Gustavo Cruz Campos

Joao Manoel Gomes Da Silva

Sophie Tarbouriech

Carlos Eduardo Pereira

Stabilisation of discrete-time systems with finite-level uniform and logarithmic quantisers

This study deals with the stabilisation of discrete-time linear systems subject to static finite-level quantisation on the control inputs. Two kinds of quantisers are considered: uniform and logarithmic. The modelling of the finite-level quantisation is obtained by the application of deadzone and saturation maps to an infinite-level quantiser. From this model, conditions for the synthesis of state feedback control laws guaranteeing the convergence of the trajectories to an attractor around the origin provided that the initial state belongs to a certain admissible set are proposed. These conditions can thus be incorporated in linear matrix inequality-based optimisation schemes to compute the stabilising gain while minimising the size of the attractor.

. In these works, based on sector-based relations, conditions to design stabilising state feedback control laws aiming at the minimisation of the attractor set are derived. In [2] the set of admissible initial states, for which the convergence to the attractor is ensured, is also characterised. It should, however, be pointed out that these papers deal only with continuous-time systems.

, we consider the effective nonlinear effect of the saturation and a formal proof regarding the

Introduction

Quantisation of signals is inherent to a digital implementation of control systems and can be considered as a hard non-linearity in the loop. Its effects may lead to undesirable phenomena such as limit cycles, multiple equilibria or chaotic behaviour, even if the controller is supposed to be a stabilising one [START_REF] Delchamps | Stabilizing a linear system with quantized state feedback[END_REF][START_REF] Tarbouriech | Control design for quantized linear systems with saturations[END_REF]. Considering the increasing implementation of Networked Control Systems (NCS), researches on quantisation regained the attention of the control community. In particular, since in NCS the control loop elements exchange information through communication channels with limited bandwidth, quantisation plays an important role in reducing the data traffic. However, the use of coarse quantisation may considerably degrade the behaviour of the controlled system.

With this motivation, the study of the effects of quantisation on control systems has attracted the attention of several researchers over the last years; see, e.g. [START_REF] Brockett | Quantized feedback stabilization of linear systems[END_REF][START_REF] Fridman | Control under quantization, saturation and delay: an LMI approach[END_REF][START_REF] Liberzon | Hybrid feedback stabilization of systems with quantized signals[END_REF][START_REF] Wang | Coding scheme based on spherical polar coordinate for control over packet erasure channel[END_REF]. It is also important to note that many of these studies have been proposed for continuous-time systems. In the context of discrete-time systems, one can cite some works dealing with the controller or observer design in the presence of uniform or logarithmic quantisers: see, e.g. [START_REF] Picasso | Stabilization of discrete-time quantized linear systems: an H ∞ /l 1 approach[END_REF][START_REF] Xia | Stability analysis of discrete-time systems with quantized feedback and measurements[END_REF]. In particular, in [START_REF] Elia | Stabilization of linear systems with limited information[END_REF], it has been shown that, for a quadratically stabilisable single-input system, a logarithmic quantiser is the optimal solution in terms of coarse quantisation density. However, it is also shown that the quantiser must have an infinite number of quantisation levels, which is not feasible in practice. Considering also infinite-level logarithmic quantisers, in [START_REF] Fu | The sector bound approach to quantized feedback control[END_REF] the authors have introduced the sector bound approach for quantised feedback systems giving simple formulae to the stabilisation problem considering state and output feedback controllers. The problem of finite-level quantisation can be addressed by using dynamic quantisers (see e.g. [START_REF] Fu | Finite-level quantized feedback control for linear systems[END_REF][START_REF] Azuma | Optimal dynamic quantizer for discrete-valued input control[END_REF][START_REF] Sawada | Synthesis of dynamic quantizers for quantized feedback systems within invariant set analysis framework[END_REF]). However, the implementation of such quantisers is in general much more complex than static ones and may not be possible in standard digital platforms.

When static finite-level quantisers are considered, an important issue regards the fact that the asymptotic convergence of the trajectories to the origin cannot be ensured for open-loop unstable systems. In this case, due to an implicit input deadzone, the trajectories will converge to either periodic or chaotic orbits around the origin. The idea, in this case, is to guarantee that the trajectories converge asymptotically to a set called attractor. Considering logarithmic quantisers, this problem is addressed in [START_REF] Elia | Stabilization of linear systems with limited information[END_REF] for singleinput systems. In that reference, for a given optimal control law, the authors focus on providing bounds on the quantiser parameters positive invariance of the attractor is provided. The proposed conditions are cast in LMI-based optimisation problems to compute the controller gain in order to minimise the size of the attractor while guaranteeing the set of admissible initial states includes some pre-specified region in the state space. A numerical example to illustrate the methodology and to compare the solutions with uniform and logarithmic quantisers is provided.

The paper is organised as follows. In Section 2, the generic framework and the problem statement are presented. Section 3 addresses the case of uniform quantisation, while Section 4 deals with the logarithmic case. Optimisation and computational issues are discussed in Section 5. A numerical example and some concluding remarks end the paper.

Notation: Throughout the paper, I denotes the identity matrix and 0 denotes the null matrix (equivalently the null vector) of appropriate dimensions. For a matrix A ∈ ℝ n × n , A′, A (i) , tr(A) denote its transpose, its ith row and its trace, respectively. The matrix diag(A 1 , A 2 , …, A n ) is the block-diagonal matrix having A 1 , A 2 , …, A n as diagonal blocks. In symmetric matrices, * stands for symmetric blocks. For a vectorx ∈ ℝ n , x (i) , x′, |x| denote its ith component, its transpose and the component-wise absolute value operator, respectively. sign(x) is the component-wise sign function, with sign(0) = 0, and ⌊x⌋ the component-wise floor operator. For two sets S 1 and S 2 , S 1 ∖S 2 denotes the set S 1 deprived of S 2 .

Problem statement

Consider the following discrete-time system:

x(k + 1) = Ax(k) + B dz(sat(q(u(k))) x(0) = x 0 (1)
where x ∈ ℝ n , u ∈ ℝ p , x 0 ∈ ℝ n are, respectively, the state, the input of the system and the initial state. A, B are real matrices of suitable dimensions and q: ℝ p → ℝ p is a component-wise infinite level quantiser. In order to model the finite-level quantisation, we consider a component-wise saturation function, sat: ℝ p → ℝ p and a component-wise deadzone function dz: ℝ p → ℝ p , i.e. for v ∈ ℝ p it follows that:

sat (i) (v) := sign(v (i) )min{u 0(i) , | v (i) | }, i = 1, …, p (2)
dz (i) (u) := 0 if -ϵ (i) < v (i) < ϵ (i) v (i) if | v (i) | ≥ ϵ (i) , i = 1, …, p (3)
In this case, ±u 0(i) represents the maximum/minimum quantisation levels for the ith input channel, while ±ϵ (i) denote the limits for which the ith input signal is coded as zero.

In order to stabilise system (1), we consider a state feedback control law

u(k) = Kx(k) ( 4
)
By defining now the functions ψ, ϕ and θ as follows:

ψ(v) := q(v) -v (5) ϕ(v) := sat(v) -v (6) θ(v) := dz(v) -v (7)
the closed-loop dynamics is given by the non-linear system

x(k + 1) = (A + BK)x(k) + Bψ(k) + Bϕ(k) + Bθ(k) ( 8
)
where for simplicity we denote

ψ(k) = ψ(u(k)), ϕ(k) = ϕ(q(u(k)))) = ϕ(u(k) + ψ(u(k))) and θ(k) = θ(sat(q(u(k))) = θ(u(k) + ψ(k) + ϕ(k)).
From ( 2)-( 4), two sets of interest in the state space can be defined

S ns = {x ∈ ℝ n : | K (i) x | ≤ u 0(i) , i = 1, …, p} (9)
S dz = {x ∈ ℝ n : | K (i) x | ≤ ϵ (i) , i = 1, …, p} (10)
The set S ns corresponds to the region where none of the inputs is saturated in the maximal or minimal quantisation level, while S dz corresponds to the region where the actual control input injected into the system is zero. If x(k) ∈ S ns ∖S dz , the closed-loop dynamics reads

x(k + 1) = Ax(k) + q(u(k)) = (A + BK)x(k) + Bψ(k) (11)
i.e. only the non-linear effect of the infinite-level quantiser q(k) is present.

On the other hand, note that if x(k) ∈ S dz , the system is in open loop. Hence, if the matrix A is not Schur-Cohn, the asymptotic stabilisation of the origin for the closed-loop system cannot be achieved even when the gain K is supposed to be a stabilising one (i.e. when (A + BK) is Schur-Cohn). Actually, in this case, either a limit cycle or a chaotic behaviour will be generated around the origin. Hence the convergence to the origin should be replaced by the convergence to an attractor set around the origin, as small as possible (i.e. referred as practical stability in [START_REF] Elia | Stabilization of linear systems with limited information[END_REF]). Moreover, if A is not Schur-Cohn, under input saturation only local (regional) stability can be achieved [START_REF] Sussmann | A general result on the stabilization of linear systems using bounded controls[END_REF][START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF].

In this work, we focus on the case in which matrix A is not Schur-Cohn, i.e. the open-loop system is exponentially unstable. Then, the problem we aim to solve can be stated as follows.

Problem 1: Given system (1), with saturation and deadzone limits given, respectively, by vectors u 0 ∈ ℝ p and ϵ ∈ ℝ p , and the control law (4), determine a gain K, a set S 0 ⊂ ℝ n and a compact set S u ⊂ ℝ n containing the origin, such that • S 0 and S u are positively invariant sets with respect to the closed-loop system (1)-( 4) (or equivalently system (8)). • For every initial condition x 0 ∈ S 0 ∖S u , the trajectories of the closed-loop system (1)-( 4) are bounded and converge in finite time to S u (which is therefore an attractor of the trajectories). • S u is as smaller as possible.

The sets in Problem 1 are depicted in Fig. 1. Note that the set S 0 is implicitly included in the region of attraction of the attractor set S u . As aforementioned, since A is assumed not to be Schur-Cohn, the global convergence to the attractor is not possible to achieve with bounded controls. On the other hand, the set S u is the region where the trajectories are ultimately bounded and, if minimised can be seen as an estimate of the periodic or chaotic orbit around the origin.

In the sequel, we address Problem 1 considering uniform and logarithmic finite-level quantisations.

Uniform quantisation

The function q(u) corresponding to a uniform quantisation is defined as follows:

q (i) (u) := ϵ (i) sign(u (i) ) |u (i) | ϵ (i) i = 1, …, p (12)
where ϵ (i) is a positive real-scalar representing the quantisation step of the ith control input. In this case, u 0(i) = N i ϵ i , where N i is the number of positive levels between u 0(i) and zero. Moreover, as it can be seen in Fig. 2, the deadzone is implicitly included in the definition of q. Hence, the equation describing the closed-loop system (8), considering a finite-level uniform quantisation can be simplified to the following one:

x(k + 1) = (A + BK)x(k) + Bψ(k) + Bϕ(k) (13) with ψ(k) = ψ(u(k)), ϕ(k) = ϕ(q(u(k))) = ϕ(u(k) + ψ(u(k))).
Before, proposing a result to address Problem 1, we recall some auxiliary conditions regarding the sector bounded functions ψ and ϕ.

Lemma 1: (see [START_REF] Ferrante | Stabilization of continuous-time linear systems subject to input quantization[END_REF] and [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF] for details) For ψ(u) as defined in (5), with q(u) defined by [START_REF] Azuma | Optimal dynamic quantizer for discrete-valued input control[END_REF], and considering ϵ = [ϵ (1) ⋯ϵ (p) ]′, the relations

ψ(u)′S 1 ψ(u) -ϵ′S 1 ϵ ≤ 0, (14)
ψ(u)′S 2 (ψ(u) + u) ≤ 0 (15)
are verified for any diagonal positive definite matrices S 1 , S 2 ∈ ℝ p × p . Actually, ( 14) corresponds to a generalisation of the condition in [START_REF] Ferrante | Stabilization of continuous-time linear systems subject to input quantization[END_REF] to cope with different quantisation parameters in each input.

Lemma 2: (see [START_REF] Ferrante | Stabilization of continuous-time linear systems subject to input quantization[END_REF] and [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF] for details) For ϕ(v) as defined in (6) and a vector w ∈ ℝ p , the relation

ϕ(v)′S 3 (ϕ(v) + v + w) ≤ 0 (16)
is verified for any diagonal positive matrix S 3 ∈ ℝ p × p , provided that w ∈ S(u 0 ) with

S(u 0 ) = {w ∈ ℝ p ; | w (i) | ≤ u 0(i) , i = 1, …, p} (17)
Note that considering v = q(u), from (5), condition [START_REF] Maestrelli | Stability analysis of input and output finite level quantized discrete-time linear control systems[END_REF] reads

ϕ(v)′S 3 (ϕ(v) + ψ(u) + u + w) ≤ 0 (18)
By using Lemmas 1 and 2, the following proposition to solve Problem 1 can be stated.

Proposition 1: If there exists a symmetric positive definite matrix

W ∈ ℝ n × n , diagonal positive matrices S 1 , S 2 , R 3 ∈ ℝ p × p , matrices
Y, Z ∈ ℝ p × n , positive scalars τ 1 , τ 2 and a scalar α, 0 ≤ α ≤ 1, satisfying the following conditions:

βW -Y′S 2 -Y′ -Z′ W A′ + Y′B′ * -S 1 -2S 2 -I B′ * * -2R 3 R 3 B′ * * * -W < 0 with β = (τ 2 -ατ 1 -1) (19
)

ϵ′S 1 ϵ + τ 1 -τ 2 ≤ 0 ( 20
)
αW Z′ (i)

Z (i) u 0(i) 2 > 0, i = 1, …, p (21)
then, for K = YW -1 , it follows that ∀x(0) ∈ S 0 ∖S u the trajectories of the system (13) converge in finite time to the set S u , with

S u = ℰ(P) = {x ∈ ℝ n : x′Px ≤ 1}, P = W -1 (22) S 0 = ℰ(αP) = {x ∈ ℝ n : x′αPx ≤ 1} (23)
Moreover, S 0 and S u are positively invariant sets with respect to [START_REF] Sawada | Synthesis of dynamic quantizers for quantized feedback systems within invariant set analysis framework[END_REF].

Proof: Considering the quadratic Lyapunov function V(x) = x′Px, with P = P′ > 0, we want to prove that

ΔV(x) = V(x(k + 1)) -V(x(k)) < -μ | | x(k) | | 2 , ( 24
)
with μ being a positive scalar, ∀x(k) ∈ S 0 ∖S u .

In this case, it follows that S 0 is a positively invariant set and ∀x(0) ∈ S 0 , the corresponding trajectory converges in finite time to S u . Note that

S 0 ∖S u = {x ∈ ℝ n : (x′αPx -1) ≤ 0 and (1 -x′Px) ≤ 0}
Now applying S-procedure along with Lemmas 1 and 2, considering u = Kx and w = Gx, with G being a free matrix variable, if

ΔV(x) -τ 1 (x′αPx -1) -τ 2 (1 -x′Px) -ψ′S 1 ψ + ϵ′S 1 ϵ -2ψ′S 2 (ψ + Kx) -2ϕ′S 3 (ϕ + ψ + Kx + Gx) < -μ | | x(k) | | 2 (25)
it follows that (24) is satisfied, provided that x ∈ S ~(u 0 ), with

S ~(u 0 ) = {x ∈ ℝ n ; | G (i) x | ≤ u 0(i) , i = 1, …, p} (26)
By using [START_REF] Sawada | Synthesis of dynamic quantizers for quantized feedback systems within invariant set analysis framework[END_REF], it is possible to re-write (25) as follows:

[x′ ψ′ ϕ′]N u x ψ ϕ + ϵ′S 1 ϵ + τ 1 -τ 2 < -μx′x (27)
with N u defined as follows:

N u = (τ 2 -ατ 1 -1)P -K′S 2 -K′S 3 -G′S 3 * -S 1 -2S 2 -S 3 * * -2S 3 + (A + BK)′ B′ B′ P (A + BK) B B ( 28
)
It is straightforward to verify that (27) will be satisfied for some μ > 0 if we guarantee [START_REF] Ferrante | Observer-based control for linear systems with quantized output[END_REF] and

N u < 0. Applying Schur complement, N u < 0 is equivalent to βP -K′S 2 -K′S 3 -G′S 3 (A + BK)′ * -S 1 -2S 2 -S 3 B′ * * -2S 3 B′ * * * -P -1 < 0 (29)
Then by pre-and post-multiplying (29) by diag(P -1 , I, S 3 -1 , I) and with the change of variables P -1 = W, KW = Y, GP -1 = Z, S 3 -1 = R 3 , we obtain relation [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF]. Furthermore, one has to prove that the set S u is positively invariant. This is proven if we guarantee that x(k + 1)′Px(k + 1) ≤ 1 whenever x(k)′Px(k) ≤ 1 (i.e. when x(k) ∈ S u ). It is possible to do that by verifying the following inequality, for some positive scalar τ 3 :

x(k + 1)′Px(k + 1) -1 -τ 3 (x(k)′Px(k) -1) ≤ 0 (30)
On the other hand, inequality (30) is true if we are able to verify that

τ 3 -1 ≤ -ϵ′S 1 ϵ (31)
and

x(k + 1)′Px(k + 1) -τ 3 x(k)′Px(k) -ψ(k)′S 1 ψ(k) -2ψ(k)′S 2 (ψ(k) + Kx(k)) -2ϕ(k)′S 3 (ϕ(k) + ψ(k) + Kx(k) + Gx(k)) ≤ 0 (32)
provided that x ∈ S ~(u 0 ). Now by choosing -τ 3 = τ 2 -ατ 1 -1, it is straightforward to verify that (32) is implied by [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF]. Furthermore, since α ≤ 1 and τ 1 , τ 2 > 0, it follows that:

ϵ′S 1 ϵ + τ 3 -1 = ϵ′S 1 ϵ + ατ 1 -τ 2 ≤ ϵ′S 1 ϵ + τ 1 -τ 2 (33)
Hence ( 20) implies (31). Then, if [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF] and ( 20) are satisfied, (30) is implicitly verified and thus S u is positively invariant with respect to system [START_REF] Sawada | Synthesis of dynamic quantizers for quantized feedback systems within invariant set analysis framework[END_REF].

Finally, the satisfaction of relation [START_REF] Amato | Stabilization of bilinear systems via linear state feedback control[END_REF] implies that the ellipsoid S 0 = ℰ(αP) is included in the polyhedral set S ~(u 0 ) defined in (26), which ensures the validity of condition [START_REF] Campos | Stability of discrete-time control systems with uniform and logarithmic quantizers[END_REF], with w = Gx, ∀x ∈ S 0 , which concludes the proof. □

Logarithmic quantisation

In the case of the logarithmic quantisation, each component of the decentralised vector q(u) is defined as

q (i) (u) := λ (i) ρ (i) j , if u (i) > 0 and λ (i) ρ (i) j (1 + δ (i) ) ≤ u (i) < λ (i) ρ (i) j (1 -δ (i) ) - λ (i) ρ (i) j , if u (i) < 0 and λ (i) ρ (i) j (1 + δ (i) ) ≤ -u (i) < λ (i) ρ (i) j (1 -δ (i) ) , i = 1, …, p, j ∈ ℤ ( 34
)
δ (i) = 1 -ρ (i) 1 + ρ (i) , ρ (i) = 1 -δ (i) 1 + δ (i) , 0 < ρ (i) < 1 (35)
The parameter ρ (i) is said to be the quantisation density (instead of quantisation step as in the uniform quantisation). The function q (i) (u) corresponding to an infinite-level quantiser is depicted in Fig. 3. Recall now that, to obtain a finite-level quantiser, saturation and deadzone effects are applied over the infinite-level quantiser q(u). In this case, λ (i) is considered as the least positive quantisation level. Actually, λ (i) is related to the deadzone limit through the equation λ (i) = ϵ (i) (1 + δ (i) ). Moreover, the saturation limit of each quantised input is given by u 0(i) = λ (i) / ρ (i)

N i -1 , where N i corresponds to the number of quantisation levels between u 0(i) and zero.

To solve Problem 1, we recall a sector condition verified by the nonlinearity ψ (it corresponds to a multi-input version of the one in [START_REF] De Souza | Stability analysis of finite-level quantized discrete-time linear control systems[END_REF]) Lemma 3: For ψ(u) = q(u) -u with q(u) defined in (34) and (35), and D = diag(δ (1) , …, δ (p) ), the relation

(ψ(u) + Du)′S 1 (ψ(u) -Du) ≤ 0 (36)
is verified for any diagonal positive definite matrix S 1 ∈ ℝ p × p . For the non-linearity θ, one can use a similar condition to that one stated in Lemma 1, as follows: Lemma 4: For every θ(v) defined in [START_REF] Picasso | Stabilization of discrete-time quantized linear systems: an H ∞ /l 1 approach[END_REF], the relation

θ(v)′S 3 θ(v) -ϵ′S 3 ϵ ≤ 0 (37)
is verified for any diagonal positive definite matrix S 3 ∈ ℝ p × p . By using Lemmas 2-4, the following proposition to solve Problem 1 can be stated.

Proposition 2: If there exists a symmetric positive definite matrix

W ∈ ℝ n × n , diagonal positive matrices R 1 , R 2 , S 3 ∈ ℝ p × p , matrices Y, Z ∈ ℝ p × n
, positive scalars τ 1 , τ 2 and a scalar α, 0 ≤ α ≤ 1, satisfying the following conditions:

-Y′ -Z′ 0 W A′ + Y′B′ Y′D * -R 1 -R 1 0 R 1 B′ 0 * * -2R 2 0 R 2 B′ 0 * * * -S 3 B′ 0 * * * * -W 0 * * * * * -R 1 < 0, with β = (τ 2 -ατ 1 -1) (38
)

ϵ′S 3 ϵ + τ 1 -τ 2 ≤ 0 (39)
αW Z′ (i)

Z (i) u 0(i) 2 > 0, i = 1, …, p (40)
then, for K = YW -1 , it follows that ∀x(0) ∈ S 0 ∖S u the trajectories of the system (8) converge in finite time to the set S u , with S u and S 0 as defined in ( 22) and ( 23). Moreover, S 0 and S u are positively invariant sets with respect to [START_REF] Xia | Stability analysis of discrete-time systems with quantized feedback and measurements[END_REF].

Proof: Applying S-procedure along with Lemmas 2-4, if

ΔV(x) -τ 1 (x′αPx -1) -τ 2 (1 -x′Px) -(ψ + DKx)′S 1 (ψ -DKx) -2ϕ′S 2 (ϕ + ψ + Kx + Gx) -θ′S 3 θ + ϵ′S 3 ϵ < -μ | | x | | 2 (41)
with μ being a positive scalar, then it follows that

ΔV(x) < -μ | | x | | 2 , ∀x ∈ S 0 ∖S u is satisfied, provided that x ∈ S ~(u 0 ).
By using [START_REF] Xia | Stability analysis of discrete-time systems with quantized feedback and measurements[END_REF] and considering ξ = [x′ ψ′ ϕ′ θ′]′, it is possible to re-write (41) as follows:

ξ′N l ξ + ϵ′S 3 ϵ + τ 1 -τ 2 < -μx′x (42)
with N l defined as follows:

N l = P(τ 2 -ατ 1 -1) + K′DS 1 DK 0 -K′S 2 -G′S 2 0 * -S 1 -S 2 0 * * -2S 2 0 * * * -S 3 + (A + BK)′ B′ B′ B′ P (A + BK) B B B ( 43
)
It is straightforward to verify that (42) will be satisfied for some μ > 0 if we guarantee (39) and N l < 0. By applying Schur complement twice, then by pre-and post-multiplying by diag(P -1 , S 1 -1 , S 2 -1 , I, I, I) and with the change of variables

P -1 = W, KP -1 = Y, GP -1 = Z, S 1 -1 = R 1 , S 2 -1 = R 2 ,

we obtain relation (38).

To prove that the set S u is positively invariant, it suffices to verify that, for some positive scalar τ 3 ,

x(k + 1)′Px(k + 1) -1 -τ 3 (x(k)′Px(k) -1) ≤ 0 ( 44
)
~Recalling Problem 1, the goal is to compute K in order to obtain an attractor set S u as small as possible. To achieve this objective, one needs to find a suitable measure for the set S u and minimise it subject to conditions of Propositions 1 and 2. In particular, it is possible to implicitly minimise the volume of S u by minimising tr(P -1 ), or equivalently, tr(W).

On the other hand, it is also important to guarantee that the set S 0 has a reasonable size or that it covers a given set of admissible initial conditions X 0 . With this aim, we can consider X 0 as a polytope in the state space described by the convex hull of its vertices, i.e. X 0 = Conv{v 1 , …, v r }. In this case, an extra condition to ensure that X 0 ⊂ S 0 is given as follows:

1 αv l ′ * αW > 0, l = 1, …, r (45)
Hence, in order to compute K to minimise the attractor while ensuring a guaranteed region of attractivity, i.e. regional stability, the following optimisation problems can be considered. Notice that conditions W > 0 and R 3 > 0 are implicitly ensured in (46) by [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF]. On the other hand, conditions W > 0, R 1 > 0, R 2 > 0, S 3 > 0 are implicitly ensured in (47) by (38).

It is important to note that conditions ( 19) and ( 21), in the uniform quantisation case, and conditions (38) and (40), in the logarithmic quantisation case, are non-linear in the decision variables (i.e. they are not LMIs), which prevents from solving directly a convex optimisation problem. Nevertheless, it is possible to overcome this problem by considering the variables τ 1 , τ 2 and α as tuning parameters. In the uniform quantisation case, S 2 must be fixed as well [START_REF] Ferrante | Observer-based control for linear systems with quantized output[END_REF]. Moreover, to guarantee feasibility, it is necessary that the following conditions are respected:

τ 2 > τ 1 ; τ 2 < 1 + ατ 1 (48)
Thus, the optimal value of ( 46) or (47) can be obtained by solving LMI-based optimisation problems on a grid with respect to α, τ 1 , τ 2 and S 2 (in the uniform case). This can be efficiently addressed by using, for instance, the Nelder-Mead simplex method (implemented in MATLAB by the fminsearch function). As aforementioned, in problems ( 46) and (47) we ensure that S 0 is sufficiently large by imposing that X 0 ⊂ S 0 . If we seek a larger S 0 , we can enlarge the shape set X 0 . This can be done, for instance, by using a scaling factor η for X 0 in order to ensure ηX 0 ⊂ S 0 , which can be translated in the LMI constraint

η ¯αv l ′ * αW > 0, l = 1, …, r (49)
where η ¯= η -2 . In this case, while tr(W) is minimised, the scalar α is implicitly adjusted to fulfil (49). However, by increasing η a side effect is that the optimal S u will also be probably increased since both sets are defined from the same matrix P. There is, in fact, a trade-off to manage. Moreover, note that for η too large the conditions may be not feasible. This comes from the fact that, since the open-loop system is exponentially unstable (by assumption), the global stabilisation is indeed not possible.

Then, by choosing -τ 3 = τ 2 -ατ 1 -1, following the same reasoning is done in the proof of Proposition 1, it follows that (38) and (39) imply (44). Finally, the satisfaction of relation (40) implies that the ellipsoid S 0 = ℰ(αP) is included in the polyhedral set S(u 0 ) defined in (26), which concludes the proof. □

Computational issues

In this section, we discuss how to compute the stabilising control law by using the results stated in Propositions 1 and 2.

On the other hand, we can formulate a converse problem in which the main objective is the maximisation of S 0 , while keeping S u sufficiently small. For this, we can consider for instance a polyhedral shape set X u represented as

X u = {x ∈ ℝ n : H (i) x ≤ 1, i = 1, …, l} (50)
where H (i) ′ ∈ ℝ n . Thus, in order to ensure that S u is sufficiently small we can force S u ⊂ X u , which is ensured by the following set of constraints:

W WH′ (i)
H (i) W 1 > 0, i = 1, …, l (51)
In this case, in order to maximise S 0 the following optimisation problems can be considered: 21

• Uniform quantisation min tr( -M) s . t . : W > αM; S 1 > 0; S 2 > 0; (19) -(

Numerical example

Consider the following discrete-time system derived from the continuous-time system treated in [START_REF] Amato | Stabilization of bilinear systems via linear state feedback control[END_REF], with a sampling period T s = 0.01 s:

A = 0.
Our objective is to compare a uniform and a logarithmic quantiser when designing a static state feedback controller that minimises the size of the attractor. With this aim, we consider that both quantisers use the same number of bits (i.e. presents the same number of quantisation levels). Considering that the most significant bit is reserved for the sign, the relation between the number of bits N bits and the number of positive levels of the quantiser N, for each input channel, is given by

N i = 2 (N bits, i -1) -1, i = 1, …, p (55)
The levels of saturation in the control channels are also considered as being the same for both quantisers and are given by vector u 0 = [12. 4 12.4]′. Fixing all these parameters, in the uniform case, the quantisation step in each channel can be calculated by ϵ (i) = u 0(i) /N i , i = 1, …, p, with N i being the number of quantisation levels in control input i. In the logarithmic case, the quantisation density ρ (i) is a degree of freedom, which will affect the smallest positive quantised value λ (i) through the following relation:

λ (i) = u 0(i) ρ (i) (N i -1) (56)
Assuming N bits = 6 for both channels (which yields N 1 = N 2 = N = 31), the two quantisation cases are analysed in the sequel. For this, we apply the optimisation problems ( 46 Notice that in this case both analysis and simulation show the superiority of logarithmic quantisers in terms of attractor size, considering the same number of quantisation levels and the same saturation limits. Nevertheless, as mentioned previously in this section, the quantisation density ρ (i) is a degree of freedom of the logarithmic quantiser and a good choice of this parameter must be made. One would expect that the closer to one ρ (i) is, a smaller S u would be obtained since quantisation would become denser and the quantisation non-linearity q (i) (u) (see Fig. 3) would lie in a closer sector (i.e. δ (i) would be close to zero). However, considering u 0 and the number of quantisation levels N i fixed, relation (56) shows that there is a trade-off between ρ (i) and λ (i) (which is proportional to ϵ (i) , Table 1 also shows that for ρ (i) = 0.8349, we obtain S u which is similar in size to the one obtained in the uniform case. Finally, for ρ (i) = 0.8935 the deadzone limit ϵ ci is approximately the same as that one of the uniform quantisation, but S u is several times larger than the one obtained in the uniform case. This indicates that, depending on the quantisation density chosen for the logarithmic quantiser, if we fix the saturation and the deadzone limits as well as the number of quantisation levels, the uniform quantiser may provide better results.

In order to illustrate the trade-off regarding the size of S 0 and S u , as discussed in Section 5, we replace now (45) by ( 49) in optimisation problems (46) and (47). Tables 2 and3 show the results obtained for different values of the scaling factor η = 1/ η for the uniform and logarithmic cases, respectively. Note that as η increases, i.e. the larger is the shape set that must be included by S 0 , the smaller is the value of tr(αP), which indicates that S 0 increases. On the other hand, it can be observed that tr(P) decreases, meaning that S u also increases as we increase η. Moreover, as expected, for η too large (η > 1.3 in the uniform case and η > 1.18 in the logarithmic one) no feasible solution (N.F.) can be found since the global stability is not achievable because the open-loop system is exponentially unstable.

Conclusion

This paper has tackled the stabilisation of (unstable) multi-input discrete-time linear systems involving input static finite quantisation. A unified framework to handle uniform and logarithmic quantisers has been proposed. The approach is based on the modelling of the finite quantisation through the appropriate application of deadzone and saturation operators. Based on the properties of the three types of non-linearities (i.e. infinite quantisation, deadzone and saturation), conditions to design a static state feedback controller that guarantees the convergence to an attractor set S u , provided the initial state belongs to a set S 0 , are developed. These conditions are incorporated in optimisation problems aiming at determining the stabilising feedback gain while minimising the size of the attractor set and guaranteeing that the set S 0 covers a pre-defined region in the state space.

Although this work considered only input quantisation, the approach can be extended to cope with state quantisation and also be used to optimally tune the quantisers parameters when the gain is given. another parameter that influences the size of S u ). In this example, the best choice for ρ = ρ (1) = ρ (2) was found numerically. Table 1 shows the results of the optimisation problem (47) that would have been obtained with different values of ρ (i) = ρ, i = 1, 2. Notice that, when ρ (i) > 0.7391, ϵ (i) becomes larger and, as expected, S u becomes larger too (i.e. tr(P) is smaller). On the other hand, when ρ (i) < 0.7391, ϵ ci becomes smaller but, since the quality of quantisation is worse (actually the sector where the q (i) (u) lies is more opened, i.e. more uncertainty is considered), S u grows.

Fig. 1 Fig. 2

 12 Fig. 1 Sets S 0 and S u

Fig. 3

 3 Fig. 3 Logarithmic quantisation

  ensures that tr(α -1 W) > tr(M), i.e. the minimisation of tr( -M) implies the maximisation of the tr(α -1 W)).

  ) or (47) with X 0 = {v 1 }, with v 1 = [2 2 2]′.Considering the uniform quantiser, for N = 31 and u 0 given above, we obtain ϵ (1) = ϵ (2) = 0.4. A grid search allows to find the best feasible values for the tuning parameters, respectively, α = 0.0026, τ 1 = 5.3059 × 10 -11 , τ 2 = 0.0198 and S 2 = diag(6.3644 × 10 -4 , 0.0015). The solution to the optimisation problem (46) in this case yields K = 0.6137 -0.3853 -10.4675 9.2395 17.0548 14.6553 and P = 15.6182 7.5034 -1.4831 7.5034 19.6851 -1.3810 -1.4831 -1.3810 51.5721which has its trace equal to 86.88.In the logarithmic quantiser case, also for N = 31 and u 0 given above, we choose ρ (1) = ρ (2) = 0.7391. Recalling that λ (i) = ϵ (i) (1 + δ (i) ), we obtain ϵ (1) = ϵ (2) = 0.0012. Comparing with the deadzone limits of the uniform quantiser case, one can expect the attractor size to be considerably smaller in the logarithmic case. A grid search allows to find the best feasible values for the tuning parameters, respectively, α = 4.2786 × 10 -5 , τ 1 = 1.21 × 10 -10 and τ 2 = 0.0014. The solution to the optimisation problem (47) in this case yields 5139 -0.6808 5.0021 which has traced its equal to 8.7736 × 10 3 .Figs.4-6show, respectively, the evolution of the state for an initial condition of x 0 = [2 -2 -1]′ in the uniform and logarithmic cases, and the ellipsoids S u obtained in both cases.

Fig. 4

 4 Fig. 4 States evolution in the unif. quantisation case

Fig. 5

 5 Fig. 5 States evolution in the log. quantisation case

Table 1

 1 Results obtained for different values of ρ (i) -

	logarithmic quantisation ρ (i) δ (i)	λ (i)	ϵ (i)	tr(P)
	0.4706	0.36	1.87 × 10 -9	1.38 × 10 -9	2569.9
	0.5385	0.3	1.07 × 10 -7	8.22 × 10 -8	5830.8
	0.7391	0.15	1.43 × 10 -3	1.24 × 10 -3	8773.6
	0.7857	0.12	8.94 × 10 -3	7.98 × 10 -3	2958.0
	0.8182	0.10	3.01 × 10 -2	2.74 × 10 -2	289.0
	0.8349	0.09	5.52 × 10 -2	5.06 × 10 -2	83.7
	0.8935	0.056	4.22 × 10 -1	4.00 × 10 -1	1.45

Table 2

 2 Influence of η over the sizes of S u and S 0 -uniform

	quantisation η	tr(W)	tr(P)	tr(αP)
	0.5	0.0299	1369.70	3.3106
	0.8	0.0685	231.77	0.6230
	1.0	0.1602	86.88	0.2259
	1.2	21.1349	2.34	0.0744
	1.3	N.F.	N.F.	N.F.

Table 3

 3 Influence of η over the sizes of S u and S 0 -

	logarithmic quantisation η tr(W)	tr(P)	tr(αP)
	0.5	1.31 × 10 -4	2.88 × 10 5	5.5860
	0.8	3.77 × 10 -4	5.22 × 10 4	1.0100
	1.0	0.0017	8.77 × 10 3	0.3754
	1.17	2.9110	36.72	0.0705
	1.18	N.F.	N.F.	N.F.

Acknowledgments

The work was supported by CAPES (STIC-AmSud project 88881.143275/2017-01) and CNPq (grants PQ-305979/2015-9 and Univ-422992/2016-0), Brazil.


Gustavo Cruz Campos

João Manoel Gomes Da Silva

Sophie Tarbouriech

Carlos Eduardo Pereira
email: cepereira@ufrgs.br

Stability of Discrete-time Control Systems with Uniform and Logarithmic Quantizers

Keywords: Quantized control, stability, attractors, Networked Control Systems (NCS)

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The use of digital controllers became very frequent in control systems in the last decade. It is well known that in digital control, some undesired side effects such as timedelays, asynchronism, saturation and quantization are implicit. For instance, saturation may not only reduce the performance of a system, but also deteriorate stability. Similarly, the effect of quantization in control systems is a known phenomenon, which may lead to limit cycles, undesired equilibria or chaotic behavior, even if the controller is a stabilizing one [START_REF] Delchamps | Stabilizing a linear system with quantized state feedback[END_REF], [START_REF] Tarbouriech | Control design for quantized linear systems with saturations[END_REF]). Now that Networked Control Systems (NCS) are becoming increasingly popular, researches on quantization regained attention of the control community. Since in NCS the control loop elements exchange information through communication channels with limited bandwidth, the control systems may become more susceptible to quantization side effects.

Then, this kind of communication constraints has attracted the attention of researchers over the last years; see, for example, [START_REF] Brockett | Quantized feedback stabilization of linear systems[END_REF], [START_REF] Fridman | Control under quantization, saturation and delay: An LMI approach[END_REF], [START_REF] Liberzon | Hybrid feedback stabilization of systems with quantized signals[END_REF]. It is also important to note that several studies have been proposed for continuous-time systems. In the context of discretetime systems, one can cite several works dealing with the controller or observer design in presence of uniform or logarithmic quantizers: see, for example, [START_REF] Picasso | Stabilization of discretetime quantized linear systems: an H ∞ /l 1 approach[END_REF] and [START_REF] Xia | Stability analysis of discrete-time systems with quantized feedback and measurements[END_REF]. In particular, in [START_REF] Elia | Stabilization of linear systems with limited information[END_REF], it has been shown that, for a quadratically stabilizable system, a logarithmic quantizer is the optimal solution in terms of coarse quantization density. However, it is also shown that the quantizer must have an infinite number of quantization levels, which is not possible to implement. Following the same idea, [START_REF] Fu | The sector bound approach to quantized feedback control[END_REF] have introduced the sector bound approach for quantized feedback systems giving simple formulae to the stabilization problem considering state and output feedback controllers. Many other works were carried out as in de [START_REF] De Souza | Stability analysis of finite-level quantized discrete-time linear control systems[END_REF], where the sector bound approach is applied to derive LMI based conditions for estimating a set of initial conditions and an attractor, such that all the state trajectories starting in the first set will enter the attractor in a finite time and remain inside it. In that work the controller and a finite logarithmic quantizer are supposed to be given and the stability analysis problem is addressed.

Despite these results obtained for logarithmic quantizers, research on uniform quantizers is not less relevant. In [START_REF] Ferrante | Stabilization of continuous-time linear systems subject to input quantization[END_REF], sector conditions are used to find a compact invariant set surrounding the origin, which attracts all the state trajectories, considering two different settings involving uniform quantization in continuous-time linear systems. The controller is designed in order to minimize the dimensions of the attractor. [START_REF] Tarbouriech | Control design for quantized linear systems with saturations[END_REF] also considers the effects of saturation to design a state feedback control law while minimizing the attractor and maximizing the set of admissible initial conditions. This paper deals with the stability analysis of discrete-time linear systems involving saturation and either an uniform or a logarithmic finite quantizer. For simplicity, only the input quantization case is considered. Considering uniform quantizers, we extend the work of [START_REF] Ferrante | Stabilization of continuous-time linear systems subject to input quantization[END_REF] and [START_REF] Tarbouriech | Control design for quantized linear systems with saturations[END_REF] to the discretetime case. Using sector conditions and the S-procedure, uniform ultimate boundedness stability is analyzed for a given static state feedback controller, while the attractor and the set of admissible initial conditions are determined. Then an LMI-based optimization problem is proposed to minimize the attractor and simultaneously maximize the set of admissible initial conditions (which can be considered as an estimate of the region of attraction of the origin). For the logarithmic quantizers, a similar analysis is carried out. After the analysis of both uniform and logarithmic quantization, we propose a method to compare the performance of quantizers, in terms of attractor size, in a NCS scenario. Finally, some simulations are presented to validate the results.

Notation. Throughout the article, I denotes the identity matrix and 0 denotes the null matrix (equivalently the null vector) of appropriate dimensions. For a matrix A ∈ R n×m , A , A (i) , tr(A) denote its transpose, its ith row and its trace respectively. The matrix diag(A 1 , A 2 , . . . , An) is the block-diagonal matrix having A 1 , A 2 , . . ., An as diagonal blocks and in symmetric matrices * stands for symmetric blocks. For a vector x ∈ R n , x (i) , x , |x| denote its ith component, its transpose and the componentwise absolute value operator respectively. sign(x) is the componentwise sign function, with sign(0) = 0, and x the componentwise floor operator. For two sets S 1 and S 2 , S 1 \S 2 denotes the set S 1 deprived of S 2 .

STABILITY WITH UNIFORM QUANTIZERS

Problem statement

Consider the following discrete-time linear system:

x(k + 1) = Ax(k) + Bsat q u(k) x(0) = x 0 (1)
where x ∈ R n , u ∈ R p , x 0 ∈ R n are respectively the state, the input of the system and the initial state. A, B are real matrices of suitable dimensions, and q(•) is a componentwise uniform quantizer, which is described by the static nonlinear function defined as

q (i) (u) := ∆sign(u (i) ) |u (i) | ∆ ( 2
)
where ∆ is a positive real scalar representing the quantization step. The saturation map sat(•) represents a symmetric saturation function defined as follows:

sat(u (i) ) = sign(u (i) ) min u 0 , |u (i) | , i = 1, . . . , p (3)
with u 0 > 0 being the symmetric bounds on the control input u (i) , i = 1, . . . , p.

Assuming that the state x is fully accessible, we want to analyze stability of system (1) subject to the following control law u = Kx. Therefore, by defining the functions

ψ(v) := q(v) -v (4) φ(v) := sat(v) -v (5)
the closed-loop system becomes:

x(k + 1) = (A + BK)x(k) + Bψ(k) + Bφ(k) x(0) = x 0 (6) with ψ(k) = ψ u(k) φ(k) = φ q u(k) = φ u(k) + ψ u(k)
The presence of the uniform quantizer defined in (2) can represent a real obstacle to the asymptotic stabilization of the closed-loop system, due to its deadzone effect [START_REF] Ferrante | Stabilization of continuous-time linear systems subject to input quantization[END_REF]). If the matrix A is not Schur-Cohn, the asymptotic stabilization of the origin for the closed-loop system cannot be achieved even when the gain K is supposed to be a stabilizing one (that is, when (A + BK) is Schur-Cohn). Moreover, if A is not Schur-stable, under input saturation, only local (regional) stability can be achieved [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]). In this work, we focus on this case. Then the problem we aim to solve can be stated as follows.

Problem 1. Given the matrices A, B and a stabilizing gain K of adequate dimensions and a positive real quantization step ∆, determine a set S 0 ⊂ R n and a compact set S u ⊂ R n containing the origin, such that

• S 0 and S u are invariant sets;

• For every initial condition x 0 ∈ S 0 \S u , the trajectories are bounded and converge in a finite time into S u (which is therefore an attractor of the trajectories).

Main results

To solve Problem 1, we recall the sector conditions used in [START_REF] Ferrante | Stabilization of continuous-time linear systems subject to input quantization[END_REF]) and [START_REF] Tarbouriech | Control design for quantized linear systems with saturations[END_REF]). Lemma 1. For every ψ(u) as defined in (4) and every control law u = Kx, u ∈ R p , the following relations are verified:

ψ S 1 ψ -tr(S 1 )∆ 2 ≤ 0 (7) ψ S 2 (ψ + Kx) ≤ 0 (8)
for any diagonal positive definite matrices S 1 , S 2 ∈ R p×p . Lemma 2. For every φ(u) as defined in (5) and every matrix G ∈ R p×n , the following relation is verified:

φ S 3 sat q(Kx) + Gx ≤ 0 (9)
for any diagonal positive matrix

S 3 ∈ R p×p , provided that x ∈ S(u 0 ) with S(u 0 ) = x ∈ R n ; -u 0 ≤ G (i) x ≤ u 0 , ∀i ∈ {1, . . . , p} (10)
Note that condition (9), combined with ( 4) and ( 5), reads:

φ S 3 (φ + ψ + Kx + Gx) ≤ 0 (11)
By using Lemmas 1 and 2, the following proposition to solve Problem 1 can be stated. Proposition 1. If there exist a symmetric positive definite matrix W ∈ R n×n , three diagonal positive matrices S 1 , S 2 , R 3 ∈ R p×p , a matrix Z ∈ R p×n , two positive scalars τ 1 , τ 2 and a scalar α, 0 ≤ α ≤ 1, satisfying the conditions ( 12)-( 14):

   W (τ 2 -ατ 1 -1) -W K S 2 -W K -Z W (A + BK) * -S 1 -2S 2 -I B * * -2R 3 R 3 B * * * -W    < 0 (12) tr(S 1 )∆ 2 + τ 1 -τ 2 ≤ 0 (13) αW Z (i) Z (i) u 2 0 > 0, i = 1, . . . , p (14)
then, it follows that ∀x(0) ∈ S 0 \S u the trajectories of system (6) converge in a finite time to the set S u , with

S u = ε(P ) = {x ∈ R n : x P x ≤ 1}, P = W -1 (15) S 0 = ε(αP ) = {x ∈ R n : x αP x ≤ 1} (16)
Moreover, S 0 and S u are positively invariant sets with respect to (6).

Proof. Considering the quadratic Lyapunov function V (x) = x P x, we want to prove that ∆V

(x) = V x(k + 1) -V x(k) < 0, ∀x ∈ S 0 \S u . (18)
In this case, it follows that S 0 is an invariant set and that ∀x(0) ∈ S 0 , the corresponding trajectory converge to S u . Note that

S 0 \S u = {x ∈ R n : (x αP x -1) ≤ 0 and (1 -x P x) ≤ 0}
Now applying S-procedure along with Lemmas 1 and 2, if 18) is satisfied, provided that x ∈ S(u 0 ). Developing (19) by using equation ( 6), it is possible to re-write (19) as follows:

∆V (x)-τ 1 (x αP x-1)-τ 2 (1-x P x)-ψ S 1 ψ+tr(S 1 )∆ 2 -2ψ S 2 (ψ + Kx) -2φ S 3 (φ + ψ + Kx + Gx) < 0 (19) it follows that (
[x ψ φ ] N u x ψ φ + tr(S 1 )∆ 2 + τ 1 -τ 2 < 0 (20)
where N u is given at the bottom of the page in (17).

It is straightforward to verify that (20) will be satisfied if we guarantee (13) and N u < 0.

Applying Schur complement, N u < 0 is equivalent to:

   P (τ 2 -ατ 1 -1) -K S 2 -K S 3 -G S 3 (A + BK) * -S 1 -2S 2 -S 3 B * * -2S 3 B * * * -P -1    < 0 (21)
Then by pre-and post-multiplying relation ( 21) by diag(P -1 , I, S -1 3 , I) and with the change of variables 12). Furthermore, one has to prove that the set S u is positively invariant. This is proven if we guarantee that x(k + 1) P x(k + 1) ≤ 1 whenever x(k) P x(k) ≤ 1 (i.e., when x ∈ S u ). It is possible to do that by verifying the following inequality, for some positive scalar τ 3 ,

P -1 = W , GP -1 = Z, S -1 3 = R 3 , we get relation (
x (k + 1)P

x(k + 1) -1 -τ 3 (x (k)P x(k) -1) ≤ 0 (22)
The inequality ( 22) is true if we are able to verify that tr(S 1 )∆ 2 + τ 3 -1 ≤ 0 (23)

N u =   P (τ 2 -ατ 1 -1) -K S 2 -K S 3 -G S 3 * -S 1 -2S 2 -S 3 * * -2S 3   +   (A + BK) B B   P [(A + BK) B B] (17)
and

x (k + 1)P x(k + 1) -τ 3 x (k)P x(k) -ψ S 1 ψ -2ψ S 2 (ψ + Kx) -2φ S 3 (φ + ψ + Kx + Gx) ≤ 0 (24)
Now by choosing -τ 3 = τ 2 -ατ 1 -1, it is straightforward to verify that ( 24) is implied by (12). From ( 23), since α ≤ 1 and τ 1 , τ 2 > 0, it follows that tr(S 1 )∆

2 + ατ 1 -τ 2 ≤ tr(S 1 )∆ 2 + τ 1 -τ 2 (25)
Therefore, ( 13) implies ( 23). Then, if ( 12) and ( 13) are satisfied, ( 22) is implicitly verified and thus S u is positively invariant.

Finally, the satisfaction of relation ( 14) implies that the ellipsoid S 0 = ε(αP ) is included in the polyhedral set S(u 0 ) defined in (10), which ensures the validity of condition ( 11) in S 0 . That concludes the proof.

2 3. STABILITY WITH LOGARITHMIC QUANTIZERS

Problem statement

Consider the following discrete-time linear system:

     x(k + 1) = Ax(k) + B • dz sat q u(k) x(0) = x 0 (26)
where x ∈ R n , u ∈ R p , x 0 ∈ R n are respectively the state, the input of the system and the initial state. A, B are real matrices of suitable dimensions, and q(•) is the logarithmic quantizer, which is described by the static nonlinear function defined as

q (i) (u) :=      λ ρ j , if λ ρ j (1 + δ) ≤ u (i) < λ ρ j (1 -δ) , j ∈ Z -q(-u (i) ), if u (i) < 0 (27) δ = 1 -ρ 1 + ρ , ρ = 1 -δ 1 + δ , 0 < ρ < 1 (28)
The parameter ρ is said to be the quantization density (instead of quantization step as in the uniform quantization), and λ is the least positive quantization level outside the deadzone. λ is related to the deadzone limit by λ = (1+δ), i.e., the deadzone is defined by its limit , > 0, and the nonlinear function given by

dz (i) (u) := 0, if -< u (i) < u (i) , if |u (i) | ≥ (29)
The saturation map sat(•) ∈ R p is defined from the symmetric saturation function having as level the positive scalar

µ = λ ρ N -1 , ∀i = 1, . . . , p: sat(u (i) ) = sign(u (i) ) min µ, |u (i) | , i = 1, . . . , p (30)
N is the number of positive levels between µ and .

Notice that we obtain a finite-level quantizer by applying saturation and deadzone effects over an infinite-level quantizer.

The functions ψ and φ related to the quantization and saturation are defined as in (4)-( 5). By defining now the function θ(v) := dz(v) -v (31) the closed-loop system becomes:

x(k + 1) = (A + BK)x(k) + Bψ(k) + Bφ(k) + Bθ(k) x(0) = x 0 (32)
Due to the deadzone, when approaching the origin, the quantizer output will be null if its input is inside (-, ). Thus, asymptotic stabilization of the origin cannot be achieved if A is not Schur-Cohn. Again we analyze ultimate boundedness stability rather than stability in the conventional sense. The following problem is addressed in this case. Problem 2. Given the matrices A, B and a stabilizing gain K of appropriate dimensions, the quantization density ρ, the deadzone limit and the saturation limit µ, determine a set S 0 ⊂ R n and a compact set S u ⊂ R n containing the origin, such that

• S 0 and S u are invariant sets;

• For every initial condition x 0 ∈ S 0 \S u , the trajectories are bounded and converge in a finite time into S u (which is an attractor of the trajectories).

Main results

To solve Problem 2, we recall a sector condition verified by the nonlinearities ψ and θ (de Souza et al. [2010]): Lemma 3. For ψ = q(u) -u with q(u) defined in ( 27)-( 28) and u = Kx ∈ R p , the following relation is verified: (ψ + δKx) S 1 (ψ -δKx) ≤ 0 (33) for any diagonal positive definite matrix S 1 ∈ R p×p .

For the sector conditions originated by the saturation, we use Lemma 2 (rewritten with S 2 ∈ R p×p instead of S 3 for clarity purposes):

φ S 2 (φ + ψ + Kx + Gx) ≤ 0 ( 34
)
Lemma 4. For every θ defined in (31), and every control law u = Kx ∈ R p , the following relation is verified: θ S 3 θ -tr(S 3 ) 2 ≤ 0 (36) for any diagonal positive definite matrix S 3 ∈ R p×p . By using Lemmas 2-4, the following proposition to solve Problem 2 can be stated. Proposition 2. If there exist a symmetric positive definite matrix W ∈ R n×n , three diagonal positive matrices R 1 ,

N l =    P (τ 2 -ατ 1 -1) + δ 2 K S 1 K 0 -K S 2 -G S 2 0 * -S 1 -S 2 0 * * -2S 2 0 * * * -S 3   +    (A + BK) B B B    P [(A + BK) B B B] (35)
R 2 , S 3 ∈ R p×p , a matrix Z ∈ R p×n , two positive scalars τ 1 , τ 2 and a scalar α, 0 ≤ α ≤ 1, satisfying the conditions (37)-( 39):

       γW 0 -W K -Z 0 W (A + BK) δW K * -R 1 -R 1 0 R 1 B 0 * * -2R 2 0 R 2 B 0 * * * -S 3 B 0 * * * * -W 0 * * * * * -R 1        < 0,
where γ = τ 2 -ατ 1 -1 ( 37)

tr(S 3 ) 2 + τ 1 -τ 2 ≤ 0 (38) αW Z (i) Z (i) µ 2 > 0, i = 1, . . . , p (39)
then, it follows that ∀x(0) ∈ S 0 \S u the trajectories of system (32) converge in a finite time to the set S u , with S u and S 0 as in ( 15)-( 16). Moreover, S 0 and S u are positively invariant sets with respect to (32).

Proof. Applying S-procedure along with Lemmas 2-4, if

∆V (x) -τ 1 (x αP x -1) -τ 2 (1 -x P x) -(ψ + δKx) S 1 (ψ -δKx) -2φ S 2 (φ + ψ + Kx + Gx) -θ S 3 θ + tr(S 3 ) 2 < 0 (40) then it follows that ∆V (x) < 0, ∀x ∈ S 0 \S u is satisfied, provided that x ∈ S(u 0 ).
Developing (40) by using equation ( 32) and considering ξ = [x ψ φ θ ] , it is possible to re-write (40) as follows:

ξ N l ξ + tr(S 3 ) 2 + τ 1 -τ 2 < 0
(41) where N l is given at the bottom of the page in (35). It is straightforward to verify that (41) will be satisfied if we guarantee (38) and N l < 0. By applying Schur complement twice, then by pre-and post-multiplying by diag(P -1 , S -1 1 , S -1 2 , I, I, I) and with the change of variables

P -1 = W , GP -1 = Z, S -1 1 = R 1 , S -1 2 = R 2 ,

we get relation (37).

To prove that the set S u is positively invariant, it suffices to verify that, for some positive scalar τ 3 ,

x (k + 1)P

x(k + 1) -1 -τ 3 (x (k)P x(k) -1) ≤ 0 (42)
Then, by chossing -τ 3 = τ 2 -ατ 1 -1, following the same reasoning done in the proof of Proposition 1, it follows that (37)-( 38) imply (42).

Finally, the satisfaction of relation (39) implies that the ellipsoid S 0 = ε(αP ) is included in the polyhedral set S(µ) defined in (10). That concludes the proof.

2

COMPUTATIONAL ISSUES

Regarding Problems 1 and 2, the objective is to obtain a set S 0 as large as possible and a set S u as small as possible. To achieve this objective, one needs to find a suitable measure for the set S u and minimize it. The largest solution for S 0 will be determined by taking the smallest value of α for which the problem remains feasible. It is possible to implicitly minimize the volume of S u by minimizing tr(P -1 ), or equivalently, tr(W ).

It is important to note that conditions ( 12) and ( 14), in the uniform quntization case, and conditions ( 37) and ( 39), in the logarithmic quntization case, are nonlinear in the decision variables, which prevents from solving directly a convex optimization problem. Nevertheless, it is possible to overcome this problem by considering the variables τ 1 , τ 2 and α as tuning parameters. In the uniform quantization case, S 2 must be fixed as well. To be feasible, it is strictly necessary that the following conditions are respected:

τ 2 > τ 1 ; τ 2 < 1 + ατ 1 (
43) Thus, the optimal value for the trace minimization can be performed by solving a convex-optimization problem on a grid. For simulation purposes, it is also interesting to require the set S 0 to be large enough to contain the initial conditions of interest. One can impose x 0 to be inside the ellipsoid of S 0 by writing x 0 αP x 0 < 1. To write it in an LMI form in terms of W , one can rearrange that statement as follows:

1 αx 0 * αW > 0 (44)

After choosing the initial conditions to be considered and fixing τ 1 , τ 2 , α, S 2 at feasible values, the convex optimization to address Problem 1 is stated as follows:

min tr(W ), s.t.: S 1 > 0; S 2 > 0; (12)-( 14); (44) (45)

Similarly, the convex optimization to address Problem 2 is stated as follows: min tr(W ), subject to: (37)-( 39); (44) (46)

Notice that conditions W > 0, R 1 > 0, R 2 > 0, R 3 > 0, S 3 > 0 are ensured by ( 12) and (37).

COMPARISON OF THE QUANTIZERS

Now we consider a scenario of a networked control system where the only element that need to be designed is the quantizer. In such scenario, the sampling period is already fixed. Thus, so are the matrices A, B of the discrete-time system. The number of bits used for each variable is also a parameter of the network, and it determines the number of positive levels of the quantizer. Considering that the most significant bit is reserved for the sign, the relation between the number of bits N bits and the number of positive levels of the quantizer N is given by

N = 2 (N bits -1) -1 (47)
The level of saturation is also considered as being the same when comparing an uniform and a logarithmic quantizer:

u 0 = µ.
Fixing all these parameters is already enough to determine an uniform quantizer. The quantization step can be calculated by ∆ = u0 N . In the logarithmic quantization case, the quantization density ρ is a degree of freedom, which will affect the smallest positive quantized value λ by the following relation:

λ = µρ (N -1) (48)
The performance of the system in the logarithmic quantizer case depends directly on a good choice of ρ. If ρ is too close to 1, λ will be not small. Thus the attractor set S 0 risks of being too large (because the deadzone will have large bounds). On the other hand, a too small ρ can deteriorate the dynamics of the control system because of a poor resolution. Furthermore, ρ should be carefully chosen even in order to keep feasibility of the optimization problem. Note that a small quantization density can lead to closedloop instability if the system is open-loop unstable.

Finally, the analysis of the two cases taking into account the scenario described should enable a comparison of the quantizers' performance in terms of the smallest attractor around the origin. Nevertheless, simulation is always necessary to confirm the results obtained, since the proposed methods can only estimate the attractor.

NUMERICAL EXAMPLE

Consider the following discrete-time system derived from the continuous-time system in [START_REF] Amato | Stabilization of bilinear systems via linear state feedback control[END_REF]), with a sampling period T s = 0.01s: Assuming N bits = 6 (which yields N = 31), the saturation level u 0 (= µ) = 12.4 and the considered initial condition as x 0 = [2 2 2] , both quantization cases are analyzed in the sequel.

A = 0.

Uniform quantizer case

Fixing N and u 0 yields ∆ = 0.4. A grid search allows to find the best feasible values for the tuning parameters, respectively, α = 0.0026, τ 1 = 5. , which has trace equal to 86.88.

Logarithmic quantizer case

In the logarithmic quantizer case, we choose to fix ρ = 0.7391 which yields = 0.0012. Comparing with ∆ = 0.4 of the uniform quantizer case, one can expect the attractor size to be considerably smaller in the logarithmic quantizer case. A grid search allows to find the best feasible values for the tuning parameters, respectively, α = 9.9218 • 10 -6 , τ 1 = 1.6985 • 10 -15 and τ 2 = 0.0083. The solution of the optimization problem (46) in this case yields Notice that both analysis and simulation show the superiority of logarithmic quantizers in terms of attractor size.

CONCLUSION

This paper has tackled the stability analysis of discretetime linear systems involving input finite quantization (uniform or logarithmic). In both cases, the attractor of the states is minimized through a quasi-LMI optimization problem and the results were compared, confirming the superiority of logarithmic quantizers. The performance of systems involving both kinds of quantizers were compared.

A simulation was carried out to confirm the results. In this work we considered only the minimization of the attractor set S 0 . The simultaneous maximization of S u and also considering S 0 and S u of different shapes (i.e. defined with different matrices P ) as done in de Souza et al. [2010], are possible extensions of this approach. Furthermore, the stabilization problem and extension of the approach to consider also quantization on the measured states is an ongoing work.

Fig. 3 .

 3 Figures (1)-(3) show, respectively, the states evolution for a initial condition of x 0 = [2 -2 -1] in the uniform and logarithmic cases, and the ellipsoids of S u obtained in the examples.

  3059 • 10 -11 , τ 2 = 0.0198 and S 2 = diag(6.3644 • 10 -4 , 0.0014). The solution of the optimization problem (45) in this case yields

		15.6182 7.5034 -1.4831
	P =	7.5034 19.6851 -1.3810
		-1.4831 -1.3810 51.5721

This work was in part supported by CAPES (projects PVE-129/2012 and STICAmSud) and CNPq (PQ 305979/2015-9 and 308605/2015-2), Brazil, and also by ANR project LIMICOS grant number 12BS0300501.


R G Novikov
email: novikov@cmap.polytechnique.fr

I A Taimanov
email: taimanov@math.nsc.ru

To S P Novikov

Darboux-Moutard transformations and Poincare-Steklov operators *

Formulas relating Poincare-Steklov operators for Schrödinger equations related by Darboux-Moutard transformations are derived. They can be used for testing algorithms of reconstruction of the potential from measurements at the boundary.

Investigations of inverse problems for two-dimensional Schrödinger operators at a given energy level were initiated within the framework of the theory of solitons by S.P. Novikov and his scientific school. In particular, in [START_REF] Dubrovin | The Schrödinger equation in a periodic field and Riemann surfaces[END_REF] the spectral data were introduced for the two-dimensional periodic Schrödinger operator, in a magnetic field, which is finite-gap at one energy level, and the inverse problem of reconstructing the operator from these data was solved; in [START_REF] Veselov | Finite-zone, two-dimensional Schrödinger operators. Explicit formulas and evolution equations[END_REF][START_REF] Veselov | Finite-zone, two-dimensional Schrödinger operators. Potential operators[END_REF], in terms of these data, potential operators were singled out and there were derived the evolutionary equations (the Novikov-Veselov equations) which preserve this class of operators and their spectra at the given energy level, and theta-functional formulas for solving the equations were obtained; in [START_REF] Grinevich | Two-dimensional "inverse scattering problem" for negative energies and generalized-analytic functions. I.Energies below the ground state[END_REF] the first results on the inverse scattering problem at the negative energy level were obtained and for the first time in the theory of inverse problems the methods of the theory of generalized analytic functions were used.

In this paper we develop an approach to direct and inverse problems, for the two-dimensional Schrödinger operator, based on the Moutard transformation. Since the Darboux transformation is a one-dimensional reduction of the Moutard transformation, our results on two-dmensional operators from §2 are extended to the case of one-dimensional Schrödinger operators (see §3).

∂ v ∂ w θ + uθ = 0
and is determined by a solution of the initial equation [START_REF] Moutard | Sur la construction des équations de la forme 1 z d 2 z dxdy = λ(x, y), qui admettent une intégrale générale explicite[END_REF]. We consider its special reduction for which the variables v and w are complex-conjugate and after renormalizations the second order equation reduces to the Schrödinger equation

Hϕ = -∆ϕ + uϕ = 0, (1)
where

∆ = ∂ 2 ∂x 2 + ∂ 2 ∂y 2
is the Laplace operator on the two-plane R 2 . Let us take a solution ω of (1) and construct by using this solution the new Schrödinger operator

H = -∆ + u with the potential u = u -2∆ log ω = -u + 2 ω 2 x + ω 2 y ω 2 . ( 2
)
Is is easy to show by straightforward computations that if a function ϕ satisfies (1), then the function θ, which is determined by the relations

(ωθ) x = -ω 2 ϕ ω y , (ωθ) y = ω 2 ϕ ω x , (3)
satisfies the equation

Hθ = -∆θ + uθ = 0, (4)
obtained from (1) by the Moutard transformation determined by the initial solution ω. We note that the relations (3) determine θ up to summands of the form C ω , C = const, the function ω -1 = 1 ω satisfies the equation

H 1 ω = 0
and determines the inverse Moutard transformation from H to H:

u ω -→ u = u -2∆ log ω 1/ω -→ u = u -2∆ log 1 ω = u + 2∆ log ω.
For a potential

u = u(x)
which depends on one variable the Moutard transformation reduces to the Darboux transformation [START_REF] Darboux | Sur une proposition relative aux équations linéarires[END_REF]. Let H 1 be a one-dimensional Schrödinger operator

H 1 = - d 2 dx 2 + u
and ω 1 be its eigenfunction:

H 1 ω 1 = Eω 1 , E 0 = κ 2 .
The Moutard transformation, of the two-dimensional Schrödinger operator, determined by the solution ω(x, y) = e κy ω 1 (x) of ( 1), maps the potential of the operator into

u(x) = u(x) -2∆ log(e κy ω 1 (x)) = u -2 d 2 dx 2 log ω 1 (x).
Therewith the corresponding one-dimensional Schrödinger operator H 1 is transformed into the one-dimensional operator

H 1 = - d 2 dx 2 + u.
This transformation is called the Darboux transformation. Its action on eigenfunctions takes the following form. Let

H 1 ϕ 1 = Eϕ 1 , ϕ 1 = ϕ 1 (x), E = µ 2 .
We put ϕ = e µy ϕ 1 (x).

It is clear that Hϕ = 0 and, by (3), we get the Moutard transformation of ϕ in the form θ = e µy θ 1 (x), [START_REF] Moutard | Sur la construction des équations de la forme 1 z d 2 z dxdy = λ(x, y), qui admettent une intégrale générale explicite[END_REF] were the second of relations (3) gives the Darboux transformation of eigenfunctions:

θ 1 = 1 µ + κ d dx - d log ω 1 dx ϕ 1 , (6)
H 1 θ 1 = - d 2 θ 1 dx 2 + uθ 1 = Eθ 1 , E = µ 2
, and the first one gives its inversion

ϕ 1 = 1 κ -µ d dx + d log ω 1 dx θ 1 . (7)
In this case, since the image of the transformation is sought in the form (5), the transformation of eigenfunctions becomes single-valued, because the addition of summands of multiple 1 ω does not preserve the form (5) (for µ = κ). The Darboux transformation was repeatedly used and was often rediscovered (see, for example, [START_REF] Crum | Associated Sturm-Liouville systems[END_REF]) for solving problems of mathematical physics and the spectral theory [START_REF] Matveev | Darboux Transformations and Solitons[END_REF][START_REF] Matveev | Darboux transformations, covariance theorems and integrable systems[END_REF] (see also the review [START_REF] Taimanov | On the Moutard transformation and its applications to spectral theory and soliton equations[END_REF]).

The Moutard transformation and its extension for solutions of the Novikov-Veselov equations [START_REF] Heng-Chun | Darboux transformation and variable separation approach: the Nizhnik-Novikov-Veselov equation[END_REF][START_REF] Taimanov | Two-dimensional Schrödinger operators with fast decaying rational potential and multidimensional L 2kernel[END_REF] in recent years has been applied to the construction of the first examples of two-dimensional Schrödinger operators with fast decaying potential and with a nontrivial kernel [START_REF] Taimanov | Two-dimensional Schrödinger operators with fast decaying rational potential and multidimensional L 2kernel[END_REF] and blowing-up solutions of the Novikov-Veselov equation with regular initial data [START_REF] Taimanov | Blowing up solutions of the Novikov-Veselov equation[END_REF][START_REF] Taimanov | Two-dimensional rational solitons and their blow-up via the Moutard transformation[END_REF] (see the numerical analysis of the negative discrete spectrum and its dynamics for these examples in [START_REF] Adilkhanov | On numerical study of the discrete spectrum of a two-dimensional Schrodinger operator with soliton potential[END_REF]), to the construction of explicit examples of two-dimensional potentials of Wigner-von Neumann type [START_REF] Novikov | Two-dimensional von Neumann-Wigner potentials with a multiple positive eigenvalue[END_REF]. In [START_REF] Taimanov | Faddeev eigenfunctions for twodimensional Schrödinger operators via the Moutard transformation[END_REF], the action of the Moutard transformation on the Faddeev eigenfunctions at the zero energy level was described, and in [START_REF] Novikov | The Moutard transformation and two-dimensional multi-point delta-type potentials[END_REF], using this transformation, Faddeev's eigenfunctions at the zero energy level for multipoint delta-like potentials were found.

In [START_REF] Taimanov | The Moutard transformation of two-dimensional Dirac operators and the Mobius geometry[END_REF][START_REF] Matuev | The Moutard transformation of two-dimensional Dirac operators and the conformal geometry of surfaces in four-space[END_REF], it was established a relation of the generalized Moutard transformation for two-dimensional Dirac operators [START_REF] Delong Yu | Darboux transformation for the modified Veselov-Novikov equation[END_REF] to the conformal geometry of surfaces in three-and four-dimensional spaces, and with this were constructed blowing-up solutions of the modified Novikov-Veselov equation with regular initial data [START_REF] Taimanov | A fast decaying solution to the modified Novikov-Veselov equation with a one-point singularity[END_REF][START_REF] Taimanov | Blowing up solutions of the modified Novikov-Veselov equation and minimal surfaces[END_REF]. A generalization of the Moutard transformation to the case of generalized analytic functions, in particular, gave an approach to constructing the theory of generalized analytic functions with contour poles [START_REF] Grinevich | Generalized analytic functions, Moutard-type transforms, and holomorphic maps[END_REF][START_REF] Grinevich | Moutard transform approach to generalized analytic functions with contour poles[END_REF][START_REF] Grinevich | Moutard transform for generalized analytic functions[END_REF][START_REF] Novikov | Moutard-type transform for matrix generalized analytic functions and gauge transforms[END_REF] and also allowed to construct a Moutard-type transformation for the conductivity equation [START_REF] Grinevich | Moutard transform for the conductivity equation[END_REF].

Poincare-Steklov operators

Let in the domain D with the boundary ∂D there is given an elliptic differential equation

Lψ = Eψ. (8)
We single out two boundary conditions, each of which, as a rule, completely determines the solution of the equation. Then the Poincare-Steklov operator, by definition, takes the value of one boundary condition into the value of another condition.

Let us consider the most well-known particular cases of Poincare-Steklov operators. Let the elliptic equation ( 8) be given by a linear differential secondorder expression L. Then

RR : Γ α ψ -→ Γ α-π/2 ψ.
As particular cases, we get the DN operator for α = 0 and the ND operator for α = π 2 .

The action of the Moutard transformation on Poincare-Steklov operators

We assume that equation (1) holds in a bounded simply-connected two-dimensional domain D ⊂ R 2 with a smooth boundary ∂D and that u is a regular function on D ∪ ∂D.

Let H and u be the operator and the potential from (4), i.e., the Moutard transformations of H and u, determined by a solution ω of (1) via formulas [START_REF] Veselov | Finite-zone, two-dimensional Schrödinger operators. Explicit formulas and evolution equations[END_REF].

For equation [START_REF] Dubrovin | The Schrödinger equation in a periodic field and Riemann surfaces[END_REF] we consider the operator

Φ u = DN
of the form ( 9) with E = 0 and the operator

Φ -1 u = N D
of the form ( 10) with E = 0, where u is the potential from (1). Let σ D (H) and σ N (H) denote the spectra of the operators defined by H = -∆ + u and the Dirichlet and Neumann boundary conditions, respectively.

For solutions ϕ of (1) we consider also the following boundary data on ∂D:

Γ τ ω ϕ = ω ϕ ω τ = ϕ τ - ω τ ω ϕ (11)
and

Γ ν ω ϕ = ω ϕ ω ν = ϕ ν - ω ν ω ϕ, ( 12
)
where ω is the fixed solution of (1) in D, ν is the outer normal to ∂D, τ is the path-length parameter on ∂D, which grows in the direction of ν ⊥ = (-ν 2 , ν 1 ) with ν = (ν 1 , ν 2 ); the lower indices τ and ν denote the derivations in τ and along the normal ν.

For simplicity, we assume that ω has no zeroes on ∂D.

The important observation is that relations (3) on ∂D can be rewritten as

Γ ν ω -1 θ = -Γ τ ω ϕ, (13)
Γ τ ω -1 θ = Γ ν ω ϕ. ( 14
)
Lemma 1 Assuming that 0 / ∈ σ D (H), the following formulas hold:

Ker Γ τ ω = {cω : c ∈ C}, Im Γ τ ω = {f : ∂D ω -1 f dτ = 0},
where Γ τ ω is considered as an operator on solutions of (1).

Lemma 1 follows from the definition of Γ τ ϕ by [START_REF] Heng-Chun | Darboux transformation and variable separation approach: the Nizhnik-Novikov-Veselov equation[END_REF] and the fact that under our assumptions the Dirichlet problem for (1) is uniquely solvable.

Lemma 2 Assuming that 0 / ∈ σ D ( H), we have

Ker Γ ν ω = {cω : c ∈ C}, (15)
Im Γ ν ω = {f : ∂D ωf dτ = 0},
where Γ ν ω is considered as an operator on solutions of (1).

Remark 1. The condition 0 / ∈ σ D ( H) we understand in the sense that the Dirichlet problem for ( 4) is uniquely solvable. That may be essential if ω has zeroes, on D, implying singularities of u.

Lemma 2 follows from relation ( 14), the fact that relations (3) determine θ from ϕ up to summands of the form const • ω -1 and ϕ from θ up to summands of the form const • ω, and from Lemma 1 applied to equation ( 4).

For equation [START_REF] Dubrovin | The Schrödinger equation in a periodic field and Riemann surfaces[END_REF] and boundary data [START_REF] Heng-Chun | Darboux transformation and variable separation approach: the Nizhnik-Novikov-Veselov equation[END_REF] and [START_REF] Taimanov | Two-dimensional Schrödinger operators with fast decaying rational potential and multidimensional L 2kernel[END_REF] we consider the following Poincare-Steklov operators Λ u,ω and Λ -1 u,ω :

Λ u,ω : Γ τ ω ϕ → Γ ν ω ϕ, 0 / ∈ σ D (H) (16)
and

Λ -1 u,ω : Γ ν ω ϕ → Γ τ ω ϕ, 0 / ∈ σ D ( H). ( 17
)
Proposition 1 Assuming that 0 / ∈ σ D (H), the operators Φ u and Λ u,ω are related by the following formulas:

Λ u,ω Γ τ ω ϕ = (Φ u ωIω -1 -ω ν Iω -1 )Γ τ ω ϕ, Φ u ϕ = (Λ u,ω Γ τ ω + ω ν ω -1
)ϕ, where ω (in the cases where ω is not a lower index), ω -1 , and ω ν denote the operators of multiplication by the corresponding functions on ∂D and

Iω -1 f (τ ) = τ 0 ω -1 (t)f (t) dt for f ∈ Im Γ τ
ω . Proposition 1 follows from the definitions given by formulas ( 9), [START_REF] Heng-Chun | Darboux transformation and variable separation approach: the Nizhnik-Novikov-Veselov equation[END_REF], and ( 16), and from straightforward computation.

Proposition 2 If 0 / ∈ σ N (H) ∪ σ D ( H)
, the operators Φ -1 u and Λ -1 u,ω are related as follows:

Λ -1 u,ω Γ ν ω ϕ = Γ τ ω Φ -1 u Id - ω ν ω Φ -1 u -1
Γ ν ω ϕ, where Id is the identity operator. Therewith,

Ker Id - ω ν ω Φ -1 u = {cω ν : c ∈ C} and the inverse operator Id -ων ω Φ -1 u -1 is defined on Im Γ ν ω up to summands of the form cω ν , c ∈ C.
Proposition 2 follows from formulas [START_REF] Taimanov | On the Moutard transformation and its applications to spectral theory and soliton equations[END_REF], [START_REF] Heng-Chun | Darboux transformation and variable separation approach: the Nizhnik-Novikov-Veselov equation[END_REF], [START_REF] Taimanov | Two-dimensional Schrödinger operators with fast decaying rational potential and multidimensional L 2kernel[END_REF], [START_REF] Adilkhanov | On numerical study of the discrete spectrum of a two-dimensional Schrodinger operator with soliton potential[END_REF], and [START_REF] Taimanov | Faddeev eigenfunctions for twodimensional Schrödinger operators via the Moutard transformation[END_REF], from the representation Γ ν ω ϕ = Id -

ω ν ω Φ -1 u ϕ ν ,
and from straightforward computations.

In the assumptions of this section on D and ω we have the following result.

Theorem 1 Let 0 / ∈ σ D (H) ∪ σ D ( H).
Then the following formuals hold:

Λ u,ω -1 = -Λ -1 u,ω , Λ -1 u,ω -1 = -Λ u,ω
. Theorem 1 follows from relations ( 13) and ( 14) and from Lemmas 1 and 2 which describe the domains of definition of the operators Λ u,ω , Λ -1 u,ω , Λ u,ω -1 , and Λ -1 u,ω -1 under the assumption of Theorem 1. Propositions 1 and 2 and Theorem 1 provide ways to find the operators Φ u and Φ -1 u from Φ u and Φ -1 u and from the restrictions of ω and ω ν onto ∂D. In this case it is required only the invertibility of operators acting on functions on ∂D. In this sense these methods are essentially simpler than the direct reconstruction of Φ u and Φ -1 u from u, when it is necessary to invert an operator acting on functions defined on the whole domain D. This effect becomes quite obvious for the Darboux transformation that we are demonstrating in the next section.

The action of the Darboux transformation on

Poincare-Steklov operators

Let us consider the Schrödinger equation

H 1 ψ = - d 2 dx 2 + u ψ = Eψ, E = µ 2 (18)
on the interval D =]a, b[⊂ R and the Schrödinger equation

H 1 ψ = - d 2 dx 2 + u ψ = E ψ, (19)
which is obtained from ( 18) by using the Darboux transformation determined by the solution ω 1 of the equation

H 1 ω 1 = κ 2 ω 1 . (20)
We assume that u(x) is a regular function of the closed interval [a, b] = D ∪ ∂D.

For an equation of type [START_REF] Novikov | The Moutard transformation and two-dimensional multi-point delta-type potentials[END_REF] we consider the operators

Q u and Q -1 u such that Q u : ψ| ∂D -→ dψ dx | ∂D , (21)
Q -1 u : dψ dx | ∂D -→ ψ| ∂D , (22)
i.e., the DN and ND operators of the form ( 9) and [START_REF] Taimanov | On the Moutard transformation and its applications to spectral theory and soliton equations[END_REF], where, for simplicity, the derivation along the out normal ν is replaced by the derivation in x.

Let σ D (H 1 ) and σ N (H 1 ) define the spectra of the operators defined by H 1 and the Dirichlet and Neumann conditions, respectively.

For solutions ψ of (18) we also consider the following boundary condition on ∂D:

Γ x ω1 ψ = ω 1 ψ ω 1 x = ψ x - ω 1,x ω 1 ψ,
where ω 1 is the given solution of (20) which determines the Darboux transformation.

We assume that ω 1 has no zeroes on ∂D.

The relations ( 6) and ( 7) on ∂D can be rewritten as follows:

1 κ -µ Γ x ω -1 1 ψ = ψ, (23)
ψ = 1 κ + µ Γ x ω1 ψ. (24)
Lemma 3 Assuming that 0 / ∈ σ D ( H 1 ) and κ = ±µ, we have

Ker Γ x ω1 = 0, Coker Γ x ω1 = 0,
where Γ x ω1 is considered as an operator on solutions of (18).

Remark 2. The condition 0 / ∈ σ D ( H 1 ) is understood in the sense similar to Remark 1.

Lemma 3 follows from ( 7), [START_REF] Grinevich | Generalized analytic functions, Moutard-type transforms, and holomorphic maps[END_REF], and the two-dimensionality of the space of functions on ∂D and the spaces of solutions of ( 18) and [START_REF] Taimanov | The Moutard transformation of two-dimensional Dirac operators and the Mobius geometry[END_REF].

Theorem 2 Assuming that 0 / ∈ σ N (H 1 ) ∪ σ D ( H 1 ) and κ = ±µ, the following formula is valid:

Q u = - ω 1,x ω 1 + (κ 2 -µ 2 )Q -1 u Id - ω 1,x ω 1 Q -1 u -1 , (25)
where ω1,x ω1 is the operator of multiplication by the corresponding function on ∂D.

The formula [START_REF] Grinevich | Moutard transform approach to generalized analytic functions with contour poles[END_REF] explicitly specifies the transformation of the DN and ND operators due to the Darboux transformation. Such formulas can be used, in particular, for testing the algorithms for reconstructing the potential from measurements at the boundary.

Proof of Theorem 2. We have the representation

Γ x ω1 ψ = Id - ω 1,x ω 1 Q -1 u ψ x . (26)
It follows from Lemma 3 and ( 26) that the operator Id-ω1,x ω1 Q -1 u is invertible. Further, using [START_REF] Delong Yu | Darboux transformation for the modified Veselov-Novikov equation[END_REF] and [START_REF] Taimanov | A fast decaying solution to the modified Novikov-Veselov equation with a one-point singularity[END_REF], we rewrite the relations ( 23) and [START_REF] Grinevich | Generalized analytic functions, Moutard-type transforms, and holomorphic maps[END_REF] in the form

Q u + ω 1,x ω 1 ψ = (κ -µ)ψ, (27)
ψ = 1 κ + µ Id - ω 1,x ω 1 Q -1 u ψ x . (28)
With the help of ( 21) and ( 28), we get

ψ x = (κ + µ) Id - ω 1,x ω 1 Q -1 u -1 ψ, ψ = (κ + µ)Q -1 u Id - ω 1,x ω 1 Q -1 u -1 ψ. (29)
Formula (25) follows from ( 27) and (29) after equating the expressions for ψ.

Q 2/x 2 = 1 µ -µ 2 + 1 x 1 x -Q 0 µ Q 0 - 1 x -1 = - 1 x -µ 2 Q 0 - 1 x -1
, which is a particular case of (25).

  The Moutard transformation constructs from solutions of a second order equation of the type∂ v ∂ w ϕ + uϕ = 0solutions ϑ of another second order equation of the same type

  1. if E is not an eigenvalue of the problem[START_REF] Matveev | Darboux Transformations and Solitons[END_REF] with the Dirichlet condition ψ| ∂D = 0, then the boundary data ψ| ∂D determine the solution of (8) uniquely and we define the DN (Dirichlet-to-Neumann) operator, which takes the values of ψ on the boundary to the values of the derivatives of ψ along the exterior normal ν to the boundary (the data of the Neumann problem): DN : ψ| ∂D -→ ∂ψ ∂ν | ∂D ; (9) 2. if E is not an eigenvalue of the problem (8) with the Neumann condition ∂ψ ∂ν | ∂D = 0, then the ND (Neumann-to-Dirichlet) operator is defined: N D : ∂ψ ∂ν | ∂D -→ ψ| ∂D ; (10) 3. the above operators are special cases of the RR (Robin-to-Robin) operator, which in the general case relates mixed boundary conditions (the Robin conditions). If E is not an eigenvalue of the problem (8) with the boundary condition cos α ψ -sin α ∂ψ ∂ν | ∂D = 0, then the RR operator maps the boundary data Γ α ψ = cos α ψ -sin α ∂ψ ∂ν | ∂D to the boundary data Γ α-π/2 ψ:

2 .

 2 Theorem 2 is proved. Example. Let us consider the operator H = -d 2 dx 2 on the interval ]a, b[ with 0 < a < b and its Darboux transformation determined by ω 1 = x. We haveu = 0, u = 2 xLet us take µ = κ, where κ = 0.A general solution of the equationHψ = -ψ = µ 2 ψ (30)has the form ψ = α cos µx + β sin µx. the DN operator for the problem (30) takes the formQ 0 = BA -1 = µ sin µ(b -a) -cos µ(b -a) 1 -1 cos µ(b -a) .It is clear that sin µ(b -a) = 0 if and only if µ 2 does not belong to the spectrum of the Dirichlet problem for H = -d 2 dx 2 . The Darboux transformation for solutions takes the form

The work was supported by the French-Russian grant (RFBR 17-51-150001 NCNI a/PRC 1545 CNRS/RFBR) and done during the visit of the second author (I.A.T.) to Centre de