MIMO H∞ Feedback Controller with Feedforward Compensator for Scanning Tunneling Microscope having 3D Cross-Coupled Piezoelectric Actuator

Scanning Tunneling Microscope (STM) is used to generate the surface image of any conducting sample surface with an atomic-scale resolution. A multi-axis 3D piezoelectric actuator is attached with the STM tip to move it in horizontal (x and y) and vertical (z) directions. The purpose of control design is to achieve precise reference tracking for horizontal 2D scanning system and to keep the tunneling current constant in the vertical direction in the presence of all possible disturbances. A usual practice is to design independent single-input-single-output (SISO) controllers for individual x, y and z axes by neglecting the cross-coupling dynamics of the multi-axis 3D piezoelectric actuator. In this paper, a complete 3D STM system, without neglecting the cross-coupling dynamics as well as the hysteresis nonlinearity of the actuator, is first mathematically modeled. The parameters of the hysteresis model are identified from the real-time experimental data by using the nonlinear least-squares curve fitting problem. A feedforward compensator is then designed without finding an inverse hysteresis model to avoid any inverse modeling complexity. Then, two control strategies (SISO and multi-input-multi-output (MIMO) H∞ feedback controllers cascaded in series with the feedforward compensator) are investigated for an overall 3D system. Three different scanning trajectories (raster, spiral and Lissajous) are considered to analyze and compare the performance of SISO and MIMO control schemes. In the presence of cross-couplings, an average improvement of 83% in reducing the variations of the tunneling current is achieved with the suggested MIMO control scheme as compared to the generally used SISO control scheme for STM.


I. INTRODUCTION
Scanning Tunneling Microscope (STM) was invented in early 1980s by Gerd Binnig and Heinrich Rohrer [1] and they received Noble prize in physics in 1986. This great invention of STM has brought about a revolution in the field of nanotechnology due to its capability of scanning the conducting surfaces with an atomic-scale resolution. STM operation is based on a quantum mechanical phenomenon of the tunneling current which was first noticed in 1927 by Friedrich Hund [2]. Briefly, if two conducting materials (a sharp metallic tip and a conducting sample surface in case of STM) are brought close to each other so that the distance between them is from 0.1 − 1 × 10 −9 m and a bias voltage is applied between them, then a potential barrier is overcome by the electrons and they tunnel through the vacuum creating a sensitive current named tunneling current. One of the applications of the tunneling current is to characterize the surface morphology with an atomic-scale resolution through STM. Other than surface imaging, tunneling current is also used to measure the accelerations down to sub-micro-g [3], [4] and to sense sub-micrometer displacements [5], [6].
Tunneling current is highly sensitive to the distance between the tip and a conducting sample surface. In STM, the aim is to keep the tunneling current constant (vertical VOLUME 4, 2016 z-direction) while scanning the sample surface (horizontal x-y directions) to generate its topographical image with an atomic-scale resolution. When the gap between the tip and the sample surface increases or decreases due to the variations in the sample topography, the control signal regulates the distance between the tip and the sample surface by actuating the tip away from or near to the sample surface in the vertical z-direction. The 3D displacements over the sample surface are viable by using the multi-axis 3D piezoelectric actuator due to its capability of expanding and contracting when a voltage is applied. However, these actuators introduce some challenges from control systems point of view, such as hysteresis, creep, cross-coupling and structural vibrations [7]. These issues must be considered at the time of control design in order to achieve the desired performance of STM. If these issues are not considered then large variations in the tunneling current may result in a surface topographic image that does not correspond to the reality [8].
For the horizontal 2D scanning system of STM, the requirement is to achieve the precise reference tracking. A lot of research work has been done in this direction to the control of piezoelectric actuators for different micronanopositioning applications [9]- [12]. The main challenges in order to achieve the precise reference tracking are the actuator nonlinearities (e.g. hysteresis and creep) and high frequency vibrations [13]. In order to deal with hysteresis nonlinearity of the piezoelectric actuator, a number of hysteresis models are investigated in the literature [14]- [17] and accordingly, their compensation techniques are also discussed [18]- [22]. However, a commonly used compensation technique is to design an inverse hysteresis model, as a feedforward compensator, which is often a challenging task due to inverse modeling complexity. [23] can be consulted for a survey on different hysteresis models and their compensation techniques. As far as creep nonlinearity is concerned, its effects are visible only when the tracking is required over extended periods of time during slow-speed operations. A number of mathematical models for creep nonlinearity and compensation techniques are analyzed in the literature [24]- [26]. However, the effects of creep nonlinearity can be neglected if the tracking is performed for a short duration of time. The issue of vibrations when tracking is performed at a high frequency is also analyzed in the literature [27]- [29]. In short, a lot of research work has already been done in order to achieve the precise reference tracking but this much effort, from control design point of view, has not been done yet to the control of piezoelectric actuator in the presence of nonlinear tunneling phenomenon (e.g. control design for the vertical tunneling current system in STM).
In most commercial equipment of STM, still the very classical proportional-integral (PI) or sometimes proportionalintegral-derivative (PID) controllers are implemented for the vertical tunneling current system. The parameters of these classical controllers are adjusted manually by the operators of STM. Due to the manual adjustment of the controller parameters, there is always a high risk of achieving a surface topographic image that does not correspond to the reality. The analysis of the vertical tunneling current system of STM with classical PI controller is presented in [30]- [32]. Closed-loop stability analysis of STM with a classical PI controller to reduce the risk of tip-sample crash is presented in [33]. A variable structure control design methodology with a classical PI controller is analyzed in [34] to avoid the tipsample collision due to any sudden change in the topography of the sample surface.
As mentioned earlier, the main objective for the vertical tunneling current system is to keep the tunneling current constant while scanning the sample surface. Other than sample surface variations and the sensor noise, the cross-coupling of the multi-axis piezoelectric actuator is also an important source of producing large variations in the tunneling current. The adverse effect of cross-coupling for the vertical tunneling current system is often neglected and independent singleinput-single-output (SISO) controllers are designed for individual axis of STM. Few control approaches have been discussed in the literature to compensate the cross-coupling effects in different scanning probe microscopy (SPM) applications. In [35], an inversion based iterative SISO control law is suggested to compensate for the x-to-z cross-coupling effect in piezo-scanners for the application of atomic force microscope (AFM). [36] further enhanced this work by using the same inversion based iterative control law to obtain the control input which is augmented, as a feedforward control, to the classical PID feedback control to cancel the crosscoupling effect in tapping-mode imaging of AFM. In [37], a SISO H ∞ feedback controller is suggested to compensate the x-to-y cross-couplings in AFM. A modified repetitive control based cross-coupling compensation approach is suggested in [38] to mitigate the coupling effect in XY piezoelectric tube scanner in AFM. An internal reference model-based optimal SISO linear quadratic Gaussian (LQG) controller with a vibration compensator is suggested in [39] to compensate the cross-couplings in AFM. It has been mentioned in [7] that the cross-coupling effects in piezoelectric tube scanners must also be investigated with different MIMO control schemes. [40] has analyzed a MIMO linear quadratic integral control with Kalman observer for STM having x/y to z cross-couplings for the application of surface reconstruction. However, the cross-coupling between x and y axes of the piezoelectric actuator is neglected. A multivariable compensation technique is suggested in [41] to reduce the cross-couplings of the multi-axis 2D piezoelectric actuator.
It has been observed by the authors of this paper that mostly SISO control methodologies are investigated to compensate the cross-coupling in different SPM applications and most of this work is related to AFM. There is a scarcity of research work about the compensation of cross-couplings through some advanced control design in STM. The idea of the presented work is to analyze the performance of STM by considering not only the cross coupling dynamics of the multi-axis 3D piezoelectric actuator and its effect on all 3 axes of the system but also the nonlinear hysteresis phenomenon of the actuator. It is always an interesting study to consider the different adverse effects, which can degrade the performance of the considered system, altogether. Crosscouplings and hysteresis are the two most significant undesirable behaviors of the multi-axis piezoelectric actuator which can not be neglected. To deal with both of these issues, a MIMO H ∞ feedback controller cascaded in series with a feedforward compensator is thoroughly analyzed in this paper in order to achieve the precise reference tracking for a 2D horizontal scanning system and to keep the tunneling current constant in the vertical z-direction, while scanning the sample surface.
The main contributions of this paper are: • A complete 3D MIMO system of STM, by considering the cross-couplings dynamics as well as the hysteresis nonlinearity of the multi-axis piezoelectric actuator, is mathematically modeled. • The parameters of the hysteresis model are identified from the real-time experimental data and the hysteresis compensator is designed without formulating an inverse hysteresis model to avoid inverse modeling complexity. • In contrast to the usual practice of designing independent SISO controllers for individual axis of STM, MIMO H ∞ feedback controller cascaded in series with the feedforward compensator is suggested and thoroughly analyzed. • The performance of the suggested MIMO control scheme is compared with the SISO control scheme by analyzing three different scanning trajectories (raster scanning, spiral scanning and Lissajous scanning).
This manuscript is organized as follows: Section II presents the working principle of STM. Mathematical modeling of the considered MIMO system, having a horizontal scanning system with hysteresis nonlinearity, cross-coupling dynam-ics and a vertical tunneling current system, is presented in Section III. Section IV discusses in detail about the controller design. This section starts with the formulation of the control problem and expresses the desired performance of the system. Then, feedforward compensator design and MIMO H ∞ feedback controller design schemes are presented in this section. Section V presents the achieved results of the horizontal scanning system as well as the tunneling current system by considering raster, spiral and Lissajous scanning. Finally, Section VI concludes this paper.

II. WORKING PRINCIPLE OF SCANNING TUNNELING MICROSCOPE
STM consists of a 2D horizontal scanning system in x and y directions and a 1D vertical tunneling current system in z direction. Overall, the block diagram of the considered 3D STM system with the suggested MIMO control strategy is presented in Fig. 1.
The working principle of STM is first to bring the sharp metallic tip, attached with a piezoelectric actuator, in the tunneling region so that the distance (d) between the tip and the conducting sample surface must be less than 1 × 10 −9 m. When the sharp metallic tip comes in the tunneling region to scan the sample surface, a bias voltage applied between the two allows the electrons to tunnel through the vacuum separating them. The direction of the resulting tunneling current (i t ) depends on the polarity of the applied bias voltage. The purpose of control design for STM is to achieve the precise reference tracking for a 2D horizontal scanning system (in x and y directions) and to keep the tunneling current constant in the vertical z-direction in the presence of all possible disturbances. The control signal which keeps the tunneling current constant in the vertical z-direction helps to generate the image of the sample surface with an atomic-scale resolution. A mutli-axis 3D piezoelectric actuator moves the tip in VOLUME 4, 2016 the appropriate direction depending on the applied voltages (v xv and v yv for the horizontal displacement in x and y directions respectively and v zv for the vertical displacement in z-direction) given by the voltage amplifier. The input voltages (u x , u y and u z ) of the voltage amplifier are adjusted by the controller.
For a 2D horizontal scanning system, the output displacements (z xp and z yp in x and y directions respectively) of the piezoelectric actuator are perturbed due to cross-couplings among these two axes. Coupling-caused positioning errors (z xy and z yx ) will be added in the actual displacements (z xp and z yp ) of the actuator and these overall displacements are sensed by the capacitive position sensors. These position sensors generate the output voltages (v x and v y ) which are used for the feedback. The feedback voltages are compared with the reference voltages (v xr and v yr for x and y directions respectively) to generate the error voltages (v xe and v ye ). For a 2D horizontal scanning system, the piezoelectric actuator consists of linear vibration dynamics as well as the nonlinear hysteresis phenomenon. The nonlinear hysteresis behavior is modeled for the horizontal scanning system as the displacements in x and y directions are in micro-scale. However, the nonlinear hysteresis behavior of the piezoelectric actuator is neglected for the vertical displacement in z-direction as the overall displacement is in sub-nano scale. The cross-coupling effect due to sub-nano scale displacement in z direction on the other two axes (x and y) can also be neglected.
For the vertical tunneling current system, the actual output displacement (z zp ) of the piezoelectric actuator is perturbed due coupling-caused positioning errors (z zx and z zy due to displacements in x and y directions respectively). Other than this overall displacement of the piezoelectric actuator, the sample surface variations (z S ) changes the distance (d) between the tip and the sample surface. Tunneling current (i t ) exponentially depends on this distance (d). Changes in distance (d) while scanning the sample surface bring changes in the tunneling current (i t ). The current sensor, which measures the tunneling current, is a current to voltage converter with a high gain. It converts the weak tunneling current (i t ) in nano-amperes into a voltage. Measurement noise (n) induces variations in the measured tunneling voltage (v z ). This tunneling voltage (v z ) is then compared with the reference input voltage (v zr ) which corresponds to the desired value of the tunneling current.
In the suggested control scheme, a MIMO H ∞ feedback controller acts on the three error voltages (v xe , v ye and v ze ) and generates the control signals (u xb , u yb and u z ). For a 2D horizontal scanning system, the feedforward compensators are also designed for x and y axes to compensate the hysteresis nonlinearity of the actuator and cascaded in series with the feedback controller.

III. MODELING OF SCANNING TUNNELING MICROSCOPE
STM has a 2D (xy) horizontal scanning system and a vertical (z) tunneling current system. The overall 3D system is coupled with each other due to cross-couplings of the piezoelectric actuator as presented in Fig. 1. The overall mathematical model of STM is presented in this section.

A. DYNAMICS OF HORIZONTAL XY SCANNING SYSTEM
The horizontal 2D scanning system consists of a voltage amplifier, piezoelectric actuator and a capacitive position sensor. The dynamics in horizontal x-direction is considered the same as in y-direction. Therefore, 5 th order mathematical model of horizontal x-direction is presented here only. The same mathematical model is considered for the horizontal ydirection.
The considered state-space mathematical model of the voltage amplifier is: where, u x and v xv are input and output voltages of the voltage amplifier respectively, ω xv the bandwidth and G xv is the gain of the amplifier. Piezoelectric actuator has linear dynamics as well as the hysteresis nonlinearity. The nonlinear hysteresis phenomenon of the piezoelectric actuator is modeled by considering the Bouc-Wen hysteresis model. All the necessary details about the Bouc-Wen hysteresis model can be found in [42], [43]. The reason for selecting the Bouc-Wen hysteresis model is its simplicity in representing a large class of hysteresis with accuracy, easy in implementation as well as in identification of its parameters. The hysteresis phenomenon of the piezoelectric actuator is modeled as: where, v xv and v xp are input and output of the hysteresis model and Γ, Φ and Ψ are the constant parameters which govern the shape of the hysteresis loop. G h is the coefficient of the piezoelectric actuator and is always positive. The linear dynamics of the piezoelectric actuator is represented by the following state-space model: where, v xp and z xp are input voltage and output displacement of the piezoelectric actuator respectively, ω xp the bandwidth, ζ xp the damping coefficient and G xp is the gain of the piezoelectric actuator. The output displacement of the piezoelectric actuator is sensed by a capacitive position sensor. The output displacement of the piezoelectric actuator in x-direction (z xp ) is disturbed due to its cross-coupling with the displacement in y-direction. The state-space mathematical model of the capacitive position sensor is: where, v x is output voltage of the capacitive sensor, z xy is coupling-caused displacement in x-direction due to displacement in y-direction, ω xc the bandwidth and G xc is the gain of the capacitive position sensor. So, the horizontal scanning system in x-direction has an overall 5 th order mathematical model with input and output voltages of u x and v x respectively. Similarly, the horizontal scanning system in y-direction has the similar 5 th order mathematical model with input and output voltages of u y and v y respectively.

B. DYNAMICS OF CROSS-COUPLING
The displacement of STM tip in x, y and z-directions are coupled with each other due to cross-couplings of the piezoelectric actuator. A number of studies have been undertaken by the research community to understand the mechanism and root causes of this unwanted behavior of the piezoelectric actuator in micro/nanopositioning stages. The primary root cause highlighted by [44] is the combination of the slip and rotation of the piezoelectric disks due to their frictional behavior, concurrent with changes in the geometry of the stage. According to [28], there are several complex reasons for the erroneous motion due to cross-couplings in the piezoactuated micro/nanopositioning stages; for instance, the eccentricity of the scanner due to improper machining, nonuniformity of the electric field across the scanner and also the error in the orthogonality of the scanning axes due to collocated structure of the scanner.
The coupling-caused displacement in one direction, let's say in y-direction, due to the displacement in other direction, let's say in x-direction, becomes more significant when the displacement in x-direction is for large range or with high speed. The coupling-caused displacement in one direction becomes insignificant when the displacement in other direction is for small range or with slow speed. This motivates to consider the dynamics of the commonly used high-pass filter as the dynamics of the cross-coupling. As in z-direction, STM tip just needs to maintain a constant distance between the tip and the sample surface, or in order words, needs to maintain a constant tunneling current, therefore the couplingcaused displacement in other directions due to the displacement in z-direction can be neglected.
The coupling-caused displacement (z xy ) in x-direction due to the displacement in y-direction (z yp ) is represented by the following model: The coupling-caused displacement (z yx ) in y-direction due to the displacement in x-direction (z xp ) is represented by the following model: The coupling-caused displacement (z zx ) in z-direction due to the displacement in x-direction (z xp ) is represented by the following model: The coupling-caused displacement (z zy ) in z-direction due to the displacement in y-direction (z yp ) is represented by the following model: Here, in (5)- (8), G cc is the gain which represents the maximum coupling-caused positioning error and ω cc is the bandwidth of the high pass filter which represents the scanning speed.

C. DYNAMICS OF VERTICAL Z TUNNELING CURRENT SYSTEM
The vertical z tunneling current system consists of a voltage amplifier, piezoelectric actuator for the precise movement of the tip in vertical z-direction and a current sensor for the measurement of tunneling current. The voltage amplifier is modeled as: where, u z and v zv are input and output voltages of the voltage amplifier respectively, ω zv the bandwidth and G zv is the gain of the amplifier. The movement of STM tip in the vertical z-direction is in sub-nano scale, therefore, 2 nd order linear dynamics is considered for the piezoelectric actuator and hysteresis nonlinearity of the actuator in neglected. The considered mathematical model of the piezoelectric actuator is: where, v zp and z zp are input voltage and output displacement of the piezoelectric actuator respectively, ω zp the bandwidth, ζ zp the damping coefficient and G zp is the gain of the piezoelectric actuator. The tunneling current (i t ) depends exponentially on the distance (d) between the tip and the sample surface. This distance can be obtained as: (11) where, d 0 is the initial distance, z S is the unknown sample surface variations, z zx and z zy are the coupling-caused displacements in z-direction due to the displacements in x and y-directions respectively and z zp is the output displacement of the piezoelectric actuator in z-direction. The change in VOLUME 4, 2016 x,y distance (d) will change the tunneling current (i t ) which can be modeled by an exponential static nonlinearity as: where, i 0 is the initial tunneling current and k is a constant which depends on the work function of the tip and the sample surface. The tunneling current equation can be linearized by using the first-order linear approximation approach around an operating point. The linearized equation of the tunneling current is: where, i p and d p are the operating points of the tunneling current and the corresponding distance between the tip and the sample surface respectively. The current sensor measures the tunneling current and generate the corresponding voltage accordingly. The current sensor is mathematically modeled as: where, v z is the output voltage of the sensor, ω zc the bandwidth, G zc the gain and n is the measurement noise. So, the vertical tunneling current system has an overall 4 th order mathematical model with input and output voltages of u z and v z respectively.

D. OVERALL DYNAMICS OF MIMO SYSTEM
The overall dynamics of the considered MIMO system has three inputs (u x , u y and u z ) and three outputs (v x , v y and v z ). This overall dynamics of a 3D MIMO system can be represented as: All parameter of the above MIMO model are defined earlier and their values are given in Table 1.

IV. CONTROLLER DESIGN
In this section, first of all, the control problem and the desired performance of STM is highlighted. Then, in order to achieve the desired performance, the feedforward compensator as well as the MIMO H ∞ feedback control design schemes are presented.

A. CONTROL PROBLEM AND DESIRED PERFORMANCE
The objective of control design for STM, when its tip moves in 3-directions (x, y and z), is to achieve the precise reference tracking in x and y-directions in order to scan the sample surface and also to maintain the tunneling current constant, while scanning the sample surface, in the vertical z-direction. Due to the cross-couplings of the piezoelectric actuator, coupling-caused displacements in each direction (x, y or z) act as a disturbance. Other than this, sample surface variations (z S ) as well as sensor noise (n) also act as external disturbances. So, the control problem is to achieve precise reference tracking in the horizontal (x and y) direction and also to maintain the tunneling current constant in the vertical (z) direction by rejecting the effects of all the disturbances, indeed by maintaining the closed-loop stability of the overall system.
The desired performance of STM is to achieve precise reference tracking in the horizontal (x and y) direction with mean-absolute-error of less than 1% of the reference trajectory and the maximum allowed variations of the tunneling current in the vertical (z) direction is ±5%. It means that if the reference tunneling current is 0.5nA then the maximum allowed variations of the tunneling current must be in between 0.475nA and 0.525nA while scanning the sample surface in the presence of all possible disturbances. Other than this, the initial overshoot of the tunneling current must also be less than 5% in order to avoid the collision between the sharp metallic tip and the sample surface.

B. FEEDFORWARD COMPENSATOR DESIGN
To deal with the hysteresis nonlinearity of the piezoelectric actuator, a feedforward compensator is designed. A usual practice to design a feedforward compensator is to design an inverse hysteresis model. Then, the inverse hysteresis model is usually cascaded in series with the nonlinear model of the piezoelectric actuator to compensate the hysteresis nonlinearity. Designing an inverse hysteresis model is often a challenging task due to the inverse-modeling and computational complexity.
In this paper, the feedforward compensator is designed without calculating the inverse hysteresis model, rather only the inverse of the piezoelectric coefficient is required, which is strictly positive, with the identified hysteresis model [45]. So, the first step is to identify the parameters of the considered hysteresis model (2)  To analyze the mismatch of the hysteresis loops, the experimental displacement as well as the simulated displacement of the piezoelectric actuator with the considered hysteresis model is presented in Fig. 4(a). A small difference between these two plots can be observed which is presented as modeling error in Fig. 4(b). It can be observed in Fig. 4(b) that the peak-to-peak modeling error is 0.48µm which is around 3.7% of the displacement of the piezoelectric actuator. A close match between the experimental and the simulated displacements of the piezoelectric actuator in Fig. 4(a) and also the experimental and the simulated hysteresis loops in Fig. 3(a) validate the nonlinear hysteresis model for the considered piezoelectric actuator. The modeling error will be dealt with the robustness of the suggested MIMO H ∞ feedback controller.
The feedforward compensator for the horizontal x-  direction, without calculating the inverse hysteresis model, is designed as follows: where, u xb and u x are input and output voltages of the feedforward compensator, G h piezoelectric coefficient and x 2 is the solution of nonlinear hysteresis model (2). It can be noticed here that there is no need to calculate the inverse of the hysteresis model for this compensator design. If this compensator is cascaded in series with the hysteresis model then the achieved result, as presented in Fig. 3(b) shows that the nonlinear hysteresis phenomenon is well compensated by the feedforward compensator. The same compensator is designed for the horizontal scanning system in y-direction. The feedforward compensator for the horizontal y-direction is designed as follows: where, u yb and u y are input and output voltages of the feedforward compensator in the horizontal y-direction respectively.

C. MIMO H∞ FEEDBACK CONTROLLER DESIGN
In order to design the MIMO H ∞ feedback controller, the linearized system model is used where the hysteresis nonlinearity is well compensated by the feedforward compensator. However, after designing the feedback controller, the validation of the controller is performed in simulations in the presence of actual nonlinear system model. The considered block diagram for the feedback controller design is presented in Fig. 5. In this block diagram, six performance weighting functions (W xe , W xt , W ye , W yt , W ze and W zt ) can be observed which are designed in order to weight the controlled outputs y x1 , y x2 , y y1 , y y2 , y z1 and y z2 respectively. In this Fig. 5, D xv (x-direction), D yv (ydirection) and D zv (z-direction) represent the dynamics of the voltage amplifier, D xp (x-direction), D yp (y-direction) and D zp (z-direction) represent the dynamics of the piezoelectric actuator and D xc (x-direction) and D yc (y-direction) represent the dynamics of the capacitive position sensor. In vertical z-direction, D zc represent the dynamics of the current sensor and D zt represent the relationship between tunneling current (i t ) and distance (d). C xy and C yx represent the cross-coupling dynamics between horizontal x and y directions. C zx represents the cross-coupling dynamics between x and z directions and C zy represents between y and z directions.
The first performance weighting function (W xe ) is designed as follows: where, for the inverse of this first performance weighting function, xe = −75 dB represents the maximum attenuation level in the low-frequency region, ω xe = 1 kHz represents the bandwidth and N xe = 6 dB represents the maximum amplification in the high frequency region. This performance weighting function (W xe ) is imposed over the error signal (v xe ) in the horizontal x-direction which weights the controlled output y x1 . The similar first order dynamic models are chosen for the other two performance weighting functions (W ye and W ze ). The dynamic model of the performance weighting function (W ye ) is given as: where, this performance weighting function weights the controlled output y y1 and is imposed over the error signal (v ye ) with constant parameters of ye = −75 dB, ω ye = 1 kHz and N ye = 6 dB. The dynamic model of the performance weighting function (W ze ) is given as: where, this performance weighting function weights the controlled output y z1 and is imposed over the error signal (v ze ) with constant parameters of ze = −75 dB, ω ze = 3.8 kHz and N ze = 6 dB.
The performance weighting function (W xt ) is designed as follows: where, for the inverse of this performance weighting function, N xt = 3.5 dB represents the amplification in the low frequency region, ω xt = 2 kHz bandwidth and xt = −45 dB represents the maximum attenuation level in the  high frequency region. The similar first order dynamic models are selected for W yt and W zt . The dynamic model of the performance weighting function (W yt ) is: where, N yt = 3.5 dB, ω yt = 2 kHz and yt = −45 dB are the constant parameters. The dynamic model of the performance weighting function (W zt ) is: where, N zt = 3.5 dB, ω zt = 4 kHz and zt = −45 dB are the constant parameters. W xt , W yt and W zt weight the controlled outputs y x2 , y y2 and y z2 respectively as shown in Fig. 5. These values are selected in order to achieve the desired performance in terms of precise reference tracking in horizontal x and y directions, maintaining tunneling current (i t ) at the desired value in the presence of all the disturbances and to achieve good robustness and stability margins. After designing the performance weighting functions, the next step is to achieve a generalized system model (P ) which is actually the combination of the overall system model with the performance weighting functions. The input vector of the generalized plant model is [v xr v yr v zr z S n u x u y u z ] T and the output vector is [y x1 y x2 y y1 y y2 y z1 y z2 v xe v ye v ze ] T as shown in where, VOLUME 4, 2016 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
If P = P up1 P up2 P lo1 P lo2 then: After establishing the generalized system model matrix (P ), the next step of H ∞ control problem is to create a closed-loop transfer matrix from the external inputs vector [v xr v yr v zr z S n] T to the controlled outputs vector [y x1 y x2 y y1 y y2 y z1 y z2 ] T . This closed-loop transfer matrix is generally referred to as a lower linear fractional transformation (LFT) and is given by: where, K represents the controller dynamics as shown in Fig. 6. The objective of the H ∞ control design is to find a controller (K) which stabilizes the generalized system model (P ) and generates the control signal which counteracts the influence of the external inputs vector on the controlled outputs vector by minimizing the H ∞ norm (γ) of the closedloop transfer matrix F l (P, K) ∞ . This H ∞ control problem is then solved by the Algebraic Riccati equation approach as presented in [46]. A 22 nd order MIMO H ∞ controller is achieved with the value of γ = 0.99. The overall closed-loop system will be stable if in H ∞ framework, the closed-loop transfer matrix between the external inputs vector and the controlled outputs vector is internally stable. Lets say, M = F l (P, K), which has the dimension of (6 × 5) and can easily be achieved from (25), is given by: 26) then the condition for the closed-loop internal stability is that all the elements of M are stable. It has been verified that there are no zeros in Re s ≥ 0 in the characteristic polynomial of each element of M to guarantee the closed-loop internal stability of the system.

V. RESULTS AND DISCUSSION
In order to analyze the performance of the presented control scheme, three different types of scanning trajectories are considered for the horizontal x and y directions: i) raster scanning ii) spiral scanning and iii) Lissajous scanning. The raster scanning is the most conventional scanning method. However, in recent years, the need for higher imaging speeds has motivated the investigation of novel scanning trajectories like spiral scanning and Lissajous scanning.
For the purpose of performance comparison, SISO (Single Input Single Output) H ∞ feedback controller is also designed, individually for each axis, by considering the same performance weighting functions as considered for the design of MIMO H ∞ feedback controller. A 6 th order SISO H ∞ feedback controller is achieved for the horizontal x as well as for y directions with γ = 0.89. For the vertical zdirection, a 6 th order SISO H ∞ feedback controller is also designed with γ = 0.72.
In order to thoroughly analyze the performance of the considered system, three different cases are examined for each different type of scanning trajectory. The considered three cases are: Case I: SISO H ∞ feedback controller with feedforward compensator without cross-couplings. Case II: SISO H ∞ feedback controller with feedforward compensator with cross-couplings. Case III: MIMO H ∞ feedback controller with feedforward compensator with cross-couplings. All analysis has been done in the presence of nonlinear system model with sensor noise (n) of 10mV /

√
Hz and sample surface variations (sinusoidal) (z S ) of amplitude 0.5 Å with the frequency of 1000 rad/sec.
For each above mentioned case, the tracking performance in horizontal x and y directions is evaluated by calculating the mean-absolute-error (MAE) where error is the difference between the reference trajectory and the actual displacement of the piezoelectric actuator. The mean-absolute-error is calculated as: where, R and Y are the reference and the actual displacements of the piezoelectric actuator respectively and N is the total number of data points. In the vertical z-direction, the objective is to maintain the tunneling current constant, while scanning the sample surface, and the performance in each case is evaluated by calculating the standard deviation (SD) of the tunneling current. The standard deviation is calculated as: where, i T is the tunneling current, m is its mean value and N is the total number of data points.

A. RASTER SCANNING
For raster scanning of the sample surface, a triangular waveform is applied to the x-axis and a smooth staircase waveform is applied to the y-axis. The sample surface of the area 1µm × 2µm is scanned in x and y directions, while keeping the tunneling current constant at 0.5nA in the vertical zdirection. First of all, the analysis is performed with the SISO H ∞ feedback controller cascaded in series with the feedforward compensator without cross-couplings (Case I). It can be observed in Fig. 7 that the precise reference tracking in x and y directions is achieved with the variations of the tunneling current (i t ) remain within the acceptable bound of ±5%. The achieved mean absolute errors are 5.23 nm and 1.42 nm in x and y directions respectively. These mean absolute errors are approximately 0.52% and 0.07% of the reference trajectories in x and y directions respectively. The achieved standard deviation of the tunneling current in the vertical zdirection is 7.18 pA. So, the SISO H ∞ feedback controller with the feedforward compensator works well and the desired performance of the system can be achieved if the crosscouplings of the piezoelectric actuator are neglected. Next, the analysis is performed with the same SISO H ∞ feedback controller cascaded in series with the feedforward compensator in the presence of cross-couplings of the piezoelectric actuator (Case II). A small increase in the tracking errors in the horizontal x and y directions can be observed in Fig. 8 but the tunneling current variations are now large and beyond the acceptable bound of ±5%. The achieved mean absolute errors are 6.31 nm (0.63% of the reference trajectory) and 1.73 nm (0.09% of the reference trajectory) in x and y directions respectively. The standard deviation of the tunneling current is 57.32 pA. This result indicates that the desired performance of the considered system can not be achieved with SISO control scheme in the presence of crosscouplings.
Finally, the analysis is performed with the suggested MIMO H ∞ feedback controller cascaded in series with the feedforward compensator in the presence of cross couplings (Case III). Fig. 9 shows that the tunneling current variations are now within the acceptable bound of ±5% even in the presence of cross-couplings. The standard deviation of the tunneling current in the vertical z-direction is 7.66 pA and the mean absolute errors in the horizontal x and y directions are 6.07 nm and 1.64 nm respectively. These mean absolute errors are actually 0.61% of the reference trajectory in xdirection and 0.08% in y-direction. All these results indicate that the desired performance of the system can be well achieved with the suggested MIMO H ∞ feedback controller cascaded in series with the feedforward compensator in the presence of cross-couplings. The achieved results in all three cases with the raster scanning are summarized in Table 2..

B. SPIRAL SCANNING
In spiral scanning, the x and y scan trajectories consist of a sinusoidal and cosine waveform of the same frequency but varying amplitude. For spiral scanning, the sample surface of the area 0.5µm × 0.5µm is scanned in the horizontal x and y directions, while keeping the tunneling current constant at 0.5 nA in the vertical z direction. For spiral scanning, first of all, the analysis is performed with the SISO H ∞ feedback controller with feedforward compensator without cross couplings (Case I). The precise reference tracking with the mean absolute errors of 1.33 nm and 1.32 nm in x and y directions respectively can be observed in Fig. 10. These achieved mean absolute errors are  actually 0.27% and 0.26% of the reference trajectories in x and y directions respectively. The reference and actual spiral scanning can also be observed in Fig. 10(e). It can also be observed that the variations of the tunneling current remain within the acceptable bound of ±5% with the standard deviation of 6.82 pA in the absence of cross-couplings. So, the desired performance in case of the spiral scanning is well achieved with the SISO control scheme in the absence of

cross-couplings.
After the analysis of the SISO control scheme in the absence of cross-coupling, next the analysis is performed with the same SISO control scheme in the presence of crosscouplings (Case II). A small increase in the tracking errors in the horizontal x and y directions can be observed in Fig.  11 as compared to Case I. So, the mean absolute error in xdirection is 1.61 nm (which is 0.32% of the scanning range in x-direction) and in y-direction is 1.56 nm (which is 0.31% of the scanning range in y-direction). In this Case II, a large variations in the tunneling current can be observed and these variations are beyond the acceptable bound of ±5%. The achieved standard deviation of the tunneling current is now 27.73 pA in the presence of cross-couplings. These results indicate that the SISO control scheme will not perform as desired in the presence of cross-couplings.
Finally, the analysis is performed with the suggested MIMO control scheme in the presence of cross-couplings (Case III). Precise reference tracking in the x and y directions with the tunneling current variations within the acceptable bound of ±5% can be observed in Fig. 12 current in the vertical z-direction is 7.62 pA. The mean absolute errors are actually 0.3% and 0.29% of the reference trajectories in the x and y directions respectively. All these results indicate that the desired performance is well achieved with the suggested MIMO feedback with feedforward control scheme in the presence of cross-couplings. The achieved results with the spiral scanning are summarized in Table 3.

C. LISSAJOUS SCANNING
In Lissajous scanning, the x and y scan trajectories consist of purely sinusoidal signals that contain slightly different frequencies. The sample surface area of 1µm × 1µm is scanned in the horizontal x and y directions, while keeping the tunneling current constant at 0.5 nA in the vertical z direction. The Lissajous scanning is first performed with the SISO H ∞ feedback controller cascaded in series with the feedforward compensator without cross-couplings (Case I). The precise reference tracking in x and y directions with reference and actual trajectories of Lissajous scanning can be observed in Fig. 13. It can also be observed in Fig. 13 that the tunneling current variations remain within the acceptable bound of ±5% in the absence of cross-couplings. The mean absolute errors of the reference tracking in x and y directions are 5.92 nm (0.59% of the reference trajectory) and 5.29 nm (0.53% of the reference trajectory) respectively. The standard deviation of the tunneling current is 6.92 pA. All these results indicate that the desired performance is well achieved with the SISO control scheme in the absence of cross-couplings.
Next, the analysis is performed with the same SISO H ∞ feedback controller with feedforward compensator in the presence of cross-couplings (Case II). It can be observed in Fig. 14 that the tracking error is now a little increased both in x and y directions. The mean absolute errors of tracking are now 7.03 nm (0.7% of the reference trajectory) and 6.42 nm (0.64% of the reference trajectory) in x and y directions respectively. A large variations in the tunneling current can also be observed in this Fig. 14 Finally, the analysis is performed for Lissajous scanning with the suggested MIMO H ∞ feedback controller cascaded in series with the feedforward compensator in the presence of cross-couplings (Case III). As compared to Case II, less tracking errors in x and y directions with acceptable variations of the tunneling current can be observed in Case III as shown in Fig. 15. The means absolute errors of tracking are now 6.77 nm (0.68% of the reference trajectory) and 6.12 nm (0.61% of the reference trajectory) in x and y directions respectively. The standard deviation of the tunneling current in the vertical z direction is now 7.67 pA. All these results with Lissajous scanning are summarized in Table 4. It is

D. TRANSIENT RESPONSE OF TUNNELING CURRENT
The transient characteristics of the tunneling current are important to analyze in order to see how fast the STM tip moves in the tunneling region (0.1 − 1 × 10 −9 m distance between the tip and the sample surface) and if there is an overshoot whenever there is a change in the reference tunneling current. Ideally, the STM tip must move very fast (settling time less than 1×10 −3 sec.) in the tunneling region to keep the distance between the tip and the sample surface constant or in other words, to keep the tunneling current constant. Other than this, ideally the overshoot must be reduced (overshoot of 5%) in order to avoid any collision between the sharp metallic tip and the sample surface. A step response of the tunneling current with the suggested MIMO H ∞ feedback controller cascaded in series with the feedforward compensator in the presence of cross-couplings is presented in Fig. 16 to analyze the transient characteristics. It can be observed in Fig. 16 that the settling time of the system is around 0.2 × 10 −3 sec. and the overshoot is almost 0%. This result indicates that a good transient performance is achieved with the suggested feedback -feedforward control scheme.

VI. CONCLUSION
In this paper, the performance of STM having a multi-axis 3D piezoelectric actuator in the presence of two control strategies (SISO and MIMO H ∞ feedback controllers cascaded in series with a feedforward compensator) is thoroughly analyzed. For controller design, a complete MIMO system is mathematically modeled in the presence of cross-couplings as well as hysteresis nonlinearity of the piezoelectric actuator. The parameters of the hysteresis model are identified from the real-time experimental data and the feedforward compensator is designed without calculating the inverse hysteresis model. Three different types of scanning trajectories (raster, Step response of the tunneling current with MIMO H∞ feedback controller cascaded in series with the feedforward compensator in the presence of cross-coupling to analyze the transient characteristics. spiral and Lissajous) are used for the performance comparison. It is evident from the achieved results that in the presence of cross-couplings, the SISO control scheme can not deliver the desired performance. A small increase in the tracking error for horizontal 2D scanning system is observed with SISO control scheme, however, the main issue is the large variations in the tunneling current (beyond the acceptable bound of ±5%) due to cross-couplings. These large variations in the tunneling current will result in a surface image that will not correspond to the reality. This issue has been resolved by the suggested MIMO control scheme. In case of the raster scanning, 86.6% improvement is observed in reducing the variations of the tunneling current with the suggested MIMO control scheme in the presence of cross-couplings, 72.5% in case of spiral scanning and 90% in case of Lissajous scanning as compared to the tunneling current variations with the generally used SISO control scheme for STM. All the presented results with different scanning trajectories indicate the efficacy of the suggested MIMO H ∞ feedback controller cascaded in series with the feedforward compensator.
It will be an interesting study in the prospective of the presented work to analyze the performance of the system by considering all the possible adverse effects of a piezoelectric actuator, like creep, hysteresis, high frequency vibrations and cross-couplings altogether with a suitable control methodology.