Nonlinear disturbance observer–based adaptive neural control for electro-hydraulic servo system with model uncertainty and full-state constraints

Abstract

The electro-hydraulic servo system (EHSS) performs model uncertainty and state constraints such that the exact model-based controller is difficult to be designed. In this work, a nonlinear disturbance observer (NDO)-based adaptive neural control (ANC) is proposed for the EHSS, in which a nonlinear transformation function is constructed to make the state constraints problem transformed into state unconstraint problem. The NDO is introduced to improve the disturbance rejection ability. The ANC is utilized to approximate unmodeled dynamics. The second-order filters are integrated with backstepping control to solve the explosion of complexity. The proposed NDO-based ANC scheme confines all states within the predefined bounds, improves the robustness of closed-loop system, and alleviates the computation burden. Moreover, the stability analysis for the closed-loop system is given within the Lyapunov framework. Simulations and experiments show that the proposed control scheme can achieve excellent control performance and robustness with regard to full-state constraints and model uncertainty.

Keywords

Electro-hydraulic servo system nonlinear disturbance adaptive control neural network full-state constraints model uncertainty

Introduction

Electro-hydraulic servo system (EHSS) has been widely used in modern industry, such as vehicle suspension, load simulator, and tunnel boring machine, due to the advantages of high power-weight ratio, large driving force, and fast response (Deng et al., 2023; Guo et al., 2022; Lee et al., 2020; Vaezi and Izadian, 2015; Wang et al., 2020b; Yang et al., 2022a). However, the nonlinearity, saturation, dead zone, hysteresis, friction, and backlash existed in hydraulic system hinder the high-precision motion control of EHSS (Wang and Na, 2020; Yin et al., 2019; Zhao et al., 2020). Furthermore, the parametric uncertainty, time-variant dynamics, load sensitivity, and unmodeled dynamics further deteriorate the EHSS control performance (Shtessel et al., 2012). Nevertheless, current EHSS in modern industry demands both high tracking performance and good anti-interference. To meet this requirement, the enhancement of the control performance of EHSS is necessary.

In recent decades, numerous control strategies have been developed to improve the control performance for EHSS, such as proportional-integration-differentiation (PID) (Van et al., 2020), adaptive control (AC) (Yao et al., 2019), sliding mode control (SMC) (Cheng et al., 2020), backstepping control (BC) (Zhuang et al., 2020), adaptive robust control (ARC; Bai et al., 2022), and adaptive neural control (ANC; Yang et al., 2022b). Due to the simple structure, PID and its various variants are commonly used for position tracking in the situations of low control response and precision. However, traditional PID with fixed gain cannot obtain the satisfactory tracking performance and robustness because of the time-varying parameters interference (Zhao et al., 2022). AC can handle the parametric uncertainty well but show degraded control performance when facing unmodeled disturbance (Liu et al., 2021a). Since the high robust to uncertainty and disturbance, SMC is widely used in the nonlinear hydraulic system (Palli et al., 2018). Nevertheless, the chattering caused by the discontinuous switching part in control law may degrade the control performance and even disrupt the system stability (Deldar et al., 2015; Liu et al., 2021b). To eliminate the chattering phenomenon, the continuous switching function (such as saturation function and hyperbolic function) was used to substitute the discontinuous sign function in SMC design procedure (Li et al., 2023). The design of BC and the combination of adaptive BC have certain advantages in dealing with nonlinear problem, especially in the control of complicated hydraulic servo system (Liu et al., 2017). However, the repeated derivative of virtual control law in each step led to the explosion of complexity problem. To remedy this issue, the dynamic surface control was integrated in the controller design in each recursive step to avoid the differentiation operation (Guo et al., 2016). In addition, the first-order filter was introduced to design the BC control scheme for the nonlinear system subjected to parameter uncertainty and external disturbance. Nevertheless, for nonlinear virtual control laws, the estimation precision of first-order filters is limited, especially when the system was subject to external disturbance (Li et al., 2020a). Interestingly, ARC took the effects of uncertainty into consideration in the control design to guarantee the stability and tracking precision in the presence of bounded uncertainty. However, its strict limit conditions hindered the application in practice. Recently, ANC was introduced into the control field due to its excellent features such as universal approximation, online self-learning capability, and ideal parallel processing structure (Li et al., 2020b; Liu et al., 2019; Wang et al., 2020a).

In recent years, the application disturbance observer–based control method has received much attention for controller design with disturbances and uncertainty (Zhou et al., 2020). It is noted that disturbance observer–based control methods estimate the current disturbance and use estimates with appropriate feedback control to counteract the effects of the disturbance. Chen et al. (2017) proposed a disturbance observer–based ANC to tackle the external disturbances and input saturation of uncertain nonlinear system, where the auxiliary system was constructed to compensate the effect of the time-varying external unknown disturbances. Qiu et al. (2022) utilized disturbance observer to estimate fuzzy approximation errors and external disturbance of strict-feedback nonlinear system for achieving finite-time prescribed performance. In spite of good results in terms of disturbance tracking, the abovementioned disturbance observer was no way to handle the estimation accuracy well. Thus, the nonlinear disturbance observer (NDO) was developed to accurately estimate the unknown external disturbance. Mohammadi et al. (2013) employed NDO to estimate the unknown payloads and unmodeled dynamics of robotic manipulators. Yang et al. proposed NDO-based robust control for airbreathing hypersonic vehicles, where NDO was applied to estimate the nonlinear cross coupling effect between propulsive and aerodynamic forces. The disturbance estimation error was handled under the assumption that the controller coefficients with known signs, upper and lower bounds, and bounded derivatives. These conservative assumptions limit the practical application of NDO. Thus, it is necessary to design a novel NDO for relaxing the restrictions and enhancing the robustness (Hamann et al., 2017; Livingstone and Cavanaugh, 2015; Maghareh et al., 2018).

Be mindful of the fact that many physical systems suffer from the effect of the constraints, such as working space for mechanical systems, temperature for the chemical reaction facility, and transmission distance of the communication equipment (Wen and Chen, 2023). These constraints are predefined in the practical control process for the purpose of safety. In many practical situations, if the state variable violates a certain range, it may degrade system performance, damage the system, or even cause serious security incidents. There are two major approaches to tackle state constraints problem. One is the barrier Lyapunov function (BLF) method, and the other is the constraint transformation method. The idea of BLF is to limit the system state indirectly by using the related error constraints. Wang et al. (2022) presented BLF based on adaptive fuzzy controller to handle the state constraints of pure-feedback nonlinear system, where integral BLF was employed to guarantee the state variables within the prescribed constraints by using the finite time function. Wang et al. (2017) designed asymmetric BLF-based adaptive controller to cope with the trajectory tracking for nonlinear pure-feedback system with time-varying state constraint, where the developed time-varying adaptive BLF relaxed the restriction on initial conditions. Different from BLF, constraint transformation method directly transforms the constrained system into an equivalent unconstrained system by using some nonlinear mapping function. Guo and Wu (2014) presented a nonlinear-mapping-based BC controller to transform the pure-feedback constrained system into the strict-feedback unconstrained system by mapping the constraint output into the real number set. Yan et al. (2023) developed an adaptive optimal controller for nonlinear strict-feedback system subjected to time-varying asymmetric state constraints, where nonlinear state-dependent function was formulated to transform the system into non-constrained system.

Motivated by the above discussion, this paper presents a NDO-ANC tracking controller for EHSS subjected to full-state constraints and model uncertainty. To convert the state constraint system into state unconstraint system, the state transformation function is designed by using first-order filter. Neural networks (NNs) are used to approximate the model uncertainty, NDO are constructed to compensate the external disturbance, and the second-order integral filter is constructed to estimate the virtual control law. Based on the Lyapunov stability analysis, it can be proven that closed-loop system is stable and the time-varying full-state constraints are never violated. Compared with the existing work, the main contributions are as follows:

The NDO is proposed for EHSS with model uncertainty and external disturbance, and the NDO estimate-based ANC is also established.

The constraint system is transformed into unconstraint system by using the nonlinear mapping function. For the transformed systems, a modified backstepping design is proposed to prevent the violation of the full-state constraint.

The second-order integral filter is designed to estimate the virtual control law, which can significantly reduce the calculation cost caused by manual work.

This paper is organized as follows. Section “Dynamic model and problem formulation” provides the dynamic model and problem formulation. The NDO and ANC design are discussed in section “Controller design.” Sections “Simulation results” and “Experimental studies” provide the simulation results and experimental studies, respectively. The conclusion is summarized in section “Conclusion.”

Dynamic model and problem formulation

In order to design the proposed controller, a mathematical model that represents the EHSS dynamics is needed. The EHSS considered in this work consists of hydraulic cylinder, servo valve, and load force. For simplicity, the load force is represented by a mass–spring–damper system. The control mechanism of EHSS is depicted in Figure 1, where P_s and P_r denote the supply pressure and return pressure, respectively.

Figure 1.

The control mechanism of EHSS.

The dynamics of the load force can be described by

m \overset{\cdot\cdot}{y} = P_{L} A - B \overset{\cdot}{y} - Ky + f (t)

(1)

where y and m are the displacement and the mass of the load, respectively. P_L = P₁-P₂ is the load pressure, where P₁ and P₂ are the pressures at each side of cylinder chamber. A is the efficient ram area. B is viscous coefficient. f(t) is the lumped unmodeled disturbances.

The cylinder dynamics is given by

\frac{V_{t}}{4 β_{e}} {\overset{\cdot}{P}}_{L} = - A \overset{\cdot}{y} - C_{t} P_{L} + Q (t) + Q_{L}

(2)

where V_t is the total control volume, β_e is the effective bulk modulus, C_t is the internal leakage coefficient, Q(t) is the unmodeled errors, and Q_L is the load flowrate.

The load flowrate can be written as

Q_{L} = C_{d} w x_{v} \sqrt{\frac{P_{s} - sgn (x_{v}) P_{L}}{ρ}}

(3)

where C_d is the discharge coefficient, w is the valve orifice area gradient, ρ is the hydraulic fluid density, and x_v is the spool position of the servo valve.

Considering the fact that the dynamic of servo valve is much faster than the remaining parts of EHSS, the relationship between spool position of the servo valve and input voltage is approximated as

x_{v} = k_{v} u

(4)

where k_v is the electrical constant and u is the control input voltage of motor.

Combining equations (1)–(4) and choosing state variable as $x_{1} = y$ , $x_{2} = \overset{\cdot}{y}$ , $x_{3} = A P_{L} / m$ , the state-space form of EHSS can be expressed as

\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = x_{3} - \frac{B}{m} x_{2} - \frac{K}{m} x_{1} + \frac{f (t)}{m} \\ {\overset{\cdot}{x}}_{3} = - \frac{4 β_{e} C_{t}}{V_{t}} x_{3} - \frac{4 A^{2} β_{e}}{m V_{t}} x_{2} + \frac{4 A β_{e}}{m V_{t}} Q (t) \\ + \frac{4 A β_{e} k_{v} C_{d} w x_{v}}{m V_{t}} \sqrt{\frac{P_{s} - sgn (u) m x_{3}}{ρ A}} \end{matrix}

(5)

In order to make the controller design more convenient, equation (5) can be rewritten as the following third-order nonlinear model

\begin{matrix} {\overset{\cdot}{x}}_{1} = f_{1} (x_{1}, x_{2}) + d_{1} (t) \\ {\overset{\cdot}{x}}_{2} = f_{2} ({\bar{x}}_{2}, x_{3}) + d_{2} (t) \\ {\overset{\cdot}{x}}_{3} = f_{3} ({\bar{x}}_{3}, u) + d_{3} (t) \end{matrix}

(6)

where $f_{1} (x_{1}, x_{2}) = x_{2}$ , $f_{2} ({\bar{x}}_{2}, x_{3}) = x_{3} - (B / m) x_{2} - (K / m) x_{1} + f (t) / m$ , $d_{2} (t) = f (t) / m$ , $f_{3} ({\bar{x}}_{3}, u) = - (4 β_{e} C_{t} / V_{t}) x_{3} - (4 A^{2} β_{e}) / (m V_{t}) x_{2} + 4 A β_{e} k_{v} C_{d} w x_{v} \sqrt{P_{s} - sgn (u) P_{L}} / (m V_{t} \sqrt{ρ A})$ , and $d_{3} (t) = 4 A β_{e} Q (t) / (m V_{t})$ .

Assumption 1: The state variables satisfy $k_{i}^{l} < x_{i} < k_{i}^{u}$ , where $k_{i}^{l}$ and $k_{i}^{u}$ are the lower and upper bound of $x_{i}$ , respectively.

Assumption 2: The d_i(t) and its derivative ${\overset{\cdot}{d}}_{i} (t)$ are bounded by $| d_{i} (t) | \leq d_{i max}$ and $| {\overset{\cdot}{d}}_{i} (t) | \leq {\overset{\cdot}{d}}_{i max}$ , where $d_{i max}$ and ${\overset{\cdot}{d}}_{i max}$ are unknown positive constants.

Assumption 3: The reference trajectory y_d is continuous and bounded.

Controller design

The adaptive controller is constructed using NDO, NN, and backstepping method to ensure that the output signal can track the reference signal as closely as possible in spite of the various model uncertainty. At the same time, the output signal is not violated.

In this part, the radial basis function (RBF) NNs are used to approximate any smooth nonlinear functions. Specially, the RBF NNs are expressed as follows

f (X_{d}) = W^{T} Φ (X_{d}) + ε (X_{d})

(7)

where X_d is the neural input vector, W is the NN weight matrix, Φ is the Gaussian radian function, and ε is the network reconstruction error.

According to equation (7), the NN-based approximated value for f(X_d) is constructed by

\hat{f} (X_{d}) = {\hat{W}}^{T} Φ (X_{d})

(8)

where $\hat{W}$ is the estimation of the weight matrix W.

To tackle the problem of state constraints, a novel nonlinear mapping function is designed to convert the constrained state into the equivalent unconstrained

x_{i} = T_{i} (s_{i}) = k_{ai} \arctan (s_{i}) + k_{bi}

(9)

where s_i is the converted state variable, T_i(s_i) is the nonlinear mapping function, $k_{ai} = (k_{i}^{u} - k_{i}^{l}) / π$ , and $k_{bi} = (k_{i}^{u} + k_{i}^{l}) / π$ . According to the strictly increasing attributes of arctangent function, the following results are obtained

lim_{s_{i} \to - \infty} T_{i} (s_{i}) = k_{i}^{l}, lim_{s_{i} \to \infty} T_{i} (s_{i}) = k_{i}^{u}

(10)

The inverse of T_i(s_i) is derived as follows

s_{i} = T_{i}^{- 1} (x_{i}) = \tan (x_{i} / k_{ai} - k_{bi} / k_{ai})

(11)

Then, the derivative of s_i can be expressed as

{\overset{\cdot}{s}}_{i} = [1 + \tan^{2} (x_{i} / k_{ai} - x_{bi} / k_{ai})] {\overset{\cdot}{x}}_{i} / k_{ai}

(12)

To convert the full-state constraints system into any stateless constraints system, the intermediate function is defined as $F_{i} (s_{i}) = (1 + s_{i}^{2}) / k_{ai}$ . Then, the augmented model of equation (6) is given by

\begin{matrix} {\overset{\cdot}{s}}_{1} = g_{1} ({\bar{s}}_{2}) + F_{1} (s_{1}) s_{2} + F_{1} (s_{1}) d_{1} (t) \\ {\overset{\cdot}{s}}_{2} = g_{2} ({\bar{s}}_{3}) + F_{2} (s_{2}) s_{3} + F_{2} (s_{2}) d_{2} (t) \\ {\overset{\cdot}{s}}_{3} = g_{3} ({\bar{s}}_{3}, u) + F_{3} (s_{3}) u + F_{3} (s_{3}) d_{3} (t) \\ \overset{\cdot}{u} = - a (u - v) \\ y_{s} = s_{1} \end{matrix}

(13)

where s_i (i = 1, 2, 3) and u are the state variable of the augmented system, $g_{1} ({\bar{s}}_{2}) = F_{1} (s_{1}) f_{1} ({\bar{x}}_{2}) - F_{1} (s_{1}) s_{2}$ , $g_{2} ({\bar{s}}_{3}) = F_{2} (s_{2}) f_{2} ({\bar{x}}_{2}) - F_{2} (s_{2}) s_{3}$ , $G_{3} (s_{3}, u) = f_{3} ({\bar{x}}_{3}, u)$ , v and y_s are the system input and output, respectively.

The system state errors z_i (i = 1, …, 4) are defined as

\begin{matrix} z_{1} = s_{1} - s_{d} \\ z_{2} = s_{2} - α_{1} \\ z_{3} = s_{3} - α_{2} \\ z_{4} = s_{4} - α_{3} \end{matrix}

(14)

where s_d is the transformed desired reference signal, α_i (i = 1,2,3) are the virtual control variables in the backstepping iteration.

Step 1: Defining the transformed tracking error as z₁ = y_s−y_d = T⁻¹(y) −T⁻¹(y_d), then the time derivative of z₁ is

{\overset{\cdot}{z}}_{1} = g_{1} ({\bar{s}}_{2}) + F_{1} (s_{1}) (z_{2} + α_{1}) + F_{1} (s_{1}) d_{1} (t) - {\overset{\cdot}{s}}_{d}

(15)

A Lyapunov function is defined as

V_{1} = \frac{1}{2} z_{1}^{2} + \frac{1}{2} {\tilde{θ}}_{1}^{2} + \frac{1}{2} {\tilde{d}}_{1}^{2}

(16)

where ${\tilde{θ}}_{1} = {\hat{θ}}_{1} - θ_{1}$ is the estimation error of neural weight, ${\hat{θ}}_{1}$ is the estimation of $θ_{1}$ , ${\tilde{d}}_{1} = {\hat{d}}_{1} (t) - d_{1} (t)$ is the estimate error of model uncertainty of $d_{1}$ with ${\hat{d}}_{1}$ being its estimate.

The time derivative of V₁ is

{\overset{\cdot}{V}}_{1} = z_{1} (g_{1} ({\bar{s}}_{2}) + F_{1} (s_{1}) (z_{2} + α_{1}) + F_{1} (s_{1}) d_{1} (t) - {\overset{\cdot}{s}}_{d}) + {\tilde{θ}}_{1} {\overset{\cdot}{\hat{θ}}}_{1} + {\tilde{d}}_{1} \overset{\cdot}{\tilde{d_{1}}}

(17)

According to equation (7), RBF NN is used to approximate the unknown function $g_{1} ({\bar{s}}_{2})$ , $g_{1} ({\bar{s}}_{2}) = W_{1}^{* T} Φ_{1} ({\bar{s}}_{2}) + ε ({\bar{s}}_{2})$ , where $W_{1}^{*}$ is the optimal weight. Applying Young’s inequality, we have

z_{1} g_{1} ({\bar{s}}_{2}) \leq \frac{1}{2 a_{1}^{2}} z_{1}^{2} θ_{1} Φ_{1}^{T} ({\bar{s}}_{2}) Φ_{1} ({\bar{s}}_{2}) + \frac{a_{1}^{2}}{2} + \frac{z_{1}^{2}}{2} + \frac{ε_{1}^{2}}{2}

(18)

where a₁ is a positive design constant.

Substituting equation (18) into equation (17), we have

\begin{matrix} {\overset{\cdot}{V}}_{1} = z_{1} (\frac{z_{1}^{2} θ_{1} Φ_{1}^{T} ({\bar{s}}_{2}) Φ_{1} ({\bar{s}}_{2})}{2 a_{1}^{2}} + F_{1} (s_{1}) (z_{2} + α_{1}) + F_{1} (s_{1}) d_{1} (t) - {\overset{\cdot}{s}}_{d}) \\ + \frac{a_{1}^{2}}{2} + \frac{z_{1}^{2}}{2} + \frac{ε_{1}^{2}}{2} + {\tilde{θ}}_{1} {\overset{\cdot}{\hat{θ}}}_{1} + {\tilde{d}}_{1} {\overset{\cdot}{\tilde{d}}}_{1} \end{matrix}

(19)

Thus, the virtual control law is designed as

α_{1} = \frac{1}{F_{1} (s_{1})} (- k_{1} z_{1} - \frac{z_{1} {\hat{θ}}_{1} Φ_{1}^{T} ({\bar{s}}_{2}) Φ_{1} ({\bar{s}}_{2})}{2 a_{1}^{2}} + {\overset{\cdot}{s}}_{d}) - {\hat{d}}_{1}

(20)

where k₁ is a positive design parameter.

To estimate the model uncertainty d₁(t), the NDO is designed as

\begin{matrix} {\overset{\cdot}{κ}}_{1} = - l_{1} (\frac{{\hat{θ}}_{1} Φ_{1}^{T} ({\bar{s}}_{2}) Φ_{1} ({\bar{s}}_{2})}{2 a_{1}^{2}} + F_{1} (s_{1}) s_{2}) - l_{1} F_{1} (s_{1}) (κ_{1} + l_{1} s_{1}) \\ {\hat{d}}_{1} = κ_{1} + l_{1} s_{1} \end{matrix}

(21)

where κ₁ is a positive design parameter.

And the adaptive control law is defined as

{\overset{\cdot}{\hat{θ}}}_{1} = - σ_{1} {\hat{θ}}_{1} + \frac{1}{2 a_{1}^{2}} z_{1}^{2} Φ_{1}^{T} ({\bar{s}}_{2}) Φ_{1} ({\bar{s}}_{2})

(22)

where σ₁ is a positive design parameter.

Note that $\overset{\cdot}{\tilde{d_{1}}} = {\hat{d}}_{1} - d_{1}$ and using Yong’s inequality, it yields

\overset{\cdot}{\tilde{d_{1}}} \leq - l_{1} F_{1} (s_{1}) {\tilde{d}}_{1} - \frac{l_{1}}{2 a_{1}^{2}} {\tilde{θ}}_{1} Φ_{1}^{T} ({\bar{s}}_{2}) Φ ({\bar{s}}_{2}) + \frac{l_{1}}{2} a_{1}^{2} + l_{1} ε_{1}^{*} - {\overset{\cdot}{d}}_{1}

(23)

Substituting equations (20), (22), and (23) into equation (19), one obtains

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq - k_{1} z_{1}^{2} + \frac{a_{1}^{2}}{2} + \frac{z_{1}^{2}}{2} + \frac{ε_{1}^{* 2}}{2} - F_{1} (s_{1}) {\tilde{d}}_{1} - σ_{1} {\tilde{θ}}_{1} {\hat{θ}}_{1} + F_{1} (s_{1}) z_{1} z_{2} \\ - l_{1} F_{1} (s_{1}) {\tilde{d}}_{1}^{2} - \frac{l_{1}}{2 a_{1}^{2}} {\tilde{θ}}_{1} {\tilde{d}}_{1} Φ_{1}^{T} ({\bar{s}}_{2}) Φ_{1} ({\bar{s}}_{2}) + ψ_{1} {\tilde{d}}_{1} \end{matrix}

(24)

where $ψ_{1} = (l_{1} / 2) a_{1}^{2} + l_{1} ε_{1}^{*} + d_{1 max}$ .

According to assumption 1 and Youn’s inequality, we have

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq - \frac{2 k_{1} - 1 - γ_{1} F_{1}^{2} (s_{1})}{2} z_{1}^{2} - (\frac{σ_{1}}{2} - \frac{l_{1} ϕ_{1}^{* 2}}{2 a_{1}^{2}}) {\tilde{θ}}_{1}^{2} \\ - (l_{1} F_{1} (s_{1}) - \frac{1}{γ_{1}} - \frac{l_{1} ϕ_{1}^{* 2}}{2 a_{1}^{2}} - \frac{β_{1}}{2}) {\tilde{d}}_{1}^{2} \\ + \frac{a_{1}^{2}}{2} + \frac{ε_{1}^{* 2}}{2} + \frac{σ_{1}}{2} θ_{1}^{2} + \frac{1}{2 β_{1}} ψ_{1}^{2} + F_{1} (s_{1}) z_{1} z_{2} \end{matrix}

(25)

where $β_{1}$ and $γ_{1}$ are positive constants.

Step 2: For $z_{2} = s_{2} - α_{1}$ , the time derivative of z₂ is calculated

{\overset{\cdot}{z}}_{2} = g_{2} ({\bar{s}}_{3}) + F_{2} (s_{2}) (z_{3} + α_{2}) + F_{2} (s_{2}) d_{2} (t) - {\overset{\cdot}{α}}_{1}

(26)

where

\begin{matrix} {\overset{\cdot}{α}}_{1} = \frac{\partial α_{1}}{\partial s_{1}} (g_{1} ({\bar{s}}_{2}) + F_{1} (s_{1}) s_{2} + F_{1} (s_{1}) d_{1} (t)) + \frac{\partial α_{1}}{\partial {\hat{θ}}_{1}} {\overset{\cdot}{\hat{θ}}}_{1} + \frac{\partial α_{1}}{\partial s_{d}^{1}} s_{d}^{2} \\ + \frac{\partial α_{1}}{\partial {\hat{d}}_{1}} {\overset{\cdot}{\hat{d}}}_{1} + \frac{\partial α_{1}}{\partial s_{2}} (g_{2} ({\bar{s}}_{3}) + F_{2} (s_{2}) s_{3} + F_{2} (s_{2}) d_{2} (t)) \end{matrix}

(27)

To reduce the computation burden of the virtual control law derivative, the second-order integral filter is adopted to estimate the ${\overset{\cdot}{α}}_{1}$

\begin{matrix} {\overset{\cdot}{\hat{ξ}}}_{1, 1} = - \frac{1}{ρ_{1, 1}} ({\hat{ξ}}_{1, 1} - α_{1}) \\ {\overset{\cdot}{\hat{ξ}}}_{1, 2} = - \frac{1}{ρ_{1, 2}} ({\hat{ξ}}_{1, 2} - {\overset{\cdot}{\hat{ξ}}}_{1, 1}) \end{matrix}

(28)

where ${\hat{ξ}}_{1, 1}$ and ${\hat{ξ}}_{1, 2}$ are the states of the filter, $ρ_{1, 1}$ and $ρ_{1, 2}$ are the positive design parameters.

Then, the virtual control law derivative ${\overset{\cdot}{α}}_{1}$ is estimated by

{\overset{\cdot}{α}}_{1} = {\hat{ξ}}_{1, 2} + η_{1}

(29)

where $η_{1}$ is the estimate error, and $| η_{1} | \leq η_{1}^{*}$ .

Substituting equation (29) into equation (26), it yields

{\overset{\cdot}{z}}_{2} = g_{2} ({\bar{s}}_{3}) + F_{2} (s_{2}) (z_{3} + α_{2}) + F_{2} (s_{2}) d_{2} (t) - {\hat{ξ}}_{1, 2} - η_{1}

(30)

Then, the Lyapunov function is defined as

V_{2} = V_{1} + \frac{1}{2} z_{2}^{2} + \frac{1}{2} {\tilde{θ}}_{2}^{2} + \frac{1}{2} {\tilde{d}}_{2}^{2}

(31)

The derivative of V₂ is computed as

\begin{matrix} {\overset{\cdot}{V}}_{2} = {\overset{\cdot}{V}}_{1} + z_{2} (g_{2} ({\bar{s}}_{3}) + F_{2} (s_{2}) (z_{3} + α_{2}) + F_{2} (s_{2}) d_{2} (t) - {\hat{ξ}}_{1, 2} - η_{1}) \\ + {\tilde{θ}}_{2} {\overset{\cdot}{\hat{θ}}}_{2} + {\tilde{d}}_{2} \overset{\cdot}{\tilde{d_{2}}} \end{matrix}

(32)

Based on $g_{2} ({\bar{s}}_{3}) = W_{2}^{* T} Φ_{2} ({\bar{s}}_{3}) + ε ({\bar{s}}_{3})$ and Young’s inequality, we have

\begin{array}{l} {\overset{\cdot}{V}}_{2} \leq {\overset{\cdot}{V}}_{1} + z_{2} (\frac{z_{2}^{2} θ_{2} Φ_{2}^{T} ({\bar{s}}_{3}) Φ_{2} ({\bar{s}}_{3})}{2 a_{2}^{2}} + F_{2} (s_{2}) (z_{3} + α_{2}) + F_{2} (s_{2}) d_{2} (t) - {\hat{ξ}}_{1, 2} - η_{1}) \\ + \frac{a_{2}^{2}}{2} + \frac{z_{2}^{2}}{2} + \frac{ε_{2}^{2}}{2} + {\tilde{θ}}_{2} {\overset{\cdot}{\hat{θ}}}_{2} + {\tilde{d}}_{2} {\overset{\cdot}{\tilde{d}}}_{2} \end{array}

(33)

where a₂ is a positive constant.

The virtual control law is designed as

α_{2} = \frac{1}{F_{2} (s_{2})} (- k_{2} z_{2} - \frac{z_{2} {\hat{θ}}_{2} Φ_{2}^{T} ({\bar{s}}_{3}) Φ_{2} ({\bar{s}}_{3})}{2 a_{2}^{2}} + {\hat{ξ}}_{1, 2} - F_{1} (s_{1}) z_{1}) - {\hat{d}}_{2}

(34)

where k₂ is a positive design parameter.

Then, the adaptive control law is given by

{\overset{\cdot}{\hat{θ}}}_{2} = - σ_{2} {\hat{θ}}_{2} + \frac{1}{2 a_{2}^{2}} z_{2}^{2} Φ_{2}^{T} ({\bar{s}}_{3}) Φ_{2} ({\bar{s}}_{3})

(35)

where σ₂ is a positive design parameter.

The following NDO is designed to estimate the ${\hat{d}}_{2}$

\begin{matrix} {\overset{\cdot}{κ}}_{2} = - l_{2} (\frac{{\hat{θ}}_{2} Φ_{2}^{T} ({\bar{s}}_{3}) Φ_{2} ({\bar{s}}_{3})}{2 a_{2}^{2}} + F_{2} (s_{2}) s_{3}) - l_{2} F_{2} (s_{2}) (κ_{2} + l_{2} s_{2}) \\ {\hat{d}}_{2} = κ_{2} + l_{2} s_{2} \end{matrix}

(36)

Noting ${\tilde{d}}_{2} = {\hat{d}}_{2} - d_{2}$ and using Young’s inequality, it yields

\overset{\cdot}{\tilde{d_{2}}} \leq - l_{2} F_{2} (s_{2}) {\tilde{d}}_{2} - \frac{l_{2}}{2 a_{2}^{2}} {\tilde{θ}}_{2} Φ_{2}^{T} ({\bar{s}}_{3}) Φ_{2} ({\bar{s}}_{3}) + \frac{l_{2}}{2} a_{2}^{2} + l_{2} ε_{2}^{*} - {\overset{\cdot}{d}}_{2}

(37)

Substituting equations (34), (35), and (37) into equation (33), it yields

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq - \frac{2 k_{1} - 1 - γ_{1} F_{1}^{2} (s_{1})}{2} z_{1}^{2} - \frac{2 k_{2} - 1 - γ_{2} F_{2}^{2} (s_{2})}{2} z_{2}^{2} + F_{1} (s_{1}) z_{1} z_{2} \\ + F_{2} (s_{2}) z_{2} z_{3} - (l_{1} F_{1} (s_{1}) - \frac{1}{2 γ_{1}} - \frac{l_{1} ϕ_{1}^{* 2}}{2 a_{1}^{2}} - \frac{β_{1}}{2}) {\tilde{d}}_{1}^{2} \\ - (l_{2} F_{2} (s_{2}) - \frac{1}{2 γ_{2}} - \frac{l_{2} ϕ_{2}^{* 2}}{2 a_{2}^{2}} - \frac{β_{2}}{2}) {\tilde{d}}_{2}^{2} - (\frac{σ_{1}}{2} - \frac{l_{1} ϕ_{1}^{* 2}}{2 a_{1}^{2}}) {\tilde{θ}}_{1}^{2} \\ - (\frac{σ_{2}}{2} - \frac{l_{2} ϕ_{2}^{* 2}}{2 a_{2}^{2}}) {\tilde{θ}}_{2}^{2} + \frac{a_{1}^{2}}{2} + \frac{a_{2}^{2}}{2} + \frac{ε_{1}^{* 2}}{2} + \frac{ε_{2}^{* 2}}{2} + \frac{σ_{1}}{2} θ_{1}^{2} \\ + \frac{σ_{2}}{2} θ_{2}^{2} + \frac{η_{1}^{* 2}}{2} + \frac{1}{2 β_{1}} ψ_{1}^{2} + \frac{1}{2 β_{2}} ψ_{2}^{2} \end{matrix}

(38)

where $ψ_{1} = (l_{1} / 2) a_{1}^{2} + l_{1} ε_{1}^{*} + d_{1 max}$ , $ψ_{2} = (l_{2} / 2) a_{2}^{2} + l_{2} ε_{2}^{*} + d_{2 max}$ , $β_{2}$ and $γ_{2}$ are positive constants.

Step 3: For $z_{3} = s_{3} - α_{2}$ , the time derivative of z₃ is calculated as

{\overset{\cdot}{z}}_{3} = g_{3} ({\bar{s}}_{4}) + F_{3} (s_{3}) (z_{4} + α_{3}) + F_{3} (s_{3}) d_{3} (t) - {\overset{\cdot}{α}}_{2}

(39)

where

\begin{matrix} {\overset{\cdot}{α}}_{2} = \frac{\partial α_{2}}{\partial s_{1}} (g_{1} ({\bar{s}}_{2}) + F_{1} (s_{1}) s_{2} + F_{1} (s_{1}) d_{1} (t)) \\ + \frac{\partial α_{2}}{\partial s_{2}} (g_{2} ({\bar{s}}_{3}) + F_{2} (s_{2}) s_{3} + F_{2} (s_{2}) d_{2} (t)) \\ + \frac{\partial α_{2}}{\partial s_{3}} (g_{3} ({\bar{s}}_{4}) + F_{3} (s_{3}) s_{4} + F_{3} (s_{3}) d_{3} (t)) \\ + \frac{\partial α_{2}}{\partial {\hat{θ}}_{1}} {\overset{\cdot}{\hat{θ}}}_{1} + \frac{\partial α_{2}}{\partial {\hat{θ}}_{2}} {\overset{\cdot}{\hat{θ}}}_{2} + \frac{\partial α_{2}}{\partial s_{d}^{1}} s_{d}^{2} \\ + \frac{\partial α_{2}}{\partial s_{d}^{2}} s_{d}^{3} + \frac{\partial α_{2}}{\partial {\hat{d}}_{1}} {\overset{\cdot}{\hat{d}}}_{1} + \frac{\partial α_{2}}{\partial {\hat{d}}_{2}} {\overset{\cdot}{\hat{d}}}_{2} \end{matrix}

(40)

Similarity, the following second-order integral filter is used to estimate the ${\overset{\cdot}{α}}_{2}$

\begin{matrix} {\overset{\cdot}{\hat{ξ}}}_{2, 1} = - \frac{1}{ρ_{2, 1}} ({\hat{ξ}}_{2, 1} - α_{2}) \\ {\overset{\cdot}{\hat{ξ}}}_{2, 2} = - \frac{1}{ρ_{2, 2}} ({\hat{ξ}}_{2, 2} - {\overset{\cdot}{\hat{ξ}}}_{2, 1}) \end{matrix}

(41)

where ${\hat{ξ}}_{2, 1}$ and ${\hat{ξ}}_{2, 2}$ are the states of the filter, $ρ_{2, 1}$ and $ρ_{2, 2}$ are the positive design parameters.

The virtual control law derivative ${\overset{\cdot}{α}}_{2}$ is estimated by

{\overset{\cdot}{α}}_{2} = {\hat{ξ}}_{2, 2} + η_{2}

(42)

where $η_{2}$ is the estimate error, and $| η_{2} | \leq η_{2}^{*}$ .

Substituting equation (42) into equation (39) yields

{\overset{\cdot}{z}}_{3} = g_{3} ({\bar{s}}_{4}) + F_{3} (s_{3}) (z_{4} + α_{3}) + F_{3} (s_{3}) d_{3} (t) - {\hat{ξ}}_{2, 2} - η_{2}

(43)

Then, the Lyapunov function is defined as

V_{3} = V_{2} + \frac{1}{2} z_{3}^{2} + \frac{1}{2} {\tilde{θ}}_{3}^{2} + \frac{1}{2} {\tilde{d}}_{3}^{2}

(44)

The derivative of V₃ is computed as

\begin{matrix} {\overset{\cdot}{V}}_{3} = {\overset{\cdot}{V}}_{2} + z_{3} (g_{3} ({\bar{s}}_{4}) + F_{3} (s_{3}) (z_{4} + α_{3}) + F_{3} (s_{3}) d_{3} (t) - {\hat{ξ}}_{2, 2} - η_{2}) \\ + {\tilde{θ}}_{3} {\overset{\cdot}{\hat{θ}}}_{3} + {\tilde{d}}_{3} \overset{\cdot}{\tilde{d_{3}}} \end{matrix}

(45)

Based on $g_{3} ({\bar{s}}_{4}) = W_{3}^{* T} Φ_{3} ({\bar{s}}_{4}) + ε ({\bar{s}}_{4})$ and Young’s inequality, we have

\begin{array}{l} {\overset{\cdot}{V}}_{3} \leq {\overset{\cdot}{V}}_{2} + z_{3} (\frac{z_{3}^{2} θ_{3} Φ_{3}^{T} ({\bar{s}}_{4}) Φ_{3} ({\bar{s}}_{4})}{2 a_{3}^{2}} + F_{3} (s_{3}) (z_{4} + α_{3}) + F_{3} (s_{3}) d_{3} (t) - {\hat{ξ}}_{2, 2} - η_{2}) \\ + \frac{a_{3}^{2}}{2} + \frac{z_{3}^{2}}{2} + \frac{ε_{3}^{2}}{2} + {\tilde{θ}}_{3} {\overset{\cdot}{\hat{θ}}}_{3} + {\tilde{d}}_{3} {\overset{\cdot}{\tilde{d}}}_{3} \end{array}

(46)

where a₃ is a positive constant.

The virtual control law is designed as

α_{3} = \frac{1}{F_{3} (s_{3})} (- k_{3} z_{3} - \frac{z_{3} {\hat{θ}}_{3} Φ_{3}^{T} ({\bar{s}}_{4}) Φ_{3} ({\bar{s}}_{4})}{2 a_{3}^{2}} + {\hat{ξ}}_{2, 2} - F_{2} (s_{2}) z_{2}) - {\hat{d}}_{3}

(47)

where k₃ is a positive design parameter.

And the adaptive control law is given by

{\overset{\cdot}{\hat{θ}}}_{3} = - σ_{3} {\hat{θ}}_{3} + \frac{1}{2 a_{3}^{2}} z_{3}^{2} Φ_{3}^{T} ({\bar{s}}_{4}) Φ_{3} ({\bar{s}}_{4})

(48)

where σ₃ is a positive design parameter.

The following NDO is designed to estimate the ${\hat{d}}_{3}$

\begin{matrix} {\overset{\cdot}{κ}}_{3} = - l_{3} (\frac{{\hat{θ}}_{3} Φ_{3}^{T} ({\bar{s}}_{4}) Φ_{3} ({\bar{s}}_{4})}{2 a_{3}^{2}} + F_{3} (s_{3}) s_{4}) - l_{3} F_{3} (s_{3}) (κ_{3} + l_{3} s_{3}) \\ {\hat{d}}_{3} = κ_{3} + l_{3} s_{3} \end{matrix}

(49)

Note that ${\tilde{d}}_{3} = {\hat{d}}_{3} - d_{3}$ and using Young’s inequality, it yields

\overset{\cdot}{\tilde{d_{3}}} \leq - l_{3} F_{3} (s_{3}) {\tilde{d}}_{3} - \frac{l_{3}}{2 a_{3}^{2}} {\tilde{θ}}_{3} Φ_{3}^{T} ({\bar{s}}_{4}) Φ_{3} ({\bar{s}}_{4}) + \frac{l_{3}}{2} a_{3}^{2} + l_{3} ε_{3}^{*} - {\overset{\cdot}{d}}_{3}

(50)

Substituting equations (47), (48), and (50) into equation (46) yields

\begin{matrix} {\overset{\cdot}{V}}_{3} \leq - \frac{2 k_{1} - 1 - γ_{1} F_{1}^{2} (s_{1})}{2} z_{1}^{2} - \frac{2 k_{2} - 1 - γ_{2} F_{2}^{2} (s_{2})}{2} z_{2}^{2} \\ - \frac{2 k_{3} - 1 - γ_{3} F_{3}^{2} (s_{3})}{2} z_{3}^{2} - (l_{1} F_{1} (s_{1}) - \frac{1}{2 γ_{1}} - \frac{l_{1} ϕ_{1}^{* 2}}{2 a_{1}^{2}} - \frac{β_{1}}{2}) {\tilde{d}}_{1}^{2} \\ - (l_{2} F_{2} (s_{2}) - \frac{1}{2 γ_{2}} - \frac{l_{2} ϕ_{2}^{* 2}}{2 a_{2}^{2}} - \frac{β_{2}}{2}) {\tilde{d}}_{2}^{2} - (l_{3} F_{3} (s_{3}) - \frac{1}{2 γ_{3}} - \frac{l_{3} ϕ_{3}^{* 2}}{2 a_{3}^{2}} - \frac{β_{3}}{2}) {\tilde{d}}_{3}^{2} \\ - (\frac{σ_{1}}{2} - \frac{l_{1} ϕ_{1}^{* 2}}{2 a_{1}^{2}}) {\tilde{θ}}_{1}^{2} - (\frac{σ_{2}}{2} - \frac{l_{2} ϕ_{2}^{* 2}}{2 a_{2}^{2}}) {\tilde{θ}}_{2}^{2} - (\frac{σ_{3}}{2} - \frac{l_{3} ϕ_{3}^{* 2}}{2 a_{3}^{2}}) {\tilde{θ}}_{3}^{2} \\ + F_{1} (s_{1}) z_{1} z_{2} + F_{2} (s_{2}) z_{2} z_{3} + F_{3} (s_{3}) z_{3} z_{4} + \frac{a_{1}^{2}}{2} + \frac{a_{2}^{2}}{2} + \frac{a_{3}^{2}}{2} \\ + \frac{ε_{1}^{* 2}}{2} + \frac{ε_{2}^{* 2}}{2} + \frac{ε_{3}^{* 2}}{2} + \frac{σ_{1}}{2} θ_{1}^{2} + \frac{σ_{2}}{2} θ_{2}^{2} + \frac{σ_{3}}{2} θ_{3}^{2} + \frac{η_{1}^{* 2}}{2} \\ + \frac{η_{2}^{* 2}}{2} + \frac{1}{2 β_{1}} ψ_{1}^{2} + \frac{1}{2 β_{2}} ψ_{2}^{2} + \frac{1}{2 β_{3}} ψ_{3}^{2} \end{matrix}

(51)

where $ψ_{1} = (l_{1} / 2) a_{1}^{2} + l_{1} ε_{1}^{*} + d_{1 max}$ , $β_{3}$ and $γ_{3}$ are positive constants, $ψ_{2} = (l_{2} / 2) a_{2}^{2} + l_{2} ε_{2}^{*} + d_{2 max}$ , $ψ_{3} = (l_{3} / 2) a_{3}^{2} + l_{3} ε_{3}^{*} + d_{3 max}$ .

Step 4: Defining $z_{4} = u - α_{3}$ and combining equation (13), its derivative is computed as

{\overset{\cdot}{z}}_{4} = - au + av - {\overset{\cdot}{α}}_{3}

(52)

To estimate the ${\overset{\cdot}{α}}_{3}$ , the following second-order filter is designed as

\begin{matrix} {\overset{\cdot}{\hat{ξ}}}_{3, 1} = - \frac{1}{ρ_{3, 1}} ({\hat{ξ}}_{3, 1} - α_{3}) \\ {\overset{\cdot}{\hat{ξ}}}_{3, 2} = - \frac{1}{ρ_{3, 2}} ({\hat{ξ}}_{3, 2} - {\overset{\cdot}{\hat{ξ}}}_{3, 1}) \end{matrix}

(53)

where ${\hat{ξ}}_{3, 1}$ and ${\hat{ξ}}_{3, 2}$ are the states of the filter, and $ρ_{3, 1}$ and $ρ_{3, 2}$ are the positive design parameters.

The virtual control law derivative ${\overset{\cdot}{α}}_{3}$ is estimated by

{\overset{\cdot}{α}}_{3} = {\hat{ξ}}_{3, 2} + η_{3}

(54)

where $η_{3}$ is the estimate error, and $| η_{3} | \leq η_{3}^{*}$ .

Substituting equation (54) into equation (52), it yields

{\overset{\cdot}{z}}_{4} = - au + av - {\hat{ξ}}_{3, 2} - η_{3}

(55)

Defining the Lyapunov function as

V_{4} = V_{3} + \frac{1}{2} z_{4}^{2}

(56)

The time derivative of V₄ is presented as

{\overset{\cdot}{V}}_{4} = {\overset{\cdot}{V}}_{3} + z_{4} (- au + av - {\hat{ξ}}_{3, 2} - η_{3})

(57)

The controller v is designed as

v = \frac{1}{a} (au + {\hat{ξ}}_{3, 2} - k_{4} z_{4} - F_{3} (s_{3}) z_{3})

(58)

where k₄ is a positive parameter.

Substituting equation (58) into equation (57) and combining equation (51), it yields

\begin{matrix} {\overset{\cdot}{V}}_{4} \leq - \sum_{i = 1}^{3} \frac{2 k_{i} - 1 - γ_{1} F_{i}^{2} (s_{i})}{2} z_{i} - \sum_{i = 1}^{3} (l_{i} F (s_{i}) - \frac{1}{γ_{i}} - \frac{l_{i} ψ_{i}^{* 2}}{2 a_{i}^{2}} - \frac{β_{i}}{2}) {\tilde{d}}_{i}^{2} \\ - \sum_{i = 1}^{3} (\frac{σ_{i}}{2} - \frac{l_{i} ψ_{i}^{* 2}}{2 a_{i}^{2}}) {\tilde{θ}}_{i}^{2} + \sum_{i = 1}^{3} [\frac{a_{i}^{2}}{2} + \frac{ε_{i}^{* 2}}{2} + \frac{η_{i}^{* 2}}{2} + \frac{σ_{i}^{2}}{2} θ_{i}^{2} + \frac{ψ_{i}^{2}}{2 β_{i}}] \\ - k_{4} z_{4}^{2} \end{matrix}

(59)

{\overset{\cdot}{V}}_{4} \leq - λ V_{4} + δ

(60)

where $δ = \sum_{i = 1}^{3} (a_{i}^{2} + ε_{i}^{* 2} + η_{i}^{* 2} + σ_{i} θ_{i}^{2} + ψ_{i}^{2} / β_{i}) / 2$ , $λ = min {k_{i}^{*}, σ_{i}^{*}, l_{i}^{*}, k_{4}}$ .

Thus, it can be claimed that

V_{4} \leq (V_{4} (0) - χ) \exp (- ρ t) + χ

(61)

where $χ = δ / λ$ .

It can be derived that the state variable remains in the constrained bounds. Noting that $z \leq \sqrt{2 χ}$ as $t \to 0$ , which implies that the closed-loop system has asymptotic stability in the sense that z₁ can converge to a small neighborhood of zero by increase of $k_{i}, a, σ_{i}$ and decrease of $ρ_{i 1}, ρ_{i 2}, a$ . More importantly, the proposed method removes the restrictions on control coefficients with known signs and bounds, and extends the feasible scope to a large extent.

Simulation results

To evaluate the position tracking performance of the designed controller, the following five controllers have been compared using MATLAB/Simulink under the same working conditions. The main parameters of the EHSS are listed in Table 1. Specially, the model uncertainties are at most 10% of the nominal values. In particular, parameters of these five controllers are well tuned for the sake of best tracking performance.

NDO-ANC: The following control gains are utilized: $k_{1}^{l} = - 80$ , $k_{1}^{u} = 70$ , $k_{2}^{l} = - 85$ , $k_{2}^{u} = 80$ , $k_{3}^{l} = - 80$ , $k_{3}^{u} = 80$ . The initial parameters are set as ${\hat{θ}}_{1} = 0$ , ${\hat{θ}}_{2} = 0$ , ${\hat{θ}}_{3} = 0$ , $x_{1} = 0$ , $x_{2} = 0$ , $x_{3} = 0$ , ${\hat{ξ}}_{11} = - 70$ , ${\hat{ξ}}_{12} (0) = 0$ , ${\hat{ξ}}_{21} (0) = - 80$ , ${\hat{ξ}}_{22} (0) = 0$ , ${\hat{ξ}}_{31} (0) = - 75$ , ${\hat{ξ}}_{32} (0) = 0$ , k₁ = 300, k₂ = 350, k₃ = 2000, k₄ = 3000, a = 1200, a₁ = 60, a₂ = 80, a₃ = 100, σ₁ = 70, σ₂ = 80, σ₃ = 85, l₁ = l₂ = 500, ρ₁₁ = ρ₂₁ = ρ₃₁ = 0.03, ρ₁₂ = ρ₂₂ = ρ₃₂ = 0.01.

BC: The control law of SMC is designed as: $u_{b} = (- k_{b 2} z_{b 2} - f_{b 2} - g_{b 1} z_{b 1} + ϑ_{b 1}) / g_{b 2}$ , where $ϑ_{b 1} = (- k_{b 1} z_{b 1} - f_{b 1} + {\overset{\cdot}{x}}_{1 r}) / g_{b 1}$ is virtual control variable, $x_{b 1} = \overset{\cdot}{y}$ and $x_{b 2} = A P_{L}$ are defined state variables, $z_{b 1} = x_{b 1} - x_{1 r}$ and $z_{b 2} = x_{b 2} - ϑ_{b 1}$ are system state errors, $f_{b 2} = - 4 β_{e} A^{2} x_{1} / V_{t} - 4 β_{e} C_{t} A x_{2} / V_{t}$ , $f_{b 1} = B x_{b 1} / m$ , $g_{b 1} = 1 / m$ , $g_{b 2} = 4 β_{e} C_{d} w k_{v} x_{v} A \sqrt{P_{s} - sign (u) x_{2} / A} / (V_{t} \sqrt{ρ})$ , $k_{b 1} = 700$ and $k_{b 2} = 500$ are control gains.

SMC: The SMC control law is set as: $u_{s} = - (g_{s 2} z_{s 2} + f_{s 2} + Δ_{s 2} - {\overset{\cdot}{α}}_{s 2} + k_{s 3} z_{s 3}) / g_{s 3}$ , where $α_{s 2} = - (k_{s 2} z_{s 2} + z_{s 1} + f_{s 2} - g_{s 2} s_{2} + Δ_{s 1} - {\overset{\cdot}{α}}_{s 1})$ , $α_{s 1} = - k_{s 1} z_{s 1} + s_{1} + {\overset{\cdot}{y}}_{d}$ , $x_{s 1} = y$ , $x_{s 2} = \overset{\cdot}{y}$ , $x_{s 3} = A P_{L}$ , $f_{s 2} = - (K x_{1} + B x_{2}) / m$ , $f_{s 2} = - 4 β_{e} A^{2} x_{2} / V_{t} - 4 β_{e} C_{t} x_{3} / V_{t}$ , $g_{s 2} = 1 / m$ , $g_{s 3} = 4 β_{e} C_{d} k_{v} w x_{v} A \sqrt{P_{s} - sign (u) x_{3} / A} / (V_{t} \sqrt{ρ})$ , $Δ_{s 1}$ and $Δ_{s 2}$ are uncertain items, $k_{s 1} = 3000$ , $k_{s 2} = 2000$ and $k_{s 3} = 100$ are control gains.

ANC: The ANC is the same as NDO-ANC but without NDO. Hence, it can be used to address the anti-interference ability of the model uncertainty in ANC. To ensure the fairness of the comparison results, the control parameters are chosen as those of the NDO-ANC controller.

PID: The controller gains are selected k_p =200, k_i = 32, k_d = 0.02, which denotes the P-gain, I-gain, and D-gain, respectively.

Table 1.

The main parameters of EHSS.

Parameters	value	Parameters	Value
m	200 kg	P_s	1.2×10⁸ Pa
A	3.14×10^-4 m²	C_t	4×10^-3
B	1.2×10³ N·s/m	C_d	0.6
K	6×10³ N/m	w	5×10^-3
V_t	8×10^-5 m³	ρ	900 kg/m³
β _e	7×10⁸ Pa	k_v	2×10^-5

The tracking result of the aforementioned controllers for the desired signal y =60arctan(sin(20πt))[1-e^-8t] mm is shown in Figure 2. Obviously, the proposed NDO-ANC achieves better control precision in comparison to the other four controllers. This is due to the developed adaptive NDO technology based on the unknown dynamic compensation of ANC, which greatly reduces the influence of external disturbance and model uncertainty on tracking performance. Comparative results of the control inputs of the five controllers are plotted in Figure 2(c). It can be found that the NDO-ANC requires less control voltage and provides smoother control signal compared with the other four controllers. This advantage is preferable in controller design, which means that the proposed control scheme generates smaller impact for device and requires less energy. The estimation results of the second-order integral filter are shown in Figure 2(d). It is clear that the estimated filter output tracks the virtual control law differential well.

Figure 2.

Compared tracking result of y = 60arctan(sin(20πt))[1-e^-8t]: (a) tracking performance, (b) tracking error, (c) compared control input, and (d) estimation of the second-order integral filter.

In order to evaluate the performance of each controller quantitatively, the maximum error M_e, the average error μ_e, and the standard deviation error σ_e are defined as performance index.

\begin{matrix} M_{e} = max_{i = 1, 2, \dots N} | e_{1} (i) | \\ μ_{e} = \frac{1}{N} \sum_{i = 1}^{N} | e_{1} (i) | \\ σ_{e} = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} {| e_{1} (i) - μ |}^{2}} \end{matrix}

(62)

The performance indexes of different controllers under same condition are calculated and listed in Table 2. It can be seen that the proposed controller can deliver the smallest tracking error. The average error of the NDO-ANC, BC, SMC, and ANC is reduced by 53%, 40%, 30%, and 15%, respectively. Thus, the superiority of the proposed controller compared with other controllers is verified, which demonstrates that the designed controller is capable of obtaining higher precision.

Table 2.

Performance indexes of different controllers.

Indexes	M _e	μ _e	σ _e
PID	2.5356	0.7605	0.6066
AC	1.7548	0.6440	0.5699
SMC	1.6404	0.5354	0.4510
BC	1.0321	0.4597	0.3508
NDO-ANC	0.9049	0.3574	0.2341

Experimental studies

The diagram of experimental setup is shown in Figure 3. The host computer was used as the user interface, which enabled the user to program the designed controller and adjust the controller parameter. The target computer executed the controller procedure in real time which was communicated with the host computer by TCP/IP. The generated control signal was converted to analog signal by D/A and processed by signal conditioner. The position sensor and pressure sensor were attached to hydraulic servo actuator to measure the feedback displacement and load pressure of actuator, respectively.

Figure 3.

The diagram of experimental setup.

The five controllers were first tested for sinusoidal signal y =60sin(2πt)mm. The tracking result and the tracking error of five controllers are shown in Figure 4. As shown in Figure 4(a), these controllers can make the actuator of the hydraulic cylinder follow the desired signal well. The maximum tracking error of the piston rod is not more than 20 mm as described in Figure 4(b), which shows that the abovementioned controllers can achieve good dynamic response performance. Specially, the tracking errors range of PID, ANC, SMC, and BC are 19 mm, 15 mm, 11 mm, and 8 mm, respectively. The tracking error of NDO-ANC is relatively very small, mostly within 6 mm, which demonstrates the superiority of the proposed control strategy in dealing with the model uncertainty and external disturbance encountered by the EHSS.

Figure 4.

Compared tracking result of sinusoidal signal: (a) tracking performance and (b) tracking error.

To validate the effectiveness of the proposed NDO-ANC method, the response results of multifrequency sinusoidal signal y =50sin(2πt) + 30sin(4πt)mm are shown in Figure 5. The experimental results demonstrate that the proposed NDO-ANC controller achieves the best tracking performances since its adaptive law and robustness for external disturbance are based on NDO. Although ANC also adopts the adaptive mechanism, its tracking performance is worse than that of NDO-ANC. SMC and BC just have some robustness against the model uncertainty, but their tracking errors are larger than that of the proposed controller.

Figure 5.

Compared tracking result of multifrequency sinusoidal signal: (a) tracking performance and (b) tracking error.

To further investigate the effectiveness of the proposed NDO-ANC, the quasi-sinusoidal signal y = 25-25cos(4πt)[1-e^-t]mm is adopted. The tracking trajectories and tracking errors of the compared five controllers are shown in Figure 6. It is shown that the proposed NDO-ANC can achieve satisfactory control precision. This is mainly because the designed controller contains the NDO and adaptive law, thus improves transient response and steady tracking performance. The tracking error of BC is smaller than ANC and SMC. In these five controllers, PID produces the largest tracking error.

Figure 6.

Compared tracking result of quasi-sinusoidal signal: (a) tracking performance and (b) tracking error.

Comparative performance indexes for the five control methods under the three desired signals are shown in Figure 7. In Figure 7, it can be clearly found that the proposed NDO-ANC yields the smallest index values among the three given performance indices, which means that the NDO-ANC provides superior performances over the other controllers in terms of dynamic tracking response performance. This implies that the NDO-ANC has better capability to withstand model uncertainty.

Figure 7.

Performance indexes of different controllers.

Conclusion

In this paper, the NDO-ANC control scheme has been proposed for the motion control of EHSS with model uncertainty and full-state constraints. The dynamic model is first derived with consideration of various model uncertainty and external disturbance. Then, NN is employed to approximate the model dynamics of EHSS for the sake of disturbance compensation in the NDO design. The core of the proposed controller is to construct the state transformation function and the second-order integral filter which are integrated in the adaptive control design. Hence, the synthesized NDO-ANC can effectively reduce the limitation of unknown control coefficient and ensure that all states of the system are in the predetermined range. Simulation and experimental comparative results have been conducted to demonstrate the effectiveness of the proposed control scheme.

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 by the Science and Technology Key Project Foundation of Henan Provincial Education Department (Grant No. 23A460014) and the High Level Talent Foundation of Henan University of Technology (Grant No. 2020BS043).

ORCID iD

Zhenshuai Wan

Data availability statement

The authors confirm that the data supporting the findings of this study are available within the article.

References

Bai

Yao

(2022) Adaptive neural network output feedback robust control of electromechanical servo system with backlash compensation and disturbance rejection. Mechatronics 84: 102794.

Chen

Shao

S-Y

Jiang

(2017) Adaptive neural control of uncertain nonlinear systems using disturbance observer. IEEE Transactions on Cybernetics 47(10): 3110–3123.

Cheng

Liu

(2020) Sliding mode observer-based fractional-order proportional-integral-derivative sliding mode control for electro-hydraulic servo systems. Proceedings of the Institution of Mechanical Engineers Part C-Journal of Mechanical Engineering Science 234(10): 1887–1898.

Deldar

Izadian

Vaezi

, et al. (2015) Modeling of a hydraulic wind power transfer utilizing a proportional valve. IEEE Transactions on Industry Applications 51(2): 1837–1844.

Deng

Zhou

, et al. (2023) Neural network-based adaptive asymptotic prescribed performance tracking control of hydraulic manipulators. IEEE Transactions on Systems Man Cybernetics-Systems 53(1): 285–295.

Guo

Chen

Shi

, et al. (2022) Synchronous control for multiple electrohydraulic actuators with feedback linearization. Mechanical Systems and Signal Processing 178: 109280.

Guo

Zhang

Celler

, et al. (2016) Backstepping control of electro-hydraulic system based on extended-state-observer with plant dynamics largely unknown. IEEE Transactions on Industrial Electronics 63(11): 6909–6920.

Guo

(2014) Backstepping control for output-constrained nonlinear systems based on nonlinear mapping. Neural Computing & Applications 25(7–8): 1665–1674.

Hamann

Hug

Rosinski

(2017) Real-time optimization of the Mid-Columbia Hydropower System. IEEE Transactions on Power Systems 32(1): 157–165.

10.

Lee

Eskilsson

(2020) Energetically passive multi-degree-of-freedom hydraulic human power amplifier with assistive dynamics. IEEE Transactions on Control Systems Technology 28(4): 1296–1308.

11.

Ban

Gong

, et al. (2020a) Extended state observer-based finite-time dynamic surface control for trajectory tracking of a quadrotor unmanned aerial vehicle. Transactions of the Institute of Measurement and Control 42(15): 1–13.

12.

Guo

Yan

, et al. (2023) Terminal sliding mode observer based-asymptotic tracking control of electro-hydraulic systems with lumped uncertainties. Transactions of the Institute of Measurement and Control 45(1): 17–26.

13.

Feng

, et al. (2020b) Neural network-based adaptive control for pure-feedback stochastic nonlinear systems with time-varying delays and dead-zone input. IEEE Transactions on Systems Man Cybernetics-Systems 50(12): 5317–5329.

14.

Liu

Y-J

Tong

(2019) Neural networks-based adaptive finite-time fault-tolerant control for a class of strict-feedback switched nonlinear systems. IEEE Transactions on Cybernetics 49(7): 2536–2545.

15.

Liu

Zhao

Liu

Y-J

, et al. (2021a) Adaptive finite-time neural network control of nonlinear systems with multiple objective constraints and application to electromechanical system. IEEE Transactions on Neural Networks and Learning Systems 32(12): 5416–5426.

16.

Liu

Lim

C-C

Shi

, et al. (2017) Backstepping fuzzy adaptive control for a class of quantized nonlinear systems. IEEE Transactions on Fuzzy Systems 25(5): 1090–1101.

17.

Liu

Zhang

Rogers

, et al. (2021b) Terminal sliding mode-based tracking control with error transformation for underwater vehicles. International Journal of Robust and Nonlinear Control 31(15): 7186–7206.

18.

Livingstone

Cavanaugh

(2015) The real reasons why hydraulic fluids fail and strategies to stop problems before they start. Tribology & Lubrication Technology 71(7): 44–51.

19.

Maghareh

Silva

Dyke

(2018) Servo-hydraulic actuator in controllable canonical form: Identification and experimental validation. Mechanical Systems and Signal Processing 100: 398–414.

20.

Mohammadi

Tavakoli

Marquez

, et al. (2013) Nonlinear disturbance observer design for robotic manipulators. Control Engineering Practice 21(3): 253–267.

21.

Palli

Strano

Terzo

(2018) Sliding-mode observers for state and disturbance estimation in electro-hydraulic systems. Control Engineering Practice 74: 58–70.

22.

Qiu

Wang

Sun

, et al. (2022) Disturbance observer-based adaptive fuzzy control for strict-feedback nonlinear systems with finite-time prescribed performance. IEEE Transactions on Fuzzy Systems 30(4): 1175–1184.

23.

Shtessel

Taleb

Plestan

(2012) A novel adaptive-gain supertwisting sliding mode controller: Methodology and application. Automatica 48(5): 759–769.

24.

Vaezi

Izadian

(2015) Piecewise affine system identification of a hydraulic wind power transfer system. IEEE Transactions on Control Systems Technology 23(6): 2077–2086.

25.

Van

Mavrovouniotis

(2020) Self-tuning fuzzy PID-nonsingular fast terminal sliding mode control for robust fault tolerant control of robot manipulators. ISA Transactions 96: 60–68.

26.

Wang

(2017) Barrier Lyapunov functions-based adaptive control for nonlinear pure-feedback systems with time-varying full state constraints. International Journal of Control Automation and Systems 15(6): 2714–2722.

27.

Wang

Song

, et al. (2022) Barrier-Lyapunov-based adaptive fuzzy finite-time tracking of pure-feedback nonlinear systems with constraints. IEEE Transactions on Fuzzy Systems 30(4): 1139–1148.

28.

Wang

, et al. (2020a) Neural-network-based adaptive funnel control for servo mechanisms with unknown dead-zone. IEEE Transactions on Cybernetics 50(4): 1383–1394.

29.

Wang

(2020) Parameter estimation and adaptive control for servo mechanisms with friction compensation. IEEE Transactions on Industrial Informatics 16(11): 6816–6825.

30.

Wang

Tao

Chen

, et al. (2020b) USDE-based sliding mode control for servo mechanisms with unknown system dynamics. IEEE-ASME Transactions on Mechatronics 25(2): 1056–1066.

31.

Wen

Chen

CLP

(2023) Optimized backstepping consensus control using reinforcement learning for a class of nonlinear strict-feedback-dynamic multi-agent systems. IEEE Transactions on Neural Networks and Learning Systems 34(3): 1524–1536.

32.

Yan

Liu

Zhang

, et al. (2023) Reinforcement learning based adaptive optimal control for constrained nonlinear system via a novel state-dependent transformation. ISA Transactions 133: 29–41.

33.

Yang

Yao

Dong

(2022a) Neuroadaptive learning algorithm for constrained nonlinear systems with disturbance rejection. International Journal of Robust and Nonlinear Control 32(10): 6127–6147.

34.

Yang

Yao

Ullah

(2022b) Neuroadaptive control of saturated nonlinear systems with disturbance compensation. ISA Transactions 122: 49–62.

35.

Yao

Sun

(2019) Adaptive RISE control of hydraulic systems with multilayer neural-networks. IEEE Transactions on Industrial Electronics 66(11): 8638–8647.

36.

Yin

Zhang

Jiang

, et al. (2019) Adaptive robust integral sliding mode pitch angle control of an electro-hydraulic servo pitch system for wind turbine. Mechanical Systems and Signal Processing 133: 1–15.

37.

Zhao

Gao

(2022) Robust tracking control of uncertain nonlinear systems with adaptive dynamic programming. Neurocomputing 471: 21–30.

38.

Zhao

Wang

Yang

, et al. (2020) Design of a novel modal space sliding mode controller for electro-hydraulic driven multi-dimensional force loading parallel mechanism. ISA Transactions 99: 374–386.

39.

Zhou

Jiang

She

, et al. (2020) Generalized-extended-state-observer-based repetitive control for DC motor servo system with mismatched disturbances. International Journal of Control Automation and Systems 18(8): 1936–1945.

40.

Zhuang

Sun

Chen

, et al. (2020) Back-stepping sliding mode control for pressure regulation of oxygen mask based on an extended state observer. Automatica 119: 109106.