Output Feedback Robust Tracking Control for a Variable-Speed Pump-Controlled Hydraulic System Subject to Mismatched Uncertainties

Abstract

In this paper, a novel simple, but effective output feedback robust control (OFRC) for achieving a highly accurate position tracking of a pump-controlled electro-hydraulic system is presented. To cope with the unavailability of all system state information, an extended state observer (ESO) was adopted to estimate the angular velocity and load-pressure-related state variable of the actuator and total matched disturbance, which enters the system through the same channel as the control input in the system dynamics. In addition, for the first time, another ESO acting as a disturbance observer (DOB) was skillfully integrated to effectively compensate for the adverse effects of the lumped mismatched uncertainty caused by parameter perturbation and external loads in the velocity dynamics. Then, a dynamic surface-control-based backstepping controller (DSC-BC) based on the constructed ESOs for the tracking control of the studied electro-hydraulic system was synthesized to guarantee that the system output closely tracks the desired trajectory and avoid the inherent computational burden of the conventional backstepping method because of repetitive analytical derivative calculation at each backstepping iteration. Furthermore, the stability of the two observes and overall closed-loop system was verified by using the Lyapunov theory. Finally, several extensive comparative experiments were carried out to demonstrate the advantage of the recommended control approach in comparison with some reference control methods.

1. Introduction

Due to some excellent characteristics such as high power density, great reliability, excellent durability, and high-accuracy control capability [1,2,3], electro-hydraulic systems (EHSs) are broadly employed in a variety of industrial and engineering applications including construction machinery, aerospace, manufacturing, agriculture, and so on [4]. According to the utilization of the control element, EHSs are classified into two main groups, which are valve-controlled and pump-controlled EHSs. With regard to the tracking control aspect, valve-controlled EHSs can be considered as a remarkable solution for high-accuracy tracking control applications; however, they are not suitable for applications that require high energy efficiency because of throttling loss at the control valves [5]. Meanwhile, variable-speed pump-controlled hydraulic systems (VSPHSs) are able to get rid of this energy dissipation and, consequently, improve the overall system efficiency [5,6]. Nonetheless, it is challenging to achieve high-precision motion control of the actuator owing to the high-order dynamics, highly nonlinear characteristics, uncertain nonlinearities, and external disturbances of the system dynamics.

Although the closed-loop system stability cannot be theoretically confirmed, the proportional–derivative–integral (PID) control, fuzzy control, and fuzzy PID hybrid control schemes are control methods that are favorable and widely adopted for nonlinear EHSs in various applications due to their simplicity in implementation without the requirement of a system model and the limited number of control parameters that need to be adjusted. For example, a self-tuning grey predictor fuzzy-PID controller [7] was developed for force control of a hydraulic load simulator. The obtained results showed that enhanced force regulation performance in both the transient and steady-state regimes was achieved in comparison with the conventional PID, fuzzy PID, and fixed-step grey predictor fuzzy PID. Besides, in [8], a fractional-order PID whose control gains were automatically tuned by fuzzy rules was developed to significantly improve the tracking performance of a VSPHS compared to the traditional PID controller. Additionally, how to obtain optimal PID gains is also an intriguing problem that has attracted great attention from the research community, recently. For instance, an improved particle swam optimization (PSO)-PID was introduced in [9] to increase the convergence speed of the basic PSO algorithm. Distinct optimization algorithms such as the beetle antennae search (BAS) algorithm [10], genetic algorithm (GA) [11], improved differential evolution (IDE) [12], etc., have also been utilized to enhance the tracking performance of EHSs. Nonetheless, it is noteworthy that due to the lack of dynamic model and disturbance compensation mechanisms, it is challenging to attain satisfactory control performance using such model-free control approaches. Therefore, model-based control strategies including sliding mode control [13,14], the backstepping approach [15,16,17], and feedback linearization control [18] can be considered as powerful tools to enhance the control performance of VSPHSs, which have been introduced in various previous studies [4,19,20,21]. Among them, the backstepping control technique is the most-appropriate to design a control strategy for high-order nonlinear systems in general and EHSs specifically based on its recursive structure. Nonetheless, this technique suffers from the problem of computational complexity as a consequence of the requirement of repetitive analytic derivative calculation of virtual control laws at each subsequent step. To overcome this obstacle, the dynamic surface control (DSC) concept [22] was developed by D. Swaroop et al. The key idea of this technique is using a set of low-pass filters for the virtual control laws to obtain their first-order derivatives. Hence, the “explosion of complexity” of the traditional backstepping is effectively avoided. However, it should be noted that the above model-based control algorithms require information on the system states, which may not always be readily accessible in practical applications.

The utilization of state estimators that are based on the system model is the key technique to implement state feedback control algorithms for nonlinear systems in the case of lacking system state information. The first concept of a state observer was developed by Luenberger [23] to construct the missing internal states of a linear time-invariant system. Subsequently, the extended Luenberger observer [24] for nonlinear systems was introduced by M. Zeitz, in which the relationships for a state transformation into the nonlinear observer canonical form were developed. However, the Luenberger observers may not be sufficiently accurate in the presence of model uncertainties. To overcome this problem, sliding mode observers can be considered as a robust state estimator based on the equivalent theory for reconstructing unmeasurable states in the presence of model perturbation [25], which have been applied to various applications including electro-hydraulic systems [26,27], electric motor drivers [28,29], wind turbine systems [30,31], and so on. However, because of the employment of the discontinuous term, the so-called “sign” function and, consequently, the chattering in the estimated values might be unavoidable. It is worth noting that the above state observers are only able to reconstruct the system states and their performance mainly depends on the accuracy of the identified system model. Then, the concept of the active disturbance rejection control (ADRC) paradigm [32], which relaxes the demand of the precise system model and requires the system order only, was introduced by J. Han. In this concept, an extended state observer (ESO) is a core component for state and disturbance estimation in the case of insufficient information on the system dynamics. Inspired by this research, several ESO-based control algorithms have been developed for numerous practical systems [20,33,34,35,36,37,38]. For instance, in [20], an ESO-based sliding mode control for an electro-hydraulic actuator was constructed based on the acceleration model. In [5], B. Helian et al. proposed an adaptive robust controller for the motion control of a directly driven EHS, which is able to cope with dynamic nonlinearities and uncertainties, and improved tracking performance was accordingly achieved. Nonetheless, some sensors must be utilized to directly measure internal system states including velocity and pressure, and the system cost increases as a result. In [38], an output feedback control scheme that employs a single ESO was presented. The mechanism was able to effectively estimate and compensate for matched disturbances in a feedforward manner; however, a negligible improvement of the control performance was achieved compared to a traditional PI controller since the mismatched uncertainties were not considered. However, the implementation of the output feedback motion control for VSPHSs using ESOs to handle both mismatched and matched uncertainties is an intriguing and unexplored control problem.

Motivated by the above analysis, in this paper, a novel output feedback robust tracking control for a pump-controlled hydraulic system in the presence of disturbances and uncertainties caused by the parameter deviations, unmodeled dynamics, and external load in a mechanical system and pressure dynamics was developed. The main contributions of this work are as follows:

• For the first time, two ESOs were sophisticatedly coordinated to address the shortage of system state measurement mechanisms and the estimation of both matched and mismatched uncertainties in the studied pump-controlled hydraulic system. This combination serves as the basis for the implementation of the robust control algorithm, which merely requires the system output information.
• Based on the nominal system parameters and information provided by the above ESOs, a novel output feedback robust control mechanism in which the uncertainties and disturbances estimated by the ESOs are feedforward compensated was constructed to guarantee a high-accuracy motion control performance in spite of the nonlinearities and uncertainties in the system dynamics.
• The theoretical stability of the ESOs and overall closed-loop system was rigorously confirmed using Lyapunov’s theory. Several comparative experiment results are given to verify the effectiveness and superiority of the proposed control methodology over the existing popular control algorithms under different working conditions.

The rest of this article is structured as follows. The overall system dynamics and control problem are provided in Section 2. According to this, the establishment of two ESOs and a control algorithm is meticulously presented in Section 3. Subsequently, Section 4 presents the experimental verification. Finally, a conclusion is given in Section 5.

2. System Modeling and Problem Formulation

The schematic of the considered pump-controlled hydraulic system is illustrated in Figure 1. A bidirectional hydraulic pump controlled by an electric motor was used to manipulate the motion of the hydraulic rotary actuator (HRA). To ensure that the system pressure does not surmount the maximal predefined pressure, two relief valves ${\mathrm{V}}_3$ and ${\mathrm{V}}_4$ were utilized. The two pilot-operated check valves ${\mathrm{V}}_1$ and ${\mathrm{V}}_2$ were employed to regulate the flow of the hydraulic oil into and out of the actuator. Meanwhile, the general check valves ${\mathrm{V}}_5$ and ${\mathrm{V}}_6$ guaranteed that there was no flow from the system to the tank. A gravitational load rigidly connected to the actuator shaft through a drum and pulley system was intentionally added to evaluate the system performance.

Remark 1. 

According to the schematic of the studied EHS, it should be pointed out that, in the normal working condition, the pilot-operated check valves ${\mathit{V}}_1$ and ${\mathit{V}}_2$ allow the flow from the hydraulic pump to a chamber of the HRA and from the remaining chamber to the suction of the hydraulic pump for lifting or lowering the gravitational load. In the non-working state, i.e., the hydraulic pump does not run, no flow is supplied by the pump to the hydraulic components; therefore, the pilot-operated check valves ${\mathit{V}}_1$ and ${\mathit{V}}_2$ are completely blocked to securely lock the load at the current position and guarantee that there is no adverse impact on the hydraulic pump.

According to Newton’s second law, the motion dynamics of the hydraulic actuator are derived by

$$J_A\ddot{\phi }=D_A\left(P_1-P_2\right)-B\dot{\phi }-A_f{S}_f\left(\dot{\phi }\right)-{\tau }_l$$

(1)

where $J_A$ denotes the total inertial moment of the actuator shaft; $\dot{\phi }$ and $\ddot{\phi }$ are the angular speed and acceleration of the hydraulic actuator, respectively; $D_A$ signifies the displacement of the actuator; B signifies the total viscous friction coefficient; $A_f$ presents the Coulomb friction coefficient; $S_f\left(\dot{\phi }\right)$ denotes the continuous approximation of the standard signum function; $P_1$ and $P_2$ are the pressures in the left and the right chambers of the actuator; ${\tau }_l$ reflects the total torque disturbance caused by the modeling errors and unknown external load.

The pressure dynamics of the two chambers of the hydraulic actuator were determined as in the following equations:

$$\begin{array}{cc}\hfill & {\dot{P}}_1=\frac{{\beta }_e}{V_{01}+D_A\phi }\left(Q_1-D_A\dot{\phi }-C_{At}\left(P_1-P_2\right)\right)\hfill \\ \hfill & {\dot{P}}_2=\frac{{\beta }_e}{V_{02}-D_A\phi }\left(Q_2+D_A\dot{\phi }+C_{At}\left(P_1-P_2\right)\right)\hfill \end{array}$$

(2)

where ${\beta }_e$ is the bulk modulus of the hydraulic oil, $V_{01}$ and $V_{02}$ denote the initial volumes of the two chambers of the actuator including the volume of the pipeline, $\phi$ is the angular position of the actuator shaft, $C_{At}$ is the leakage coefficient of the actuator, and $Q_1$ and $Q_2$ are the flow rates into the hydraulic actuator chambers, which are determined by

$$\left\{\begin{array}{c}{Q}_1=Q_{P1}+Q_{v5}-Q_{v3}\hfill \\ Q_2=Q_{P2}+Q_{v6}-Q_{v4}\hfill \end{array}\right.$$

(3)

The flow rates supplied to the chambers of the hydraulic actuator are presented as

$$\left\{\begin{array}{c}{Q}_{P1}=D_P{\omega }_P-C_{Pt}\left(P_1-P_2\right)\hfill \\ Q_{P2}=-D_P{\omega }_P+C_{Pt}\left(P_1-P_2\right)\hfill \end{array}\right.$$

(4)

where $D_P$ is the pump displacement, ${\omega }_P$ denotes the speed of the pump, and $C_{Pt}$ signifies the total leakage coefficient of the pump.

Assuming that the system pressure does not surmount the setting pressure of the relief valve, hence, we have

$$\begin{array}{cc}\hfill & Q_{v_3}=0\hfill \\ \hfill & Q_{v_4}=0\hfill \end{array}$$

(5)

Combining (2), (3), (4), and (5) yields

$$\begin{array}{cc}\hfill & {\dot{P}}_1=\frac{{\beta }_e}{V_{01}+A\phi }\left(D_P{\omega }_P-C_t\left(P_1-P_2\right)+Q_{v5}-A\dot{\phi }\right)\hfill \\ \hfill & {\dot{P}}_2=\frac{{\beta }_e}{V_{02}-A\phi }\left(-D_P{\omega }_P+C_t\left(P_1-P_2\right)+Q_{v6}+A\dot{\phi }\right)\hfill \end{array}$$

(6)

where $C_t=C_{At}+C_{Pt}$ is the total equivalent leakage coefficient.

Remark 2. 

Since the system parameters $J_A$, $D_A$, B, $A_f$, ${\beta }_e$, $V_{01}$, $V_{02}$, $D_P$, and $C_t$ cannot be exactly known, their nominal values ${\overline{J}}_A$, ${\overline{D}}_A$, $\overline{B}$, ${\overline{A}}_f$, ${\overline{\beta }}_e$, ${\overline{V}}_{01}$, ${\overline{V}}_{02}$, ${\overline{D}}_P$, and ${\overline{C}}_t$, respectively, were experimental identified, which were adopted to design the observers and control law in the following sections. The effects of parameter deviations, external load, and unmodeled dynamics were lumped into total mismatched and matched disturbances.

Defining $\mathbf{x}={\left[x_1,x_2,x_3\right]}^T={\left[\phi ,\dot{\phi },{\overline{D}}_A/{\overline{J}}_A(P_1-P_2)\right]}^T$, according to (1) and (6), the overall system dynamics can be derived in the strict feedback form as

$$\begin{array}{cc}\hfill & {\dot{x}}_1=x_2\hfill \\ \hfill & {\dot{x}}_2=x_3+f_2\left(x_2\right)+d_1\left(t\right)\hfill \\ \hfill & {\dot{x}}_3=g_3\left(x_1\right)u+f_3\left(x_2,x_3\right)+d_2\left(t\right)\hfill \end{array}$$

(7)

where $u={\omega }_P$ is the control input, and the dynamical functions are given by

$$\begin{array}{cc}\hfill & f_2\left(x_2\right)=-\frac{\overline{B}}{{\overline{J}}_A}{x}_2-\frac{{\overline{A}}_f{S}_f\left(x_2\right)}{{\overline{J}}_A};d_1\left(t\right)=-\frac{{\tau }_l}{{\overline{J}}_A}+\Delta f_2\left(x_2\right)\hfill \\ \hfill & f_3(x_2,x_3)=-\left(\frac{{\overline{\beta }}_e}{{\overline{V}}_{01}+{\overline{D}}_A{x}_1}+\frac{{\overline{\beta }}_e}{{\overline{V}}_{02}-{\overline{D}}_A{x}_1}\right)\left({\overline{C}}_t{x}_3+\frac{{\overline{D}}_A^2}{{\overline{J}}_A}{x}_2\right)\hfill \\ \hfill & g_3\left(x_1\right)=\left(\frac{{\overline{\beta }}_e}{{\overline{V}}_{01}+{\overline{D}}_A{x}_1}+\frac{{\overline{\beta }}_e}{{\overline{V}}_{02}-{\overline{D}}_A{x}_1}\right)\frac{{\overline{D}}_A{\overline{D}}_P}{{\overline{J}}_A}\hfill \\ \hfill & d_2\left(t\right)=\left(\frac{{\overline{\beta }}_e}{{\overline{V}}_{01}+{\overline{D}}_A{x}_1}{Q}_{v5}-\frac{{\overline{\beta }}_e}{{\overline{V}}_{02}-{\overline{D}}_A{x}_1}{Q}_{v6}\right)\frac{{\overline{D}}_A}{{\overline{J}}_A}+\Delta g_3\left(x_1\right)u+\Delta f_3(x_2,x_3)\hfill \end{array}$$

(8)

where $\Delta f_2\left(x_2\right)$, $\Delta f_3(x_2,x_3)$, and $\Delta g_3\left(x_1\right)$ are uncertain functions due to the parameter perturbation and modeling errors.

Remark 3. 

Considering the system dynamics (7), since the nominal system parameters were assume to be known, the dynamical functions $f_2\left(x_2\right)$, $f_3(x_2,x_3)$, and $g_3\left(x_1\right)$ can be defined according to system states. The influences of parametric uncertainties, unmodeled dynamics, and external disturbances on the mechanical and pressure dynamics were treated as the lumped terms $d_1\left(t\right)$ and $d_2\left(t\right)$, the so-called lumped mismatched and matched disturbances, respectively, which were to be online estimated and then adequately compensated in the control laws.

The main aim of this paper was to propose a model-based control strategy that can guarantee that the system output $x_1$ tracks the desired trajectory $x_{1d}$ as closely as possible in the case that only the angular position of the actuator is measurable and in the presence of both mismatched and matched uncertainties. To facilitate the control design, some practical assumptions are made as follows:

Assumption 1. 

The desired position trajectory $x_{1d}$ is sufficiently smooth up to the third-order derivative, that is $x_{1d}\in C^3$.

Assumption 2. 

The function $f_2\left(x_2\right)$ is globally Lipschitz with respect to $x_2$ and $f_3(x_2,x_3)$ is Lipschitz with respect to $x_2$ and $x_3$.

Assumption 3. 

The lumped disturbances $d_1\left(t\right)$ and $d_2\left(t\right)$ were assumed to be differentiable, and their first-order derivatives $h_1\left(t\right)$ and $h_2\left(t\right)$, respectively, were bounded by unknown positive constants, i.e., $\left|h_1\left(t\right)\right|\le {\overline{h}}_1$ and $\left|h_2\left(t\right)\right|\le {\overline{h}}_2$. In addition, the lumped mismatched disturbance $d_1\left(t\right)$ is also supposed to be bounded by an unknown nonnegative constant, that is $\left|d_1\left(t\right)\right|\le {\overline{d}}_1$.

Remark 4. 

It should be admitted that the torque generation ability of a hydraulic system is restricted in a specific range because of the maximal working pressure that the pumping system can provide and the safety constraints. The load-carrying capacity of it is limited as a result. Hence, the assumption that the lumped mismatched uncertainty is bounded is practically reasonable.

3. Control System Design

The control structure of the proposed method is illustrated in Figure 2. As shown, an ESO is established to reconstruct not only the internal states, including the angular velocity and state variable relating to load pressure, but also the total matched disturbances resulting from modeling errors and parametric uncertainties. It plays a crucial role in implementing the output feedback control mechanism for accurate motion tracking of the studied VSPHS. In addition, to improve the control performance, a mismatched disturbance observer that estimates the lumped mismatched disturbances produced by uncertain nonlinearities and unknown external loads, was constructed. Finally, to guarantee high-accuracy tracking performance, a DSC-based backstepping controller was designed. The DSC technique was adopted to overcome the computational burden of traditional backstepping, which requires the repetitive analytical derivative calculation of the virtual control laws. The design of the above observers and control strategy will be meticulously presented in the subsequent sections.

3.1. Extended State Observer Design

Considering the system dynamics (7) and defining $x_{e1}\stackrel{\Delta }{=}{d}_2\left(t\right)$ as an extended state, the augmented system dynamics can be obtained as

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2\hfill \\ {\dot{x}}_2=x_3+f_2\left(x_2\right)+d_1\left(t\right)\hfill \\ {\dot{x}}_3=g_3\left(x_1\right)u+f_3\left(x_2,x_3\right)+x_{e1}\hfill \\ {\dot{x}}_{e1}=h_2\left(t\right)\hfill \end{array}\right.$$

(9)

According to this, the ESO for estimating not only the internal states $x_2$ and $x_3$, but also the lumped matched disturbance is constructed as follows:

$$\left\{\begin{array}{c}{\dot{\widehat{x}}}_1={\widehat{x}}_2+4{\omega }_1{\tilde{x}}_1\hfill \\ {\dot{\widehat{x}}}_2={\widehat{x}}_3+f_2\left({\widehat{x}}_2\right)+6{\omega }_1^2{\tilde{x}}_1\hfill \\ {\dot{\widehat{x}}}_3=g_3\left(x_1\right)u+f_3\left({\widehat{x}}_2,{\widehat{x}}_3\right)+{\widehat{x}}_{e1}+4{\omega }_1^3{\tilde{x}}_1\hfill \\ {\dot{\widehat{x}}}_{e1}={\omega }_1^4{\tilde{x}}_1\hfill \end{array}\right.$$

(10)

where ${\tilde{x}}_1$ denotes the error between the measured and estimated values of the system output, i.e., ${\tilde{x}}_1=x_1-{\widehat{x}}_1$; ${\widehat{x}}_2$, ${\widehat{x}}_3$, and ${\widehat{x}}_{e1}$ are the estimates of $x_2$, $x_3$, and $x_{e1}$, respectively; ${\omega }_1$ signifies the bandwidth of the observer (10).

Combining (9) and (10), the error dynamics of the observer are derived as

$$\left\{\begin{array}{c}{\dot{\tilde{x}}}_1=-4{\omega }_1{\tilde{x}}_1+{\tilde{x}}_2\hfill \\ {\dot{\tilde{x}}}_2=-6{\omega }_1^2{\tilde{x}}_1+{\tilde{x}}_3+{\tilde{f}}_2\left(x_2,{\widehat{x}}_2\right)+d_1\left(t\right)\hfill \\ {\dot{\tilde{x}}}_3=-4{\omega }_1^3{\tilde{x}}_1+{\tilde{x}}_{e1}+{\tilde{f}}_3\left(x_2,x_3,{\widehat{x}}_2,{\widehat{x}}_3\right)\hfill \\ {\dot{\tilde{x}}}_{e1}=-{\omega }_1^4{\tilde{x}}_1+h_2\left(t\right)\hfill \end{array}\right.$$

(11)

where ${\tilde{x}}_2$, ${\tilde{x}}_3$, and ${\tilde{x}}_{e1}$ are the estimation errors of $x_2$, $x_3$, and $x_{e1}$, respectively, i.e., ${\tilde{x}}_2=x_2-{\widehat{x}}_2$, ${\tilde{x}}_3=x_3-{\widehat{x}}_3$, and ${\tilde{x}}_2=x_{e1}-{\widehat{x}}_{e1}$. Error functions ${\tilde{f}}_2(x_2,{\widehat{x}}_2)$ and ${\tilde{f}}_3(x_2,x_3,{\widehat{x}}_2,{\widehat{x}}_3)$ are given by

$$\begin{array}{cc}\hfill & {\tilde{f}}_2\left(x_2,{\widehat{x}}_2\right)=f_2\left(x_2\right)-f_2\left({\widehat{x}}_2\right)\hfill \\ \hfill & {\tilde{f}}_3\left(x_2,x_3,{\widehat{x}}_2,{\widehat{x}}_3\right)=f_3\left(x_2,x_3\right)-f_3\left({\widehat{x}}_2,{\widehat{x}}_3\right)\hfill \end{array}$$

(12)

According to Assumption 2, one obtains

$$\begin{array}{cc}\hfill & \left|{\tilde{f}}_2\left(x_2,{\widehat{x}}_2\right)\right|\le L_{f_2}\left|{\tilde{x}}_2\right|\hfill \\ \hfill & \left|{\tilde{f}}_3\left(x_2,x_3,{\widehat{x}}_2,{\widehat{x}}_3\right)\right|\le L_{f_{31}}\left|{\tilde{x}}_2\right|+L_{f_{32}}\left|{\tilde{x}}_3\right|\hfill \end{array}$$

(13)

where $L_{f_2}$, $L_{f_{31}}$, and $L_{f_{32}}$ are known Lipschitz constants.

Let $\mathbf{\epsilon }={[{\epsilon }_1,{\epsilon }_2,{\epsilon }_3,{\epsilon }_4]}^T={[{\tilde{x}}_1,{\tilde{x}}_2/{\omega }_1,{\tilde{x}}_2/{\omega }_1^2,{\tilde{x}}_{e1}/{\omega }_1^3]}^T$ denote the scaled estimation error vector, then the error dynamics (11) can be transformed into

$$\dot{\mathit{\epsilon }}={\omega }_1{\mathit{A}}_1\mathbf{\epsilon }+{\mathit{B}}_1\frac{{\tilde{f}}_2\left(x_2,{\widehat{x}}_2\right)+d_1\left(t\right)}{{\omega }_1}+{\mathit{C}}_1\frac{{\tilde{f}}_3\left(x_2,x_3,{\widehat{x}}_2,{\widehat{x}}_3\right)}{{\omega }_1^2}+{\mathit{D}}_1\frac{h_2\left(t\right)}{{\omega }_1^3}$$

(14)

where the matrices ${\mathit{A}}_1$, ${\mathit{B}}_1$, ${\mathit{C}}_1$, and ${\mathit{D}}_1$ are given by

$${\mathit{A}}_1=\left[\begin{array}{cccc}-4& 1& 0& 0\\ -6& 0& 1& 0\\ -4& 0& 0& 1\\ -1& 0& 0& 0\end{array}\right];{\mathit{B}}_1=\left[\begin{array}{c}0\\ 1\\ 0\\ 0\end{array}\right];{\mathit{C}}_1=\left[\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\right];{\mathit{D}}_1=\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right]$$

(15)

Since the matrix ${\mathit{A}}_1$ is negative definite, for an arbitrary symmetric positive definite ${\mathit{Q}}_1$, there always exists a symmetric positive definite ${\mathit{P}}_1$ satisfying the following Lyapunov equation:

$${\mathit{A}}_1^T{\mathit{P}}_1+{\mathit{P}}_1{\mathit{A}}_1=-{\mathit{Q}}_1$$

(16)

Theorem 1. 

Considering the system dynamics (7) and the observer (10), the estimation errors converge to an arbitrarily small bounded region whose bound depends on the selection of the observer bandwidth ${\omega }_1$ when time goes to infinity.

Proof of Theorem 1.

Consider the following candidate Lyapunov function:

$$V_{\epsilon }={\mathbf{\epsilon }}^T{\mathit{P}}_1\mathbf{\epsilon }$$

(17)

Taking the derivative of it and combining with (14) yield

$$\begin{array}{cc}\hfill {\dot{V}}_{\epsilon }& ={\dot{\mathbf{\epsilon }}}^T{\mathit{P}}_1\mathbf{\epsilon }+{\mathbf{\epsilon }}^T{\mathit{P}}_1\dot{\mathbf{\epsilon }}\hfill \\ \hfill & =-{\omega }_1{\mathbf{\epsilon }}^T{\mathit{Q}}_1\mathbf{\epsilon }+2{\mathbf{\epsilon }}^T{\mathit{P}}_1{\mathit{B}}_1\frac{{\tilde{f}}_2\left(x_2,{\widehat{x}}_2\right)+d_1\left(t\right)}{{\omega }_1}+2{\mathbf{\epsilon }}^T{\mathit{P}}_1{\mathit{C}}_1\frac{{\tilde{f}}_3\left(x_2,x_3,{\widehat{x}}_2,{\widehat{x}}_3\right)}{{\omega }_1^2}+2{\mathbf{\epsilon }}^T{\mathit{P}}_1{\mathit{D}}_1\frac{h_2\left(t\right)}{{\omega }_1^3}\hfill \\ \hfill & \le -{\omega }_1{\mathbf{\epsilon }}^T{\mathit{Q}}_1\mathbf{\epsilon }+2{\mathbf{\epsilon }}^T{\mathit{P}}_1{\mathit{B}}_1\frac{L_{f2}\left|{\tilde{x}}_2\right|+\left|d_1\left(t\right)\right|}{{\omega }_1}+2{\mathbf{\epsilon }}^T{\mathit{P}}_1{\mathit{C}}_1\frac{L_{f_{31}}\left|{\tilde{x}}_2\right|+L_{f_{32}}\left|{\tilde{x}}_3\right|}{{\omega }_1^2}+2{\mathbf{\epsilon }}^T{\mathit{P}}_1{\mathit{D}}_1\frac{\left|h_2\left(t\right)\right|}{{\omega }_1^3}\hfill \end{array}$$

(18)

Applying Young’s inequality leads to

$$\begin{array}{cc}\hfill {\dot{V}}_{\epsilon }& \le -\left({\omega }_1{\lambda }_{min}\left\{{\mathit{Q}}_1\right\}-4-\frac{{\mathit{B}}_1^T{\mathit{P}}_1^T{\mathit{P}}_1{\mathit{B}}_1{L}_{f2}^2}{{\omega }_1^2}-{\mathit{C}}_1^T{\mathit{P}}_1^T{\mathit{P}}_1{\mathit{C}}_1\frac{\left(L_{f31}^2+L_{f32}^2\right)}{{\omega }_1^4}\right){∥\mathbf{\epsilon }∥}^2\hfill \\ \hfill & +{\mathit{B}}_1^T{\mathit{P}}_1^T{\mathit{P}}_1{\mathit{B}}_1\frac{d_1^2\left(t\right)}{{\omega }_1^2}+{\mathit{D}}_1^T{\mathit{P}}_1^T{\mathit{P}}_1{\mathit{D}}_1\frac{h_2^2\left(t\right)}{{\omega }_1^6}\hfill \\ \hfill & \le -\left({\omega }_1{\lambda }_{min}\left\{{\mathit{Q}}_1\right\}-4-\frac{{\mathit{B}}_1^T{\mathit{P}}_1^T{\mathit{P}}_1{\mathit{B}}_1{L}_{f2}^2}{{\omega }_1^2}-{\mathit{C}}_1^T{\mathit{P}}_1^T{\mathit{P}}_1{\mathit{C}}_1\frac{\left(L_{f31}^2+L_{f32}^2\right)}{{\omega }_1^4}\right){∥\mathbf{\epsilon }∥}^2\hfill \\ \hfill & +{\mathit{B}}_1^T{\mathit{P}}_1^T{\mathit{P}}_1{\mathit{B}}_1\frac{{\overline{d}}_1^2}{{\omega }_1^2}+{\mathit{D}}_1^T{\mathit{P}}_1^T{\mathit{P}}_1{\mathit{D}}_1\frac{{\overline{h}}_2^2}{{\omega }_1^6}\hfill \end{array}$$

(19)

where ${\lambda }_{\min }\left\{\mathit{X}\right\}$ and ${\mathit{\lambda }}_{\max }\left\{\mathit{X}\right\}$ are the maximal and minimal eigenvalues of the matrix $\mathit{X}$.

According to (17), one attains

$${\lambda }_{min}\left\{\mathit{P}\right\}{∥\mathbf{\epsilon }∥}^2\le V_{\epsilon }\le {\lambda }_{max}\left\{\mathit{P}\right\}{∥\mathbf{\epsilon }∥}^2$$

(20)

Based on this, (19) can be rewritten as

$${\dot{V}}_{\epsilon }\le -{\Gamma }_{\epsilon }{V}_{\epsilon }+{\mathrm{\Pi }}_{\epsilon }$$

(21)

where

$$\begin{array}{cc}\hfill & {\Gamma }_{\epsilon }=\frac{1}{{\lambda }_{max}\left\{{\mathit{P}}_1\right\}}\left({\omega }_1{\lambda }_{min}\left\{{\mathit{Q}}_1\right\}-4-\frac{{\mathit{B}}_1^T{\mathit{P}}_1^T{\mathit{P}}_1{\mathit{B}}_1{L}_{f2}^2}{{\omega }_1^2}-{\mathit{C}}_1^T{\mathit{P}}_1^T{\mathit{P}}_1{\mathit{C}}_1\frac{\left(L_{f31}^2+L_{f32}^2\right)}{{\omega }_1^4}\right)\hfill \\ \hfill & {\mathrm{\Pi }}_{\epsilon }={\mathit{B}}_1^T{\mathit{P}}_1^T{\mathit{P}}_1{\mathit{B}}_1\frac{{\overline{d}}_1^2}{{\omega }_1^2}+{\mathit{D}}_1^T{\mathit{P}}_1^T{\mathit{P}}_1{\mathit{D}}_1\frac{{\overline{h}}_2^2}{{\omega }_1^6}\hfill \end{array}$$

(22)

According to (21) and (22), it can be seen that the estimation errors are constrained in a region whose bound depends on the selection of the observer bandwidth ${\omega }_1$, which is defined by ${\mathrm{\Pi }}_{\epsilon }/{\Gamma }_{\epsilon }$ when time goes to infinity.

This completes the proof of Theorem 1. □

3.2. Mismatched Disturbance Observer

It can be seen from the above section that only lumped matched disturbance was estimated. The system performance may be seriously deteriorated owing to the inherent existence of mismatched uncertainties caused by modeling errors and external load in the velocity dynamics.

Consider the velocity dynamics of the actuator:

$${\dot{x}}_2=x_3+f_2\left(x_2\right)+d_1\left(t\right)$$

(23)

Let $x_{e2}=d_1\left(t\right)$ denote the extended state; (23) can be transformed into

$$\left\{\begin{array}{c}{\dot{x}}_2=x_3+f_2\left(x_2\right)+x_{e2}\hfill \\ {\dot{x}}_{e2}=h_1\left(t\right)\hfill \end{array}\right.$$

(24)

According to Theorem 1, the immeasurable system state $x_2$ and $x_3$ can be exactly estimated with arbitrarily small estimation errors; hence, the second derivatives of these estimation errors are also bounded by unknown positive constants. Their estimates were employed to the establish disturbance observer, whose dynamics are given as

$$\left\{\begin{array}{c}{\dot{z}}_2={\widehat{x}}_3+f_2\left({\widehat{x}}_2\right)+{\widehat{x}}_{e2}+2{\omega }_2\left({\widehat{x}}_2-z_2\right)\hfill \\ {\dot{\widehat{x}}}_{e2}={\omega }_2^2\left({\widehat{x}}_2-z_2\right)\hfill \end{array}\right.$$

(25)

where $z_2$ and ${\widehat{x}}_{e2}$ are the estimated values of ${\widehat{x}}_2$ and $x_{e2}$; ${\omega }_2$ is the observer gain to be designed.

In accordance with (24) and (25), the error dynamics can be obtained as

$$\left\{\begin{array}{c}{\dot{\tilde{z}}}_2=-2{\omega }_2{\tilde{z}}_2+{\tilde{x}}_{e2}+{\tilde{x}}_3+{\tilde{f}}_2\left(x_2,{\widehat{x}}_2\right)+{\ddot{\tilde{x}}}_2\hfill \\ {\dot{\tilde{x}}}_{e2}=-{\omega }_2^2{\tilde{z}}_2+h_1\left(t\right)\hfill \end{array}\right.$$

(26)

Defining the scaled estimation error vector $\mathbf{\eta }={[{\eta }_1,{\eta }_2]}^T={[{\tilde{z}}_2,{\tilde{x}}_{e2}]}^T$, the error dynamics (26) can be rewritten as

$$\dot{\mathbf{\eta }}={\omega }_2{\mathit{A}}_2\mathbf{\eta }+{\mathit{B}}_2\frac{{\tilde{x}}_3+{\tilde{f}}_2\left(x_2,{\widehat{x}}_2\right)+{\ddot{\tilde{x}}}_2}{{\omega }_2}+{\mathit{C}}_2\frac{h_1\left(t\right)}{{\omega }_2^2}$$

(27)

where the matrices ${\mathit{A}}_2$, ${\mathit{B}}_2$, and ${\mathit{C}}_2$ are given by

$${\mathit{A}}_2=\left[\begin{array}{cc}-2& 1\\ -1& 0\end{array}\right];{\mathit{B}}_2=\left[\begin{array}{c}1\\ 0\end{array}\right];{\mathit{C}}_2=\left[\begin{array}{c}0\\ 1\end{array}\right]$$

(28)

Since the matrix ${\mathit{A}}_2$ is Hurwitz, for a symmetric positive definite ${\mathit{Q}}_2$, there exists a symmetric positive definite ${\mathit{P}}_2$ satisfying the following Lyapunov equation:

$${\mathit{A}}_2^T{\mathit{P}}_2+{\mathit{P}}_2{\mathit{A}}_2=-{\mathit{Q}}_2$$

(29)

Theorem 2. 

Consider the system dynamics (24); the observer (25) is capable of exactly estimating the lumped mismatched uncertainty $d_1\left(t\right)$ with a small estimation error, whose value decreases if the observer bandwidth increases.

Proof  of Theorem 2.

A candidate Lyapunov function was chosen as $V_{\eta }={\mathbf{\eta }}^T{\mathit{P}}_2\mathbf{\eta }$. Taking time derivative of it and combining with (27) and (29) yield

$$\begin{array}{cc}\hfill V_{\eta }& ={\dot{\mathbf{\eta }}}^T{\mathit{P}}_2\mathbf{\eta }+{\mathbf{\eta }}^T{\mathit{P}}_2\dot{\mathbf{\eta }}\hfill \\ \hfill & =-{\omega }_2{\mathbf{\eta }}^T{\mathit{Q}}_2\mathbf{\eta }+2{\mathbf{\eta }}^T{\mathit{P}}_2{\mathit{B}}_2\frac{{\tilde{x}}_3+{\tilde{f}}_2\left(x_2,{\widehat{x}}_2\right)+{\ddot{\tilde{x}}}_2}{{\omega }_2}+2{\mathbf{\eta }}^T{\mathit{P}}_2{\mathit{C}}_2\frac{h_1\left(t\right)}{{\omega }_2^2}\hfill \end{array}$$

(30)

Applying Young’s inequality leads to

$$\begin{array}{cc}\hfill V_{\eta }& \le -\left({\omega }_2{\lambda }_{min}\left\{{\mathit{Q}}_2\right\}-3\right){∥\mathbf{\eta }∥}^2+{\mathit{B}}_2^T{\mathit{P}}_2^T{\mathit{P}}_2{\mathit{B}}_2\frac{{\tilde{x}}_3^2+L_{f_2}{\tilde{x}}_2^2+{\left|{\ddot{\tilde{x}}}_2\right|}_{max}}{{\omega }_2^2}+{\mathit{C}}_2^T{\mathit{P}}_2^T{\mathit{P}}_2{\mathit{C}}_2\frac{{\overline{h}}_1^2}{{\omega }_2^4}\hfill \\ \hfill & =-{\Gamma }_{\eta }{V}_n+{\Pi }_{\eta }\hfill \end{array}$$

(31)

where

$${\Gamma }_{\eta }=\frac{{\lambda }_{min}\left\{{\mathit{Q}}_2\right\}-3}{{\lambda }_{max}\left\{{\mathit{P}}_2\right\}};{\Pi }_{\eta }={\mathit{B}}_2^T{\mathit{P}}_2^T{\mathit{P}}_2{\mathit{B}}_2\frac{{\tilde{x}}_3^2+L_{f_2}{\tilde{x}}_2^2+{\left|{\ddot{\tilde{x}}}_2\right|}_{max}}{{\omega }_2^2}+{\mathit{C}}_2^T{\mathit{P}}_2^T{\mathit{P}}_2{\mathit{C}}_2\frac{{\overline{h}}_1^2}{{\omega }_2^4}$$

(32)

According to Theorem 1, Assumption 3, and (31), the estimation error ${\tilde{x}}_{e2}$ reaches a small region whose boundary decreases if the observer gain ${\omega }_2$ increases and vice visa.

Hence, Theorem 2 is completely verified. □

3.3. Observer-Based Control Strategy Design

Based on the states and disturbances estimation, a backstepping controller was synthesized to guarantee the high-accuracy tracking performance of the closed-loop system as follows:

Reconsider the original system dynamics:

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2\hfill \\ {\dot{x}}_2=x_3+f_2\left(x_2\right)+d_1\left(t\right)\hfill \\ {\dot{x}}_3=g_3\left(x_1\right)u+f_3\left(x_2,x_3\right)+d_2\left(t\right)\hfill \end{array}\right.$$

(33)

The system output tracking error is defined as $e_1=x_1-x_{1d}$. To ensure that the system output tracks the desired trajectory, the virtual control law is synthesized as

$${\alpha }_1={\dot{x}}_{1d}-k_1{e}_1$$

(34)

where $k_1>0$ is to be designed later.

The dynamic surface control approach was adopted to avoid repeatedly differentiating ${\alpha }_1$ by using a first-order filter as

$${\tau }_1{\dot{\alpha }}_{1f}+{\alpha }_{1f}={\alpha }_1;{\alpha }_{1f}\left(0\right)={\alpha }_1\left(0\right)$$

(35)

where ${\tau }_1$ is a positive constant and ${\alpha }_{1f}$ denotes the filtered signal of ${\alpha }_1$.

According to this, the dynamic surface function is the error between the filtered signal ${\alpha }_{1f}$ and the virtual control law (34) in this step, that is ${\rho }_1={\alpha }_{1f}-{\alpha }_1$, and the first-order derivative of the filtered control law is given as

$${\dot{\alpha }}_{1f}=-\frac{{\rho }_1}{{\tau }_1}$$

(36)

Following the structure of the applied first-order filter (34) and (35), the error dynamics can be determined as

$${\dot{\rho }}_1=-\frac{{\rho }_1}{{\tau }_1}+{ϵ}_1$$

(37)

where ${ϵ}_1=-{\dot{\alpha }}_1$ is continuous and bounded, and its maximum value is defined as ${ϵ}_{1M}$.

A candidate Lyapunov function was selected as $V_1=e_1^2/2$. Taking the derivative of it and combining with (33), one obtains

$$\begin{array}{cc}\hfill {\dot{V}}_1& =e_1{\dot{e}}_1\hfill \\ \hfill & =-k_1{e}_1^2+e_1{e}_2+{\rho }_1{e}_1\hfill \end{array}$$

(38)

where $e_2=x_2-{\alpha }_{1f}$.

Applying Young’s inequality yields

$$\begin{array}{cc}\hfill {\dot{V}}_1& \le -k_1{e}_1^2+e_1{e}_2+\frac{1}{2}{\rho }_1^2+\frac{1}{2}{e}_1^2\hfill \\ \hfill & =-\left(k_1-\frac{1}{2}\right)e_1^2+\frac{1}{2}{\rho }_1^2+e_1{e}_2\hfill \end{array}$$

(39)

From the definition of $e_2$, we have

$$\begin{array}{cc}\hfill {\dot{e}}_2& ={\dot{x}}_2-{\dot{\alpha }}_{1f}\hfill \\ \hfill & =x_3+f_2+d_1-{\dot{\alpha }}_{1f}\hfill \end{array}$$

(40)

Choose a candidate Lyapunov function as

$$V_2=V_1+\frac{1}{2}{e}_2^2+\frac{1}{2}{\rho }_1^2$$

(41)

Differentiating (41), then combining with (37) and (40), one obtains

$$\begin{array}{cc}\hfill {\dot{V}}_2& ={\dot{V}}_1+e_2{\dot{e}}_2+{\rho }_1{\dot{\rho }}_1\hfill \\ \hfill & ={\dot{V}}_1+e_2\left({\rho }_2+{\alpha }_2+e_3+f_2+d_1-{\dot{\alpha }}_{1f}\right)+{\rho }_1\left(-\frac{{\rho }_1}{{\tau }_1}+{ϵ}_1\right)\hfill \end{array}$$

(42)

where $e_3=x_3-{\alpha }_{2f}$ is the virtual tracking error of the subsequent step; the dynamic surface function of this step ${\rho }_2={\alpha }_{2f}-{\alpha }_2$ with ${\alpha }_{2f}$ as the filtered signal through a first-order filter of virtual control law ${\alpha }_2$, which is formulated as

$${\tau }_2{\dot{\alpha }}_{2f}+{\alpha }_{1f}={\alpha }_2;{\alpha }_{2f}\left(0\right)={\alpha }_2\left(0\right)$$

(43)

where ${\tau }_2$ is a small positive constant to be designed.

The dynamics of the dynamic surface function is given by

$${\dot{\rho }}_2=-\frac{{\rho }_2}{{\tau }_2}+{ϵ}_2$$

(44)

where ${ϵ}_2=-{\dot{\alpha }}_2$ is also continuous and upper bounded by ${ϵ}_{2M}>0$.

Since the angular velocity and lumped mismatched disturbance are immeasurable, the estimates of these terms were utilized; according to (42), the virtual control law is designed as

$${\alpha }_2=-f_2\left({\widehat{x}}_2\right)-{\widehat{x}}_{e2}+{\dot{\alpha }}_{1f}-k_2{e}_2-e_1$$

(45)

where $k_2$ is a non-negative constant to be selected.

Substituting (39) and (44) into (42), then applying Young’s inequality, we have

$$\begin{array}{cc}\hfill {\dot{V}}_2& \le -\left(k_1-\frac{1}{2}\right)e_1^2+\frac{1}{2}{\rho }_1^2+e_1{e}_2-k_2{e}_2^2-e_1{e}_2+e_2{\rho }_2+e_2{e}_3+e_2{\tilde{f}}_2+e_2{\tilde{x}}_{e2}+{\rho }_1\left(-\frac{{\rho }_1}{{\tau }_1}+{ϵ}_1\right)\hfill \\ \hfill & \le -\left(k_1-\frac{1}{2}\right)e_2^2-\left(k_2-\frac{3}{2}\right)e_2^2-\left(\frac{1}{{\tau }_1}-\frac{1}{2a}{ϵ}_{1M}^2-\frac{1}{2}\right){\rho }_1^2+e_2{e}_3+\frac{1}{2}{\tilde{f}}_2^2+\frac{1}{2}{\tilde{x}}_{e2}^2+\frac{1}{2}{\rho }_2^2+\frac{a}{2}\hfill \end{array}$$

(46)

where a is a positive constant, ${\tilde{f}}_2=f\left(x_2\right)-f\left({\widehat{x}}_2\right)$, and $e_3=x_3-{\alpha }_{2f}$.

According to the definition of the virtual tracking error $e_3$, one obtains

$$\begin{array}{cc}\hfill {\dot{e}}_3& ={\dot{x}}_3-{\dot{\alpha }}_{2f}\hfill \\ \hfill & =g_3\left(x_1\right)u+f_3\left(x_2,x_3\right)+d_2-{\dot{\alpha }}_{2f}\hfill \end{array}$$

(47)

A candidate Lyapunov function is selected as

$$V_3=V_2+\frac{1}{2}{e}_3^2+\frac{1}{2}{\rho }_2^2$$

(48)

Taking the derivative of it, then combining with (44) and (47) yield

$${\dot{V}}_3={\dot{V}}_2+e_3\left(g_{3}u+f_3+d_2-{\dot{\alpha }}_{2f}\right)+{\rho }_2\left(-\frac{{\rho }_2}{{\tau }_2}+{ϵ}_2\right)$$

(49)

The actual control law is constructed as

$$u=\frac{1}{g_3}\left(-{\widehat{f}}_3-{\widehat{x}}_{e1}+{\dot{\alpha }}_{2f}-k_3{e}_3-e_2\right)$$

(50)

where $k_3>0$ is to be chosen and ${\widehat{f}}_3=f_3\left({\widehat{x}}_2,{\widehat{x}}_3\right)$.

According to (49) and (50) and applying Young’s inequality, one obtains

$$\begin{array}{cc}\hfill {\dot{V}}_3& ={\dot{V}}_2+e_3\left({\tilde{f}}_3+{\tilde{x}}_{e1}-k_3{e}_3-e_2\right)+{\rho }_2\left(-\frac{{\rho }_2}{{\tau }_2}+{ϵ}_2\right)\hfill \\ \hfill & \le {\dot{V}}_2-\left(k_3-1\right)e_3^2-\left(\frac{1}{{\tau }_2}-\frac{1}{2a}{ϵ}_{2M}^2\right){\rho }_2^2-e_2{e}_3+\frac{1}{2}{\tilde{f}}_3^2+\frac{1}{2}{\tilde{x}}_{e1}^2+\frac{a}{2}\hfill \end{array}$$

(51)

Substituting (46) into (51) leads to

$$\begin{array}{cc}\hfill {\dot{V}}_3& \le -\left(k_1-\frac{1}{2}\right)e_1^2-\left(k_2-\frac{3}{2}\right)e_2^2-\left(\frac{1}{{\tau }_1}-\frac{1}{2a}{ϵ}_{1M}^2-\frac{1}{2}\right){\rho }_2^2\hfill \\ \hfill & -\left(k_3-1\right)e_3^2-\left(\frac{1}{{\tau }_2}-\frac{1}{2a}{ϵ}_{2M}^2\right){\rho }_2^2+\frac{1}{2}{\tilde{f}}_2^2+\frac{1}{2}{\tilde{f}}_3^2+\frac{1}{2}{\tilde{x}}_{e1}^2+\frac{1}{2}{\tilde{x}}_{e2}^2+a\hfill \end{array}$$

(52)

The inequality (52) can be rewritten as

$${\dot{V}}_3\le -\Gamma V_3+\Pi$$

(53)

where $\Gamma$ and $\Pi$ are determined as

$$\begin{array}{cc}\hfill & \Gamma =min\left\{2k_1-1;2k_3-2;2k_3-2;\frac{2}{{\tau }_1}-\frac{{\epsilon }_{1M}^2}{a}-1;\frac{2}{{\tau }_2}-\frac{{\epsilon }_{2M}^2}{a}\right\}\hfill \\ \hfill & \Pi =\frac{{\tilde{f}}_2^2}{2}+\frac{{\tilde{f}}_3^2}{2}+\frac{{\tilde{x}}_{e1}^2}{2}+\frac{{\tilde{x}}_{e2}^2}{2}+a\hfill \end{array}$$

(54)

Based on Theorems 1 and 2, and (53), by using the control laws (34), (45), and (50), a bounded tracking performance of the closed-loop system is guaranteed. The output tracking error reduces as the controller gains $k_1$, $k_2$, and $k_3$ increase and vice versa.

Remark 5. 

The time constants of the low-pass filters (35) and (43) should be selected sufficiently small to guarantee not only the closed-loop system dynamics, but also the small errors between the virtual control laws and their filtered signals, and consequently, the control performance can be significantly improved. Although only bounded stability can be achieved, the employment of the DSC technique effectively avoids the analytic derivative calculation of the virtual control laws in the conventional backstepping control framework; hence, the computational complexity can be substantially reduced.

4. Experiment Verification

4.1. Experiment Setup

The real apparatus that is used to study the control problem and verify the advantage of the suggested control strategy is shown in Figure 3. As shown, it consisted of a compact hydraulic power unit, a vane rotary actuator, two pressure transmitters, and an encoder. The hydraulic power pack manufactured by Bosch Rexroth includes a bidirectional gear pump that is driven by a DC motor. To control the speed of the DC motor, a medium-power motor driver MD-03 was provided by Robot Electronics company. Besides, to measure the pressure of the actuator, two pressure sensors of Model KOBOLD SEN-8700/2A095 were adopted. In addition, an encoder E40H8-5000-3-V-5 made by Autonics company was employed to measure the angular position of the actuator.

The nominal system parameters of the real VSPHS are listed in

Table 1. The nominal parameter of the studied VSPHS.

Parameter	Notation	Value	SI Unit
Viscous friction coefficient of the actuator	${\overline{B}}_f$	50	$\mathrm{N}·\mathrm{m}·{\mathrm{rad}}^{-1}·\mathrm{s}$
Coulomb friction coefficient of the actuator	${\overline{A}}_f$	45	$\mathrm{N}·\mathrm{m}$
Moment of inertia of the actuator	${\overline{J}}_A$	$0.15$	$\mathrm{kg}·{\mathrm{m}}^2$
Hydraulic actuator displacement	${\overline{D}}_A$	$5.8442\times {10}^{-6}$	${\mathrm{m}}^3·{\mathrm{rad}}^{-1}$
Hydraulic pump displacement	${\overline{D}}_P$	$0.1544\times {10}^{-7}$	${\mathrm{m}}^3·{\mathrm{rad}}^{-1}$
Effective bulk modulus of the hydraulic oil	${\overline{\beta }}_e$	$1.5\times {10}^9$	$\mathrm{N}·{\mathrm{m}}^{-2}\mathrm{or}\mathrm{Pa}$
Total leakage coefficient	${\overline{C}}_t$	$4.267\times {10}^{-12}$	${\mathrm{m}}^3·{\mathrm{s}}^{-1}·{\mathrm{Pa}}^{-1}$
Initial control volume of the forward chamber	${\overline{V}}_{01}$	$1.25\times {10}^{-5}$	${\mathrm{m}}^3$
Initial control volume of the reverse chamber	${\overline{V}}_{02}$	$2.27\times {10}^{-5}$	${\mathrm{m}}^3$

, which were used to design the observers and control laws of the proposed control method.

To demonstrate the advantage of the recommended control approach, the following controllers were compared:

(a)

DESO-OFRBC: The proposed controller whose control gains were chosen as $k_1=80$, $k_2=15$, $k_3=30$. In addition, the bandwidths of the observers (10) and (24) were ${\omega }_1=50$ and ${\omega }_2=50$, respectively.

(b)

SESO-OFRBC: The output feedback control using a single ESO, which has similar structure to the proposed method. The parameters of the controller were also selected the same as the above controller with the observer bandwidth $\omega =50$.

(c)

VF-PID: The proportional–integral controller with the velocity feedforward mechanism is mathematically presented as

$$u=K_{P}e\left(t\right)+K_I\underset{0}{\overset{t}{\int }}e\left(\tau \right)d\tau +K_D\frac{de\left(t\right)}{dt}+K_V{\dot{x}}_{1d}\left(t\right)$$

(55)

where $e\left(t\right)=x_{1d}\left(t\right)-x_1\left(t\right)$ and ${\dot{x}}_{1d}\left(t\right)$ is the first derivative of the desired trajectory. As far we know, the PID is the most-popular control algorithm that is adopted in industrial applications due to the simple implementation and limited tuning parameters. In particular, it does not require the system model, and only the position of the actuator is required; hence, it can be treated as the reference controller for comparison. The parameters of the VF-PID controller were meticulously manually tuned to achieve an acceptable tracking performance and guarantee the system stability as $K_P=3.5\times {10}^4$, $K_I=3.5\times {10}^3$, $K_D=5\times {10}^2$, and $K_V=2\times {10}^3$. The selection of bigger parameters would cause the closed-loop system to be unstable.

To measure the effectiveness of each control approach, two performance indexes [2,38] including the maximum and standard deviation of the tracking errors were employed. These indexes are defined as follows:

(i)

The maximal absolute value of the tracking errors is given by

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

(56)

(ii)

The standard deviation performance index is defined as

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

(57)

where ${\mu }_e$ denotes the average tracking error, which is calculated as ${\mu }_e=\frac{1}{N}{\displaystyle \sum _{i=1}^N}\left|e_1\left(i\right)\right|$.

4.2. Experimental Results

4.2.1. Case Study 1

Firstly, a slow-motion desired trajectory was employed to evaluate the reference-following capability of the three considered controllers, which is mathematically presented as $x_{1d}\left(t\right)=5+30\left(1-cos\left(0.1\pi t\right)\right){\left(1-exp\left(-t\right)\right)}^{\circ }$, with a load of 10 kg.

The tracking performance of the proposed control method in comparison with those of other controllers is demonstrated in Figure 4 and Figure 5. As shown in Figure 4, all controllers guarantee that the system outputs of the hydraulic actuator are able to follow the reference trajectory. Figure 5 indicates the tracking accuracy of each controller. The peaks of the tracking errors happen when the motion direction changes because of the external load and the friction inside the hydraulic actuator. From this figure, it can be seen that the maximal tracking errors of the VF-PID controller and SESO-OFRBC controller were slightly different with the tracking errors as $1.{0033}^{\circ }$ and $0.{9717}^{\circ }$ in the steady state, as shown in

Table 2. Performance indexes in the slow-motion reference trajectory case.

Controller	${\mathit{M}}_{\mathit{e}}$ (Degree)	${\mathit{\sigma }}_{\mathit{e}}$ (Degree)
VF-PID Controller	$1.0033$	$0.1329$
SESO-OFRBC Controller	$0.9717$	$0.1615$
DESO-OFRBC Controller	$0.8354$	$0.0960$

, respectively. Meanwhile, the DESO-OFRBC control approach outperformed the other controllers, and the maximal tracking error was reduced almost down to $0.{8354}^{\circ }$ since the effect of uncertainties and external load in the velocity dynamics was effectively compensated by using the dual ESOs compared to the single ESO employed in the SESO-OFRBC control scheme.

The estimations of the angular velocity, load-pressure-related term, and lumped mismatched and matched disturbances are depicted in Figure 6 and Figure 7, respectively. According to these estimates, the state feedback control can be realized. In the DESO-OFRBC control structure, the lumped mismatched and matched disturbances caused by parametric uncertainties, unmodeled dynamics, and external disturbances were estimated based on the system output only and then compensated, resulting in better tracking performance being achieved compared to the two remaining control methods.

4.2.2. Case Study 2

For further examination of the tracking performance of the three controllers, a four-times faster reference trajectory was adopted as $x_{1d}\left(t\right)=5+20\left(1-cos\left(0.4\pi t\right)\right){\left(1-exp\left(-t\right)\right)}^{\circ }$, and the load was 15 kg.

The tracking performances of the three controllers are presented in Figure 8. As shown, similar to the above case study, all comparative controllers were able to ensure that the system output could follow the reference trajectory in the presence of the model uncertainty and external load.

In addition, the tracking capabilities of the three controllers are explicitly illustrated in Figure 9. Under the faster reference trajectory and heavier load, the tracking errors achieved by these controllers considerably increased. The two performance indexes in this case are presented in

Table 3. Performance indexes in the faster motion reference trajectory case.

Controller	${\mathit{M}}_{\mathit{e}}$ (Degree)	${\mathit{\sigma }}_{\mathit{e}}$ (Degree)
VF-PID Controller	$1.2260$	$0.1971$
SESO-OFRBC Controller	$1.1438$	$0.2079$
DESO-OFRBC Controller	$1.0134$	$0.1726$

. Due to the lack of nonlinear model dynamics and disturbance compensation mechanisms, the absolute maximal tracking error in the steady state obtained by the VF-PID controller went up to $1.{2260}^{\circ }$. Meanwhile, this performance index attained by the SESO-OFRBC controller was $1.{1438}^{\circ }$. It should be noted that the smallest tracking error was achieved by the DESO-OFRBC controller ($1.{0134}^{\circ }$).

According to the above analysis, the proposed control approach was capable of achieving better tracking performance in both case studies. With a simple, but effective control structure with limited tuning parameters, it can be considered as a useful technique for realizing an output feedback control scheme where both mismatched and matched uncertainties are approximated. Hence, the effects of these uncertainties in the mechanical dynamics and pressure dynamics on the tracking performance were effectively mitigated, and improved control accuracy could be achieved.

5. Conclusions

This paper presented a novel active disturbance rejection control for position tracking of a variable-speed pump-controlled hydraulic system. To cope with the shortage of system state information and the online estimate of the lumped matched disturbance, a linear ESO was adopted with a single tuning parameter. In addition, to alleviate the effect of the lumped mismatched disturbance originating from the parametric uncertainties, unmodeled dynamics, and external load on the mechanical dynamics, a mismatched disturbance observer was constructed. According to this, an observer-based approach was established by using the backstepping control technique. Moreover, to avoid the computational burden of the conventional backstepping concept, the dynamic surface control was integrated. Furthermore, the stability of not only the observers, but also the closed-loop system was verified by using the Lyapunov theory. Finally, several experiments were carried out on the real VSPHS to demonstrate the effectiveness of the recommended method in comparison with some reference methods. Two case studies were considered, and the results showed that smaller maximal absolute tracking errors were achieved by the suggested controller ($0.{8534}^{\circ }$ in the slow-motion case and $1.{0134}^{\circ }$ in the faster-motion case) compared to these values obtained by the SESO-OFRBC controller ($0.{9717}^{\circ }$ and $1.{1438}^{\circ }$, respectively) and VF-PID controller ($1.{0033}^{\circ }$ and $1.{2260}^{\circ }$, respectively). Some advanced estimation and control techniques for improving the tracking performance of VSPHSs will be investigated in future work.