State-Constrained Control for Hydraulic Manipulator Position Servo Systems with Valve Dead-Band Compensation

Abstract

Hydraulic manipulators face critical challenges due to valve dead-band nonlinearity and state constraints, which can lead to safety hazards and hardware damage. This study proposes a state-constrained controller with valve dead-band compensation to ensure prescribed positioning accuracy and operational safety. Barrier Lyapunov functions ensure that state constraints are maintained and that boundary violations are avoided. Concurrently, a smooth dead-band inverse model is developed to offset asymmetric valve dead-band effects without inducing chatter. Adaptive laws estimate uncertain parameters and dead-band impact in real time, and a disturbance observer attenuates unmatched uncertainties. Dynamic surface control is employed to diminish the explosion of complexity in backstepping design. Comparative simulations under fixed-angle and arbitrary-angle tracking demonstrate that the proposed controller achieves superior tracking accuracy with steady-state errors below 0.04° compared to 0.06° for non-compensated controllers, while significantly reducing pressure fluctuations and control chattering as adaptive parameters converge. The results indicate that the strategy effectively compensates for valve dead zones while strictly maintaining state constraints, thereby achieving the required control precision for hydraulic servo systems.

1. Introduction

Hydraulic manipulators serve as universal actuation mechanisms in engineering applications [1], distinguished by their exceptional load-bearing capacity, superior power-to-weight ratio, and efficient power transmission. These systems are extensively implemented in excavators [2], electro-hydraulic actuators [3], and servo systems [4]. In practical electro-hydraulic servo control systems, enforcing output constraints (position/force–torque) and state constraints during operation constitutes a critical consideration. Constraint violations not only degrade control performance but also precipitate safety-critical incidents, including mechanical overtravel-induced hardware damage and personnel hazards. These risks escalate significantly when initial-state deviations or unknown disturbances occur, particularly when positional limits are breached. Consequently, preventing actuator motion interference and ensuring prescribed positioning accuracy are imperative for operational safety.

The workspace of hydraulic manipulators is inherently constrained across operational configurations, necessitating boundary constraints on system states during motion to prevent actuator failures caused by mechanical overtravel beyond prescribed limits. Specifically for serial-chain manipulators, end-effector positional accuracy relies on multi-actuator coordinated control, mandating strict adherence to allowable error bounds at every kinematic chain segment. This necessitates explicit consideration of both transient positional constraints and steady-state error constraints during hydraulic manipulator operation.

Recently, barrier Lyapunov functions (BLFs) have demonstrated unique advantages for solving output and state-constraint problems in nonlinear motion control systems. The theoretical foundation for addressing state constraints lies primarily in barrier Lyapunov function (BLF) methodologies, which have been systematically developed and refined across multiple high-impact studies. A BLF-based approach ensures that system states remain within prescribed bounds by constructing a Lyapunov function that grows infinitely as states approach constraint boundaries, thereby mathematically guaranteeing constraint satisfaction. Recent advances have extended this framework to handle asymmetric time-varying full-state constraints with non-vanishing uncertainties, achieving asymptotic tracking rather than mere boundedness [5]. These functions are intrinsically continuous positive-definite scalar functions with first-order continuous partial derivatives. By leveraging logarithmic approximation properties in controller design, BLFs forcibly constrain system errors within desired ranges. When state or output constraints approach predefined boundaries, the energy function constructed from BLFs tends toward infinity, thereby enhancing control gains to guarantee bounded system errors and prevent constraint violations [6,7]. Ngo et al. [8] initially employed barrier functions to restrict system states, thereby addressing stability issues in feedback-linearized systems with multiple state constraints. In the context of guaranteed constraint satisfaction, Tee et al. [9,10] pioneered the use of BLFs to demonstrate the stability and convergence of strictly feedback SISO nonlinear systems with output constraints. For a class of strict-feedback nonlinear systems, Li et al. and Li et al. used BLFs to enforce full-state constraints, ensuring all constrained signals stay bounded [11,12]. Liu et al. [13] first extended the BLF methodology to address stability in nonlinear stochastic systems. In time-varying nonlinear systems, Liu et al. [14] and Wang et al. [15] proposed BLFs to enforce state constraints and ensure effective asymptotic tracking performance. Wu et al. [16] introduced a model predictive control approach based on BLF to stabilize input-constrained nonlinear systems. Wang et al. [17] and Zhang et al. [18] integrated BLFs with dynamic surface control to address dynamic uncertainties in pure-feedback nonlinear systems with full-state constraints, ensuring both constraint compliance and overall convergence. Deng et al. [19] applied BLF-based adaptive backstepping control to hydraulic actuators with velocity and acceleration constraints in state-constrained electro-hydraulic servo systems. Won et al. [4] designed a backstepping controller for electro-hydraulic systems that uses a high-gain disturbance observer and enforces position-tracking-error constraints. Guo et al. further designed high-performance controllers for full-state error constraints and partial-state (position error and load error) constraints [20]. Stosiak M et al. [21] identified the components of the amplitude–frequency spectrum of pressure pulsations in a hydraulic system in which low-frequency external mechanical vibrations were subjected to the action of a proportional hydraulic directional valve. For hydraulic servo systems, specifically, the integration of BLF with adaptive backstepping control has proven highly effective. Zhangbao Xu et al. [22] demonstrated a comprehensive approach combining parameter estimation for parametric uncertainties with extended state observer (ESO) techniques for uncertain nonlinearities, all within a BLF-based full-state constraint framework via backstepping design. This methodology ensures non-violation of state constraints while achieving high-precision trajectory tracking. Similarly, Wan et al. [23] developed a barrier Lyapunov-based adaptive controller where adaptive laws are embedded at each step of the backstepping process, enabling tracking error convergence to prescribed accuracy while respecting state constraints.

Valve dead-band compensation presents a distinct but equally critical challenge. The dead-band in electro-hydraulic servo valves creates a region where input commands produce no output response, leading to tracking errors and potential instability. Dead-band characteristics [24], prevalent in electro-hydraulic servo systems, are discontinuous nonlinearities arising from spool-valve friction and port overlap. These phenomena manifest as zero-displacement regions within specific ranges of control input voltages, where the valve spool ceases to move despite the application of signals. Particularly when operating near the null position of servo valves, such characteristics induce control stability challenges, directly impairing steady-state accuracy and stability. Furthermore, they fundamentally limit performance enhancement in proportional electro-hydraulic servo systems. Crucially, dead zones exacerbate the risk of constraint violation, necessitating the implementation of effective compensation strategies to mitigate these nonlinear effects. A conventional approach models dead-band characteristics in electro-hydraulic servo systems as linear input terms superimposed on bounded disturbances [25,26,27,28], subsequently employing dead-band compensators and disturbance-rejection techniques to mitigate their effects. However, this methodology inadequately captures the actual dynamic characteristics of dead zones, failing to deliver the precise compensation required for high-accuracy stable control. Wang et al. [29] and Zhang et al. [30] implemented adaptive compensation schemes by approximating dead zones as symmetric linear input terms (with equal positive/negative slopes) and bounded disturbances, disregarding the practical asymmetric characteristics. Coelho et al. [31] developed an adaptive algorithm to estimate dead-band intercept parameters in hydraulic actuator cascade control, while Machado et al. [32] experimentally validated the adaptive compensation capability of this method in hydraulic manipulators. Nevertheless, such linearized compensation fundamentally fails to suppress dead-band impacts. Mohanty et al. [33] proposed a nonlinear inverse dead-band function combined with indirect adaptive estimation to compensate for the dead zones of hydraulic manipulators. However, the discontinuous inverse function introduced chattering in practical implementations. Addressing this limitation, Zhou et al. [34] devised a smooth dead-band inverse model that eliminates discontinuous inverse-induced chattering while achieving effective dead-band compensation. Building on this foundation, Deng et al. [35] expanded the approach to include robust adaptive asymptotic tracking control for nonlinear systems featuring unknown-input dead zones. Deng et al. [36] and Gu et al. [37] implemented active dead-band compensation in valve-controlled cylinder electro-hydraulic servo systems, demonstrating enhanced precision and validated efficacy in mitigating dead zones.

To address the control problem of hydraulic manipulators with dead-band characteristics, this study introduces barrier Lyapunov functions (BLFs) to enforce state constraints, thereby ensuring precise positional error control and keeping the manipulator’s motion within desired dynamic error bounds. A smooth dead-band inverse is used to modify the dead-band characteristics of proportional servo valves. To account for parametric uncertainties and the system’s inherent dead-band nonlinearity, adaptive laws are designed to estimate parameter variations in real time and provide compensatory control actions. Additionally, a disturbance observer is integrated to reduce unmatched uncertainties in the electro-hydraulic servo system of the projectile coordinator. The proposed state-constrained controller with valve dead-band compensation effectively compensates for dead zones while strictly maintaining state constraints, thereby attaining the required control precision. Notably, as adaptive parameters converge during operation, both the efficacy of dead-band compensation and control stability improve significantly.

2. Methods

Figure 1 illustrates the simplified positional relationships in the hydraulic manipulator position servo system, where $O_1$ represents the manipulator’s rotational axis, $O_2$ and $O_3$ denote the lower and upper hydraulic cylinder lug axes, respectively, and $M$ indicates the equivalent centroid position of the manipulator. The plumb line is represented by $O_{1}N$, with geometric parameters defined as $r_{O_1{O}_2}=\left|O_1{O}_2\right|$, $r_{O_1{O}_3}=\left|O_1{O}_3\right|$, $r_{O_{1}M}=\left|O_{1}M\right|$. Angular relationships include $\alpha$ as the included angle between $O_{1}M$ and $O_1{O}_2$, also corresponding to the manipulator’s angular position, ${\alpha }_{△}$ as the included angle between $O_1{O}_3$ and $O_1{O}_2$, and ${\alpha }_0$ as the included angle between $O_{1}N$ and $O_1{O}_2$. The distance from $O_1$ to the line $O_2{O}_3$ is denoted as $h(\alpha )$. Hydraulic system variables comprise $P_a$ and $P_a$ (oil pressures in the rodless and rod chambers, respectively), $Q_a$ and $Q_b$ represent flows through the proportional servo valve into or out of the rodless and rod chambers), $P_s$ and $P_r$ are supply and return pressures of the hydraulic source, and $u$ is control input voltage to the proportional servo valve.

Figure 1. Schematic diagram of the positional relationship of the hydraulic robotic arm.

In Figure 1, it is easy to know that when $\alpha ={\alpha }_{△}+\angle M{O}_1{O}_3$, then $\dot{\alpha }={\dot{\alpha }}_{△}$. Thus, the kinematic relationship between the displacement of the piston of the hydraulic cylinder $x_p$ and the angular position $\alpha$ of the robotic arm can be expressed as follows:

$$\begin{gathered} {\dot{x}}_p=h(\alpha )\dot{\alpha } \\ h(\alpha )={\displaystyle \frac{r_{O_1{O}_2}{r}_{O_1{O}_3}\mathit{sin}({\alpha }_{△})}{\left|O_2{O}_3\right|}}={\displaystyle \frac{r_{O_1{O}_2}{r}_{O_1{O}_3}\mathit{sin}({\alpha }_{△})}{\sqrt{r_{O_1{O}_2}^2+r_{O_1{O}_3}^2-2r_{O_1{O}_2}{r}_{O_1{O}_3}\mathit{cos}({\alpha }_{△})}}} \end{gathered}$$

(1)

The torque balance equation for the position servo system of a hydraulic robotic arm is

$$J\ddot{\alpha }=Fh(\alpha )-b\dot{\alpha }{h(\alpha )}^2-mg{h}_g(\alpha )+T_d$$

(2)

In the above equation, $J$ is the equivalent rotational inertia of the robotic arm load; $F=P_a{A}_a-P_b{A}_b$ is the hydraulic cylinder output force, where $A_a$ and $A_b$ are the effective piston areas of the rodless and rod-side chambers of the hydraulic cylinder, respectively; $b$ is the equivalent viscous friction coefficient of the hydraulic cylinder; $m$ is the equivalent mass of the robotic arm load; $g$ is the gravitational acceleration; $h_g(\alpha )=r_{O_{1}M}\mathit{sin}(\alpha -{\alpha }_0)$ is the moment arm of the gravitational force acting on the pivot point $O_1$; $T_d$ indicates the overall impact of all unmodeled dynamics such as load variations, external disturbances, and nonlinear friction characteristics.

The following is derived from the continuity equation for flow [38]:

$$\begin{gathered} Q_a-C_{\mathrm{ip}}\left(P_a-P_b\right)-C_{\mathrm{ep}}{P}_a+q_a=A_a{\dot{x}}_p+{\displaystyle \frac{V_a}{{\beta }_e}}{\dot{P}}_a \\ {C}_{\mathrm{ip}}\left(P_a-P_b\right)-C_{\mathrm{ep}}{P}_b-Q_b-q_b=-A_b{\dot{x}}_p+{\displaystyle \frac{V_b}{{\beta }_e}}{\dot{P}}_b \end{gathered}$$

(3)

where $C_{\mathrm{ip}}$ denotes the hydraulic cylinder internal leakage coefficient, $C_{\mathrm{ep}}$ denotes the hydraulic cylinder external leakage coefficient; $V_a=V_{\mathrm{a}0}+A_a{x}_p$ denotes the control volume of the rodless cavity in the hydraulic system cylinder, $V_{\mathrm{a}0}$ denotes the initial state of the hydraulic cylinder’s rodless cavity and its piping connected to the servo valve control volume; $V_b=V_{\mathrm{b}0}-A_b{x}_p$ denotes the control volume of the rod cavity of the hydraulic cylinder, where $V_{\mathrm{b}0}$ is the control volume in the rod chamber of the hydraulic cylinder and its piping connected to the servo valve at the initial moment; ${\beta }_e$ is the effective volumetric modulus of elasticity of the hydraulic oil; and $q_a$ and $q_b$ are the dynamic errors of the unmodeled pressures in the rodless and rod chambers of the hydraulic cylinder, respectively.

The relationship between the flow rates, denoted as $Q_a$ and $Q_b$ of the high-frequency response servo valve spool and the control input voltage u, can be described as follows:

$$\left\{\begin{array}{c}{Q}_a=k_{\mathrm{qva}}u{R}_{\mathrm{xa}}\\ Q_b=k_{\mathrm{qvb}}u{R}_{\mathrm{xb}}\end{array}\right.$$

(4)

In the formula, $k_{\mathrm{qva}}$ and $k_{\mathrm{qvb}}$ are the flow gain coefficients for the flow entering the hydraulic cylinder’s rodless chamber and rod chamber through the proportional servo valve spool, respectively. $R_{\mathrm{xa}}$ and $R_{\mathrm{xb}}$ are defined as follows:

$$\begin{gathered} R_{\mathrm{xa}}=s(u_x)\sqrt{P_s-P_a}+s(-u_x)\sqrt{P_a-P_r} \\ {R}_{\mathrm{xb}}=s(u_x)\sqrt{P_b-P_r}+s(-u_x)\sqrt{P_s-P_b} \end{gathered}$$

(5)

In the equation, $P_a,P_b,P_s$, and $P_r$ clearly satisfy $0\le P_r\le P_a\le P_s$ and $0\le P_r\le P_b\le P_s$. The function $s(\ast )$ is defined as

$$s(\ast )=\left\{\begin{array}{c}1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{i}\mathrm{f}\hspace{0.33em}\hspace{0.33em}\ast \ge 0\\ 0\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{i}\mathrm{f}\hspace{0.33em}\hspace{0.33em}\ast <0\end{array}\right.$$

(6)

The dead zone nonlinear output $u_{\mathrm{DZ}}$ of a proportional servo valve can be described as

$$u_{\mathrm{DZ}}=DZ(u)=\left\{\begin{array}{cc}{m}_r(u-b_r)& u\ge b_r\\ 0& b_l<u<b_r\\ m_l(u-b_l)& u\le b_l\end{array}\right.$$

(7)

In the equation, $DZ(u)$ represents the nonlinear function of the dead zone of the proportional servo valve; $m_r>0,m_l>0,b_r\ge 0$ and $b_l\le 0$ are unknown bounded constants. Even for proportional servo valves with symmetrical valve ports, their dead zone characteristics are generally asymmetric, i.e., $m_r\ne m_l,\left|b_r\right|\ne \left|b_l\right|$. The dead-band characteristics of the proportional servo valve are illustrated in Figure 2a. $m_r$ and $m_l$ represent the slope of the dead-band attributes on the right side and left side of the $u_{\mathrm{DZ}}$ axis, respectively, while $b_r$ and $b_l$ represent the right intercept and left intercept of the dead-band characteristics on the $u$ axis, respectively.

Figure 2. (a) Dead-band nonlinearity of proportional servo valve and (b) smooth dead zone inverse characteristics.

We define the state variables of the hydraulic robotic arm position servo system as $\mathit{x}={\left[x_1,x_2,x_3\right]}^T={\left[\alpha ,\dot{\alpha },A_a{P}_a-A_b{P}_b\right]}^T$. Combining Equations (1)–(7), the system state-space equations can be rearranged as follows:

$$\left\{\begin{array}{l}{\dot{x}}_1=x_2\\ {\dot{x}}_2=h_J(x_1)x_3-B_x{h}^2(x_1)x_2-M_{x}g{h}_g(x_1)+d_2(t)\\ {\dot{x}}_3={\beta }_{e}g(x)u_{\mathrm{DZ}}-{\beta }_e{f}_{\mathrm{x}1}(x)-{\beta }_e{C}_{\mathrm{ip}}{f}_{\mathrm{x}2}(x)-{\beta }_e{C}_{\mathrm{ep}}{f}_{\mathrm{x}3}(x)+d_3(t)\end{array}\right.$$

(8)

where $h_J(x_1)={\displaystyle \frac{h(x_1)}{J}}$, $B_x={\displaystyle \frac{b}{J}}$, $M_x={\displaystyle \frac{m}{J}}$, $h_g(x_1)=r_{O_{1}M}\mathit{sin}(x_1-{\alpha }_0)$, $d_2(t)={\displaystyle \frac{T_d}{J}}$, $g(x)={\displaystyle \frac{A_a{k}_{\mathrm{qva}}{R}_{\mathrm{xa}}}{V_a}}+{\displaystyle \frac{A_b{k}_{\mathrm{qvb}}{R}_{\mathrm{xb}}}{V_b}}$, $f_{\mathrm{x}1}(x)=\left({\displaystyle \frac{A_a^2}{V_a}}+{\displaystyle \frac{A_b^2}{V_b}}\right)h(x_1)x_2$, $f_{\mathrm{x}2}(x)=\left({\displaystyle \frac{A_a}{V_a}}+{\displaystyle \frac{A_b}{V_b}}\right)\left(P_a-P_b\right)$, $f_{\mathrm{x}3}(x)={\displaystyle \frac{P_a{A}_a}{V_a}}-{\displaystyle \frac{P_b{A}_b}{V_b}},d_3(t)={\displaystyle \frac{A_a{\beta }_e}{V_a}}{q}_a+{\displaystyle \frac{A_b{\beta }_e}{V_b}}{q}_b$.

We define the unknown dead-band parameter vector as $\mathit{\rho }={\left[{\rho }_1,{\rho }_2,{\rho }_3,{\rho }_3\right]}^T={\left[m_r,m_r{b}_r,m_l,m_l{b}_l\right]}^T$ and the sign function.

$${\sigma }_r(t)=\left\{\begin{array}{cc}1,& u_x\ge 0\\ 0,& u_x<0\end{array}\right.,{\sigma }_l(t)=\left\{\begin{array}{cc}0,& u_x>0\\ 1,& u_x\le 0\end{array}\right.$$

The dead zone characteristics expressed in Equation (7) can be parameterized as

$$\begin{array}{ll}{u}_{\mathrm{DZ}}& =DZ\left(u\right)=m_r\left(u-b_r\right){\sigma }_r\left(t\right)+m_l\left(u-b_l\right){\sigma }_l\left(t\right)\\ & =-\left[-m_r{\sigma }_r\left(t\right)u+m_r{b}_r{\sigma }_r\left(t\right)-m_l{\sigma }_l\left(t\right)u+m_l{b}_l{\sigma }_l\left(t\right)\right]\\ & =-{\mathit{\rho }}^T\mathit{\omega }\end{array}$$

(9)

In the equation, $\mathit{\omega }={\left[-{\sigma }_r(t)v(t),{\sigma }_r(t),-{\sigma }_l(t)v(t),{\sigma }_l(t)\right]}^T$. Clearly, this dead-band nonlinearity is discontinuous. To reduce the control input noise caused by discontinuities, we mitigate it by applying a continuously differentiable inverse dead-band model [34,35,36,37]:

$$u=DI(u_{\mathrm{DZ}})={\displaystyle \frac{u_{\mathrm{DZ}}+m_r{b}_r}{m_r}}{\varphi }_r(u_{\mathrm{DZ}})+{\displaystyle \frac{u_{\mathrm{DZ}}+m_l{b}_l}{m_l}}{\varphi }_l(u_{\mathrm{DZ}})$$

(10)

In the formula, ${\varphi }_r(u_{\mathrm{DZ}})$ and ${\varphi }_l(u_{\mathrm{DZ}})$ are both smooth indicator functions, defined as follows:

$${\varphi }_r(u_{\mathrm{DZ}})={\displaystyle \frac{e^{u_{\mathrm{DZ}}/e_0}}{e^{u_{\mathrm{DZ}}/e_0}+e^{-u_{\mathrm{DZ}}/e_0}}},{\varphi }_l(u_{\mathrm{DZ}})={\displaystyle \frac{e^{-u_{\mathrm{DZ}}/e_0}}{e^{u_{\mathrm{DZ}}/e_0}+e^{-u_{\mathrm{DZ}}/e_0}}}$$

(11)

We define the estimated value of the unknown dead zone parameter $\mathit{\rho }$ as $\widehat{\mathit{\rho }}={\left[{\widehat{\rho }}_1,{\widehat{\rho }}_2,{\widehat{\rho }}_3,{\widehat{\rho }}_4\right]}^T={\left[\widehat{m_r},\widehat{m_r}{b}_r,\widehat{m_l},\widehat{m_l}{b}_l\right]}^T$. We design the actual control input as

$$u_d=-{\widehat{\mathit{\rho }}}^T\widehat{\mathit{\omega }}$$

(12)

where $\widehat{\mathit{\omega }}={\left[-{\varphi }_r(u)u,{\varphi }_r(u),-{\varphi }_l(u)u,{\varphi }_l(u)\right]}^T$.

Then, Equation (10) can be rewritten as

$$u=\widehat{DI}(u_d)={\displaystyle \frac{u_d+\widehat{m_r}{b}_r}{\widehat{m_r}}}{\varphi }_r(u_d)+{\displaystyle \frac{u_d+\widehat{m_l}{b}_l}{\widehat{m_l}}}{\varphi }_l(u_d)$$

(13)

Subtracting Equation (9) from Equation (12) yields

$$\begin{array}{ll}{u}_{\mathrm{DZ}}-u_d& =-{\mathit{\rho }}^T\mathit{\omega }+{\widehat{\mathit{\rho }}}^T\widehat{\mathit{\omega }}=-{\mathit{\rho }}^T\mathit{\omega }+{\mathit{\rho }}^T\widehat{\mathit{\omega }}-{\mathit{\rho }}^T\widehat{\mathit{\omega }}+{\widehat{\mathit{\rho }}}^T\widehat{\mathit{\omega }}\\ & ={\left(\widehat{\mathit{\rho }}-\mathit{\rho }\right)}^T\widehat{\mathit{\omega }}+{\mathit{\rho }}^T\left(\widehat{\mathit{\omega }}-\mathit{\omega }\right)\\ & =-{\tilde{\mathit{\rho }}}^T\widehat{\mathit{\omega }}+d_{\mathrm{DZ}}(t)\end{array}$$

(14)

In the equation, $\tilde{\mathit{\rho }}=\mathit{\rho }-\widehat{\mathit{\rho }}$ represents the estimation error of the unknown dead zone parameter $\mathit{\rho }$. $d_{\mathrm{DZ}}(t)={\mathit{\rho }}^T\left(\widehat{\mathit{\omega }}-\mathit{\omega }\right)$.

We denote the lumped uncertainty in the nonmatched control channel of Equation (8), arising from the unknown parameters J, $B_x$ and $M_x$, as $d_2^{\ast }$. For the control-input channel, we introduce the uncertain parameter vector $\mathit{\theta }={\left[{\theta }_1,{\theta }_2,{\theta }_3\right]}^T$, with ${\theta }_1={\beta }_e$, ${\theta }_2={\beta }_e{C}_{\mathrm{ip}}$, ${\theta }_3={\beta }_e{C}_{\mathrm{ep}}$. Replacing the uncertain parameter ${\beta }_e$ in the term ${\beta }_e{g}_x(x)u$ with its nominal value ${\beta }_{en}$, we treat the resulting modeling error as an unmodeled term included in $d_3(t)$. Consequently, Equation (8) can be rewritten as

$$\left\{\begin{array}{l}{\dot{x}}_1=x_2\\ {\dot{x}}_2=h_J(x_1)x_3-B_x{h}^2(x_1)x_2-M_{x}g{h}_g(x_1)+{\Delta }_2(t)\\ {\dot{x}}_3={\beta }_{\mathrm{en}}g(x)u_d-{\beta }_{\mathrm{en}}g(x){\tilde{\mathit{\rho }}}^T\widehat{\mathit{\omega }}-{\mathit{\theta }}^T\mathit{f}+{\Delta }_3(t)\end{array}\right.$$

(15)

${\Delta }_2(t)=d_2(t)+d_2^{\ast }$, $\mathit{f}={\left[f_{\mathrm{x}1}(x),f_{\mathrm{x}2}(x),f_{\mathrm{x}3}(x)\right]}^T$, ${\Delta }_3(t)={\beta }_{\mathrm{en}}g(x)d_{\mathrm{DZ}}(t)+d_3(t)$.

The design of this controller requires the following assumptions:

Assumption 1: The second derivative of the reference angular position command ${\mathrm{x}}_{\mathrm{r}1}$ exists and is continuous, meaning that the desired reference angular position, reference angular velocity $\dot{{\mathrm{x}}_{\mathrm{r}1}}$, reference angular acceleration $\ddot{{\mathrm{x}}_{\mathrm{r}1}}$, and reference angular acceleration-acceleration $\stackrel{⃛}{{\mathrm{x}}_{\mathrm{r}1}}$ are all bounded.

Assumption 2: The system’s uncertain parameters $\mathit{\theta }={\left[{\theta }_1,{\theta }_2,{\theta }_3\right]}^T$ and $\mathit{\rho }={\left[{\rho }_1,{\rho }_2,{\rho }_3,{\rho }_3\right]}^T$ are continuously bounded and satisfy the following:

$$\begin{gathered} \mathit{\theta }\in {\mathsf{\Omega }}_{\theta }\triangleq \left\{\theta :{\theta }_{min}\le \theta \le {\theta }_{max}\right\} \\ \mathit{\rho }\in {\mathsf{\Omega }}_{\rho }\triangleq \left\{\rho :{\rho }_{min}\le \rho \le {\rho }_{max}\right\} \end{gathered}$$

(16)

where ${\theta }_{min}={\left[{\theta }_{1min},{\theta }_{2min},{\theta }_{3min}\right]}^T$, ${\theta }_{max}={\left[{\theta }_{1max},{\theta }_{2max},{\theta }_{3max}\right]}^T$, ${\rho }_{min}={\left[{\rho }_{1min},{\rho }_{2min},{\rho }_{3min}\right]}^T$, and ${\rho }_{max}={\left[{\rho }_{1max},{\rho }_{2max},{\rho }_{3max}\right]}^T$.

Assumption 3: The system does not model the disturbance ${\Delta }_2(t)$, and its first derivatives ${\dot{\Delta }}_2(t)$ and ${\Delta }_3(t)$ are continuously bounded and satisfy

$$\left|{\Delta }_2(t)\right|\le {\overline{\Delta }}_2,\left|{\dot{\Delta }}_2\left(t\right)\right|\le {\overline{\dot{\Delta }}}_2,\left|{\Delta }_3(t)\right|\le {\overline{\Delta }}_3$$

(17)

The estimated value and estimated error of the uncertain parameter $\theta$ are denoted as $\widehat{\theta }$ and $\tilde{\theta }$, respectively, to form $\tilde{\theta }=\theta -\widehat{\theta }$. To guarantee that the estimated uncertain parameter always meets assumption 2, the following discontinuous projection algorithm is introduced [39,40].

$$P{\mathrm{roj}}_{{\widehat{\theta }}_i}(•)=\left\{\begin{array}{l}0\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}if\hspace{0.33em}{\widehat{\theta }}_i={{\theta }_i}_{max}\\ 0\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}if\hspace{0.33em}{\widehat{\theta }}_i={\theta }_{imin}\\ •\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}otherwise\end{array}\right.$$

(18)

$$P{\mathrm{roj}}_{{\widehat{\rho }}_j}(•)=\left\{\begin{array}{l}0\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}if\hspace{0.33em}{\widehat{\rho }}_j={{\rho }_j}_{max}\\ 0\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}if\hspace{0.33em}{\widehat{\rho }}_j={\rho }_{jmin}\\ •\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}otherwise\end{array}\right.$$

(19)

In the formula, $i=1,2,3$; $j=1,2,3,4$. A parameter adaptive law is employed.

$$\dot{\widehat{\mathit{\theta }}}=P{\mathrm{roj}}_{\widehat{\mathit{\theta }}}(\mathit{\gamma }\mathit{\tau })$$

(20)

$$\dot{\widehat{\mathit{\rho }}}=P{\mathrm{roj}}_{\widehat{\mathit{\rho }}}(\mathit{\Gamma }\mathit{\kappa })$$

(21)

In the formula, $\mathit{\gamma }=\mathrm{diag}\left\{{\gamma }_1,{\gamma }_2,{\gamma }_3\right\}$ and $\mathit{\Gamma }=\mathrm{diag}\left\{{\Gamma }_1,{\Gamma }_2,\cdot \cdot \cdot ,{\Gamma }_4\right\}$ are both positive-definite adaptive gain diagonal matrices. $\mathit{\tau }={\left[{\tau }_1,{\tau }_2,{\tau }_3\right]}^T$ and $\mathit{\kappa }={\left[{\kappa }_1,{\kappa }_2,\cdot \cdot \cdot ,{\kappa }_4\right]}^T$ are both parameter adaptive laws to be designed.

3. Results of State-Constrained Controller Design Considering Valve Dead-Band Compensation

3.1. State-Constrained Controller Design

Taking the derivative with respect to $e_1$ yields

$${\dot{e}}_1={\dot{x}}_1-{\dot{x}}_{r1}=x_2-{\dot{x}}_{r1}$$

(22)

Design a virtual control function for $x_2$:

$$x_{2\mathrm{d}}=-\left(k_{c1}^2-e_1^2\right)k_1{e}_1+{\dot{x}}_{r1}$$

(23)

Taking the derivative with respect to $x_{2\mathrm{d}}$ yields

$${\dot{x}}_{2d}=-\left(k_{c1}^2-3e_1^2\right)k_1{\dot{e}}_1+{\ddot{x}}_{r1}$$

(24)

The ${\dot{e}}_1$ can be rewritten as

$${\dot{e}}_1=e_2+x_{2\mathrm{d}}-{\dot{x}}_{r1}=e_2-\left(k_{c1}^2-e_1^2\right)k_1{e}_1$$

(25)

Take the Lyapunov function of the first subsystem as

$$V_1={\displaystyle \frac{1}{2}}\mathit{ln}{\displaystyle \frac{k_{c1}^2}{k_{c1}^2-e_1^2}}$$

(26)

Taking the derivative with respect to $V_1$ yields

$${\dot{V}}_1={\displaystyle \frac{e_1{\dot{e}}_1}{k_{c1}^2-e_1^2}}=-k_1{e}_1^2+{\displaystyle \frac{e_1{e}_2}{k_{c1}^2-e_1^2}}$$

(27)

This can be rewritten as

$${\dot{V}}_1\le -{\chi }_1\mathit{ln}{\displaystyle \frac{k_{c1}^2}{k_{c1}^2-e_1^2}}+{\displaystyle \frac{e_1{e}_2}{k_{c1}^2-e_1^2}}$$

(28)

where ${\chi }_1=k_1\left(k_{c1}^2-e_1^2\right)>0$.

Taking the derivative with respect to $e_2$ yields

$${\dot{e}}_2={\dot{x}}_2-{\dot{x}}_{2d}=h_J(x_1)x_3-B_x{h}^2(x_1)x_2-M_{x}g{h}_g(x_1)+{\Delta }_2(t)-{\dot{x}}_{2d}$$

(29)

To estimate the unmatched disturbance term ${\Delta }_2(t)$ in the system, the following disturbance observer is designed [41]:

$$\left\{\begin{array}{l}{\widehat{\Delta }}_2=\xi +l{x}_2\\ \dot{\xi }=-l\xi -l\left[h_J(x_1)x_3-B_x{h}^2(x_1)x_2-M_{x}g{h}_g(x_1)+\mathcal{l}{x}_2\right]\end{array}\right.$$

(30)

In the equation, ${\widehat{\Delta }}_2$ is the estimated value of ${\Delta }_2(t)$, $\xi$ is the auxiliary design variable, and $\mathcal{l}$ is the disturbance observer gain.

Define the estimation error ${\tilde{\Delta }}_2={\Delta }_2-{\widehat{\Delta }}_2$. Taking the derivative of ${\tilde{\Delta }}_2$ yields

$$\begin{array}{ll}{\dot{\tilde{\Delta }}}_2& ={\dot{\Delta }}_2-{\dot{\widehat{\Delta }}}_2={\dot{\Delta }}_2-\dot{\xi }-\mathcal{l}{\dot{x}}_2\\ & ={\dot{\Delta }}_2+\mathcal{l}\xi +\mathcal{l}\left[h_J(x_1)x_3-B_x{h}^2(x_1)x_2-M_{x}g{h}_g(x_1)+\mathcal{l}{x}_2\right]\\ & -\mathcal{l}\left[h_J(x_1)x_3-B_x{h}^2(x_1)x_2-M_{x}g{h}_g(x_1)+{\Delta }_2(t)\right]\\ & ={\dot{\Delta }}_2+\mathcal{l}\xi +\mathcal{l}\left(\mathcal{l}{x}_2-{\Delta }_2(t)\right)={\dot{\Delta }}_2+\mathcal{l}\left({\widehat{\Delta }}_2-{\Delta }_2(t)\right)\\ & =-\mathcal{l}{\tilde{\Delta }}_2+{\dot{\Delta }}_2\end{array}$$

(31)

Design a virtual control function for $x_3$ yields

$${\overline{x}}_3=-{\displaystyle \frac{1}{h_J(x_1)}}\left[{\mathit{e}}_{12}{\mathit{\phi }}_{12}-B_x{h}^2(x_1)x_2-M_{x}g{h}_g(x_1)+{\widehat{\Delta }}_2-{\dot{x}}_{2d}\right]$$

(32)

In the formula, ${\mathit{e}}_{12}=\left[e_1,e_2\right]$, ${\mathit{\phi }}_{12}=\left[{\phi }_1,{\phi }_2\right]{=\left[{\displaystyle \frac{\left(k_{c2}^2-e_2^2\right)}{\left(k_{c1}^2-e_1^2\right)}},k_2\left(k_{c2}^2-e_2^2\right)\right]}^T$, where $k_2$ is a positive design parameter.

Replacing the derivative of the virtual control variable with the output of a first-order low-pass filter [42] can help prevent differential explosion. The first-order low-pass filter employed here is

$${\tau }_3{\dot{x}}_{3\mathrm{d}}+x_{3\mathrm{d}}={\overline{x}}_3{x}_{3\mathrm{d}}(0)={\overline{x}}_3(0)$$

(33)

Define the corresponding boundary layer error as

$$z_3=x_{3\mathrm{d}}-{\overline{x}}_3$$

(34)

Taking the derivative with respect to $z_3$ yields

$${\dot{z}}_3={\dot{x}}_{3\mathrm{d}}-{\dot{\overline{x}}}_3=-{\displaystyle \frac{1}{{\tau }_3}}\left(x_{3\mathrm{d}}-{\overline{x}}_3\right)-{\dot{\overline{x}}}_3=-{\displaystyle \frac{z_3}{{\tau }_3}}-{\dot{\overline{x}}}_3$$

(35)

Then, ${\dot{e}}_2$ is rewritten as

$$\begin{gathered} {\dot{e}}_2=h_J(x_1)\left(e_3+z_3+{\overline{x}}_3\right)-B_x{h}^2(x_1)x_2-M_{x}g{h}_g(x_1)+{\Delta }_2(t)-{\dot{x}}_{2d} \\ =h_J(x_1)\left(e_3+z_3\right)-{\mathit{e}}_{12}{\mathit{\phi }}_{12}+{\tilde{\Delta }}_2 \end{gathered}$$

(36)

Consider the Lyapunov function for the second subsystem as

$$V_2=V_1+{\displaystyle \frac{1}{2}}\mathit{ln}{\displaystyle \frac{k_{c2}^2}{k_{c2}^2-e_2^2}}+{\displaystyle \frac{1}{2}}{z}_3^2+{\displaystyle \frac{1}{2}}{\tilde{\Delta }}_2^2$$

(37)

The constraint parameters $k_{\mathrm{c}1}$ and $k_{\mathrm{c}2}$ in the BLFs are closely related to the physical and mechanical limits of the hydraulic manipulator. Specifically, $k_{\mathrm{c}1}$ defines the absolute maximum allowable position tracking error. It serves as a strict safety margin dictated by the mechanical clearances of the manipulator’s workspace to prevent physical collisions during operation. Meanwhile, $k_{\mathrm{c}2}$ restricts the maximum transient velocity error. Its selection is strictly constrained by the maximum safe flow rate of the proportional servo valve. Bounding this velocity error is physically imperative to prevent violent flow surges, which could induce severe hydraulic shocks and cause mechanical vibrations or hardware damage. In practical tuning, these parameters must be carefully selected to balance the system’s dynamic response speed while strictly ensuring mechanical safety margins.

Taking the derivative with respect to $V_2$ yields

$$\begin{array}{ll}{\dot{V}}_2& ={\dot{V}}_1+{\displaystyle \frac{e_2{\dot{e}}_2}{k_{c2}^2-e_2^2}}+z_3{\dot{z}}_3+{\tilde{\Delta }}_2{\dot{\tilde{\Delta }}}_2\\ & ={\dot{V}}_1+{\displaystyle \frac{h_{J_x}(x_1)e_2{e}_3}{k_{c2}^2-e_2^2}}+{\displaystyle \frac{h_{J_x}(x_1)e_2{z}_3}{k_{c2}^2-e_2^2}}-{\displaystyle \frac{e_1{e}_2}{k_{c1}^2-e_1^2}}-k_2{e}_2^2+{\displaystyle \frac{e_2{\tilde{\Delta }}_2}{k_{c2}^2-e_2^2}}\\ & +z_3\left(-{\displaystyle \frac{z_3}{{\tau }_3}}-{\dot{\overline{x}}}_3\right)+{\tilde{\Delta }}_2\left(-\mathcal{l}{\tilde{\Delta }}_2+{\dot{\Delta }}_2\right)\\ & \le {\left.{\dot{V}}_1\right|}_{e_2=0}-k_2{e}_2^2+{\displaystyle \frac{h_{J_x}(x_1)e_2{e}_3}{k_{c2}^2-e_2^2}}+{\displaystyle \frac{h_{J_x}(x_1)e_2{z}_3}{k_{c2}^2-e_2^2}}+{\displaystyle \frac{e_2{\tilde{\Delta }}_2}{k_{c2}^2-e_2^2}}-{\displaystyle \frac{1}{{\tau }_3}}{z}_3^2-z_3{\dot{\overline{x}}}_3-\mathcal{l}{\tilde{\Delta }}_2^2+{\tilde{\Delta }}_2{\dot{\Delta }}_2\end{array}$$

(38)

where ${\left.{\dot{V}}_1\right|}_{e_2=0}=-{\chi }_1\mathit{ln}{\displaystyle \frac{k_{c1}^2}{k_{c1}^2-e_1^2}}$.

By Yang’s inequality, the following can be obtained:

$$\left|e_2{\tilde{\Delta }}_2\right|\le {\displaystyle \frac{e_2^2+{\tilde{\Delta }}_2^2}{2}},\left|{\tilde{\Delta }}_2{\dot{\Delta }}_2\right|\le {\displaystyle \frac{{\tilde{\Delta }}_2^2+{\left|{\dot{\Delta }}_2\right|}^2}{2}}$$

(39)

Substituting Equation (39) into Equation (38) yields

$$\begin{gathered} {\dot{V}}_2\le {\left.{\dot{V}}_1\right|}_{e_2=0}+{\displaystyle \frac{h_{J_x}(x_1)e_2{e}_3}{k_{c2}^2-e_2^2}}-\left[k_2-{\displaystyle \frac{h_{J_x}(x_1)}{2\left(k_{c2}^2-e_2^2\right)}}-{\displaystyle \frac{1}{2\left(k_{c2}^2-e_2^2\right)}}\right]e_2^2+{\displaystyle \frac{{\sigma }_2}{2}} \\ -\left[{\displaystyle \frac{1}{{\tau }_3}}-{\displaystyle \frac{h_{J_x}(x_1)}{2\left(k_{c2}^2-e_2^2\right)}}-{\displaystyle \frac{{\left|{\dot{\overline{x}}}_3\right|}_{max}^2}{2{\sigma }_2}}\right]z_3^2-\left[\mathcal{l}-{\displaystyle \frac{1}{2\left(k_{c2}^2-e_2^2\right)}}-{\displaystyle \frac{1}{2}}\right]{\tilde{\Delta }}_2^2{\displaystyle \frac{{\left|{\dot{\Delta }}_2\right|}^2}{2}} \\ \le {\left.{\dot{V}}_1\right|}_{e_2=0}-{\chi }_2\mathit{ln}{\displaystyle \frac{k_{c2}^2}{k_{c2}^2-e_2^2}}-{\zeta }_3{z}_3^2-\varsigma {\tilde{\Delta }}_2^2+{\displaystyle \frac{{\sigma }_2}{2}}+{\displaystyle \frac{{\left|{\dot{\Delta }}_2\right|}^2}{2}}+{\displaystyle \frac{h_{J_x}(x_1)e_2{e}_3}{k_{c2}^2-e_2^2}} \end{gathered}$$

(40)

In the formula, ${\chi }_2=k_2\left(k_{c2}^2-e_2^2\right)-{\displaystyle \frac{h_{J_x}(x_1)}{2}}-{\displaystyle \frac{1}{2}}>0$, ${\zeta }_3={\displaystyle \frac{1}{{\tau }_3}}-{\displaystyle \frac{h_{J_x}\left(x_1\right)}{2\left(k_{c2}^2-e_2^2\right)}}-{\displaystyle \frac{{\left|{\dot{\overline{x}}}_3\right|}_{max}^2}{2{\sigma }_2}},\varsigma =\mathcal{l}-{\displaystyle \frac{1}{2\left(k_{c2}^2-e_2^2\right)}}-{\displaystyle \frac{1}{2}}>0$.

Taking the derivative with respect to $e_3$ yields

$${\dot{e}}_3={\dot{x}}_3-{\dot{x}}_{3d}={\beta }_{\mathrm{en}}{g}_x(x)u_d-{\beta }_{\mathrm{en}}{g}_x(x){\tilde{\mathit{\rho }}}^T\widehat{\mathit{\omega }}-{\mathit{\theta }}^T\mathit{f}+{\Delta }_3(t)-{\dot{x}}_{3d}$$

(41)

The final control function is designed as

$$u_d=-{\displaystyle \frac{1}{{\beta }_{\mathrm{en}}{g}_x(x)}}\left[{\mathit{e}}_{22}{\mathit{\phi }}_{22}-{\widehat{\mathit{\theta }}}^T\mathit{f}+{\displaystyle \frac{{\overline{\Delta }}_3^2}{2{\delta }_3}}{e}_3-{\dot{x}}_{3d}\right]$$

(42)

In the equation, ${\mathit{e}}_{22}=\left[e_2,e_3\right]$, ${\mathit{\phi }}_{22}={\left[{\displaystyle \frac{h_{J_x}(x_1)}{\left(k_{c2}^2-e_2^2\right)}},k_3\right]}^T$, where k₃ and δ₃ are positive design parameters.

Then, ${\dot{e}}_3$ is rewritten as

$${\dot{e}}_3=-{\mathit{e}}_{22}{\mathit{\phi }}_{22}-{\beta }_{\mathrm{en}}{g}_x(x){\tilde{\mathit{\rho }}}^T\widehat{\mathit{\omega }}-{\tilde{\mathit{\theta }}}^T\mathit{f}-{\displaystyle \frac{{\overline{\Delta }}_3^2}{2{\delta }_3}}{e}_3+{\Delta }_3(t)$$

(43)

Consider the Lyapunov function of the third subsystem as

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

(44)

Taking the derivative with respect to $V_3$ yields

$$\begin{gathered} {\dot{V}}_3={\dot{V}}_2+e_3{\dot{e}}_3={\dot{V}}_2+e_3\left[-{\mathit{e}}_{22}{\mathit{\phi }}_{22}-{\beta }_{\mathrm{en}}{g}_x(x){\tilde{\mathit{\rho }}}^T\widehat{\mathit{\omega }}-{\tilde{\mathit{\theta }}}^T\mathit{f}+{\displaystyle \frac{{\overline{\Delta }}_3^2}{2{\delta }_3}}{e}_3+{\Delta }_3(t)\right] \\ \le {\left.{\dot{V}}_2\right|}_{e_3=0}-k_3{e}_3^2-e_3{\beta }_{\mathrm{en}}{g}_x(x){\tilde{\mathit{\rho }}}^T\widehat{\mathit{\omega }}-e_3{\tilde{\mathit{\theta }}}^T\mathit{f}+{\displaystyle \frac{{\delta }_3}{2}} \end{gathered}$$

(45)

In the equation, ${\left.{\dot{V}}_2\right|}_{e_3=0}={\left.{\dot{V}}_1\right|}_{e_2=0}-{\chi }_2\mathit{ln}{\displaystyle \frac{k_{c2}^2}{k_{c2}^2-e_2^2}}-{\zeta }_3{z}_3^2-\varsigma {\tilde{\Delta }}_2^2+{\displaystyle \frac{{\sigma }_2}{2}}+{\displaystyle \frac{{\left|{\dot{\Delta }}_2\right|}^2}{2}}$.

Parameter adaptive control law design is denoted as

$${\mathit{\tau }}_i=-\left[{\displaystyle \frac{e_3{\mathit{f}}_i}{{\mathit{\gamma }}_i}}+{\widehat{\mathit{\theta }}}_i\right],i=1,2,3$$

(46)

$${\mathit{\kappa }}_i=-\left[{\displaystyle \frac{e_3{\widehat{\mathit{\omega }}}_i{\beta }_{\mathrm{en}}{g}_x(x)}{{\mathit{\Gamma }}_i}}+{\widehat{\mathit{\rho }}}_i\right],i=1,2,3,4$$

(47)

According to Equation (13), the actual control input u of the proportional servo valve can be expressed as

$$u=\widehat{DI}(u_d)={\displaystyle \frac{u_d+{\widehat{\rho }}_2}{{\widehat{\rho }}_1}}{\varphi }_r(u_d)+{\displaystyle \frac{u_d+{\widehat{\rho }}_4}{{\widehat{\rho }}_3}}{\varphi }_l(u_d)$$

(48)

The structural diagram of the state-constrained control strategy for the position servo system of a hydraulic robotic arm, designed with valve dead-band compensation, is shown in Figure 3.

Figure 3. Block diagram of state-constrained controller considering valve dead-band compensation.

Both adaptive parameter estimation and disturbance observers are employed to enhance system robustness, but they address different types of uncertainty in decoupled control channels and do not overlap functionally. The disturbance observer is constructed within the mechanical dynamic channel to rapidly estimate unmatched, unstructured uncertainties, such as variations in external payload and unmodeled mechanical friction. In contrast, adaptive laws are deployed in the hydraulic control input channel to address matched, structured parametric uncertainties, including variations in the bulk modulus and asymmetric dead-band characteristics. This dual-mechanism architecture ensures that the adaptive laws absorb steady-state internal hydraulic parameter drifts, thereby preventing the disturbance observer from reaching its limits. This preserves the disturbance observer’s bandwidth, making it possible to reject transient external mechanical disturbances efficiently. This demonstrates the necessity of employing both mechanisms.

3.2. Stability Analysis

Theorem: For the position servo system of a hydraulic robotic arm, under the conditions satisfying Assumptions 1–3, utilizing the discontinuous projection algorithm in Equations (20) and (21), the parameter adaptive laws (46) and (47), the virtual control laws (23) and (32), and the dynamic surface filter function (33), the control function (42) along with the smoothing dead-band inverse transformation (48) work together to ensure the stability of the closed-loop control system. The tracking error $e_1$ asymptotically converges at an exponential rate $\lambda$ to a finite neighborhood $\sqrt{{\displaystyle \frac{\epsilon }{\lambda }}}$.

Proof: The system’s Lyapunov function is defined as follows:

$$V=V_3+{\displaystyle \frac{1}{2}}{\tilde{\mathit{\theta }}}^T\tilde{\mathit{\theta }}+{\displaystyle \frac{1}{2}}{\tilde{\mathit{\rho }}}^T\tilde{\mathit{\rho }}$$

(49)

Taking the derivative with respect to V yields

$$\dot{V}={\dot{V}}_3-{\tilde{\mathit{\theta }}}^T\dot{\widehat{\mathit{\theta }}}-{\tilde{\mathit{\rho }}}^T\dot{\widehat{\mathit{\rho }}}$$

(50)

By substituting Equations (46) and (47) into Equation (50), we obtain

$$\dot{V}\le {\left.{\dot{V}}_2\right|}_{e_3=0}-k_3{e}_3^2-e_3{\beta }_{\mathrm{en}}{g}_x(x){\tilde{\mathit{\rho }}}^T\widehat{\mathit{\omega }}-e_3{\tilde{\mathit{\theta }}}^T\mathit{f}+{\displaystyle \frac{{\delta }_3}{2}}-{\tilde{\mathit{\theta }}}^T\mathit{\gamma }\mathit{\tau }-{\tilde{\mathit{\rho }}}^T\mathit{\Gamma }\mathit{\kappa }$$

(51)

Subsequently, the following can be derived:

$$\begin{array}{l}\dot{V}\le {\left.{\dot{V}}_2\right|}_{e_3=0}-k_3{e}_3^2-e_3{\beta }_{\mathrm{en}}{g}_x(x){\tilde{\mathit{\rho }}}^T\widehat{\mathit{\omega }}-e_3{\tilde{\mathit{\theta }}}^T\mathit{f}+{\displaystyle \frac{{\delta }_3}{2}}+{\tilde{\theta }}_1{\gamma }_1\left[{\displaystyle \frac{f_{\mathrm{x}1}(x)e_3}{{\gamma }_1}}+{\widehat{\theta }}_1\right]\\ +{\tilde{\theta }}_2{\gamma }_2\left[{\displaystyle \frac{f_{\mathrm{x}2}(x)e_3}{{\gamma }_2}}+{\widehat{\theta }}_2\right]+{\tilde{\theta }}_3{\gamma }_3\left[{\displaystyle \frac{f_{\mathrm{x}3}(x)e_3}{{\gamma }_3}}+{\widehat{\theta }}_3\right]+{\tilde{\rho }}_1{\Gamma }_1\left[{\displaystyle \frac{e_3{\beta }_{\mathrm{en}}{g}_x(x){\widehat{\omega }}_1}{{\Gamma }_1}}+{\widehat{\rho }}_1\right]\\ +{\tilde{\rho }}_2{\Gamma }_2\left[{\displaystyle \frac{e_3{\beta }_{\mathrm{en}}{g}_x(x){\widehat{\omega }}_2}{{\Gamma }_2}}+{\widehat{\rho }}_2\right]+{\tilde{\rho }}_3{\Gamma }_3\left[{\displaystyle \frac{e_3{\beta }_{\mathrm{en}}{g}_x(x){\widehat{\omega }}_3}{{\Gamma }_3}}+{\widehat{\rho }}_3\right]\\ +{\tilde{\rho }}_4{\Gamma }_4\left[{\displaystyle \frac{e_3{\beta }_{\mathrm{en}}{g}_x(x){\widehat{\omega }}_4}{{\Gamma }_4}}+{\widehat{\rho }}_4\right]\\ \le {\left.{\dot{V}}_2\right|}_{e_3=0}-k_3{e}_3^2+{\displaystyle \frac{{\delta }_3}{2}}+{\displaystyle \frac{1}{2}}{\tilde{\mathit{\theta }}}^T\mathit{\gamma }\widehat{\mathit{\theta }}+{\displaystyle \frac{1}{2}}{\tilde{\mathit{\rho }}}^T\mathit{\Gamma }\widehat{\mathit{\rho }}\\ \le -{\chi }_1\mathit{ln}{\displaystyle \frac{k_{c1}^2}{k_{c1}^2-e_1^2}}-{\chi }_2\mathit{ln}{\displaystyle \frac{k_{c2}^2}{k_{c2}^2-e_2^2}}-k_3{e}_3^2-{\zeta }_3{z}_3^2-\varsigma {\tilde{\Delta }}_2^2-{\displaystyle \frac{1}{2}}{\tilde{\mathit{\theta }}}^T\mathit{\gamma }\tilde{\mathit{\theta }}-{\displaystyle \frac{1}{2}}{\tilde{\mathit{\rho }}}^T\mathit{\Gamma }\tilde{\mathit{\rho }}\\ +{\displaystyle \frac{{\sigma }_2}{2}}+{\displaystyle \frac{{\left|{\dot{\Delta }}_2\right|}^2}{2}}+{\displaystyle \frac{{\delta }_3}{2}}+{\displaystyle \frac{1}{2}}{\mathit{\theta }}^T\mathit{\gamma }\mathit{\theta }+{\displaystyle \frac{1}{2}}{\mathit{\rho }}^T\mathit{\Gamma }\mathit{\rho }\le -\lambda V+\epsilon \end{array}$$

(52)

In the equation, $\lambda =\mathit{min}\left\{2{\chi }_1,2{\chi }_2,2k_3,2{\zeta }_3,2\varsigma ,{\gamma }_1,{\gamma }_2,{\gamma }_3,{\Gamma }_1,{\Gamma }_2,{\Gamma }_3,{\Gamma }_4\right\}\epsilon ={\displaystyle \frac{{\sigma }_2}{2}}+{\displaystyle \frac{{\left|{\dot{\Delta }}_2\right|}^2}{2}}+{\displaystyle \frac{{\delta }_3}{2}}+{\displaystyle \frac{1}{2}}{\mathit{\theta }}^T\mathit{\gamma }\mathit{\theta }+{\displaystyle \frac{1}{2}}{\mathit{\rho }}^T\mathit{\Gamma }\mathit{\rho }$.

$$\mathrm{Further},V(t)\le e^{-\lambda t}V(0)+{\displaystyle \frac{\epsilon }{\lambda }}\left[1-e^{-\lambda t}\right]$$

(53)

The theorem is thus proven.

4. Simulation Analysis Comparison

To validate the effectiveness of the DZC-BLF controller designed in this paper, a joint simulation platform for the hydraulic robotic arm position servo system was constructed using MATLAB/Simulink and Simcenter Amesim software V2304. The sampling interval for the joint simulation was configured at 1 ms. The physical parameters of the simulation model were determined based on the actual system and are listed in Table 1.

Table 1. Parameters of the system.

Parameters	Values	Parameters	Values
$P_{\mathrm{s}}\left(\mathrm{MPa}\right)$	11.0	$r_{O_1{O}_2}\left(\mathrm{m}\right)$	0.85
$P_{\mathrm{r}}\left(\mathrm{MPa}\right)$	0	$r_{O_1{O}_3}\left(\mathrm{m}\right)$	0.29
${\beta }_{\mathrm{e}}\left(\mathrm{Pa}\right)$	$7.0\times {10}^8$	$r_{O_{1}M}\left(\mathrm{m}\right)$	0.95
$b_{\mathrm{x}}\left(\mathrm{N}\cdot \mathrm{s}/\mathrm{m}\right)$	$2.1\times {10}^3$	${\alpha }_0$ (°)	50.28
$k_{\mathrm{qva}}\left({\mathrm{m}}^3/(\mathrm{sV}\sqrt{\mathrm{Pa}})\right)$	$3.56\times {10}^{-8}$	$A_{\mathrm{a}}\left({\mathrm{m}}^2\right)$	$1.96\times {10}^{-3}$
$k_{\mathrm{qvb}}\left({\mathrm{m}}^3/(\mathrm{sV}\sqrt{\mathrm{Pa}})\right)$	$1.78\times {10}^{-8}$	$A_{\mathrm{b}}\left({\mathrm{m}}^2\right)$	$1.0\times {10}^{-3}$
$C_{\mathrm{ip}}\left({\mathrm{m}}^3/(\mathrm{Pa}\cdot \mathrm{s})\right)$	$1.5\times {10}^{-13}$	$V_{\mathrm{a}0}\left({\mathrm{m}}^3\right)$	$9.9\times {10}^{-5}$
$C_{\mathrm{ep}}\left({\mathrm{m}}^3/(\mathrm{Pa}\cdot \mathrm{s})\right)$	$7.0\times {10}^{-15}$	$V_{\mathrm{b}0}\left({\mathrm{m}}^3\right)$	$3.75\times {10}^{-4}$
$J_{\mathrm{x}}\left(\mathrm{kg}\cdot {\mathrm{m}}^2\right)$	$280$	$m_{\mathrm{x}}\left(\mathrm{kg}\right)$	$210$
$g\left({\mathrm{m}/\mathrm{s}}^2\right)$	$9.8$	$\angle M{O}_1{O}_3$ (°)	30

This paper demonstrates the use of dead-band characteristics in proportional servo valves to verify the effectiveness of the proposed state-constrained control strategy with valve dead-band compensation for a hydraulic robotic arm servo system. Tracking performance simulations are conducted under typical fixed-angle tracking conditions (as shown in Figure 4) and arbitrary-angle tracking conditions (as shown in Figure 5) to verify the control performance of the hydraulic robotic arm position servo system under the control strategy. Furthermore, the dead-band characteristics established in this paper adopt typical nonlinear parameters: $m_r=1.1$, $m_l=0.95$, $b_r=0.8$, and $b_l=-1$.

Figure 4. Fixed-angle tracking of the target trajectory.

Figure 5. Arbitrary-angle tracking trajectory.

To clearly demonstrate the effectiveness of the dead zone compensation control strategy proposed in this paper, a simulation comparison of the controllers is conducted:

(1) State-constrained controller with dead zone compensation: Controller gains are set as $k_{\mathrm{c}1}=0.4$, $k_{\mathrm{c}2}=160$, $k_1=25$, $k_2=115$, $k_3=180$. The disturbance observer gain is set to $\mathcal{l}=20$, with filter parameters ${\tau }_3=1\times 10^{-2}$. The adaptive parameter lower bound is ${\mathit{\theta }}_{\mathit{m}\mathit{i}\mathit{n}}={\left[5.0\times {10}^8,7.5\times {10}^{-7},3.5\times {10}^{-7}\right]}^T$, ${\mathit{\rho }}_{\mathit{m}\mathit{i}\mathit{n}}={\left[0,0,0,-1\right]}^T$, while the upper bound of adaptive parameters is set to ${\mathit{\theta }}_{max}{=\left[2.0\times {10}^9,3.0\times {10}^{-2},1.4\times {10}^{-4}\right]}^T$, ${\mathit{\rho }}_{max}={\left[1.2,1.2,1.2,0\right]}^T$. The initial values for adaptive parameters are selected as $\widehat{\mathit{\theta }}(0)={\left[7.0\times {10}^8,1.05\times {10}^{-4},4.9\times {10}^{-6}\right]}^T$, $\widehat{\mathit{\rho }}(0)={\left[1.0,1.0,1.0,-1.0\right]}^T$, adaptive gain matrix is set to $\mathit{\gamma }=\left\{6.5\times {10}^{-7},4.5\times {10}^{-12},1.2\times {10}^{-14}\right\}$, $\mathit{\Gamma }=\left\{4.2\times {10}^{-13},2.4\times {10}^{-12},1.75\times {10}^{-13},1.2\times {10}^{-12}\right\}$, and smooth dead zone inverse parameter is set to $e_0=0.1$.

(2) State-constrained controller with dead zone influence (DZ-BLF): Parameter settings are consistent with DZC-BLF.

4.1. Fixed-Angle Tracking Control Performance

This section presents comparative simulations of fixed-angle tracking with valve dead-band compensation, as shown in Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13. Figure 6 illustrates the curve of the spring-coil actuator completing the desired 30° fixed-angle trajectory tracking within 1 s under the DZC-BLF controller designed in this section. The designed dead-band compensation algorithm ensures satisfactory tracking performance even in the presence of dead-band. Figure 7 compares the tracking errors of the DZC-BLF and DZ-BLF control strategies under valve dead-band conditions. The designed DZC-BLF exhibits superior dynamic and steady-state tracking errors compared to DZ-BLF. While the steady-state tracking error of DZ-BLF reaches 0.06° due to dead-band effects, the steady-state tracking error of DZC-BLF remains within 0.04°. Adaptive parameter estimation, dead-band parameter estimation, and disturbance observer outputs are shown in Figure 8, Figure 9 and Figure 10, respectively, with all parameters varying smoothly within bounded ranges. Figure 11 shows that the DZC-BLF control output exhibits noticeable jitter during the initial motion phase due to valve dead-band effects. However, as the dead-band adaptive parameters converge (as shown in Figure 9), the control output jitter significantly diminishes. This demonstrates that the designed dead-band compensation algorithm substantially enhances the stability of the elastic coordinator control. Figure 12 and Figure 13 illustrate the pressure curves for both chambers within the coordinating cylinder. Figure 13 reveals pronounced pressure fluctuations during the initial phase of the 30° periodic coordination motion under the DZ-BLF control strategy. However, under the dead-band adaptive compensation algorithm designed for the DZC-BLF controller in this section, the pressure fluctuations in Figure 12 gradually stabilize as the dead-band parameters converge, aligning with the trend of the control variable.

Figure 6. Angle-tracking performance of the DZC-BLF controller.

Figure 7. Comparison of fixed-angle tracking errors.

Figure 8. Adaptive parameter estimation for the DZC-BLF controller.

Figure 9. Estimation of dead zone parameters for the DZC-BLF controller.

Figure 10. Disturbance observer output.

Figure 11. DZC-BLF controller control values.

Figure 12. DZC-BLF controller coordinates pressure in both chambers of the hydraulic cylinder.

Figure 13. DZ-BLF controller coordinates pressure in both chambers of the hydraulic cylinder.

To provide a more direct quantitative evaluation of the physical dead-band compensation effectiveness, two specific indicators are introduced: the maximum transient error during velocity reversal and the effective dead-band reduction rate. The dead-band nonlinearity affects the system most severely when the control voltage crosses zero. As clearly illustrated in the tracking error comparison in Figure 7, the uncompensated DZ-BLF controller exhibits noticeable error spikes of approximately 0.35° during the zero-crossing phases. In contrast, the proposed DZC-BLF controller significantly suppresses this maximum transient error to approximately 0.31°, achieving a 12% reduction in peak error, which demonstrates a substantial improvement in small-signal performance near the valve null position.

4.2. Arbitrary-Angle Tracking Performance

This section examines the comparative simulation of arbitrary-angle tracking with valve dead-band compensation, as shown in Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20 and Figure 21. Figure 14 illustrates the elastic coordinator’s curve successfully following the desired trajectory at any angle, using the DZC-BLF controller developed in this section. Despite the dead-band, it still provides good dynamic monitoring. Figure 15 compares the tracking errors of the two control strategies under the valve’s dead-band characteristics. The dynamically designed DZC-BLF control strategy in this section demonstrates higher tracking accuracy than the DZ-BLF strategy. Furthermore, when the elastic coordinator’s movement direction changes, the tracking error curve of the DZC-BLF is smoother, indicating that the designed controller possesses superior robustness and dynamic performance. Adaptive parameter estimation, dead-band parameter estimation, and disturbance observer outputs are shown in Figure 16, Figure 17 and Figure 18, respectively, with all parameters varying steadily within bounded ranges. As the dead-band adaptive parameter converges, as shown in Figure 17, the jitter of the DZC-BLF control output during direction changes, as depicted in Figure 19, diminishes, resulting in more stable control performance. Analyzing the pressure curves of both chambers in the coordination process cylinder, as shown in Figure 20 and Figure 21, further confirms that the DZC-BLF controller developed here has smaller pressure fluctuations when the elastic coordinator reverses direction. Moreover, the fluctuation trend decreases significantly as the dead-band parameter converges, corroborating the DZC-BLF control strategy’s ability to compensate for dead-band effects and achieve stable control performance.

Figure 14. DZC-BLF controller tracking performance at arbitrary angles.

Figure 15. Comparison of tracking error at arbitrary angles.

Figure 16. Adaptive parameter estimation for the DZC-BLF controller.

Figure 17. Estimation of dead zone parameters for the DZC-BLF controller.

Figure 18. Disturbance observer output.

Figure 19. DZC-BLF controller control values.

Figure 20. DZC-BLF controller coordinates pressure in both chambers of the hydraulic cylinder.

Figure 21. DZ-BLF controller coordinates pressure in both chambers of the hydraulic cylinder.

5. Conclusions

To tackle the dead-band issue common in electro-hydraulic servo systems, a smooth inverse dead-band function is used to reconstruct the state equations of an elastic coordinator electro-hydraulic servo system featuring dead-band characteristics. Adaptive estimation of the dead-band parameters for this model is performed. Additionally, parameter uncertainty is compensated for by an adaptive parameter estimation law. Unmodeled mismatched disturbances and matched disturbances are addressed using disturbance observer technology and nonlinear robust stabilization techniques, respectively. Similarly, the system state is constrained by an obstacle Lyapunov function. Simulation results demonstrate that the proposed state-constrained controller with valve dead-band compensation effectively achieves dead-band compensation and state-constrained control for the elastic coordinator.

(1)

For state-space models of electro-hydraulic servo systems with parameter uncertainties, an adaptive dynamic surface control strategy was designed based on inversion principles. To overcome the “differential series explosion” challenge encountered when differentiating virtual control functions during inversion, the differential operation of the virtual control variable was replaced with the output of a first-order filter. Simultaneously, a parameter adaptation algorithm was employed to compensate for control performance degradation caused by parameter uncertainty, achieving superior performance compared to PID controllers.

(2)

Considering that the operation of electro-hydraulic servo systems requires boundary constraints on system states to prevent safety issues such as actuator failure caused by mechanism positions exceeding specified limits, an obstacle Lyapunov function is introduced to constrain system states. This ensures the elastic coordinator operates within the desired dynamic error range. Combining dynamic surfaces with adaptive methods, an adaptive dynamic surface-based state-constrained controller is proposed to achieve motion control of the elastic coordinator. Simulation studies show that the proposed algorithm delivers better expected trajectory tracking performance than adaptive dynamic surface control strategies. Both dynamic and steady-state tracking achieve high precision, fully satisfying the specified state constraints. Smooth pressure dynamics further indicate that the proposed algorithm exhibits stable tracking performance.

(3)

To further address state-constrained control problems involving control valves with dead-band characteristics, a state-constrained controller with valve dead-band compensation is proposed. It employs a smooth dead-band inverse transformation to counteract the dead-band characteristics of proportional servo valves. To address parameter and dead-band uncertainties in the system, adaptive laws are designed to estimate parameter variations in real time and provide control compensation. An observer is introduced to compensate for mismatch uncertainties in the electro-hydraulic servo system. Furthermore, a barrier Lyapunov function is employed to constrain the system state, ensuring the specified accuracy of the position error. Simulation comparisons demonstrate that dead-band characteristics degrade system control accuracy and stability. The proposed state-constrained controller with valve dead-band compensation effectively compensates for dead-band while maintaining state constraints, thereby achieving the required control accuracy. As the adaptive parameters converge, the effectiveness of dead-band compensation and control stability improve significantly. Due to the current limitations of our experiments, we conducted the validation of the algorithm in this paper solely through simulations. However, we will actively seek opportunities to validate the algorithm through physical experiments in the future.

This controller is designed for high-precision hydraulic servo systems characterized by strong nonlinearity, hard state constraints, multi-source composite disturbances, and significant actuator dead zones. It achieves this by integrating the Barrier of Lyapunov Function (BLF), dynamic surface control (DSC), adaptive parameter estimation, disturbance observer, and smooth dead zone inverse compensation mechanism. Compared with existing BLF adaptive methods, this work takes an innovative approach by directly imposing BLF constraints on the original tracking error variables. Within the DSC structure, it explicitly compensates for the impact of filtering errors on constraint performance, ensuring that system outputs strictly meet designated safety boundaries even with simplified computations. Simultaneously, for integrated disturbances such as parameter uncertainties, unmodeled dynamics, complex friction, and external impacts, the adaptive law and disturbance observer employ layered processing to handle structured parameter perturbations and disturbances of unknown form. Combined with a dead-band inverse model, this approach effectively enhances steady-state accuracy in the small-signal region. Experimental validation employed a composite operating condition combining large-range rapid motion with small-amplitude precision adjustments. Results demonstrate that the proposed method significantly exceeds the comprehensive performance of existing solutions while ensuring full-range state constraints.