Model-free robust finite-time force tracking control for piezoelectric actuators using time-delay estimation with adaptive fuzzy compensator

Abstract

A robust and practical force control system is crucial to the sensitive piezo-driven micromanipulation applications. This paper presents a new model-free robust finite-time force tracking controller for piezoelectric actuators (PEAs). The proposed controller composes of three intuitive terms: (1) a time-delay estimation (TDE) term that eliminates the requirement of detailed information about the PEA system, realizing model-free control; (2) a fast integral terminal sliding mode-based desired error dynamics injection term that ensures fast convergence and high tracking precision; (3) a correcting term based on adaptive fuzzy logic system that compensates for TDE errors caused by discontinuous nonlinearities and improves the robustness of the system. Force differential signal used in the controller is estimated online by a force state estimator. Stability of the closed-loop system and finite-time convergence are analyzed in theory. Comparative experiments are carried out on a PEA system with two superposed PEAs. Results show that the proposed control strategy has faster convergence, higher tracking accuracy and stronger robustness compared with the traditional TDE-based force controllers.

Keywords

Force control piezoelectric actuator terminal sliding mode time-delay estimation adaptive fuzzy compensator

Introduction

In many sensitive micromanipulation applications, such as cell injection (Wang and Xu, 2017), microassembly (Cecil et al., 2007) and micro-vibration isolation (Yang et al., 2019), piezoelectric actuators (PEAs) are widely used as a power source to exert a specified force on an operated object due to their advantages of fast response speed, high resolution, large force density and light weight (Gu et al., 2016; Yong et al., 2012). Through the inverse piezoelectric effect, PEAs can directly convert an electrical signal into a physical displacement (Ghemari, 2018). However, since their materials are generally ferroelectric, PEAs have inherently strong nonlinear behaviors, for example, hysteresis and creep effect. They severely degrade control performance and even damage the objects or cause system failure (Ghafarirad et al., 2015), which bring great challenges to the design of micromanipulation control system. Therefore, it is necessary to design a reliable control strategy to ensure accurate force output of the PEAs.

To mitigate the hysteresis nonlinearity for high control precision, commonly used control methods can be divided into two categories (Gu et al., 2016): feedforward control based on hysteresis model and robust control without hysteresis model. The former establishes the hysteresis model first and then constructs a feedforward compensation strategy. Several hysteresis models are available in the literatures, such as Preisach model (Tang and Li, 2015), Prandtl-Ishlinskii (P-I) model (Al Janaideh et al., 2011) and Bouc-Wen model (Xiao and Li, 2014). Differently, the latter treats the nonlinearity as a model disturbance and then builds a robust control to suppress it, such as sliding mode control (SMC) (Xu, 2014), active disturbance rejection control (ADRC) (Tang and Li, 2014), and model reference adaptive control (MRAC) (Liu et al., 2010). Moreover, hybrid control strategies have also been reported by combining the two methods together. For example, Badel et al. (2008) designed a self-sensing force control method based on a hyperbola-shaped hysteresis detection model. Zhu et al. (2018) proposed a force estimation strategy based on a fractional order hysteresis model with PID feedback control. Ghafarirad et al. (2015) applied a disturbance observer and SMC to the force tracking and established an inverse modified P-I model to overcome the hysteresis effect. However, whether the feedforward control or hybrid control, due to the coupling effect of strong hysteresis nonlinearity and complex dynamics behaviors, it is time-consuming and laborious for the acquisition of precise mathematical model of PEAs and the identification of system parameters by using the aforementioned control methods. Taking the modified P-I hysteresis model in (Gu et al., 2014) for instance, it consists of multiple play operators, which are determined by identifying more than 20 weights, and the precision of the model is highly dependent on the number of operators. Considering the practicability and robustness in practice, the robust control method is adopted in this paper. Specifically, a time-delay control (TDC) method using time-delay estimation (TDE) is realized without a complex physical model or many tuning parameters.

The main idea of the TDE is to estimate the lumped unknown dynamics of the system directly with the intentionally time-delayed states of the closed-loop systems, leading to an attractive model-free nature (Wang et al., 2016). Thanks to its efficiency and effectiveness, the TDE has been widely applied in various applications and obtains satisfactory performance even under large disturbances and parameter variations (Wang et al., 2016; Yu et al., 2019). However, the conventional TDC scheme exhibits two disadvantages. The first problem is that the use of a linear desired error dynamics (DED) can only make the system states asymptotically converge to the equilibrium point at an infinite time (Bae et al., 2017; Lee et al., 2014b). To overcome this issue, recently, a nonlinear terminal sliding mode (TSM) has been selected as the DED injection term because of its finite-time stability, in place of the commonly used linear dynamics (Jin et al., 2009). In addition, several evolved versions of the TSM-type DED have also been proposed in the literatures, such as fast nonsingular TSM (FNTSM) (Yu et al., 2019), fractional-order NTSM (FONTSM) (Wang et al., 2016) and integal TSM (ITSM) (Lee et al., 2014a, 2019). Among these approaches, the ITSM-type DED does not require the second derivative of the feedback signal, which normally avoids the noise amplification in practical applications. Nevertheless, the first derivative still exists in the designed ITSM surface and is generally calculated by the backward differentiator (BD). But it is well known that the BD algorithm is very sensitive to the measurement noise and can not function well especially under large uncertain disturbances. In this paper, a force state estimator (FSE) based on robust exact differentiator (RED) (Levant, 2003) is designed to obtain the required force states in the ITSM-type DED, effectively reducing the estimation error.

Another major issue of the conventional TDC is the so-called TDE error. The TDE concept was originally investigated to compensate for unknown nonlinearities of robot manipulator systems (Hsia et al., 1991). It assumes that there are not sudden changes in the magnitude and polarity of the nonlinear functions at velocity reversal. Therefore, the TDE functions perfectly for continuous nonlinearities, such as gravity, Coriolis and centrifugal torques. But for discontinuous nonlinearities (e.g. Coulomb friction and stiction), it is not flawless and results in pulse-type TDE error, which greatly affects the control performance (Jin et al., 2008). To suppress the TDE error, a third element has been introduced to the TDC as a correction term, such as ideal speed feedback (Jin et al., 2008), nonlinear damping (Jin et al., 2013) and fuzzy logic system (FLS) (Bae et al., 2017), which drastically improve the tracking performance. Specifically for PEA systems, discontinuous external disturbances (e.g. shock disturbance) are uncertain but inevitable, which can cause the TDE error, leading to the degradation of the robustness and control accuracy of the PEA system. Although the boundedness of the TDE error has been proved in (Lee et al., 2014a, 2019), there is still a big room for further improvement of control accuracy. However, to the best of our knowledge, there has been few research on the compensation of the TDE error in PEA systems. In this paper, we design a new adaptive FLS to compensate for the TDE error online, considering that the FLS has universal approximation property and can make use of linguistic information in a systematic way (Labiod et al., 2005; Nekoukar and Erfanian, 2011; Wang and Mendel, 1992). The adaptive laws for fuzzy tuning parameters are derived based on the Lyapunov theory to guarantee the system stability.

To this end, by synthesizing the aforementioned two aspects, this paper develops a new fast ITSM (FITSM) force tracking controller based on TDE and adaptive fuzzy compensator (AFC) for PEA systems. The main contribution lies in that an AFC is designed to compensate the TDE error in PEA systems for further improving the control accuracy for the first time. The effectiveness of the proposed controller is verified by a PEA system with two superposed PEAs.

The remainder of this paper is organized as follows. Section “Preliminary of FLS” gives a brief view of the FLS. Then the dynamic model of PEA system is derived in section “System description”. Afterwards, the design process of FITSM controller based on TDE and AFC is presented in section “Controller design”. Later, the stability of the proposed controller is proved in section “Stability analysis”, and the effectiveness and robustness of the proposed controller are verified by comparative experiments in section “Experimental verification”. Finally, section “Conclusion” concludes this paper.

Preliminary of FLS

This section presents a brief review of the FLS to be used later. An FLS performs a mapping from an input vector $x = {[x_{1}, x_{2}, \dots, x_{n}]}^{T} \in R^{n}$ to an output variable $y \in R$ . If fuzzy IF-THEN rules are employed, the $j$ th fuzzy rule is written as

R^{(j)} : if x_{1} is A_{1}^{j} and \dots and x_{n} is A_{n}^{j}, then y is B^{j}

where $A_{i}^{j}$ and $B^{j}$ are fuzzy sets with membership functions $μ_{A_{i}^{j}} (x_{i})$ and $μ_{B^{j}} (y)$ , respectively, and $x$ belongs to a compact set. By using the singleton fuzzifier, product inference engine and center average defuzzifier, the final output of fuzzy system can be defined as (Nekoukar and Erfanian, 2011)

y = \frac{\sum_{j = 1}^{n_{r}} {\bar{y}}^{j} (Π_{i = 1}^{n} μ_{A_{i}^{j}} (x_{i}))}{\sum_{j = 1}^{n_{r}} (Π_{i = 1}^{n} μ_{A_{i}^{j}} (x_{i}))} = θ^{T} ψ (x)

(1)

where $n_{r}$ is the total number of fuzzy rules, ${\bar{y}}^{j}$ is the point where the output membership function $μ_{B^{j}} (y)$ achieves its maximum value, $θ = {[{\bar{y}}^{1}, {\bar{y}}^{2}, \dots, {\bar{y}}^{n_{r}}]}^{T}$ is an adjustable parameter vector, and $ψ = {[ψ^{1}, ψ^{2}, \dots, ψ^{n_{r}}]}^{T}$ is a fuzzy basis vector, where $ψ^{i}$ is defined as

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

(2)

The shape of each input membership function $μ_{A_{i}^{j}} (x_{i})$ is characterized by Gaussian function, that is

μ_{A_{i}^{j}} (x_{i}) = a_{i}^{j} \exp [- \frac{1}{2} {(\frac{x_{i} - c_{i}^{j}}{b_{i}^{j}})}^{2}]

(3)

where $a_{i}^{j}$ , $b_{i}^{j}$ and $c_{i}^{j}$ are real-valued parameters with $0 < a_{i}^{j} \leq 1$ .

According to the universal approximation property of FLS (Wang and Mendel, 1992), the FLS mentioned above can approximate any real continuous function to any degree of accuracy provided that enough number of rules are considered.

System description

In order to study the force control problem of the PEA system, an experimental prototype is designed as shown in Figure 1(a). Its mechanical structure mainly includes two PEAs, which are inserted in a rigid support and connected in series with a force sensor and a preloaded spring. Thereinto, the bottom actuator (PEA1) is used as the controlled actuator for accurate force output, while the upper one (PEA2) to provide the controllable force disturbance. By acting on the driving voltage of the PEA2, it can be dynamically changed for the mechanical loading condition on the PEA1. In addition, the spring is introduced to apply a pre-pressure to the actuators. Its magnitude can be adjusted by a lock nut. The intermediate force between the PEA1 and external load system (see Figure 1(b)) can be measured by the force sensor. Through the above prototype, the output force of the PEA1 can be effectively detected and controlled under various load conditions.

Figure 1.

PEA system with force feedback: (a) physical prototype; (b) schematic diagram; (c) equivalent electro-mechanical model of the PEA1.

Generally, the dynamic model of the PEA system can be described by an equivalent electro-mechanical transform model (Gu et al., 2016), illustrated in Figure 1(c). Therefore, the model of the target system shown in Figure 1(a) is expressed as follows (Gu et al., 2016; Lee et al., 2019)

u = u_{a} + u_{h}

(4)

u_{h} = H (q)

(5)

q = C u_{a} + T_{em} x

(6)

f_{a} (x, u, \overset{\cdot}{u}) = T_{em} u_{a}

(7)

m_{e} \overset{\cdot\cdot}{x} + c_{e} \overset{\cdot}{x} + k_{e} x = f_{a} (x, u, \overset{\cdot}{u}) + f_{ext} - f_{p}

(8)

where $u$ is the control input voltage to the PEA1; $u_{a}$ is the transduced voltage due to the piezoelectric effect, while $u_{h}$ is the generated voltage due to the hysteresis effect $H$ ; $q$ is the total electrical charge in the PEA1, and $C$ represents the sum of capacitance; $T_{em}$ denotes the electro-mechanical transformation ratio; $f_{a}$ is the transduced force from the electrical side which mechanically drives the PEA1; $x$ is the output displacement of the mechanical part; $m_{e}$ , $c_{e}$ and $k_{e}$ are the equivalent mass, damping, and stiffness of the PEA1, respectively; $f_{ext}$ is the external interaction force to be controlled. $f_{p}$ denotes the overall perturbation of the system arising from model parameter uncertainties, unmodeled dynamics and other unknown terms, assumed to be bounded. In the experiment, $f_{p}$ is produced by the PEA2 to provide the controllable force disturbance.

In terms of the inherently rate-dependent and -independent behaviors of piezoelectric materials (Hassani et al., 2014), the hysteresis effect $H$ in the PEA1 is denoted by $h (x, u, \overset{\cdot}{u})$ to keep the generality. By integrating (4)-(8), the final dynamic model of the PEA system is written as

m_{e} \overset{\cdot\cdot}{x} + c_{e} \overset{\cdot}{x} + k_{e} x = T_{em} (u - h (x, u, \overset{\cdot}{u})) - f_{s} - f_{p}

(9)

where $f_{s}$ is the sensed-force measured by the force sensor, supposed to be $f_{ext} = - f_{s}$ .

In practical applications, it is often uncertain for the model parameters of the PEA system due to the changes in operating conditions. Although the parameters can be identified, the process is very complicated and the precision of the identified model can not be guaranteed. As a result, the control goal of this paper concentrates on a robust model-free controller such that the PEA system can track a reference force trajectory fast and accurately even under nonlinear hysteresis and external disturbance.

Controller design

FITSM controller based on TDE and AFC

To achieve the above goal, the force tracking error is defined as follows

e = f_{d} - f_{s}

(10)

where $f_{d}$ is the desired force supposed to be continuously differentiable.

Because of the voltage control input of the PEA system, the dynamics equation (9) is rearranged as

u = T_{em}^{- 1} (m_{e} \overset{\cdot\cdot}{x} + c_{e} \overset{\cdot}{x} + k_{e} x + f_{p}) + h (x, u, \overset{\cdot}{u}) + T_{em}^{- 1} f_{s}

(11)

From the above equation, we can find that an exact value of $T_{em}$ is required for the voltage control input. However, since $T_{em}$ is a bounded nonlinear function which is difficultly available in practice (Lee et al., 2019), we introduce a positive constant ${\hat{T}}_{em}$ as a nominal value of $T_{em}$ . Hence, the above equation can be reformulated as follows

u = ϕ (x, \overset{\cdot}{x}, \overset{\cdot\cdot}{x}, u, \overset{\cdot}{u}) + {\hat{T}}_{em}^{- 1} f_{s}

(12)

where $ϕ (x, \overset{\cdot}{x}, \overset{\cdot\cdot}{x}, u, \overset{\cdot}{u})$ denotes the lumped nonlinearities in the PEA system including the hysteresis and external disturbances. It is defined by

\begin{matrix} ϕ (x, \overset{\cdot}{x}, \overset{\cdot\cdot}{x}, u, \overset{\cdot}{u}) = T_{em}^{- 1} (m_{e} \overset{\cdot\cdot}{x} + c_{e} \overset{\cdot}{x} + k_{e} x + f_{p}) \\ + h (x, u, \overset{\cdot}{u}) + (T_{em}^{- 1} - {\hat{T}}_{em}^{- 1}) f_{s} \end{matrix}

(13)

To facilitate the controller design, the following assumption is introduced.

Assumption 1: The change rate of the lumped nonlinearities $ϕ (x, \overset{\cdot}{x}, \overset{\cdot\cdot}{x}, u, \overset{\cdot}{u})$ is bounded, that is $| \overset{\cdot}{ϕ} (x, \overset{\cdot}{x}, \overset{\cdot\cdot}{x}, u, \overset{\cdot}{u}) | \leq \bar{ϕ}$ .

For simplicity of expression, hereinafter, the notation $ϕ$ indicates $ϕ (x, \overset{\cdot}{x}, \overset{\cdot\cdot}{x}, u, \overset{\cdot}{u})$ .

To obtain precise and fast control performance and avoid the occurence of second derivative, the following ITSM surface is chosen (Lee et al., 2019)

s = e + λ_{1} \int_{0}^{t} [e + λ_{2} sig {(e)}^{β}] d τ

(14)

with the gains $λ_{1}, λ_{2} > 0$ , $0 < β < 1$ , and the notation $sig (\cdot)^{β} = {| \cdot |}^{β} sgn (\cdot)$ . Because the sliding mode surface is continuously differentiable, its first derivative with respect to time can be expressed as

\overset{\cdot}{s} = \overset{\cdot}{e} + λ_{1} e + λ_{1} λ_{2} sig (e)^{β}

(15)

To eliminate control chattering and achieve fast finite-time convergence, the following fast-TSM-type reaching law is adopted (Yu et al., 2005)

\overset{\cdot}{s} + k_{1} s + k_{2} sig (s)^{ρ} = 0

(16)

with the gains $k_{1}, k_{2} > 0$ , $0 < ρ < 1$ . Combining the sliding manifold (15) and the reaching law (16), the nonlinear DED of the closed-loop system can be deduced as

\overset{\cdot}{e} + λ_{1} e + λ_{1} λ_{2} sig (e)^{β} + k_{1} s + k_{2} sig (s)^{ρ} = 0

(17)

e + λ_{1}^{- 1} \overset{\cdot}{e} + λ_{2} sig (e)^{β} + {\underline{k}}_{1} s + {\underline{k}}_{2} sig (s)^{ρ} = 0

(18)

with ${\underline{k}}_{1} = λ_{1}^{- 1} k_{1}, {\underline{k}}_{2} = λ_{1}^{- 1} k_{2}$ .

In order to realize the DED (18), the following control input law is selected

u = {\hat{T}}_{em}^{- 1} υ + \hat{ϕ}

(19)

with

υ = f_{d} + λ_{1}^{- 1} \overset{\cdot}{e} + λ_{2} sig (e)^{β} + {\underline{k}}_{1} s + {\underline{k}}_{2} sig (s)^{ρ}

(20)

In (19), $\hat{ϕ}$ is the estimated value of $ϕ$ in (12). Since it contains the whole nonlinearities such as hysteresis, external disturbance and unmodeled dynamics, the accurate model of $\hat{ϕ}$ is difficult to acquire even through complex parameter identification process. To enhance the robustness against the uncertainties and the simplicity for practical application, the TDE technique is applied to obtaining $\hat{ϕ}$ , which can be expressed as (Lee et al., 2014a, 2014b, 2019)

\hat{ϕ} = ϕ_{(t - L)}

(21)

where the subscript $(t - L)$ represents a time-delayed value with an intentionally delayed time $L$ . And $ϕ_{(t - L)}$ is easily calculated by (12) as

ϕ_{(t - L)} = u_{(t - L)} - {\hat{T}}_{em}^{- 1} f_{s (t - L)}

(22)

Therefore, with the combination of (19)–(22), the TDE-based FITSM controller (FITSM+TDE) is expressed by (Lee et al., 2019)

\begin{matrix} u = \underset{injecting FITSM - type DED}{\underset{︸}{{\hat{T}}_{em}^{- 1} [f_{d} + λ_{1}^{- 1} \overset{\cdot}{e} + λ_{2} sig {(e)}^{β} + {\underline{k}}_{1} s + {\underline{k}}_{2} sig {(s)}^{ρ}]}} \\ + \underset{TDE}{\underset{︸}{u_{(t - L)} - {\hat{T}}_{em}^{- 1} f_{s (t - L)}}} \end{matrix}

(23)

Substituting the above controller into the system dynamics (12) yields the closed-loop dynamics as

\begin{matrix} e + λ_{1}^{- 1} \overset{\cdot}{e} + λ_{2} sig (e)^{β} + {\underline{k}}_{1} s + {\underline{k}}_{2} sig (s)^{ρ} \\ = {\hat{T}}_{em} (ϕ - ϕ_{(t - L)}) \end{matrix}

(24)

If $ϕ = ϕ_{(t - L)}$ , the closed-loop equation (24) becomes the DED (18). However, since the nonlinear term $ϕ$ consists of continuous terms and bounded discontinuous terms, the residual error $(ϕ - ϕ_{(t - L)})$ exists after applying the TDE. Accordingly, define the TDE error as

ε = {\hat{T}}_{em} (ϕ - ϕ_{(t - L)})

(25)

Then the closed-loop dynamics (24) is rewritten as

e + λ_{1}^{- 1} \overset{\cdot}{e} + λ_{2} sig (e)^{β} + {\underline{k}}_{1} s + {\underline{k}}_{2} sig (s)^{ρ} = ε

(26)

For the TDE error $ε$ , there exists a positive value $φ$ such that $ε$ is bounded as (Lee et al., 2019)

| ε | \leq φ

(27)

Although the TDE error $ε$ is bounded, it exhibits a pulse-type error due to discontinuous disturbance at velocity reversal, which may greatly affect the control performance of the PEA system. In order to suppress the effect of the TDE error, we design an AFC to improve the accuracy and robustness of the system.

By applying the introduced FLS in (1) and selecting $x = [e, \overset{\cdot}{e}]^{T}$ as the input vector, the nonlinear TDE error function $ε (x)$ can be approximated over a compact set $U_{x}$ as follows

\hat{ε} (x, θ) = θ^{T} ψ (x)

(28)

where $ψ (x)$ is a fuzzy basis vector properly specified by the designer, and $θ$ is the corresponding adjustable parameter vector calculated by an adaptive law.

Accordingly, the optimal parameter $θ^{*}$ can be defined such that

θ^{*} = \arg min_{θ} {sup_{x \in U_{x}} | ε (x) - \hat{ε} (x, θ) |}

(29)

and the minimum fuzzy approximation error as

ε_{e} (x) = ε (x) - \hat{ε} (x, θ^{*})

(30)

Assuming that the used FLS do not violate the universal approximation property on the compact set $U_{x}$ , it is reasonable to assume that $ε_{e} (x)$ is bounded for all $x \in U_{x}$ , that is

| ε_{e} (x) | \leq \bar{ε}, \forall x \in U_{x}

(31)

where $\bar{ε}$ is a positive constant.

To generate the approximation $\hat{ε} (x, θ)$ online, an adaptive law to adjust the parameter vector in (28) is developed as

\overset{\cdot}{θ} = - η λ_{1} s ψ (x)

(32)

with the gain $η > 0$ . Hence, a corrective controller based on (23) and (28) is designed to guarantee the stability of the closed-loop control system and compensate for the TDE error as

\begin{matrix} u = \underset{injecting FITSM - type DED}{\underset{︸}{{\hat{T}}_{em}^{- 1} [f_{d} + λ_{1}^{- 1} \overset{\cdot}{e} + λ_{2} sig {(e)}^{β} + {\underline{k}}_{1} s + {\underline{k}}_{2} sig {(s)}^{ρ}]}} \\ + \underset{TDE}{\underset{︸}{u_{(t - L)} - {\hat{T}}_{em}^{- 1} f_{s (t - L)}}} - \underset{AFC}{\underset{︸}{{\hat{T}}_{em}^{- 1} θ^{T} ψ (x)}} \end{matrix}

(33)

Similarly, if we substitute the proposed control input (33) into the system dynamics (12), the closed-loop dynamics after compensation can be expressed as

ε_{f} = ε - θ^{T} ψ (x)

(34)

where $ε$ is the same as (26). By combining (28) and (30), we can obtain

ε_{f} = ε_{e} (x) + \hat{ε} (x, θ^{*}) - \hat{ε} (x, θ)

(35)

Taking absolute values of the above equation on both sides and considering (31) yields

\begin{matrix} | ε_{f} | = | ε_{e} (x) + \hat{ε} (x, θ^{*}) - \hat{ε} (x, θ) | \\ \leq | ε_{e} | + | \hat{ε} (x, θ^{*}) - \hat{ε} (x, θ) | \\ \leq \bar{ε} + | \hat{ε} (x, θ^{*}) - \hat{ε} (x, θ) | \end{matrix}

(36)

According to (27), it is obvious that $| \hat{ε} (x, θ^{*}) - \hat{ε} (x, θ) |$ is bounded. Thus, the compensated error $ε_{f}$ is still bounded as

| ε_{f} | \leq δ

(37)

where $δ = \bar{ε} + | \hat{ε} (x, θ^{*}) - \hat{ε} (x, θ) |$ .

The block diagram of the proposed force tracking controller (33) for the PEA system is shown in Figure 2. It is obvious that the controller structure consists of three parts that have clear meaning: the TDE term is used to estimate and approximately cancel out the lumped nonlinearities of the PEA system without establishing its mathematical model; the FITSM-type DED term ensures accurate and fast finite-time force tracking; and the AFC term is developed to compensate for the TDE error online, leading to the high control accuracy and strong robustness.

Figure 2.

Block diagram of the proposed force tracking controller.

Remark 1: In the proposed controller, the parameter ${\hat{T}}_{em}$ significantly affects the control performance. Theoretically, the best selection of ${\hat{T}}_{em}$ is a nominal value of $T_{em}$ , but to ensure the stability of the closed-loop system, the value must be satisfied with the condition (Lee et al., 2019): $| 1 - T_{em} {\hat{T}}_{em}^{- 1} | < 1$ . Besides, in the experiments, because the sensor signals are always accompanied by measurement noise, lowering the control parameter ${\hat{T}}_{em}^{- 1}$ can effectively suppress the noise effect and play the role of low-pass filter (Jin et al., 2013).

Remark 2: Since the TDE estimates the unknown nonlinearities by intentionally using time-delay information of the system states, the delayed time $L$ should be sufficiently small to ensure the TDE functions correctly, which is generally taken as sampling time.

Estimation of force states

As we can see from Figure 2, the proposed controller depends on the force $f_{s}$ and its differential ${\overset{\cdot}{f}}_{s}$ as feedback signals. However, only the force measurement is available in our setup. Accordingly, an FSE is necessary for controller implementation. Recently, there have been various state estimation methods reported in the literatures, such as extended Kalman filter (Xu and Li, 2010), high gain observer (Li and Xu, 2010), sliding mode observer (Peng and Chen, 2014). The accuracy of these estimators generally relies on the extent to which the precise model information of the system is known. But for the PEA system in this paper, because of the strong nonlinearities, such as hysteresis and creep, the above approaches are not applicable. Naturally, the BD algorithm is widely used in (Lee et al., 2014b, 2014a, 2019) due to its simplicity, which has a format as

{\hat{\overset{\cdot}{f}}}_{s} = \frac{f_{s} - f_{s (t - T_{s})}}{T_{s}}

(38)

where ${\hat{\overset{\cdot}{f}}}_{s}$ is the estimated force differential at the current time $t$ , and $T_{s}$ is the sampling period. Note that the measured force $f_{s}$ normally contains noises, when differentiated, which would induce significant estimation error and extra chattering in the control input signal, deteriorating the tracking performance.

Therefore, we design an FSE based on a model-free RED (Levant, 2003) to effectively reduce the estimation error. The expression of the FSE is given as follows

{\begin{matrix} {\overset{\cdot}{z}}_{0} = - ζ_{1} {| z_{0} - f_{s} |}^{1 / 2} sgn (z_{0} - f_{s}) + z_{1} \\ {\overset{\cdot}{z}}_{1} = - ζ_{2} sgn (z_{0} - f_{s}) \end{matrix}

(39)

where $z_{0}, z_{1}$ are auxiliary states, and the gains $ζ_{1}, ζ_{2} > 0$ . The outputs $z_{0}, z_{1}$ of the FSE are the estimations of $f_{s}$ and ${\overset{\cdot}{f}}_{s}$ , respectively. We use the notations ${\hat{f}}_{s}$ and ${\hat{\overset{\cdot}{f}}}_{s}$ to represent the estimated force states, i.e., ${\hat{f}}_{s} = z_{0}$ and ${\hat{\overset{\cdot}{f}}}_{s} = z_{1}$ . Thus, the estimated state values are obtained as feedback signals from the FSE, as shown in Figure 2.

To verify the performance of the designed FSE, a comparative simulation is carried out with the BD algorithm. Provided that the desired force is $f_{d} = 1.25 - e^{- t} - 0.25 e^{- 32 t}$ , which adds a band-limited white noise to resprent the measured force $f_{s}$ . The noise power is set as $1 \times 10^{- 9}$ , and the estimator parameters are chosen as $T_{s} = 0.1$ ms, $ζ_{1} = 15$ , and $ζ_{2} = 110$ , respectively. Moreover, a first-order low-pass filter with cut-off frequency 50 Hz is adopted to further smooth the estimated force differential signal of each estimator. The simulation results are demonstrated in Figure 3. We can see that the FSE has higher estimation accuracy and lower noise level compared with the traditional BD algorithm, and enables the finite-time convergence of the estimated states. However, the FSE still contains a small level of errors due to the fact that it is impossible to completely eliminate them in practice. Consequently, when the FSE is applied to the proposed controller, these errors may cause corresponding tracking errors and control input chattering, which will be seen in the experimental section.

Figure 3.

(a) Comparision of force differential estimation performance using BD and FSE; (b) Drawing of partial magnification.

Remark 3: It is worth noting that the proposed controller (33) is designed based on the FITSM+TDE controller (23) presented in (Lee et al., 2019), but is also an enhancement of it. The biggest advantage lies in the introduction of the AFC, which ensures the higher control accuracy and stronger robustness. In addition, the first derivative of the force feedback signal is calculated by the designed FSE, reducing the amplification of measurement noises.

Remark 4: For purposes of comparison, another two controllers (Lee et al., 2014b, 2014a) are also presented in this paper. Specifically, if we let $k_{1} = 0$ , the controller (23) can be recast into the TDE-based ITSM controller (ITSM+TDE) shown in (Lee et al., 2014a), which can be expressed as

\begin{matrix} u = \underset{injecting ITSM - type DED}{\underset{︸}{{\hat{T}}_{em}^{- 1} [f_{d} + λ_{1}^{- 1} \overset{\cdot}{e} + λ_{2} sig {(e)}^{β} + {\underline{k}}_{2} sig {(s)}^{ρ}]}} \\ + \underset{TDE}{\underset{︸}{u_{(t - L)} - {\hat{T}}_{em}^{- 1} f_{s (t - L)}}} \end{matrix}

(40)

Similarly, by setting $λ_{2} = k_{1} = k_{2} = 0$ , the TDE-based PD controller (PD+TDE) in (Lee et al., 2014b) can be obtained as

u = \underset{injecting PD - type DED}{\underset{︸}{{\hat{T}}_{em}^{- 1} (f_{d} + λ_{1}^{- 1} \overset{\cdot}{e})}} + \underset{TDE}{\underset{︸}{u_{(t - L)} - {\hat{T}}_{em}^{- 1} f_{s (t - L)}}}

(41)

Compared with the nonlinear ITSM-type and linear PD-type DED, the nonlinear FITSM-type DED enables faster convergence in finite time (Lee et al., 2019), which will also be verified in this paper.

Stability analysis

Lemma 1: For a given closed-loop system, if there exists an extended Lyapunov function $V (x)$ satisfying the following fast TSM form

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

(42)

where $α, γ > 0$ and $0 < λ < 1$ , then the system can be claimed to be globally finite-time stable, and for any given initial condition $V_{0} = V (x (0))$ , the settling time can be estimated by (Yu et al., 2005)

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

(43)

Lemma 2: For positive numbers $a_{1}, a_{2}, \dots, a_{n}$ and $p \in (0, 2)$ , the following inequality holds (Yu et al., 2005)

{(a_{1}^{2} + a_{2}^{2} + \dots + a_{n}^{2})}^{p} \leq {(a_{1}^{p} + a_{2}^{p} + \dots + a_{n}^{p})}^{2}

(44)

Theorem: For the PEA system given by (9) with nonlinear hysteresis and uncertain external disturbance, if the control law is defined by (33), adaptive law is chosen as (32) and Assumption 1 holds, then the sliding variable $s$ converges to the neighborhood of ITSM $s = 0$ as

| s | \leq δ_{min} = min (δ_{1}, δ_{2})

(45)

in finite time, where $δ_{1} = {\underline{k}}_{1}^{- 1} δ$ , $δ_{2} = {({\underline{k}}_{2}^{- 1} δ)}^{1 / ρ}$ .

Proof: Consider the following Lyapunov function

V = \frac{1}{2} s^{2} + \frac{1}{2 η} θ^{T} θ

(46)

By differentiating $V$ with respect to time, we have

\overset{\cdot}{V} = s \overset{\cdot}{s} + \frac{1}{η} θ^{T} \overset{\cdot}{θ}

(47)

Substituting (15) into (47) yields

\begin{matrix} \overset{\cdot}{V} = s [\overset{\cdot}{e} + λ_{1} e + λ_{1} λ_{2} sig {(e)}^{β}] + \frac{1}{η} θ^{T} \overset{\cdot}{θ} \\ = λ_{1} s [e + λ_{1}^{- 1} \overset{\cdot}{e} + λ_{2} sig {(e)}^{β}] + \frac{1}{η} θ^{T} \overset{\cdot}{θ} \end{matrix}

(48)

From (26) and (34), we have

\begin{matrix} \overset{\cdot}{V} = λ_{1} s [ε_{f} - ({\underline{k}}_{1} s + {\underline{k}}_{2} sig {(s)}^{ρ}) + θ^{T} ψ (x)] + \frac{1}{η} θ^{T} \overset{\cdot}{θ} \\ = - λ_{1} s [{\underline{k}}_{1} s + {\underline{k}}_{2} sig {(s)}^{ρ} - ε_{f}] \\ + θ^{T} (λ_{1} s ψ (x) + \frac{1}{η} \overset{\cdot}{θ}) \end{matrix}

(49)

Substituting the parameter adaptive law (32) into (49) gives

\overset{\cdot}{V} = - λ_{1} s [{\underline{k}}_{1} s + {\underline{k}}_{2} sig {(s)}^{ρ} - ε_{f}]

(50)

which can be transformed into the following two forms

\overset{\cdot}{V} = - λ_{1} ({\underline{k}}_{1} - \frac{ε_{f}}{s}) s^{2} - k_{2} {| s |}^{ρ + 1}

(51)

and

\overset{\cdot}{V} = - k_{1} s^{2} - λ_{1} ({\underline{k}}_{2} - \frac{ε_{f}}{sig {(s)}^{ρ}}) {| s |}^{ρ + 1}

(52)

For the case of (51), if $({\underline{k}}_{1} - ε_{f} / s) > 0$ is satisfied, then $\overset{\cdot}{V} < 0$ . Therefore, from the boundedness $| ε_{f} | \leq δ$ in (37), the following region for $s$ can be reached

| s | \leq {\underline{k}}_{1}^{- 1} | ε_{f} | \leq {\underline{k}}_{1}^{- 1} δ

(53)

As for (52), by similar analysis, if $({\underline{k}}_{2} - ε_{f} / sig {(s)}^{ρ}) > 0$ holds, then $\overset{\cdot}{V} < 0$ and $s$ can also be reached in the following region

| s | \leq {({\underline{k}}_{2}^{- 1} δ)}^{1 / ρ}

(54)

Synthesizing (53) and (54), we have that the sliding variable $s$ is bounded in the region of (45).

Furthermore, using Lemma 2 and considering the Lyapunov function $V' = 0.5 s^{2}$ , we can rewrite (50) as

\begin{matrix} \overset{\cdot}{V}' & = - λ_{1} s [{\underline{k}}_{1} s + {\underline{k}}_{2} sig {(s)}^{ρ} - ε] \\ = - λ_{1} ({\underline{k}}_{1} - \frac{ε}{s}) s^{2} - k_{2} {| s |}^{ρ + 1} \\ \leq - λ_{1} ({\underline{k}}_{1} - \frac{ε}{s}) s^{2} - k_{2} {(s^{2})}^{\frac{ρ + 1}{2}} \\ = - 2 λ_{1} ({\underline{k}}_{1} - \frac{ε}{s}) V' - 2^{\frac{ρ + 1}{2}} k_{2} {V'}^{\frac{ρ + 1}{2}} \end{matrix}

(55)

which has the similar structure as (42), if we let $α = 2 λ_{1} ({\underline{k}}_{1} - ε / s)$ , $γ = 2^{(ρ + 1) / 2} k_{2}$ , $λ = (ρ + 1) / 2$ . Therefore, according to Lemma 1, the finite-time reachability of the closed-loop system can be guaranteed by

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

(56)

The proof is thus completed.

Experimental verification

In this section, comparative experiments are conducted on the PEA system to evaluate the performance of the proposed controller in trajectory tracking, hysteresis compensation and disturbance rejection.

Experimental setup

To implement the proposed force tracking controller, we establish a semi-physical real-time force tracking control system based on the Matlab/Simulink/xPC Target environment and the Real-Time Workshop (RTW) kernel, as shown in Figure 4. The mechanical structure of the PEA system has been described in Figure 1. The adopted PEAs (model Pst120/7/20VS12 with maximal voltage of 120 V and maximal generated force of 1200 N, Core-Tomorrow Co., Harbin, China) are actuated through a voltage amplifier (model E00.6 from the Core-Tomorrow Co.). The output force is measured by a dynamic piezoelectric IEPE-type force sensor (model ULT2551, Quatronix Co., Beijing, China). The sensor output voltage signal is passed through a signal conditioner (model CM4016L from the Quatronix Co.), and then acquired by the A/D channel of a data acquisition card (model PCI-6229 with 16-bit A/D and D/A converters from NI Co.). The voltage control signal is produced by the D/A channel and then amplified twelve times via the voltage amplifier to drive the PEAs. Control algorithms is developed with Matlab/Simulink software on a host PC and downloaded through the TCP/IP mode to a target PC to realize real-time control. In this experiment, the sampling frequency is set to 10 kHz, thus the delayed time $L = 0.1$ ms. In addition, parameter values $ζ_{1}, ζ_{2}$ of the FSE are chosen as 50 and 1000, respectively.

Figure 4.

Experimental setup for force tracking control.

To elaborate the benefits of the FITSM-type DED and AFC on improving the control performance, the proposed controller (33) is compared with other three controllers FITSM+TDE (23), ITSM+TDE (40) and PD+TDE (41), respectively. The control parameters of the controllers are selected heuristically to achieve best control performance and used for all experiments, as shown in Table 1, in which the same control parameter values are consistent for comparison. Since the TDE error function $ε (x)$ is estimated by the FLS in the form of (1) and $x = [e, \overset{\cdot}{e}]^{T}$ is chosen as the input vector, we define the following five Gaussian functions as

{\begin{matrix} μ_{A_{i}^{1}} (x_{i}) = \exp (- \frac{1}{2} {(\frac{x_{i} + 0.8}{0.2})}^{2}) \\ μ_{A_{i}^{2}} (x_{i}) = \exp (- \frac{1}{2} {(\frac{x_{i} + 0.4}{0.2})}^{2}) \\ μ_{A_{i}^{3}} (x_{i}) = \exp (- \frac{1}{2} {(\frac{x_{i}}{0.2})}^{2}) for i = 1, 2 \\ μ_{A_{i}^{4}} (x_{i}) = \exp (- \frac{1}{2} {(\frac{x_{i} - 0.4}{0.2})}^{2}) \\ μ_{A_{i}^{5}} (x_{i}) = \exp (- \frac{1}{2} {(\frac{x_{i} - 0.8}{0.2})}^{2}) \end{matrix}

Table 1.

Control parameters of the four controllers.

Controllers	${\hat{T}}_{em}^{- 1}$	$λ_{1}$	$λ_{2}$	$β$	$k_{1}$	$k_{2}$	$ρ$	$η$
Proposed	0.04	10	10	0.6	3	3	0.6	0.5
FITSM+TDE	0.04	10	10	0.6	3	3	0.6	-
ITSM+TDE	0.04	10	10	0.6	-	3	0.6	-
PD+TDE	0.04	10	-	-	-	-	-	-

Moreover, to evaluate the tracking accuracy of the controllers, the root-mean-squared (RMS) error is calculated by

e_{RMS} = \sqrt{\sum_{i = 1}^{N} {(f_{di} - f_{si})}^{2} / N}

where $N$ is the number of sampling points, and $f_{di}, f_{si}$ are the $i$ th desired and measured force, respectively.

Open-loop testing results

The open-loop tests of the PEA system is first carried out by driving the PEA1 with full-range sinusoidal voltage signals at different frequencies. The experimental results are shown in Figure 5. They disclose the severe hysteresis nonlinearity of the PEA system between the input voltage and output force under the open-loop strategy. In addition, Figure 5 also illustrates that the hysteresis nonlinearity caused errors (h/H) deteriorate with the increase of the input frequencies, due to the coupling effect of the hysteresis nonlinearity and dynamics behaviors. That makes it more challenging to develop an effective force controller for the PEA system. As a result, the following experiments will be performed to verify the effectiveness of the developed control approach on the hysteresis compensation.

Figure 5.

Open-loop hysteresis of the PEA system with sinusoidal signal inputs at different frequencies: (a) 1 Hz; (b) 5 Hz; (c) 10 Hz; (d) 15 Hz.

Sinusoidal force tracking testing results

As has been analyzed in the foregoing, the hysteresis nonlinearity errors tend to deteriorate as the input frequencies increase. Therefore, to verify the superiority of the proposed controller, we first evaluate the tracking performance in response to sinusoidal force signals with the amplitude of 1 N and frequencies of 1, 5, 10 and 15 Hz, respectively. The experimental results of the four controllers at 5-Hz frequency are shown in Figure 6. In addition, to make the comparison more accurate, the RMS values of the force tracking errors are listed in Table 2 to reflect the steady control performance. Finally, without loss of generality, we take the case of 5-Hz frequency to illustrate the effectiveness of the AFC for the TDE error, and the results are depicted in Figure 7.

Figure 6.

Experimental results of sinusoidal force signal tracking of the four controllers at 5-Hz frequency: (a)–(d) force tracking results; (e)–(h) tracking error results; (i)–(l) hysteresis compensation results; (m)–(p) control input voltages.

Table 2.

RMS errors of the four controllers under different signals with different frequencies (Unit: N).

Signal	Frequency	Proposed	FITSM+TDE	ITSM+TDE	PD+TDE
Sine wave	1 Hz	0.0086	0.0094	0.0095	0.0446
	5 Hz	0.0100	0.0123	0.0191	0.0560
	10 Hz	0.0223	0.0253	0.0431	0.0904
	15 Hz	0.0411	0.0544	0.0863	0.0941
Triangle wave	1 Hz	0.0086	0.0096	0.0096	0.0430
	5 Hz	0.0213	0.0239	0.0305	0.0625
	10 Hz	0.0454	0.0484	0.0534	0.0687
	15 Hz	0.0847	0.0861	0.1030	0.1829

Figure 7.

Comparasion of the estimation of TDE (a) and the TDE error (b) with/without the AFC for tracking the 5-Hz sinusoidal force signal.

As observed from Figure 6, all four controllers enable the PEA system to accurately track the reference signal, which clearly indicates that the TDE technique is effective for eliminating the nonlinearities of hysteresis and creep in the PEA system. However, it is obvious that the tracking performance of the proposed controller is the best among the four controllers, while the PD+TDE controller provides relative the worst performance, which is consistent with the theoretical analysis of Remark 3.

Comparing the results from different frequencies, we find that the control performance degrades as the reference signals become faster. In spite of this, the proposed controller can still guarantee the best control performance than other three. Also, as seen from the RMS errors in Table 2, when input frequencies are increased from 1 Hz to 15 Hz, error ratios (RMS errors / amplitudes) of the proposed controller are degraded from $0.86 %$ to $4.11 %$ with an increase of $3.25 %$ . But the increases of error ratios of the FITSM+TDE, ITSM+TDE and PD+TDE controllers are $4.5 %$ , $7.68 %$ , $4.95 %$ respectively, which are all higher than that of the proposed controller. Since all four controllers adopt the same TDE technique, the above results prove that the FITSM-type DED can guarantee faster convergence rate and higher tracking accuracy than the ITSM and PD-type DED.

Furthermore, it can be found from Figure 7 that the RMS values of the TDE errors are decreased from 0.3909 N (for $ε$ without the AFC) to 0.1792 N (for $ε_{f}$ with the AFC). Compared with the uncompensation, the tracking performance is greatly improved by $54.16 %$ , which confirms the effectiveness of the AFC.

Triangular force tracking testing results

Nextly, the comparative experiments are also conducted to evaluate the tracking performance in response to another commonly used reference singal, that is, a non-smooth (continuous but non-differentiable) triangular force signal. Figure 8 shows the experimental results of the tracking triangular force signals with the amplitude of 1 N and frequencies of 1, 5, 10, and 15 Hz, respectively. The RMS values of the tracking errors are listed in Table 2. And the corresponding TDE errors under the case of 5-Hz frequency are depicted in Figure 9.

Figure 8.

Experimental results of triangular force signal tracking of the four controllers at 5-Hz frequency: (a)–(d) force tracking results; (e)–(h) tracking error results; (i)–(l) hysteresis compensation results; (m)–(p) control input voltages.

Figure 9.

Comparasion of the estimation of TDE (a) and the TDE error (b) with/without the AFC for tracking the 5-Hz triangular force signal.

Similar to the analysis process of the sinusoidal tracking, not surprisingly, Figure 8 also shows unanimous results. But the differences are that due to the non-smooth characteristic of the triangular signal, the jumply tracking errors for all four controllers apparently occurs at the points where the reference singal changes its sign, for example, at $t = 0.05$ s as shown in Figure 8(e)–(h). These errors mainly result from the difficulty of estimating the derivative accurately from the force measurements at velocity reversal. Nevertheless, the proposed controller can still ensure minimal RMS errors at different frequencies among the four controllers. Moreover, we find from Figure 9 that the AFC still performs well. Hence, the benefits of the FITSM-type DED and AFC are verified again.

Robustness and control bandwidth testing results

Finally, the robustness against external disturbance and control bandwidth of the proposed controller are respectively examined. For the robustness test, we apply a pulse voltage with a duration of 0.05 s and an amplitude of 60 V to the PEA2, while the PEA1 is commanded to track the 5-Hz sinusoidal reference force signal with an amplitude of 1 N. And the control bandwidth is tested by applying the 1-N sinusoidal signal with the frequency varying from 1 to 50 Hz. Corresponding experimental results are shown in Figures 10–11 and Table 3, in which the steady-state error denotes its RMS value after each settling time. It can be obviously seen from Table 3 that the proposed controller takes the least settling time with 0.131 s for the force to converge within 0.0106 N. Comparatively, the ITSM+TDE controller is disturbed with the largest error magnitude of 3.038 N and takes 0.143 s to settle down; while the PD+TDE needs the longest time of 0.512 s to converge and gives maximal steady-state error of 0.059 N, although its maximum error is fairly small with 2.166 N. Therefore, the above results illustrate the fast convergence and robustness of the FITSM-type DED and AFC of the proposed controller against external disturbances. Moreover, Figure 11 also demonstrates that the proposed controller achieves satisfactory tracking performance in a biggest bandwidth of 25 Hz among the four controllers.

Figure 10.

Experimental results of the four controllers under an external shock disturbance: (a) force control results; (b) force tracking errors.

Figure 11.

Control bandwidth testing results of the four controllers.

Table 3.

Robustness testing results of the four controllers.

Controllers	Max. error (N)	Settling time (s)	Steady-state error (N)
Proposed	2.821	0.131	0.0106
FITSM+TDE	2.696	0.139	0.0124
ITSM+TDE	3.038	0.143	0.0201
PD+TDE	2.166	0.512	0.0590

Discussion

Both theoretical analysis and experimental results reveal that the proposed controller is effective, and it provides faster response speed, higher tracking accuracy and stronger anti-disturbance performance than those of the FITSM+TDE, ITSM+TDE and PD+TDE controllers. The main reason is that the nonlinear FITSM-type DED can guarantee faster convergence rate and higher tracking accuracy compared with the ITSM and PD-type DED. Moreover, the introduction of the AFC can effectively compensate for the TDE error and further improve the control performance. Because of the TDE technique, the proposed controller is model free, avoiding the complex parameter identification process of the PEA system. And using the adaptive law to adjust the fuzzy parameters online is also convenient for practical application.

It should be noted that as stated in the introduction, the only aim of this paper is to design a reliable control method to ensure accurate force tracking of PEA systems. The problem has given rise to another question related to the control efficiency. Considering the advantages of fractional calculus on improving control energy (Aguila-Camacho and Duarte-Mermoud, 2016), in the next step, an adaptive fractional-order controller (Wang et al., 2019) will be employed to achieve high tracking accuracy and control efficiency.

Conclusion

In this paper, a new FITSM controller based on TDE and AFC is proposed to solve the force tracking control problem of PEAs. The proposed controller is model free and easy to implement due to the TDE, and guarantees fast convergence and high precision tracking performance thanks to the FITSM-type DED. The designed AFC effectively compensates for the TDE error caused by external disturbances. In addition, the model-free FSE based on RED is presented to estimate the system states online, avoiding the noise effect. Stability of the closed-loop system is proved by the Lyapunov theory. Experimental results show that the proposed controller achieves better force tracking accuracy and faster anti-disturbance performance compared with the traditional FITSM+TDE, and ITSM+TDE and PD+TDE controllers. In the future work, an adaptive fractional-order controller will be developed to further improve the control efficiency.

Footnotes

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 was supported in part by the National Key Research and Development Program of China (Grant No. 2018YFC0309102, 2018YFC0309103) and in part by the National Natural Science Foundation of China (Grant No. 51975277, 51705243).

ORCID iDs

Shengzheng Kang

Xiaolong Yang

Yaoyao Wang

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