Adaptive Control of a Piezo-Positioning Mechanism With Hysteresis and Input Saturation Using Time Delay Estimation

In this paper, based on backstepping technique and time delay estimation (TDE) technique, an adaptive time delay compensation control scheme is developed for a class of piezoelectric positioning mechanical systems with Bouc-Wen hysteresis and input saturation constraint. The nonlinear part of the Bouc-Wen model is estimated online by TDE, and the TDE error introduced by TDE is compensated online by an adaptive law. Furthermore, an auxiliary variable system is used to deal with the input saturation constraint. Based on the Lyapunov method, the stability of the closed-loop system is analyzed and proved. Two simulation examples are given to demonstrate the effectiveness of the proposed control scheme.


I. INTRODUCTION
Hysteresis widely exists in modern electromechanical systems, especially in the systems that contain actuators and sensors made of intelligent materials, such as precision piezoelectric positioning mechanism [1], atomic force microscopy [2], hydraulic piezoelectric valves [3], fast cutting tool servo system [4] and so on. Without hysteresis compensation, it will seriously affect the performance of electromechanical systems, and even cause instability of the systems. For example, in an open-loop control, the error caused by hysteresis can reach up to 10%-15% [5]. Therefore, in recent years, modeling and compensation control of hysteresis have attracted significant attention [6]- [8]. There are four main methods for hysteresis modeling. The first is the physics-based modeling method, which mainly includes Jiles-Atherton model [9], Maxwell-slip model [10] and so on; The second method is based on differential equations, including Bouc-Wen model [11], Duhem model [12], and so on; The third one is based on operators, which includes Preisach model [13], Prandtl-Ishlinskii model [14], etc. The The associate editor coordinating the review of this manuscript and approving it for publication was Feiqi Deng . last one is intelligent modeling method based on computational intelligence, such as neural network model [15] and support vector machine model [16].
Hysteresis has the properties of multi-valued mapping, memory, rate-dependent, etc. Therefore, the control of systems with hysteresis is still an open question, which has attracted many scholars' interest [17]- [22]. Generally speaking, there are three methods for compensation control of hysteresis, the first is feed-forward compensation control, the second is feedback-feedforward compensation control, and the last is feedback compensation control.
Feedforward compensation method is a very effective and low-cost hysteresis compensation control method, which constructs a hysteresis inverse in the forward channel to eliminate the impact of hysteresis on systems. However, due to its open-loop mode, it completely depends on the accuracy of hysteresis modeling and parameter identification and is sensitive to external disturbances and model uncertainties. Therefore, this method has rarely been used in practical application.
Feedback-feedforward compensation control method is to construct an inverse model in the forward path to eliminate the hysteresis effects and design a controller in the feedback path to further improve the performance of the system and enhance the robustness of the system. A Bouc-Wen hysteresis inverse model is proposed in [17], which can eliminate the effect of the Bouc-Wen hysteresis, and an adaptive controller with hysteresis inverse is designed. The simulation results show that the control strategy is still effective in the presence of parameter perturbations of the hysteresis model. In [19], the continuous Prandtl-Ishlinskii model is decomposed into a finite number of discrete Prandtl-Ishlinskii operators, and the hysteresis is compensated by using the analytical inverse hysteresis model. For the error caused by the continuous model being converted into a discrete model, an adaptive control technique is used to compensate online. The feedbackfeedforward compensation control method not only needs to construct hysteresis inverse models (constructing the hysteresis inverse model is a very difficult thing in itself), but also needs to know the exact parameters of the hysteresis model, and the number of these parameters is generally not small.
Feedback compensation control method does not need to construct the hysteresis inverse model or approximate inverse model but treats the nonlinear part of the hysteresis as a disturbance, and then uses sliding mode control, robust control and other methods to design feedback controllers. The Bouc-Wen hysteresis is decomposed into a linear term and a bounded nonlinear hysteresis term in [18]. Then, the nonlinear hysteresis term can be treated as a bounded disturbance. In [20], the hysteresis is directly regarded as a part of disturbance, and then the disturbance is eliminated by an adaptive sliding mode controller. The stability of the closed-loop system is proved theoretically. Although the feedback compensation control method does not need to construct a hysteresis inverse but treats the nonlinear term in the hysteresis as a bounded disturbance, it will affect the control accuracy to a certain extent.
Due to the limitations of physical conditions and sensors, not all system states are available in the practical systems. Therefore, state observer has become one of the main solutions. In [23], [24], state observers based on fuzzy logic system are designed for nonlinear systems. In [25], a nonlinear state observer is proposed for a stochastic nonlinear strictfeedback system. In [26], a high gain observer with updated gain and homogeneous correction terms is used to estimate the unknown states. Compared with the traditional observer, the high-gain observer not only has the performance of the state feedback controller when the gain is high enough, but also can suppress the disturbance to a certain extent as long as the gains design is reasonable [27]. Therefore, the high-gain observer has been widely used in many control problems.
In addition, due to physical constraints, in practical control systems, input saturation constraints are often encountered. Saturation often leads to excessive overshoot and large tracking error, which greatly limits the performance of the system. How to solve the problem of saturation nonlinearity is still a challenging task. Auxiliary variable method [28], [29], neural networks [30], [31], composite nonlinear feedback method [32]- [34] are used to deal with the input saturation constrains. Moreover, the problems of different control systems with input saturation have also attracted much attention [35], [36]. However, to the best of authors' knowledge, few results are available for adaptive time delay control of electromechanical systems with hysteresis.
The Bouc-Wen model is widely used in modelling hysteresis. And after the boundedness of the nonlinear hysteresis term was proved in [18], applications of Bouc-Wen hysteresis in control systems have received considerable attention. Based on time delay estimation(TDE) technique [37]- [39], a novel feedback control method is proposed for a class of electromechanical systems with Bouc-Wen hysteresis and input saturation constraint. In this control method, the TDE is used to estimate the nonlinear part of the Bouc-Wen model, and then an adaptive controller is designed by backstepping design method. The TDE error introduced by TDE is estimated and compensated by the adaptive law, so that the control performance of the system is further improved. The main contributions of this paper can be summarized as follows: (i) A new method of hysteresis compensation control is proposed. Compared with [18], [20], this method uses the TDE technique to realize the on-line estimation of the nonlinear term of hysteresis, which can be directly used in the controller design, instead of treating the hysteresis nonlinearity as a bounded disturbance. Compared with the feedbackfeedforward control in [17], [19], the proposed method does not need to construct the hysteresis inverse, and only one parameter of the hysteresis is used in the controller, which effectively reduces the calculation.
(ii) An adaptive method is proposed for the electromechanical system, which does not require knowledge of system dynamics.
(iii) Compared with [17]- [22], the input saturation constraint is considered in this paper, which makes the control method proposed in this paper can be better used in practical engineering systems.
The rest of this paper is organized as follows. The problem to be tackled is stated and the control objective is given in Section II. The proposed control scheme is given in Section III. In Section IV, simulation results are presented. Finally, conclusions are provided in Section V.

II. PROBLEM STATEMENT
Consider a piezoelectric positioning mechanic system [40], [41] preceded by an actuator with input hysteresis and input saturation: where x,ẋ andẍ are the position, velocity and the acceleration, M , D and F denote the unknown mass, damping, and stiffness coefficients, u is the applied voltage to the piezoelectric positioning platform, H [u](t) denotes the Bouc-Wen hysteresis nonlinearity, and the specific parameters are known. VOLUME 8, 2020 The expression of H [u](t) is given as follows [17], [18]: where 0 < µ < 1 is a weighting parameter, κ is stiffness coefficient, µ 1 and µ 2 are constants with the same sign, and ϑ is given by the following nonlinear first-order differential equation:θ where parameters β and χ describe the shape and amplitude of the hysteresis, respectively, n governs the smoothness of the transition from the initial slope to the slope of the asymptote, and β > |χ|,n ≥ 1.
Consider the voltage actuator input constraint, and u is given by: where v is the control signal to be designed, and u max is the known saturation limit. Let x 1 = x, x 2 =ẋ, and = µ 2 ϑ, using (2), system (1) can be rewritten as: The control objective is to design a control scheme for the system (1)-(4) such that the displacement x can track the desired trajectory x d .
Remark 1: The boundedness of the ϑ(t) means that it can be considered as a bounded disturbance in the system. And the initial value is usually set to ϑ(0) = 0.
Remark 2: Although the piezoelectric positioning mechanic system in [40], [41] is discussed herein, the control method can also be used for the systems represented as Equation (1), such as the piezo-actuated stage described in [42] and the mechanical system in [43].
Assumption 1: The desire trajectory x d and its first and second derivatives are known and bounded.
Lemma 2 [44]: Assume the function y(t) and its first n derivatives are bounded, thus |y (k) | < Y k for k = 0, . . . , n, where Y k are positive constants. Consider the following linear system: where is any small positive constant and the parameters λ 1 , . . . , λ n−1 are chosen so that the polynomial s n−1 + λ 1 s n−2,..., + 1 is Hurwitz. Then there exist positive constants l k for k = 2, . . . , n and t * , such that for all t > t * , we have

III. CONTROL DESIGN
In this paper, two cases are investigated for the piezoelectric positioning mechanic system (5): (i) full state feedback control design, that is, x 1 and x 2 are known; and (ii) output feedback control design, that is, only x 1 is known. For the second case where x 2 cannot be directly measured, a high-gain observer is proposed to estimate x 2 .

A. FULL STATE FEEDBACK CONTROL
Before using the backstepping technique to design the control law, the following change of coordinates is made: where α 1 is a virtual control law to be designed.
Step 1: From (5) and (8), we havė then the virtual control law α 1 is designed as: where c 1 is a positive parameter to be designed. From (8)-(10), we obtain Step 2: From (5) and (8), we get The following auxiliary design system is used to deal with the input saturation: where u = u − v, ξ is the state of the auxiliary design system, K is a positive parameter to be designed, and parameter σ is a small positive constant. From (13), we have Then, considering the input saturation effect, we design the following control law: where parameters c 2 and K v are positive constants to be designed,D,F andM are the estimates of the parameters D, F and M ,B is the estimate the bound B of the TDE error which will be introduced in the following.
Unlike traditional hysteresis feedback compensation control design methods, instead of treating as a bounded disturbance, we estimate it using the TDE technique. Letˆ denote the estimation of , then, with the help of (1) and (5), we can obtainˆ using the TDE schemê where h is an adequate small delay time. In practice, h is set to one unit of sampling time. According to the principle of TDE technique, as long as h is small enough, then (t) ∼ = (t − h) =ˆ (t). However, there still be a TDE error Using B to denote the bound of the TDE error, then we have Remark 3: From Lemma 1, it is reasonable to assume that e d (t) is bounded. Furthermore, in practice,ˆ is computed according to (16). Since u is given by the controller, u(t − h) can be assumed to be known. If signal w is available, then we can obtainˆ from to estimate the parameters D, F and M , then get the estimationˆ .
The parameters updating laws are designed aṡ Using (11) and (12), take the time derivative of V , we obtaiṅ Note that u = v + u and z 2 e d ≤ |z 2 |B, and substituting (15) into (22), we obtaiṅ Substituting (14) and (17)- (20) into (23), we havė Considering the facts: Remark 4: To ensure the stability of the close-loop system, the control parameters should satisfy the following condition: Thus,V will be negative definite.

B. OUTPUT FEEDBACK CONTROL
The proposed control law (15) is designed under the assumption that all outputs are measurable. However, some output information may not be measurable due to practical VOLUME 8, 2020 issues such as cost and dimensions. In this section, a high gain observer is used to estimate the unmeasurable term x 2 . According to Lemma 2, the high gain observer for the system (5) is considered with n = 2, and the unmeasurable state x 2 =ẋ can be approximated bŷ Then, the unmeasurable backstepping state signal z 2 can be estimated by: where the dynamics of ω 2 are given as follows: According to Lemma 2, there exist constants t * and l 2 such that ∀t > t * , we have Note thatx From (30), we havez 2 2 ≤ ( l 2 ) 2 = β (31) Now the control law and parameter updating laws are designed as follows: where c 21 , d 1 , K v1 , γ D1 , γ F1 , γ M 1 , γ B1 , σ D , σ F , σ M and σ B are positive constants to be designed. And the auxiliary design system is designed as follows: where ξ 1 is the state of the auxiliary design system, K 1 is a positive constant to be designed, and parameter σ 1 is a small positive constant. Considering a Lyapunov function candidate as Note that 1 2z 2 2 ≤ 1 2 β, therefore, we only need to determine the stability of Substituting (33) into (40), we obtaiṅ According to (34)- (38), from (41), we havė Thus, the following inequality holds Similarly, we haveDD Considering the (3-5)th terms of (42), we have From (8)-(10), we havė Considering the terms of the second line of (42), we have Considering the terms of the third line of (42), we obtain Considering the terms of the fourth and fifth lines of (42), we obtain Considering the terms of the last line of (42), we get According to (46) and (48)-(51), (42) can be rewritten aṡ where Remark 5: In order to guarantee thatV 2 is negative definite, the parameters of the controller need to fulfil the following criteria: Considering the plant (1)-(4) under Assumption 1, the adaptive output time delay control scheme, consisting of the control laws (10) and (33), auxiliary system (38), the updated laws (34)- (37), and the TDE (16), guarantees that the closed-loop system is semi-globally stable in the sense that all the closed-loop signals are bounded. Furthermore, the tracking error signal z 1 converges asymptotically to the compact set defined by where Z = 2(V 2 (0) + C 1 κ 1 ) with C 1 , κ 1 are given in (53) and (54).
Proof: From (25), and following steps in [44], we can can be concluded that z 1 converges to the compact set defined by (55). By using the method of the proof for Theorem 1, it is easily proved that all the signals in the closed-loop system are bounded, and hence is omitted. Remark 6: To avoid the possible chattering problem caused by the sign function, the sign function in (15) and (33) can be replaced by a hyperbolic tangent function as in [19]: where ς is a small positive constant. Using the inequality 0 ≤ |z| − ztanh( z 2 ς ) ≤ 0.2785ς, it is easy to conclude that Theorem 2 is still true and that the asymptotic tracking performance in Theorem 1 would change to asymptotically converge to a compact set as in Theorem 2.
Remark 7: The transient performance of the system can be improved under the condition of ensuring the stability of the system by the following tuning methods: (i) Large c 1 , c 2 , c 21 can improve the tracking error of the system, but it may lead to system oscillation and large control energy. Therefore, c 1 , c 2 and c 21 should not be chosen too large when the tracking error is guaranteed. (ii) The adaptive scaling factors γ D , γ F , γ M , γ B , γ D1 , γ F1 , γ M 1 , γ B1 are usually designed to be small, and the other constants (such as K , σ B and so on) are designed to meet the requirements in Remark 4 and Remark 5, mainly to ensure the stability of the system. Then, the fine-tuning of these parameters is carried out to improve the transient performance of the system.
Remark 8: Both the control laws and the adaptive update laws have the same complexity as the classical adaptive backstepping controller in [45]. The auxiliary variable system consists of only a first order differential equation, and the state of the auxiliary variable system remains unchanged when the system enters a steady state (i.e. ξ < σ 1 ). The TDE is calculated from the system state and parameters at time (t − h). Because the system state and parameters at time (t − h) are known, the implementation is also convenient and simple. Therefore, in general, the control scheme proposed in this paper has clear objectives of each component and is easy to implement.

IV. SIMULATION STUDY
In this section, we will provide two cases to demonstrate the effectiveness of our proposed controller (15) and (33) for   To illustrate the effectiveness of the proposed control scheme, the schemes in [18], [46] and the classic PID controller are also applied to the system (1)-(4). In [18], the nonlinear part of the hysteresis is considered as a bounded disturbance. In [46], the sliding mode control with perturbation estimation is used to deal with the hysteresis. Fig. 1 shows the tracking performance using the proposed method under state feedback control. It can be clearly seen that the output of the system can track the desired trajectory well. The control input under state feedback control is shown in Fig.2. And Fig.3 shows the tracking error under the state feedback control. Obviously, the proposed scheme is better than the other three schemes under state feedback control. As for output feedback control simulation, Fig.4-Fig.6 show the tracking performance, control input and tracking error, respectively. It can be concluded that the proposed scheme is effective and gives better performance than the other three control methods in comparison.
It should be mentioned that simulations for several different desired trajectories with different initial conditions have also been conducted. Results show that they all have similar behaviours as the one shown in this paper. The simulations for h = 0.01, which means larger delay time for TDE, give almost identical results. This further demonstrates the effectiveness of the proposed control scheme.

V. CONCLUSION
In this paper, a new time delay feedback control strategy with hysteresis compensation is proposed for a class of secondorder electromechanical systems with Bouc-Wen hysteresis and input saturation constraint. The controller is designed by backstepping design method. In the controller design process, the TDE is used to estimate the nonlinear part of the Bouc-Wen model, the adaptive law is applied to eliminate the TDE error, and the auxiliary variable system is used to deal with the input saturation problem. Based on Lyapunov direct method, the corresponding controllers are designed for the full state feedback control and the output feedback control, respectively, and the stability of the closed-loop system is analyzed. Finally, the effectiveness of the control scheme is further demonstrated by simulations.
In the proposed control scheme, only a classic backstepping feedback controller is designed with the help of the time delay control. In the future, fractional order controller, sliding mode controller, reinforcement learning controller can be combined with the time delay control to compensate the hysteresis. In addition, the hysteresis compensation control scheme based on TDE can be applied to the controller design of other systems with hysteresis, such as piezoelectric positioning stages, atomic force microscope, etc.