Jonathan Cailliez
email: cailliez@isir.upmc.fr

Mokrane Boudaoud
email: mokrane.boudaoud@isir.upmc.fr

Shuai Liang
email: liang@isir.upmc.fr

Stéphane Régnier
email: stephane.regnier@upmc.fr

Robust Hybrid Control of an Atomic Force Microscope for the Characterization of Interaction Force Regions at the Nanoscale

Keywords: Atomic force microscope (AFM), hybrid control, nanorobotics, piezoelectric actuator

This article deals with a novel hybrid control of an atomic force microscope (AFM) for the robust characterization of interaction tip/sample force regions. The hybrid structure is composed of several position and force controllers and a specific switch compensation structure. A single selected controller is online at a time, and a robust bumpless switch between different control laws is designed. The objectives are the robustness of each controller and switch compensator with respect to uncertain parameters of the AFM. The method uses discrete points in the range of uncertainties. For each operating point, the hybrid controller is designed by an eigenstructure assignment. The method is generalized by a multimodel approach considering only uncertainties defined by a worst-case analysis, thus reducing the conservatism. The order of the controller is related to the number of observers selected by the user. This offers the possibility to design a low-order controller depending on the control specifications. The experimental results demonstrate the effectiveness of the hybrid controller for a fully automated landing procedure of the AFM tip on a sample surface and for force distance curve cycle characterization. Thanks to the robustness of the control method, landing and cycles are reproducible despite actuator uncertainties, friction variation due to aging, and instrumental issues such as the real-time data acquisition.

I. INTRODUCTION

T HE atomic force microscope (AFM) is a key tool to explore, image, and manipulate a variety of samples at the nanometer scale [START_REF] Binnig | Atomic force microscope[END_REF]. Almost all AFMs require accurate feedback control schemes to achieve the high-quality and speed imaging performances [START_REF] Abramovitch | A tutorial on the mechanisms, dynamics, and control of atomic force microscopes[END_REF], [START_REF] Yong | Invited review article: High-speed flexure-guided nanopositioning: Mechanical design and control issues[END_REF]. Control methods can be used to either land precisely the AFM cantilever on the sample surface or for the scanning. The latter issue is the key technology for the high-resolution and reliable imaging [START_REF] Fleming | A new method for robust damping and tracking control of scanning probe microscope positioning stages[END_REF], [START_REF] Ragazzon | Lyapunov estimator for high-speed demodulation in dynamic mode atomic force microscopy[END_REF]. The former one greatly reduces the AFM tip or sample damage but relies mainly on the operator experience and semiautomated methods [START_REF] Belikov | Automatic probe landing in atomic force microscopy resonance modes[END_REF].

This article will mainly focus on the Z control of an AFM tip for a safe and fast landing procedure. The literature shows that landing the tip of an AFM or more generally a micrometer-sized probe tool on a sample is often done manually using a haptic interface [START_REF] Bolopion | Haptic teleoperation for 3-D microassembly of spherical objects[END_REF] or a remote control [START_REF] Abrahamians | A nanorobotic system for in situ stiffness measurements on membranes[END_REF]- [START_REF] Ru | Automated four-point probe measurement of nanowires inside a scanning electron microscope[END_REF] or using some homemade automated toolboxes but without a specific dynamic contact force control [START_REF] Mick | Combined nanorobotic AFM/SEM system as novel toolbox for automated hybrid analysis and manipulation of nanoscale objects[END_REF]. For automated procedures, a dual stage composed of a coarse actuator and a fine actuator mounted in serial can be used [START_REF] Belikov | Automatic probe landing in atomic force microscopy resonance modes[END_REF], [START_REF] Fleming | Dual-stage vertical feedback for high-speed scanning probe microscopy[END_REF], [START_REF] Belikov | Digital q-control and automatic probe landing in amplitude modulation phase imaging AFM mode[END_REF]. In [START_REF] Belikov | Automatic probe landing in atomic force microscopy resonance modes[END_REF], the work deals with the definition of the most appropriate signal to use for a reliable contact detection depending on the operating mode of the AFM. For instance, in amplitude modulation AFM, since the phase velocity is proportional to the force gradient near the contact, it provides an accurate information about the contact. An iterative method for a safe landing of the AFM tip is proposed in [START_REF] Belikov | Digital q-control and automatic probe landing in amplitude modulation phase imaging AFM mode[END_REF]. First, a fine stage holding the AFM cantilever is driven with a closed-loop force controller. If the fine actuator reaches its maximal extension, the force controller is disconnected, and the coarse actuator is moved forward with an amplitude lower than the maximum extension of the fine actuator. Hence, the tip-sample contact can only happen during the motion of the fine positioner which ensures a secure contact detection. The operation is repeated until the contact is detected. This procedure can take a very long time depending on the initial tip-sample distance and the extension range of the fine actuator. A similar method has been proposed in the previous work [START_REF] Boudaoud | A voltage/frequency modeling for a multi-DOFs serial nanorobotic system based on piezoelectric inertial actuators[END_REF] using stick-slip type actuators. A faster method is proposed in [START_REF] Malovichko | Development and application of an AFM probe soft approach method[END_REF]. The AFM tip is charged to generate the electrostatic forces between the tip and the sample. Such forces can be detected further from the sample than intermolecular forces. The coarse positioner is moved forward until electrostatic forces are detected, and then, the method described in [START_REF] Belikov | Digital q-control and automatic probe landing in amplitude modulation phase imaging AFM mode[END_REF] can be used when the tipsample distance is very small which reduces the detection time. In [START_REF] Fleming | Dual-stage vertical feedback for high-speed scanning probe microscopy[END_REF], a force controller is used, and its output is filtered by a low-pass filter to drive the coarse actuator and a highpass filter to drive the fine actuator. This method only allows to control the tip-sample force and not the tip position, thus complicating the characterization of interaction force regions. Most of the methods reported in the literature do not discuss the switching dynamic of the AFM actuator when the contact happens and the robustness issues of the closed-loop control.

In this article, AFM tip landing is studied through a new robust hybrid control methodology (Fig. 1). The hybrid controller is composed of multiple controllers and a supervisor with a set of rules to switch between controllers. The issue is that at the time of switching, the controller output is not guaranteed to be smooth, meaning that the derivative of the control signal at the time of switching is very high. This undesired behavior can lead to plant instability, output saturation, or system damage. Many structures have been proposed in the field of hybrid control to ensure bumpless switching between controllers, such as integral structure between the controllers and the plant [START_REF] Lemos | Observer dynamics in switching control: Application to neuromuscular blockade[END_REF], feedback loop to control the output of the offline controllers [START_REF] Qin | LQ bumpless transfer between two tracking controllers[END_REF], multicontroller structure [START_REF] Yamé | On bumps and reduction of switching transients in multicontroller systems[END_REF], resetting approach of the state of the offline controller at the time of the switch [START_REF] Pasamontes | A switching control strategy applied to a solar collector field[END_REF], and antiwindup scheme [START_REF] Peng | Anti-windup, bumpless, and conditioned transfer techniques for PID controllers[END_REF]. An integral action after the switch [START_REF] Lemos | Observer dynamics in switching control: Application to neuromuscular blockade[END_REF], [START_REF] Lourenco | Learning in multiple model adaptive control switch[END_REF] is easy to implement and guarantees zero signal error, but it constraints the controller design. Controller resetting [START_REF] Kinnaert | State resetting for bumpless switching in supervisory control[END_REF] consists in computing a state for the next online controller compatible with the current state of the system and its input. This strategy requires a state-space representation of each controller. Furthermore, it constraints the time between switches to be at least equal to the controlled plant response time. The feedback control is the most flexible technique. The working principle is that a compensator is designed for each controller. The goal is that the output of each offline controller follows the output of the online one. As this strategy reduces the bumpless transfer problem to a simple tracking problem, many strategies can be used such as proportional controller [START_REF] Steenis | Bumpless control for reduced THD in power factor correction circuits[END_REF], LQ controller [START_REF] Qin | LQ bumpless transfer between two tracking controllers[END_REF], and sliding mode controller [START_REF] Qi | Robust bumpless transfer design using adaptive sliding mode approach[END_REF]. The LQ strategy involves a high conservatism and requires an accurate knowledge of the plant. Bumpless transfer through a sliding mode strategy is a robust technique against the parametric variations of the plant, but its main drawback is the shattering. The proportional controller induces a tradeoff between the closed loop bandwidth and the stability margin. Some other methods can also be used to deal with robustness issues [START_REF] Lopez Martinez | High performance teleoperation by bumpless transfer of robust controllers[END_REF]. None of these methods have the property of been robust and low conservative.

The proposed methodology of this article aims to design a robust and low conservative hybrid controller. The basis of this approach has been reported in [START_REF] Döll | A robust self-scheduled missile autopilot: Design by multi-model eigenstructure assignment[END_REF] and [START_REF] Gorrec | Controller parametric robustification using observer-based formulation and multimodel design technique[END_REF] and has been studied in the case of microrobotics in the previous works [START_REF] Boudaoud | Gain scheduling control of a nonlinear electrostatic microgripper: Design by an eigenstructure assignment with an observer-based structure[END_REF], [START_REF] Boudaoud | Robust microscale grasping through a multimodel design: Synthesis and real time implementation[END_REF]. This methodology is extended here for hybrid control issues. The hybrid structure is composed of position and force controllers and a specific switch compensator. A robust bumpless switch between different control laws is designed. For the synthesis of each subcontroller, discrete points in the range of the plant uncertainties are used. For each operating point, the hybrid controller is designed by an eigenstructure assignment (ESA) and a multimodel approach to deal with uncertainties defined by a worst-case analysis.

This article is organized as follows. Section II describes the experimental setup and the AFM modeling with uncertainties consideration. The robust hybrid control issue is presented in Section III, and the description of the proposed method is detailed in Sections IV and V. The controller is validated in Section VI considering the specific case of AFM tip landing. The robust characterization of interaction force regions is described in Section VII.

II. EXPERIMENTAL SETUP AND UNCERTAIN SYSTEM MODELING

A. Experimental Setup

The AFM system is composed of three piezoelectric inertia actuators assembled in series to from a XYZ stage and an Akiyama tuning fork probe [START_REF] Akiyama | Symmetrically arranged quartz tuning fork with soft cantilever for intermittent contact mode atomic force microscopy[END_REF] mounted on the Z actuator. The Z actuator has a travel range of 12 mm in the coarse operating mode and a travel range of ∼1.5 μm in the fine operating mode. It moves the tuning fork probe in the vertical direction. A cantilever with a sharp tip is fixed on the two prongs of the tuning fork as shown in Fig. 2.

The tuning fork probe is made up of two coupled resonators (Fig. 3). The silicon cantilever can be described by the resonator (m 1 , k 1 , and c 1 ) and the tuning fork by the resonator (m 2 , k 2 , and c 2 ). The typical resonance frequency of the silicon cantilever is ∼50 kHz and that of the tuning fork before the cantilever was glued is ∼32 kHz.

In order to track the resonance frequency of the tuning fork probe, the following steps are followed. First, a frequency analysis without tip/sample contact is performed to get the natural resonance frequency of the tuning fork probe as well as the associated phase shift. Then, a phase-locked loop (PLL) is programed to keep the phase shift at a constant value and the tuning fork probe oscillations at its resonance frequency. The input voltage of the PLL voltage-controlled oscillator (VCO) is recorded as a measurement of the frequency shift. There is a linear relationship between frequency shift f and the voltage measurement V sh , i.e., f = K tf V sh , where

K tf = f max U r (1)
where f max is the frequency shift measurement span specified by the user and U r is the input voltage span of the VCO, here 10 V. When the tip-sample distance is of few nanometers, the attractive and repulsive interaction forces are detected, which cause a shift of the resonance frequency of the tuning fork probe. For force measurement or for imaging, this resonance frequency is tracked by the PLL. A feedback control of the tip-sample distance is used to keep this resonance frequency at a constant reference value. In this article, the tip-sample distance is controlled by the piezoelectric inertia Z actuator whose resonance frequency is around 6 kHz [START_REF] Boudaoud | A voltage/frequency modeling for a multi-DOFs serial nanorobotic system based on piezoelectric inertial actuators[END_REF]. Initially, the tip-sample distance can be of several millimeters. The tip can be moved toward the sample by means of mixed coarse/fine positioning strategy [START_REF] Boudaoud | A voltage/frequency modeling for a multi-DOFs serial nanorobotic system based on piezoelectric inertial actuators[END_REF]. In the final fine step positioning (i.e., just before the tip-sample contact), a secure and robust contact detection is required to avoid damaging the AFM tip. Indeed, the radius of the Akiyama cantilever tip is lower than 15 nm. This radius can be highly increased if the cantilever tip abruptly touches the sample. This will cause a loss of the AFM imaging resolution.

B. Uncertain Model G()

The dynamic state-space model G() of the AFM system is described by the following equation:

G () : Ẋ = A()X + B()U v z = C()X (2)
where X ∈ R n is the state vector, A ∈ R n×n is the state matrix, B ∈ R n×m is the input matrix, and C ∈ R p×n is the output matrix.

PARAMETERS UNCERTAINTIES OF G()

The input of the model is the driving voltage U of Z actuator. The maximum admissible value of U is 100 V. The output v z of G() is the displacement in the vertical direction of the Z actuator, on which the tuning fork probe is fixed. This displacement is measured with an integrated optical encoder sensor of 5-nm resolution.

The dynamic of the tuning fork probe is neglected because its resonance frequency is well above that of the piezoelectric inertia actuator. Therefore, the model G() of the AFM describes exclusively the dynamic of the Z inertia actuator that holds the tuning fork probe.

The uncertainty vector includes three uncertain parameters (1), [START_REF] Abramovitch | A tutorial on the mechanisms, dynamics, and control of atomic force microscopes[END_REF], and (3) describing, respectively, the uncertain fundamental resonance frequency of Z actuator, the uncertain static gain of Z actuator, and the measurement delay of the actuator displacement as shown in Table I. [START_REF] Binnig | Atomic force microscope[END_REF] is included only in the matrix A, (2) is included only in the matrix B, and (3) is included only in the matrices A and C.

1) Uncertainty (1): The working principle of piezoelectric inertia actuators is based on a friction force produced by a piezoelectric element to drive a slider. The dynamic of such actuators is related to the parameters of its internal structure (i.e., dimensions, mass, and so on) and may also have a relation with the friction parameters [START_REF] Boudaoud | A voltage/frequency modeling for a multi-DOFs serial nanorobotic system based on piezoelectric inertial actuators[END_REF]. All these parameters are highly uncertain due to the fabrication process. These actuators are also sensitive to aging due to the friction-based working principle. The first uncertain parameter is the uncertainty of the resonance frequency of Z actuator. The fundamental resonance frequency of three different actuators of the same reference (SmarAct SLC-1720) has been considered to define the range of variation of (1) as shown in Table I. The values of the resonance frequencies have been identified in [START_REF] Boudaoud | A voltage/frequency modeling for a multi-DOFs serial nanorobotic system based on piezoelectric inertial actuators[END_REF].

2) Uncertainty (2): Piezoelectric inertia actuators are nonlinear due to a hysteresis. It is the relation between the displacement of the slider and the input voltage of the piezoelectric element. In a previous work [START_REF] Oubellil | Mixed stepping/scanning mode control of stick-slip SEM-integrated nano-robotic systems[END_REF], the hysteresis of the three actuators has been modeled through a multilinear approximation and identified experimentally. It consists of an approximation of the static hysteresis by several straight lines around operating points. The slope of each straight line can be considered as a static gain of the actuator. The effect of the hysteresis will then be considered in this article as an uncertainty (2) of the static gain. Step responses of all the considered LTI models.

3) Uncertainty (3):

The third uncertain parameter (3) is the measurement delay T r induced by the optical encoder and the measurement chain, i.e., interpolation, counter, and so on. To model this delay, a Padé approximation is added to the model of ( 2) by means of the following transfer function:

H (s) = 1 -T r 2 × s 1 + T r 2 × s ( 3
)
where s is the Laplace variable.

The actuator dynamics and the static hysteresis have been modeled and identified in a previous work [START_REF] Oubellil | Mixed stepping/scanning mode control of stick-slip SEM-integrated nano-robotic systems[END_REF]. The results are shown in Table I.

C. Definition of a Finite Number of LTI Models From G()

Instead of considering all the values of each uncertainty in the range of variation (see Table I), the discrete points in the range of uncertainties are defined to build a set of linear time-invariant (LTI) models from G().

The uncertainty of the resonance frequency is taken into account by considering the dynamics of the three actuators SLC-1720. The fundamental resonance frequency of the actuators 1-3 are, respectively, 2.2, 1.5, and 5.95 kHz.

For each actuator dynamics, and therefore, for each resonance frequency, six different values of the static gain equally separated in the range [12.4-32.4 nm/V] are considered.

For each actuator dynamics and each value of the static gain, five values of the measurement delay equally separated in the range (0.05-0.25 ms) are considered.

Finally, taking into account the combination of the dynamics (i.e., resonance frequency) of each of the three actuators, the six different values of the static gain and the five different values of the measurement delay, 3 × 6 × 5 = 90 LTI models, are defined to form a set of models describing G().

Each LTI model has seven states, i.e., six states to describes three eigenmodes and one state for the Padé approximation. The step responses of all the 90 open-loop models are shown in Fig. 4.

III. HYBRID CONTROL ISSUE

The methodology for a safe contact detection in the fine positioning mode of the AFM tip is proposed through the robust hybrid control. The inputs of the hybrid control (Fig. 1) Fig. 5. Hybrid control scheme of the AFM treated here. The design of the position controllers 1 and 2 is presented in Section VI-A. The design of the force controller is presented in Section VII-B. The detailed internal structure of the force controller is shown in Fig. 20(a). An SEM image of the AFM tuning fork is presented in Fig. 2. Fig. 6. General hybrid control scheme of an uncertain system G(). The more specific control scheme treated in this article including position and force control is shown in Fig. 5.

are the position reference of the Z actuator V z ref and its realtime position (i.e., actual position) v z , the desired shift of the resonance frequency of the tuning fork fref , and its real-time value f . Its output is the control voltage U that feeds the Z actuator.

The hybrid control structure includes in addition to the position controllers, a force controller that will be described in Sections VI-A and VII-B, respectively. The control diagram is shown in Fig. 5, where

V z ref = [v z ref1 v z ref2 ] T .
Let us consider the control structure of Fig. 6, where an uncertain system G() is controlled by a set of k controllers providing control signals U 1 , U 2 , . . ., and U k . At any time, only one controller is activated by a supervisor, it is said to be online, and the others are said to be offline. The aim is that all the closed-loop systems of the hybrid structure satisfy the robustness specifications with respect to (w.r.t) uncertain parameters of . The switch between the different control signals must be achieved at any time with the small possible discontinuity of U .

The switch compensation structure must be also robust. It must be designed to track the output of the online controller with the output of the offline ones.

Control Method: Each controller and compensator of the hybrid control scheme of Fig. 6 can be designed by many robust control methods to deal with the system uncertainties. Classical robust controllers such as those based on the H ∞ with LMI constraints take into account a continuum of values in the range of variation of and are highly conservative. The multimodel approach proposed in this article uses a different approach by considering only a finite number of selected uncertainties to obtain a robust controller. This is a progressive method that is first designed on a nominal model and then adds robustness step by step based on a worst case analysis. It does not require the treatment of a continuum of models in the range of variation of to satisfy the robust performances with respect to the uncertainties but just a wellselected number of relevant worst case models. This approach has many advantages that will be discussed in Section VIII.

This multimodel method has been first developed to deal with control issues in the field of aeronautics [START_REF] Döll | A robust self-scheduled missile autopilot: Design by multi-model eigenstructure assignment[END_REF], [START_REF] Gorrec | Controller parametric robustification using observer-based formulation and multimodel design technique[END_REF]. It has then been applied with an experimental validation in the field of microsystems, and especially, for those involving nonlinear mechanical suspensions [START_REF] Boudaoud | Gain scheduling control of a nonlinear electrostatic microgripper: Design by an eigenstructure assignment with an observer-based structure[END_REF], [START_REF] Boudaoud | Robust microscale grasping through a multimodel design: Synthesis and real time implementation[END_REF]. This method has, nevertheless, not been designed to deal with hybrid control issues. Therefore, the first contribution of this article is the generalization of the method in the case of hybrid control as described in Section V. This generalization has been first presented in the conference [START_REF] Cailliez | A multi-model design for robust hybrid control of non-linear piezoelectric actuators at the micro/nano scales[END_REF], and it is generalized here, where switch conditions are not only defined by position measurement of a piezoelectric inertia actuator but also by data from a tuning fork-probe of the AFM. The application of this method for robust AFM tip landing and the characterization of interaction force regions are the second contribution of this article. It is the first time that such a low conservative control method is applied in the field of AFM whose advantages are discussed in Section VIII.

The hybrid control structure includes in addition to the position controllers, a force controller with a feedback signal from the tuning fork-based sensor as shown in Fig. 5. This generalization is of importance for a secure and robust landing of the AFM tip. A precise characterization of attractive and repulsive tip-sample interaction forces regions is thereafter proposed and validated experimentally using the proposed control method.

IV. ROBUST CONTROL BY ESA AND MULTIMODEL FORMULATION

The aim of this section is to define the basis of the multimodel control with an ESA and an observer-based structure.

Let us consider the dynamic model of the AFM G() with uncertain parameters included in a vector as described in [START_REF] Abramovitch | A tutorial on the mechanisms, dynamics, and control of atomic force microscopes[END_REF]. The objective is to assign a set of eigenstructures (i.e., eigenvalues and eigenvectors) for the closed-loop system. The variations of must be taken into account during the control design to satisfy the robustness specifications.

In the sequel, the control design will be described for a frozen value of , and then, the method will be generalized considering the variation of the uncertain parameters.

A. Closed-Loop Control for a Frozen Value of = 1

The classical ESA methodology is based on the following Lemma [START_REF] Moore | Eigenvector selection in the linear regulator problem: Combining modal and optimal control[END_REF]. Let us consider the triplet (λ i , ν i , ω i ) with λ i an eigenvalue, ν i an eigenvector, and ω i an input direction, satisfying

A( 1 ) -λ i I -B( 1 ) ν i ω i = 0. ( 4
)
This triplet is assigned in closed loop by the constant gain K if and only if

K C( 1 )ν i = ω i . ( 5
)
With the classical ESA, it is not possible to assign more eigenstructures than the number of measurable outputs of the plant (2), and this is the degree of freedom constraint [START_REF] Döll | A robust self-scheduled missile autopilot: Design by multi-model eigenstructure assignment[END_REF]. Moreover, the assignment does not guarantee the closed-loop robustness against the variations of [START_REF] Boudaoud | Gain scheduling control of a nonlinear electrostatic microgripper: Design by an eigenstructure assignment with an observer-based structure[END_REF], [START_REF] Boudaoud | Robust microscale grasping through a multimodel design: Synthesis and real time implementation[END_REF].

The robustness issue can be solved by a multimodel design as explained in Section IV-B. To deal with the degree of freedom, the dedicated observers can be designed. One possible solution for the observation is to estimate a linear combination of the states X through the following model:

⎧ ⎨ ⎩ dz i dt = π i z i -t π i v z + u π i B( 1 )U z i = u π i X (6)
where z i is the output of the observer. This model is based on a modal approach for observer design [START_REF] Magni | An observer based multimodel control design approach[END_REF]. It shows that a linear combination of the states u π i X can be estimated by an observer increasing the number of outputs. It allows assigning as many more eigenstructures as the number of observations. The observer is obtained through the vector u π i ∈ C n , the vector t π i ∈ C p , and a complex number π i ∈ C satisfying

u π i A( 1 ) + t π i C( 1 ) = π i u π i . ( 7
)
where n = 6 and p = 1. π i is the pole of the observer that defines its dynamics.

The pole π i must be different from the poles of G( 1 ).

Let q be the number of eigenstructures that have to be assigned and p be the number of nonredundant outputs of the plant (2). If q > p, n 0 = qp observers have to be designed.

The following notations will be used to define a set of n 0 observers:

U o = ⎡ ⎢ ⎣ u π 1 . . . u π no ⎤ ⎥ ⎦ T o = ⎡ ⎢ ⎣ t π 1 . . . t π no ⎤ ⎥ ⎦ o = ⎡ ⎢ ⎣ π 1 . . . 0 . . . . . . . . . 0 . . . π n o ⎤ ⎥ ⎦ U o ∈ C n o ×n , T o ∈ C n o × p , and o ∈ C n o ×n o . ( 8
)
Now, it is possible to define an augmented system G( 1 ) with a set of n o observers to have additional outputs Z with

Z = ⎡ ⎢ ⎢ ⎢ ⎣ z 1 z 2 . . . z n o ⎤ ⎥ ⎥ ⎥ ⎦ . ( 9
)
Using ( 2) and ( 6) with the notation of ( 8) and ( 9), the statespace representation of the augmented model is given as follows:

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ Ẋ Ż = A( 1 ) 0 n×n o -T o C( 1 ) o X Z + B( 1 ) U o B( 1 ) U v z Z = C( 1 ) U o X. ( 10
)
Thanks to the separation principle, the control problem consists, now, of finding the static gain K to assign the eigenstructures to the system of equations

⎧ ⎪ ⎨ ⎪ ⎩ Ẋ = A( 1 )X + B( 1 )U v z = C( 1 )X Z = U o X. ( 11
)
To deal with precision requirements, i.e., zero closed-loop static error, an integral action can be added. In this case, the number of the states of the model ( 11) is increased by one.

The control law becomes

U = K i (v z ref -v z )dt -K y v z -K z Z (12)
where v z ref is the closed-loop input reference, and the parameters K i , K y , and K z are the gains of the controller such as K = K z K y K i . It is not needed to add an observer to the system to deal with the integral action.

B. Robust Control by a Multimodel Design

The discrete values of in the set min , max are considered. This allows the definition of g LTI models G( 1 ), G( 2 ), and, …, G( g ) from (2). If needed, the number of values of can be increased around particular points.

In this case, r i ESAs are necessary for each model G( i ). Therefore, if g LTI models are treated, g i=1 r i constraints induced by these assignments must satisfy the following set of equations [see [START_REF] Ragazzon | Lyapunov estimator for high-speed demodulation in dynamic mode atomic force microscopy[END_REF]]:

K C int ( 1 )ν 1, 1 = ω 1, 1 . . . K C int ( 1 )ν r 1 , 1 = ω r 1 , 1 . . . K C int ( g )ν 1, g = ω 1, g . . . K C int ( g )ν r g , g = ω r g , g . ( 13
)
C int defines the output matrix of the state-space model (2) with an integrator.

To design a set of observers for a model G( i ), ( 6) and ( 7) can be used taking into account the uncertain vector i . Moreover, by refereeing to [START_REF] Magni | An observer based multimodel control design approach[END_REF], each assignment is related to a vector γ j, i , where

γ j, i = (λ j, i I n o ×n o -o ) -1 ×(U o B( i )ω j, i -T o C int ( i )ν j, i ) . ( 14
)
I n o ×n o is the identity matrix. For each i , 1 < j < r i . γ j, i is fundamental for the computation of the gain K . The closed-loop system fits the constraints (13) if and only if [START_REF] Magni | An observer based multimodel control design approach[END_REF]

K = ⎡ ⎢ ⎣ ω 1, 1 . . . ω r g , g ⎤ ⎥ ⎦ T γ 1, 1 • • • γ r g , g C int ( 1 )ν 1, 1 • • • C int ( g )ν r g , g -1 (15)
where

K = [K y K z K i ].
If the models G( i ) that have been used for the control design are worst case models, the computed gain K can lead to a robust controller with respect to . The block diagram of the output feedback controller is shown in Fig. 7.

The robust control design can be performed iteratively following three main steps [START_REF] Döll | A robust self-scheduled missile autopilot: Design by multi-model eigenstructure assignment[END_REF]: initialization, analysis, and robustification (Fig. 8).

1) Initialization: Definition of g LTI models G( i ). The selection of any model G( i ) in the defined set and application of the assignment constraints of (13) in accordance with control specifications. If needed, a set of observers can be designed. Computation of the gain K using ( 14) and (15). 2) Analysis: Simulation of the pole map of the g closedloop models with the observers and the computed gain K . If every model fits the control specifications, then finish. Else define a worst case model G( i+1 ) and go to the robustification step. The worst case model is the one whose one of the performances is the farthest from the specification. In the sequel, the worst case model will be the one whose closed-loop pole has the highest real part value. 3) Robustification: Definition of new assignment constraints for the current worst case model and addition of these new constraints to the previous ones to increase the number of equations in [START_REF] Belikov | Digital q-control and automatic probe landing in amplitude modulation phase imaging AFM mode[END_REF]. If needed, new observers can be computed for the current value of . Computation of a new gain K using ( 14) and ( 15) considering the assignment constraints of all the treated models.

If needed, quasi-similar assignments can be eliminated.

Those assignments tend to lead to unrealistic gains. Then go to the analysis step.

V. NEW FORMULATION FOR ROBUST HYBRID CONTROL

The goal is to define a hybrid structure so that each subcontroller and each compensator can be designed by a robust ESA. This way, each controller and compensator as well as the switch between different control laws will be robust against the variations of , and its synthesis will have a low conservatism.

The compensator is a controller that controls the output of the offline controller to follow the output of the online one. As such at the switching time, a bumpless transition of the plant's control signal is satisfied. For instance, in Fig. 9, the compensator C p1 is active when the controller 1 is online. It must ensure a bumpless transfer from U 1 to U 2 at the time of the switch. Therefore, it is used to keep U 2 stable when U 1 is controlling the plant so that when switching between U 1 and U 2 happens, there is as little discontinuity of the control signal driving the plant as possible.

Let us consider the structure of Fig. 6, providing that the controllers are independent. Each controller can be assigned with an independent compensator. Each controller can be designed independently with different control specifications. Each compensator controls the output signal of its associated offline controller to track the output of the online controller. In any case, only one controller is active.

Without the loss of generality, the structure of Fig. 6 will be treated for k = 2. The hybrid scheme will be composed of two dynamic controllers, namely, controllers 1 and 2 and two compensators C p1 and C p2 as shown in Fig. 9. The supervisor is a high-level control that is based on predefined conditions, i.e., the conditions for which the control law U switches from U 1 to U 2 and vice versa.

The internal structure of the controllers 1 and 2 is the same as in Fig. 7 except that the control input is not v z ref , but the signals v z ref1 + U c2 for the controller 1 and v z ref2 + U c1 for controller 2.

The observers used to design the controller 1 will be designated as "set of observers 1" with

U o 1 ∈ C n o 1 ×n , T o 1 ∈ C n o 1 × p , and o 1 ∈ C n o 1 ×n o 1 .
Those of controller 2 will be designated as "set of observers 2" with U o 2 ∈ C n o 2 ×n ,

T o 2 ∈ C n o 2 × p , and o 2 ∈ C n o 2 ×n o 2 .
The static control gain of controller 1 and controller 2 will be expressed, respectively, as

K 1 = K z 1 K y 1 K i 1 and K 2 = K z 2 K y 2 K i 2 .
In the sequel, the proposed structure of the compensators will be explained considering C p1 . The same principle is used to compute C p2 .

The compensator C p1 is active when the controller 1 is online. It must ensure a bumpless transfer from U 1 to U 2 at the time of the switch. As shown in Fig. 9, the inputs of C p1 are the signals U and U 2 , and its output is the signal U c1 . The latter is used to control the dynamic of U 2 so that it tracks the active signal U 1 . The output v z and the input reference v z ref2 are considered in the proposed approach as disturbances. Then, the tracking issue can be solved by a robust ESA, providing that a state-space representation of the dynamic system of input U c1 and output U 2 is defined.

Let us consider the following state vector:

X c2 = Z 2 v z 2 int . ( 16
)
With Z 2 , the output vector of the set of observers 2 and

v z 2 int = (v z ref 2 -v z + U c1 )dt.
Let us now take into account the g models of G() used for the synthesis of controller 2. For each model, r i ESAs have been defined. In this case, n 0 2 = ( g i=1 r i )p, where n 0 2 is the number of observers in the set of observers 2, and p is the number of nonredundant outputs of the plant G().

For a model of uncertainty i , the output of the observer is Z 2 i , and its parameters are (U o 2 i , T o 2 i , and o 2 i ). Therefore, the output and the parameters of the set of observers 2 can be defined as follows:

Z 2 = Z 2 1 Z 2 2 • • • Z 2 g U o 2 = U T o 2 1 U T o 2 2 • • • U T o 2g T T o 2 = T T o 2 1 T T o 2 2 • • • T T o 2g T o 2 = diag(o 2 1 o 2 2 • • • o 2g ). ( 17
)
Using ( 6), for each value of i , the derivative of Z 2 i is

Ż2 i = U o 2 i B( i )U 2 -T o 2 i v z + o 2 i Z 2 i . ( 18
)
U 2 can be expressed as

U 2 = K z 2 K i 2 X c2 + K y 2 v z . ( 19
)
The derivative of v z 2 int can be calculated trivially

vz 2 int = v z ref2 -v z + U c1 . ( 20
)
By substituting the result of ( 19) in [START_REF] Yamé | On bumps and reduction of switching transients in multicontroller systems[END_REF], one gets

Ż2 i = U o 2 i B( i ) K z 2 K i 2 X c2 +(U o 2 i B( i )K y 2 -T o 2 i )v z + o 2 i Z 2 i . (21)
Now, it is possible to combine the results of ( 20) and [START_REF] Lourenco | Learning in multiple model adaptive control switch[END_REF]. One is able to get the derivative of X c2 , thus getting the desired state-space representation

⎧ ⎪ ⎨ ⎪ ⎩ Ẋc2 = A c2 X c2 + B c2 U c1 + B c2 v z + B c2 v z ref 2 U 2 = C c2 X c2 + K y 2 v z (22)
with

A c2 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ U o 2 1 B( 1 )K z 2 U o 2 1 B( 1 )K i 2 U o 2 2 B( 2 )K z 2 U o 2 2 B( 2 )K i 2 . . . . . . U o 2g B( g )K z 2 U o 2g B( g )K i 2 0 1×n o 2 0 1×1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ + M o M o = o 2 0 n o 2 ×1 0 1×n o 2 0 1×1 B c2 = 0 n o 2 ×1 1 , B c2 = ⎡ ⎢ ⎢ ⎢ ⎣ U o 2 1 B( 1 )K y 2 -T o 2 1 . . . U o 2g B( g )K y 2 -T o 2g -1 ⎤ ⎥ ⎥ ⎥ ⎦ B c2 = ⎡ ⎢ ⎢ ⎢ ⎣ 0 . . . 0 1 ⎤ ⎥ ⎥ ⎥ ⎦ C c2 = K z 2 K i 2 .
Using this representation, it is possible to seize the problem caused by this type of controller for switching. A c2 has an eigenvalue equal to zero due to the integrator, so if the references for both controllers 1 and 2, i.e., v z ref1 and v z ref2 are not the same, U 2 will diverge if the compensator C p1 is not added to the control scheme. This remark is also valid for U 1 and the compensator C p2 .

Let us recall that, here, v z and v z ref2 are considered as disturbances in the model. As such, using the separation principle, they are considered for the compensator design as equal to zero. The state-space representation of the tracking problem becomes

Ẋc2 = A c2 X c2 + B c2 U c1 U 2 = K z 2 K i 2 X c2 . ( 23
)
This model contains n 0 2 + 1 states.

VI. ROBUST HYBRID CONTROL FOR AFM TIP LANDING

The goal of this section is to design the hybrid control method including controller 1, controller 2, compensator 1, and compensator 2 of Fig. 5. The force control part will be treated in Section VII.

The controllers 1 and 2 will be designed with the procedure described in Section IV-B based on the set of LTI models derived from Section II-B. The uncertainty includes the three uncertainties of Table I. Let us recall that (1) is included only in the matrix A, (2) is included only in the matrix B, and (3) is included only in the matrices A and C of the state-space model (2) of the AFM.

Each compensator will be designed with an ESA using the model of [START_REF] Steenis | Bumpless control for reduced THD in power factor correction circuits[END_REF].

Two protocols I and II will be defined to set the switching rules of the supervisor of Fig. 5.

A. Hybrid Control Design

Based on the LTI models G( i ) 1 < i < 90, the controller of Fig. 5 will be designed. The controllers 1 and 2 and the force controller will have different closed-loop specifications.

The position control specifications are listed as follows:

1) The controller 1 must satisfy in closed loop no static error, an overshoot less than 5%, and a settling time lower than 150 ms. 2) The controller 2 must satisfy in closed loop no static error, an overshoot less than 5%, and a settling time lower than 6 ms, i.e., 25 times faster and the closedloop system with the controller 1. Indeed, as shown in Fig. 16, the controller 1 will drive the system when the AFM tip is not in the region of interaction forces. A slow response time is preferable to avoid approaching the force regions too fast. However, controller 2 will drive the system when the AFM tip is within the region of interaction forces. A higher closed-loop bandwidth is needed for AFM interaction force characterization.

1) Design of Controller 1:

The controller is designed using the methodology of Section IV-B. The nominal model G( 1) is chosen to be that of the actuator 3 ( 1 (1) = 5.95 kHz) with a static gain equal to 1 (2) = 12.4 nm/Volts and a sensor delay of 1 (3) = 0.25 ms. For this nominal model, four eigenstructures are assigned in closed loop with the poles -20, -300, -2.2 × 10 2 , and -5 × 10 3 . The model G( 1) has one measurable output only. Therefore, two observers are designed. Using [START_REF] Malovichko | Development and application of an AFM probe soft approach method[END_REF], the gain K 1 is computed taking into account four assignment constraints. The worst case analysis is performed for the 90 LTI closed-loop models with the computed gain. The pole map of the closed-loop models is shown in Fig. 10. Almost, all the closed-loop models are unstable except for three of them, one is the closed-loop nominal model (i.e., that used for the first synthesis) and the two others do not fit the specifications. An experimental test has been performed with the controller, and the results are shown in Fig. 11. As soon as the reference goes to 1 μm, the controller diverges. This leads to a saturation of the amplifier, whose output changes abruptly from 0 to 100 V. A slip of the actuator slider occurs.

This result shows that a robustification step is necessary. The worst case model is chosen to be the model of actuator 3 ( 1 (1) = 5.95 kHz) with a static gain equal to 1 (2) = 32.4 nm/V and a sensor delay of 1 (3) = 0.25 ms. For this model, two assignments are added with the poles at -30 and -5.2 × 10 2 . The new gain K 1 is computed again using [START_REF] Malovichko | Development and application of an AFM probe soft approach method[END_REF]. Now, six assignment constraints are considered, and they include the four assignments of the previous nominal model and the two assignments of the worst case model. The step responses of all the closed-loop models with the new computed gain (i.e., after the robustification) are shown in Fig. 12. All the closed-loop models fit the specifications. The procedure is stopped, and the values for both the static gain and the observer's parameters of controller 1 are kept. This controller is robust with respect to the variations of .

2) Design of Controller 2:

The procedure is the same as for controller 1 except for the specifications. Two iterations have been also necessary to satisfy the control specifications in a robust way. The step responses of the closed-loop LTI models with the robust controller 2 are also shown in Fig. 12.

3) Design of the Compensators C p1 and C p2 and Simulation of the Hybrid Position Controller: The model of ( 23) is used for the design of the compensator. For each compensator, two ESAs are performed with the poles -5 × 10 4 and -4 × 10 4 . The robustness against the variations of is not needed here as none of the state-space matrices of the model ( 23) are dependent on the variation of . They are frozen matrices. However, if the controllers 1 and 2 were system dependent, uncertain, or time varying, the robustness would have been possible with the proposed methodology.

The robust hybrid controller including the controllers 1 and 2 and the compensators C p1 and C p2 has been first tested by simulations for each of the 90 LTI models G( i ). For each model, the supervisor is defined with the following protocol.

Protocol I:

1) Controller 1 is first online with a step input reference of 1 μm. 2) Once the Z actuator reaches 0.8 μm, the controller 2 comes online with a step input reference of 0.7 μm plus a sine with increasing amplitude. 3) Once the actuator reaches 0.85 μm, the controller 1 is back online with a step input reference of 0 μm.

This protocol is of importance for the robust AFM tip landing and for the characterization of attractive and repulsive tip-sample forces as explained in detail in Section VI-B2.

The simulation results of the closed-loop step responses of the 90 LTI models are shown in Fig. 13. The Z actuator follows well the Protocol I in a robust way. The closedloop specifications of the controllers 1 and 2 are satisfied. Closed-loop displacements of all LTI models with the hybrid controller. The switch from the controller 1 to controller 2 and vice versa is achieved with no overshoot or no vibrations. The vibrations observed around 0.07 s are due to the design constraints of controller 2, and their amplitude is below 5 nm.

B. Experiments 1) Experimental Implementation of the Hybrid Controller and Analysis:

The experimental setup includes a processor board with analog-to-digital converters (ADCs) and digital-toanalog converters (DACs), a counter, and an interpolator for the optical encoders of the piezoelectric actuators, a real-time interface, and the three piezoelectric inertia actuators of the same reference (SmarAct SLC-1720). The hybrid controller has been designed using MATLAB/Simulink software and has been implemented, using the Simulink model, to the processor board. First, the controllers 1 and 2 have been experimentally tested on the three actuators to validate their robustness with respect to the dynamic variations of the friction and the resonance frequency. Different input references have been applied, namely 100 nm, 500 nm, 1 μm, and 1.5 μm. This is done to validate the robustness of the controllers with respect to the hysteresis. The measurement delay is inherent to the acquisition chain, i.e., interpolation, counter, and so on. The experimental closed-loop responses normalized to unity (i.e., divided by the input reference) using the controllers 1 and 2 are shown in Fig. 14(a) and (b), respectively. The curves with noise are those for a step reference of 100 nm. These results validate experimentally the robustness of the controllers 1 and 2.

In the third experiment, the protocol I has been followed. The experimental results are shown in Fig. 15. They show a bumpless switch before the steady state of the closed-loop systems with the controller 1 (i.e., switch from U 1 to U 2 ) and a bumpless switch from U 2 to U 1 .

The results of Figs. 13 and 15 show that the hybrid control method can be efficiently applied for a large batch of AFM systems despite: 1) parameters variations due to the fabrication process; 2) friction parameters changing due to aging; 3) hysteresis; and 4) measurement delay.

2) Protocol for Robust AFM Tip Landing: Let us consider Fig. 16 to illustrate the protocol of robust AFM tip landing. This figure is that of Fig. 15 with an inversion of the coordinate axis "Z actuator position (μm)" to highlight the AFM tipsample distance with an origin to zero distance. The protocol for robust AFM tip landing is, therefore, defined as follows.

Protocol II:

1) The controller 1 is first online with a step input reference of 1 μm. 2) When the tuning fork probe detects an attractive force, the controller 2 is switched ON with a reference equal to the position of the actuator at the switch time minus 0.1 μm plus a sine with an increasing amplitude. An attractive force is a force whose effect is to pull the AFM cantilever toward the sample, inducing a negative frequency shift of the tuning fork. Such forces can be electrostatic or van der Waals forces for instance. The supervisor informs the hybrid controller about the contact when a shift of the resonance frequency f of the tuning fork is measured. The sine with an increasing amplitude reference allows a characterization of attractive and repulsive force regions in a smooth way. 3) When f reaches a threshold value corresponding to a repulsive force due to a real tip-sample contact, the controller 1 is back online with a step input reference of 0 μm. As shown in Fig. 17, the limit between the free space region and the periodic contact region is at the inflection point that occurs at -0.8 Hz. The condition considered here is for the threshold value f = -0.4 Hz. This protocol will be expanded with more constraints in Section VII to consider also the force control.

The efficiency of the switch from the controller 1 to controller 2 when the condition is provided by the tuning fork has been tested experimentally. The results are shown in Fig. 18. The dashed curves are those of the controlled three actuator positions when the switch condition is that when the amplitude displacement is equal to 0.8 μm. The continuous curve is that of the displacement of the Z actuator within the AFM when the switch condition is provided by f . For visual comparison Fig. 18. Experimental closed-loop displacements of the three actuators at the time of the switch from the controller 1 to controller 2. The continuous curve is that of the displacement of the Z actuator within the AFM when the switch condition is provided by the contact detection, thanks to the measurement of f . reasons, the continuous curve is shifted in X -and Y -axes so that the maximum of the curve is equal to 0.8 μm and the switch time is close to that of the other curves. The experimental results show that even in the presence of forces exerted on the AFM tip, the switch is achieved in a bumpless way.

VII. ROBUST CHARACTERIZATION OF INTERACTION FORCE REGIONS AND CONSTANT FREQUENCY-SHIFT CONTROL

A. Force Regions Characterization With the Controller 2

The characterization starts when the AFM tip is within the region of interaction forces. For that purpose, the first steps of protocol II are applied, i.e., the controller 1 is first online with a step input reference of 1 μm, and when the tuning fork-probe detects an attractive force, controller 2 is switched ON with a reference equal to the position of the actuator at the switch time minus 0.1 μm. The state of the system at that moment is the initial conditions for the force characterization. Now, only the controller 2 is active. A sinusoidal input position reference with a fixed frequency and amplitude is applied. The AFM tip is in a noncontact mode, and it is oscillating around its initial state. During the motion of the Z actuator, the frequency shift of the tuning fork is measured. The contact is considered when the frequency shift is at -0.4 Hz. The frequency shift/Z actuator position is extracted experimentally for two different angular frequencies of the sinusoidal input reference, namely, 0.5 and 5 rad/s, and for three different amplitudes, namely, 1 μm, 200 nm, and 150 nm.

Before obtaining exploitable results, the slow drift of the frequency shift has been managed by a posttreatment of the signal. This treatment consists of a statistical analysis of the signal and a removal of the linear tendency of the time series. The results after the treatment for the two angular frequencies are shown in Fig. 19(a) and (b). The approach-retract characteristic exhibits a hysteresis for an angular frequency of 5 rad/s, while no hysteresis is observed for 0.5 rad/s.

B. Force Control

Sections VI and VII-A have shown that the position of Z actuator holding the tuning fork-probe can be controlled efficiently by the robust hybrid controller, and the interaction Fig. 19. Tuning fork frequency shift against Z actuator position for sine wave amplitudes of 1 μm (red), 200 nm (dark), and 150 nm (blue). The results are shown for sinusoidal angular frequencies of (a) 0.5 and (b) 5 rad/s. For visual comparison reasons, the curves related to 200-and 150-nm range displacements have been shifted in y-axis so that their x-coordinates for -0.4 Hz frequency shift are the same as for the curves related to 1-μm range displacement.

forces near the contact and in contact can be efficiently characterized in a robust way despite the AFM parametric uncertainties. The goal, now, is to complete the protocol II with the aim of a frequency modulation AFM by the force controller of Fig. 5. The detailed internal structure of the force controller is shown in Fig. 20(a). In its initial state, the AFM tip is within the region of interaction forces.

The frequency modulation AFM consists of keeping the frequency shift at a desired value with a feedback control on the Z actuator. To do so, the human operator can decide to switch from U 2 to U 3 by acting on the selector of Fig. 20(a). When the force controller is offline and the controller 2 is online, the selector is in position 2, and the memory function is a unit gain. When the force controller is online, the selector is in position 1, and the memory function keeps the value of v ref2 used for controller 2 at the time of the switch. When the force controller is online, its inputs are the measured frequency shift f , the frequency shift reference f ref , and the measured Z actuator position v z . Its output is the control voltage U 3 .

The internal structure of the force controller is composed of the controller 2 and a PID controller. The strategy consists of tracking the desired shift of frequency of the tuning fork probe f ref with the PID controller to define the reference position for the Z actuator. The latter is the output of the PID controller. Then, the robust controller 2 will drive the Z actuator to reach this reference position which corresponds to the desired f ref . In this way, the issues related to AFM uncertainties and measurement delays are avoided by the same way as before, thanks to the robustness of the controller 2.

To define the parameters of the PID controller, the specifications are a closed-loop response time higher than 0.1 s and a phase margin of 90 o . The specification of the response time is done to avoid being affected by the hysteresis as shown in Fig. 19(b).

The function PID tuner of MATLAB [START_REF] Åström | ENS Cachan, Cachan, France, and the master's degree in industrial informatics from Université Pierre et Marie Curie (UPMC)[END_REF] has been used to find the parameters of the PID controller, which meet the specifications. To this end, the model of Fig. 20(b) has been built, where G( 1) is the nominal model used in Section VI-A1. To define the relation between f and v z , the results of Fig. 19(a) are used. The curve of Fig. 19(a) is divided into several straight lines around operating points. The slope of each straight line can be considered as a static gain. The highest static gain is considered for the synthesis of the PID controller [Fig. 20(b)].

The bumpless transfer between the controller 2 and the force controller is very critical here because the AFM tip is very close to the sample. It is satisfied by choosing the error input of the PID controller to be 0 when it is not active. When switched ON, the previous reference for the controller 2 is stored and added to the controller output as shown in Fig. 20(a). For the experimental validation, a scan of the attractive force region is first made using the controller 2 with a sinusoidal input reference as in Section VII-A. Then, at an arbitrary time, the user can decide to switch to a force control to keep the frequency shift at the current value, i.e., the value of f at the time of switching. The results are shown in Fig. 21.

Thanks to these results, one can define a complete new protocol for robust AFM tip landing and force characterization as follows.

Protocol III:

1) Detection of the Interaction Force Region: Initially, the AFM tip can be several hundred micrometers away from the sample. The tip can be moved toward the sample by means of mixed coarse/fine positioning strategy with the Z inertia actuator [START_REF] Boudaoud | A voltage/frequency modeling for a multi-DOFs serial nanorobotic system based on piezoelectric inertial actuators[END_REF] as explained in Section I. The controller 1 is applied for each fine positioning.

In the final fine step positioning, (i.e., just before the tip-sample contact), the hybrid control scheme of Fig. 5 can be used.

2) Characterization of the Tuning Fork Probe Frequency

Shift Against Z Actuator Position: when the tuning fork probe detects an attractive force, the controller 2 is switched on with a reference equal to the switch position minus 0.1 μm plus a sine with an increasing amplitude and an angular frequency around 0.5 rad/s to avoid the hysteresis of Fig. 19(b). In this way, the relation f /v z as in Fig. 19(a) can be defined. 3) Force Control for Frequency Modulation AFM: The PID substructure of the force controller can be tuned online taking into account the highest slope of the relation f /v z . Then, the user can decide to switch from U 2 to U 3 by acting on the selector of Fig. 20(a). A frequency modulation AFM can then be started.

VIII. DISCUSSION

The control method presented in this article has advantages and drawbacks. The main drawback is that it relies on the experience of the designer on the system to be controlled, especially for the definition of the set of g LTI models to be treated and the selection of the worst case models which are not known a priori and are defined step by step. The number g of LTI models will define the resolution of the multimodel synthesis. The more the value of g is high, the highest will be the models considered for the worst case analysis. It is mainly related to the number of discrete values selected between the lower and the upper bounds of each uncertain parameter.

The proposed control method has, however, several advantages that are as follows.

1) The method only considers closed-loop specifications and design constraints imposed by the user, thus leading to a higher potential of specifications in closed loop with a controller order that can be set by the user, therefore potentially of low order. The controller order depends on the number of observers selected by the user. This is the main difference with traditional robust and conservative controllers that often lead to unnecessary very high order controllers because the control design itself does not consider only the specifications given by the user but many others that are not mastered. This is typically the case with the H ∞ control design [START_REF] Oubellil | Mixed stepping/scanning mode control of stick-slip SEM-integrated nano-robotic systems[END_REF]. In this article, we show a new potential of robustness in AFM with a robust method, where the control constraints are really mastered by a wise choice of the pole assignment and the choice of the worst case models.

2) The gain matrix K obtained from (15) can be structured.

Each gain corresponds to an ESA related to a specific LTI model. The designer can master each feedback by adding for instance dynamic filter (e.g., low-pass filter, bandstop filter, and so on) to some feedback if necessary. In [START_REF] Oubellil | Mixed stepping/scanning mode control of stick-slip SEM-integrated nano-robotic systems[END_REF], a H ∞ controller is designed to control, in scanning mode, the same piezoelectric inertia actuator than those used in this article. The control specifications were almost the same than those specified for the design of the controller 2. In [START_REF] Oubellil | Mixed stepping/scanning mode control of stick-slip SEM-integrated nano-robotic systems[END_REF], the control specifications were: maximal closed-loop response time lower than 5 ms, vibrations damping with no overshoot, and maximal static error lower than 2%. With the H ∞ design, the order of the controller is equal to the order of the open-loop system + the order of weighting functions. In the multimodel design, the order of the controller is equal to the number of the observers +1. Therefore, the same closed-loop performances have been obtained with a controller of ninth order in [START_REF] Oubellil | Mixed stepping/scanning mode control of stick-slip SEM-integrated nano-robotic systems[END_REF] with the H ∞ approach, while here with the controller 2, a fiveorder controller has been enough. If the H ∞ approach is used to design the entire hybrid control structure of Fig. 9, this will lead to a significant computing time. Here, the multimodel design for hybrid control is a real advantage for real-time implementation issues.

IX. CONCLUSION

A novel approach for robust hybrid control and its application to atomic force microscopy has been presented in this article. The main advantage of the control strategy is its robustness and its low conservatism. The structure of the hybrid controller has been presented, and each control part has been detailed. The control method has shown its ability to do in an automated and robust way, a complete landing procedure of AFM tip, and a precise characterization of interaction force regions with a bumpless switching under dynamic constraints. The results have shown the robustness of the closed-loop systems against the hysteresis, the uncertainties of the dynamic parameters due to the fabrication process and system aging, and instrumental issues such as the real-time data acquisition. This capability suggests a promising potential in view of using the controller in an industrial context. More generally, the new hybrid control method can be applied to any uncertain and/or nonlinear system that can be approximated by a family of linear statespace models and for which switch constraints are needed.

Fig. 1 .

 1 Fig. 1. AFM based on a tuning fork with a hybrid control for closed-loop landing procedure.

Fig. 2 .

 2 Fig. 2. View of the AFM tuning fork probe inside a scanning electron microscope.

Fig. 3 .

 3 Fig. 3. Simplified model of an Akiyama tuning fork probe. F tf is the driving force generated by piezoelectric effect of the tuning fork.

Fig. 4 .

 4 Fig. 4.Step responses of all the considered LTI models.

Fig. 7 .

 7 Fig. 7. Robust output feedback control scheme of the uncertain system G() with a set of n o observers and an integrator.

Fig. 8 .

 8 Fig. 8. Flowchart of the robust control synthesis.

Fig. 9 .

 9 Fig. 9. (a) Hybrid control scheme of an uncertain system G(). Each controller and compensator are designed by an ESA and a multimodel formulation. (b) Control part including controller 1 and compensator 2 with a detailed view of the internal structure of controller 1.

Fig. 10 .

 10 Fig. 10. Pole map of all the closed-loop models with controller 1 at the first synthesis.

Fig. 11 .

 11 Fig. 11. Step response of one actuator with controller 1 at the first synthesis (experimental result).

Fig. 12 .

 12 Fig. 12. Normalized to unity step responses of all closed-loop models with controllers 1 and 2 at the second synthesis.

Fig. 13 .

 13 Fig.[START_REF] Belikov | Digital q-control and automatic probe landing in amplitude modulation phase imaging AFM mode[END_REF].Closed-loop displacements of all LTI models with the hybrid controller.

Fig. 14 .

 14 Fig. 14. Normalized to step responses of the three actuators controlled with (a) controller 1 and (b) controller 2 at the second synthesis. The step input references are 100 nm, 500 nm, 1 μm, and 1.5 μm. The step response of the reference closed-loop model is shown.

Fig. 15 .

 15 Fig. 15. Experimental closed-loop displacements of the three actuators with the robust hybrid control.

Fig. 16 .Fig. 17 .

 1617 Fig.[START_REF] Lemos | Observer dynamics in switching control: Application to neuromuscular blockade[END_REF]. Experimental closed-loop displacements of the three actuators and protocol for robust AFM tip landing.

Fig. 20 .

 20 Fig. 20. (a) Proposed structure for frequency modulation AFM with the force controller. When the force controller is online, the selector is in position 1 and the memory function keeps the value of v ref used for controller 2 at the time of the switch. (b) Closed-loop system used to tune the parameters of the PID controller by simulation.

Fig. 21 .

 21 Fig. 21. (a) Experimental Z actuator position and (b) experimental frequency shift of the tuning fork probe during the position/force control.

TABLE I

 I

ACKNOWLEDGMENT

The authors would like to thank Vincent Cailliez for his statistical analysis of the tuning fork signal.

This work was supported in part by the research program Investissements d'avenir through the Robotex Equipment of Excellence under Grant ANR-10-EQPX-44 and in part by the research program ANR JCJC Robine under Grant S19JRAR086. Recommended by Associate Editor G. Pin.


A L Ramon

J R Bertrand

C Zandanel

C Vauthier

C Malvy

BIODISTRIBUTION OF siRNA DELIVERED WITH CHITOSAN-DECORATED POLY(ISOBUTYLCYANOACRYLATE) NANOPARTICLES

INTRODUCTION

In previous works, we showed that chitosan-decorated poly(isobutylcyanoacrylate) (PIBCA) nanoparticles were suitable carriers for intravenous delivery of active siRNA to xenografted model tumors implanted in mice. A spectacular reduction of the tumor volume accompanied by a reduction of the amount of the target mRNA acknowledged the specific effect of the siRNA delivered with the nanoparticles in a model of the papilloma thyroid carcinoma [START_REF] De Martimprey | New core-shell nanoparticles for the intravenous delivery of siRNA to experimental thyroid papillary carcinoma[END_REF] . A similar result was reported on a model of Ewing sarcoma, another cancer caused by the expression of a junction oncogene. Although tumor volume reduction was less spectacular, the activity of the siRNA was highlighted by the reduction of the amount of the target mRNA in the tumor [START_REF] Ramon | Nouvelle approche ciblée pour le traitement des tumeurs de la famille du sarcome d'Ewing[END_REF] . To elucidate the delivery of the siRNA to the tumor using these nanoparticles, the present work was aimed to investigate the biodistribution of the siRNA and that of the nanoparticles after an intravenous administration to mice xenografted with a model tumor of Ewing sarcoma.

MATERIALS AND METHODS Preparation and characterization of the fluorescent labeled nanoparticles

Fluorescent nanoparticles were synthesized by radical emulsion polymerization (40°C, 1h) of isobutylcyanoacrylate and polyfluor ® 570 in 200 mM nitric acid containing 16 mM cerium ammonium nitrate and 3% pluronic F68. Nanoparticles were dialysed. Targeted nanoparticles were obtained by grafting an anti-CD99 biotinylated antibody [START_REF] Segura-Sanchez | Synthesis and characterization of functionalized poly(gamma-benzyl-L-glutamate) derivatives and corresponding nanoparticles preparation and characterization[END_REF] . Characterization included measurements of size and zeta potential, evaluation of the activation of the C3 component of the complement system and study of the association of the siRNA (adsorption adsorption, isothermal titration calorimetry). All characteristics were compared with those of unlabelled nanoparticles.

siRNA-nanoparticle association and biodistribution studies

siRNA was labeled on the 5' end with T4 polynucleotide kinase and [γ 32 P] ATP. For the adsorption isotherm, siRNA was incubated with nanoparticles (10 min, NP/siRNA mass ratio = 0 to 300). The radioactivity was measured in the supernatant obtained after centrifugation (45000 rpm, 1h, 4°C). For biodistribution studies, the siRNA was associated with the nanoparticles to a NP/siRNA mass ratio of 50. All animal experiments were approved by the ethic committee on animal experimentation of the Institute Gustave Roussy (Villejuif, France). siRNA-loaded nanoparticles (non targeted nanoparticles) were injected in the tale vein of mice ( 24µg/kg siRNA, 547300 cpm). Mice were sacrificed after 24 h using a CO 2 chamber. Tumors and organs were retrieved, grounded and centrifuged. The radioactivity of the supernatant was measured using a Wallac 1409, Pegasus Scientific Inc., USA). The biodistribution of the fluorescent nanoparticles (non-targeted and targeted nanoparticles) was determined using a live animal imaging system (IVIS 50, Caliper).

Author Manuscript published in: Abstract book of the 16 th International Symposium on Recent Advances in Drug Delivery Systems. Salt Lake City, UT, USA, 3-6 February 2013. Abstract #66

RESULTS

All characteristics of the fluorescent-labeled nanoparticles were identical to those of the non labeled nanoparticles [START_REF] Zandanel | Characterization of fluorescent poly(isobutylcyanoacrylate) nanoparticles obtained by copolymerization of a fluorescent probe during redox radical emulsion polymerization (RREP)[END_REF] . The biodistribution study of the siRNA revealed that the amount of radioactivity reaching the tumor was at the same level than that found in the liver and in the spleen (Figure 1). Quite large amounts of radioactivity were also found in other organs. This indicated that the nanoparticles were stealth and could deliver siRNA to the tumor. After intravenous administration, the nanoparticles accumulated importantly in the tumor (Figure 2). As expected, the targeted nanoparticles accumulated more in the tumor than the non targeted nanoparticles (Figure 3).

Figure 2 :Figure 1 :

 21 Figure 2: Imaging of the mice and isolated organs 24h after iv injection of the fluorescent nanoparticles.

Figure 3 :

 3 Figure 3: Fluorescence in tumor after iv injection of siRNAloaded targeted (white) and nontargeted (black) nanoparticles.

ACKNOWLEDGEMENTS

This work was supported by the Foundation Innabiosanté project NANOINTERFERENCE and the Association Leyla for providing a fellowship to ALR.


Davide Lo Presti
email: davide.lopresti@nottingham.ac.uk

Sara Bressi

João Santos

Stefanie Brodie

James Bryce

Veronique Cerezo

Tony Parry

A SUSTAINABILITY PERFORMANCE-BASED ASSESSMENT TOOL FOR ECODESIGN OF ROADS AND RAILWAYS

INTRODUCTION:

Addressing the sustainability of transport infrastructures requires exploring the environmental, social, and economic impacts of technological options while balancing the often conflicting priorities of different stakeholders, at an early design phase of the infrastructure delivery process. That is a typical multi-criteria decision-making (MCDM) problem, in which the decision-makers need to measure the sustainability through a set of meaningful, representative and quantifiable criteria, balance the relative importance of those criteria and determine the sustainability sequence of multiple alternative technologies for fostering transportation sustainability.

In order to help the decisions makers to efficiently address this challenging task, a decision support toll (DST) was developed in the scope of the FP7 Marie Cruie training-through-research programme Sustainable Pavements & Railways Initial Training Network (www.superitn.eu). It consists of a computational platform that implements a conceptual framework developed to quantify sustainability. It comes with a set of sustainability indicators tailored to both road and railway systems as well as several objective and subjective weighting methods. Amongst those belonging to the last category, the DST includes a set of default weights derived from an Analytical Hierarchy Process (AHP)-based survey that engaged stakeholders from different sectors and from several European countries. At last, the Preference Ranking Organization Methodology of Enrichment Evaluation II (PROMETHEE-II) MCDM method is employed for prioritizing alternative road pavement and railway tracks solutions at the design stage. [START_REF] Brans | Multiple criteria analysis -State of the art surveys[END_REF]. This was implemented, enabling a quicker ranking and thereby identification of the most sustainable pavement and railways track construction and management solution. It was developed using C#, on Visual Studio 2015. The database engine used is the SQL Server 2014 Management Studio, where all the user's and default data are stored. The SUPR&R ITN DST framework is displayed in Figure 1.

OVERVIEW OF THE TOOL: SUPR&R ITN DST is based on the multi-criteria decision making (MCDM) approach. PROMETHEE II -Preference Ranking Organization Methodology of Enrichment Evaluation

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The first step consists of selecting the set of indicators that will rationally prioritize alternatives. The DST comes with a set of default indicators defined according to a methodology developed under the framework of the SUP&R ITN project [START_REF] Bressi | Indicators selection for the sustainability assessment of road pavements and rail infrastructures[END_REF], with the final aim of maximizing their significance to the three Wellbeing dimensions. Nonetheless, the user is enabled to add and consider other indicators.

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In the second step the evaluation matrix is formulated. It corresponds to assess the performance of each alternative with regard to each indicator. This information is inserted by the user and might be both qualitative and quantitative, depending on the type of indicator.

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In the third step the matrix undergoes a filtering process. First, a dominance analysis is performed, meaning that the various alternatives are compared for each indicator individually. It allows to identify if a given alternative is better than the others for all the considered indicators. If so, that alternative might be excluded from the analysis depending on the user's intention. Next, a correlation analysis is undertaken by calculating the Pearson's correlation coefficients to determine if there is a correlation between the selected indicators. The DST displays in matrixes the Pearson's correlation coefficients values as well as the results of their statistical significance. This procedure is done by calculating the t-statistic value of each correlation coefficient at the level of significance of α=0.05 and the respective p-value. Additionally, multivariate descriptive statistics (i.e., covariance matrix) and univariate descriptive statistics (i.e., mean, standard deviation, variance, maximum value, minimum value and range value) are also made available to the user in matrixes.

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In the step 4 the user has to decide about the relative importance (weights) of the different indicators. For that purpose, two different weighting approaches were considered, namely objective and subjective approaches, and for each one several methods were made available. The set of objective methods comprise the Entropy, the Criteria Importance through Inter-Criteria Correlation (CRITIC), the Mean Weights and the Standard Deviation methods, whereas the set of subjective methods include the AHP-based SUP&R ITN weights and the Manually Definition method. The AHP weights were obtained from a survey conducted within the scope of the SUP&R ITN research project. It engaged stakeholders of different sectors and from several European countries. 

In the step 5 the user is asked to define the PROMETHEE-II parameters, what means selecting preference functions (PF) and respective parameters (preference, indifference and eventually Gaussian thresholds). PROMETHEE-II methodology uses PF, which is a function of the difference between two alternatives for any indicator. Six types of PF based on the notions of indicators can be selected. Regarding the definition of the parameters, two alternative approaches were made available: an absolute value-based and a relative value-based approach. The relative value-based approach was considered with the objective of helping the users who do not have sufficient knowledge about the decision-making methodology. According to this approach, the parameters values are calculated as a user defined percentage of the difference between the highest and lowest evaluation for each indicator.

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In the step 6 the multi-criteria decision analysis results are displayed. They consist of the ranking of alternatives as well as several other PROMETHEE intermediate and final outputs (e.g., preference indexes, incoming, outgoing, and net flows).

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In the step 7 the user is given the possibility of conducting an uncertainty analysis to investigate the influence of modified input data and methodological choices on the calculated results and stability of the solution found as the most sustainable. The set of input data and methodologies allowed to be changed comprise the evaluation matrix, the weighting method and the PROMETHEE PF and parameters.

CONCLUSIONS:

The SUP&R ITN DST is a freely available tool and can be used at professional level, by professionals interested in advancing sustainability in transportation, as well as for educational purposes, to provide knowledge and educate on the use sustainability concepts and on what are the important issues to consider during the sustainable transportation decision-making process. The methodology is flexible enough to be adapted to any system

ACKNOWLEDGEMENTS:

The research presented in this paper was carried out as part of the Marie Curie Initial Training Network (ITN) action, FP7-PEOPLE-2013-ITN. This project has received funding from the European Union's Seventh Framework Programme for research, technological development and demonstration under grant agreement number 607524.