]y j (s), ( 22
)
where the degree of each polynomial Ψ i,j [s] is less than n je .

We can now define the non regular feedback u(t) = F x(t)+ Gv(t) by:

F =   Fp-k×n 0 k×n F 0 m-p×n   and G =   0 p-k×k 0 p-k×p-k I k 0 k×p-k 0 m-p×k G 0m-p×p-k   .
Here rank

F = p -k and rank G 0 = p -k ≤ m -p.
The possibly non regular feedback (F 0 , G 0 ) acts only on R * to create the p -k independant chains of integrators. Noting B r = ImB ∩ R * and x R * (t) trajectories in R * , this part of feedback is such that:

F 0 x R * (t)+B r G 0 v * (t) =    f 1 a1 (•) . . . f p-k a1 (•)   , where v * (t) =    v p-k+1 (t) . . . v p (t)   .
Then, as y

(fi) i (t) = u i (t) = f i e (t) for i = 1, ..., p -k, we obtain: L(y (nie) i (t)) = f i e x(s)s ∆i = f i a (t) = f i a1 (•) + f i a2 (•) + p-k j=1 Ψ i,j [s]y j (s) with ∂Ψ i,j [s] < n je .
Then, outputs y(s) and new entries v(s) are linked by:

H 1,1 [s] (0) p-k×k H 2,1 [s] H 2,2 [s] y(s) = V (s) = V 1 (s) V 2 (s) , (23)
where 1-The p -k × p -k polynomial matrix H 1,1 is given by:

H 1,1 =        s n1e -Ψ 1,1 . . . (-Ψ i,j )
(ϕ e mi,j -Ψ i,j ) . . .

s n (p-k)e -Ψ p-k,p-k        . 2-H 2,1 = ϕ e mi,j [s] is a k × p -k polynomial matrix 3-H 2,2 [s] = diag{s n (p-k+1)e , ..., s npe } k . 4-V 1 (s) =    f 1 a1 (•) + f 1 a2 (•) . . . f p-k a1 (•) + f p-k a2 (•)   and V 2 (s) =    v p-k+1 (s) = u p-k+1 (s) . . . v p (s) = u p (s)    .
As the polynomial of highest degree of each column of H[s] is on the diagonal the matrix, the infinite structure coincides with essential orders. This system is then regularly decouplable without increasing the essential orders.This ends the proof of Lemma 1. Remark 2. Functions f i a (t) cannot contain derivatives ou entries of R * . If that were the case, for exemple for the first chain, the effective lenght of this chain would be less than ∆ 1 . Indeed, suppose that this chain of lenght ∆ 1 is activated by u p+1 (t) and by u

(1)

p+j (t), for j = 1, ..., m - p. Then f 1 e (∆1) (t) = f 1 a (t) = u p+1 (t) + u (1) p+j (t) and f 1 e (∆1-1 )(t) = f 1 a (t)dt = u p+1 (t)dt + u (1) p+j (t)dt = f (x(t)) + u p+j (t).
The effective lenght of the chain is ∆ 1 -1. Remark 3. Inputs u 1 (t), ..., u p-k (t) are suppressed, while inputs u p-k+1 (t), ..., u p (t) are preserved. Remark 4. If functions f i a (t) do not include derivatives of inputs u i (t), i = 1, ..., p -k, the system is decoupled.

Lemma 1 will help us to derive conditions on Σ m for ensure that such p -k independant decoupling chains exist. To characterize the maximal lenghts of the chains of R * which are not actived by derivatives of entries u j (t) of order higher or equal to ∆ j , j = 1, ..., p, it is possible to apply Theorem 4.1 in Herrera H and [START_REF] Herrera | New results about the Morgan's problem[END_REF] taking into account the following remark: Remark 5. The chains of R * satisfying items (b) and (c) of Lemma1 for Σ m are the same as the chains of R * which are not actived by derivatives of entries w 1 , ..., w p-k u p-k+1 , ..., u p for Σ S m .

Theorem 4.1 in Herrera H and [START_REF] Herrera | New results about the Morgan's problem[END_REF] becomes: Proposition 4. Let Σ S m be given. Let {c i } m the controllability indices of (A S m , B S m ). Then the set {α i } m-p of maximal lenghts of subchains of R * not activated by derivatives of inputs w 1 , ..., w p-k , u p-k+1 , ..., u p is

α i = c i , for i = p + 1, ..., m. (24
where {α i } sup αi the dual list of {α i } (m-p) and {γ i } sup ∆i is the dual list of {∆ i } (p-k) .

Proof 2. Sufficiency:

Theorem 1 cannot be applied directly because it transforms globally a list {n i } p-k into a list {n ie } p-k , and for decoupling, the transformations will be done done term by term. If this is was not the case, the essential orders would be changed. But, as mentionned in [START_REF] Lafay | Minimal decoupling indices for linear systems In IFAC joint TDS-SSSC-FDA 2013 Conference[END_REF] 26) of Theorem 2 is a necessary and sufficient condition for the general Morgan's problem for a system Σ when m -p = k.

COMPARISONS

At our knowledge, the more advanced structural results on static Morgan's problem are in [START_REF] Herrera | New results about the Morgan's problem[END_REF], [START_REF] Zagalak | On a special case of the Morgan problem[END_REF] and [START_REF] Lafay | Minimal decoupling indices for linear systems In IFAC joint TDS-SSSC-FDA 2013 Conference[END_REF].

5. 1 Herrera H and Lafay [1993] In this paper, SRMP is solved when k = p-1. The solution was stated as follows, with the notations adopted here: Theorem 3. Let the right invertible system Σ m be given with R * =V * . Suppose k = p -1. Let δ 1 be the (nonzero)infinite structure of the interactor Π 1m [s], and {α r,i } m-p the controllability indices of the pair (A m , [B s |B r ]) related with the columns of B r . Then the SRMP has a solution if and only if:

m-p j=1 α r,j ≥ δ 1 . ( 26
) Le Σ m = (C m , A m , B m ) and note B m = [b 1 |B s |B r ]
as in Section 4. In this particular case, the list of decoupling indices contains only one term δ 1 = p i=1 n ie -p i=1 n i . Then this list is minimal. A static solution will exist if and only if the sum of lenghts of the subchains of R * not actived by derivatives of entries (u 2 , ..., u p ) is greather than or equal to δ 1 which is strictly less than n 1e . Consider now the shifted system associated with Σ m . Following equations ( 13) and ( 14), we have:

A S m = A m A S 1 (0) J ∆1 where (27)
• J δ1 is the upper δ 1 × ∆ 1 Jordan bloc, and matrix A S i are given by A

S 1 = b 1(n×1) (0) n×∆1-1 . • B S m 1 = (0) (n×1) B s(n×p-1) B r (n×m-p) b a(δ 1 ×1) (0) (0) with (28) b a = [0 . . . 0 1] T .
Let us compare the controllability indices {α i } m-1 of the the pair (A m , [B s |B r ]) and the controllability c S i m of Σ S m . Using the Brunovsky's procedure, [START_REF] Brunovski | A classification of linear multivariable system[END_REF], we have c S 1 ≥ n 1e , then

• if α i ≤ n 1e , then α i = c S i+1 , and • if α i > n 1e , we can have c S i+1 < α i but c S i+1
> n 1e . Then as δ 1 ≤ n 1e the two Theorems are equivalents for the existence of a solution for SRMP.

Zagalak et al. [1998]

In this paper, the authors consider systems Σ with special conditions on dimensions as m = 2p and p i=1 δ i = p i=1 σ i , and mainly without couplings between R * and the blocks of infinite structure, ie Φ e m2 = Φ e m2,1 Φ e m2,2 =[0] in the extended interactor of Σ m . The problem was however difficult althought these assumptions seem very simplistic. Certainly, the authors propose factorizations which must have a link with the minimal list of decoupling indices but, unless I have not properly understood their approach, this structural information does never appear explicitly.

CONCLUSION

In this paper, we propose a necessary and sufficient conditions for the row by row decoupling problem without modifying the essential orders. Even in this special form, the problem was recognized as structurally difficult. It remains to solve the general problem which is much more difficult because we do not know yet how to define minimal structures for the decoupled system.

Appendix A. MINIMAL LIST OF DECOUPLING INDICES

We give here the procedure to obtain the minimal list

{∆ i } p-k ; Let the interactor Φ[s] = Φ 1 [s] (0) Φ 2 [s] Φ 3 [s] = [ϕ i,j [s]] of Σ
that is supposed to be in the form (4) .

For columns 1, 2, ..., p-k, we note p ie the maximum degree of the ith column of Φ 1 [s] , γ i = n ie -p ie and define the index ∆ = min {γ} p-k . We will first prove that ∆ does not depend on the permutation Π such as the interactor of Σ Π is under the form (4) and in a second step that it is the smallest decoupling index possible for Σ Π .

According to Proposition 2, the polynomial of highest Except this permutation, the other polynomials of the pth row of B 1 (s)ΦΠ -1 are not modified. Note that Φ1 (s) and Φ2 (s) are no longer necessarily polynomial matrices. Consider now the other columns of B 1 (s)Φ[s]Π -1 : Lemma 2. For j = 2, ..., p -k, the highest degree the jth columns of B 1 (s)ΦΠ -1 and Φ[s] is not changed and the polynomials of degree n je are in Φ2 [s]. Proof 3. The degrees of columns are the same because B 1 (s) is biproper. In equation (A.3), ϕ i,j is transformed as φi,j =ϕ i,j -ϕi,1 ϕp,1 ϕ p,j . For i and j = 1, ..., p -k, ∂ϕ i,j < n je . As ∂ϕ i,1 < ∂ϕ p,1 and ∂ϕ p,j ≤ n je , we have ∂ ϕi,1 ϕp,1 ϕ p,j < n je , which ends the proof of Lemma 2.

We will now prove the following Lemma: Lemma 3. : ∆ is the same for Φ[s] and for B 1 (s)ΦΠ -1 . Proof 4. Let B 1 (s)ΦΠ -1 and n ie -∂ φi,j = γi , for i = 1, ..., p -k 1-In (A.3), each polynomial φi,1 of Φ1 [s] is given by: φi,1 = -ϕ p,p ϕ i,1 ϕ p,1 , (A.4) with ∂ϕ p,p = n pe and ∂ϕ p,1 = n 1e . Then n pe -∂ φi,j = n 1e -∂ϕ i,j . So γ1 = γ 1 . Especially, if we had γ 1 = ∆,

  4. THE SOLUTION OF SRMPLet Σ m (C m , A m , B m ) as described in subsection 2.3. We note B m = B b B s B r ,where B b = [b 1 , ..., b p-k ],B s = [b p-k+1 , ..., b p ] and B r = B ∩ R * = [b p+1 , ..., b m ]. The associated entries are u(t) = u b (t) u S (t) u r (t) T .Define the m × m polynomial matrix Φ

) 4. 3

 3 Main result: the solution of SRMP Theorem 2. Let the right invertible system Σ be given with R * = V * and k the rank at infinity of the proper part of its interactor. Let Σ m deducted by regular state feedback from Σ such that the infinite structure {∆ i } (p-k) of the proper part of the interactor of Σ m is the minimal list of decoupling indices of Σ. Let {α i } (m-p) the controllability indices of the entries of R * for the shifted system Σ S m associated wit Σ m . Then SRMP has a solution if and only if, for all i ≥ 1,

  a biproper matrix B Π (s) such that Φ Π [s] = B Π (s)Φ[s]Π -1 is the interactor of Σ Π .First we cancel by a left biproper operation B 1 (s) the polynomials ϕ j,1 , j = 1, ..., p -1, of the pth column of (A.1). B 1 (s) always exists since ∂ϕ p,1 [s] = n 1e ≥ ∂ϕ j,1 [s], ∀j. Choosing s] is a p -k × p -k matrix, and where Φ3 [s] and Φ 3 [s] differ only by φp,p = ϕ p,1 while 1 φp,1 = s npe =ϕ p,p .

  it suffices to apply Theorem 1 choosing for list {n i } p-k the list {1} p-k and try to turn it into the list {1 + ∆ i } p-k . This amounts to build, from the m-p chains of integrators of lengths σ i of R * , p -k independent chains of lengths {∆ i }. Note that condition (10) is still always true. So there remains only conditions (11). Let us now return to SRMP. It is sufficient to replace the dynamic extension of Subsection 4.1 by p -k independant chains of lenghts {∆ i } (p-k) extracted from R * and satisfying Lemma 1. This is equivalent to create, from Σ S m , p -k independant chains extracted from R * not actived by derivatives of entries w 1 , ..., w p-k , u p-k+1 , ..., u p (cf Remark 5). By Proposition 4, such chains should be built from the maximal subchains of R * which are not undividually actived by derivatives of w 1 , ..., w p-k , u p-k+1 , ..., u p .According to the construction of the shifted system, and Remark 5, integers {α i } m-p represent equivalently the maximal lenghts of subchains of R * not activated by derivatives of inputs u p-k+1 , ..., u p and for i = 1, ..., p -k not activated by derivatives of inputs u i of order greather than or equal to ∆ i .

	Applying Proposition 1 as men-
	tionned at the beginning of this proof, we obtain condition
	(26) which is sufficient to solve SRMP.
	Necessity: The necessity of this condition comes from
	two facts. First: by Proposition 2, if there is a solution
	to SRMP, there are some which which require only (p-
	k) increases of infinite structure. Conditions (11 there-
	fore requires that the minimum list of decoupling indices
	contain only (p-k)termes. Secondly: the list of minimal
	decoupling indices contains (p-k) terms and "minore" all
	the other lists of decoupling indices. In other words, if
	there is no solution for these indices, so with the order of
	outputs of Σ m , it does not exist solution for any other list
	if decoupling indices coming from other permutations of
	outputs of Σ. This ends the proof .
	Corollary 1. The condition (

  Its degree is n je . Let Π -1 the permutation of the tth and jth columns of Φ[s], which corresponds to the permutation Π of the tth and jth outputs of (1). Φ[s]Π -1 is no longer an interactor.A.1 ∆ is not modifiedTo simplify the notations and without loss of generality, we assume that j = 1 and t = p. Noting (only for the next equation) α = p -k, Φ[s]Π -1 is given by:

		0	0	. .	.	.	0	ϕ 1,1
	        	0 . 0 0 ϕ α+1,2 . . ϕ α+1,α+1 . ϕ 2,2 . . . . . . . . . ϕ α,2 . ϕ α,α . . . . . . 0 . 0 ϕ p-1,2 . . 0 0 ϕ p-1,p-1 . 0 ϕ 2,1 . . 0 . . . . .
		ϕ p,p ϕ p,2 . ϕ p,α	0	0	0	ϕ p,1

degree of each column of Φ 1 [s] Φ 2 [s] belongs to Φ 2 [s]. Let ϕ t,j [s], with j ≤ p -k and t > p -k, this polynomial for the jth column of Φ[s].

We could always suppose that ϕ p,1 is normalized as s n 1e

this index is transmitted in the first column of B 1 (s)ΦΠ -1 . But still nothing proves here that ∆ = min {γ i } p-k . 2 -For the other columns of Φ1 [s], φi,j = ϕ i,j -ϕ p,j ϕi,1 ϕp,1 , and then: ∂ φi,j ≤ max (∂ϕ i,j , ∂ϕ p,j ϕi,1 ϕp,1 ). Then or γj = min i {n je -∂ϕ i,j } p-k = γ j , or γj = min i {n je -∂ϕ p,j -∂ϕ i,1 + ∂ϕ p,1 } p-k : in this last case, as ∂ϕ p,1 -∂ϕ i,1 ≥ γ 1 , we have: n je -∂ φi,j ≥ n je -∂ϕ p,j + γ 1 = γ 1 + c, where c ≥ 0. So, γj ≥ min {γ j , γ 1 + c}. If γ j = ∆ = γ 1 , there may be cancellation of the terms of highest degree of the polynomial and then γj > ∆. But, in this case, ∆ is still in the first column, as we have seen in item 1. The consequence is that, for each column j, γj ≥ min (γ j , γ 1 ) (A.5) and ∆ = min {γ i } p-k , which ends the proof of Lemma 3.

A.2 ∆ is the smallest decoupling index of Σ Suppose that, for the rth column of B 1 (s)ΦΠ -1 , n rep re = ∆ and ∂ φs,r = p re . Permute (by Π r -1 ) the rth and the (p -k)th columns of B 1 (s)Φ[s]Π -1 . We obtain

There exists a biproper transformation B 2 (s) such that the interactor of the new permuted system is given by Φ

), B 2 (s) has the following structure:

where B 2,1 (s) is a biproper p -k × p -k matrix such that B 2,1 (s) Φ1,r (s)Π r -1 is structured as an interactor. As B 2 (s) and B 2,1 (s) are biproper, the maximum degrees of each column j of B 2 (s)B 1 (s)ΦΠ -1 Π r -1 and of B 2,1 (s) Φ1,r [s] are respectively equal to n je and p ie . Then φrp-k,p-k = s pe , φri,p-k = 0 for i = 1, ..., p -k -1, and the monomials of degree less than or equal to pe have been canceled by B 2,2 (s) in the polynomials φri,p-k = 0 for i = p -k + 1, ..., p . So, the decoupling index of the new (p -k)th output, noted ∆ p-k is ∆ p-k = ∆. It is the smallest possible for SRMP from Lemmas 1 and 3. Remark 6. From (A.5) and as B 2,1 (s) is biproper, the new quantities γri computed for the interactor Φ r [s] are greater than or equal to ∆ p-k = ∆.

A.3 the minimal list of decoupling indices

where Φ1 [s] has the structure of a p -k -1 × p -k -1 interactor. We apply the same procedure as before with regard to the columns 1, 2, ..., p -k + 1 of Φ r : we obtain a second decoupling index ∆ p-k-1 ≥ ∆ p-k which is as small as possible at this step. This procedure is iterated until obtaining the unique list

The minimality comes from the fact that at each iteration i, the increase of infinite structure ∆-i is the smallest that we can obtain by permutation of the outputs of Σ. Then this list "minore" all the other lists at the sense of Definition 2.


Adam Sochacki
email: adam.sochacki@polsl.pl

Joanna Surmacz-Górska

Olivier Faure
email: ofaure@emse.fr

Bernard Guy
email: bguy@emse.fr

Polishing of synthetic electroplating wastewater in microcosm upflow constructed wetlands: Metals removal mechanisms

Keywords: electroplating wastewater, metals, cyanides, constructed wetlands, upflow, mechanism

This paper presents the study of the metals removal mechanisms in microcosm upflow constructed wetlands used for polishing of synthetic electroplating wastewater. Four types of columns were used: with and without plants and with different bed media (peat or gravel). The main design goal of the columns with peat was to promote precipitation of metals with biogenic sulfides produced by sulfate-reducing bacteria. The feed of the system contained mainly Cu, Ni, Pb, Zn, sulfates and cyanides. The substrate from the microcosm was sampled after cessation of the experiment (53 weeks) from the bottom and top layers of selected columns and was analyzed using sequential extraction procedure and scanning electron microscopy-energy-dispersive X-ray spectroscopy method. Additionally, solids retained on filter surface after filtering water samples withdrawn from the water layer covering the bed media in selected columns were analyzed by the latter method. The obtained results showed that the intended process for metals removal, which was precipitation of metal sulfides, was responsible for binding only a minor fraction of metals. The major portion of metals was present as exchangeable and reducible fraction. The presented study suggests that the onset of the conditions favorable for precipitation of sulfides may not occur during shortterm experiment. It may stem from the specific characteristics of the experiment. It is suggested that short-term studies should not be extrapolated to predict long-term behavior of the system.

Introduction

The concept of using constructed wetlands (CWs) for cost-and energy-efficient treatment of industrial wastewaters laden with metals has been demonstrated with a high degree of success in the world [START_REF] Haarstad | Organic and metallic pollutants in water treatment and natural wetlands: a review[END_REF][START_REF] Kosolapov | Microbial processes of heavy metal removal from carbon-deficient effluents in constructed wetlands[END_REF]. The application of CWs for the treatment of electroplating wastewater was reported scarcely in the literature. On the Web Of Knowledge [START_REF]Thomson Reuters Web of Knowledge SM[END_REF] a query including 'electroplating' and 'wetland*' (as topics) yielded only 6 results. Five of which dealt with interactions between plants and metals, and one regarding mobility of Cr in natural wetlands.

Noteworthy, none of these articles reported application of CWs for the treatment of electroplating wastewater. The challenge of treating or polishing electroplating wastewater in CWs is that it contains high concentrations of metals and cyanides (often on the order of hundreds of mg/L) and is deficient in organic matter [START_REF] Bartkiewicz | Oczyszczanie ścieków przemysłowych (Industrial wastewater treatment)[END_REF]. Indeed, the treatment of raw electroplating wastewater appears unfeasible to the toxic effect on the plants and microorganisms and the necessity of frequent exchange of the bed medium due to exhaustion of its sorption properties [START_REF] Lesage | Effects of sorption, sulphate reduction, and Phragmites australis on the removal of heavy metals in subsurface flow constructed wetland microcosms[END_REF]. Constructed wetlands can be used for the treatment (polishing) of pretreated electroplating wastewater, in which relatively low concentrations of metals and cyanides are present. Metals because of their nature, unlike cyanides or numerous organic compounds, cannot be degraded in CWs to simpler substances. For this reason, the retention mechanism of metallic pollutants is crucial as it determines their mobility in the system [START_REF] Ranieri | Clogging influence on metals migration and removal in subsurface flow constructed wetlands[END_REF][START_REF] Kosolapov | Microbial processes of heavy metal removal from carbon-deficient effluents in constructed wetlands[END_REF].

The removal of metals in subsurface-flow CWs is dominated by four mechanisms: adsorption, precipitation of metal sulfides or (oxy)hydroxides, and deposition of suspended solids. It should be noted that a minor fraction of metals is removed by hydrophytes, which unlike terrestrial plants, cannot hyperaccumulate metals [START_REF] Marchand | Metal and metalloid removal in constructed wetlands, with emphasis on the importance of plants and standardized measurements: A review[END_REF]. Preciptation of metals with biogenic sulfides is the crucial removal process because metal sulfides can remain in the sediments as long as anoxic conditions occur and solubility and mobility is very low as compared to other retention mechanisms [START_REF] Kosolapov | Microbial processes of heavy metal removal from carbon-deficient effluents in constructed wetlands[END_REF]. This process is mediated by sulfate-reducing bacteria (SRB), whose growth requires: anoxic conditions, presence of simple organic compounds (as electron donors), sulfates (as electron acceptors) [START_REF]Engineering guidelines for the passive remediation of acidic and/or metalliferrous mine drainage and similar wastewaters[END_REF][START_REF] Reddy | Biogeochemistry of Wetlands: Science and Applications[END_REF]. Considering the polishing of electroplating wastewater in CWs it can be stated that the required availability of sulfates can be easily met because this type of wastewater contains elevated concentrations of sulfates (added as sulfuric acid to adjust pH) [START_REF] Pümpel | Removal of nickel from plating rinsing water with a moving-bed sand filter inoculated with metal sorbing and precipitating bacteria[END_REF]. Anoxic conditions can occur in the CWs with saturated bed media as horizontal-flow systems or vertical-flow system with saturated bed (e.g. upflow (UF) systems). A proper carbon source should be provided to cause other microorganisms to remove the oxygen from the environment but also to stimulate the growth of SRB [START_REF] Kosolapov | Microbial processes of heavy metal removal from carbon-deficient effluents in constructed wetlands[END_REF]. The aforementioned carbon deficiency of electroplating wastewater necessitates the use of organic carbon sources for SRB when this wastewater is to be treated in CWs. Carbon source for SRB can be of internal or external origin. The latter can be added as aqueous solution to the wastewater and the former is present in the system as a solid form. Both types of carbon source can be used in CWs [START_REF] Liamleam | Electron donors for biological sulfate reduction[END_REF]. Constructed wetlands were found to efficiently remove metals by their precipitation with sulphides, but this process is dependent on numerous factors such as: type of wastewater, type of carbon source, retention time, maturity of the system etc. [START_REF] Lesage | Effects of sorption, sulphate reduction, and Phragmites australis on the removal of heavy metals in subsurface flow constructed wetland microcosms[END_REF][START_REF] Sheoran | Heavy metal removal mechanism of acid mine drainage in wetlands: A critical review[END_REF][START_REF] Chagué-Goff | Assessing the removal efficiency of Zn, Cu, Fe and Pb in a treatment wetland using selective sequential extraction: a case study[END_REF][START_REF] O'sullivan | Accumulation and fate of contaminants (Zn, Pb, Fe and S) in substrates of wetlands constructed for treating mine wastewater[END_REF]. The predominant removal mechanisms should be evaluated for a specific case.

The efficiency of CWs for wastewater treatment is often based on influent-effluent comparison, in which the treatment system is regarded as a "black box" [START_REF] Kosolapov | Microbial processes of heavy metal removal from carbon-deficient effluents in constructed wetlands[END_REF]. The knowledge of the major removal mechanisms allows prediction of the future performance of a system and its impact on the environment after its operation is terminated. Importantly, the information furnished by the small-scale studies could enhance the design of large-scale system by emphasizing those features of a system that promote the dominance of the most advantageous mechanism of removal.

The goal of the experiment presented in this paper was to study the removal mechanisms of metallic pollutants in microcosm upflow CWs polishing synthetic wastewater. The obtained results were also discussed in light of the treatment efficiency of the microcosm system presented in [START_REF] Sochacki | Polishing of synthetic electroplating wastewater in microcosm upflow constructed wetlands: Effect of operating conditions[END_REF]. The detailed description of the experimental system and the performance results were presented in [START_REF] Sochacki | Polishing of synthetic electroplating wastewater in microcosm upflow constructed wetlands: Effect of operating conditions[END_REF].

Methods

Microcosm constructed wetlands

The experimental system included four types of microcosm upflow CWs depending on the type of bed media and the presence of vegetation: vegetated columns with an equivolume mixture of peat and gravel (hereafter 'peat') (UF-PP columns), unvegetated columns with peat (UF-PU), vegetated columns with gravel (UF-GP) and unvegetated columns with gravel (UF-GU). Each type of column was duplicated, therefore 8 columns were used in total. The goal of the columns with peat (UF-PP and UF-PU) was the removal of metals by their precipitation with biogenic sulfides produced by SRB. Peat was used to serve as a carbon source for SRB on the assumption of its hydrolysis and fermentation to produce compounds utilized by SRB (e.g. acetates or lactates). The columns with gravel were used a reference systems, to which no carbon was added, neither as solid material (e.g. peat) nor dissolved substances present in the feed, before week 44 of the experiment. In week 44 lactates were added to stimulate SRB in the gravel columns and to compare the effect of internal (peat) and external (lactates) carbon source on the performance of the microcosm CWs. The feed with lactates were added only to the gravel columns. The electron acceptor for SRB was sulfates present in the feed.

The columns were fed for 53 weeks (the whole experiment) with synthetic electroplating wastewater. In the end of the experiment the substrate from the columns was analyzed to elucidate metal removal mechanisms. The composition of the feed was modified during the experiment and was presented in detail in [START_REF] Sochacki | Polishing of synthetic electroplating wastewater in microcosm upflow constructed wetlands: Effect of operating conditions[END_REF]. The median composition of wastewater (with minimum -maximum values in the brackets) for the whole experiment (53 weeks) was as follows:

-For the feed without lactates: Cu The feed without lactates was used during the whole experiment (weeks 1-53) as the influent of the columns with peat and in weeks 1-43 as the influent of the columns with gravel. In weeks 44-53 the feed with lactates was used as the influent of the columns with gravel. The results presented in this paper are for the planted columns with peat (UF-PP), planted columns with gravel (UF-GP), and to lesser extent for the unplanted columns filled with gravel (UF-GU). The UF-PU columns will not be further discussed in this article.

Sampling procedure

Substrate samples were taken from the bottom and top of columns after cessation of the experiment (53 weeks). It was assumed that these zones were represented by 150 mm layer from the bottom or top surface. The samples were than homogenized and subdivided into the subsamples for sequential extraction procedure (SEP) and scanning electron microscopyenergy-dispersive X-ray spectroscopy (SEM-EDS) analyses. The additional samples were:

filters through which water layer over the bed medium of the UF-GU and UF-PP columns were filtered. To this end 1.5 L wastewater was withdrawn from the water layer of the UF-PP and UF-GU columns and was filtered through a 0.45 µm filter.

Sequential extraction procedure

The manner in which metals were bound to substrate in the experimental system was assessed by the SEP based on the BCR2 protocol [START_REF] Rauret | Improvement of the BCR three step sequential extraction procedure prior to the certification of new sediment and soil reference materials[END_REF]. This procedure consists of three steps devised to extract three operationally defined fractions. In the first step exchangeable, water-and acidsoluble fraction was targeted and 0.11 M acetic acid was used. The second step was to extract reducible phase by the use of 0.5 M hydroxylammonium hydrochloride at pH 1.5. The target phase in step 3 was reducible fraction and reagents used were 8.8 M H 2 O 2 at 85°C and then 1

M ammonium acetate at pH 2.0. As an internal check on the procedure, the residue from step 3 was digested in aqua regia (step 4) and the amount of metals extracted (i.e., sum of step 1, step 2, step 3 and residue) was compared with that obtained by independent aqua regia digestion of a separate sample of the sediment. The SEP according to the BCR2 protocol is presented in Tab. 1 including operationally-defined fractions and the corresponding nominal target phases. Noteworthy, metals bound as sulfides are extracted from the analyzed substrate in step 3 and are regarded as oxidisable fraction.

Table 1

The BCR2 sequential extraction scheme (adapted from [START_REF] Rauret | Improvement of the BCR three step sequential extraction procedure prior to the certification of new sediment and soil reference materials[END_REF][START_REF] Bacon | Is there a future for sequential chemical extraction[END_REF]) The samples were air-dried before analysis for 48 hours. The SEP was conducted with approx. 2 g of substrate accurately weighed in 150 mL polypropylene centrifuge tubes. The increased mass of samples (compared to the protocol) was taken because of their poor homogeneity due to the presence of coarse gravel. The extracting solutions were added in a volume twice as large as in the BCR2 protocol, with the exception of step 3. The aqua regia digestion was based on the ISO 11466:1995 norm [START_REF]Soil quality -Extraction of trace elements soluble in aqua regia[END_REF] using 50 mL of the extracting solution.

Reagent

Each step (1-3) involved shaking the samples with extracting solution for 16 h on the endover-end shaker. The extract from the solid residue was separated by centrifugation at 3000 g for 20 min and the supernatant was decanted into a PE container, filtered and stored at 4°C for analyses. The residue was washed by adding 20 mL of distilled water, shaking for 15 min, centrifuging for 20 min, and then discarding the supernatant. The amount of metals extracted in the independent aqua regia digestion was assumed to represent pseudo-total concentration of metals in substrate. The extracts were analyzed for Cu, Ni, Pb, Zn using flame atomic absorption spectroscopy (AA Scan 1 Thermo Jarrell Ash apparatus). The recoveries between the sum of steps 1-4 and single-step aqua regia digestion of substrate samples were: Cu 109.56%, Ni 129.92%, Pb 123.75%, Zn 101.84%.

Scanning electron microscopy-energy-dispersive X-ray spectroscopy

Surface morphology of the samples was carried out using a scanning electron microscope (SEM) HITACHI S-3400N with backscattered electron detector (BSE). The analysis of chemical composition of the samples was performed using energy-dispersive X-ray spectroscopy (EDS) Thermo Noran System (System Six). EDS analysis is a semi-quantitative method with the limit of detection of ca. 0.2 atomic % depending on an element. SEM imaging and chemical microanalysis were performed in a low vacuum (50 Pa) and at accelerating voltage of 15 kV. The analyzed samples were placed on titanium sample mounts covered with carbon tape.

Results and discussion

The fractionation of metals in the substrate of CWs is highly important as unstable forms of metals may be easily leached from the system and may compromise the treatment efficient in the long-term operation. The substrates from the experimental were analysed by the SEP to elucidate the fractionation of metals in the experimental systems. The SEM-EDS analysis was used to verify the results of SEP and to examine the hypothesis that elemental sulfur was precipitated in the columns of the experimental system.

Operationally-defined metal fractions in the spent substrate

The fractionation of metals in the substrate of CWs is highly important as unstable forms of metals may be easily leached from the system. The upward flow direction and constant saturation of the bed media was believed, when designing the experiment, to promote stable anoxic conditions and to remove majority of the contaminants in the bottom part of the column. To verify this hypothesis the bed media from the planted columns were analyzed by the SEP and aqua regia extraction. The substrates from the UF-PP and UF-GP columns were analyzed by the SEP to elucidate the fractionation of metals in the experimental system. The results for the UF-PP column are presented in Fig. 1a and for the UF-GP column in Fig. 1b. On comparing the results presented in Fig. 1 it can be observed that the propotion of various binding mechanisms varies between the columns with peat (UF-PP) and gravel (UF-GP). The exchangeable fraction is higher in the UF-GP column than in the UF-PP column for all the metals in the bottom and top layers, except for Zn in the top layer. The exchangeable fraction is 43% higher for Cu (both in top and bottom layers), 33% and 8% higher for Ni (bottom and top layers, respectively), 44% and 6% for Pb (bottom and top layers, respectively), and 8% for Zn in the bottom layer. Only the exchangeable fraction of Zn in the top layer of the UF-GP column is 5% lower than in the UF-PP column. In contrast to the observations for the exchangeable fraction, the reducible fraction is higher in the UF-PP column than in the UF-GP column with only one exception, which is Ni in the top layer of the bed media higher by 39% in the UF-GP column. In comparison to the UF-GP column the reducible fraction of Cu in the UF-PP column is 22% and 24% higher (bottom and top layers, respectively), 18% higher for Ni in the bottom layer, 3% and 28% for Pb (bottom and top layers, respectively), and 4% and 5% for Zn (bottom and top layers, respectively). As compared to the exchangeable and reducible fractions in both the UF-PP and UF-GP columns the oxidisable fraction is lower for most metals in both top or bottom layers. The oxidisable fraction is higher in the UF-PP column for Cu by 16% and 41% (bottom and top layers, respectively), 46% for Ni in the top layer, and 19% for Pb in the bottom layer. It is almost equal for Zn in either layers, as the difference is 1 -2%. In the case of Ni in the bottom layer and Pb in the top layer, the oxidisable fraction is higher in the UF-GP columns by 5% and 8%, respectively.

It was found that Cu and Ni were bound with the less mobile fractions, and that ZN was bound mostly with the mots easily leached fractions. Similar observations were presented in [START_REF] Chagué-Goff | Assessing the removal efficiency of Zn, Cu, Fe and Pb in a treatment wetland using selective sequential extraction: a case study[END_REF] for the surface-flow CW treating domestic wastewater.

The operatinally-defined oxidisable fraction (nomial fractions: organic matter and sulfides [START_REF] Bacon | Is there a future for sequential chemical extraction[END_REF]) was found to be the major fraction only for Ni (66.4%) and Cu (48.8%) in the top layers of the UF-PP columns. Interestingly, the removal of these elements was the highest in the UF-PP columns (above 90%) [START_REF] Sochacki | Polishing of synthetic electroplating wastewater in microcosm upflow constructed wetlands: Effect of operating conditions[END_REF]. The obtained results indicate that desired process of metals precipitation with biogenic sulfides did not play a major role in removal of the metallic contaminants in the UF-PP columns, with the exception of Cu and Ni in the top layers of these columns. There are two groups of factors that could be responsible for this observation.

The first group includes operational factors: deficiency of organic carbon available for SRB, presence of oxidized forms of Fe and Mn, brevity of the experiment or short hydraulic retention time (HRT). The second group includes analytical factors, which are related to the manner in which the samples were analyzed. The performance results indicated low or negative removal of sulfates in the peat and gravel columns, respectively. The addition of lactates allowed achieving positive removal (approx. 4%) of sulfates in the columns with gravel in the last 10 weeks of the experiment [START_REF] Sochacki | Polishing of synthetic electroplating wastewater in microcosm upflow constructed wetlands: Effect of operating conditions[END_REF]. The brevity of the experiment (53 weeks)

was not sufficient for establishing efficient sulfate-reducing reactors. The results presented in this article indicate that adsorption and oxic processes were predominant in the studied columns. Oxidized forms of Fe and Mn, which are more favorable in SRB metabolism than sulfates, could serve as electron acceptors for SRB until they were depleted. Also the relatively short HRT [START_REF] Sochacki | Polishing of synthetic electroplating wastewater in microcosm upflow constructed wetlands: Effect of operating conditions[END_REF] could be hindering the onset of conditions conducive to sulfate reduction. Some researchers also did not found sulfides precipitation to be the major sink for metals. For example, Neculita et al. [START_REF] Neculita | Effectiveness of sulfate-reducing passive bioreactors for treating highly contaminated acid mine drainage: II. Metal removal mechanisms and potential mobility[END_REF] found that only up to 15% of metals were removed by the precipitation with sulfides in planted passive bioreactors with HRT 7.3 d and 10 d on analyzing spent substrate withdrawn from the reactors in week 44 of the experiment.

The artifacts introduced during the analysis of the samples could have been introduced by oxidizing the reduced forms of metals and extracting oxidisable fraction by hydroxylammonium chloride, which was used in the applied SEP [START_REF] Ngiam | Speciation patterns of heavy metals in tropical estuarine anoxic and oxidized sediments by different sequential extraction schemes[END_REF][START_REF] Peltier | Metal speciation in anoxic sediments: When sulfides can be construed as oxides[END_REF]. Ngiam and Lim [START_REF] Ngiam | Speciation patterns of heavy metals in tropical estuarine anoxic and oxidized sediments by different sequential extraction schemes[END_REF] found that anoxic sediments were oxidized despite the fact that oxygen-free atmosphere were maintained during the analysis. Peltier et al. [START_REF] Peltier | Metal speciation in anoxic sediments: When sulfides can be construed as oxides[END_REF] found that the application of hydroxylammonium chloride may lead to great underestimation of the sulfidic fraction in anoxic sediments. These shortcomings practically preclude inter-experimental comparison of SEP results, however, it is possible to employ SEP to compare the fraction of metals when uniform conditions are ensured during the analysis (as in this study).

It was expected that the sulfidic fraction would not be the major sink for metals in the UF-GP columns due to the organic carbon deficiency in the bed medium and in the feed. The UF-GP columns were used as reference columns in terms of the effect of organic bed medium (peat).

Noteworthy, the UF-GP columns were amended by adding lactates to the feed to stimulate the activity of SRB in the last 10 weeks of the experiment. The addition of lactates was found to cause significant increase in the removal of metals and sulfates [START_REF] Sochacki | Polishing of synthetic electroplating wastewater in microcosm upflow constructed wetlands: Effect of operating conditions[END_REF], but the short period of lactate dosing did not cause the oxidisable fraction to be a major sink for any of the studied metals in any of the analysed layers.

Pseudo-total concentrations of metals in the substrate

The relative content of the metal fractions does not indicate the quantitative distribution in the columns and the concentration which in concert with speciation can furnish complete information on the toxicological risk associated with the contaminants bound in the CWs. The pseudo-total concentration of metals in the tested substrates is represented by the amount of metals extracted in the aqua regia digestion of the substrate (Tab. 2). The presentation of the absolute concentrations of the fractions was omitted in this article, as large difference between the concentrations of metals would diminish the clarity of the data presentation regarding the fractionation of metals. The main difference between the UF-PP and UF-GP columns consisted in the quantitative distribution of metals. In the UF-PP columns higher concentration of all the metals was observed in the top part of the columns in contrast to the UF-GP column. It is difficult to elucidate the cause of different behavior of metals in the UF-PP and the UF-GP columns. The probably explanation could be the acidifying effect of peat (pH 3.5-4.5, [START_REF] Sochacki | Polishing of synthetic electroplating wastewater in microcosm upflow constructed wetlands: Effect of operating conditions[END_REF]) that enhanced mobility of metals entering the system and deeper penetration into the column with subsequent binding in the top part of the column at the interface between anoxic and oxic layers where various processes could play a role in the removal. The concentration of metals in both parts of the planted columns was in the decreasing order of Zn > Ni > Cu > Pb.

SEM-EDS analysis of the spent substrate and filtered solids

The application of SEP for anoxic sediments has been widely discussed in the literature. It was suggested that commonly applied SEPs may underestimate the amount of sulfidic fraction in those sediments. Those chemical analyses should be verified using, among other methods, X-ray absorption spectroscopy or SEM-EDS [START_REF] Peltier | Metal speciation in anoxic sediments: When sulfides can be construed as oxides[END_REF].

SEM-EDS analysis of the substrate from the UF-PP

The SEM-BSE images of the samples taken from the bottom and top parts of the UF-PP column are presented in Fig. 2 and the elemental composition of the selected areas is listed in Tab. 3. The colors of the rectangles indicating the selected area may be ignored. White or black colors were used to increase visibility. It can be seen that the grain of gravel had some peat attached to it, in which small bright particles were present, indicating their higher mean atom mass resulting probably from the presence of Fe. The amount of S is small suggesting that sulfides were not abundant in the top Both samples had similar composition characterized by elevated amounts of Fe accompanied by much lower amounts of Mn. It was also observed that the increased content of Fe corresponded to elevated amounts of S, which may indicate the presence of sulfides (e.g. area 7 of sample UF-GP-bot( 1)). This may also suggest that ferrous iron competes for sulfide with other cations, which were present in the feed as was observed by Song [START_REF] Song | Mechanisms of lead and zinc removal from lead mine drainage in constructed wetland[END_REF]. Some of the areas observed on the analyzed material seemed to be native to the substrate, e.g. area 2 of sample UF-GP-bot(1), and areas 4-5 of sample UF-GP-bot [START_REF] Kosolapov | Microbial processes of heavy metal removal from carbon-deficient effluents in constructed wetlands[END_REF]. The EDS analysis of the sample from the top part of the UF-GP column indicated the presence of the metals of concern: Ni, Cu and Zn. Area 2 of sample UF-GP-top(2) contains elevated amounts of S and Zn, which may suggest the presence of sulfides. Also area 4 of the UF-GP-top(2) may contain sulfides, however, the amount of S is small. The composition of area 1 of sample UF-GP-top( 2) is untypical as it contains chromium Cr, which was not added to the feed and it is likely that it was native to the substrate.

UF-PP-top(1)_ar6

SEM-EDS analysis of the raw gravel

The above-presented results of the SEM-EDS analysis of the spent substrate contained several areas in which the elemental composition was much different from the other analyzed areas.

In this situation it was assumed that the detected elements were of native origin. This means that they were present in the media before filling the columns. The SEM-EDS analysis was performed to furnish more information on the forms of native contaminants in the raw media.

The SEM-BSE images of the samples of the raw gravel used in the microcosm system are presented in Fig. 4 and the elemental composition of the selected areas is listed in Tab. 5.

Figure 4

SEM-BSE image of the raw gravel used as a bed medium

Table 5

Elemental (atomic %) composition of the areas (ar#) indicated in the image in Fig. 4 It can be observed that the surface of the gravel is rough. This property of the material could be essential as it provides advantageous substrate for bacterial attachment and biofilm formation. The elemental composition indicates that the gravel contained more impurities that could have been leached into the wastewater. Noteworthy, Fe was found in all the analyzed areas, Ni was found in two areas, and Co was found once. It was not analyzed if the native elements could be leached from the gravel. Selection of bed medium for the full-scale system should include analysis of the material as elevated concentration of metals in the medium may pose environmental risk. Additionally, metals of the native origin may be construed as the removed from the wastewater when a spent substrate is analyzed.

Analysis of suspended solids retained on filters

The concentration of contaminants was only analyzed in filtered samples of the influent and effluent of the experimental system [START_REF] Sochacki | Polishing of synthetic electroplating wastewater in microcosm upflow constructed wetlands: Effect of operating conditions[END_REF]. This is a common practice, however, certain metals species (e.g. aggregated hydroxides or sulfides) may not pass through filter, markedly overestimating the removal efficiency of the system. This analysis is not fully representative as the sediments on the surface of the bed media in the columns were resuspended while sampling the wastewater above the surface of the bed. Resuspension of the sediments was aimed at analyzing the white precipitates easily noticeable in the UF-GU column and presumed to be elemental S. The white precipitates were only observed in the columns with gravel (UF-GP and UF-GU), which were fed with the influent containing lactates from week 44 on. Noteworthy, the white precipitates were absent in these columns before the addition of to the feed. It was hypothesized that lactates stimulated bacterial production of sulfides, which subsequently were oxidized to elemental S on the anoxic-oxic interface in the columns with gravel.

The lack of peat particles in the UF-GU columns allowed observation of dilute skim milk appearance in this column, which was caused probably by elemental S (Fig. 5). It was hypothesized that the occurrence of elemental S was the effect of sulfur-oxidizing bacteria as

Beggiatoa that incompletely oxidize sulfide to S [23].

Figure 5

Outflow of the UF-GU column with visible white precipitates

The SEM-BSE images of the filter through which the wastewater form the UF-GU column was filtered are presented in Fig. 6 and the elemental composition of the selected areas is listed in Tab. 6. It can be seen that all the analyzed areas contained high amounts of S. It was mostly in amorphous form but could also be present in the crystalline form (area 1 of sample UF-GUfilter(2)). Also the metals of concern including Ni, Cu and Zn were present on the filter as species that are difficult to determine, however, the presence of sulfides cannot be excluded.

In the UF-GU-filter(4) sample more crystals of sulfur were observed. The metals of concern, especially Cu and Zn were present in higher concentration that in the previous two samples.

The white tarnish was also observed on the topmost grains of gravel in the UF-GU column.

This layer of gravel was collected and analyzed to confirm if the white precipitates found on the filter were also present on the gravel. The SEM-BSE images of topmost gravel layer from the UF-GU column are presented in Fig. 7 and the elemental composition of the selected areas is listed in Tab. 7. Elemental (atomic %) composition of the areas (ar#) indicated in the images in Fig. 7 determined using EDS The EDS analysis of the gravel confirmed the presence of S in the UF-GU column, however, in quantities smaller than on the filter. This could be caused, amongst others, by the composition of the gravel itself and the drying of gravel, which could promote precipitation of sulfates of, for example, Ca and Na. However, most of the forms observed on the gravel are not crystalline, whereas sulfates of Ca and Na form crystalline forms. It is noticeable that the amount of the S precipitates is smaller than on the filter and that it forms cluster of different sizes. The largest cluster of S (area 1 of sample UF-GU-top( 2)) contained 80% of this element. The crystalline forms in sample UF-GU-top(1) that had the appearance of a rosette consisted mainly of S and Ca and were probably gypsum.

The precipitates occurring in the columns with gravel were not observed in the UF-PP and UF-PU columns, which could be attributed to the entrapment of the precipitates in a thick layer of sediments (originating from the peat) in these columns. The main reason might be, however, that the columns with peat (UF-PP and UF-PU) were not fed with the influent containing lactates. The SEM-BSE images of the filter through which the wastewater form the UF-PP column was filtered are presented in Fig. 8 and the elemental composition of the selected areas is listed in Tab. 8. The elemental composition of the solids retained on the filter indicates elevated concentration of S and two of the metals of concern: Ni and Zn (Tab. 7). The presence of elemental S cannot be excluded, however, the S species present in the sample could be rather sulfate and sulfide.

Sulfate was probably present in area 2 of the UF-PP-filter(1) sample as calcium sulfate. This compound could form when the substrate was dried for the analysis. Interestingly, Cu was not detected in the two areas analyzed on the filter, as in the other samples of substrate taken from this column (Section. 3.3.1). The properties of the sediments in the UF-PP columns (peat)

hinder observation and analysis of small particles such as elemental S or sulfides, however, neither naked-eye observation nor SEM-EDS analysis indicated the presence of elemental S.

The hypothesis on the presence of elemental S cannot be corroborated nor rejected. The eggshaped objects in Fig. 8b were probably cysts of Daphnia or Oligochaeta.

It should also be mentioned that the white precipitates observed in the UF-GU column cannot be regarded as S with absolute certainty. This is because some of the heavy metals have X-ray spectra that overlap the light element X-ray spectra and create difficulties with analysis. In this study it is of particular importance that the M series of Pb at 2.346 and 2.502 overlaps the Kα line of S [START_REF] Echlin | Handbook of Sample Preparation for Scanning Electron Microscopy and X-ray Microanalysis[END_REF]. Similar appearance as is assumed to be caused by elemental sulfur can be also cause by the precipitates of lead sulfate. Hence, the hypothesized presence of sulfur should be confirmed using other analytical method.

The unsightly appearance of the dilute skim milk phenomena, whether it is elemental S or not, should be prevented in the full-scale systems.

Conclusions

The target process for the removal of metals in the columns filled with peat was precipitation of metals with biogenic sulfide being the product of the activity of SRB. In general, this process was found not to play a major role in the removal of metals. The major portion of metals was bound to exchangeable or reducible fractions. It allows conclusion that adsorption and oxic removal processes are mainly responsible for the removal of metals in a short-time experiment, which was described in this paper. Peat was found, however, to promote precipitation of metal sulfides to higher extent than gravel. The oxidisable fraction (including sulfidic fraction) was found to be the major sink for Cu and Ni in the top layers of substrate in the UF columns filled with peat. Zn was bound with exchangeable fraction in both gravel and peat media and was prone to be easily leached from the system.

The commonly available tools as SEP and SEM-EDS analysis, however, tend to underestimate the contribution of the metal sulfide precipitation. According to the statement that the metal retention mechanism evolves during the life of a system [START_REF] Neculita | Effectiveness of sulfate-reducing passive bioreactors for treating highly contaminated acid mine drainage: II. Metal removal mechanisms and potential mobility[END_REF] it cannot be ruled out that CWs can effectively remove metals from pretreated electroplating wastewater by sulfide precipitation. However, relatively brief microcosm experiment might not be suitable, therefore a pilot-scale study using real electroplating should be performed over a longer period. The presented study suggests that the onset of the conditions favorable for precipitation of sulfides may not occur after a relatively long period of operation. It may stem from the fact that peat (as internal carbon source), and all the more low carbon concentration in electroplating wastewater, are not sufficient for SRB. It should be further studied how the onset of sulfide precipitation could be accelerated. In the first part of this paper [START_REF] Sochacki | Polishing of synthetic electroplating wastewater in microcosm upflow constructed wetlands: Effect of operating conditions[END_REF] lactates were added to stimulate the activity of SRB, however, dosing started only in the last 10 weeks of the experiment and did not affect the binding mode of metals in the extent that could be represented by the SEP or SEM-EDS results.

As it was reported in [START_REF] Sochacki | Polishing of synthetic electroplating wastewater in microcosm upflow constructed wetlands: Effect of operating conditions[END_REF] both gravel and peat columns were efficient in removal of Cu, Ni and Zn. The presented results show, however, that the mechanisms responsible for the retention of metals vary. It is suggested that short-term studies should not be extrapolated to predict long-term behavior of the system. High efficiency of the CWs filled with gravel treating carbon-deficient wastewater under anoxic conditions in short-time experiments might be misleading as adsorption and oxic phases are responsible for the removal of metals. The former process may be terminated when the sorption capacity is exhausted. Oxic phases, on the other hand, can be dissolved on the onset of anoxic conditions e.g. when the system is amended with external carbon source.

5

1

 1 Metal fractions in the bottom (bot) and top layers of the UF-PP column (a) and UF-GP column (b)

Figure 2

 2 Figure 2 SEM-BSE images of wetland substrates sampled from the bottom (bot) and top of the UF-PP column

  part of the UF-PP column. The UF-PP-bot(2) sample contained elevated amount of Ni but it was probably of native origin. Both UF-PP-top samples contain the amounts of Ca and S suggesting the presence of gypsum (CaSO 4 •2H 2 O). The increased Fe content was observed in the area 7 of the UF-PP-top(1) sample. The characteristics of the adjacent areas may suggest that Fe is not present in sulfide form, but rather as oxidized species, namely (oxy)hydroxides.

Figure 3 SEM

 3 Figure 3

Figure 6

 6 Figure 6SEM-BSE images of the filter surface through which the effluent from the UF-GU column was passed

Figure 7

 7 Figure 7SEM-BSE images of wetland substrates sampled from the top of the UF-GU column

Figure 8

 8 Figure 8SEM-BSE images of the filter surface through which the effluent from the UF-PP column was passed

  Although not officially a step in sequential extraction, it is recommended that the residue at the end of Step 3 be digested with aqua regia and the sum of the four fractions be compared with the results of a separate aqua regia digestion of the sample

			Fraction label	Nominal target
				phase(s)
	Step 1	Acetic acid, 0.11 mol/L	Exchangeable,	Soluble and
			water and acid-	exchangeable cations
			soluble	and carbonates
	Step 2	Hydroxylammonium hydrochloride	Reducible	Fe-Mn
		0.5 mol/L at pH 1.5		oxyhydroxides
	Step 3	Hydrogen peroxide (85°C), 300	Oxidisable	Organic matter and
		mg/g, 8.8 mol/L then		sulphides
		ammonium acetate, 1.0 mol/L
	Step 4	Aqua regia 1	Residual
	1

Table 2

 2 Pseudo-total concentrations of metals (mg/kg DW) in the bottom and top layers of the substrate sampled from the planted UF columns

	Column	Sample	Cu	Ni	Pb	Zn
		location
	UF-PP	bottom 5.61	12.10	2.40	24.20
		top	11.74	66.65	7.47	103.9
	UF-GP	bottom 59.91	91.1	20.14	132.61
		top	9.00	24.13	3.54	29.89

Table 3 Elemental

 3

	Sample_Area	Atomic % 1
		Na Mg Al	Si	P	S	Cl K	Ca	Ti	Mn Fe
	UF-PP-bot(1)_ar1 2.9 2.5 16.2 56.7 -	2.7 -	10.2 3.4	-	5.3
	UF-PP-bot(1)_ar2 3.4 1.2 14.1 59.1 -	1.5 -	3.8 1.0 0.6 -	15.3
	UF-PP-bot(1)_ar3 2.4 1.7 16.6 64.2 -	2.4 -	7.5 2.4 -	-	2.7
	UF-PP-bot(1)_ar4 8.6 0.8 16.6 69.7 -	-	-	2.2 0.9 -	1.2
	UF-PP-bot(2)_ar1 -	-	3.9 78.8 -	-

(atomic %) composition of the areas (ar#) indicated in the images in Fig.

2

determined using EDS

Table 4

 4 Elemental (atomic %) composition of the areas (ar#) indicted in the images in Fig.3determined using EDS

	Sample_Area	Atomic % 1
		Na Mg Al	Si	P	S	K	Ca	Ti	Cr	Mn Fe	Ni Cu Zn
	UF-GP-bot(1)_ar1 7.2 0.6	24.7 59.0 -	0.9	3.3	1.9	-	-	-	2.4	-	-	-
	UF-GP-bot(1)_ar2 1.9 7.3	20.7 35.1 -	6.1	2.0	1.1	1.9 -	-	23.9 -	-	-
	UF-GP-bot(1)_ar3 3.8 1.0	15.1 40.9 -	1.3	6.1	5.7	4.6 -	-	21.6 -	-	-
	UF-GP-bot(1)_ar4 3.2 4.1	16.9 35.1 0.5 8.9	2.7	2.4	2.9 -	-	23.3 -	-	-
	UF-GP-bot(1)_ar5 6.7 1.3	23.2 54.0 -	2.4 3.2	4.4	-	-	-	4.9	-	-	-
	UF-GP-bot(1)_ar6 6.1 2.5	13.9 36.1 1.1 9.6	2.3	16.2 0.9 -	-	11.3 -	-	-
	UF-GP-bot(1)_ar7 3.9 1.3	8.0	20.4 0.6 26.9 0.9	5.2	-	-	0.5 32.1 -	-	-
	UF-GP-bot(2)_ar1 0.8 10.1 19.8 29.7 -	4.1	1.0	1.2	0.8 -	0.8 31.8 -	-	-
	UF-GP-bot(2)_ar2 1.0 6.3	14.8 26.4 -	4.2	4.9	1.3	2.6 -	0.7 37.9 -	-	-
	UF-GP-bot(2)_ar3 1.3 7.3	17.4 29.8 -	5.8	3.6	1.7	2.8 -	0.5 29.7 -	-	-
	UF-GP-bot(2)_ar4 1.2 7.7	16.9 26.8 0.4 4.6	2.2	1.9	1.3 -	0.7 36.3 -	-	-
	UF-GP-bot(2)_ar5 0.7 6.4	16.1 26.9 -	2.9	4.2	1.2	3.1 -	0.5 37.9 -	-	-
	UF-GP-bot(2)_ar6 -	0.7	9.6	79.0 -	1.4	3.4	1.3	-	-	-	4.6	-	-	-
	UF-GP-bot(2)_ar7 0.9 7.1	15.7 28.7 -	9.1	2.5	1.8	1.5 -	-	32.7 -	-	-
	UF-GP-top(1)_ar1 8.4		19.1 61.3 -	2.5	3.3	2.1	-	-	-	3.3	-	-	-
	UF-GP-top(1)_ar2 1.9 1.0	28.7 52.2 0.7 1.1	11.4 -	0.2 -	-	2.8	-	-	-
	UF-GP-top(1)_ar3 3.4 0.4	5.8	82.1 0.8 1.7	1.5	1.2	-	-	-	2.1	-	-	-
	UF-GP-top(1)_ar4 4.2 5.6	17.2 47.5 0.6 2.2	5.0	1.6	1.3 -	-	13.2 0.8 -	-
	UF-GP-top(2)_ar1 3.0 0.7	8.8	23.1 -	0.6	1.9	0.7	-	10.9 0.8 44.5 5.1 -	-
	UF-GP-top(2)_ar2 -	0.8	9.6	45.2 1.5 14.3 1.8	2.1	-	-	-	6.8	-	0.8 17.1
	UF-GP-top(2)_ar3 2.5 3.7	16.5 51.1 0.7 1.1	7.3	1.1	1.2 -	-	14.9 -	-	-
	UF-GP-top(2)_ar4 2.4 1.5	26.1 53.1 0.7 0.9	10.9 -	0.2 -	-	4.2	-	-	-
	1 Compositions normalized to 100%

Table 6

 6 Elemental (atomic %) composition of the areas (ar#) indicated in the images in Fig.6determined using EDS

	Sample_Area	Atomic % 1
		Na	Mg Al	Si	P	S	K	Ca	Fe	Ti	Ni	Cu Zn
	UF-GU-filter(1)_ar1 -	-	-	0.9	-	99.1	-

Table 8

 8 Elemental (atomic %) composition of the areas (ar#) indicated in the images in Fig.8determined using EDS

	Sample_Area	Atomic % 1
		Na Mg Al	Si	P	S	K	Ca	Ti	Mn Fe	Ni	Zn
	UF-PP-filter(1)_ar1 2.2 0.8 1.7 8.1 0.5 18.1 0.6 5.5 0.2 0.8 34.6 17.8 9.1
	UF-PP-filter(1)_ar2	0.1 0.7 3.4 0.4 47.1		37.8		5.0 2.1 3.4
	UF-PP-filter(1)_ar3 2.1 0.9 2.0 8.5 0.9 44.4 0.7 10.8		0.4 15.4 6.6 7.3

Compositions normalized to 100%

Acknowledgements

The project was supported by Grant N N523 561938 from the Polish Ministry of Science and Higher Education. The lead author was granted scholarship in the Project "SWIFT (Scholarships Supporting Innovative Forum of Technologies)" POKL.08.02.01-24-005/10 cofinanced by the European Union under the European Social Fund.

SEM-EDS analysis of the substrate from the UF-GP

The SEM-BSE images of the samples taken from the bottom and top parts of the UF-GP column are presented in Fig. 3 and the elemental composition of the selected areas is listed in Tab. 4. (b) Sample UF-GP-bot(1) (b) Sample UF-GP-bot(2) (c) Sample UF-GP-top(1) (d) Sample UF-GP-top(2)