Piezoelectric stages use piezoelectric actuators and flexure hinges as driving and amplifying mechanisms, respectively. These systems have high positioning accuracy and high-frequency responses, and they are widely used in various precision/ultra-precision positioning fields. However, the main challenge with these devices is the inherent hysteresis nonlinearity of piezoelectric actuators, which seriously affects the tracking accuracy of a piezoelectric stage. Inspired by this challenge, in this work, we developed a Hammerstein model to describe the hysteresis nonlinearity of a piezoelectric stage. In particular, in our proposed scheme, a feedback-linearization algorithm is used to eliminate the static hysteresis nonlinearity. In addition, a composite controller based on equivalent-disturbance compensation was designed to counteract model uncertainties and external disturbances. An analysis of the stability of a closed-loop system based on this feedback-linearization algorithm and composite controller was performed, and this was followed by extensive comparative experiments using a piezoelectric stage developed in the laboratory. The experimental results confirmed that the feedback-linearization algorithm and the composite controller offer improved linearization and trajectory-tracking performance.

## ARTICLE HIGHLIGHTS

A Hammerstein model is proposed that uses a static modified Prandtl–Ishlinskii part and a second-order linear system in combination.

A feedback-linearization algorithm is used to compensate for the hysteresis nonlinearity in a piezoelectric stage.

An equivalent-disturbance compensator is implemented to suppress model uncertainties and external disturbances.

## I. INTRODUCTION

With the development of micro- and nanoscale positioning technology, piezoelectric stages driven by actuators with high resolution and fast response speeds are widely used in precision-engineering applications, such as atomic force microscopes,^{1} piezoelectric-driven hydraulic valves,^{2} and transfer-printing technology.^{3} However, the inherent hysteresis nonlinearity of piezoelectric actuators greatly reduces the tracking accuracy of piezoelectric stages; furthermore, this nonlinearity is very complex, with asymmetric and rate-dependent properties.^{4} Therefore, hysteresis modeling and compensation control for piezoelectric stages is a notable challenge.

A large amount of research has been devoted to the modeling and control of hysteresis nonlinearity in piezoelectric stages.^{5,6} Surveying the modeling of hysteresis nonlinearity, it can be established that the models are generally classified into three categories: physical hysteresis models, phenomenological hysteresis models, and intelligent hysteresis models.^{7–9} Among these, the Prandtl–Ishlinskii (PI) model has a simple structure and possesses a specific analytic inverse, meaning that it is widely used in hysteresis modeling. However, the classical PI model can only describe a static and symmetric hysteresis structure. Therefore, a number of modified PI models have been proposed to describe the hysteresis nonlinearity of piezoelectric stages.^{10–12}

In Refs. 10–12, PI models based on a dynamic threshold function, a dynamic weight function, and a dynamic envelope function are respectively proposed. Unfortunately, these models have more complex structures, making the mathematical derivation of the model inversion difficult or even impossible; the Hammerstein model was proposed to overcome this problem. The Hammerstein model consists of a nonlinearity model and a linear transfer-function model:^{13–15} the former describes the static hysteresis nonlinearity, while the latter is employed to describe the dynamic characteristics of the system.

Based on the hysteresis model of a piezoelectric stage, a suitable control strategy should be designed to eliminate nonlinearity and improve precision. A large number of control methods have been proposed, such as feedforward control,^{16} feedback control,^{17} and feedforward–feedback control.^{18,19} Feedforward–feedback control combines the advantages of both the former approaches, and it is the most widely used control method. In Ref. 18, the design of a feedforward control system based on an inverse rate-dependent model is described, and a feedback-control system considering model uncertainties based on the interval technique is presented. Similarly, Ref. 19 presents the design of a polynomial-based rate-dependent PI model and the construction of an inverse model as feedforward control; in this system, a disturbance observer is used to handle bounded disturbances. However, the mathematical form of this inverse model is difficult to derive, and the designed controller is complex. Therefore, it is important to study the control method without inverse hysteresis model. Moreover, model inaccuracies and variations in the external environment can also affect the accuracy of piezoelectric-stage control schemes.

To address these challenges, in this paper, we propose a feedback-linearization algorithm based on a Hammerstein model and a composite controller. First, the Hammerstein model is used to accurately describe the hysteresis nonlinearity of the piezoelectric stage, and its parameters are estimated. A feedback-linearization algorithm is then implemented based on this Hammerstein model. This does not require the solution of the inverse of the hysteresis model, but it can nonetheless transform the nonlinear system into a linear system and eliminate the static hysteresis nonlinearity. A linear composite control strategy can then be used to analyze and control the piezoelectric stage, which is combined by a feedforward controller, a feedback controller, and an equivalent-disturbance compensation (EDC) controller.

The feedforward controller uses the reference signal and its derivatives, and the feedback controller takes a proportional–integral–derivative (PID) form using the pole-assignment method. Importantly, the unmeasured model uncertainties and external disturbances are comprehensively described as an equivalent disturbance, and this can be suppressed by the proposed optimal equivalent-disturbance compensator. The equivalent-disturbance compensator was designed in the Laplace domain with a second-order low-pass filter, obtaining a state-space implementation. The effectiveness, performance, and bounded-input, bounded-output (BIBO) stability of the composite controller were analyzed and experimentally validated.

The main contributions of this paper are as follows. A feedback-linearization algorithm is proposed to transform the nonlinear system into a linear system. An equivalent-disturbance compensation controller is proposed to compensate for unmeasured model uncertainties and external disturbances.

The remainder of the paper is organized as follows. Section II presents the hysteresis model for the piezoelectric stage. Section III presents the feedback-linearization and disturbance-compensation control strategy. The experimental validation of the proposed control strategy is then presented in Sec. IV. Finally, Sec. V gives a summary of the conclusions of this research.

## II. HYSTERESIS MODEL FOR PIEZOELECTRIC STAGE

### A. Hysteresis phenomena

Hysteresis nonlinearity is the most notable problem exhibited by stages driven by piezoelectric actuators. Therefore, this subsection discusses the hysteresis phenomena of a piezoelectric stage. A piezoelectric stage consists of piezoelectric actuators and a flexible amplification mechanism. The hysteresis characteristics of piezoelectric actuators mean that piezoelectric stages suffer from severe amplitude- and rate-dependent hysteresis nonlinearity, as shown in Fig. 1. Therefore, establishing an accurate hysteresis model provides the basis for control synthesis.

### B. Hammerstein model

To describe the hysteresis nonlinearity of a piezoelectric stage, the Hammerstein model is used to analyze its dynamics. The structure of the Hammerstein model is presented in Fig. 2, in which *u* is the input voltage, *v* is the intermediate hysteresis, and *y* is the displacement of the piezoelectric stage. The Hammerstein model consists of a static hysteresis nonlinearity model *H* and a linear transfer-function model *G*(*s*); these can be cascaded together to accurately describe the hysteresis nonlinearity.

*H*; this is expressed

*u*(

*t*) is the input voltage,

*v*(

*t*) is the intermediate hysteresis,

*a*

_{0}and

*a*

_{i}are weight parameters,

*r*

_{i}is a threshold value,

*i*is the serial number of the threshold, and $Fri[u](t)$ is the play operator.

*u*(

*t*) is used as the input, the output value is

*u*(0) is the initial value of the input, and $Fri[u](0)$ is the initial output value of the play operator.

*m*

_{1}

*u*(

*t*)

^{3}is thus introduced into the PI model to obtain a modified PI (MPI) model, which can describe the asymmetric hysteresis curve.

^{20}The MPI model is expressed as

*m*

_{1}is a weight parameter.

*G*(

*s*) in the Hammerstein structure describes the linear dynamics of the system. In a piezoelectric stage,

*G*(

*s*) can be expressed by a second-order oscillation element

^{4}as

*ζ*and

*ω*

_{n}are the damping ratio and natural frequency of the flexible mode, respectively.

*v*(

*t*) and

*y*(

*t*) can be obtained from (5) as

### C. Parameter identification for Hammerstein model

Before designing the controller, the parameters of the Hammerstein model should be identified. However, the multiple variables and high nonlinearity mean that this is a challenge. Fortunately, a well-developed particle-swarm optimization (PSO) algorithm can provide a good solution.^{21}

First, based on the input and output data of the piezoelectric stage at low frequencies, the PSO algorithm is used to identify the parameters in the MPI model. Next, a set of sinusoidal swept signals is used to drive the piezoelectric stage, and a discrete set of *u*(*t*) and *y*(*t*) are obtained. The value of *v*(*t*) is calculated based on the MPI model obtained from the PSO algorithm. Finally, the linear second-order system parameters are identified using the least-squares method.

## III. FEEDBACK-LINEARIZATION AND EQUIVALENT-DISTURBANCE COMPENSATION CONTROL STRATEGY

### A. Feedback-linearization algorithm

A feedback-linearization algorithm is proposed to linearize the static hysteresis nonlinearity, and this is expressed by the above-described Hammerstein model. The relationship between *u*(*t*) and *y*(*t*) in the Hammerstein model is nonlinear; thus, a feedback-linearization algorithm is used to convert this nonlinear system into a linear system. The linear system should have a simple and linear relationship, as shown in Fig. 3. Therefore, a suitable control input *w*(*t*) was designed to control the tracking properties of the output *y*(*t*).

*u*(

*t*) is designed by a feedback-linearization algorithm with the expression

*w*(

*t*) is the new input of the feedback-linearization algorithm and

*f*(

*t*) is the nonlinear part of (8), whose expression is

*y*(

*t*) and the new input

*w*(

*t*) is

Clearly, the relationship between the output *y*(*t*) and the new input *w*(*t*) is simply linear. Therefore, the static hysteresis nonlinearity of the piezoelectric stage is eliminated. A linear control methodology can then be used to analyze and design the linear system according to (12).

### B. Equivalent-disturbance analysis

*b*

_{1}= 2

*ζω*

_{n}, $b2=\omega n2$. Furthermore: $b\u03031$, $b\u03032$, $m\u03031$, $a\u03030$, and $a\u0303i$ are the theoretical values;

*b*

_{1},

*b*

_{2},

*m*

_{1},

*a*

_{0}, and

*a*

_{i}are the nominal values; Δ

*b*

_{1}, Δ

*b*

_{2}, Δ

*m*

_{1}, Δ

*a*

_{0}, and Δ

*a*

_{i}are the parametric uncertainties; and

*q*

_{1}(

*t*) is an external disturbance.

### C. Composite controller design

In this section, a composite controller is designed to eliminate model uncertainties and external perturbations to further improve the tracking accuracy of the piezoelectric stage.

*r*(

*t*) is the reference signal,

*y*(

*t*) is the displacement of the piezoelectric stage, and

*e*(

*t*) is the error.

The composite controller is designed based on the above error-state differential equation. This consists of three parts: nominal feedforward control, nominal PID-based feedback control, and EDC control, as shown in Fig. 4.

*w*(

*t*) can be expressed as

*w*

_{FF}(

*t*) is the feedforward control signal,

*w*

_{FB}(

*t*) is the feedback control signal, and

*w*

_{DC}(

*t*) is the EDC control signal. A block diagram of this composite control is shown in Fig. 4. The three controllers will be designed separately.

#### 1. Feedforward controller design

*q*(

*t*) in (15), we design the feedforward control input

*w*

_{FF}(

*t*) under the nominal model as

#### 2. Feedback controller design

*w*

_{FF}(

*t*), the nominal error system can be expressed as

**(**

*E**t*) as

*E*

_{2}(

*t*) =

*e*(

*t*), $E3(t)=e\u0307(t)$. Thus, (19) becomes

*d*

_{c}(

*s*) is defined as

*p*

_{i}is the

*i*-th root of the eigenequation and

*s*is the complex frequency variable. The parameters

**are then obtained by the pole-placement method.**

*K*^{22}

#### 3. EDC control

*q*(

*t*). From (16), the desired EDC is

^{23}The equivalent-disturbance compensator can then be designed as

*F*(

*s*) is a second-order low-pass filter, which is implemented by two first-order low-pass filters in series, and

*f*

_{1}and

*f*

_{2}are two positive constants that decide the bandwidth of the low-pass filter. If

*f*

_{1}and

*f*

_{2}are sufficiently large, the gain of the robust filter will be approximately equal to 1.

*s*domain), (27) can be rewritten as

*z*

_{1}(

*s*) and

*z*

_{2}(

*s*) as

*w*

_{DC}(

*t*) can be realized with states

*z*

_{1}(

*t*) and

*z*

_{2}(

*t*) as

### D. Controller performance analysis

This subsection presents an analysis of the stability of the proposed composite controller. Because the model uncertainties, control inputs, disturbances, and reference trajectory are not infinite in real physical systems, the following assumptions are made, which are reasonable in practical engineering.

*Assumption 1*: The model uncertainties are bounded.*Assumption 2*: The control input*w*(*t*) is bounded.*Assumption 3*: The equivalent disturbance*q*(*t*) is bounded.*Assumption 4*: The reference signal*r*(*t*), and its first- and second-order derivatives are piecewise uniformly bounded. The following lemma and theorem can then be obtained.

*A*

_{c}is the Hurwitz matrix,

*f*

_{1}and

*f*

_{2}are sufficiently large, and

*f*

_{1}≫

*f*

_{2}> 0, there exists a positive constant

*λ*with no relation to

*f*

_{1}and

*f*

_{2}, satisfying

*m*×

*n*-dimensional matrix.

*w*(

*t*) in the error (16) is expressed by (17), for any given small positive number

*ɛ*, it is always possible to find positive values of $f1*$, $f2*$, and

*T*. If $f1\u2265f1*$, $f2\u2265f2*$, and

*f*

_{1}≥

*f*

_{2}> 0, then all states involved are bounded; i.e.,

*E*(0) = 0, then

The theorem illustrates the BIBO stability of the proposed composite control strategy. If the boundedness of assumptions 1–4 hold, then the error states must be bounded. When *f*_{1} and *f*_{2} are large enough, there exists a positive constant time; when the time is larger than this, the error states converge to a sufficiently small field.

## IV. EXPERIMENTAL VALIDATION

This section presents an experimental demonstration of the proposed feedback-linearization and EDC control strategy in a laboratory-developed piezoelectric stage.

### A. Experimental setup

The experimental piezoelectric stage system was set up in our laboratory, as shown in Fig. 5. This consists of a piezoelectric stage, a drive amplifier, a capacitive displacement sensor and its conditioner, a real-time control system, and a host computer. Primarily, the piezoelectric stage^{25} consists of a piezoelectric actuator and a flexure-hinge amplification mechanism, which can provide nanoscale positioning. The maximum stroke of the piezoelectric actuator is 15 μm, and the displacement amplification ratio of the flexure-hinge amplification mechanism is 6.1. The capacitive displacement sensor (sensitivity = 20 μ/V; resolution = 2.5 nm) and its conditioner are employed to measure the displacement of the piezoelectric stage, which is used as feedback to the real-time control system. A Speedgoat real-time control system is used to sample the measured signal and implement the control algorithm at a sampling and control frequency of 10 kHz. The control system outputs the control signal, which is magnified 15 times by the drive amplifier to drive the piezoelectric stage. The host computer provides the compilation and human–computer interaction environment.

### B. System identification

In the experimental system, the parameters of the Hammerstein model are obtained by the identification method described in Sec. II C. First, the MPI model selects six play operators, and the parameters are identified using a PSO algorithm. These parameters were found to be *a*_{0} = 0.4931, *a*_{1} = 0.1165, *a*_{2} = 0.0189, *a*_{3} = 0.0677, *a*_{4} = 0.0063, *a*_{5} = 0.0890, *a*_{6} = − 0.0349, and *m*_{1} = − 4.7548 × 10^{−6}. Next, the linear second-order oscillation system parameters were found to be *ζ* = 0.8623 and *ω*_{n} = 5734.9; these were identified using the least-squares method.

Figure 6 shows a comparison of the Hammerstein-model hysteresis curves and the actual hysteresis curves with inputs of sinusoidal signals of 10, 50, and 100 Hz, and a composite sinusoidal signal. The composite sinusoidal signal was composed of 20, 40, 60, 80, and 100 Hz sinusoidal signals. Table I shows the rms and maximum errors for the Hammerstein hysteresis modeling at different frequencies. These results show that the developed Hammerstein model can accurately fit the hysteresis nonlinearity of the piezoelectric stage.

Frequency (Hz) . | Max error (μm) . | Rms error (μm) . |
---|---|---|

10 Hz | 1.6960 | 0.6859 |

50 Hz | 1.8088 | 0.9289 |

100 Hz | 2.5495 | 1.3010 |

(20, 40, 60, 80, 100) Hz | 2.9233 | 0.8861 |

Frequency (Hz) . | Max error (μm) . | Rms error (μm) . |
---|---|---|

10 Hz | 1.6960 | 0.6859 |

50 Hz | 1.8088 | 0.9289 |

100 Hz | 2.5495 | 1.3010 |

(20, 40, 60, 80, 100) Hz | 2.9233 | 0.8861 |

### C. Feedback linearization

### D. Implementation of composite controller

The composite controller consists of a feedforward controller, a feedback controller, and an equivalent-disturbance compensator.

#### 1. Feedforward controller

#### 2. Feedback controller

*K*

_{P}= 0.086,

*K*

_{I}= 286.7, and

*K*

_{D}= 8.7 × 10

^{−6}. The feedback controller signal

*w*

_{FB}is thus

#### 3. Equivalent-disturbance compensator

*f*

_{1}and

*f*

_{2}. As stated in Sec. III D, if

*f*

_{1}and

*f*

_{2}are large enough, the error states will converge to a sufficiently small field. In this work, the parameters of the EDC controller were chosen as

*f*

_{1}= 11 304 and

*f*

_{2}= 1884. Therefore, the EDC control signal

*w*

_{DC}(

*t*) can be realized as

### E. Experimental results and discussion

Comparison experiments were carried out for different references between the proposed control method (FF+FB+EDC) with two other control methods, as follows:

FF+FB: a feedforward controller and a feedback controller with no EDC.

FF+FB+ESO: a feedforward controller and a feedback controller with an extended state observer (ESO).

#### 1. Tracking of single-frequency sinusoidal signal

*f*is the frequency.

The results of the three control strategies are shown in Fig. 8, and their maximum and rms errors are listed in Table II. From these results, it can be seen that the tracking performance is significantly improved by disturbance compensation, including both ESO and EDC; however, the performance of the proposed EDC outperforms the commonly used ESO compensation algorithm. In particular, the trajectory-tracking error performance advantage of the proposed method becomes more apparent as the reference frequency increases.

Reference . | Index . | FF+FB . | FF+FB+ESO . | Proposed . |
---|---|---|---|---|

10 Hz | Max error (μm) | 0.5009 | 0.2603 | 0.1689 |

Rms error (μm) | 0.2092 | 0.0924 | 0.0440 | |

50 Hz | Max error (μm) | 1.0441 | 0.8490 | 0.5788 |

Rms error (μm) | 0.4592 | 0.3760 | 0.1981 | |

100 Hz | Max error (μm) | 2.3097 | 1.8839 | 0.9434 |

Rms error (μm) | 1.2756 | 0.8611 | 0.3868 | |

Composite frequency | Max error (μm) | 1.4411 | 1.2860 | 0.6636 |

Rms error (μm) | 0.4780 | 0.3650 | 0.1878 | |

Fourth-order trajectory | Max error (μm) | 0.4159 | 0.2572 | 0.1944 |

Rms error (μm) | 0.1398 | 0.0623 | 0.0559 |

Reference . | Index . | FF+FB . | FF+FB+ESO . | Proposed . |
---|---|---|---|---|

10 Hz | Max error (μm) | 0.5009 | 0.2603 | 0.1689 |

Rms error (μm) | 0.2092 | 0.0924 | 0.0440 | |

50 Hz | Max error (μm) | 1.0441 | 0.8490 | 0.5788 |

Rms error (μm) | 0.4592 | 0.3760 | 0.1981 | |

100 Hz | Max error (μm) | 2.3097 | 1.8839 | 0.9434 |

Rms error (μm) | 1.2756 | 0.8611 | 0.3868 | |

Composite frequency | Max error (μm) | 1.4411 | 1.2860 | 0.6636 |

Rms error (μm) | 0.4780 | 0.3650 | 0.1878 | |

Fourth-order trajectory | Max error (μm) | 0.4159 | 0.2572 | 0.1944 |

Rms error (μm) | 0.1398 | 0.0623 | 0.0559 |

#### 2. Tracking of composite-frequency sinusoidal signal

#### 3. Tracking of fourth-order trajectory

It is necessary to verify the performance of the system with the s-curve trajectory, which is commonly used in engineering practice, to meet the requirements of practical applications. Therefore, a fourth-order trajectory was examined to further verify the performance of the composite controller, and the results are shown in Fig. 9(b). It can be seen that the proposed EDC method obtains the best tracking performance, with an rms error of 0.0559 μm and a maximum error of 0.1944 μm.

## V. CONCLUSION

In this work, to eliminate the hysteresis nonlinearity of a piezoelectric stage, an EDC composite control method based on a feedback-linearization algorithm was developed. In this approach, the feedback-linearization algorithm is used to transform the nonlinear system into a linear system, and the errors caused by model uncertainties and external disturbances are compensated using EDC control. This method has the following advantages: (1) static hysteresis nonlinearity is eliminated without the need for an inverse hysteresis model; (2) a composite EDC controller further improves the tracking accuracy of the piezoelectric stage.

It should be noted that the coupling of high-frequency hysteresis and mechanical resonance was not considered in this paper, and only a single-degree-of-freedom platform was studied. In future work, a controller with high-frequency hysteresis and mechanical resonance will be designed based on the proposed method, and a multi-degree-of-freedom piezoelectric stage will be studied to further verify the effectiveness and feasibility of the proposed method.

## ACKNOWLEDGMENTS

This work was supported by the National Key R&D Program of China (Grant No. 2022YFB3206700), the Independent Research Project of the State Key Laboratory of Mechanical Transmission (Grant No. SKLMT-ZZKT-2022M06), and the Innovation Group Science Fund of Chongqing Natural Science Foundation (Grant No. cstc2019jcyj-cxttX0003).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

**Huang Tao** received a B.Eng. degree in automation from Southeast University, Nanjing, China, in 2008; an M.Eng. degree in electrical engineering and electronics from University of Electronic Science and Technology of China, Chengdu, China, in 2011; and a Ph.D. degree in mechanical engineering from Tsinghua University, Beijing, China, in 2017. He is currently an Associate Professor at the State Key Laboratory of Mechanical Transmission for Advanced Equipment and the College of Mechanical and Vehicle Engineering, Chongqing University, Chongqing, China. His research interests include dynamic modeling and control of mechatronic systems and ultra-precision motion control.

**Yingbin Wang** received a B.Eng. degree from Three Gorges University, Yichang, China, in 2022. He is currently studying for a master’s degree in mechanical engineering at the College of Mechanical and Vehicle Engineering, Chongqing University, with a research interest in ultra-precision motion control.

**Zhihong Luo** received a B.Eng. degree from Chongqing University of Technology, Chongqing, China, in 2020, and an M.Eng. degree from Chongqing University, Chongqing, China, in 2023. He is an engineer at the Hongdu Aviation Industry Group, Nanchang, China. His research interests include ultra-precision motion control, nonlinear systems modeling and identification, and multi-degree-of-freedom composite controller design.

**Huajun Cao** received a B.Eng. degree from Chongqing University, Chongqing, China, in 1999, and a Ph.D. degree from Chongqing University in 2004. He is currently a Professor at the State Key Laboratory of Mechanical Transmission for Advanced Equipment and the College of Mechanical and Vehicle Engineering, Chongqing University, Chongqing, China. His research interests include advanced manufacturing technology, green manufacturing and equipment, and manufacturing systems engineering.

**Guibao Tao** received a B.Eng. degree from Chongqing University, Chongqing, China, in 1989; an M.Eng. degree from Chongqing University in 1992; and a Ph.D. degree from Chongqing University in 2003. He is currently an Associate Professor at the State Key Laboratory of Mechanical Transmission for Advanced Equipment and the College of Mechanical and Vehicle Engineering, Chongqing University, Chongqing, China. His research interests include advanced manufacturing technology, intelligent manufacturing and equipment, and mechatronics technology.

**Mingxiang Ling** (Member, IEEE) received a B.Eng. degree from Xi’an Jiaotong University, Xi’an, China, in 2009; an M.Eng. degree from Harbin Institute of Technology, Harbin, China, in 2011; and a Ph.D. degree from Xi’an Jiaotong University in 2019. He is currently a Professor with the Robotics and Microsystems Center, Soochow University, Suzhou, China. His research interests include compliant mechanisms and piezoelectric acoustic intelligent sensing.