Abstract
Nanopositioning stages are widely used in high-precision positioning applications. However, they suffer from an intrinsic hysteretic behavior, which deteriorates their tracking performance. This study proposes an adaptive conditional servocompensator (ACS) to compensate the effect of the hysteresis when tracking periodic references. The nanopositioning system is modeled as a linear system cascaded with hysteresis at the input side. The hysteresis is modeled with a modified Prandtl–Ishlinskii (MPI) operator. With an approximate inverse MPI operator placed before the system hysteresis operator, the resulting system takes a semi-affine form. The design of the ACS consists of two stages: first, we design a continuously implemented sliding mode control (SMC) law. The hysteresis inversion error is treated as a matched disturbance, and an analytical bound on the inversion error is used to minimize the conservativeness of the SMC design. The second part of the controller is the ACS. Under mild assumptions, we establish the well-posedness and periodic stability of the closed-loop system. In particular, the solution of the closed-loop error system will converge exponentially to a unique periodic solution in the neighborhood of zero. The efficacy of the proposed controller is verified experimentally on a commercial nanopositioning device under different types of periodic reference inputs, via comparison with multiple inversion-based and inversion-free approaches.
1 Introduction
Micro/nanopositioning stages driven by piezoelectric actuators are widely used in applications with precision requirements, such as atomic force microscopes [1,2], micromanipulators [3,4], ultraprecision grinding operation [5], mechanical nanomanufacturing system for nanomilling [6], high-precision electrochemical etching-based micromachining [7], nanofabrication of materials [8], and investigation of biological systems over scales ranging from single-molecules to whole cells [9,10]. The critical requirement of precision for these applications poses a challenge of dealing with intrinsic nonlinearities like creep and hysteresis of piezoelectric actuators [11]. Particularly, ignoring hysteresis nonlinearity could lead to either poor tracking performance [12] or unstable responses [13]. Therefore, to ensure the desired accuracy, it is needed to compensate such nonlinearities to remove undesired harmonics in the closed-loop system [14].
A number of techniques and methodologies have been developed in the literature for modeling and control of hysteric behavior. Many mathematical models have been developed to depict the hysteresis phenomenon. Examples of these models include Duhem model [15], Maxwell resistive capacitor model [16], Bouc–Wen model [17], Prandtl–Ishlinskii (PI) model [18], and Preisach model [19,20]. One popular approach to design control systems involves the use of inverse hysteresis models in feedforward open-loop scheme to mitigate the effect of the hysteresis [16,20,21].
Despite the reasonable tracking performance achieved using open-loop inverse compensation, it has been shown that it is necessary to ensure system robustness against model uncertainties and external disturbances [22]. Therefore, the inversion-based feedforward control scheme is often combined with a feedback control law to achieve robustness and enhance tracking performance. In that sense, the inversion error is considered as a matched disturbance, and the feedback control is designed to mitigate its effect on the system performance. Many methodologies along this line have been reported in the literature, including, for example, proportional-integral-derivative-based controller [20,23,24], control [25], iterative control [26,27], model reference adaptive inverse control [28], internal model-based servocompensator [29,30], sliding mode control (SMC) [31–34], and hysteretic perturbation estimation [35–38]. A disturbance observer combined with hysteresis inversion is presented in Ref. [39]. The disturbance observer utilizes an internal model-based estimation of the exogenous disturbances, with an assumption that the internal model dynamics have at least an eigenvalue at the origin. Another approach combines the disturbance observer with repetitive control [40], where a low-pass filter is designed to behave approximately as the nominal dynamics. However, for all these inversion-based approaches, the achieved tracking precision will depend mainly on the smallness of the hysteresis inversion error, which is highly dependent on the accuracy of the hysteresis model and its identified parameters.
Another direction in the control of such hysteretic systems is to implement feedback control without explicit hysteresis inversion. For example, an implicit (pseudo) inverse approach with adaptive sliding mode control was introduced in Ref. [41]. In a similar fashion, the authors of Ref. [42] proposed an implicit inverse approach combined with model reference adaptive control.
Yet another class of approaches treats the hysteretic disturbance and other uncertainties as a lumped matched disturbance, which is estimated and compensated accordingly. Examples of work taking this methodology include disturbance observer combined with sliding mode control [43], uncertainty and disturbance estimator [44], active disturbance rejection control [45,46], extended high-gain observer [47], and dynamic inversion based on extended high-gain observer [48,49]. The hysteresis estimation approach does not require necessarily an inverse hysteresis model to be used in the design of the closed-loop control system.
In this paper, we focus on the design of an adaptive conditional servocompensator (ACS) for a class of systems with hysteresis to track periodic references with high precision. We note that tracking periodic references has broad applications for nanopositioners, for example, in atomic force microscopy. The hysteresis is assumed to be modeled by a modified Prandtl–Ishlinskii (MPI) hysteresis operator [50], which outperforms the classical PI operator with its ability to incorporate asymmetric hysteretic behavior. The paper has the following major contributions:
The hysteresis inversion error is analyzed, and an analytical bound on the inversion error is derived and used in the controller design. As observed in experiments, the controller shows that using this analytical bound gives less conservative results as compared to the case when a constant bound is used.
An output feedback controller is designed using adaptive conditional servocompensator by assuming that the residual disturbance due to imperfect hysteresis inversion is composed of a finite number of unknown frequencies. The unmeasured states are estimated by a high-gain observer.
Periodic stability analysis is conducted using contraction mapping arguments by following the stability analysis framework introduced in Ref. [51]. This approach is useful in establishing the periodic stability in a less conservative manner as compared to Lyapunov-based stability arguments under the smallness assumption for the hysteretic inversion perturbation. However, the main challenge is that our closed-loop control system does not fit exactly the system model assumed in Ref. [51] due to the inclusion of nonsmooth terms in our control law. We are able to establish, under mild assumptions, that the closed-loop system solution will converge exponentially to a unique periodic solution when the inversion error is sufficiently small.
The proposed control approach is experimentally evaluated on a commercial piezoelectric nanopositioner. It shows superior precision tracking performance as compared to for other control approaches implemented on the same apparatus. The four competing control approaches used for comparison are sliding mode control [34], single harmonic servocompensator (SHSC) and multiharmonic servocompensator (MHSC) [29], the dynamic inversion based on extended high-gain observer [49], and a classical PI controller without hysteresis inversion.
It is worth pointing out that part of this work included our preliminary results in Ref. [52]. In this paper, we have extended the work to more thorough and in-depth theoretical analysis by establishing the well-posedness and periodic stability results. In addition, we have added more experiments considering sawtooth and van der Pol references to challenge the controller. In addition, we have prepared a comparative study with other control approaches. We have another relevant publication [53], in which we designed an inversion-free adaptive conditional servocompensator. The main difference between Ref. [53] and the current paper is that Ref. [53] does not require an explicit hysteresis inversion. However, Ref. [53] requires a restrictive assumption on the MPI operator, which compromises its generality in representing the asymmetric hysteresis behavior. The approach in Ref. [53] also requires a low-pass filter with a PI operator at its input. One value added in this work as compared to Ref. [53] is that we are able to establish the periodicity of the closed-loop system variables. This could not be achieved in Ref. [53] because of large hysteretic perturbations in the absence of inverse compensation.
The remaining sections are organized as follows. In Sec. 2, the problem formulation, the system model with hysteresis, and the derivation of the analytical bound of the inversion error are presented. Section 3 explains the adaptive conditional compensator design. In Sec. 4, the periodic stability analysis for the closed-loop system with hysteresis perturbation is discussed. Experimental results are given in Sec. 5. Finally, our conclusion is provided in Sec. 6.
2 Problem Formulation
2.1 System Model With Hysteresis Nonlinearity.
where , ai's and b > 0 are the system parameters. The objective is to make the system output track a desired reference input , which is assumed to obey the following assumption.

Schematic of the class of systems considered, with linear dynamics proceeded by a hysteresis operator
Assumption 1. The desired reference yd and its time-derivatives up to order n are piecewise continuous in t, bounded for all, and T-periodic (i.e.,) for some.
2.2 Modified Prandtl–Ishlinskii Hysteresis Operator.
where ubk is the output of the backlash operator, and is the initial state. In essence, the PI operator consists of weighted integral of a continuum of backlash operators, which makes it an infinite-dimensional operator [55]. Due to practical consideration, a finite-dimensional PI operator is often considered, which is represented by a weighted sum of a finite number of backlash operators. Accordingly, the following assumption is made for the MPI operator.
Assumption 2. The hysteresis nonlinearity Fh is modeled with a finite-dimensional MPI operator, which consists of (q + 1) backlash operators and () one-sided deadzone operators, where.
where u1, u2 are two different inputs in the set .
By combining the two inequalities (16) and (17), we get Eq. (15).▪
2.3 Inversion of the Modified Prandtl–Ishlinskii Operator.
As shown in Fig. 3, the hysteresis inversion is achieved by cascading an inverse MPI operator with the MPI hysteresis operator Fh. Let denotes the approximated forward MPI operator resulted from the model identification.
Assumption 3. For the MPI operator (12), only the values of the radii vector r and the thresholds vector d are known.
where and are the individual inverse PI and deadzone opertaors, respectively, and and are the vectors of the inverse PI operator and inverse deadzone operator vectors, respectively. The variables and are the vectors of the inverse PI operator state and its initial state, respectively. And and are the weights for the individual inverse backlash and deadzone operators, respectively, where and . Let and be the thresholds vectors of the inverse PI operator and inverse deadzone operator, respectively.
Remark 2. Due to space limitation, the procedure of how to calculate the inverse MPI operators parameters' vectors , and has been omitted. For more details about these calculations, the readers may consult Ref. [50].
where denotes the operator resulted from the inversion process. In the following proposition, we will show that the output δinv of the operator obeys a growth condition, whose upper bound is a linear function of the input uin, and it can be used later to design a less conservative controller as compared to the case when the inversion error is bounded by a constant such that , where is some positive constant.
whereis the maximum perturbation of the weight vectors, andis the infinity norm.
Finally, by inserting Eq. (32) back into Eq. (31) and arranging the terms, we can obtain Eq. (23).▪
The smallness of the inversion error bound depends directly on the maximum perturbation , which depends on how accurate the MPI hysteresis model is. Therefore, the following assumption is made to characterize the smallness of the inversion error in the closed-loop system.
where, and the hysteresis operatoris the composite MPI operator due to the inversion.
3 Adaptive Output Feedback Controller Design
3.1 Continuously Implemented Sliding Mode Control Law Design.
where and .
then the control law (36) achieves a nonzero steady-state error. In other words, if μ is chosen small enough, the closed-loop system trajectory will reach the boundary layer and will stay in for all future time. The purpose of using the saturation function instead of the function in the switching control law (38) is to avoid the chattering of the control action; however, the drawback is that the error e will be instead of zero. To mitigate the residual error, we use the adaptive conditional servomechanism [54,61], which will be discussed in Sec. 3.2.
where is the estimated state of ei, ε is a very small positive design parameter, and hi is the estimation gain for the ith state, where the gains are chosen such that the polynomial is Hurwitz. Accordingly, the desired control law uin(36) and the surface function ξc(35) are modified by replacing e with .
In order to avoid the effect of observer peaking, the observer states are saturated before being plugged into the control law. This remedy, suggested in Ref. [62], will make the control law globally bounded in its arguments in the domain of interest.
3.2 Adaptive Conditional Servocompensator Design.
In the boundary-layer phase, due to the hysteresis inversion error, there is a nonvanishing matched perturbation δinv. In theory, the disturbance could contain an infinite number of harmonics of the reference signal frequency yd. However, for practical reasons and based on the adaptive conditional servomechanism theory [54,61], we will assume a finite number of frequencies to be estimated by the conditional servocompensator. In particular, the spectra of tracking errors in our experiments (see Sec. 5) are well approximated by a few harmonic elements, providing support for the assumption below.
where the matrix is Hurwitz. Therefore, if is will be also.
The purpose of the function is to keep constant outside the boundary set .
Figure 5 provides a more comprehensive picture of the overall closed-loop system including the control law (36), the observer (42), and the adaptation law (47).
3.3 Output Feedback Closed-Loop System Dynamics.
Notice that the closed-loop system dynamics (48) include some nonlinearities, for example, the saturation function in the control law (38) and the projection function in adaptation control law (47). For the purpose of conducting the analysis in Sec. 4, the following assumption is needed. This is mainly due to the necessity to have a unique T-periodic solution for the hysteresis-free closed-loop system, which cannot be easily established for such nonlinear system. In addition to that, Lemma 1, in which the exponential stability of the hysteresis-free closed-loop system is established, is needed in Sec. 4 analysis (in particular, the proof of Theorem 2).
Assumption 6. If the desired reference input yd is T-periodic, then there exists a unique T-periodic solutionfor the hysteresis-free closed-loop systemin Eq. (48).
where. Then there issuch that for every, there issuch that for everyandand for all initial conditions, ifis persistently exciting, then the closed-loop variables vectoris boundedand the hysteresis-free closed-loop systemhas an exponentially stable equilibrium point at, whereis the regression vector, andis the servocompensator state vector in the boundary-layer stage.
The proof of this lemma is carried out by repeating the steps of the proof of Theorem 1 in Ref. [54] for the hysteresis-free closed-loop system .
4 Well-Posedness and Periodic Stability of the Closed-Loop System With Hysteresis Inversion Perturbations
Before proving the existence of an exponentially stable, periodic solution of the closed-loop system dynamics under hysteresis inversion (48), we need to establish that the system (48) is well-posed. By establishing well-posedness, we mean establishing the existence and uniqueness of the solution of the closed-loop system (48).
is satisfied for anyand, where tu > 0. Then there exists, such that the system (48) has a unique solutionfor allandover the time interval.
The proof of this theorem can be found in the Appendix. By establishing the well-posedness of the hysteretic closed-loop system (48), we are now prepared to prove its periodic stability. Define and . Under Assumptions 1–6, will be T-periodic and vT will also be T-periodic, but after some transient period of time. To prove the existence of exponentially stable periodic solution of the hysteretic closed-loop system (48), we need to establish the existence of a contraction property for the composite hysteresis operator resulted from the inversion. This property can be established for a T-periodic reference input if and satisfy the following condition.
whereand.
Theorem 2. (Periodic stability of the hysteretic closed-loop system) Consider the hysteretic closed-loop system (48). Let Assumptions 1–7 be satisfied and letandbe compact sets. Assume. Under T-periodic desired referenceand under the exponential stability of the hysteretic-free closed-loop system (49), there exists, such that for all the initial conditionsand, the solutionof the hysteretic closed-loop system (48) will converge exponentially to a unique periodic solution.
where and are positive constants and are function of the constants and of inequality (23). From Lemma 1, under the persistency of excitation of the regressor vector , we can show that the hysteresis-free closed-loop system (49) is T-convergent about [51]. From Theorem 1, we have established the existence and uniqueness (well-posedness) of the solution of the hysteretic closed-loop system (48). Therefore, by following similar steps to those of Theorem (2.1) of Ref. [51], we can establish that the solution of the hysteretic closed-loop system (48) will converge exponentially to a unique periodic solution when εh is sufficiently small.▪
In Theorem 2, we have established that the solution of the closed-loop system converges exponentially to a periodic solution provided the inversion error is sufficiently small. Furthermore, it can be shown that an ultimate bound on the tracking error can be reduced by reducing the controller parameters μ and ε. The ultimate boundedness can be established by following similar steps in the proofs of Theorems 1 and 2 of Ref. [53]. The first step is to show that the closed-loop systems variables in the reaching phase will converge exponentially to a positively invariant set that is parameterized by the parameters (μ and ε), which can shrink to zero if these two parameters are pushed to zero. The second step is to establish that the closed-loop system variables in the boundary-layer phase are ultimately bounded by a bound that depends on the controller parameters (μ and ε) and the inversion error perturbation .
5 Experimental Results
In this section, we examine the performance of the proposed control scheme by implementing tracking experiments on a commercial piezo-actuated nanopositioner stage (Nano-OP65) shown in Fig. 6. This platform, manufactured by Mad City Labs Inc., provides a practical tool to benchmark our controller in handling hysteretic disturbances. Position measurement is provided by a built-in capacitive sensor, where the travel range of the nanopositioner is . The power amplifier unit (Nano-Drive, Mad City Labs, Inc., Madison, WI) drives the piezo actuator and has a gain of 15. In the system setup, the manipulated control input is the one to the power amplifier not the actual voltage input to the piezo actuator. For real-time implementation, the controller is deployed to dSPACE (DS1104) platform using matlab/simulink real-time coder tools.

Experimental setup of the nanopositioner system. (a) The complete setup including the nanopositioner stage Nano-OP65, Nano-Drive power amplifier unit, and the dSPACE DS1104 data acquisition interface unit and (b) magnified picture of the nanopositioner stage Nano-OP65.
The linear part of the model is identified using frequency-based identification methods, and its parameters are found to be , and [65]. The identified model is found well-representative of the system behavior with the first resonant frequency of and bandwidth of . Note that due to the high resonant frequency of the system, the identified parameters are very large.
To ensure periodic stability, one major assumption of Theorem 2 is . To comply with this assumption, let , then by using formulas (24) and (25), the error bound (23) constants can be computed as and . The rest of the switching function (41) parameters are chosen as . We chose the sliding function (35) constant and the boundary-layer width constant μ = 1000. The boundary-layer width μ is chosen by gradually reducing it until we reach a point where the surface function starts to chatter. We then take a value above this threshold to maintain the device safety. Notice that the high-gain observer parameter ε is chosen smaller than the parameter μ such that the high-gain observer is the fastest portion of the dynamics.
For the adaptive servocompensator, we assume that the residual disturbance in the boundary-layer phase (due to hysteresis inversion) has only in its frequency spectrum the fundamental frequency of the desired reference input. Moreover, we assume that there is an additional bias disturbance term alongside the periodic disturbance terms. As a result, the internal model will be a third-order model (namely, a second-order model augmented with an integrator state).
with frequency . It is worth mentioning that we published part of our evaluation results using sinusoidal reference in Ref. [52]. Therefore, we are not going to repeat these results in this paper. In Tables 1 and 2, we conduct a comparison between the achieved tracking error accuracy of our proposed approach in the boundary-layer phase as compared to other control approaches proposed in previous projects implemented on the same experimental setup. Those approaches are (a) the inversion-based sliding mode controller proposed in Ref. [34] and it will be abbreviated as SMC, (b) the results of Ref. [29], in which both of SHSC and MHSC are designed and implemented, (c) the results obtained from combining extended high-gain observer and the dynamic inversion (EHGO–DI) approaches, which appear in Ref. [49], and (d) a classical PI controller without hysteresis inversion implemented experimentally by the authors and its gains are chosen to yield the best possible performance.
Percentage of mean tracking error (mean ) with respect to the maximum peak-to-peak value of the reference under sinusoidal reference input for the proposed controller versus competing methods
Frequency (Hz) | SMC | SHSC | MHSC | EHGO–DI | PI | Inv-B ACS |
---|---|---|---|---|---|---|
5 | 0.0595 | 0.3245 | 0.1355 | 0.0672 | 0.0736 | 0.0016 |
25 | 0.3100 | 0.3535 | 0.1340 | 0.0665 | 0.0939 | 0.0041 |
50 | 0.3300 | 0.3850 | 0.1420 | 0.0686 | 0.1498 | 0.0101 |
100 | 0.4150 | 0.4075 | 0.1760 | 0.1026 | 0.2897 | 0.0148 |
Frequency (Hz) | SMC | SHSC | MHSC | EHGO–DI | PI | Inv-B ACS |
---|---|---|---|---|---|---|
5 | 0.0595 | 0.3245 | 0.1355 | 0.0672 | 0.0736 | 0.0016 |
25 | 0.3100 | 0.3535 | 0.1340 | 0.0665 | 0.0939 | 0.0041 |
50 | 0.3300 | 0.3850 | 0.1420 | 0.0686 | 0.1498 | 0.0101 |
100 | 0.4150 | 0.4075 | 0.1760 | 0.1026 | 0.2897 | 0.0148 |
Percentage of peak tracking error (max ) with respect to the maximum peak-to-peak value of the reference under sinusoidal reference input for the proposed controller versus competing methods
Frequency (Hz) | SMC | SHSC | MHSC | EHGO–DI | PI | Inv-B ACS |
---|---|---|---|---|---|---|
5 | 0.4750 | 0.8600 | 0.4495 | 0.1153 | 0.7894 | 0.0083 |
25 | 0.8500 | 0.9250 | 0.4405 | 0.1383 | 0.8422 | 0.0212 |
50 | 1.1250 | 0.9650 | 0.5050 | 0.1821 | 1.0058 | 0.0465 |
100 | 1.3750 | 1.1900 | 0.7850 | 0.3333 | 1.5185 | 0.0610 |
Frequency (Hz) | SMC | SHSC | MHSC | EHGO–DI | PI | Inv-B ACS |
---|---|---|---|---|---|---|
5 | 0.4750 | 0.8600 | 0.4495 | 0.1153 | 0.7894 | 0.0083 |
25 | 0.8500 | 0.9250 | 0.4405 | 0.1383 | 0.8422 | 0.0212 |
50 | 1.1250 | 0.9650 | 0.5050 | 0.1821 | 1.0058 | 0.0465 |
100 | 1.3750 | 1.1900 | 0.7850 | 0.3333 | 1.5185 | 0.0610 |
In Tables 1 and 2, we show the percentage of the maximum absolute tracking error and the percentage of the mean absolute tracking error with respect to the maximum peak-to-peak value of the reference. It can be noticed in both tables that the inversion-based adaptive conditional servocompensator (Inv-B ACS) approach greatly outperforms the other five approaches in reducing the tracking errors for all frequencies. The trend in both tables shows that the next best tracking performance comes from the approach using the EHGO–DI. Note that the mean absolute error for the EHGO–DI approach is higher than the Inv-B ACS one by almost sevenfolds for the 100 Hz frequency.
The second round of experiments are done using a sawtooth desired reference with the same frequencies used with the sinusoidal reference (frequencies ). A second-order prefilter is inserted to smooth out the signal to avoid spiking impulses at the signal edges. The measured output displacement under the inversion-based ACS control method for the 100 Hz frequency reference case is shown in Fig. 7. In Fig. 8, the tracking error is presented. The magnified subfigure to the left side shows the tracking error response for the first 0.03 s. Notice that the tracking error converges quickly in around 0.003 s. Another magnified subfigure added to the right shows a time interval of the tracking error in the boundary-layer phase (7.0–7.1 s). Notice that the error is not increasing in the period (0.03–10 s).

Measured displacement versus sawtooth desired reference with 100 Hz frequency using the inversion-based ACS
In Fig. 9, we show the frequency spectral content of the tracking error in the boundary-layer phase (0.03–10 s) for the 100 Hz reference. It is noticed that we have nine harmonics shown in the spectrum with odd harmonics being relatively stronger than the even harmonics, which are barely noticeable. It can be seen that the first harmonic (the fundamental) has magnitude of less than 5.5 nm, and the rest of harmonics are lower than this magnitude.

Frequency spectrum of the tracking error with a 100 Hz sawtooth reference in the boundary-layer phase
Another set of experiments are conducted using the van der Pol oscillator output as desired reference input to the system. In Fig. 10, the measured output displacement is shown for the 100 Hz frequency. In Fig. 11, we demonstrate the tracking error performance. Similar to the sawtooth reference case, it can be noticed that the tracking error converges around 0.003 s. However, the tracking error magnitudes are a little bit higher than those in the sawtooth reference case. This can be seen clearly in Fig. 12, in which we demonstrate the spectrum content of the tracking error frequency. It can be seen that the fundamental harmonic has magnitude of 14 nm as compared to less than 5.5 nm in the sawtooth reference case. We can see in Fig. 12 that we have nine harmonics in the signal spectrum similar to the sawtooth reference, but with higher amplitudes.

Measured displacement versus van der Pol desired reference with 100 Hz frequency using the inversion-based ACS

Frequency spectrum of the tracking error with a 100 Hz van der Pol reference in the boundary-layer phase
Finally, in Table 3, we offer a comparison in absolute percentage tracking errors for the sawtooth and van der Pol references' cases. It can be observed in the table that for all the tested frequencies, the van der Pol reference case has higher tracking errors as compared to the sawtooth case. For instant, in the 100 Hz frequency case, we can see that the mean absolute error in the van der Pol reference case is larger by more than three times compared to the sawtooth reference case. This is an indication that the van der Pol reference stimulates strongly the disturbance odd harmonics as compared to the previous two references.
Percentage of tracking errors for the proposed controller with respect to the reference maximum peak under sawtooth and van der Pol desired references
Sawtooth | van der Pol | |||
---|---|---|---|---|
Frequency (Hz) | Mean | Max | Mean | Max |
5 | 0.0016 | 0.0133 | 0.0017 | 0.0210 |
25 | 0.0054 | 0.0481 | 0.0075 | 0.0560 |
50 | 0.0104 | 0.1027 | 0.0202 | 0.1291 |
100 | 0.0226 | 0.2584 | 0.0707 | 0.3695 |
Sawtooth | van der Pol | |||
---|---|---|---|---|
Frequency (Hz) | Mean | Max | Mean | Max |
5 | 0.0016 | 0.0133 | 0.0017 | 0.0210 |
25 | 0.0054 | 0.0481 | 0.0075 | 0.0560 |
50 | 0.0104 | 0.1027 | 0.0202 | 0.1291 |
100 | 0.0226 | 0.2584 | 0.0707 | 0.3695 |
Notice that we have done the same set of experiments to the controller of Ref. [53], and the tracking error results came comparable to this work. The tracking errors of this controller can be made smaller if a more accurate hysteresis model is used.
It is worth mentioning that we tried replacing the switching function designed based on the analytical error bound by a constant gain taken as the maximum of the switching function for . However, the results obtained from the fixed switching gain have shown that the tracking errors have increased with more aggressive control actions for all the considered references.
In the implementation of the adaptive servocompensator (45), to avoid the case when the matrix S has very large eigenvalues, we utilized the technique suggested in Ref. [66] to scale down the internal model matrices such that and , where is chosen as a Hurwitz matrix with eigenvalues (−1, −1.5, −2) and the pair () is in controllable canonical form, is the scaling factor and is chosen to be . This technique helps in making the adaptation parameters in reasonably small.
6 Conclusion
This paper focuses on designing an adaptive conditional servocompensator for a class of hysteretic systems. Under this control law, the behavior of the closed-loop system has two stages. The first one is a reaching phase, where the controller is a continuously implemented sliding mode control law. For this stage, we have designed the control law to accommodate hysteric inversion error perturbations by deriving an analytical bound on these perturbations, which is used to design the switching control component. Then, we designed the switching part of the control law using the analytical bound to reduce the conservativeness of the sliding mode controller. The second stage starts when the sliding variable enters the boundary layer and stays therein forever. In particular, at this time, the adaptation law is activated “conditionally” to handle the residual hysteretic perturbations.
Aside from the proposed new control algorithms with experimentally proven performance, this work makes contributions to the theory of systems with hysteresis. Our stability analysis embodied in Theorems 1 and 2 establishes well-posedness and periodic stability for the closed-loop system. The theoretical framework used to prove the periodic stability was originally presented in Ref. [51]; however, some of the assumptions in Ref. [51] are not satisfied due to nonsmooth terms in both the control and the adaptation laws. In our work, we have been able to establish the periodic stability of the closed-loop system under mild conditions. Experimental validation of our proposed control algorithm, including both advanced inversion-based and inversion-free algorithms and the traditional PI controller, confirms its superiority as compared with competing control algorithms implemented on the same device. We note that the proposed control algorithm is computationally efficient, as it does not involve solving sophisticated optimization problems or require online estimation of hysteresis parameters.
Funding Data
National Science Foundation (CMMI 1301243, Funder ID: 10.13039/100000001).
Appendix: Proof of Theorem 1
where . From the above inequality, we have established the boundedness of the function . Hence, by choosing , we can ensure that .
which means that cannot leave the set for all time , and this implies that any solution lies in , from which the uniqueness of the solution is established in the space .