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Abstract

Piezoelectric micropositioning systems (PMSs) have been widely utilized in the high-precision manipulation applications, but are also subjected to undesired nonlinearities, like hysteresis, and parameter uncertainties. To solve this problem, this paper proposes a new robust sliding mode control scheme for a class of nonlinear PMSs with time-varying uncertainties. Different from the conventional sliding mode control (SMC), the proposed controller further combines the Fourier series-based function estimation technique, fuzzy logic system and adaptive learning algorithm to realize online estimation and compensation of system time-varying uncertainties without their boundary information. The adaptive laws of Fourier coefficients and fuzzy adjustable parameters are obtained via the Lyapunov stability theory. Compared with the existing SMC methods, the proposed control effectively eliminates the control chattering problem, and guarantees the convergence of the tracking error in finite time in the presence of time-varying uncertainties. Theoretical analysis and numerical simulation results show that the proposed control strategy can meet the high-speed, high-precision robust tracking performance requirements of PMSs for micro/nano-manipulation applications.

Keywords

Piezoelectric devices micropositioning robust control terminal sliding mode time-varying uncertainties

Introduction

With the rapid growth of high-precision requirements for manufacturing industry, micro/nano-manipulation technology has been paid much attention. Piezoelectric micropositioning systems (PMSs), as typical devices in this field, have widely used in the ultra-high precision positioning applications due to its compact structure, fast response speed, large output force and high resolution, such as scanning probe microscopy, biological cell manipulation, MEMS assembly.^1–3 However, due to the adverse nonlinearities including hysteresis, parameter uncertainties of PMSs, it is a great challenge to achieve high-speed and high-precision control performance.³ Therefore, it is necessary to design an effective and feasible control strategy to overcome these nonlinearities.

Hysteresis is a nonlinear behavior between the input voltage and output displacement of a piezoelectric actuator. As one of the main reasons for affecting the performance of PMSs, if it is not controlled, serious open-loop positioning errors will occur, especially at high frequencies.⁴ To suppress this nonlinearity, a straightforward approach is to build a feedforward compensator based on an inverse hysteresis model. Commonly used hysteresis models include Preisach model,⁵ Prandtl-Ishlinskii model,⁶ and Bouc-Wen model.⁷ However, in practice, the hysteresis cannot be completely eliminated because the feedforward control depends on the identification accuracy of the hysteresis model. To achieve high-precision tracking performance, feedforward plus feedback control methods are often used to suppress hysteresis nonlinearity and other uncertainties.^8–11 In addition, to avoid the complex process of hysteresis modeling, independent feedback control methods are also widely used for tracking control of PMSs. Such methods generally treat hysteresis as an unmodeled uncertainty, and then construct a robust controller to compensate, such as sliding mode control,^12–17 iterative control,¹⁸ and adaptive control.¹⁹

As a simple and effective nonlinear control method, the sliding mode control (SMC) is popular because of its powerful ability to deal with uncertain problems of the system. However, in the traditional SMC design, the control law is generally discontinuous, which will cause chattering problems, and the linear sliding mode surface can only ensure asymptotic stability and cannot converge in finite time. To this end, many improved versions have emerged. The terminal sliding mode (TSM) control adopts nonlinear sliding mode surface, and can ensure finite-time convergence.²⁰ However, the existence of negative fractional powers may lead to singularities at the equilibrium point.²¹ Thus, the nonsingular TSM (NTSM) control was provided,²² but the chattering problem still existed. The so-called fast NTSM (FNTSM) control was first proposed in Yu et al.,²³ which not only keeps the advantages of NTSM control, but also avoids the chattering because the control law itself is continuous. Furthermore, by utilizing a fast TSM-type reaching law, the system can be stabilized faster in finite time. However, no matter it is SMC methods or its improved forms, the boundary value of uncertainties needs to be known, which is difficult to realize in practice.

To overcome the above shortcoming and improve the robustness of the system, many estimation techniques have been reported to estimate system uncertainties online, such as disturbance observer (DO),²⁴ fuzzy logic systems (FLS),^25–27 perturbation estimation (PE),^28,29 and function estimation technique (FET).^30,31 In the design process of the DO, the bandwidth and order of the low-pass filter play a decisive role in the stability of the system, so it needs careful design and is difficult to implement. Although FLS can approximate any nonlinear function with arbitrary precision, it is difficult to guarantee the stability and robustness of the system because the selection of fuzzy rules depends on expert experience, and then needs to combine with other robust controllers. The PE methods are simple in design, but requires full state feedback. Compared with the above methods, the FET does not require full state feedback, and has the characteristics of determinacy and least mean square approximation. In view of these advantages, this paper will use the FET to observe the system uncertainties.

To this end, the objective of this paper is to propose an enhanced FNTSM control method to further improve the robust performance of a class of PMSs with time-varying uncertainties. In particular, the FET based on Fourier series is firstly used to estimate the uncertainties of the system, and then the FLS is applied to compensate the estimated error, and the corresponding adjustable parameters are updated online with the designed adaptive laws via Lyapunov stability. With the help of FET and FLS, the proposed control not only overcomes the limitation of the FNTSM control relying on prior boundary information, but also ensures the convergence of the system tracking error in finite time, which obviously improves the tracking accuracy. Finally, a series of simulations are conducted and verify the effectiveness and superiority of the proposed control. The main contributions of this paper include: (1) an FET based on Fourier series and an FLS are combined for the first time to estimate the time-varying uncertainties of the piezoelectric micropositioning system; (2) new Lyapunov theory based adaptive laws are designed to update the adjustable parameters of the controller online.

The organization of this paper is as follows: Section 2 presents the dynamic model of a class of PMSs, and briefly describes the Fourier series-based FET and FLS; after that, the design process of the proposed controller and its stability analysis are carried out in Section 3; then, the simulation verification and discussions are conducted in Sections 4 and 5, respectively; finally, Section 6 summarizes this paper.

Problem statement and preliminary

System description

Considering the fact that there often exist time-varying uncertainties such as external disturbances and hysteresis in the actual PMSs, without loss of generality, here we take a class of single-input single-output (SISO) PMSs for example, and the dynamic model can be described as follows²⁷:

\overset{\cdot\cdot}{x} (t) = f (X, t) + g (X, t) H {u (t)} + f_{d} (t)

(1)

where $X = {[x, \overset{\cdot}{x}]}^{T} \in R^{2}$ is the system state vector, $x (t) \in R$ is the output displacement, $u (t) \in R$ is the input control voltage, $f_{d} (t) \in R$ is an unknown but bounded external disturbance, $f (X, t), g (X, t) \in R$ are bounded time-varying functions and satisfy that

f (X, t) = f_{0} (X, t) + Δ f (X, t),

g (X, t) = g_{0} (X, t) + Δ g (X, t),

where $f_{0} (X, t)$ and $g_{0} (X, t)$ are the nominal model of functions $f (X, t)$ and $g (X, t)$ , while $Δ f (X, t)$ and $Δ g (X, t)$ are the unmodeled dynamics. $H {u (t)}$ represents the nonlinear hysteresis of the system, which can generally be decomposed into a linear term and a bounded nonlinear term,³² that is, $H {u (t)} = u (t) + h (t)$ . Therefore, the dynamic model (1) can be described as

\overset{\cdot\cdot}{x} (t) = f_{0} (X, t) + g_{0} (X, t) u (t) + F_{d} (t)

(2)

where $F_{d} = Δ f + Δ g \cdot u + g \cdot h + f_{d}$ is the lumped system uncertainty including external disturbances, hysteresis, and unmodeled dynamics, and it satisfies the following assumption:

Assumption 1: The system uncertainty $F_{d} (t)$ is bounded, and there is an unknown constant $\bar{D}$ such that $| F_{d} (t) | \leq \bar{D}$ .

Due to the existence of time-varying uncertainty $F_{d} (t)$ , the traditional SMC methods and their improved forms cannot be directly applied to the PMSs described by equation (2). In order to achieve robust and precise motion control of such systems and overcome the limitations of the FNTSM control relying on prior boundary information, this paper will use the FET to estimate $F_{d} (t)$ online, and use the FLS to compensate the estimated error. Next, the Fourier series-based FET is briefly introduced.

FET based on Fourier series

For any piecewise continuous function $f (t)$ , if it satisfies the Dirichlet condition, then it can be represented by a generalized Fourier series expansion defined on the interval $[0, T_{f}]$ , as follows:

f (t) = a_{0} + \sum_{i = 1}^{\infty} (a_{i} \cos ω_{i} t + b_{i} \sin ω_{i} t)

(3)

where $ω_{i} = 2 i π / T_{f} (i \in N)$ is the frequency of the sine function, $a_{0}$ , $a_{i}$ and $b_{i}$ are the Fourier coefficients.

Define

z (t) = {[\begin{matrix} 1 & \cos ω_{1} t & \sin ω_{1} t & \begin{matrix} \dots & \cos ω_{N} t & \sin ω_{N} t \end{matrix} \end{matrix}]}^{T}

(4)

w = {[\begin{matrix} a_{0} & a_{1} & b_{1} & \begin{matrix} \dots & a_{N} & b_{N} \end{matrix} \end{matrix}]}^{T}

(5)

Then, equation (3) can be rewritten as

f (t) = w^{T} z (t) + ε (t)

(6)

where $z (t)$ is the orthogonal basis function vector, $w$ is the weight coefficient vector, and $ε (t)$ is the estimated error. When $N \to \infty$ , the error $ε (t) \to 0$ . Hence, as long as $N$ is large enough, $f (t)$ can be approximately expressed as

f (t) ≅ w^{T} z (t)

(7)

Remark 1: No matter whether the function $f (t)$ is a periodic or aperiodic function, as long as a sufficiently large $T_{f}$ is selected, it can be expanded into the form of equation (3), and when $N$ is large enough, the function can also be estimated by equation (7).³

The advantage of using equation (7) to estimate the system uncertainty is that it converts the unknown time-varying function into a time-varying basis function vector $z (t)$ and constant vector $w$ . And according to the definitions in equations (4) and (5), $z (t)$ is known and $w$ is unknown. Therefore, the unknown constant vector $w$ can be updated online by choosing a suitable Lyapunov function to obtain an adaptive law.

It can be seen from equation (6) that the estimated error $ε (t)$ depends on the number $N$ of terms of the Fourier series, and it is difficult to select for the specified accuracy requirements. Although the estimation accuracy can be improved by increasing $N$ , the accompanying computation cost will also increase. That is, the value of $N$ is limited in practice, and the estimated error always exists. In order to further reduce the estimated error, a compensator needs to be designed. Therefore, the FLS is introduced as a compensator in this paper to improve the robustness of the system.

Overview of FLS

The rule base of the FLS used to compensate the estimated error can be regarded as the mapping from the input variable $χ = {[χ_{1}, χ_{2}, \dots, χ_{n}]}^{T} \in R^{n}$ to the output variable $ε \in R$ . This paper adopts the IF-THEN rule with multiple input and single output, and the jth fuzzy rule can be expressed as:

R^{(j)} : : i f χ_{1} is A_{1}^{j} and \dots and χ_{n} is A_{n}^{j} then ε is B^{j}

(8)

where $A_{i}^{j}$ and $B^{j}$ are the fuzzy sets corresponding to the input and output variables in the jth fuzzy rule, whose membership functions are $μ_{A_{i}^{j}}$ and $μ_{B^{j}}$ , respectively. If a single-valued fuzzer, a product inference engine, and a center-averaged defuzzer are used to design an FLS, the system output is

ε = \frac{\sum_{j = 1}^{n_{r}} {\bar{ε}}^{j} (Π_{i = 1}^{n} μ_{A_{i}^{j}})}{\sum_{j = 1}^{n_{r}} (Π_{i = 1}^{n} μ_{A_{i}^{j}})} = θ_{ε}^{T} ψ_{ε}

(9)

where $n_{r}$ is the number of fuzzy rules, ${\bar{ε}}^{j}$ is the point corresponding to the maximum value of the membership function $μ_{B^{j}}$ , that is $μ_{B^{j}} ({\bar{ε}}^{j}) = 1$ , $μ_{A_{i}^{j}}$ is the Gaussian membership function of the fuzzy input variable $χ$ , $θ_{ε} = {[{\bar{ε}}^{1}, {\bar{ε}}^{2}, \dots, {\bar{ε}}^{n_{r}}]}^{T}$ is the adjustable parameter vector, $ψ_{ε} = {[ψ_{ε}^{1}, ψ_{ε}^{2}, \dots, ψ_{ε}^{n_{r}}]}^{T}$ is the fuzzy basis vector, and $ψ_{ε}^{j}$ is defined as follows:

ψ_{ε}^{j} = \frac{Π_{i = 1}^{n} μ_{A_{i}^{j}}}{\sum_{j = 1}^{n_{r}} (Π_{i = 1}^{n} μ_{A_{i}^{j}})}

(10)

Controller design

The control goal of this paper is to design a robust controller that can quickly and accurately track the reference displacement command even when the PMS has model uncertainty and external disturbances. To this end, the controller design is mainly divided into three steps: first, the FET is used to estimate the uncertainty of the system, and the FLS is then adopted to compensate the estimated error; after that, the proper adaptive laws are selected through the Lyapunov stability theory to update the adjustable parameters online; finally, the control law is derived by improving the FNTSM control.

Control law

The tracking error is defined as

e \overset{Δ}{=} x_{r} - x

(11)

where $x_{r}$ is the reference displacement, and $x$ is the output displacement of the PMS. Therefore, a sliding mode variable can be defined as

s = e + β sig (\overset{\cdot}{e})^{γ}

(12)

where $β > 0$ , $1 < γ < 2$ , and the expression $sig (x)^{a}$ is denoted as $sig (x)^{a} = {| x |}^{a} sign (x)$ . It should be noted that the TSM surface defined by equation (12) is continuous and nonsingular.²³ When the sliding mode variable is equal to zero, we have

s = e + β sig (\overset{\cdot}{e})^{γ} = 0

(13)

For any given initial condition, the above equation can converge to zero in finite time, and the stable time can be expressed as

t_{s} = \frac{β^{- 1 / γ}}{1 - 1 / γ} {| e (0) |}^{1 - 1 / γ}

(14)

Taking the derivative of equation (13) with respect to time, we can get

\overset{\cdot}{s} = \overset{\cdot}{e} + β γ {| \overset{\cdot}{e} |}^{γ - 1} \overset{\cdot\cdot}{e} = \overset{\cdot}{e} + β γ {| \overset{\cdot}{e} |}^{γ - 1} ({\overset{\cdot\cdot}{x}}_{r} - \overset{\cdot\cdot}{x}) = 0

(15)

Ignoring external disturbance and system uncertainty, the equivalent control input is obtained by substituting equation (2) into equation (15) as

u_{eq} = g_{0}^{- 1} [- f_{0} + {\overset{\cdot\cdot}{x}}_{r} + β^{- 1} γ^{- 1} sig (\overset{\cdot}{e})^{2 - γ}]

(16)

In order to improve the robustness of the system and realize fast and continuous control, the following fast TSM reaching law is selected²³:

\overset{\cdot}{s} = - k_{1} s - k_{2} sig (s)^{p}

(17)

Then the control input can be given as

u_{re} = g_{0}^{- 1} [- F_{d} + k_{1} s + k_{2} sig (s)^{p}]

(18)

where $k_{1}, k_{2} > 0$ , $0 < p < 1$ .

It can be seen from equation (18) that the FNTSM control law depends on the boundary information of $F_{d}$ , so in order to realize the above control input, it is necessary to estimate $F_{d}$ . According to the analysis in Section 2.2, equation (7) can be used to estimate the unknown term $F_{d}$ , and the estimated error can be compensated according to equation (9), so we have

F_{d} = w^{T} z

(19)

{\hat{F}}_{d} = {\hat{w}}^{T} z + θ_{ε}^{T} ψ_{ε}

(20)

where ${\hat{F}}_{d}$ is the estimated value of the time-varying uncertainty term $F_{d}$ , and $\hat{w}$ is the estimated value of the weight coefficient vector $w$ . In order to realize the online estimation of the time-varying nonlinear function $F_{d}$ of PMS and avoid repeated trial and error to select the estimation function, it is necessary to design adaptive laws to adjust the coefficients $\hat{w}$ of the Fourier series in equation (20) and the adjustable parameter vector $θ_{ε}$ of the FLS, according to the Lyapulov stability theory, the following adaptive laws are selected:

\overset{\cdot}{\hat{w}} = - η_{1} β γ {| \overset{\cdot}{e} |}^{γ - 1} s z

(21)

{\overset{\cdot}{θ}}_{ε} = - η_{2} β γ {| \overset{\cdot}{e} |}^{γ - 1} s ψ_{ε}

(22)

where $η_{1}, η_{2} > 0$ . Substituting equation (20) into (18) gets

u_{re} = g_{0}^{- 1} [- {\hat{w}}^{T} z - θ_{ε}^{T} ψ_{ε} + k_{1} s + k_{2} sig (s)^{p}]

(23)

To ensure the control stability of the closed-loop system and overcome the shortcomings of traditional methods relying on disturbance boundary information, the following improved FNTSM control input is selected:

u = u_{eq} + u_{re}

(24)

where $u_{eq}$ and $u_{re}$ are given by equations (16) and (23), respectively.

Stability analysis

Lemma 1: If a Lyapunov function $V (x)$ satisfies the following first-order nonlinear differential inequality:

\overset{\cdot}{V} (x) + α V (x) + ρ V^{λ} (x) \leq 0

(25)

where $α, ρ > 0$ and $0 < λ < 1$ , then for any given initial condition $V (x (0)) = V_{0}$ , the stabilization time is²⁶

t_{s} \leq \frac{1}{α (1 - λ)} \ln \frac{α V_{0}^{1 - λ} + ρ}{ρ}

(26)

Theorem 1: For a class of PMSs with time-varying uncertainty and unknown boundary as described in equation (2), if the control law in equation (24) and the adaptive laws in equations (21) and (22) are used, then the following tracking performances are guaranteed:

(i) all signals in the closed-loop system are bounded;

(ii) if the nonlinear function $F_{d}$ is equal to its estimated value ${\hat{F}}_{d}$ , the tracking error $e$ and its velocity $\overset{\cdot}{e}$ converge to zero in finite time; otherwise, the tracking error $e$ and its velocity $\overset{\cdot}{e}$ converge to the following region in finite time:

| e | \leq 2 Δ = 2 \min (Δ_{1}, Δ_{2})

(27)

| \overset{\cdot}{e} | \leq {(\frac{Δ}{β})}^{1 / γ}

(28)

with $Δ_{1} = \frac{| {\hat{F}}_{d} - F_{d} |}{k_{1}}, Δ_{2} = {(\frac{| {\hat{F}}_{d} - F_{d} |}{k_{2}})}^{1 / p}$ .

Proof:

(i) Considering the following Lyapunov function:

V = \frac{1}{2} s^{2} + \frac{1}{2 η_{1}} {\tilde{w}}^{T} \tilde{w} + \frac{1}{2 η_{2}} θ_{ε}^{T} θ_{ε}

(29)

with $\tilde{w} = w - \hat{w}$ . Since $w$ is a constant vector, we have $\dot{\tilde{w}} = - \dot{\hat{w}}$ . The derivative of equation (29) with respect to time can be obtained by

\overset{\cdot}{V} = s \overset{\cdot}{s} - \frac{1}{η_{1}} {\tilde{w}}^{T} \overset{\cdot}{\hat{w}} + \frac{1}{η_{2}} θ_{ε}^{T} {\overset{\cdot}{θ}}_{ε}

(30)

Substituting equation (2) into (15) and combining equation (24) gets

\overset{\cdot}{s} = \overset{\cdot}{e} + β γ {| \overset{\cdot}{e} |}^{γ - 1} ({\overset{\cdot\cdot}{x}}_{r} - f_{0} - g_{0} u_{eq} - g_{0} u_{re} - F_{d})

(31)

And substituting equations (16), (23), and (19) into (31) yields

\overset{\cdot}{s} = - {\underline{k}}_{1} s - {\underline{k}}_{2} sig (s)^{p} - β γ {| \overset{\cdot}{e} |}^{γ - 1} {\tilde{w}}^{T} z + β γ {| \overset{\cdot}{e} |}^{γ - 1} θ_{ε}^{T} ψ_{ε}

(32)

with ${\underline{k}}_{i} = β γ {| \overset{\cdot}{e} |}^{γ - 1} k_{i} (i = 1, 2)$ . Multiply s at both sides of equation (32) to get

\begin{matrix} s \overset{\cdot}{s} = - {\underline{k}}_{1} s^{2} - {\underline{k}}_{2} s sig (s)^{p} - β γ {| \overset{\cdot}{e} |}^{γ - 1} s {\tilde{w}}^{T} z \\ + β γ {| \overset{\cdot}{e} |}^{γ - 1} s θ_{ε}^{T} ψ_{ε} \end{matrix}

(33)

Next, substitute equation (33) into (30) to get

\overset{\cdot}{V} = {\overset{\cdot}{V}}_{1} + {\overset{\cdot}{V}}_{2}

(34)

where

{\overset{\cdot}{V}}_{1} = - {\underline{k}}_{1} s^{2} - {\underline{k}}_{2} s sig (s)^{p}

(35)

{\overset{\cdot}{V}}_{2} = - {\tilde{w}}^{T} (β γ {| \overset{\cdot}{e} |}^{γ - 1} s z + \frac{1}{η_{1}} \overset{\cdot}{\hat{w}}) + θ_{ε}^{T} (\begin{matrix} β γ {| \overset{\cdot}{e} |}^{γ - 1} s ψ_{ε} \\ + \frac{1}{η_{2}} {\overset{\cdot}{θ}}_{ε} \end{matrix})

(36)

According to the adaptive laws (21)−(22) and (36), we can get

{\overset{\cdot}{V}}_{2} = 0

(37)

it is obvious that

\overset{\cdot}{V} = {\overset{\cdot}{V}}_{1} = - {\underline{k}}_{1} s^{2} - {\underline{k}}_{2} s sig (s)^{p} \leq 0

(38)

Therefore, all signals in the closed-loop system are bounded.

(ii) Choose the Lyapulov function $V = \frac{1}{2} s^{2}$ , then its derivative with respect to time is $\overset{\cdot}{V} = s \overset{\cdot}{s}$ . Substitute equation (16) and (23) into (31) to get

\overset{\cdot}{s} = \overset{\cdot}{e} + β γ {| \overset{\cdot}{e} |}^{γ - 1} (\begin{matrix} - β^{- 1} γ^{- 1} sig (\overset{\cdot}{e})^{2 - γ} + {\hat{F}}_{d} - k_{1} s \\ - k_{2} sig (s)^{p} - F_{d} \end{matrix})

(39)

Case 1: if $F_{d} = {\hat{F}}_{d}$ , it can be deduced from equation (39) that

\overset{\cdot}{V} = s \overset{\cdot}{s} = - {\underline{k}}_{1} s^{2} - {\underline{k}}_{2} s sig (s)^{p} = - {\underline{k}}_{1} s^{2} - {\underline{k}}_{2} {| s |}^{p + 1}

(40)

Since all signals in the closed-loop system are bounded and ${\underline{k}}_{1}, {\underline{k}}_{2}$ are positive, if $\overset{\cdot}{e} \neq 0$ , there exists constants $α_{1}, ρ_{1} > 0$ such that

\begin{matrix} \overset{\cdot}{V} = - {\underline{k}}_{1} s^{2} - {\underline{k}}_{2} {| s |}^{p + 1} \\ \leq - α_{1} s^{2} - ρ_{1} {| s |}^{p + 1} \\ = - 2 α_{1} V - 2^{(p + 1) / 2} ρ_{1} V^{(p + 1) / 2} \end{matrix}

(41)

Obviously, equation (41) satisfies the finite-time stability condition in Lemma 1. Therefore, the TSM surface is reachable in finite time, and the convergence time is

t_{s} \leq \frac{1}{α_{1} (1 - p)} \ln [1 + \frac{α_{1}}{ρ_{1}} {(2 V_{0})}^{(1 - p) / 2}]

(42)

Then, it can be obtained that the tracking error $e$ and its velocity $\overset{\cdot}{e}$ can converge to zero in finite time.

Case 2: if $F_{d} \neq {\hat{F}}_{d}$ , it can be deduced from equation (39)

\begin{matrix} \overset{\cdot}{V} = s \overset{\cdot}{s} = - {\underline{k}}_{1} s^{2} - {\underline{k}}_{2} s sig (s)^{p} - β γ {| \overset{\cdot}{e} |}^{γ - 1} s (F_{d} - {\hat{F}}_{d}) \\ = β γ {| \overset{\cdot}{e} |}^{γ - 1} ({\hat{F}}_{d} - F_{d}) s - {\underline{k}}_{1} s^{2} - {\underline{k}}_{2} {| s |}^{p + 1} \end{matrix}

(43)

In order to deduce that equation (43) satisfies the finite-time stability condition in Lemma 1, equation (43) is rewritten into the following two forms:

\overset{\cdot}{V} = - [{\underline{k}}_{1} - β γ {| \overset{\cdot}{e} |}^{γ - 1} ({\hat{F}}_{d} - F_{d}) s^{- 1}] s^{2} - {\underline{k}}_{2} {| s |}^{p + 1}

(44)

\overset{\cdot}{V} = - {\underline{k}}_{1} s^{2} - [{\underline{k}}_{2} - β γ {| \overset{\cdot}{e} |}^{γ - 1} ({\hat{F}}_{d} - F_{d}) sig {(s)}^{- p}] {| s |}^{p + 1}

(45)

For equation (44), if ${\underline{k}}_{1} - β γ {| \overset{\cdot}{e} |}^{γ - 1} ({\hat{F}}_{d} - F_{d}) s^{- 1} > 0$ and $\overset{\cdot}{e} \neq 0$ , then there exists $α_{2}, ρ_{2} > 0$ such that

\begin{matrix} \overset{\cdot}{V} = - [{\underline{k}}_{1} - β γ {| \overset{\cdot}{e} |}^{γ - 1} ({\hat{F}}_{d} - F_{d}) s^{- 1}] s^{2} - {\underline{k}}_{2} {| s |}^{p + 1} \\ \leq - α_{2} s^{2} - ρ_{2} {| s |}^{p + 1} \\ = - 2 α_{2} V - 2^{(p + 1) / 2} ρ_{2} V^{(p + 1) / 2} \end{matrix}

(46)

It can be found that the structure of equation (46) is the same as that of equation (41). Therefore, in order to realize the stability of equation (46) in finite time, it is necessary to ensure that the condition ${\underline{k}}_{1} - β γ {| \overset{\cdot}{e} |}^{γ - 1} ({\hat{F}}_{d} - F_{d}) s^{- 1} > 0$ is established, that is, it needs to be satisfied in the arrival phase

| s | > \frac{| β γ {| \overset{\cdot}{e} |}^{γ - 1} ({\hat{F}}_{d} - F_{d}) |}{{\underline{k}}_{1}} = \frac{| {\hat{F}}_{d} - F_{d} |}{k_{1}} = Δ_{1}

(47)

In other words, the sliding mode variable s can converge to the following region in finite time

| s | \leq Δ_{1}

(48)

Likewise, the conditions for achieving finite-time stabilization for equation (45) are

{\underline{k}}_{2} - β γ {| \overset{\cdot}{e} |}^{γ - 1} ({\hat{F}}_{d} - F_{d}) sig (s)^{- p} > 0

and $\overset{\cdot}{e} \neq 0$ . Therefore, the sliding mode variable s can also converge to the following region in finite time

| s | \leq {(\frac{| β γ {| \overset{\cdot}{e} |}^{γ - 1} ({\hat{F}}_{d} - F_{d}) |}{{\underline{k}}_{2}})}^{1 / p} = {(\frac{| {\hat{F}}_{d} - F_{d} |}{k_{2}})}^{1 / p} = Δ_{2}

(49)

Combining equations (48) and (49), it can be obtained that the sliding mode variable s converges to the region $| s | \leq Δ = \min (Δ_{1}, Δ_{2})$ in finite time. It is worth noting that, for the above two cases, there is no effect on the arrival phase when $\overset{\cdot}{e} = 0$ . Substituting equation (24) into (2), we can get

\overset{\cdot\cdot}{e} = - β^{- 1} γ^{- 1} sig (\overset{\cdot}{e})^{2 - γ} - k_{1} s - k_{2} sig (s)^{p} + {\hat{F}}_{d} - F_{d}

(50)

Then, for any $\overset{\cdot}{e} = 0$ , we have

\overset{\cdot\cdot}{e} = {\begin{matrix} - (k_{1} - \frac{{\hat{F}}_{d} - F_{d}}{s}) s - k_{2} sig {(s)}^{p} \neq 0, for | s | > Δ_{1} \\ - k_{1} s - (k_{2} - \frac{{\hat{F}}_{d} - F_{d}}{sig {(s)}^{p}}) sig {(s)}^{p} \neq 0, for | s | > Δ_{2} \end{matrix}

(51)

It shows that $\overset{\cdot}{e} = 0$ does not affect $| s | > Δ_{1}$ or $| s | > Δ_{2}$ . Therefore, the sliding mode variable s can still be guaranteed to converge to the region $| s | \leq Δ = \min (Δ_{1}, Δ_{2})$ in finite time.

To obtain the results in equations (27) and (28), the TSM surface (12) is rewritten as

e + (β - \frac{s}{sig {(\overset{\cdot}{e})}^{γ}}) sig (\overset{\cdot}{e})^{γ} = 0

(52)

When $β - s / sig {(\overset{\cdot}{e})}^{γ} > 0$ , equation (52) is consistent with the form of TSM surface (12). Since $| s | \leq Δ$ , the tracking error velocity $\overset{\cdot}{e}$ can converge to the region in finite time

| \overset{\cdot}{e} | \leq {(\frac{Δ}{β})}^{1 / γ}

(53)

According to equations (52) and (53), it can also be deduced that the tracking error e converges to the region within a finite time

| e | \leq β {| \overset{\cdot}{e} |}^{γ} + | s | \leq 2 Δ

(54)

Thus, the proof of Theorem 1 is completed.

Remark 2: According to equation (27), on the one hand, if the parameters $k_{1}, k_{2}$ are selected large enough to make the boundary value $Δ$ very small, the tracking performance of the controller will be further improved. However, increasing $k_{1}, k_{2}$ will also cause the control input to increase, making it difficult to achieve in practice. On the other hand, the boundary value $Δ$ also depends on the value of $| {\hat{F}}_{d} - F_{d} |$ . Based on equations (19) and (20), the boundary of $| {\hat{F}}_{d} - F_{d} |$ can be convergent as small as possible under the action of FLS compensator, which proves the benefit of the introduction of FLS.

In order to compare the superiority of the proposed controller with the adaptive fuzzy sliding mode with function approximation (AFSMFA) controller in Chen and Liang,³¹ the control input and adaptive laws of the AFSMFA controller based on the boundary layer principle are also given as follows:

\begin{matrix} u = g_{0}^{- 1} [- f_{0} - {\hat{w}}^{T} z - θ_{ε}^{T} ψ_{ε} + {\overset{\cdot\cdot}{x}}_{r} + λ_{1} \overset{\cdot}{e} + λ_{2} sat (s / δ)] \\ \overset{\cdot}{\hat{w}} = - η_{1} s z, {\overset{\cdot}{θ}}_{ε} = - η_{2} s ψ_{ε} \end{matrix}

(55)

where $λ_{1}, λ_{2} > 0$ , the sliding mode variable $s = \overset{\cdot}{e} + λ_{1} e$ , and other parameters are the same as those in this paper; the saturation function $sat (s / δ)$ is defined as

sat (s / δ) = {\begin{matrix} sign (s / δ), for | s | > δ \\ s / δ, for | s | \leq δ \end{matrix}

(56)

where $δ > 0$ is the thickness of the boundary layer, and the selection of its value requires a trade-off between the control input chattering and the tracking error. Using equation (56) to replace the discontinuous sign function can reduce the control chattering. Obviously, since the control law (55) adopts a linear sliding mode surface, the system tracking error cannot be guaranteed to converge in finite time, and the error interval is related to the thickness of the boundary layer.²²

Figure 1 shows the control block diagram of the proposed robust controller. It is worth noting that if the system uncertainty $F_{d}$ is set to zero, the stand-alone FNTSM control can be obtained as follows:

u = u_{eq} + g_{0}^{- 1} [k_{1} s + k_{2} sig (s)^{p}]

(57)

where $u_{eq}, s$ are the same as those in equations (16) and (18), respectively. Although the FNTSM controller can also ensure that the tracking error converges in finite time, when there is an unknown bounded disturbance in the system, the error convergence interval will increase and is proportional to the upper bound of the disturbance.²³

Figure 1.

Control block diagram of the proposed robust controller for PMSs.

Numerical simulation

Controller setup

In order to verify the performance of the designed robust controller, without loss of generality, this paper takes the PMS in Li and Xu²⁹ for example. Since the positioning platform is well decoupled in the XY directions, each axial movement can be regarded as a single-input single-output system. Considering the X-axis motion, the dynamic model of the system can be expressed as follows:

m \overset{\cdot\cdot}{x} + b \overset{\cdot}{x} + kx = k (du - h) + f_{d}

(58)

\overset{\cdot}{h} = ζ_{1} d \overset{\cdot}{u} - ζ_{2} | \overset{\cdot}{u} | h - ζ_{3} \overset{\cdot}{u} | h |

(59)

where x is the displacement, m, b, k represent the mass, damping, and stiffness in the X-axis direction of the system, respectively; d is the piezoelectric coefficient, u represents the input voltage, and h represents the displacement hysteresis loop, whose shape is determined by the parameters $ζ_{i} (i = 1, 2, 3)$ ; besides, $f_{d}$ represents the external disturbance of the system.

As reported in Li and Xu,²⁹ the model parameter values used in the simulation are given as

m = 0.1283 kg, b = 1.58 \times 10^{3} Ns / m, k = 1.9567 \times 10^{5} N / m,

\begin{matrix} d = 1.7339 \times 10^{- 6} m / V, ζ_{1} = 0.3575, ζ_{2} = 0.0364, \\ ζ_{3} = 0.0272 . \end{matrix}

Considering that the model parameters may vary with the operating environment in the actual PMS, it is necessary to consider the uncertainty of the parameters. Assuming that the nominal values of the parameters $m, b, k, d$ are $m_{0} = 0.1219 kg, b_{0} = 1.501 \times 10^{3} Ns / m, k_{0} = 1.8589 \times 10^{5} N / m$ , $d_{0} = 1.6472 \times 10^{- 6} m / V$ , respectively, the dynamic model (58) can also be rewritten in the form of equation (2) as

\overset{\cdot\cdot}{x} (t) = f_{0} (X, t) + g_{0} (X, t) u (t) + F_{d} (t)

(60)

where $f_{0} = - b_{0} / m_{0} \overset{\cdot}{x} - k_{0} / m_{0} x$ , $g_{0} = k_{0} d_{0} / m_{0}$ , $F_{d}$ is the lumped system uncertainty including external disturbances, hysteresis, and unmodeled dynamics. In the simulation process, $F_{d}$ is assumed to be an unknown time-varying nonlinear function.

Since $F_{d}$ is assumed to be unknown, it needs to be estimated using the Fourier function (3). By weighing the computation cost and estimation accuracy, the number of terms of the Fourier series is selected as $N = 50$ , and the period is $T_{f} = 2$ . In addition, to implement the FLS (9), we select the sliding mode variable s and its first derivative $\overset{\cdot}{s}$ as the input variables, and define the following five Gaussian membership functions for the input vector $χ = {[s, \overset{\cdot}{s}]}^{T}$ :

\begin{matrix} μ_{A_{i}^{1}} = {(1 + \exp (5 \times 10^{6} (χ_{i} + 2.5 \times 10^{- 6})))}^{- 1}, \\ μ_{A_{i}^{2}} = \exp (- \frac{1}{2} \times 10^{12} {(\frac{χ_{i} + 1.5 \times 10^{- 6}}{0.75})}^{2}), \\ μ_{A_{i}^{3}} = \exp (- \frac{1}{2} \times 10^{12} {(\frac{χ_{i}}{0.75})}^{2}), i = 1, 2 \\ μ_{A_{i}^{4}} = \exp (- \frac{1}{2} \times 10^{12} {(\frac{χ_{i} - 1.5 \times 10^{- 6}}{0.75})}^{2}), \\ μ_{A_{i}^{5}} = {(1 + \exp (- 5 \times 10^{6} (χ_{i} - 2.5 \times 10^{- 6})))}^{- 1} \end{matrix}

According to the analysis in the previous section, it can be found that the proposed controller has obvious advantages over the traditional SMC methods. However, without exception, this method also needs to select appropriate control parameters to effectively exert the performance of the controller. According to the parameter selection method of the FNTSM controller in Zheng et al.,³³ for the nominal system (60), the control parameters are selected as follows: $β = 0.01, γ = 1.5, k_{1} = 20850,$ $k_{2} = 20850, p = 0.5, η_{1} = 0.01, η_{2} = 0.01$ . Moreover, the initial conditions of the PMS are $x (0) = 0, \overset{\cdot}{x} (0) = 0$ , and the initial values of the estimated parameters $\hat{w}$ and $θ_{ε}$ are set to 0.

Since multi-frequency, triangular, and swept-frequency signals are often used as reference signals in PMSs,^4,28,33 the above signals are also adopted here to verify the tracking performance of the proposed controller. Besides, its robustness is also tested by time-varying parameters and external shock disturbances. For comparisons, the AFSMFA and FNTSM controllers are implemented with the same conditions. Table 1 lists the control simulation conditions for different performance tests, where the values of the system parameter m, b, k, d are randomly changed within $\pm 50 %$ of their nominal values to reflect the time-varying uncertainties in the actual PMS.

Table 1.

Control simulation conditions.

Tests	Reference signal ( $μ m$ )	Parameter uncertainty	Simulation time (s)
1	Multi-frequency sine: $x = 4 - \cos (200 π t) - \cos (500 π t) - \cos (800 π t) - \cos (1000 π t)$	$\pm 50 %$	0.02
2	Triangular: frequency=100Hz, P-P amplitude = 10	$\pm 50 %$	0.04
3	Swept-frequency sine: frequency = 1–100 Hz, P-P amplitude = 10	$\pm 50 %$	0.1
4	External shock disturbance	−	0.2

Different from literature,^28,33,34 this paper not only considers the influence of time-varying uncertainty of system parameter m, but also analyzes time-varying uncertainties of parameters b, k, d and high-frequency external disturbances. Furthermore, the highest frequency of specified reference signals are up to 500 Hz, which will strongly demonstrate the superiority of the proposed controller.

Multi-frequency trajectory tracking test

Multi-frequency trajectories are often used in scanning probe microscopes as reference signals for tracking the shape of uniform or non-uniform surface profiles.⁴ In these applications, especially in high frequency operation, the performance requirements of the controller are very demanding. Therefore, it is necessary to conduct Test 1 to evaluate the tracking performance of the controller. In this case, the proposed controller (24), the AFSMFA (55) and FNTSM (57) controllers are implemented according to the simulation conditions in Section 4.1. Figure 2 shows the time responses of positions, tracking errors, and control inputs for the three controllers in the PMS with/without time-varying uncertainties, respectively. It can be easily seen from the simulation results that when the system has no time-varying uncertainties, as shown in Figure 2(a) to (c), although all three controllers can accurately track the reference signal, the tracking error of the proposed controller is smaller compared with the other two controllers, and its maximum error is $0.27 μ m$ , which is only $4.5 %$ of the reference amplitude.

Figure 2.

Multi-frequency trajectory tracking results: (a–c) without uncertainties, (d–f) with uncertainties.

Moreover, for the case with time-varying uncertainties, it can be found from Figure 2(e) that the tracking errors of the PMS increase for all three controllers. Nevertheless, the proposed controller can still guarantee the minimum tracking error margin $max (| e |) = 0.33 μ m$ , which exactly confirms the previous theoretical analysis results. In addition, it can be seen from Figure 2(f) that compared with the AFSMFA controller, the control input signals of the proposed controller and the FNTSM controller are smoother and have no chattering, the reason of which lies in that the sizes of control chattering and tracking error of the AFSMFA controller depend on the selection of the boundary layer $δ$ , and the robust performance of the system cannot be guaranteed for a constant value $δ$ .

Triangular trajectory tracking test

Since the proposed controller considers continuous differentiable trajectories in the design process, it is very challenging to follow continuous but non-differentiable triangular trajectories. Thus, to further verify the tracking performance of the proposed controller, Test 2 is carried out. Similarly, three controllers are also implemented according to the simulation conditions in Section 4.1, and the simulation results are shown in Figure 3. It can be seen that the proposed controller can accurately track the triangular signal, and guarantee minimum error boundaries, which are $0.1 μ m$ and $0.2 μ m$ for the case without and with time-varying uncertainties, respectively. In comparison, the AFSMFA controller has the largest tracking error and obvious control chattering, especially at the positions where the reference velocity direction changes (such as the point at t = 5 ms). It is worth noting that the maximum tracking errors of all controllers occur at the positions, due to the discontinuity of the reference signal. For this reason, Bashash et al.³⁵ and Bazaei et al.³⁶ have done detailed research on this discontinuous trajectory and proposed proper control methods, but it is undeniable that the proposed controller is also an optional method.

Figure 3.

Triangular trajectory tracking results: (a–c) without uncertainties, (d–f) with uncertainties.

Swept-frequency trajectory tracking test

It is known that the hysteresis nonlinearity of PMSs is related to the input voltage frequency, and as the input frequency increases, the hysteresis effect becomes more serious. In order to verify the tracking performance of the proposed controller for variable frequency trajectories, the swept-frequency trajectory of Test 3 is carried out. In this case, three controllers are implemented with the same initial conditions in Section 4.1. Unsurprisingly, from the tracking results in Figure 4, it can be found that compared with AFSMFA and FNTSM control methods, the proposed controller can achieve the best tracking performance for both two cases. Although the tracking error of the proposed controller has slight chattering when the reference frequency increases, it is acceptable for high-speed operation with a bandwidth of up to 100 Hz. It also shows that the proposed controller can overcome the rate-dependent hysteresis nonlinearity and guarantee the tracking performance requirements of such trajectories.

Figure 4.

Swept-frequency trajectory tracking results: (a–c) without uncertainties, (d–f) with uncertainties.

Anti-disturbance test

Finally, to test the fast convergence and robustness performance of the proposed controller, an impulse-type disturbance with an interval of 2 ms and an amplitude of 30 N is applied to the PMS (58)−(59), and the initial simulation conditions are the same as those described in Section 4.1. The final results are shown in Figure 5. It can be clearly seen that the proposed controller requires the shortest settling time of 4 ms, and the tracking error can converge to $0.06 μ m$ . In contrast, although the settling time of the FNTSM controller is similar to the proposed controller, the steady-state error is larger. For the AFSMFA controller, it takes the longest time of 20 ms to reach stability, and the steady-state error is also the largest about $0.48 μ m$ . In addition, the fast convergence performance of the proposed controller can also be illustrated by the time responses of the sliding mode variables in Figure 5(c)–(e).

Figure 5.

Tracking responses of external shock disturbance: (a) position responses, (b) error responses, (c–e) sliding variable responses.

Discussions

According to the previous simulation results, the maximum (max) and root-mean-square (RMS) tracking errors of the three controllers are drawn in Figure 6 for intuitive comparison. It can be seen that the proposed controller can achieve the minimum $max (e)$ and $RMS (e)$ under all test conditions, no matter whether there exists the time-varying uncertainties or not. In contrast, the AFSMFA and FNTSM controllers offer a worse tracking performance in terms of robustness and response speed. It is notable that the proposed controller is not limited to the test conditions listed in Table 1, and its tracking performance and robustness can also be guaranteed for other types of reference signals.

Figure 6.

Comparisons of the max and RMS errors: (a and b) without uncertainties, (c and d) with uncertainties.

Although the advantages of the proposed controller are obvious over the AFSMFA and FNTSM controllers, it should be noted that the tracking performance of the designed controller is still limited by three aspects: on the one hand, the choice of terms of Fourier series needs to balance computational cost and estimation accuracy, so as to achieve better real-time control; on the other hand, the design of fuzzy rules and membership functions depends on the experience of the designer; finally, the choice of control gains requires a trade-off between control chattering and tracking error. For the first two aspects, there is no universal method at present, and the parameters with the optimal performance can only be found through trial-and-error way. For the third aspect, although the proposed control method is chattering free in nature, it still has slight chattering phenomenon for high-speed and serious time-varying uncertainties. One reason may be that it is difficult to balance chattering and tracking error at the same time. Thus, multi-objective optimization algorithms³⁷ can be considered to adjust the control gains to obtain better tracking performance.

Conclusions

In this paper, a robust controller composed of FNTSM controller, function estimator and fuzzy compensator was proposed for PMSs with the nonlinear and time-varying uncertainties. Based on Lyaplov stability principle, the adaptive laws of Fourier coefficients and fuzzy adjusting parameters were designed, and the online estimation and compensation of time-varying uncertainties of the system were realized. The proposed controller not only ensured that the tracking error converged to zero in finite time, but also did not require the boundary information of system uncertainties. In addition, the proposed controller was continuous without singularity in nature, leading to chattering free. In the simulations, three typical signals were selected as reference trajectories to test the tracking performance. Compared with the AFSMFA and FNTSM controllers, the proposed controller can achieve the best tracking performance regardless of whether the system has time-varying uncertainties or not. In the future work, the experiments will be conducted to further verify the effectiveness and superiority of the proposed control.

Footnotes

Acknowledgements

And the authors are grateful as well to the anonymous reviewers who improved the quality of the investigation.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the Key Project of Natural Science Research Project of Anhui Universities (Grant no. KJ2021A0418&KJ2020A0288). we would like to thank Anhui University of Science and Technology Research Start-up Fund for High-level Talents Introduction(13200391).

ORCID iD

Long Li

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Robust high-precision tracking control for a class of nonlinear piezoelectric micropositioning systems with time-varying uncertainties

Abstract

Keywords

Introduction

Problem statement and preliminary

System description

FET based on Fourier series

Overview of FLS

Controller design

Control law

Stability analysis

Numerical simulation

Controller setup

Multi-frequency trajectory tracking test

Triangular trajectory tracking test

Swept-frequency trajectory tracking test

Anti-disturbance test

Discussions

Conclusions

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

ORCID iD

References