Robust μ-Synthesis With Dahl Model Based Feedforward Compensator Design for Piezo-Actuated Micropositioning Stage

In this article, a combined feedback-feedforward control design scheme is presented to enhance the tracking performance of a piezo-actuated micropositioning stage by compensating the nonlinear hysteretic behavior of the piezoelectric actuator and model uncertainties of the system. Detailed investigation of the presented control scheme is performed not only in simulation by analyzing the robust stability and robust performance but also in real-time with motion trajectories of multiple frequencies. To design the presented control scheme, first of all, the dynamic model of the system is identified from the real-time experimental data by using the recursive least squares parameter adaptation algorithm. Then, Dahl hysteresis model is considered to represent the nonlinear hysteretic behavior of the piezoelectric actuator. To deal with this hysteresis nonlinearity, Dahl feedforward compensator is designed without involving inverse model calculations to avoid any computational complexity. This feedforward compensator is then combined with $\mu $ -synthesis robust feedback controller which is designed in the presence of model uncertainties of the system. The presented control scheme ensures the boundedness of the closed-loop signals and the desired tracking performance of the considered micropositioning stage. Finally, experimental tests are conducted with motion trajectories of multiple frequencies for the validation of the control scheme. An average improvement of 95% in compensating the hysteresis nonlinearity and 80% in reducing the tracking error is achieved which demonstrates the efficacy of the presented control scheme.


I. INTRODUCTION
Over the past decades, the rapid advancements have been witnessed in the field of ultra-precise micro/nanopositioning stages. It has been widely acknowledged in the literature that the piezo-actuated micro/nanopositioning stages have remarkable advantages in terms of ultra-high positioning precision, nanometer or subnanometer resolution, large mechanical force, compact design, less power consumption and fast response time [1]. Due to these numerous advantages, piezoactuated positioning stages are commonly used in many applications, e.g. in scanning probe microscopy [2], advanced lithography tools for the fabrication of semiconductor The associate editor coordinating the review of this manuscript and approving it for publication was Tao Wang . integrated circuits [3], servo system of hard disk-drives [4], optical alignment systems [5], manipulation of nanoscale biological process like DNA analysis [6] and also in the manufacturing of small objects [7]. In all these applications, ultra-precise positioning with high speed and long positioning range is desired. However, there are certain challenges involved in order to achieve the desired performance of the piezo-actuated positioning stages. One of these challenges comes from the inherent hysteresis and creep nonlinearities of the piezoelectric actuator. The desired performance of the piezo-actuated positioning stages suffers or even the control system may become unstable if these nonlinearities of the piezoelectric actuator are not compensated with a suitable control methodology [8]. The hysteresis nonlinearity of the piezoelectric actuator depends on the amplitude of the applied VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ input voltage. The creep nonlinearity is related to the drift phenomenon of the output displacement of the piezoelectric actuator when subjected to a constant input voltage. However, the effects of creep phenomenon become noticeable only when the tracking is performed over extended periods of time during slow-speed operations. Another main challenge, particularly while working at micro or nano scale, is the change in the operating conditions, like ambient temperature, humidity etc., which introduces the uncertainty in the system model. Therefor, control design based on a single system model may not achieve the desired performance while working in realtime over an experimental platform [9]. Model uncertainty necessitates the design of a robust control methodology in order to achieve not only nominal stability and nominal performance but also robust stability and robust performance. All these challenges need to be addressed in order to achieve the desired performance of the piezo-actuated positioning stages.
Lots of research has been done in modeling and controlling of piezo-actuated positioning stages so far. A number of nonlinear models to represent hysteresis nonlinearity of piezoelectric actuator are investigated in the literature. Some of these models are differential based models, like Duhem model [10], Bouc-Wen model [11], Dahl model [12], and some models are operator based models, like Prandtl-Ishlinskii model [13], Preisach model [14], Maxwell model [15] and Krasnosel'skii-Pokrovskii model [16] etc. To compensate the hysteresis nonlinearity, two different control strategies are generally adopted in the literature. The first control strategy is to design a feedforward compensator as an inverse hysteresis model and then to cascade it with piezoelectric actuator in open-loop configuration [17]- [19]. This first control strategy has to deal with computational complexity while performing inversion of the actual hysteresis model. To avoid this computational complexity, a few approaches are presented in the literature which avoid calculating the inverse hysteresis model for the feedforward compensator [20], [21]. Real-time implementation of these feedforward compensators may not achieve the desired performance in the presence of modeling error and unknown disturbances. The second control strategy is an integrated design approach, where a feedback controller is generally combined with a feedforward compensator. Different feedback control algorithms, like fuzzy control [22], adaptive control [23], model predictive control [24], sliding mode control [25], robust and optimal control [26], [27] and classical proportional-integral-derivative control [28] are investigated in the literature in the presence of feedforward compensators. Other than these two control strategies, another approach to compensate the hysteresis nonlinearity is to use the chargedriven piezoelectric actuators [29]. However, this method needs to deal with increased hardware complexity. To deal with creep nonlinearity of the piezoelectric actuator, a number of models with control strategies [30]- [32] are discussed in the literature. However, working in closed-loop for a short interval of time suppresses the creep nonlinearity.
Hence, creep compensation is not discussed in this article. Regarding performance analysis of the piezo-actuated positioning stage with model uncertainties, most of the research work in the literature is based on just a single parameter variation which is the resonant frequency of the positioning stage [33]. The most effective solution to deal with this uncertainty is to damp the resonant mode of the positioning stage with a suitable damping controller [34]. An adaptive fuzzy fractional-order nonsingular terminal sliding mode controller (AFFO-NTSMC) is investigated to analyze the tracking performance of a second-order uncertain nonlinear dynamic system [35]. The presented simulation results with AFFO-NTSMC show small tracking errors with well attenuation of the chattering phenomenon, which is very common in conventional sliding mode control algorithm, in the presence of model uncertainties. Adaptive fuzzy control is also discussed in [36]- [38] to analyze the tracking problem of the uncertain nonlinear systems. The performance of the robust H ∞ feedback controller is investigated in [39] for linear systems with polytopic uncertainties. It has been observed by the authors of this article that there is a scarcity of research work about the detailed analysis in terms of robust stability and robust performance of the piezo-actuated positioning stages in the presence of model uncertainties with hysteresis nonlinearity.
In this article, the hysteresis nonlinearity of the considered piezoelectric actuator is first modeled by considering the Dahl hysteresis model and then its compensator is designed to use it as a feedforward compensator. The Dahl feedforward compensator is designed without calculating the inverse hysteresis model to avoid any computational complexity. After linearizing the hysteresis nonlinearity with the feedforward compensator, the µ-synthesis robust feedback controller is designed, in the presence of model uncertainties of the system, to enhance the tracking performance of the considered piezo-actuated micropositioning stage. Therefore, other than closed-loop nominal stability and nominal performance of the system, robust stability and robust performance analysis is also presented in this article. The main contributions of this article are as follows: • A combined feedback-feedforward control design scheme is presented, where the Dahl feedforward compensator is designed without formulating the inverse hysteresis model and the µ-synthesis robust feedback controller is designed in the presence of model uncertainties of the system. According to the authors' knowledge, the presented control scheme in this article has not been analyzed in the literature of micropositioning stages so far.
• Experimental validation of the system model as well as the presented control scheme with motion trajectories of multiple frequencies is also a part of this article. The remainder of this article is organized as follows: Section II describes a brief description of the considered piezo-actuated micropositioning stage with all the necessary details about the experimental setup, used for the real-time validation of the model and the control scheme. Modeling of the piezo-actuated micropositioning system, which includes linear time-invariant (LTI) dynamic model, uncertainty model and hysteresis model, is presented in Section III. Section IV of this article deals with the controller design. Closedloop nominal stability and nominal performance as well as robust stability and robust performance are also discussed in this section. Experimental results are thoroughly analyzed in Section V. Finally, concluding remarks are presented in Section VI.

II. DESCRIPTION OF PIEZO-ACTUATED MICROPOSITIONING STAGE
The complete block diagram of the considered piezo-actuated micropositioning system, in the presence of Dahl feedforward compensator and µ-synthesis robust feedback controller, is presented in Fig. 1. The considered micropositioning system consists of a voltage amplifier, piezoelectric actuator, capacitive position sensor, data acquisition module, and a host computer for the implementation of the control algorithm. The block diagram of the micropositioning experimental loop is presented in Fig. 2. The presented control scheme is the combination of Dahl feedforward compensator and the µ-synthesis robust feedback controller. Dahl feedforward compensator is designed to mitigate the effects of nonlinear hysteresis phenomenon of the piezoelectric actuator. The µ-synthesis robust feedback controller is designed to achieve the desired tracking performance with robustness and stability. The presented control approach ensures the boundedness of the closed-loop signals and the desired tracking performance.
The considered 1D all-ceramic insulated piezo-actuated micropositioning stage (P − 752.21 produced by Physik Instrumente GmbH & Co.) is having an integrated capacitive displacement sensor (D − 015). Flexure-guide based piezo-actuated positioning stage with all-ceramic insulation offers high travel accuracy, rapid response, high load capacity and longer lifetime than conventional polymer-insulated actuators. Flexure-guide based positioning stages have been widely used in different precision engineering applications from one degree of freedom to six degrees of freedom [40]- [42]. The travel range of the considered piezoelectric actuator is 35 µm with the resolution of 0.2 nm and maximum load capacity of 30 N. The specifications of the considered piezo-actuated micropositioning stage are listed in Table 1. The displacement (x a ) of the piezoelectric actuator is sensed by the capacitive sensor (D − 015) which can measure with subnanometer resolution. Generally, the capacitive sensor is widely used as a displacement sensor for precise measurement with subnanometer resolution and high bandwidth. This sensor has an extended measuring range of 45 µm with a resolution of 0.01 nm. The bandwidth of this sensor is 10 kHz and the analog output voltage range is from 0 to 10 V. This analog output voltage (v y ) is given to the host computer through  In the host computer, the analog output voltage (v y ), given by the I/O module, is compared with the reference input voltage (v r ) in order to find the error signal (v e ). The reference input voltage (v r ) corresponds to the desired displacement of the piezo-actuated micropositioning stage. In the presented control scheme, the feedforward compensator acts on the reference voltage (v r ), whereas the feedback controller acts on the error voltage (v e ) as shown in Fig. 1. The control action of the feedforward compensator (u f ) is summed up with the control action of the feedback controller (u b ) to generate the control input (u c ). The voltage amplifier (E − 505) receives the control input through I/O module and amplifies it with a fixed gain of 10 to derive the piezoelectric actuator. The bandwidth of the voltage amplifier is 3 kHz. The input voltage range of this amplifier is from −2 to +12 V and the output voltage range is from −30 to +130 V. The mathematical modeling of the considered piezo-actuated micropositioning stage is presented in the next section.

III. MODELING OF PIEZO-ACTUATED MICROPOSITIONING STAGE
In this section, LTI dynamic model, uncertainty model and Dahl hysteresis model of the considered piezoelectric actuator with the experimental validations are presented.

A. LINEAR-TIME-INVARIANT (LTI) DYNAMIC MODEL
The LTI dynamic model of the piezo-actuated micropositioning system is achieved from the real-time experimental data. Four main steps in order to achieve the model from the experimental data are presented in Fig. 3.
Step I: To achieve the LTI dynamic model, a sinusoidal chirp excitation signal of increasing frequency and small magnitude is applied to the piezo-actuated positioning stage in an open-loop configuration. This excitation signal has a rich frequency spectrum in order to cover the bandwidth of the system to be identified and a small magnitude to suppress the nonlinear hysteresis of the actuator.
Step II: The model complexity is selected by trial and error with six poles and three zeros. The reason for selecting the 6 th order LTI dynamic model for the considered piezo-actuated micropositioning system is to capture the high frequency vibrations while working at high speeds.
Step III: A recursive least squares (RLS) parameter adaptation algorithm is used for estimating the parameters of the LTI dynamic model. The basic idea of this algorithm is to minimize the error (prediction error) between the experimental output and the output predicted by the model, by modifying the model parameters at each sampling instant. Due to the recursive structure of the algorithm, the new value of the estimated parameters is equal to the previous value plus a correction term that will depend on the most recent measurements. Letˆ (t) is a vector of the estimated model parameters at time t, θ(t) is a measurement vector or plant model regressor vector at time t, then a priori prediction error e 0 (t + 1) can be computed at the instant t + 1 after the acquisition of the output measurement y(t + 1) as: Now, according to the recursive least squares parameter adaptation algorithm, the vector of estimated model parameters at the instant t + 1 can be computed with the help ofˆ (t) plus a correction term, as: where, W (t + 1) is a time varying gain matrix which can be computed as: Further details about the parameter adaption algorithm can be found in [43]. The achieved 6 th order LTI dynamic model G(s) of the considered piezo-actuated micropositioning system is given as follows: The identified parameters of the model are: a 0 = 2.05×10 23 , a 1 = 5.79 × 10 19 , a 2 = 1.65 × 10 16 , a 3 = 1.98 × 10 12 , Step IV: The identified LTI dynamic model is validated if the prediction error between the experimental output and the output predicted by the model asymptotically tends towards white noise. This will result in a close match between the simulated response of the identified model and the experimental response. The experimental and simulated chirp responses are compared in Fig. 4 for the validation of the model. A close match between these two plots validates the achieved LTI dynamic model of the considered system. A small difference between these plots can also be observed, which will be dealt with the robustness of the presented robust feedback controller in the presence of model uncertainties.

B. UNCERTAINTY MODEL
The characteristic equation of the identified LTI dynamic model is s 6 + a 5 s 5 + a 4 s 4 + a 3 s 3 + a 2 s 2 + a 1 s + a 0 = 0. The parameters (a 0 , a 1 , · · · , a 5 ) of this characteristic equation may vary due to changes in the operating conditions like ambient temperature, humidity, etc., which introduces the uncertainty in the system model. To account for model uncertainty, the dynamic behavior of the piezo-actuated micropositioning system is represented by a set of possible LTI models which are developed with the variation of ±5% in each parameter of the characteristic equation. By considering the input multiplicative uncertainty, the uncertain system model can be represented as: where, G ∈ is a nominal plant model, G ∈ uncertain plant model, H m multiplicative uncertainty weighting function and is any stable transfer function, satisfying ∞ < 1. The block diagram of the uncertain plant model for the piezo-actuated micropositioning system is shown in Fig. 5. According to this figure, the measured output  voltage (v y ) can be computed as: where, u f is the control action by the Dahl feedforward compensator and u b is the control action by the feedback controller. In case of the considered multiplicative uncertainty, the relative error function E m can be computed as: The identified parameters of the characteristic equation are considered to have three possible values: a nominal value and two possible values with ±5% variations around the nominal value. The total of 729 possible combinations of the relative error function are presented in Fig. 6 (dotted lines). The multiplicative uncertainty weighting function (H m ) is the bound on the multiplicative uncertainty and must be chosen in such a way that it satisfies the condition |H m (jω)| ≥ E m (ω), ∀ω. In order to satisfy this condition, the multiplicative uncertainty weighting function (H m ) of 5 th order is designed as follows: This transfer function is achieved by connecting two normalized inverse notch filters' transfer functions with a low pass filter transfer function in series. The parameter values of the uncertainty weighting function are presented in Table 2.
The achieved multiplicative uncertainty weighting function is presented in Fig. 6 (solid line). It is clear from this Fig. 6 VOLUME 8, 2020 that the proposed uncertainty weighting function satisfies the desired condition of |H m (jω)| ≥ E m (ω), ∀ω.

C. DAHL HYSTERESIS MODEL
As mentioned earlier in Section I that there are several hysteresis models which have been investigated in the literature. Some of the most commonly used hysteresis models are operator based models and some are differential based models. A common practice, to compensate hysteresis nonlinearity, is first to find out the inverse of these hysteresis models and then to cascade the inverse model, as a feedforward compensator, with the actuator. Calculating the inverse hysteresis model is often a challenging task due to its computational complexity. Other than computational complexity of inverse model calculation, real-time implementation, accuracy as well as identification process of finding parameters of the hysteresis model are other factors which must be considered for selecting a proper hysteresis model. In this article, the differential based model, particularly the Dahl hysteresis model, is considered as calculating the inverse hysteresis model, as a feedforward compensator, for an operator based model involves computational inverse modeling complexity. Dahl hysteresis model is based on the differential equation like Bouc-Wen or Duhem hysteresis models. The reason for opting Dahl hysteresis model for the considered piezoelectric actuator is its simplicity in implementation and also good accuracy in representing a large class of hysteresis. According to [12], the Dahl hysteresis model can have better accuracy than the Bouc-Wen hysteresis model with the same number of unknown model parameters. As far as Duhem hysteresis model is concerned, obtaining an inverse hysteresis model as well as the identification process of Duhem hysteresis model is not easy. All necessary details about the considered Dahl hysteresis model can be found in [44]. The Dahl hysteresis model for the piezoelectric actuator can be represented with a second-order state-space model. The state equations are: where, z 1 (t) and z 2 (t) are two intermediate state variables, x a (t) displacement of the piezoelectric actuator and α 1 , α 2 and γ a are constant parameters. The nonlinear hysteretic effect H (t) of the actuator can be represented by the following output equation: where, β 1 and β 2 are constant parameters. All constant parameters of the Dahl hysteresis model govern the shape and amplitude of the hysteresis loop. The solution of the Dahl state-space model will be used to determine the actual displacement of the piezoelectric actuator, which can be described as: where, g a is a piezoelectric coefficient and v a (t) is the applied input voltage to the piezoelectric actuator. All parameters of the Dahl hysteresis model with piezoelectric coefficient are identified from the real-time experimental data. For this purpose, triangular input voltages of different amplitudes, in order to achieve multiple positioning ranges, and frequency of 10 Hz are applied to the piezoelectric actuator. In order to identify the parameters of the Dahl hysteresis model, a nonlinear curve-fitting problem is solved in a least-square sense by using the nonlinear optimization toolbox in the MATLAB. The identified parameters of Dahl hysteresis model with piezoelectric coefficient are: α 1 = 1.76 × 10 3 , α 2 = 3.46 × 10 7 , β 1 = 0, β 2 = 3.11 × 10 7 , γ a = 30 and g a = 0.925. The experimental and simulated hysteresis loops are presented in Fig. 7(a) to Fig. 7(d).
In all these figures, a close match between the experimental and simulated hysteresis loops validates the Dahl hysteresis model for the considered piezoelectric actuator.

IV. CONTROL DESIGN FOR PIEZO-ACTUATED MICROPOSITIONING STAGE
In this section, first of all, the Dahl feedforward compensator is designed to deal with hysteresis nonlinearity of the piezoelectric actuator. Next, µ-synthesis robust feedback controller is designed, for uncertain system model, to achieve precise reference tracking. After the controller design, nominal as well as robust stability and robust performance of the closedloop system is analyzed.

A. DAHL FEEDFORWARD COMPENSATOR DESIGN
A common practice to compensate the effects of nonlinear hysteresis phenomenon is to design an inverse hysteresis model and then to use it as a feedforward compensator. Calculation of inverse hysteresis model is often cumbersome which introduces inverse modeling complexity. To avoid this complexity, a new simple approach of hysteresis compensation is presented here. According to this approach, there is no need to perform inversion of the hysteresis model, rather than only inverse of the piezoelectric coefficient (which is strictly positive) is required. The output voltage (u f (t)) of Dahl hysteresis feedforward compensator can be computed as: where, x r (t) is the desired displacement of the piezoelectric actuator. The desired displacement of the piezoelectric actuator is related to the reference input voltage (v r (t)) through a gain of 3 µm/V. For example, in order to achieve the desired displacement of 12 µm, the reference input voltage of 4 V must be applied. This output voltage (u f (t)) of Dahl hysteresis feedforward compensator will be added with the output voltage (u b (t)) of the robust feedback controller to generate the control input (u c (t)) for the micropositioning system. The simulation results of hysteresis compensation with Dahl feedforward compensator is presented in Fig. 8. This achieved result indicates the perfect compensation of nonlinear hysteresis phenomenon with the presented feedforward compensator approach. Slight hysteresis nonlinearity still can be observed in the real-time experimental results due to some mismatch between the experimental and simulated hysteresis loops.

B. µ-SYNTHESIS ROBUST FEEDBACK CONTROLLER DESIGN
To enhance the tracking performance of the considered piezo-actuated micropositioning stage, a µ-synthesis robust feedback controller is designed for LTI dynamic model in the presence of model uncertainties. As hysteresis nonlinearity is well compensated by the proposed Dahl feedforward compensator, hysteresis nonlinearity as well as its compensator is not considered in the design of the feedback controller.
Desired Tracking Performance: The objective of the presented control scheme is to achieve not only nominal stability and nominal performance, but also the robust stability and the robust performance. The desired performance of the considered piezo-actuated micropositioning stage is to achieve precise reference tracking of 12 µm. As the tracking error increases with the increase of the tracking frequency, the peak-to-peak tracking error must be less than 1% for the tracking frequency till 10 Hz and less than 5% for the maximum tracking frequency of 50 Hz.
The general control configuration, used for µ-synthesis robust feedback controller design, is presented in Fig. 9. In order to achieve the desired performance specifications, two performance weighting functions (H e and H y ) have been imposed in the control loop: H e over the error voltage (v e ) and H y over the output measured voltage (v y ), as shown in Fig. 9, where y 1 and y 2 are two controlled outputs. The performance weighting function (H e ) is designed by considering the desired performance in terms of small tracking error, large bandwidth and good robustness w.r.t. model uncertainties. H y is designed in order to achieve good reference tracking, better noise rejection and to limit the overshoots. The designed performance weighting functions are:   [y y 1 y 2 v e ] T , as shown in Fig. 9, can be described as: If P(s) = P 11 (s) P 12 (s)  (17) Matrix M (s), having input vector [u v r ] T and output vector [y y 1 y 2 ] T can be described as: Now, in order to analyze the robust stability and robust performance of the closed-loop system, the structure of matrix is defined as: (21) Here, the first block f of structured uncertainty represents the fictitious block with inputs [y 1 y 2 ] T and output v r . The second block r of structured uncertainty represents the real parametric uncertainties with input y and output u . The necessary conditions for nominal as well as robust stability and performance are presented in Table 3.
The design of performance weighting functions (H e and H y ) is an important step for the design of the considered µ-synthesis robust feedback controller. The formula (15) for H e has been proposed by keeping in mind that the inverse of this performance weighting function imposes the upper bound (template) over the closed-loop output sensitivity function S (as mentioned for the nominal performance condition in Table 3). Ideally, S must provide maximum attenuation in the low frequency area in order to minimize the tracking error, large bandwidth (corresponds to a faster rise time) and S ∞ less than 6 dB in order to achieve good robustness w.r.t. model uncertainties.
The performance weighting function H e has three variables ( e , M e and ω e ) in (15) and the values of these variables help to adjust the template in order to achieve the best possible shape of the closed-loop output sensitivity function S. The value of e moves the template upward or downward in low frequency, M e moves the template upward or downward in high frequency and ω e moves the template left or right to achieve the best possible bandwidth. Similarly, the inverse of performance weighting function H y imposes the upper bound (template) over the closed-loop complementary sensitivity function T (as mentioned for the nominal performance condition in Table 3). Ideally, T must remain at 0 dB in low frequencies in order to achieve unity feedback gain, must provide maximum attenuation in high frequencies for better noise rejection and T ∞ less than 3.5 dB in order to limit the overshoots. The values of three variables ( y , M y and ω y ) of the performance weighting function H y in (15) help to adjust the template over T in order to achieve its desired shape. DK-iteration method is used in order to design the µ-synthesis robust feedback controller. The starting point of this iterative method is the upper bound on µ in terms of the scaled maximum singular value (σ ) as: where, D is the set of matrices with the property that D = D for every D ∈ D . Now, the idea of µ-synthesis robust feedback controller is to minimize, over all stabilizing controllers K (s), the peak value over frequency of this upper bound of µ: Further necessary details about DK-iteration procedure can be found in [45]. The achieved controller transfer function is of 36 th order with upper bound of µ is 0.71. For the real-time implementation, the order of the controller transfer function is reduced. For this purpose, first the Hankel singular values are computed and then the states with relatively small Hankel singular values are discarded. Finally, the achieved 6 th order controller transfer function is given as follows: K (s) = n 5 s 5 +n 4 s 4 +n 3 s 3 +n 2 s 2 +n 1 s+n 0 s 6 +d 5 s 5 +d 4 s 4 +d 3 s 3 +d 2 s 2 +d 1 s+d 0 (24) where, n 0 = 4.  (S, T, KS and GS) are presented in Fig. 10. This figure demonstrates that all four closed-loop sensitivity functions are stable, which is the necessary condition for having the closed-loop nominal stability as mentioned in Table 3. The step response (Fig. 10(a)) of the output sensitivity function (S) shows the behavior of the error voltage (v e ) w.r.t. reference voltage (v r ). It is evident from this step response that v e → 0 as time t → ∞. The step response ( Fig. 10(b)) of the complementary sensitivity function (T) shows the behavior of the output voltage (v y ) w.r.t. reference input voltage (v r ). It is evident from this step response that the closed-loop system has unity feedback gain with good transient characteristics.
The step response (Fig. 10(c)) of the input sensitivity function (KS) shows the behavior of the control input (u c ) w.r.t. reference voltage (v r ). The step response ( Fig. 10(d)) of plant sensitivity function (GS) shows the behavior of the output voltage (v y ) w.r.t. any possible disturbance at the system input. It shows that any input disturbance will be well rejected by the presented control scheme.

2) CLOSED-LOOP NOMINAL PERFORMANCE
In order to analyze the nominal performance of the closedloop system, the conditions mentioned in Table 3 must be verified. The desired conditions of nominal performance are well achieved as H e S ∞ = 0.81 and H y T ∞ = 0.92. As the achieved H e S ∞ < 1 and H y T ∞ < 1, it means |S(jω)| < 1/|H e (jω)|, ∀ω and |T(jω)| < 1/|H y (jω)|, ∀ω which is also evident from the Fig. 11(a) and Fig. 11(b), respectively. This figure shows that the two sensitivity functions S and T remain under the templates which were imposed by the inverse of the performance weighting functions H e and H y respectively.

3) CLOSED-LOOP ROBUST STABILITY
In order to analyze the robust stability of the closed-loop system with model uncertainty, the µ plot, as per the condition mentioned in Table 3, is presented in Fig. 12(a). According to VOLUME 8, 2020  this figure, the desired condition of robust stability is satisfied with the maximal value of µ is 0.48 which must be less than unity. It means that the closed-loop stability is guaranteed for all perturbations with ∞ < 1/0.48 = 2.08.

4) CLOSED-LOOP ROBUST PERFORMANCE
Robust performance of the closed-loop system is also achieved as the desired condition, mentioned in Table 3, is satisfied and presented in Fig. 12(b). In this case, the achieved maximal value of µ is 0.71. It means that the robust performance of the closed-loop system is ensured for all perturbations with ∞ < 1/0.71 = 1.41.

V. EXPERIMENTAL RESULTS
In this section, real-time experimental results are presented for the validation of the presented control scheme. All experiments are performed with the sampling frequency of 10 kHz. The triangular reference trajectories are selected in this article for an extensive real-time experimental analysis to verify the performance of the presented control scheme. Unlike sinusoidal trajectories, which are smooth trajectories, the triangular trajectories are non-smooth trajectories and contain high frequency harmonic components, increasing the difficulty in control. The triangular trajectories are widely utilized in many applications and a very common application is the scanning probe microscopy where a sample surface is scanned to generate its topographic image with an atomicscale resolution.

A. HYSTERESIS COMPENSATION
In this sub-section, the inherent nonlinear hysteresis phenomenon of the piezoelectric actuator is analyzed in openloop (without any compensator and with Dahl feedforward compensator) as well as in closed-loop with the presented control scheme (µ-synthesis robust feedback controller combined with the Dahl feedforward compensator). This analysis is performed with the real-time experimental results by applying multiple triangular input voltages of 10 V, 20 V, 30 V and 40 V to the piezoelectric actuator with tracking frequency of 10 Hz. The reason for selecting multiple input voltages is to demonstrate the effectiveness of the presented control scheme in compensating the hysteresis nonlinearity. Hysteresis percentage is calculated in order to analyze the performance w.r.t. hysteresis compensation. To calculate the hysteresis percentage, first, the midpoint of the applied input voltage (V M ) to the piezoelectric actuator is calculated by using the following midpoint formula as shown in Fig. 13: where, V MAX and V MIN are the applied maximum and minimum voltages to the piezoelectric actuator respectively. The displacements (X UP and X LO ) of the piezoelectric actuator, as shown in Fig. 13, corresponding to the midpoint of the applied input voltage (V M ) is noted which is used to calculate the hysteresis percentage as: where, X MAX and X MIN are the displacements of the piezoelectric actuator corresponding to applied input voltages of V MAX and V MIN respectively. Hysteresis percentages of 6.19%, 9.08%, 11.16% and 12.44% are observed in the open-loop configuration without any compensator, as shown in Fig. 14(a) to Fig. 14(d), whereas 0.45%, 0.81%, 1.23% and 2.12% are achieved with Dahl feedforward compensator, as presented in Fig. 14(e) to Fig. 14(h), corresponding to input voltages of 10 V, 20 V, 30 V and 40 V respectively to the piezoelectric actuator. These results demonstrate the effectiveness of the Dahl feedforward compensator as an improvement of 92.73%, 91.07%, 88.98% and 82.96% is achieved in compensating the hysteresis nonlinearity as compared to the open-loop case without any compensator, corresponding to input voltages from 10 V to 40 V respectively. To further compensate the hysteresis nonlinearity, analysis is finally performed in closed-loop with the presented closed-loop control scheme. In this case, hysteresis percentages of 0.31%, 0.37%, 0.41% and 0.44% are observed corresponding to the input voltages of 10 V, 20 V, 30 V and 40 V respectively, as shown in Fig. 14(i) to Fig. 14   the piezoelectric actuator better than the Dahl feedforward compensator only. The summary of the achieved results is presented in Table 4.

B. TRACKING ERROR ANALYSIS
In this sub-section, the real-time tracking error between the desired triangular displacement of 12 µm with multiple tracking frequencies of 1 Hz, 10 Hz, 20 Hz, 30 Hz, 40 Hz and 50 Hz and the actual displacement of the piezo-actuated micropositioning stage is analyzed. The experiments were performed with tracking frequency from 1 Hz to 50 Hz in order to demonstrate the capability of the suggested control scheme that it works effectively with motion trajectories of multiple frequencies. The parameters of the performance weighting functions (H e and H y ) are chosen in such a way that the tracking error of less than 5% is achieved with the maximum considered tracking frequency of 50 Hz. Indeed, it is possible to work in a higher frequency but then the parameters of the performance weighting functions need to be modified first in order to achieve the desired performance specifications. Working in higher frequency may also introduce the phenomenon of high frequency vibrations which motivates to incorporate some damping controller [46], [47] in order to damp the first resonant mode of the system and accordingly to suppress the high frequency vibrations.
The proposed control scheme (the combination of µ-synthesis robust feedback controller with Dahl feedforward compensator) is now analyzed to further enhance the real-time tracking performance. Now, the tracking errors of 0.51%, 0.94%, 1.46%, 2.39%, 3.71% and 4.58% are achieved with corresponding tracking frequencies from 1 Hz to 50 Hz respectively, as shown in Fig. 15(m) to Fig. 15(r). These achieved results show significant improvement in reducing the tracking error with the presented control scheme. The improvements of 93.04%, 90.07%, 86.17%, 79.61%, 69.41% and 64.71%, as compared to the open-loop results with no compensator, and 85.59%, 78.09%, 70.39%, 54.73%, 35.14% and 26.01%, as compared to the Dahl feedforward compensator, with corresponding tracking frequencies from 1 Hz to 50 Hz, are achieved respectively. These experimental results show that the desired tracking performance is achieved with the presented control scheme. The summary of achieved experimental results in terms of tracking error and the improvements are presented in Table 5.
It has been observed from the achieved experimental results that the feedforward compensator can reduce hysteresis percentage significantly but the desired tracking performance can not be achieved with only a feedforward compensator, particularly in the presence of model uncertainties and external disturbances. A feedback controller is indeed required to further enhance the tracking performance. It has also been observed that the tracking errors increase gradually as the tracking frequency increases. High frequency vibrations may also appear in the output displacement with high tracking speeds. Therefor, the control strategy needs to be adjusted accordingly. The desired tracking performance of the considered piezo-actuated positioning stage has been well achieved with the presented control scheme. The achieved real-time tracking performance in this article is now compared with some other combined feedforwardfeedback control schemes presented in the literature [13], [28], [48]- [55]. A comparative study has been performed with those papers in the literature having real-time experimental results with similar tracking frequencies as presented in this article. Figure 16 shows the outcome of this comparison. Overall, it can be observed in Fig. 16 that the achieved peak-to-peak tracking error in this article is either VOLUME 8, 2020 smaller than or comparable to the tracking errors presented in other papers.

VI. CONCLUSION
In this article, the real-time tracking performance of the piezo-actuated micropositioning stage has been analyzed in open-loop (without any compensator and with Dahl feedforward compensator) as well as in closed-loop with the proposed control scheme which combines the µ-synthesis robust feedback controller with the Dahl feedforward compensator. The plant model uncertainties are considered during the controller design in order to achieve the robust stability and the robust performance. Real-time experimental results demonstrate, for the input applied voltages from 10 V to 40 V to the piezoelectric actuator, an average improvement of 95.93% in terms of hysteresis compensation with the proposed feedback control scheme, as compared to the open-loop system without compensator, and 57.84% as compared to the Dahl feedforward compensator. Similarly, for the tracking frequency from 1 Hz to 50 Hz, an average improvement of 80.5%, in terms of tracking precision, is achieved with the proposed feedback control scheme, as compared to the open-loop system without compensator, and 58.33% as compared to the Dahl feedforward compensator. All the achieved experimental results demonstrate the effectiveness of the presented control scheme.
Further experimental analysis with circular or even random displacements as well as with some other advanced control algorithms will be performed in the prospective work. Control design for a 2D/3D micropositioning system, to deal with a major corresponding control issue of cross-coupling, will also be considered in the perspective work.