K Zimenko

A Polyakov

D Efimov

On Finite-Time Robust Stabilization via Nonlinear State Feedback

Keywords: Non-asymptotic stabilization, Implicit Lyapunov Function, Homogeneous systems

. The control law is presented in an explicit form and allows to find the values of all control parameters, that make the solution one of the most constructive. The theoretical results are supported by numerical example.

INTRODUCTION

Frequently, strict quality requirements are imposed for design, operation and control of complex technical processes. One such requirement is completion of all transients in a finite time. In particular, this issue is relevant in control design for a variety of robotic and mechatronic devices, safety evaluation, aerospace applications, vehicles control systems, etc. Therefore the problems of finite-time control and finite-time observation are intensively studied for many years (see, for example, [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF]- [START_REF] Moulay | Finite-time stability and stabilization: State of the art[END_REF] and [START_REF] Engel | A continuous-time observer which converges in finite time[END_REF]-[8], [START_REF] Shen | Semi-global finite-time observers for nonlinear systems[END_REF] correspondingly).

Solutions of the problem of control design for chain of integrators are quite sought-after. This is motivated by the fact that control scheme for chain of integrators can be easily extended to larger classes of systems (see, for example, [START_REF] Polyakov | Robust Stabilization of MIMO Systems in Finite/Fixed Time[END_REF]). Moreover, in many applications the nominal models have the form of chain of integrators, for instance, mechanical planar systems. There is a number of research results on the finite-time stabilization or regulation of a chain of integrators (see, for example, [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF], [START_REF] Bernuau | Finite-time output stabilization of the double integrator[END_REF]). However, most of them do not find an application in practice because of constructive and computational complexity. For instance, the methods mainly do not suggest constructive algorithms of parameter tuning and ensure only existence of suitable control parameters.

One of effective methods for the finite-time stability analysis is the method of Implicit Lyapunov Function (ILF) [START_REF] Polyakov | Robust Stabilization of MIMO Systems in Finite/Fixed Time[END_REF], [START_REF] Adamy | Soft variable-structure controls: a survey[END_REF], [START_REF] Korobov | A general approach to synthesis problem[END_REF]. This method allows to check all stability conditions directly using implicitly defined Lyapunov function which is a solution of some algebraic equation. Note that for such a stability analysis there is no need to solve this equation. Another issue that can be useful in analysis of finite-time stability is property of homogeneity. For instance, if an asymptotically stable system is homogeneous of negative degree, then it is finite-time stable.

The paper [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF] is related to development of method for finite-time control of multiple integrators together with implicit Lyapunov function of closed-loop system. Conditions of finite-time stability are formulated in the form of LMI that essentially simplifies the process of tuning of control parameters. However, the practical realization of the ILF control algorithm requires development of a special numeric procedure for on-line computation of the ILF value at the current state (the control law depends on ILF value explicitly). Thus, in the case of extension of this method on a class of technical systems, the need of additional on-line computations may be a stumbling block for implementation of this method.

The present paper addresses the problem of a control design for the finite-time stabilization of a chain of integrators. The paper represents an extension and simplification of results in [START_REF] Zimenko | Stabilization of Chain of Integrators with Arbitrary Order in Finite-time[END_REF], [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF]. The developed finite-time control law does not depend on implicitly defined Lyapunov function that allows to avoid any additional on-line procedures. The result is obtained by using the method of Implicit Lyapunov Function with applying properties of homogeneous systems. As in [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF] parameter tuning is presented in LMI form.

The paper is organized as follows. Notation used in the paper is given in the next section. Problem statement is considered in Section III. Some background notions about finite-time stability, ILF method and homogeneity are introduced in Section IV. Control design algorithm and some remarks about proposed control scheme are presented in Section V. Section VI contains simulation results for effectiveness demonstration of proposed control. Finally, concluding remarks and possible directions for future research are presented in Section VII.

NOTATION

Through the paper the following notation will be used:

(i) R + = {x ∈ R : x > 0}, where R is the set of real number; (ii) • is the Euclidian norm in R n , i.e. x = x 2 1 + • • • + x 2 n for x ∈ R n ;
(iii) the positive (negative) definiteness (semi-definiteness) of a symmetric matrix P = P T ∈ R n×n is denoted by P > 0 (P < 0, P ≥ 0, P ≤ 0); (iv) for a symmetric matrix P = P T the minimal and maximal eigenvalues are denoted by λ min (P ) and λ max (P ), respectively; (v) a continuous function σ : R + ∪ {0} → R + ∪ {0} belongs to the class K if σ(0) = 0 and the function is strictly increasing; (vi) diag {λ i } n i=1 is the diagonal matrix with the elements λ i on the main diagonal;

(vii) for any x ∈ R n the upper directional derivative DV (x) of a locally Lipschitz continuous function V : R n → R + in the direction of ν ∈ R n is defined as follows:

DV (x)ν = lim sup y→x t→0 + V (y + tν) -V (y) t ,
where lim sup denotes the upper limit.

PROBLEM STATEMENT

Consider the control system

ẋ = Ax + bu + d(t, x), (1)
where x ∈ R n is the state vector, u ∈ R is the control input, d(t, x) : R n+1 → R n describes the system uncertainties and disturbances,

A =          0 1 0 • • • 0 0 0 1 • • • 0 . . . . . . . . . . . . . . . 0 0 0 • • • 1 0 0 0 • • • 0          and b =          0 0 . . . 0 1         
, which describes a chain of integrators for d(t, x) ≡ 0.

It is assumed that the disturbance d(t, x) may be non-Lipschitz or discontinuous. This assumption makes presented system attractive for a number of control applications, for example, mechanical systems with dry friction, friction electromechanical system [START_REF] Orlov | Finite Time Stability and Robust Control Synthesis of Uncertain Switched Systems[END_REF], cart-pendulum system [START_REF] Orlov | Discontinuous systems: Lyapunov analysis and robust synthesis under uncertainty conditions[END_REF], etc. Moreover, the controllable single-input nonlinear systems can be transformed to canonical form like (1) [START_REF] Isidori | Nonlinear Control Systems II[END_REF].

The aim of the paper is to develop the finite-time stabilizing control algorithm to the system (1), which is robust with respect to disturbances of certain class (to be defined below), has the form of the explicit full-state feedback (i.e. it does not require additional on-line computations as in [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF]) and admits the simple LMI-based scheme for parameters tuning.

PRELIMINARIES

Finite-time Stability

Consider the system of the form

ẋ = f (t, x), x(0) = x 0 , (2)
where x ∈ R n is the state vector, f ∈ R + × R n → R n is a nonlinear vector field, which can be discontinuous with respect to the state variable. In this case the solutions x(t, x 0 ) of the system (2) are understood in the sense of Filippov [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF].

According to [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF] an absolutely continuous function x(t, x 0 ) is called a solution of the Cauchy problem associated to (2) if x(0, x 0 ) = x 0 and for almost all t > 0 it satisfies the following differential inclusion

ẋ ∈ K[f ](t, x) = co ε>0 µ(N )=0 f (t, B(x, ε)\N ), (3)
where co(M ) defines the convex closure of the set M , B(x, ε) is the ball with the center at x ∈ R n and the radius ε, the equality µ(N ) = 0 means that the measure of N ⊂ R n is zero. The solution existence is guaranteed due to the following multi-valued function properties (standart assumptions): the right-hand side of (3) has non-empty, compact, convex values and it is upper semi-continuous [START_REF] Bernuau | On homogeneity and its application in sliding mode control[END_REF].

Assume that the origin is an equilibrium of the system (2) (i.e. 0 ∈ K[f ](t, 0)). Denote the set of all solutions of Cauchy problem (3) by Φ(x 0 ), i.e. x(t, x 0 ) ∈ Φ(x 0 ). The system (2) may have nonunique solutions for a given x 0 ∈ R n and may admit both weak (a property holds for a solution) and strong stability (a property holds for all solutions) (see, for example, [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF], [START_REF] Roxin | On finite stability in control systems[END_REF]). This paper deals only with the strong stability properties, which ask for stable behavior of all solutions of the system (2).

Definition 1 ([2], [START_REF] Roxin | On finite stability in control systems[END_REF], [START_REF] Polyakov | Stability notions and Lyapunov functions for sliding mode control systems[END_REF], [START_REF] Orlov | Finite Time Stability and Robust Control Synthesis of Uncertain Switched Systems[END_REF]):

The origin of system ( 2) is said to be globally finite-time stable if:

(i) Finite-time attractivity: there exists so-called settling time function T : R n \{0} → R + such that lim t→T (x0) x(t, x 0 ) = 0 for all x 0 ∈ R n \{0} and all x(t, x 0 ) ∈ Φ(x 0 ). (ii) Lyapunov stability: there exists a function δ ∈ K such that x(t, x 0 ) ≤ δ( x 0 ) for all

x 0 ∈ R n and all x(t, x 0 ) ∈ Φ(x 0 ).

Homogeneity

Homogeneity [START_REF] Zubov | Methods of A.M. Lyapunov and Their Applications[END_REF]- [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF] is an intrinsic property whereby objects such as functions or vector fields remain consistent with respect to some scaling operation called a dilation. For fixed r i ∈ R + , i ∈ {1, ..., n} called weights and λ > 0 one can define:

(i) the vector of weights r = (r 1 , ..., r n ) T ;

(ii) the dilation matrix

D(λ) = diag{λ ri } n i=1 and correspondingly D(λ)x = (λ r1 x 1 , ..., λ rn x n ) T for x = (x 1 , ..., x n ) T ∈ R n .
Definition 2 [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF]:

A function g : R n → R (vector field f : R n → R n ) is said to be r-homogeneous of degree m iff g(D(λ)x) = λ m g(x) (f (D(λ)x) = λ m D(λ)f (x)) for all λ > 0 and x ∈ R n .
Let us introduce the homogeneous norm x r of x ∈ R n defined by

x r = n i=1 |x i | ρ r i 1 ρ (4)
for ρ ≥ max i r i . Note, that homogeneous norm ( 4) is not a norm in classical sense, since the triangular inequality is not satisfied.

Obviously, each homogeneous norm is r-homogeneous of degree 1 and positive definite. For any x ∈ R n there are y ∈ S r = {x ∈ R n : x r = 1} and λ = x r such that x = D(λ)y [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF].

Theorem 1 [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF] Let f be a r-homogeneous of degree m continuous vector field on R n such that the system (2) is a locally asymptotically stable. Then it is globally asymptotically stable and there exists a continuously differentiable r-homogeneous of degree > -m Lyapunov function V for [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF].

Note that by definition of homogeneity there exist constants c 1 , c 2 > 0 such that

c 1 x r ≤ V (x) ≤ c 2 x r . (5)
Theorem 2 [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF] Let f : R n → R n be defined on R n and be a continuous r-homogeneous vector field with a negative degree. If the origin of the system (2) is locally attractive then it is globally finite time stable.

The following theorem and corollary extend presented results to differential inclusions.

Definition 3 A set-valued map F : R n ⇒ R n is said to be r-homogeneous of degree m ∈ R if for all x ∈ R n we have F (D(λ)x) = λ m D(λ)F (x), ∀x ∈ R n \ {0}, ∀λ ∈ R + . (6)
The system

ẋ ∈ F (x), x(0) = x 0 ∈ R n , (7)
is r-homogeneous of degree m if the set-valued map F is homogeneous of degree m.

Theorem 3 [START_REF] Bernuau | On an Extension of Homogeneity Notion for Differential Inclusions[END_REF] Let F be a r-homogeneous set-valued map with degree m, satisfying the standard assumptions. Then the next claims are equivalent: 1. The system [START_REF] Perruquetti | Finite-time observers: Application to secure communication[END_REF] is strongly globally asymptotically stable. 2. For all k > max(-m, 0), there exist a pair (V, W ) of continuous functions, such that:

V ∈ C ∞ (R n ) is positive definite and homogeneous with degree k; W ∈ C ∞ (R n \ {0}
) is strictly positive outside the origin and homogeneous with degree k + m; max ν∈F (x) DV (x)ν ≤ -W (x) for all x = 0.

Corollary 1 [START_REF] Bernuau | On an Extension of Homogeneity Notion for Differential Inclusions[END_REF] Let F be a r-homogeneous set-valued map with degree m < 0, satisfying the standard assumptions. Assume also that F is strongly globally asymptotically stable. Then F is strongly globally finite-time stable and the settling-time function is continuous at zero and locally bounded.

Implicit Lyapunov Function Method

The theorem given below presents the ILF method ( [START_REF] Adamy | Soft variable-structure controls: a survey[END_REF], [START_REF] Korobov | A general approach to synthesis problem[END_REF]) for finite-time stability analysis.

5

Theorem 4 [START_REF] Polyakov | Robust Stabilization of MIMO Systems in Finite/Fixed Time[END_REF], [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF] If there exists a continuous function

Q : R + × R n → R (V, x) → Q(V, x) such that C1) Q(V, x) is continuously differentiable for ∀x ∈ R n \{0} and ∀V ∈ R + ; C2) for any x ∈ R n \{0} there exist V -∈ R + and V + ∈ R + : Q(V -, x) < 0 < Q(V + , x); (8) C3) for Ω = {(V, x) ∈ R n+1 : Q(V, x) = 0} lim x→0 (V,x)∈Ω V = 0 + , lim V →0 + (V,x)∈Ω x =0, lim x →∞ (V,x)∈Ω V= +∞; C4) the inequality -∞ < ∂Q(V, x) ∂V < 0 holds for ∀V ∈ R + and ∀x ∈ R n \{0}; C5) the inequality ∂Q(V, x) ∂x f (x) ≤ ηV 1-µ ∂Q(V, x) ∂V holds ∀(V, x)
∈ Ω, where 0 < µ ≤ 1 and η > 0 are some constants.

Then the origin of the system (2) is globally finite-time stable with the following settling time estimate

T (x 0 ) ≤ V µ 0 ηµ ,
where

V 0 ∈ R + : Q(V 0 , x 0 ) = 0.
The conditions of Theorem 4 present in an implicit form the requirements on finite-time stability [START_REF] Moulay | Finite-time stability and stabilization: State of the art[END_REF]. The conditions C1, C2, C4 and the implicit function theorem [START_REF] Courant | Introduction to calculus and analysis[END_REF] imply that the equation Q(V, x) = 0 implicitly defines a unique function V : R n \ {0} → R + such that Q(V (x), x) = 0 for all x ∈ R n \ {0}. Due to the condition C3 the function V can be continuously prolonged at the origin by setting V (0) = 0. In addition, it is radially unbounded and positive definite. In conjunction with the condition C5, Theorem 4 guarantees fulfillment of sufficient conditions of finite-time stability mentioned, for example, in [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF], [START_REF] Moulay | Finite-time stability and stabilization: State of the art[END_REF].

MAIN RESULT

Introduce the ILF function

Q(V, x) := x T D(V -1 )P D(V -1 )x -1, (9) 6
where P = P T ∈ R n×n is a symmetric positive definite matrix, D(λ) is the dilation matrix of the form

D(λ) = diag{λ 1+(n-i)µ } for 0 < µ ≤ 1.
Note that for r = (1 + (n -1)µ, 1 + (n -2)µ, ..., 1) T the Lyapunov function V (x) is rhomogeneous of degree 1 as Q(V, D(λ)x) = Q(λ -1 V, x), i.e. V (D(λ)x) = λV (x) and the inequality (5) for = 1 takes the form

c 1 x r ≤ V (x) ≤ c 2 x r . ( 10
)
Let us denote the diagonal matrices

H µ = diag {1 + (n -i)µ} n i=1 , H(λ) = diag λ (n+1-i)µ n i=1 . Proposition 1
If the following statements hold: 1) the system of matrix inequalities

AX + XA T + by + y T b T + L by y T b T -γX < 0, ( 11
)
1 ν L ≥ XH µ + H µ X > 0, X > 0, (12)
Z ≤ 1 γ X, Z > 0, (13)
βL X(I -H(θ))

(I -H(θ))X Z ≥ 0, ∀θ ∈ [θ min , θ max ] (14)
is feasible for some µ ∈ (0, 1], β ∈ (0, 1), γ, ν, θ min , θ max ∈ R + , L, Z, X ∈ R n×n , y ∈ R 1×n ;

2) there exists a c u ∈ R + that satisfies the inequality

c 2 θ max ≤ c u ≤ c 1 θ min , (15)
where the coefficients c 1 and c 2 correspond to the inequality [START_REF] Moulay | Finite-time stability and stabilization: State of the art[END_REF].

Then the control of the form

u(x) = (c u x r ) 1-µ kD((c u x r ) -1 )x, (16)
where k = yX -1 stabilizes the origin of the system (1) in a finite time for the case d(t, x) = 0 and the settling time function is bounded as follows

T (x 0 ) ≤ c µ 2 x 0 µ r νµ(1 -β) . ( 17
)
Proof of Proposition 1 According to [START_REF] Polyakov | Robust Stabilization of MIMO Systems in Finite/Fixed Time[END_REF] the system (1) is finite-time robustly stable with some disturbance function d(t, x) if:

1) the following system of inequalities is feasible

AX + XA T + by + y T b T + L + R ≤ 0, 1 ν L ≥ XH µ + H µ X > 0, X > 0 (18)
for some µ ∈ (0, 1], ν ∈ R + and R ∈ R n×n , R > 0;

2) the conrol has the form

u(V, x) = V 1-µ kD(V -1 )x, (19)
where

V ∈ R + : Q(V, x) = 0 and Q(V, x)
presented by ( 9) with P = X -1 ;

3) the disturbance function d(t, x) satisfies the following inequality

d T (t, x)D(V -1 )R -1 D(V -1 )d(t, x) ≤ βV -2µ x T D(V -1 )P LP D(V -1 )x (20)
for some β ∈ (0, 1).

The settling time function estimate has the form

T (x 0 ) ≤ V µ 0 νµ(1 -β) . (21)
All proofs of function Q(V, x) compliance with the conditions C1)-C5) can be found in [START_REF] Polyakov | Robust Stabilization of MIMO Systems in Finite/Fixed Time[END_REF]. Let us substitute c u x r for the function V (x) in ( 19) and represent difference of controls as a disturbance in the equation ẋ = Ax + bu + d(t, x):

d(t, x) = -b V 1-µ kD(V -1 ) -(c u x r ) 1-µ kD((c u x r ) -1 ) x = -V 1-µ bk I n -H V cu x r D(V -1 )x. ( 22
)
Choose the matrix R in the form

R = I n-1 0 0 r nn ,
where , r nn ∈ R + , can be arbitrarily small. Then, substituting [START_REF] Shen | Semi-global finite-time observers for nonlinear systems[END_REF] into [START_REF] Korobov | A general approach to synthesis problem[END_REF] and taking into account

that b T D(V -1 )R -1 D(V -1 )b = V -2 r -1
nn it is easy to obtain:

V 2-2µ x T D(V -1 ) In -H V cu x r k T b T D(V -1 )R -1 D(V -1 )bk In -H V cu x r D(V -1 )x ≤ βV -2µ x T D(V -1 )P LP D(V -1 )x ⇓ 1 rnn x T D(V -1 ) In -H V cu x r k T k In -H V cu x r D(V -1 )x ≤ βx T D(V -1 )P LP D(V -1 )x ⇓ 1 rnn I -H V cu x r k T k I -H V cu x r ≤ βP LP. 8 Let r nn = 1 kP -1 k T . In this case R = r nn bb T + In-1 0 0 0 = 1 γ byX -1 y T b T + In-1 0 0 0
and the first inequality in ( 18) is equivalent to the inequality

AX + XA T + by + y T b T + L + In-1 0 0 0 by y T b T -γX ≤ 0
(for given R it can be obtained from the first inequality in (18) by using Schur complement). Since can be chosen arbitrary small this inequality can be replaced by strong inequality [START_REF] Bernuau | Finite-time output stabilization of the double integrator[END_REF]. Since

1 r nn P -1/2 k T kP -1/2 = γ P -1/2 k T kP -1/2 kP -1 k T = γ s T s ss T ≤ γI n ⇐⇒ 1 r nn k T k ≤ γP,
where s = kP -1/2 , then introducing the inequality γP ≤ Z -1 (equivalent to [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF] with P = X -1 ) we yield

I -H V c u x r Z -1 I -H V c u x r ≤ βP LP. ( 23
)
Let θ = V cu x r , where θ ∈ [θ min , θ max ]. Then, using Schur complement and multiplying both sides of obtained expression by diag{P -1 , I} the inequality ( 14) can be derived. Thus, both conditions [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF], [START_REF] Korobov | A general approach to synthesis problem[END_REF] are satisfied.

On the other hand, according to (10) θ ∈ c1 cu , c2 cu and the parameter c u should satisfy to the inequality [START_REF] Zubov | Methods of A.M. Lyapunov and Their Applications[END_REF].

Finally, the settling time function can be estimated as follows

T (x 0 ) ≤ V µ 0 νµ(1 -β) ≤ c µ 2 x 0 µ r νµ(1 -β) ,
where

V 0 ∈ R + : Q(V 0 , x 0 ) = 0.
In order to apply Proposition 1 in practice we need to solve the parametrized system of nonlinear matrix inequalities ( 11)-( 14) with respect to variables X, y, Z, L, θ min and θ max for a given γ, ν ∈ R + , µ ∈ (0, 1], and β ∈ (0, 1). By fixing the value θ ∈ [θ min , θ max ], the system (11)-( 14) becomes a system of LMIs, which can be solved using any suitable software (e.g. MATLAB, YALMIP, etc.). However, the LMIs have to be checked for any θ ∈ [θ min , θ max ]. Due to the smoothness of H(θ) with respect to θ ∈ R + , this can be done on a proper grid constructed over the interval [θ min , θ max ]. The following proposition provides sufficient feasibility condition of the inequality [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF].

Proposition 2

The parametric matrix inequality [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF] holds if for some µ ∈ (0, 1], β ∈ (0, 1) and θ min ,

θ max ∈ R +    βL X H X(I -H(θ i )) HX 1 (θi-θi-1) M 0 (I -H(θ i ))X 0 Z    ≥ 0, i = 1, . . . , N, (24)
Z H + HZ ≥ M , M > 0, (25)
where

θ min = θ 0 < θ 1 < • • • < θ N = θ max , H = diag {(n + 1 -i)µ} n i=1 .

Proof of Proposition 2 Taking into account the left-hand side of the inequality (23) let us define

W (θ) = z T (I -H(θ))Z -1 (I -H(θ))z for some z ∈ R n , then ∂W ∂θ = (I -H(θ))z 1 θ z T 1 θ ( HZ -1 + Z -1 H) -Z -1 -Z -1 θ 2 M -1 (I -H(θ))z 1 θ z -z T HM -1 Hz,
for some M > 0.

The matrix inequality

1 θ ( HZ -1 + Z -1 H) -Z -1 -Z -1 θ 2 M -1 > 0 is equivalent to HZ + Z H - 1 θ M > 0, (26)
that can be shown by using Schur complement and multiplying both sides by Z. Since θ ≥ c1 cu the inequality M = cu c1 M ≥ 1 θ M > 0 holds. Then, according to [START_REF] Efimov | Supervisory Acceleration of Convergence for Homogeneous Systems[END_REF] the inequality ( 26) holds. In this case ∂W ∂θ ≥ -z T HM -1 Hz. Integrating this inequality on interval [θ i-1 , θ i ] we obtain

W (θ i-1 ) ≤ W (θ i ) + (θ i -θ i-1 )z T HM -1 Hz
or according to ( 23)

(I -H(θ i ))Z -1 (I -H(θ i )) + (θ i -θ i-1 ) HM -1 H ≤ βP LP.
Finally, using Schur complement and multiplying both sides of obtained matrix by diag{P -1 , I, I} obtain the following inequality

   βL X H X(I -H(θ i )) HX c1 cu (θ i -θ i-1 ) -1 M 0 (I -H(θ i ))X 0 Z    ≥ 0. ( 27
)
The value of c1 cu for simplicity can be chosen equal to 1 since it affects only on number N and size of the sampling step (θ i -θ i-1 ). Thus, if inequalities [START_REF] Levant | Accelerated high-order MIMO sliding mode control[END_REF], [START_REF] Efimov | Supervisory Acceleration of Convergence for Homogeneous Systems[END_REF] are satisfied for all sampling steps (θ i -θ i-1 ), i = 1, . . . , N , the inequality ( 14) holds.

Proposition 2 allows to solve the parametrized system of matrix inequalities ( 11)-( 14) using the following algorithm with fixed µ, θ min and θ max .

Algorithm 1

Initialization: p = 0, N = 1, θ 0 = θ min , θ N = θ max , 0 = {θ 0 , θ N }.
Loop: While the system of LMIs ( 11), ( 12), ( 13), ( 24), ( 25) with θ i ∈ p is not feasible, do

p+1 = p ∪ θi-1+θi 2 N i=1
with θ i ∈ p , N ← 2N and p ← p + 1. For any fixed µ, ν, γ ∈ R + the inequalities (11), ( 12) are feasible at least for sufficiently small µ ∈ (0, 1] and ν ∈ R + [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF]. Moreover, it is obvious that ( 13), [START_REF] Efimov | Supervisory Acceleration of Convergence for Homogeneous Systems[END_REF] are feasible and since for µ → 0 : (I -H(θ)) → 0, then the presented algorithm always finds the required solution for sufficiently small µ. Thus, one can always find such µ for which the system (1) is finite-time stable with control in the form [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF]. 10 Since in addition to solving system of LMIs ( 11)-( 13), [START_REF] Levant | Accelerated high-order MIMO sliding mode control[END_REF], [START_REF] Efimov | Supervisory Acceleration of Convergence for Homogeneous Systems[END_REF] it is necessary to satisfy the inequality [START_REF] Zubov | Methods of A.M. Lyapunov and Their Applications[END_REF], a particular attention should be paid to selecting the values of the parameters θ min , θ max . Taking into account that according to [START_REF] Shen | Semi-global finite-time observers for nonlinear systems[END_REF] d(t, x) = 0 for θ = 1 it is preferable to choose 0 < θ min < 1 and θ max > 1 to satisfy the inequality [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF]. On the other hand, to fulfil the inequality [START_REF] Zubov | Methods of A.M. Lyapunov and Their Applications[END_REF] it is necessary to solve the inequalities ( 11)-( 14) with sufficiently large value of θmax θmin . Thus, to select the parameters, the following strategy can be used along with Algorithm 1: starting with θ max = θ min = 1 it is necessary to solve the system of inequalities ( 11)-( 13), ( 24), (25) using Algorithm 1 with slightly increasing the parameter θ max and/or decreasing θ min until the inequality θmax ≤ c1 θmin is satisfied. Then the parameter c u should be chosen according to [START_REF] Zubov | Methods of A.M. Lyapunov and Their Applications[END_REF]. Let us make some remarks about presented control scheme:

(i) For practical implementation of the control ( 16) feasibility of the inequality ( 15) is required.

Therefore, it is necessary to estimate the coefficients c 1 and c 2 .

To this end consider the set

D = {x ∈ R n |V (x) = 1}. Since V ∈ R + : Q(V, x) = 0
for the set D one can obtain x T P x = 1 and thus the following inequality holds

1 λ max (P ) ≤ n i=1 |x i | 2 ≤ 1 λ min (P ) . (28)
Hence

|x i | ≤ 1 λ 1 2
min (P )

.

For P ≥ I (λ min (P ) ≥ 1) we have

x r = n i=1 |x i | ρ r i 1 ρ ≤   n i=1 1 λ ρ 2r i min (P )   1 ρ ≤ n λ ρ 2rmax min (P ) 1 ρ (29)
and for P ≥ I (λ min (P ) < 1)

x r = n i=1 |x i | ρ r i 1 ρ ≤   n i=1 1 λ ρ 2r i min (P )   1 ρ ≤   n λ ρ 2r min min (P )   1 ρ . ( 30
)
On the other hand, rewrite [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF] for the set D:

c 1 x r ≤ 1 ≤ c 2 x r . (31)
Then basing on ( 29), [START_REF] Bernuau | On an Extension of Homogeneity Notion for Differential Inclusions[END_REF] the estimation of c 1 can be taken as follows

c 1 = min    λ 1 2rmax min (P ) n 1 ρ , λ 1 2r min min (P ) n 1 ρ    . ( 32
)
For estimation of c 2 one can obtain

1 λ max (P ) ≤ n i=1 |x i | 2 ≤ n i=1 x 2ri r ≤ n x 2rmin r , if x r < 1, x 2rmax r , if x r ≥ 1,
and basing on (31) the following estimation of c 2 can be taken

c 2 = max (nλ max (P )) 1 2rmax
, (nλ max (P ))

1 2r min . ( 33
)
To get better estimations of c 1 and c 2 , which is needed for relaxing the inequality (15), the following procedure can be applied. Let us denote the diagonal matrix J = diag

1 √ Pii n i=1
. Rewrite ( 9) and [START_REF] Courant | Introduction to calculus and analysis[END_REF] in the next forms

Q(V, x J ) := x T J D(V -1 ) P D(V -1 )x J -1, c 1 x J r ≤ 1 ≤ c 2 x J r ,
where x J = J -1 x and P = JP J. Note that the matrix P has the values 1 on the main diagonal and eigenvalues of P are close to 1 (i.e. λ min ( P ) ≤ 1 and λ max ( P ) ≥ 1). In this case we can obtain the estimations

c 1 = λ 1 2r min min ( P ) n 1 ρ (34)
and c 2 = nλ max ( P )

1 2r min , (35)
which in most cases are located closer to each other (and closer to the value 1) in comparison with (32), (33). This issue allows the inequality (15) to be relaxed. (ii) The parameters β and ν are tuning parameters for the upperbound of the settling time of the closed-loop system (see, the estimate ( 17)).

(iii) The control ( 16) is bounded with

|u(x)| ≤ (c u x 0 r ) 1-µ ξ n i=1 c -2ri u
, where ξ ∈ R + : k T k ≤ ξI (can be rewritten in LMI form as

X y T y ξ 1 ≥ 0, ξ 2 I ≤ X, where ξ = ξ1 ξ2 ).
Indeed,

u 2 (x) = (c u x r ) 2-2µ x T D((c u x r ) -1 )k T kD((c u x r ) -1 )x ≤ ξ (c u x r ) 2-2µ x T D 2 ((c u x r ) -1 )x ≤ ξ (c u x 0 r ) 2-2µ 1 1 • • • 1 D 2 (c -1 u ) 1 1 • • • 1 T = ξ (c u x 0 r ) 2-2µ n i=1 c -2ri u
It is easy to see, if µ = 1 then the control ( 16) is globally bounded by ξ n i=1 c -2ri u . For µ ∈ (0, 1) the control ( 16) is continuous in the state variable x. If µ → 0 then the feedback ( 16) becomes a linear u = kx. If µ = 1 the control is discontinuous at the origin and continuous outside. For practical realization of the control, the discontinuous feedback law can be replaced with a high-gain linear feedback if the system state is close to the origin as in [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF].

(iv) The presented analysis can be performed for other control laws where instead of the homogeneous norm in [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF] another positive homogeneous function of degree 1 is used. Search of such functions, for which the inequalities ( 14), ( 15) become less conservative, may be one of the directions of future research.

(v) In comparison with the of the work [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF] the control [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF] does not require any additional computational power for on-line computation of the ILF value at the current state. This issue significantly simplifies the practical realization of the developed finite-time control scheme.

The system (1) with the control ( 16) is r-homogeneous of degree -µ with the vector of weights r = (1 + (n -1)µ, 1 + (n -2)µ, ..., 1) T . Indeed, since homogeneous norm is r-homogeneous of degree 1, i.e. D(λ)x r = λ x r :

u(D(λ)x) = (c u D(λ)x r ) 1-µ kD((c u D(λ)x r ) -1 )D(λ)x = λ 1-µ x r kD(λ -1 (c u x r ) -1 )D(λ)x = λ 1-µ u(x)
and AD(λ)x + Bu(D(λ)x) = λ -µ D(λ)(Ax + Bu(x)). Thus, following [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF] the closed-loop system is robust with respect to measurement noises and exogenous perturbations.

If the system (1) has additional disturbances d, then obtained result can be easily extended.

Theorem 5

If the following statements hold: 1) the system of matrix inequalities ( 12), ( 13) is feasible with

AX + XA T + by + y T b T + L 1 by y T b T -γX < 0, (36)
β 1 L X(I -H(θ)) (I -H(θ))X Z ≥ 0, ∀θ ∈ [θ min , θ max ] , (37)
(

-) AX + XA T + by + y T b T + L 2 + δI ≤ 0, 1
L 1 > 0, L 2 ≥ αX (38)
for some µ ∈ (0, 1], ∈ (0, 1), β 1 ∈ (0, 1), α, γ, ν, δ, θ min , θ max ∈ R + , X ∈ R n×n , y ∈ R 1×n ;

2) there exists a c u ∈ R + that satisfies the inequality (15); 3) the disturbance function d(t, x) satisfies the following inequality

d T (t, x)D 2 (c 1 x r ) -1 d(t, x) ≤ β 2 δα (c 2 x r ) -2µ , (40)
where β 2 ∈ (0, 1).

Then the control of the form ( 16) stabilizes the origin of the system (1) in a finite time and the settling time function is bounded as follows

T (x 0 ) ≤ c µ 2 x 0 µ r νµ(2 -β 1 -β 2 ) .
The proof of the theorem is omitted since it repeats the arguments given for Proposition 1. The numeric procedure presented in Proposition 2 can also be adapted to the conditions of Theorem 5.

The inequality (40), that restricts the system disturbances, is presented in not appropriate form for practice. Since the inequality (40) is explicitly rewritten form of ( 20 The solution of the system (11)-( 13), ( 24), (25) was performed using the MATLAB and YALMIP software with N = 32 and sampling step (θ i -θ i-1 ) = 0.0597, i = 1, . . . , N .

Taking into account estimations of the coefficients c In order to compare the obtained results with control algorithm [START_REF] Adamy | Soft variable-structure controls: a survey[END_REF] presented in [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF] the corresponding numerical simulations have been done using the same values of x 0 , P , k, µ. The simulations results are shown in Fig. 3, Fig. 4.

The simulations results with the control (42) for the same parameter set and λ = 5 are shown in Fig. 5, Fig. 6. The convergence rate is accelerated by

c µ 2 λ cµ

CONCLUSION

The paper presents control algorithm for finite-time stabilization of the system (1). The control design procedure is based on application of the ILF method and some properties of homogeneous This work is an extension of the paper [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF] and in comparison with it the obtained control u does not require on-line procedure for computation of the ILF value at the current state. This issue allows the practical realization of the control scheme to be simplified.

There are many possible directions for further research. For instance, studying robustness with respect to delays or extension of the present result for MIMO and nonlinear systems, extension to the case of output stabilization based on using of finite-time observers (for example, [START_REF] Perruquetti | Finite-time observers: Application to secure communication[END_REF], [START_REF] Shen | Semi-global finite-time observers for nonlinear systems[END_REF]), etc.

  ), then we can use the result of Proposition 16 in[START_REF] Polyakov | Robust Stabilization of MIMO Systems in Finite/Fixed Time[END_REF]: obtain the next values of matrix P ∈ R 3×3 and the vector k ∈ R 1×3 0.1103 -0.5461 -0.9999 .

1

 1 and c 2 (34), (35) we can select c 1 = 0.2604, c 2 = 2.6455 and coefficient c u due to (15) should satisfy the following expression 1.3703 ≥ c u ≥ 1.2598. Numerical simulations have been done using the coefficient c u = 1.27 and following initial conditions x 1 (0) = 5, x 2 (0) = -6, x 3 (0) = 2. The simulation results are shown in Fig. 1, Fig. 2.

Figure 1 .

 1 Figure 1. Transients of the system states for the finite-time control

Figure 2 .

 2 Figure 2. The system input u

Figure 3 .

 3 Figure3. Transients of the system states for the finite-time control[START_REF] Adamy | Soft variable-structure controls: a survey[END_REF]

Figure 4 .Figure 5 .Figure 6 .

 456 Figure4. The system input u for the finite-time control presented in[START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF]

Figure 7 .

 7 Figure 7. Transients of the system states for the finite-time control

Figure 9 .Figure 10 .

 910 Figure 9. of the system states for the finite-time control (42)

Contract/grant sponsor: Russian Science Foundation; contract/grant number: 17-19-01422

= 6.8743 times.

Proposition 3

Let X ∈ R n×n be a solution of the system [START_REF] Polyakov | Stability notions and Lyapunov functions for sliding mode control systems[END_REF], [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF], (36)-(39) with P = X -1 . If

for some β 2i ∈ R + : β 2 = β 21 + . . . + β 2n < 1 then the inequality (40) holds.

In the next section, it is shown by numerical simulation that for obtained control the convergence rate may be slower compared to the result of [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF]. To accelerate the convergence rate one can refer to results of the papers [START_REF] Levant | Finite-time stabilization of uncertain MIMO systems[END_REF], [START_REF] Levant | Accelerated high-order MIMO sliding mode control[END_REF], [START_REF] Efimov | Supervisory Acceleration of Convergence for Homogeneous Systems[END_REF]. The following corollary is proved in a way similar to [START_REF] Efimov | Supervisory Acceleration of Convergence for Homogeneous Systems[END_REF].

Corollary 2

stabilizes the system (1) in a finite time with the settling time function not exceeding T Proof of Corollary 2 Consider the system (1) with the control

The system (44) is finite-time stable since Z λ (t, z 0 ) = Z(λt, z 0 ), where Z λ (t, z 0 ) is a solution of (44) and Z(λt, z 0 ) is a solution of the system ż = Az + bu(z). Then the settling time function is bounded by

, where V z corresponds to the following equation

Rewrite this equation as x T D(V -1 z ) P D(V -1 z )x = 1 and obtain the estimation (43) for the inequality

Thus, the settling time function is bounded by

The same result can be obtained for the case d(t, x) = 0.

NUMERICAL EXAMPLE

Consider the system (1) for n = 3 in the disturbance free case (d(t, x) = 0). By applying Algorithm 1 for µ = 0.2, ρ = 2(1 + (n -1)µ) = 2.8, γ = 1, ν = 0.0035, θ min = 0.19 and θ max = 2.1 we 14


Christos Bikis

Georg Schulz

Pierre Paleo

Alessandro Mirone

Alexander Rack

Bert Müller
email: bert.mueller@unibas.ch

Peter Thalmann

Stefan Siegrist

Emre Cörek

Jörg Huwyler

Removing ring artefacts from synchrotron radiation-based hard x-ray tomography data

Keywords: ring artefacts, tomography, 2D-2D registration, in-line tomography, high-pass filter, phase reconstruction

INTRODUCTION

In hard X-ray microtomography, ring artefacts are a common unwanted phenomenon that can severely decrease the obtained image quality. Frequently, they originate from incorrectly functioning pixel elements on the detector unit or from particles and scratches on the scintillator. The high level of sensitivity achieved by nowadays beamline setups does not only allow to be more sensitive to density variations within the specimen, but also turns the imaging setups more susceptible to inhomogeneities introduced on the impinging wavefronts. In detail, inadequacies of the X-ray optical elements like windows, attenuators and distortions of the wavefront by the sample itself can cause a partial failure of simple flatfield correction. Thus far, many ring artefacts removal algorithms are applied on the sinogram 1 or the reconstruction slice 2 itself. A better approach would be however to not only to remove the artefacts themselves, but also to maintain the radiographs true information.

All specimens investigated were composed of low-absorbing tissues, such that typical hard X-ray micro computed tomography in absorption contrast mode does not provide sufficient contrast to adequately resolve the inner structure of the specimen investigated. However, light propagation through a medium can be described by the complex refractive index n(x, y, z) = 1δ(x, y, z)

+ iβ(x, y, z), ( 1
)
where the imaginary part, β, accounts for the attenuation and the refractive index decrement, δ, accounts for the phase shift of the impinging wave respectively. [START_REF] Attwood | Soft X-Rays and Extreme Ultraviolet Radiation: Principles and Applications[END_REF] It is well known, that for materials composed of low-Z elements, as is the case for the specimens investigated, the total phase-shift cross section can be up to three orders of magnitude higher then the total absorption cross section depending on the photon energy, 4, 5 making phase tomography the preferential method for the tree-dimensional investigation of such specimens. There are several techniques for the phase retrieval, as for instance grating interferometry, [START_REF] Weitkamp | X-ray phase imaging with a grating interferometer[END_REF][START_REF] Thalmann | Single and double gratingbased x-ray microtomography using synchrotron radiation[END_REF] in-line single distance and holotomograhy. [START_REF] Zanette | Holotomography versus x-ray grating interferometry: A comparative study[END_REF][START_REF] Lang | Experimental comparison of grating-and propagation-based hard x-ray phase tomography of soft tissue[END_REF] The motivation behind developing such a ring-artefact removal approach lies on the fact that single-distance phase retrieval, i.e. our X-ray tomography modality of choice for the three-dimensional investigation of physically soft tissues, including nervous tissue, is more prone to such artefacts than its alternatives (e.g. grating interferometry). Nevertheless, the comparative advantages offered, as will be described in what follows, compensate for this specific shortcoming, which we now prove that can be tackled in an efficient manner.

The reason for single distance in-line, propagation-based tomography being the method of choice lies in the superior spatial resolution provided, simpler instrumentation and very efficient scanning times that are absolutely necessary for biomedical applications requiring several replicates to be scanned for a specific experiment. In in-line propagation-based phase tomography the recorded projections contain both absorption and phase information, where the phase information can be interpreted as the Laplacian of the phase of the wavefront after transversing the sample. In 2002, Paganin et al. [START_REF] Paganin | Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object[END_REF] derived an algorithm based on the transport of intensity equation (TIE) [START_REF] Teague | Deterministic phase retrieval: a green's function solution[END_REF] assuming a single-component specimen and that the object and the detector plane fulfill the near-field condition. and requiring only one single distance. Despite the violation of the single-component assumption, the algorithm has proven itself to be robust also for the case of biological specimens, and soft tissue in particular. [START_REF] Zanette | Holotomography versus x-ray grating interferometry: A comparative study[END_REF][START_REF] Lang | Experimental comparison of grating-and propagation-based hard x-ray phase tomography of soft tissue[END_REF] It also presents the advantage of requiring only one measurement at a single distance.

Here, we thus introduce an improved ring artefact removal algorithm for high-resolution micro computed tomography. The performance of the algorithm is illustrated using synchrotron radiation-based micro computed tomography (SRμCT) in phase contrast mode.

MATERIALS AND METHODS

Specimen preparation

Mouse brain tumor

For the experiment a nude mouse (CD-1-Foxn1nu, Charles River Laboratories, France) was used. A stereoactic frame for head fixation was used for the injection of murine glioma cells (2104 GL261 cells) into the right frontal lobe. In-vivo magnetic resonance experiments were performed using a PharmaScan 47/16 MRI system equipped with a 4.7 T magnet with 16 cm bore diameter and a cryogenic quadrature RF surface transmit/receive coil (Bruker BioSpin Ettlingen, Germany). During in-vivo experiments, the mouse was placed on a customized support equipped with a stereotactic system for head fixation and a face mask for anaesthetic administration (isofluorane (1.5 % to 2 %). For the administration of contrast agent a cannula was inserted into the right tail vein. All the experiments were approved by local authorities (license ZH 168/2010) and performed in strict adherence to the Swiss law for animal protection. The mouse was euthanized by cervical dislocation at day 15 following tumor inoculation. A specimen containing brain and tumor was explanted using augmented reality assistance [START_REF] Schneider | Augmented reality assisted brain tumor extraction in mice[END_REF] and was fixed in buffered formalin immediately after extraction. For the tomography measurements, a dedicated plastic container was used.

Human nerve

A human peripheral nerve was obtained post-mortem from a donated body, with informed consent for scientific use provided beforehand. All procedures were conducted in accordance with the Declaration of Helsinki and approved by the Ethikkommission Nordwestschweiz. Following standard pathology procedure, the peripheral nerve was extracted from the donated body and fixed in 4% histological-grade buffered formalin. It was subsequently dehydrated and finally embedded in a paraffin/plastic polymer mixture. For the tomography measurement, a cylindrical sample with a diameter of 6 mm was extracted from the paraffin block, using a metal punch.

Zebra fish embryo

Experiments were conducted using the zebrafish wild type strain AB/Tbingen (AB/TU). Adult zebrafish were kept in aerated tanks tempered to 28.0 • C at a 10:14 hour light/dark cycle. After collection, eggs were cleaned, sorted and transferred into petri dishes with 0.5 X embryo-medium (E2 medium, [START_REF] Sieber | Immobilization of enzymes on plga sub-micrometer particles by crosslinked layer-by-layer deposition[END_REF] 5.0 mM NaCl, 0.25 mM KCl, 0.5 mM MgSO 4 , 0.15 mM KH 2 PO 4 , 0.05 mM Na 2 HPO 4 , 0.5 mM CaCl 2 , 0.71 mM NaHCO 3 , 1.0 l water (Millipore 18-textohm), 0.001 % (w/v) methylene blue, pH 7.4). Petri dish containing eggs were stored in an incubator (Aqualytic, Dortmund, Germany) at 28.0 • C. E2 medium was renewed daily. Zebrafish embryos were dechorionated 24 h post fertilization (hpf) by removal of the chorion with acupuncture needles (0.20 mm × 15 mm, Wandrey, Berlin, Germany). The dechorionated embryos were immobilized 48 hpf by adding 1 ml 25 X tricaine stock solution to the medium (400.0 mg tricaine methanesulfonate, 97.9 ml water (Millipore 18-textohm), 2.1 ml trisaminomethane, pH=9). Immobilized zebrafish embryos were fixed in 4.0 % paraformaldehyde and dehydrated in ethanol. Fixed zebrafish embryos were aligned in the channel (diameter 0.8 mm) of a two-pieced custom-made aluminum mold (3.0 cm x 2.0 cm x 2.0 cm). Aligned embryos were air dried for one minute to ensure complete evaporation of excess ethanol. Thereafter, the mold was placed vertically into liquid paraffin wax (80.0°, Sigma-Aldrich, Buchs, Switzerland) allowing the channel containing the zebrafish embryo to fill with paraffin due to capillary forces. After cooling, the newly formed paraffin rod containing the embedded embryo was removed from the socket of the mold with a blunt syringe. The paraffin rods were stored at 4.0 • C until further use.

Single-distance phase retrieval

All specimens were measured using in-line single distance phase tomography with the experiments being carried out at the ID19 beamline (ESRF, Grenoble, France). Phase retrieval was performed using the algorithm proposed by Paganin et al. [START_REF] Paganin | Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object[END_REF] using ANKAphase. [START_REF] Weitkamp | ANKAphase: software for single-distance phase retrieval from inline X-ray phase-contrast radiographs[END_REF] A list of the acquisition and reconstruction parameters is given in Table 1. The phase projections were reconstructed using the standard filtered back-projection algorithm [START_REF] Kak | Principles of computerized tomographic imaging[END_REF] implemented in MATLAB (2016a, The MathWorks, Inc., Natick, Massachusetts USA).

RESULTS AND DISCUSSION

Figure 1 shows selected phase reconstruction slices for for the three specimen, before ring artefact correction. The slices contain severe ring artifacts. As already stated, usually, these artefacts arise from incorrectly working scintillator or pixel elements. In such cases they can be removed by using a stripe removing filter on the sinogram 1 or directly on the reconstruction slice. 2 However, measuring low-absorbing specimens in phasecontrast mode requires high sensitivity, provided by nowadays beamline setups. The high sensitivity achieved makes the system also more prone to disturbances of the impinging wavefronts. Under these circumstances simple flat-field correction is insufficient.

To better elucidate why this happens, consider for example the projection of the mouse brain tumor, given in Figure 2(a). Numerous donut-like artefacts can be observed, that are clearly artificial. Having a closer look on the line plot through such a structure given in 2(b) reveals a sub-pixel mismatch of the flat-field (red) with its projection (blue), as the plots slightly deviate where the curve is falling and rising. Shifting the flat-field to its corresponding projection (green) resulted in almost precise match of the two images as can also be seen in Figure 2(b), while the shift in x -direction was around 0.2 pixels and in the y-direction less than 0.1 pixels.

The effect of using translation registration to correct for the mismatch is also illustrated in Figure 3. While for the mouse brain specimen measured with a pixel size of 1.87 μm translation registration mainly resulted in the removal of several donut-like structures, its effect with decreasing pixel size was much more pronounced. For example, for the zebra fish embryo measurement, a translation over a pixel resulted in the removal of prominent cloud-like structures on the projection and thereby resulted in projection containing many more details. However, for all measurements, image registration alone could not remove all ring artefacts, as can be seen in the sinogram comparison Figure 3 (c) as well as the corresponding reconstructed slices in Figure 4(b).

It should also not be overlooked that image registration can readily increase reconstruction time. For instance, finding the best out of 100 flat-field images for 5120 radiographs would require 512000 registrations, which is an extremely computation-intensive task and can be immensely time-consuming for commercialy available workstations. Therefore, we tested different registration approaches, in an effort to decrease computation time. The parameters investigated were the registration method itself, the metric, the optimizer as well as the number of registrations per radiograph. All registrations were performed using the Insight Segmentation and Registration Toolkit (ITK). [START_REF] Itk | The Insight Segmentation and Registration Toolkit (ITK)[END_REF] In terms of method, neither the affine nor the similarity registration yielded better results than the Translation registration in terms of the obtained similarity. For instance, the final rotation after Similarity registration remained below 0.001°. In terms of speed and quality the Mattes mutual information in combination with the Powell optimizer was found to be the most suitable choice. Furthermore, taking the median over a flat-field block to register to any given projection resulted in an increase of the similarity value, compared to taking a random flat-field. Additionally, the weighting of nearest flat-field blocks can increase similarity, but the optimal weighting did not necessarily correlate with the theoretical one. With the suggested parameters one single registration took about 1 s on a workstation using six Intel(R) Xeon(R) CPU E5-2620 v3 @ 2.40GHz processors. Hence, assuming five registrations per projection to find the optimal weighting, the entire process takes about 8 h to 24 h, depending on the projection contrast, for 5120 radiographs. So far non-rigid registration has not been tested. The same holds for finding the optimally acquired flat-field as the algorithm has not been yet adapted for a cluster.

As already mentioned, registration by itself was insufficient to completely remove the ring artifacts. Thus, in a second step, we applied a frequency filter approach proposed by Mirone et al. [START_REF] Mirone | The pyhst2 hybrid distributed code for high speed tomographic reconstruction with iterative reconstruction and a priori knowledge capabilities[END_REF] For every single projection the median over a user-defined number of nearest projections is calculated. Then a high frequency filter is applied to the median averaged projection. The resulting high frequency image is then finally subtracted from the registration-corrected projection.

Figure 1 .

 1 Figure 1. The images show characteristic slices of the phase-reconstructed specimens with standard flat-field correction applied: (a) brain tumor, (b) human nerve and (c) zebra fish embryo. All slices contain more or less strong ring artifacts. Scale bar length is 500 μm.

Figure 2 .

 2 Figure 2. (a) The image shows a selected region of interest (ROI) of a representative projection of the mouse brain tumor specimen. The read arrowheads indicate the prominent donut-like artefacts that persist after standard flat-field correction. A zoom-in of the region inside the dashed box containing such a donut-like structure prior to and after having shifted the flat-field appropriately is given in (b1) and (b2) respectively. (c) A line plot through the donut-like artefact indicated by the red line in figure (b1) clearly shows the sub-pixel mismatch that occurs when simple flat-field correction is used.

Figure 3 .

 3 Figure3. This figure illustrates the beneficial effect of flat-field registration on the projection quality. For the mouse brain tumour (a) most of the donut-like artifacts were removed and for the zebra fish embryo (b) flat-field registration was able to remove the dominant cloud-like artefacts, resulting in a much more detailed projection. A representative sinogram of the zebra fish embryo, given in (c), shows the improved efficiency of flat-field registration in removing all but the thinnest sinogram artefact lines. The first 1260 projections were produced using simple flat-field correction and the second 1260 projections were achieved using flat-field registration prior to the standard correction.

Figure 4 .

 4 Figure 4. The effects of the ring artifact removal steps applied for (a) the mouse brain tumor, (b) the human nerve and (c) the zebra fish embryo, respectively. The top row shows representative slices for the specimens using simple flat-field correction. The middle row shows the same slices using translation registration prior to the flat-field correction and the bottom row shows the reconstructed slices obtained after the added step of high-pass filtering the flat-field corrected projections. Scale bar length is 500 μm.

Table 1 .

 1 Experimental and reconstruction parameters, where δ/β denotes the assumed ratio of refractive index decrement over the imaginary part of the refractive index for each specimen, d the sample-detector distance, N the total number of projections acquired, and t the exposure time per projection

	Specimen	Pixel size [μm] δ/β d [mm]	N	t [s] off-axis preparation camera
	Brain tumor	1.87	1546	202	2000 0.2	no	formalin	FReLoN 2K
	Human nerve	0.66	1500	30	5120 0.1	yes	paraffin	PCO Edge
	Zebrafish	0.33	2000	10	5120 0.1	yes	paraffin	PCO Edge

ACKNOWLEDGMENTS

The authors thank Prof. Michele Bernasconi and Dr. Zsofia Kovacs from the University Children's Hospital in Zurich for providing the mouse brain tumor sample. The authors also thank Gabriel Schweighauser, Jürgen Hench and Stephan Frank from the Neuropathology Department of the Basel University Hospital for providing the human peripheral nerve sample and assisting in the special preparation required for the tomography measuremts. The authors also acknowledge the support of Dr. Irene Zanette from the Diamond Light Source during beamtime.

FUNDING

The financial contribution of the Swiss National Science Foundation (Project Nos. 144535 and 147172) and NanoReg2 (European Union Horizon 2020 research agreement 646221, 15.0200-3) is gratefully acknowledged as well as the provision of beamtime at beamline ID19 under experiment number MD-860 and MD-981 from the European Synchrotron Radiation Facility (ESRF).

Our proposed algorithm exploits the complementarity between the registration process and the high frequency filter subtraction. In detail, the former is particularly efficient for removing thicker rings caused by bigger artefacts in the projection, while the later is especially effective for removing the remaining, thinner, ring artefacts.

CONCLUSION

We have implemented and tested a two-step procedure for ring artefact removal, based on the combination of flatfield translation registration and high-pass filtering of projections. This approach has been proven particularly effective for the visualization of several types of low absorbing samples, even using phase tomography modalities such as single distance phase retrieval, which are generally prone to ring artefacts. The implementation is both robust and time-efficient, allowing reconstruction of up to one dataset per day on a standard workstation. The main remaining challenge is optimizing the process to decrease computation time, as well as develop a userfriendly interface to allow for processing large series of datasets required for biomedical experiments with several specimen replicates.