Extended Sliding Mode Observer-Based Output Feedback Control for Motion Tracking of Electro-Hydrostatic Actuators

Abstract

This paper develops a novel output feedback control scheme for the motion-tracking problem of an electro-hydrostatic actuator (EHA) in the presence of model uncertainties and external disturbances. Firstly, a simplified third-order system model of the studied EHA is established using theoretical methods. For the first time, an extended sliding mode observer (ESMO) is introduced to simultaneously account for the shortage of unknown system states and modeling imperfections. Based on this, a robust nonlinear controller is developed using the backstepping control framework to stabilize the closed-loop system. This controller integrates estimates of immeasurable system states and lumped disturbances to deal with their adverse impacts. Moreover, the dynamic surface control (DSC) technique is employed to effectively mitigate the computational burden of the traditional backstepping framework. An ultimately uniformly bounded (UUB) performance is assured by using the recommended method. Furthermore, the stability of not only the observer but also the closed-loop system is concretely analyzed by using the Lyapunov theory. Finally, experiment results under various working scenarios are given to convincingly demonstrate the advantage of the suggested method in comparison with some reference control approaches.

1. Introduction

By means of some noteworthy features including high power density, great endurance, smooth operation, less maintenance, and precise control capability [1] compared with electric actuator-based drive systems, hydraulic actuators have become a great solution for heavy-duty industrial applications [2,3,4]. Being considered as one of the configurations of hydraulic systems, valve-operated actuation electro-hydraulic systems (VEHSs) or electro-hydraulic servo systems (EHSSs) have some advantages, including fast response, smooth motion, and high controllability [5]. Nonetheless, there are some drawbacks of EHSSs, such as low energy efficiency because of high flow throttling loss, considerable system construction and maintenance cost, and they require more space for installation compared to the so-called pump-controlled EHSs or electro-hydrostatic actuators (EHAs) [6]. In addition, the pumping system in EHSSs must supply constant pressure to the system regardless of the working conditions, which contributes to electrical extravagance. Therefore, recently, EHAs with simple structures have been broadly adopted in various applications such as exoskeleton robots [7,8], aerospace applications [9,10], hydraulic robot manipulators [11], and so on [12] to avoid the aforementioned shortcomings of the EHSSs. However, the dynamic behaviors of EHAs are extremely complex and highly nonlinear. Achieving a high-accuracy tracking performance with such EHAs is a challenging control problem, particularly in the presence of modeling errors, unmodeled dynamics, and unknown external load.

To achieve a satisfactory tracking performance, numerous controllers have been introduced in the literature. It can be observed that proportional-integral-derivative (PID) controllers are the most commonly applied model-free controller in industrial applications due to its simplicity, easy implementation, and limited tuning parameters. In addition, the requirement for an accurate system model is mitigated and the stability of the closed-loop system can be guaranteed by applying this control law. However, the desired performance may not be satisfied because of the highly nonlinear factors such as complicated friction and pressure dynamics of EHAs, and unmodeled dynamics. To this end, advanced control techniques have been introduced to overcome this problem and achieve a high-accuracy tracking performance for pump-controlled EHSs. For example, in [13], an adaptive robust control using a backstepping framework was established for a pump-controlled hydraulic system. In this control scheme, a deterministic model of pump flow rate at low speeds based on experimental data was originally constructed and compensated in the controller design, and consequently, improved tracking performance was achieved. However, in practice, the parameters of the pump flow rate model can change depending on different working conditions; hence, the accuracy of this model can be reduced. To overcome this drawback, an adaptive nonlinear pump flow model [14] was formulated, and better tracking performance was obtained as a result. For further improvement of tracking accuracy, in [15], Chen Li et al. developed an adaptive robust control framework of an independent metering hydraulic system by introducing a nonlinear valve flow model. To enhance the convergence speed of the adaptation parameter, a composite learning adaptive motion following control for electro-hydraulic servo systems was constructed [16]. Nonetheless, it should be noted that there are two major disadvantages of the above-mentioned control approaches. Firstly, all system states are assumed to be directly measured, many sensors are required, and system cost and structure complexity increase as a consequence. In addition, uncertainties are compensated by adaptive methods under the assumption that the exact model of the considered hydraulic system is available.

Disturbance and uncertainty attenuation for motion tracking problems has attracted great attention from the research community to improve the control accuracy and the robustness of the closed-loop control system of EHSs. To cope with state-dependent unstructured uncertainties, adaptive approximators based on neural network or fuzzy logic system approaches have been utilized [17,18,19,20]. Nevertheless, due to the complexity of excessive tuning parameters, expert knowledge requirements, and heavy computational burden, these approaches are difficult to implement in practical systems. On the other hand, they are no longer available for state-independent uncertainties. As one of the valuable solutions to this problem, disturbance observers (DOBs) have been carefully studied and successfully applied to EHSs to lessen the adverse effects of model uncertainties and external disturbances. Specifically, in [21,22,23], high-gain disturbance observers (HGDOBs) were developed for estimating lumped uncertainties in an EHSS, then their estimated values were fed back into backstepping control laws to compensate for their influences on the position-tracking capability. Furthermore, a coupled DOB [24] was designed to attenuate the independent and coupled elements of external load on hydraulic actuators of multi-degree-of-freedom manipulators. Based on that, the tracking errors were constrained within a prescribed steady-state level. In addition, Hamid Razmjooei et al. developed estimation error-based DOBs to counteract uncertainties and estimation errors of state observers and guarantee finite-time tracking performance [25]. However, it is worth noting that all system states are required to be accessible to implement the above control algorithms, and this requirement may complicate the control system structure and increase the system cost as a result.

Considering the shortage of system states and negative effects of model uncertainties, inherited from the Luenberger observer [26,27], a linear extended state observer (LESO), which was first introduced by Jingqing Han [28], can be considered a great solution. The convergence of an ESO for nonlinear systems subject to uncertainties was rigorously proven in [29]. ESO does not require an exact system model, and the burden of an offline identification process can be reduced [30]. Because of such a favorable feature, ESO-based control approaches, which are known as active disturbance rejection control (ADRC) schemes, have been widely employed in various engineering systems in recent years. However, for nonlinear systems subject to the so-called mismatched uncertainties, ESO-based control algorithms are no longer available. To this end, in [31], a generalized ESO-based control for systems with mismatched uncertainties was introduced. To improve the convergence time of the conventional ESO, a novel ESO [32,33], employing nonlinear and/or switching terms that ensure the estimation errors converge to a neighborhood of the origin in finite time, was put forward. Although conventional ESOs have been successfully adopted in various applications in recent years, they have several disadvantages, such as the peaking phenomenon, insufficient estimation accuracy, and long convergence time. Inheriting from sliding mode observer (ESO), recently, ESMOs have been successfully implemented in several applications. For instance, in [34], an ESMO was constructed for current control of permanent magnet synchronous motor systems to estimate the stator current and lumped disturbance. Jinyong Yu et al. proposed a fault-tolerant control based on an ESMO that is introduced to estimate states and faults of descriptor stochastic systems [35]. However, as far as the authors know, the development of an ESMO for high-order pump-controlled EHSs to estimate not only immeasurable states but also modeling uncertainties has not been considered in the literature.

Inspired by the above observations, in this paper, a novel output feedback control scheme for motion tracking of an EHA suffering from model uncertainties, unknown dynamics, and external load was presented. The main innovations of this study are summarized as follows:

(1)

For the first time, an ESMO based on a system model was constructed to estimate immeasurable system states, i.e., angular velocity and load pressure, and lumped matched disturbance in the pressure dynamics caused by parameter deviations and modeling errors.

(2)

Based on the designed ESMO, a novel output feedback robust control scheme using the backstepping control framework and dynamic surface control technique is synthesized for the motion-tracking problem of the studied EHA.

(3)

The stability of the ESMO and the overall closed-loop system is theoretically verified by the Lyapunov theory. Experiments on the real test bench are conducted to illustrate the practicability and advantage of the suggested controller in comparison with some reference methods under various working scenarios.

The rest of the paper is organized as follows. In Section 2, the system model of the considered EHA and the problem formulation are presented. Designs of the ESMO and observer-based output feedback control approach are provided in Section 3. Section 4 presents comparative experiment results of the proposed control algorithm and several reference controllers. Finally, a conclusion is given in Section 5.

2. System Modeling and Problem Statement

The schematic of the studied EHA is depicted in Figure 1. As shown, the motion of the hydraulic rotary actuator is directly controlled by a bidirectional fixed displacement hydraulic pump driven by an electric motor. Two pilot-operated check valves ${\mathrm{V}}_1$ and ${\mathrm{V}}_2$ keep the load in position if there is no flow supplied by the pump. Under normal working conditions, these valves allow flows in both directions and they are controlled by pressures created by the hydraulic pump. Check valves ${\mathrm{V}}_5$ and ${\mathrm{V}}_6$ are used to supply supplemental oil to the actuator according to the required motion of the actuator. Meanwhile, two relief valves ${\mathrm{V}}_3$ and ${\mathrm{V}}_4$ are employed to ensure that the working pressure in the system does not exceed the predefined maximum pressure, and consequently, guarantee safety conditions.

By means of Newton’s second law of rotation, the motion dynamics of the actuator can be determined as

$$J_a\ddot{\theta }=D_a{P}_L-B_a\dot{\theta }-A_f{S}_f\left(\dot{\theta }\right)+{\tau }_d$$

(1)

where $J_a$ and $D_a$ are the moment of inertia and the displacement of the actuator, respectively. $\dot{\theta }$ and $\ddot{\theta }$ stand for the angular velocity and acceleration of the actuator shaft, respectively. $P_L=P_1-P_2$ denotes the load pressure, with $P_1$ and $P_2$ being pressures in two chambers of the actuator, $B_a$ signifies the viscous friction coefficient, $A_f$ is the Coulomb friction magnitude, $S_f(·)$ is a smooth function describing some complicated behaviors of the friction force inside the actuator to be designed later, and ${\tau }_d$ represents the torque disturbance caused by unmodeled dynamics and external loads.

Under the assumption that there is no external leakage, according to the definition of the effective bulk modulus, the pressure dynamics in the two chambers of the actuator are given by

$$\begin{array}{cc}\hfill & {\dot{P}}_1=\frac{\beta }{V_1}\left(Q_1-D_a\dot{\theta }-C_a{P}_L\right)\hfill \\ \hfill & {\dot{P}}_2=\frac{\beta }{V_2}\left(Q_2+D_a\dot{\theta }+C_a{P}_L\right)\hfill \end{array}$$

(2)

where $\beta$ is the bulk modulus of the hydraulic oil; $V_1$ and $V_2$ denote the total control volumes of the actuator chambers, with $V_{01}$ and $V_{02}$ being the initial volumes, which include the volumes of the hoses and dead volumes of the actuator, and $\theta$ being the displacement of the actuator; $Q_1$ and $Q_2$ are in/out flow rates of the two chambers; and $C_a$ signifies the internal leakage coefficient of the actuator.

Since the working pressures inside the two chambers of the actuator are always smaller than the setting pressures of the relief valves $V_3$ and $V_4$ in normal working conditions, the flow rates of the two chambers of the actuator are computed as [6]

$$\begin{array}{cc}\hfill & Q_1=D_p{n}_p-C_p{P}_L+Q_{v5}\hfill \\ \hfill & Q_2=-D_p{n}_p+C_p{P}_L+Q_{v6}\hfill \end{array}$$

(3)

where $D_p$ and $n_p$ are the displacement and the speed of the hydraulic pump; $Q_{v3}$, $Q_{v4}$, $Q_{v5}$, and $Q_{v6}$ are flow rates through valves ${\mathrm{V}}_3$, ${\mathrm{V}}_4$, ${\mathrm{V}}_5$, and ${\mathrm{V}}_6$, respectively.

Substituting (3) into (2), we have

$$\begin{array}{cc}\hfill & {\dot{P}}_1=\frac{\beta }{V_1}\left(D_p{n}_p-D_a\dot{\theta }-C_t{P}_L+Q_{v5}\right)\hfill \\ \hfill & {\dot{P}}_2=\frac{\beta }{V_2}\left(-D_p{n}_p+D_a\dot{\theta }+C_t{P}_L+Q_{v6}\right)\hfill \end{array}$$

(4)

where $C_t=C_a+C_p$ is the total internal leakage coefficient.

From (4), the load pressure dynamics can be obtained as

$${\dot{P}}_L=\left(\frac{\beta }{V_1}+\frac{\beta }{V_2}\right)\left(D_p{n}_p-D_a\dot{\theta }-C_a{P}_L\right)+\frac{\beta }{V_1}{Q}_{v5}-\frac{\beta }{V_2}{Q}_{v6}$$

(5)

Defining $\mathbf{x}={\left[x_1,x_2,x_3\right]}^T={\left[\theta ,\dot{\theta },D_a{P}_L/J_a\right]}^T$ as the system state vector, according to (1) and (5), the overall system dynamics of the studied EHA are presented in state-space form as follows:

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2\hfill \\ {\dot{x}}_2=x_3+f_1\left(x_2\right)+d_1(\mathbf{x},t)\hfill \\ {\dot{x}}_3=f_2\left(\mathbf{x}\right)+g_2\left(x_1\right)u+d_2(\mathbf{x},t)\hfill \end{array}\right.$$

(6)

where the control input u is the rotational speed of the pump and dynamical functions are given by

$$\begin{array}{cc}\hfill & f_1\left(x_2\right)=-\frac{B_a}{J_a}{x}_2-\frac{A_f{S}_f\left(x_2\right)}{J_a};d_1\left(\mathbf{x},t\right)=\frac{T_d}{J_a}+\mathsf{\Delta }{f}_1\left(x_2\right)\hfill \\ \hfill & f_2\left(\mathbf{x}\right)=-\left(\frac{\beta }{V_{01}+D_a{x}_1}+\frac{\beta }{V_{02}-D_a{x}_1}\right)\left(\frac{D_a^2}{J_a}{x}_2+C_t{x}_3\right)\hfill \\ \hfill & g_2\left(x_1\right)=\left(\frac{\beta }{V_{01}+D_a{x}_1}+\frac{\beta }{V_{02}-D_a{x}_1}\right)\frac{D_a{D}_p}{J_a}\hfill \\ \hfill & d_2\left(\mathbf{x},t\right)=\frac{D_a}{J_a}\left(\frac{\beta }{V_{01}+D_a{x}_1}{Q}_{v5}-\frac{\beta }{V_{02}-D_a{x}_1}{Q}_{v6}\right)+\mathsf{\Delta }{f}_2\left(\mathbf{x}\right)+\mathsf{\Delta }{g}_2\left(x_1\right)u\hfill \end{array}$$

Remark 1.

Considering the system dynamics (6), although the dynamical functions $f_1\left(x_2\right)$ and $f_2\left(\mathbf{x}\right)$ are known and nominal system parameters can be experimentally identified, these functions cannot be directly computed due to the lack of all system states. In addition, the effects of parameter deviations, unmodeled uncertainties, and external disturbances in the mechanical subsystem and pressure dynamics are lumped into the so-called total mismatched and matched disturbances $d_1(\mathbf{x},t)$ and $d_2(\mathbf{x},t)$, respectively.

This study focuses on designing a control law that guarantees a demanded reference-following capability only using the system output information for a pump-controlled hydraulic system in the presence of model uncertainties and external disturbances. To facilitate the control design, some reasonable assumptions are made, as follows:

Assumption 1

([36,37,38]). The lumped disturbance $d_1(\mathbf{x},t)$ is assumed to be bounded, i.e., $\left|d_1\left(\mathbf{x},t\right)\right|\le$ ${\delta }_1$. $d_2(\mathbf{x},t)$ is assumed to be differential and its first derivative $h_2(\mathbf{x},t)$ is bounded by unknown constants, i.e., $\left|h_2\left(\mathbf{x},t\right)\right|\le {\gamma }_2$.

Assumption 2

([39,40]). The function $f_1\left(x_2\right)$ is globally Lipschitz with respect to $x_2$ and $f_2(x_1,x_2$, $x_3)$ is Lipschitz with respect to $x_1$, $x_2$, and $x_3$.

Assumption 3.

The desired trajectory $x_{1d}\left(t\right)$ is sufficiently smooth and its first-order derivative is bounded.

Remark 2.

Because of the restriction on the pressures inside the actuator for safety operation, the load capacity of a given system is always limited. Therefore, the lumped mismatched disturbance $d_1(\mathbf{x},t)$ should be bounded to ensure a desired tracking performance. The other assumptions have been proven to be reasonable and practical, and they could be found in many previous studies [36,37,38,39,40] focusing on observer-based control design for EHSs.

3. Observer-Based Output Feedback Control Design

The proposed control methodology is illustrated in Figure 2. In this control scheme, an ESMO is designed to estimate immeasurable states and lumped disturbance simultaneously. Based on the state estimates of the ESMO, an observer is then developed to reconstruct the lumped mismatched disturbance. To achieve desired tracking performance, an output feedback controller is designed using the backstepping control technique combined with the DSC approach, in which estimated values of immeasurable states and disturbances are fed back into the system, and consequently their influences on the system output tracking qualification are effectively attenuated.

3.1. Extended Sliding Mode Observer

Considering the system dynamics (6), by defining $x_e=d_2(\mathbf{x},t)$ as an extended state, augmented system dynamics are obtained as

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2\hfill \\ {\dot{x}}_2=x_3+f_1\left(x_2\right)+d_1\left(\mathbf{x},t\right)\hfill \\ {\dot{x}}_3=f_2\left(\mathbf{x}\right)+g_2\left(x_1\right)u+x_e\hfill \\ {\dot{x}}_e=h_2\left(\mathbf{x},t\right)\hfill \end{array}\right.$$

(7)

An extended sliding mode observer is constructed as

$$\left\{\begin{array}{c}{\upsilon }_{eq}=\eta \mathrm{sign}(x_1-{\widehat{x}}_1)\hfill \\ {\dot{\widehat{x}}}_1={\widehat{x}}_2+{\upsilon }_{eq}\hfill \\ {\dot{\widehat{x}}}_2={\widehat{x}}_3+f_1\left({\widehat{x}}_2\right)+3\omega {\upsilon }_{eq}\hfill \\ {\dot{\widehat{x}}}_3=f_2\left(\widehat{\mathbf{x}}\right)+g_2\left(x_1\right)u+{\widehat{x}}_e+3{\omega }^2{\upsilon }_{eq}\hfill \\ {\dot{\widehat{x}}}_e={\omega }^3{\upsilon }_{eq}\hfill \end{array}\right.$$

(8)

where $\eta >0$ is the switching gain, $\omega >0$ is the bandwidth of the observer to be selected, and $\mathrm{sign}(·)$ is the standard signum function. ${\widehat{x}}_1$, ${\widehat{x}}_2$, ${\widehat{x}}_3$, and ${\widehat{x}}_e$ are the estimates of $x_1$, $x_2$, $x_3$, and $x_e$, respectively.

Theorem 1.

Noting the system dynamics (6), by appropriately choosing the switching gain η, the proposed ESMO (10) guarantees that the estimation errors, i.e., ${\tilde{x}}_1=x_1-{\widehat{x}}_1$, ${\tilde{x}}_2=x_2-{\widehat{x}}_2$, ${\tilde{x}}_3=x_3-{\widehat{x}}_3$, and ${\tilde{x}}_e=x_e-{\widehat{x}}_e$, exponentially converge to an arbitrarily small bounded region depending on the selection of the observer bandwidth ω.

Proof of Theorem 1.

Choose a candidate Lyapunov function:

$$V_{{\tilde{x}}_1}=\frac{1}{2}{\tilde{x}}_1^2$$

(9)

Taking the time derivative of it leads to

$${\dot{V}}_{{\tilde{x}}_1}={\tilde{x}}_1\left({\dot{x}}_1-{\dot{\widehat{x}}}_1\right)$$

(10)

Combining with (7) and (8), (10) can be transformed into

$$\begin{array}{cc}\hfill {\dot{V}}_{{\tilde{x}}_1}& ={\tilde{x}}_1\left({\tilde{x}}_2-\eta \mathrm{sign}\left({\tilde{x}}_1\right)\right)\hfill \\ \hfill & \le -(\eta -\left|{\tilde{x}}_2\right|)\left|{\tilde{x}}_1\right|\hfill \\ \hfill & =-{\eta }_0\sqrt{2}{V}_{{\tilde{x}}_1}^{1/2}\hfill \end{array}$$

(11)

From (11), it can be observed that if the switching gain $\eta$ is chosen such that $\eta ={\eta }_0+\left|{\tilde{x}}_2\left(0\right)\right|$ with ${\eta }_0$ as a small positive constant, the estimation error ${\tilde{x}}_1$ converges to the origin in finite time and it remains at the origin thereafter.

Hence, in the sliding mode, i.e., ${\tilde{x}}_2={\upsilon }_{eq}$, the reduced-order dynamics of the observer (8) can be obtained as

$$\left\{\begin{array}{c}{\dot{\widehat{x}}}_2={\widehat{x}}_3+f_1\left({\widehat{x}}_2\right)+3\omega {\tilde{x}}_2\hfill \\ {\dot{\widehat{x}}}_3=f_2\left(x_1,{\widehat{x}}_2,{\widehat{x}}_3\right)+g_2\left(x_1\right)u+{\widehat{x}}_e+3{\omega }^2{\tilde{x}}_2\hfill \\ {\dot{\widehat{x}}}_e={\omega }^3{\tilde{x}}_2\hfill \end{array}\right.$$

(12)

For the sake of concise expression, $d_1$, $d_2$, and $h_2$ are used for upcoming mathematical transformations instead of using $d_1(\mathbf{x},t)$, $d_2(\mathbf{x},t)$, and $h_2(\mathbf{x},t)$. Noting the system dynamics (7) and the reduced-order dynamics (12), the error dynamics are derived as

$$\left\{\begin{array}{c}{\dot{\tilde{x}}}_2=-3\omega {\tilde{x}}_2+{\tilde{x}}_3+{\tilde{f}}_1+d_1\hfill \\ {\dot{\tilde{x}}}_3=-3{\omega }^2{\tilde{x}}_2+{\tilde{x}}_e+{\tilde{f}}_2\hfill \\ {\dot{\tilde{x}}}_e=-{\omega }^3{\tilde{x}}_2+h_2\hfill \end{array}\right.$$

(13)

where

$$\left\{\begin{array}{c}{\tilde{f}}_1=f_1\left(x_2\right)-f_1\left({\widehat{x}}_2\right)\hfill \\ {\tilde{f}}_2=f_2(x_1,x_2,x_3)-f_2(x_1,{\widehat{x}}_2,{\widehat{x}}_3)\hfill \end{array}\right.$$

(14)

Letting $\mathit{\epsilon }={\left[{\epsilon }_1,{\epsilon }_2,{\epsilon }_3\right]}^T={\left[{\tilde{x}}_2,{\tilde{x}}_3/\omega ,{\tilde{x}}_e/\omega \right]}^T$ as the scaled estimation error vector, (13) can be rewritten as

$$\dot{\mathit{\epsilon }}=-\omega \mathit{A}\mathit{\epsilon }+\mathit{B}({\tilde{f}}_1+d_1)+\mathit{C}\frac{{\tilde{f}}_2}{\omega }+\mathit{D}\frac{h_2}{{\omega }^2}$$

(15)

where the matrices $\mathit{A}$, $\mathit{B}$, $\mathit{C}$, and $\mathit{D}$ are defined by

$$\mathit{A}=\left[\begin{array}{ccc}-3& 1& 0\\ -3& 0& 1\\ -1& 0& 0\end{array}\right];\mathit{B}=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right];\mathit{C}=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right];\mathit{D}=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$$

(16)

Since matrix $\mathit{A}$ is Hurwitz, for any positive definite matrix $\mathit{Q}$, there always exists a positive definite matrix $\mathit{P}$ satisfying the following Lyapunov function:

$${\mathit{A}}^T\mathit{P}+\mathit{P}\mathit{A}=-2\mathit{Q}$$

(17)

Selecting a candidate Lyapunov function $V_{ESMO}=\frac{1}{2}{\mathit{\epsilon }}^T\mathit{P}\mathit{\epsilon }$, then taking the time derivative of it yields

$$\begin{array}{cc}\hfill {\dot{V}}_{ESMO}& =\frac{1}{2}{\dot{\mathit{\epsilon }}}^T\mathit{P}\mathit{\epsilon }+\frac{1}{2}{\mathit{\epsilon }}^T\mathit{P}{\dot{\mathit{\epsilon }}}^T\hfill \\ \hfill & =-\omega {\mathit{\epsilon }}^T\mathit{Q}\mathit{\epsilon }+{\mathit{\epsilon }}^T\mathit{P}\mathit{B}({\tilde{f}}_1+d_1)+{\mathit{\epsilon }}^T\mathit{P}\mathit{C}\frac{{\tilde{f}}_2}{\omega }+{\mathit{\epsilon }}^T\mathit{P}\mathit{D}\frac{h_2}{{\omega }^2}\hfill \end{array}$$

(18)

Applying the Young’s inequality, one has

$$\begin{array}{c}{\mathit{\epsilon }}^T\mathit{P}\mathit{B}{\tilde{f}}_1\le \frac{1}{2}{\mathit{\epsilon }}^T\mathit{\epsilon }+\frac{1}{2}{\mathit{B}}^T{\mathit{P}}^T\mathit{P}\mathit{B}{\tilde{f}}_1^2\hfill \\ {\mathit{\epsilon }}^T\mathit{P}\mathit{B}{d}_1\le \frac{1}{2}{\mathit{\epsilon }}^T\mathit{\epsilon }+\frac{1}{2}{\mathit{B}}^T{\mathit{P}}^T\mathit{P}\mathit{B}{d}_1^2\hfill \\ {\mathit{\epsilon }}^T\mathit{P}\mathit{C}\frac{{\tilde{f}}_2}{\omega }\le \frac{1}{2}{\mathit{\epsilon }}^T\mathit{\epsilon }+\frac{1}{2}{\mathit{C}}^T{\mathit{P}}^T\mathit{P}\mathit{C}\frac{{\tilde{f}}_2^2}{{\omega }^2}\hfill \\ {\mathit{\epsilon }}^T\mathit{P}\mathit{D}\frac{{\tilde{h}}_2}{{\omega }^2}\le \frac{1}{2}{\mathit{\epsilon }}^T\mathit{\epsilon }+\frac{1}{2}{\mathit{D}}^T{\mathit{P}}^T\mathit{P}\mathit{D}\frac{h_2^2}{{\omega }^4}\hfill \end{array}$$

(19)

From Assumption 2 and combining with the definition of scaled estimation errors, we have

$$\left\{\begin{array}{c}\left|{\tilde{f}}_1\right|\le \frac{C_1}{\omega }∥\mathit{\epsilon }∥\hfill \\ \left|{\tilde{f}}_2\right|\le \left(\frac{C_2}{\omega }+\frac{C_3}{{\omega }^2}\right)∥\mathit{\epsilon }∥\hfill \end{array}\right.$$

(20)

where $C_1$, $C_2$, and $C_3$ are Lipschitz constants.

Hence, (19) is transformed into

$$\begin{array}{cc}\hfill {\dot{V}}_{ESMO}& \le -\left(\omega {\lambda }_{\min }\left\{\mathit{Q}\right\}-2-\frac{{\lambda }_1{C}_1^2}{2{\omega }^2}-\frac{{\lambda }_2}{2}{\left(\frac{C_2}{\omega }+\frac{C_3}{{\omega }^2}\right)}^2\right){\mathit{\epsilon }}^T\mathit{\epsilon }+\left(\frac{{\lambda }_2{\delta }_1^2}{\omega }+\frac{{\lambda }_3{\gamma }_2^2}{{\omega }^4}\right)\hfill \\ \hfill & =-{\mathsf{\Gamma }}_{ESMO}{V}_{ESMO}+{\mathsf{\Pi }}_{ESMO}\hfill \end{array}$$

(21)

where ${\lambda }_1={∥\mathit{P}\mathit{B}∥}^2$, ${\lambda }_2={∥\mathit{P}\mathit{C}∥}^2$, ${\lambda }_3={∥\mathit{P}\mathit{D}∥}^2$ with $∥\mathbf{\vartheta }∥$ as the Euclidean norm of $\mathbf{\vartheta }$, and

$$\begin{array}{c}{\mathsf{\Gamma }}_{ESMO}=-\frac{1}{{\lambda }_{\max }\left\{\mathit{P}\right\}}\left(\omega {\lambda }_{\min }\left\{\mathit{Q}\right\}-2-\frac{{\lambda }_1{C}_1^2}{2{\omega }^2}-\frac{{\lambda }_2}{2}{\left(\frac{C_2}{\omega }+\frac{C_3}{{\omega }^2}\right)}^2\right)\hfill \\ {\mathsf{\Pi }}_{ESMO}=\frac{{\lambda }_2{\delta }_1^2}{\omega }+\frac{{\lambda }_3{\gamma }_2^2}{{\omega }^4}\hfill \end{array}$$

where ${\lambda }_{\min }\left\{\mathbf{\chi }\right\}$ and ${\lambda }_{\max }\left\{\mathbf{\chi }\right\}$ denote the maximal and minimal eigenvalues of matrix $\mathbf{\chi }$, respectively.

According to (21), the estimation errors exponentially converge to the neighborhood of the origin defined by ${\mathsf{\Pi }}_{ESMO}/{\mathsf{\Gamma }}_{ESMO}$ when time goes to infinity. The upper bound of this region decreases if the observer bandwidth decreases and vice versa.

This completes the proof of Theorem 1. □

Remark 3.

To reduce the chattering in the estimated values of both unmeasured states and lumped disturbance caused by unmodeled dynamics and modeling errors, a hyperbolic tangent function is adopted to replace the discontinuous “sign" function in the design of the observer. In addition, the observer bandwidth should be meticulously selected to achieve a trade-off between accurate estimation and reducing negative effects of measurement noise.

3.2. Observer-Based Control Design

We define tracking error as

$$\left\{\begin{array}{c}{e}_1=x_1-x_{1d}\hfill \\ e_2={\widehat{x}}_2-{\alpha }_{1f}\hfill \\ e_3={\widehat{x}}_3-{\alpha }_{2f}\hfill \end{array}\right.$$

(22)

where $x_{1d}$ is the reference motion trajectory, and ${\alpha }_{1f}$ and ${\alpha }_{2f}$ are filtered signals of virtual control laws ${\alpha }_1$ and ${\alpha }_2$ to be designed using the traditional backstepping control framework, respectively, through the following low-pass filters.

The command filters are constructed as

$$\left\{\begin{array}{c}{T}_1{\dot{\alpha }}_{1f}+{\alpha }_{1f}={\alpha }_1;{\alpha }_{1f}\left(0\right)={\alpha }_1\left(0\right)\hfill \\ T_2{\dot{\alpha }}_{2f}+{\alpha }_{2f}={\alpha }_2;{\alpha }_{2f}\left(0\right)={\alpha }_2\left(0\right)\hfill \end{array}\right.$$

(23)

where $T_1$ and $T_2$ are the time constants defining the cut-off frequencies of the designed filters.

Defining ${\tilde{\alpha }}_i={\alpha }_{if}-{\alpha }_i$ with $i=1,2$ as the deviations between the original virtual control laws and their filtered signals, the dynamics of the filtering errors are obtained as

$${\dot{\tilde{\alpha }}}_i=-\frac{1}{T_i}({\tilde{\alpha }}_i+{\rho }_i)$$

(24)

where ${\rho }_i=-{\dot{\alpha }}_i$ is assumed to be bounded by an unknown constant, i.e., $\left|{\rho }_i\right|\le {\rho }_{iM}$.

Remark 4.

By applying a series of first-order low-pass filters (23), influences of measurement noise can be mitigated. In addition, the derivative of the filtered signals can be directly computed; hence, the requirement of computing analytic derivations of virtual control laws at backstepping iterations is effectively relaxed.

The control laws for stabilizing the closed-loop system are designed as

$$\begin{array}{c}{\alpha }_1=-k_1{e}_1+{\dot{x}}_{1d}-{\upsilon }_{eq}\hfill \\ {\alpha }_2=-k_2{e}_2-e_1-f_1\left({\widehat{x}}_2\right)+{\dot{\alpha }}_{1f}-3\omega {\upsilon }_{eq}\hfill \\ u=\frac{1}{g_2\left(x_1\right)}(-k_3{e}_3-e_2-f_2(x_1,{\widehat{x}}_2,{\widehat{x}}_3)-{\widehat{x}}_e-3{\omega }^2{\upsilon }_{eq})\hfill \end{array}$$

(25)

Noting the system dynamics (6), definition of tracking errors (22), and synthesized control laws (25), the tracking error dynamics of the closed-loop system are given by

$$\left\{\begin{array}{c}{\dot{e}}_1=-k_1{e}_1+e_2+{\tilde{\alpha }}_1\hfill \\ {\dot{e}}_2=-k_2{e}_2-e_1+e_3+{\tilde{\alpha }}_2\hfill \\ {\dot{e}}_3=-k_3{e}_3-e_2\hfill \end{array}\right.$$

(26)

3.3. Closed-Loop Stability Analysis

Theorem 2.

Considering the system dynamics (6) under Assumptions 1, 2, and 3, an ultimately uniformly bounded (UUB) tracking performance is guaranteed by using control laws (25) and observers (12), that is, the tracking errors converge to an arbitrarily small vicinity of the origin whose upper bounds depend on the selection of control gains $k_1$, $k_2$, and $k_3$; observer bandwidth ω; and time constants $T_1$ and $T_2$ as time goes to infinity.

Proof of Theorem 2.

For analyzing the stability of the closed-loop system, a candidate Lyapunov function is chosen as

$$V_c=\frac{1}{2}{\mathit{e}}^T\mathit{e}+\frac{1}{2}{\tilde{\mathbf{\alpha }}}^T\tilde{\mathbf{\alpha }}+\frac{1}{2}{\mathit{\epsilon }}^T\mathit{P}\mathit{\epsilon }$$

(27)

where $\mathit{e}={[e_1,e_2,e_3]}^T$ and $\tilde{\mathbf{\alpha }}={[{\tilde{\alpha }}_1,{\tilde{\alpha }}_2]}^T$.

Taking the time derivative of (27) then combining with (24) and (26), we have

$${\dot{V}}_c=-k_1{e}_1^2-k_2{e}_2^2-k_3{e}_3^2-\frac{1}{T_1}{\tilde{\alpha }}_1^2-\frac{1}{T_2}{\tilde{\alpha }}_2^2+e_1{\tilde{\alpha }}_1+e_2{\tilde{\alpha }}_2+{\alpha }_1{\rho }_1+{\alpha }_2{\rho }_2+{\dot{V}}_{ESMO}$$

(28)

Applying the Young’s inequality, one has

$$\begin{array}{cc}\hfill {\dot{V}}_c\le & -\left(k_1-\frac{1}{2}\right)e_1^2-\left(k_2-\frac{1}{2}\right)e_2^2-\left(k_3-\frac{1}{2}\right)e_3^2\hfill \\ \hfill & -\left(\frac{1}{T_1}-1\right){\tilde{\alpha }}_1^2-\left(\frac{1}{T_2}-1\right){\tilde{\alpha }}_2^2+\frac{{\rho }_{1M}^2}{2}+\frac{{\rho }_{2M}^2}{2}+{\dot{V}}_{ESMO}\hfill \end{array}$$

(29)

Substituting (21) into (29) leads to

$$\begin{array}{cc}\hfill {\dot{V}}_c& \le -{\mathit{e}}^T\mathit{K}\mathit{e}-{\tilde{\mathbf{\alpha }}}^T\mathit{T}\tilde{\mathbf{\alpha }}-\frac{{\mathsf{\Gamma }}_{ESMO}}{2}{\mathit{\epsilon }}^T\mathit{P}\mathit{\epsilon }+\frac{{\rho }_{1M}}{2}+\frac{{\rho }_{2M}}{2}+{\mathsf{\Pi }}_{ESMO}\hfill \\ \hfill & =-{\mathsf{\Gamma }}_c{V}_c+{\mathsf{\Pi }}_c\hfill \end{array}$$

(30)

where the matrices $\mathit{K}$ and $\mathit{T}$ are defined by

$$\begin{array}{c}\mathit{K}=\mathrm{diag}\left\{k_1-\frac{1}{2};k_2-\frac{1}{2};k_3-\frac{1}{2}\right\}\hfill \\ \mathit{T}=\mathrm{diag}\left\{\frac{1}{T_1}-1;\frac{1}{T_2}-1\right\}\hfill \end{array}$$

and

$$\begin{array}{c}{\mathsf{\Gamma }}_c=\min \left\{2{\lambda }_{\min }\left\{\mathit{K}\right\};2{\lambda }_{\min }\left\{\mathit{T}\right\};{\mathsf{\Gamma }}_{ESMO}\right\}\hfill \\ {\mathsf{\Pi }}_c={\mathsf{\Pi }}_{ESMO}+\frac{{\rho }_{1M}^2}{2}+\frac{{\rho }_{2M}^2}{2}\hfill \end{array}$$

From (30), it can be observed that when time goes to infinity, the Lyapunov function (27) converges to a region defined by ${\mathsf{\Pi }}_c/{\mathsf{\Gamma }}_c$ as time goes to infinity, and consequently a UUB performance can be achieved. The upper bound of this region depends on the selection of observer bandwidth, controller gains, and time constants of the low-pass filters.

Hence, the proof of Theorem 2 is completed. □

4. Experiment Validation

4.1. Experiment Setup

For the performance evaluation of the recommended control method, a test rig for an electro-hydrostatic actuator was built, which is illustrated in Figure 3. It includes a compact hydraulic power pack, a rotary actuator, and an incremental encoder. The rotary actuator is provided by KNR company. To control the motion of the actuator, a bidirectional hydraulic pump was adopted and it is driven by a DC motor with the motor driver MD03, whose output voltage can be controlled by various types of input signals or communication. An incremental encoder E40H8-5000-3-V-5 with the resolution as 5000 pulses per revolution was installed so as to precisely observe the position of the actuator. Pilot check valves and relief valves are integrated into a center block to manipulate the oil flow into or out of the actuator and ensure that the system pressure inside the actuator does not exceed the predefined maximum pressure, respectively. In addition, a pulley system fixed to the frame of the test bed was directly attached to the actuator shaft and a load stand. Based on this, loads can be easily changed to evaluate the tracking performance of comparative controllers under different load conditions.

The system parameters which were experimentally identified are given in

Table 1. The parameters of the studied EHA.

Parameter	Notation	Value	SI Unit
Moment of inertia of the actuator	$J_a$	$0.25$	$\mathrm{kg}·{\mathrm{m}}^2$
Hydraulic actuator displacement	$D_a$	$5.8442\times {10}^{-6}$	${\mathrm{m}}^3·{\mathrm{rad}}^{-1}$
Viscous friction coefficient of the actuator	$B_a$	10	$\mathrm{N}·\mathrm{m}·{\mathrm{rad}}^{-1}·\mathrm{s}$
Coulomb friction coefficient of the actuator	$A_f$	10	$\mathrm{N}·\mathrm{m}$
Total leakage coefficient	$C_t$	$4.267\times {10}^{-12}$	${\mathrm{m}}^3·{\mathrm{s}}^{-1}·{\mathrm{Pa}}^{-1}$
Effective bulk modulus of the hydraulic oil	$\beta$	$1.5\times {10}^9$	$\mathrm{N}·{\mathrm{m}}^{-2}\mathrm{or}\mathrm{Pa}$
Hydraulic pump displacement	$D_p$	$0.1544\times {10}^{-7}$	${\mathrm{m}}^3·{\mathrm{rad}}^{-1}$
Initial control volume of the forward chamber	$V_{01}$	$1.25\times {10}^{-5}$	${\mathrm{m}}^3$
Initial control volume of the reverse chamber	$V_{02}$	$2.27\times {10}^{-5}$	${\mathrm{m}}^3$

. These parameters were used to design not only the main controller but also the observer. A smooth friction model is employed as $S_f\left(x_2\right)=\tanh \left({\tau }_1{x}_2\right)-\tanh \left({\tau }_2{x}_2\right)$ that is able to capture the Stribeck effect and static friction at the very low motion speed of the actuator with ${\tau }_1=100$ and ${\tau }_2=5$.

To verify the effectiveness of the proposed method, the following controllers are adopted for comparison:

(1)

ESMOBC: The proposed controller, whose control gains are chosen as $k_1=80$, $k_2=505$, $k_3=30$. The time constants of the low-pass filters are $T_1=T_2=0.01$ and the observer bandwidth is $\omega =120$.

(2)

PID: Proportional-derivative-integral controller (PID), whose controller gains are ultimately selected as $K_p=2.25\times {10}^3$, $K_i=1.25\times {10}^2$, and $K_d=11.5$. The larger gains would cause the closed-loop system to be unstable due to measurement noise and unmodeled dynamics.

(3)

PIDVFF [6]: Velocity feed-forward-based proportional-derivative-integral controller (PIDVFF), whose PID gains are chosen as the same as the above PID controller, and the velocity feed-forward coefficient is selected as $K_v=1.15\times {10}^2$.

(4)

STW [41]: Super-twisting-based controller, whose structure is designed as

$$\left\{\begin{array}{c}u=l_1{\left|x_1-x_{1d}\right|}^{1/2}\mathrm{sign}(x_1-x_{1d})+v\hfill \\ \dot{v}=l_2\mathrm{sign}(x_1-x_{1d})\hfill \end{array}\right.$$

(31)

where $l_1=4.58\times {10}^2$ and $l_2=15$.

Remark 5.

For a fair comparison, controller gains of the compared model-free controllers are first selected to guarantee that the closed-loop system is stable. As far as we know, the PID controller can be considered as the most popular controller in real-life applications because of its simplicity and limited tuning parameters. Hence, it can be treated as a reference controller for comparison. In addition, to improve the tracking performance by using the traditional PID controller, PIDVFF was commonly used for motion tracking problems by virtue of a feed-forward term which is proportional to the derivative of the desired trajectory. Furthermore, an STW controller was proven to be an effective control algorithm for systems subject to uncertainties and disturbances [42].

Performance indexes, including the maximum, average, and standard deviation of the tracking errors, are employed to assess the tracking accuracy of each control algorithm in the steady state. Their mathematical formulas are given as follows:

(1)

Maximal tracking error is defined as

$$M_e=\underset{i=1,\dots ,N}{max}\left\{\left|e_1\left(i\right)\right|\right\}$$

(32)

where N is the number of samples that are used for evaluation.

(2)

Average tracking error is computed as

$${\mu }_e=\frac{1}{N}\sum _{i=1}^N\left|e_1\left(i\right)\right|$$

(33)

(3)

The standard deviation of the tracking errors is formulated as

$${\sigma }_e=\sqrt{\frac{1}{N}\sum _{i=1}^N{\left(\left|e_1\left(i\right)\right|-{\mu }_e\right)}^2}$$

(34)

4.2. Experiment Results

4.2.1. Slow-Motion Reference Trajectory under Light-Load Condition

To demonstrate the performance of the proposed controller, the four considered controllers were first evaluated with the slow-motion reference trajectory that is mathematically formulated as $x_{1d}\left(t\right)=30sin\left(0.1\pi t-\pi /2\right)+35\left({}^{\circ }\right)$ and a gravitational load as $m=1$ kg.

The angular positions of the hydraulic actuator and desired trajectory are illustrated in Figure 4. It can be seen from this figure that the proposed controller and high-gain model-free control approaches are able to guarantee that the system output tracks the reference trajectory at an acceptable level despite the nonlinearities and model uncertainties that naturally exist in the dynamics of the considered EHA. The tracking errors caused by the compared controllers are depicted in Figure 5. As shown, the tracking errors significantly increase when the actuator changes the motion direction due to the friction between the vanes of the actuator and its frame and the load force applied to the actuator shaft. In the transient time, because of the deviations between the actual states and their estimated values, the peaking value of the tracking error generated by the ESMOBC is remarkably higher than the others as a consequence. However, in the steady-state regime, it is worth noting that due to the highly nonlinear behaviors of the EHA and the lack of model compensation mechanisms, the model-free controllers generate considerably larger tracking errors compared to the ESMOBC controller.

To quantitatively indicate the reference-tracking capability of the compared controllers, performance indexes, including maximal tracking error, average tracking error, and standard deviation, in the steady-state regime of all four controllers are presented in

Table 2. Performance indexes in the slow-motion trajectory and low-load condition.

Controller	${\mathit{M}}_{\mathit{e}}{(}^{°})$	${\mathit{\mu }}_{\mathit{e}}{(}^{°})$	${\mathit{\sigma }}_{\mathit{e}}{(}^{°})$
PID Controller	$1.6129$	$0.8859$	$0.2336$
PIDVFF Controller	$1.3412$	$0.5819$	$0.1639$
STW Controller	$1.6223$	$0.2721$	$0.2625$
ESMOBC Controller	$0.4426$	$0.1075$	$0.1340$

. From this table, the PID controller exhibited the worst performance with the three indexes, with the maximal tracking error, average tracking error, and standard deviation as ${1.6129}^{°}$, ${0.8859}^{°}$, and ${0.2336}^{°}$, respectively. Meanwhile, by using the additional velocity feed-forward term, the tracking performance was substantially enhanced by the PIDVFF controller in all performance indexes. Compared to the traditional PID controller, the maximal tracking error, average, and standard deviation of the tracking errors were reduced to ${1.3412}^{°}$, ${0.5819}^{°}$, and ${0.1639}^{°}$ by using the PIDVFF controller. Regarding the STW controller, the convergence speed of the tracking error can be improved by the employment of the fractional-order component and the discontinuous “sign” function. Hence, compared to PID and PIDVFF controllers, although both the maximal tracking error and standard deviation performance indexes are higher, the average tracking error obtained by the STW controller is significantly smaller (${0.2721}^{°}$). Nonetheless, it should be noted that with the help of compensation mechanisms by using the ESMO, the proposed method presented the best performance in terms of all performance indexes, with the maximal, average, and standard deviation of the tracking errors being ${0.4426}^{°}$, ${0.1075}^{°}$, and ${0.1340}^{°}$, respectively.

4.2.2. Slow-Motion Reference Trajectory under Heavy-Load Condition

For evaluating the tracking performance of all considered controllers under the heavy-load condition, a gravitational load $m=21\mathrm{kg}$ is applied to the actuator with the same reference trajectory as the previous test case. The tracking errors of all controllers are displayed in Figure 6 and the three performance indexes during the last cycle are presented in

Table 3. Performance indexes in the slow-motion trajectory under heavy-load condition.

Controller	${\mathit{M}}_{\mathit{e}}{(}^{\circ })$	${\mathit{\mu }}_{\mathit{e}}{(}^{\circ })$	${\mathit{\sigma }}_{\mathit{e}}{(}^{\circ })$
PID Controller	$1.8048$	$0.9508$	$0.3183$
PIDVFF Controller	$1.5000$	$0.6471$	$0.2392$
STW Controller	$2.1260$	$0.3496$	$0.3381$
ESMOBC Controller	$0.6899$	$0.1254$	$0.0978$

. It can be seen from Figure 6, similar to the above test case, according to the inherent model compensation feature of the backstepping technique with the support of the proposed ESMO, that the ESMOBC controller still performed better, with smaller tracking errors compared to the PID, PIDVFF, and STW controllers. From Table 3, it is worth noting that the tracking performance indexes slightly increase in comparison with these values in the light-load condition, which indicates the robustness of all four controllers against heavy-load conditions. In particular, the absolute maximal tracking errors derived by the PID, PIDVFF, STW, and ESMOBC controllers increased by ${0.1919}^{°}$, ${0.1588}^{°}$, ${0.5037}^{°}$, and ${0.2473}^{°}$, respectively. In addition, the smaller values of the remaining performance indexes belong to the proposed control algorithm. It indicates the advantages of the proposed method compared to the model-free controllers.

4.2.3. Fast-Motion Reference Trajectory under Light-Load Condition

In this case study, the four considered controllers were further tested with a fast-motion reference trajectory that is analytically represented as $x_{1d}\left(t\right)=20sin\left(\pi t-\pi /2\right)+25\left({}^{\circ }\right)$ under the light-load condition $m=1\mathrm{kg}$. The tracking errors under the compared controllers are illustrated in Figure 7. As shown in this figure, the tracking performances obtained by the four controllers were significantly degraded compared to the previous case studies under high-speed reference trajectory. The three performance indexes of the controllers are presented in

Table 4. Performance indexes in the slow-motion trajectory and low-load condition.

Controller	${\mathit{M}}_{\mathit{e}}{(}^{\circ })$	${\mathit{\mu }}_{\mathit{e}}{(}^{\circ })$	${\mathit{\sigma }}_{\mathit{e}}{(}^{\circ })$
PID Controller	$3.4388$	$1.7721$	$0.7923$
PIDVFF Controller	$2.4396$	$1.1953$	$0.6883$
STW Controller	$4.6298$	$2.6314$	$1.4136$
ESMOBC Controller	$1.5766$	$0.4831$	$0.2977$

. It can be seen that, similar to the above scenarios, compared to the other control approaches, the STW controller exhibited the worst tracking performance in all three performance indexes, with the maximal, average, and standard deviation of the tracking errors in the steady state as ${4.6298}^{\circ }$, ${2.6314}^{\circ }$, and ${1.4136}^{\circ }$, respectively. For the time being, the proposed control approach achieved better performance, with the corresponding performance indexes as ${1.5766}^{\circ }$, ${0.4831}^{\circ }$, and ${0.2977}^{\circ }$ compared to other controllers.

The estimated values of the angular speed of the actuator’s shaft, load pressure-related term, and lumped disturbance in the pressure dynamics are illustrated in Figure 8, Figure 9 and Figure 10, respectively. By using the proposed ESMO, the output feedback control based on the backstepping control technique can be realized in the absence of speed and pressure sensors. In addition, the ESMO plays an important role in reducing the system cost and improving the tracking performance of the EHA, since it is able to estimate not only immeasurable system states but also disturbance caused by parameter deviations and unmodeled dynamics.

The control input signal of the recommended control approach is presented in Figure 11. As shown, a relatively smooth control input was obtained by using the ESMOBC. Due to the influence of the gravitational load, a higher control effort has to be generated when lifting the load, and vice versa. To track the sinusoidal-like reference trajectory, a periodical control input signal was constructed with the same frequency as the reference trajectory. The fluctuation of the control input in case of motion direction change compensates for the effects of complicated friction characteristics inside the actuator and guarantees a high-accuracy tracking performance.

Based on the obtained experimental results in distinct working conditions, the proposed control approach was capable of achieving better tracking performance in comparison with some reference controllers. Hence, with the simple control structure, the suggested control approach provided a valuable solution to the tracking problem of the EHAs in practical applications for enhancing control accuracy. However, when designing an observer, it is essential to pick the switching gain carefully to prevent severe chattering that may arise from the selection of an excessive value. Furthermore, the observer bandwidth should be designed to make a trade-off between convergence time and the occurrence of the peaking phenomenon. Finally, advanced control algorithms will be further studied to increase the robustness and tracking performance of electro-hydrostatic actuators in future work.

5. Conclusions

This paper presented a novel output feedback control of a pump-controlled EHA subject to model uncertainties. An ESMO was developed to estimate not only immeasurable states but also lumped-matched disturbance attributed to parametric uncertainties and unmodeled dynamics in the pressure dynamics of the actuator. A backstepping controller based on the DSC technique employing estimated values of states and lumped disturbance was designed to stabilize the closed-loop system and ensure a UUB tracking performance. Furthermore, the stability of the overall closed-loop system was rigorously demonstrated through the Lyapunov theory. Finally, experimental results under various working conditions in comparison with some reference model-free control approaches are obtained to demonstrate the superiority of the proposed method. The dynamics of the pump and electric motor system will be carefully considered in future work to improve the tracking capability of the proposed method.