Adaptive Parameter Identification Based Tracking Control of Servo Systems with Unknown Actuator Backlash Compensation

Abstract

This paper presents a robust tracking control strategy for servo systems with unknown backlash, employing adaptive parameter identification to address performance degradation caused by backlash nonlinearities. In high-precision positioning and rapid-response applications, backlash significantly compromises system performance. To address this challenge, a servo system model incorporating backlash nonlinearities is developed, and a novel adaptive inverse function is introduced for backlash compensation. The estimation error of unidentified parameters is indirectly obtained through the design of a state observer. Minimizing the estimation error facilitates the accurate identification of model parameters, encompassing those associated with backlash. Additionally, an adaptive law is designed to estimate the unknown upper bounds of disturbance dynamics. Then, a robust tracking controller is proposed, which dynamically adjusts control inputs in real time based on identified backlash parameters to counteract backlash-induced adverse effects. Theoretical analysis and simulation results demonstrate that the proposed strategy significantly improves tracking performance in servo systems with unknown backlash.

1. Introduction

In the modern industrial automation and precision machinery sectors, servo systems serve as pivotal components of actuation mechanisms, with their performance directly determining the systems control accuracy and response speed [1,2,3]. Despite their critical role, servo systems in practical applications are often compromised by backlash phenomena [4]—a mechanical issue arising from minute gaps between components, such as gears or couplings, in the transmission chain due to manufacturing tolerances, wear, or assembly errors [5]. The presence of backlash introduces nonlinear characteristics into the system, leading to unpredictable behavior and degrading positioning accuracy and dynamic performance. This issue becomes particularly pronounced in high-precision applications, where even minor backlash can cause oscillations, delays, or inaccuracies in positioning, ultimately undermining the system’s ability to meet stringent performance requirements [6,7].

To mitigate the negative effects of actuator backlash, an adaptive smoothing backlash inverse model was developed in [8], which uses a smooth function to approximate and compensate for the discontinuous effects of backlash. This method, first introduced in [9] and further extended in [10,11,12], effectively reduces performance degradation caused by actuator gaps. However, it increases system complexity and faces challenges such as slow convergence during parameter tuning and potential instability in dynamic environments. To address these limitations, a Barrier Lyapunov Function (BLF) was proposed in [13], incorporating an additional term in the control law design that creates a virtual barrier to prevent the system state from entering nonlinear regions induced by backlash [14]. While effective in some cases, the BLF approach can lead to instability in systems with complex or uncertain dynamics due to its reliance on precise knowledge of system parameters and nonlinearities. Building on these insights, this paper introduces a novel adaptive inverse function that dynamically adjusts to system uncertainties and backlash variations, ensuring robust performance without compromising stability [15,16].

In traditional adaptive control approaches, the primary objective is to ensure that the system can accurately follow a predetermined reference trajectory [17]. However, in many cases, due to insufficient excitation signals, the estimation errors of the system parameters may not converge completely to zero [18]. Achieving complete convergence of parameter estimation errors is crucial for enhancing the stability and robustness of closed-loop adaptive control systems [19,20]. To address this challenge, various advanced methods have been proposed. Five sliding mode observers were introduced in [21] for dual-inertia servo systems to accurately identify mechanical parameters, thereby preventing servo resonance suppression failures caused by mechanical parameter mismatches. Additionally, an improved adaptive genetic identification method was investigated in [22] for electro-hydraulic servo systems to perform offline identification of the dynamic and static parameters of the LuGre friction model. However, this algorithm cannot reflect real-time variations in friction parameters. Furthermore, a parallel-cascaded extended sliding-mode observer was proposed in [23] for PMSM servo systems to obtain multiple mutually coupled parameters, offering the advantages of low computational complexity and easy implementation.

In addition, in the design and analysis of control systems, the presence of disturbances is inevitable [24,25]. These disturbances may arise from changes in the external environment, internal system uncertainties, or model inaccuracies [26,27]. Such disturbances pose significant challenges to system performance and stability, particularly in applications that demand high precision and reliability. In response to these challenges, advanced disturbance observation techniques have been developed. An adaptive refined disturbance observer was designed in [28] for gimbal servo systems to address the degradation of tracking performance caused by multiple disturbances. This observer consists of a traditional disturbance observer and an adaptive sliding mode observer, which are used to estimate rotor imbalance disturbance and residual disturbances with unknown change rates, respectively. Additionally, a novel disturbance identification observer was investigated in [29] to simultaneously identify and estimate dynamic rotor imbalance disturbances. In this method, the disturbance value and internal parameters are updated in real time by iteratively solving the state estimation and model identification problems within an expectation–maximization framework.

This paper proposes an adaptive sliding mode control method incorporating adaptive parameter identification for robust control of servo systems with nonlinear backlash. By introducing a novel adaptive inverse function, the system model with backlash is linearly parameterized. A state observer is then designed to obtain system parameter estimation error, including backlash parameters, while an online parameter estimation law provides real-time parameter updates. To enhance robustness, an adaptive law estimates the upper bound of disturbances, enabling effective compensation that reduces their impact on tracking performance. Finally, an adaptive sliding mode controller is developed, utilizing the parameter identification results and disturbance bound estimation to significantly improve both tracking performance and system robustness.

2. System Description

This section presents the system model. Based on the literature [16], the dynamic model of the inertial load in a servo system can be expressed as follows:

$$m\ddot{y}=K_{i}Bl\left(u\right)-B\dot{y}+f(y,\dot{y},t)$$

(1)

where m represents the moment of inertia, y denotes the angular displacement, $\dot{y}$ indicates the angular velocity, $K_i$ is the torque constant, B represents the viscous friction coefficient of the servo motor, and $f(y,\dot{y},t)$ is other system disturbances, including inertial load and various uncertainties and their effects. $Bl\left(u\right)$ signifies the unknown characteristics of the brake backlash. The relationship between the control input u and the backlash characteristics is given by the following equation:

$$Bl\left(u\right)=\left\{\begin{array}{cc}b(u\left(t\right)-s_r)\hfill & \mathrm{if}\dot{u}\left(t\right)>0\hfill \\ b(u\left(t\right)-s_l)\hfill & \mathrm{if}\dot{u}\left(t\right)<0\hfill \\ Bl\left(u\left(t_{-}\right)\right)\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(2)

where $b>0$ represents the slope of the backlash model, $s_r$ is a positive parameter, $s_l$ is a negative parameter, and $Bl\left(u\left(t_{-}\right)\right)$ indicates the backlash output value from the previous time step.

Based on Dynamic Model (1), the state vector x is defined as $x={\left[\begin{array}{cc}{x}_1& x_2\end{array}\right]}^T={\left[\begin{array}{cc}y& \dot{y}\end{array}\right]}^T$, where $x_1$ and $x_2$ denote the position signal and the velocity signal of the motor, respectively. The system model can be expressed in the following state-space formulation:

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2\hfill \\ m{\dot{x}}_2=K_{i}Bl\left(u\right)-B{x}_2+f(x_1,x_2,t)\hfill \end{array}\right.$$

(3)

To further eliminate or reduce the impact of backlash present in mechanical transmission systems on overall system performance, it is essential to ensure that the servo motor can accurately and stably follow the command signal during reverse motion, thereby enhancing the system’s precision, stability, and dynamic response capabilities. This paper addresses the backlash model within the system framework and proposes a novel adaptive inverse function to solve the compensation issue associated with backlash.

We let $V_I(•)$ denote the new backlash inverse function, which is expressed as follows [15,16]:

$$V_I\left(Bla\right)={\displaystyle \frac{1}{b}}Bla+s_r·rr+s_l·rl$$

(4)

where $Bla$ is a smooth and differentiable function of Model (2), $rr={\displaystyle \frac{\pi +2arctan\left(n_1\dot{Bla}\right)}{2\pi }}$, $rl={\displaystyle \frac{\pi -2arctan\left(n_1\dot{Bla}\right)}{2\pi }}$ is defined such that the smooth arctangent function ensures its continuity. $n_1$ is a positive tuning parameter.

Remark 1.

Both $rr$ and $rl$ satisfy the following properties:

• $rr$ and $rl$ are both continuously differentiable;
• For any $t>0$, $rr+rl=1$;
• For any $t>0$, $\left|rr\right|\le 1$ and $\left|rl\right|\le 1$;
• When $\dot{Bla}\to \infty$, then $rr\to 1$ and $rl\to 0$;
• When $\dot{Bla}\to -\infty$, then $rr\to 0$ and $rl\to 1$.

Further considering the unmeasurable nature of the unknown parameters b, $s_r$, $s_l$, and $Bl\left(u\right)$, the backlash nonlinearity can be linearized and parameterized as follows:

$$Bla=bu-rr·b·s_r-rl·b·s_l$$

(5)

The system model can also be further expressed as

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2\hfill \\ {\dot{x}}_2={\displaystyle \frac{b}{\theta }}u-v_2{x}_2-v_3·rr-v_4·rl+d\left(t\right)\hfill \end{array}\right.$$

(6)

where ${\displaystyle \frac{b}{\theta }}={\displaystyle \frac{K_i}{m}}b$, $v_2={\displaystyle \frac{B}{m}}$, $v_3={\displaystyle \frac{K_i}{m}}b·s_r$, $v_4={\displaystyle \frac{K_i}{m}}b·s_l$, $d\left(t\right)={\displaystyle \frac{f(x_1,x_2,t)}{m}}$.

Remark 2.

The proposed adaptive inverse function dynamically adjusts to system uncertainties while enabling flexible constraint handling. By reformulating the backlash model, adaptive parameter estimation suppresses backlash-induced performance loss. Simultaneously, inverting the backlash operator $Bl\left(u\right)$ to derive u simplifies controller design and lowers computational costs.

Remark 3.

For system Equation (1) with backlash nonlinearity, solution existence follows from two physical realities: (i) the equation models an implementable physical system, and (ii) the system’s controllability and stability (via Lyapunov analysis) ensure input–output correspondence. Thus, solutions exist for physically realizable reference trajectories.

3. Main Results

This section primarily focuses on adaptive parameter identification, controller design, and stability analysis, which collectively demonstrate the feasibility of this paper.

3.1. Adaptive Parameter Identification

During the operation of the servo system, real-time adjustments of system parameters are implemented to account for both intrinsic and extrinsic variations, thereby enhancing system performance and ensuring stable and efficient operation. Specifically, the system parameters may encompass physical properties such as the motor’s moment of inertia, damping coefficient, and stiffness, all of which significantly influence the dynamic response and steady-state precision of the servo system [30]. Disturbance boundary parameters pertain to various external or internal uncertainties that impact system performance, including load fluctuations, frictional forces, and thermal variations. The threshold values of these factors are critical for sustaining stable system operation. Through adaptive parameter identification, the system is equipped to detect real-time variations in these parameters and disturbance boundaries, autonomously modifying control strategies to ensure optimal system performance [27,31,32,33].

We define $\widehat{•}$ to represent the estimate of •; $\widetilde{•}=•-\widehat{•}$ denote the parameter estimation error. To separate the uncertain parameters from the control input ${\displaystyle \frac{b}{\theta }}u$, we introduce an online estimation signal $\widehat{g}$ to represent the estimate of $g={\displaystyle \frac{\theta }{b}}$. Thus, the actual controller u can be expressed as

$$u=\widehat{g}\overline{u}$$

(7)

where $\overline{u}$ represents an intermediate variable. Then, ${\displaystyle \frac{b}{\theta }}u$ in Equation (6) can be written as

$${\displaystyle \frac{b}{\theta }}u={\displaystyle \frac{b}{\theta }}\widehat{g}\overline{u}={\displaystyle \frac{b}{\theta }}(g-\tilde{g})\overline{u}=\overline{u}-{\displaystyle \frac{b}{\theta }}\tilde{g}\overline{u}$$

(8)

The system model can be further expressed as

$${\dot{x}}_2=\overline{u}+\Phi \left(x\right)v+d\left(t\right)$$

(9)

where $\Phi \left(x\right)=\left[\begin{array}{cccc}\overline{u}& -x_2& -rr& -rl\end{array}\right]$, $v={\left[\begin{array}{cccc}-{\displaystyle \frac{b}{\theta }}\tilde{g}& {\displaystyle \frac{B}{m}}& {\displaystyle \frac{K_{i}b}{m}}{s}_r& {\displaystyle \frac{K_{i}b}{m}}{s}_l\end{array}\right]}^T$. We define $v_1=-{\displaystyle \frac{b}{\theta }}\tilde{g}$, and $d\left(t\right)$ is a lumped disturbance.

Assumption: The lumped disturbance $d\left(t\right)$ is bounded within a unknown range such that $d\left(t\right)<{\delta }_0+{\delta }_1\left|e\right|+{\delta }_2\left|\dot{e}\right|$ [27], where ${\delta }_0$, ${\delta }_1$, ${\delta }_2$ are normal constants representing the upper boundary parameter for lumped disturbances. e denotes the system’s tracking error, defined as $e=x_1-x_d$, where $x_d$ is the desired trajectory.

We define ${\widehat{x}}_2$ as the state estimation of Equation (9), which is structured as follows:

$${\dot{\widehat{x}}}_2=\overline{u}+\Phi \left(x\right)\widehat{v}+k_w(x_2-{\widehat{x}}_2)+w\dot{\widehat{v}}$$

(10)

where $\widehat{v}$ is the parameter estimate generated through automatic update laws $\dot{\widehat{v}}$, $k_w$ is a positive number, and w is the output of the following filter:

$$\dot{w}=\Phi \left(x\right)-k_{w}w,w\left(t_0\right)=0$$

(11)

The parameter estimation error can be obtained as follows:

$${\dot{\tilde{x}}}_2=\Phi \left(x\right)\tilde{v}-k_w{\tilde{x}}_2-w\dot{\widehat{v}}$$

(12)

We define the auxiliary variable $\eta$:

$$\eta =(x_2-{\widehat{x}}_2)-w\tilde{v}$$

(13)

Then, $\dot{\eta }$ can be generated:

$$\begin{array}{cc}\hfill \dot{\eta }& =({\dot{x}}_2-{\dot{\widehat{x}}}_2)-\dot{w}\tilde{v}-w\dot{\tilde{v}}\hfill \\ \hfill & ={\dot{\tilde{x}}}_2-\dot{w}\tilde{v}+w\dot{\widehat{v}}\hfill \end{array}$$

(14)

Substituting (11)–(13) into above equation yields

$$\begin{array}{cc}\hfill \dot{\eta }& =\Phi \left(x\right)\tilde{v}-k_w(x_2-{\widehat{x}}_2)-\dot{w}\tilde{v}\hfill \\ \hfill & =-k_w(x_2-{\widehat{x}}_2)+k_{w}w\tilde{v}\hfill \\ \hfill & =-k_w(x_2-{\widehat{x}}_2-w\tilde{v})\hfill \\ \hfill & =-k_w\eta ,\eta \left(t_0\right)=x_{20}\left(t_0\right)-{\widehat{x}}_{20}\left(t_0\right)\hfill \end{array}$$

(15)

where $x_{20}\left(t_0\right)$ and ${\widehat{x}}_{20}\left(t_0\right)$ are the initial values and initial estimates of state $x_2$, and $t_0$ is the initial time of the system.

Based on (10), (11), and (15), the following conclusion can be derived.

We let $Q\in R^{n\theta \times n\theta }$ and $C\in R^{n\theta }$ be dynamically generated by the following dynamics, where $n\theta$ represents the dimensions of $\Phi \left(x\right)$ and v, and $n\theta =4$.

$$\dot{Q}=w^{T}w,Q\left(t_0\right)=0$$

(16)

$$\dot{C}=w^T(w{v}^0+(x-\widehat{x})-\eta ),C\left(t_0\right)=0$$

(17)

$$v^c\stackrel{\Delta }{=}Q{\left(t_c\right)}^{-1}C\left(t_c\right)$$

(18)

where $t_c$ represents the finite convergence time, meaning the estimated parameters reach their true values $v_c$ when $t>t_c$.

Then, the adaptive laws can be generated as follows [6]:

$$\dot{\widehat{v}}=\Gamma (C-Q\widehat{v}),\widehat{v}\left(t_0\right)=v^0$$

(19)

where $\Gamma =diag\left\{\begin{array}{cccc}{r}_1& r_2& r_3& r_4\end{array}\right\}$ is the adaptive rate matrix. Derived from Equation (18), $C=Qv$; thus, $C-Q\widehat{v}=Q\tilde{v}$.

Remark 4.

The state $x_2$ in System (9) is directly measurable. However, the state observer (10) is not designed to estimate $x_2$; instead, its purpose is to derive the parameter estimation error required for adaptive parameter identification. Specifically, when the parameter estimation error approaches zero, the estimated parameters converge to their true values. Notably, while System (9) includes a bounded lumped disturbance $d\left(t\right)$, this disturbance is ignored in the design of the observer (10). Consequently, under the influence of $d\left(t\right)$, the estimated parameters finally converge to a small neighborhood around the true values.

Remark 5.

The time-varying upper bound assumption for $d\left(t\right)$ is motivated by the periodic disturbances inherent in servo systems with repetitive motion. Unlike a constant bound, associating the disturbance bound with the tracking error e accounts for motion-dependent variations, yielding a more accurate and more practical estimate. This method is consistent with prior work in [27].

Remark 6.

Suppose there exists a time $t_c$ and a constant $c_1>0$ such that $Q\left(t_c\right)$ is invertible, i.e., $Q\left(t_c\right)={\int }_{t_0}^{t_c}{w}^t\left(\tau \right)w\left(\tau \right)d\tau \succ c_{1}I$; then, $v^c=Q{\left(t_c\right)}^{-1}C\left(t_c\right)$ for all $t\ge t_c$ [31].

3.2. Controller Design

This section presents the design of the controller based on the system’s characteristics and performance requirements. By employing meticulously crafted control strategies and parameter optimization, precise regulation of the system’s behavior is achieved. This ensures stable operation, enhances response speed, increases anti-interference capability, and meets the desired control accuracy.

We define the tracking error as

$$e=x_1-x_d$$

(20)

In the equation, $x_1$ denotes the actual position of the system while $x_d$ represents the desired trajectory. The derivative of e is given by

$$\dot{e}=x_2-{\dot{x}}_d$$

(21)

We define a sliding mode surface $s=\dot{e}+\lambda e$, and its derivative is obtained as

$$\begin{array}{cc}\hfill \dot{s}& ={\dot{x}}_2-{\ddot{x}}_d+\lambda \dot{e}\hfill \\ \hfill & =\overline{u}+\Phi \left(x\right)v+d\left(t\right)-{\ddot{x}}_d+\lambda \dot{e}\hfill \end{array}$$

(22)

Subsequently, the control law $\overline{u}$ can be analyzed and obtained as

$$\begin{array}{cc}\hfill \overline{u}=& -\Phi \left(x\right)\widehat{v}+{\ddot{x}}_d-\lambda \dot{e}-{\widehat{\delta }}_{0}tanh\left({\displaystyle \frac{s}{{\mu }_1}}\right)\hfill \\ \hfill & -\left|e\right|{\widehat{\delta }}_{1}tanh\left({\displaystyle \frac{s\left|e\right|}{{\mu }_2}}\right)-\left|\dot{e}\right|{\widehat{\delta }}_{2}tanh\left({\displaystyle \frac{s|\dot{e}|}{{\mu }_3}}\right)-k_{1}s\hfill \end{array}$$

(23)

where $\lambda >0,k_1>0$ represents the control gain, ${\widehat{\delta }}_0$, ${\widehat{\delta }}_1$, ${\widehat{\delta }}_2$ represent the estimated value of the disturbance boundary coefficient ${\delta }_0$, ${\delta }_1$ and ${\delta }_2$.

The improved adaptive parameter identification is investigated as follows:

$$\dot{\widehat{v}}=s\Gamma \Phi {\left(x\right)}^T+\Gamma (C-Q\widehat{v})$$

(24)

The adaptive law design for perturbation parameters is

$${\dot{\widehat{\delta }}}_0=c_1\left(\right|s|tanh({\displaystyle \frac{\left|s\right|}{3{\mu }_1}})-c_4{\widehat{\delta }}_0)$$

(25)

$${\dot{\widehat{\delta }}}_1=c_2\left(\right|s|·\left|e\right|tanh({\displaystyle \frac{\left|s\right|·\left|e\right|}{3{\mu }_2}})-c_5{\widehat{\delta }}_1)$$

(26)

$${\dot{\widehat{\delta }}}_2=c_3\left(\right|s|·|\dot{e}|tanh({\displaystyle \frac{\left|s\right|·|\dot{e}|}{3{\mu }_3}})-c_6{\widehat{\delta }}_2)$$

(27)

where $c_i>0,i=1,2,3,4,5,6$, ${\mu }_i>0,i=1,2,3$ are the adjustment parameters.

3.3. Stability Analysis

To demonstrate the stability of the system, a lemma and theorem are given as follows:

Lemma 1

([34]). For any real number $x\in R$, with $\mu >0$ and $k=0.2785$, the following relationship holds:

$$0<\left|x\right|-xtanh\left(\mu x\right)\le {\displaystyle \frac{k}{\mu }}$$

(28)

Theorem 1.

For a system with backlash as given in (1), the tracking error can be driven to a uniformly ultimately bounded (UUB) region through deriving a system controller (23), an adaptive parameter estimation law (24), and parameter adaptive laws with disturbance upper bounds (25)–(27).

Proof. 

The following Lyapunov function is devised:

$$\begin{array}{cc}\hfill V=& {\displaystyle \frac{1}{2}}{s}^2+{\displaystyle \frac{1}{2r_1}}{\tilde{v}}_1^2+{\displaystyle \frac{1}{2r_2}}{\tilde{v}}_2^2+{\displaystyle \frac{1}{2r_3}}{\tilde{v}}_3^2+{\displaystyle \frac{1}{2r_4}}{\tilde{v}}_4^2\hfill \\ \hfill & +{\displaystyle \frac{1}{2c_1}}{\tilde{\delta }}_0^2+{\displaystyle \frac{1}{2c_2}}{\tilde{\delta }}_1^2+{\displaystyle \frac{1}{2c_3}}{\tilde{\delta }}_2^2\hfill \end{array}$$

(29)

By differentiating (29), we obtain

$$\begin{array}{cc}\hfill \dot{V}=& s\dot{s}-{\displaystyle \frac{1}{r_1}}{\tilde{v}}_1{\dot{\widehat{v}}}_1-{\displaystyle \frac{1}{r_2}}{\tilde{v}}_2{\dot{\widehat{v}}}_2-{\displaystyle \frac{1}{r_3}}{\tilde{v}}_3{\dot{\widehat{v}}}_3-{\displaystyle \frac{1}{r_4}}{\tilde{v}}_4{\dot{\widehat{v}}}_4\hfill \\ \hfill & -{\displaystyle \frac{1}{c_1}}{\tilde{\delta }}_0{\dot{\widehat{\delta }}}_0-{\displaystyle \frac{1}{c_2}}{\tilde{\delta }}_1{\dot{\widehat{\delta }}}_1-{\displaystyle \frac{1}{c_3}}{\tilde{\delta }}_2{\dot{\widehat{\delta }}}_2\hfill \end{array}$$

(30)

From Equations (22)–(27), we can further derive

$$\begin{array}{cc}\hfill \dot{V}=& -{\tilde{v}}^T{\Gamma }^{-1}\dot{\widehat{v}}-{\displaystyle \frac{1}{c_1}}{\tilde{\delta }}_0{\dot{\widehat{\delta }}}_0-{\displaystyle \frac{1}{c_2}}{\tilde{\delta }}_1{\dot{\widehat{\delta }}}_1-{\displaystyle \frac{1}{c_3}}{\tilde{\delta }}_2{\dot{\widehat{\delta }}}_2\hfill \\ \hfill & +s(\Phi \left(x\right)\tilde{v}+d\left(t\right)-{\widehat{\delta }}_{0}tanh({\displaystyle \frac{s}{{\mu }_1}})-|e|{\widehat{\delta }}_{1}tanh({\displaystyle \frac{s\left|e\right|}{{\mu }_2}})\hfill \\ \hfill & -|\dot{e}|{\widehat{\delta }}_{2}tanh({\displaystyle \frac{s|\dot{e}|}{{\mu }_3}})-k_{1}s)\hfill \\ \hfill =& -{\tilde{v}}^{T}Q\tilde{v}-{\tilde{\delta }}_0\left(\right|s|tanh({\displaystyle \frac{\left|s\right|}{3{\mu }_1}})-c_4{\widehat{\delta }}_0)-{\tilde{\delta }}_1\left(\right|s|·\left|e\right|\hfill \\ \hfill & \times tanh({\displaystyle \frac{\left|e\right|·\left|s\right|}{3{\mu }_2}})-c_5{\widehat{\delta }}_1)-{\tilde{\delta }}_2\left(\right|s|·|\dot{e}|tanh({\displaystyle \frac{\left|\dot{e}\right|·\left|s\right|}{3{\mu }_3}})\hfill \\ \hfill & -c_6{\widehat{\delta }}_2)-s{\widehat{\delta }}_{0}tanh({\displaystyle \frac{s}{{\mu }_1}})-s\left|e\right|{\widehat{\delta }}_{1}tanh({\displaystyle \frac{s\left|e\right|}{{\mu }_2}})\hfill \\ \hfill & -s|\dot{e}|{\widehat{\delta }}_{2}tanh({\displaystyle \frac{s|\dot{e}|}{{\mu }_3}})+sd\left(t\right)-k_1{s}^2\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill \dot{V}\le & -{\tilde{v}}^{T}Q\tilde{v}-k_1{s}^2-{\tilde{\delta }}_0\left(\right|s|tanh({\displaystyle \frac{\left|s\right|}{3{\mu }_1}})-c_4{\widehat{\delta }}_0)\hfill \\ \hfill & -{\tilde{\delta }}_1\left(\right|s|·\left|e\right|tanh({\displaystyle \frac{\left|e\right|·\left|s\right|}{3{\mu }_2}})-c_5{\widehat{\delta }}_1)\hfill \\ \hfill & -{\tilde{\delta }}_2\left(\right|s|·|\dot{e}|tanh({\displaystyle \frac{\left|\dot{e}\right|·\left|s\right|}{3{\mu }_3}})-c_6{\widehat{\delta }}_2)-s{\widehat{\delta }}_{0}tanh({\displaystyle \frac{s}{{\mu }_1}})\hfill \\ \hfill & -s\left|e\right|{\widehat{\delta }}_{1}tanh({\displaystyle \frac{s\left|e\right|}{{\mu }_2}})-s\left|\dot{e}\right|{\widehat{\delta }}_{2}tanh({\displaystyle \frac{s|\dot{e}|}{{\mu }_3}})\hfill \\ \hfill & +\left|s\right|({\delta }_0+{\delta }_1|e|+{\delta }_2|\dot{e}\left|\right)\hfill \end{array}$$

(32)

From Lemma 1, we can derive the following relationship:

$$-s{\widehat{\delta }}_{0}tanh({\displaystyle \frac{s}{{\mu }_1}})\le -\left|s\right|{\widehat{\delta }}_0+3{\widehat{\delta }}_0{\mu }_{1}k$$

(33)

$$-s\left|e\right|{\widehat{\delta }}_{1}tanh({\displaystyle \frac{s\left|e\right|}{{\mu }_2}})\le -\left|e\right|·\left|s\right|{\widehat{\delta }}_1+3{\widehat{\delta }}_1{\mu }_{2}k$$

(34)

$$-s\left|\dot{e}\right|{\widehat{\delta }}_{2}tanh({\displaystyle \frac{s|\dot{e}|}{{\mu }_2}})\le -\left|\dot{e}\right|·\left|s\right|{\widehat{\delta }}_2+3{\widehat{\delta }}_2{\mu }_{3}k$$

(35)

Subsequently, Equation (32) can be derived as follows:

$$\begin{array}{cc}\hfill \dot{V}& \le -{\tilde{v}}^{T}Q\tilde{v}-k_1{s}^2-{\tilde{\delta }}_0\left(\right|s|tanh({\displaystyle \frac{\left|s\right|}{3{\mu }_1}})-c_4{\widehat{\delta }}_0)\hfill \\ \hfill & -{\tilde{\delta }}_1\left(\right|s|·\left|e\right|tanh({\displaystyle \frac{\left|e\right|·\left|s\right|}{3{\mu }_2}})-c_5{\widehat{\delta }}_1)\hfill \\ \hfill & -{\tilde{\delta }}_2\left(\right|s|·|\dot{e}|tanh({\displaystyle \frac{\left|\dot{e}\right|·\left|s\right|}{3{\mu }_3}})-c_6{\widehat{\delta }}_2)\hfill \\ \hfill & +\left|s\right|{\delta }_0+\left|s\right|·\left|e\right|{\delta }_1+\left|s\right|·\left|\dot{e}\right|{\delta }_2-\left|s\right|{\widehat{\delta }}_0\hfill \\ \hfill & -\left|s\right|·\left|e\right|{\widehat{\delta }}_1-\left|s\right|·\left|\dot{e}\right|{\widehat{\delta }}_2+3{\widehat{\delta }}_0{\mu }_{1}k+3{\widehat{\delta }}_1{\mu }_{2}k\hfill \\ \hfill & +3{\widehat{\delta }}_2{\mu }_{3}k\hfill \\ \hfill & \le -{\tilde{v}}^{T}Q\tilde{v}-k_1{s}^2-{\tilde{\delta }}_0\left|s\right|tanh({\displaystyle \frac{\left|s\right|}{3{\mu }_1}})\hfill \\ \hfill & -{\tilde{\delta }}_1\left|s\right|·\left|e\right|tanh({\displaystyle \frac{\left|e\right|·\left|s\right|}{3{\mu }_2}})\hfill \\ \hfill & -{\tilde{\delta }}_2\left|s\right|·\left|\dot{e}\right|tanh({\displaystyle \frac{|\dot{e}|·|s|}{3{\mu }_3}})\hfill \\ \hfill & +c_4{\tilde{\delta }}_0{\widehat{\delta }}_0+c_5{\tilde{\delta }}_1{\widehat{\delta }}_1+c_6{\tilde{\delta }}_2{\widehat{\delta }}_2+\left|s\right|{\tilde{\delta }}_0+\left|s\right|·\left|e\right|{\tilde{\delta }}_1\hfill \\ \hfill & +\left|s\right|·\left|\dot{e}\right|{\tilde{\delta }}_2+3k({\widehat{\delta }}_0{\mu }_1+{\widehat{\delta }}_1{\mu }_2+{\widehat{\delta }}_2{\mu }_3)\hfill \\ \hfill & \le -{\tilde{v}}^{T}Q\tilde{v}-k_1{s}^2+3{\tilde{\delta }}_0{\mu }_{1}k+3{\tilde{\delta }}_1{\mu }_{2}k+3{\tilde{\delta }}_2{\mu }_{3}k\hfill \\ \hfill & +c_4{\tilde{\delta }}_0{\widehat{\delta }}_0+c_5{\tilde{\delta }}_1{\widehat{\delta }}_1+c_6{\tilde{\delta }}_2{\widehat{\delta }}_2+3k({\widehat{\delta }}_0{\mu }_1+{\widehat{\delta }}_1{\mu }_2+{\widehat{\delta }}_2{\mu }_3)\hfill \\ \hfill & \le -v^{T}Qv-k_1{s}^2-{\delta }_0-{{\widehat{\delta }}_0}^2-{\delta }_1-{{\widehat{\delta }}_1}^2\hfill \\ \hfill & -{\delta }_2-{{\widehat{\delta }}_2}^2+\Delta \hfill \end{array}$$

(36)

where $\Delta ={\displaystyle \frac{c_4{\delta }_0^2}{4(c_4-1)}}+{\displaystyle \frac{c_5{\delta }_1^2}{4(c_5-1)}}+{\displaystyle \frac{c_6{\delta }_2^2}{4(c_6-1)}}+3k({\mu }_1{\delta }_0+{\mu }_2{\delta }_1+{\mu }_3{\delta }_2)$. Equation (36) can be rewritten as follows:

$$\begin{array}{cc}\hfill \dot{V}& \le -{\tilde{v}}^{T}Q\tilde{v}-k_1{s}^2-2c_1({\displaystyle \frac{1}{2c_1}}{\tilde{\delta }}_0^2)\hfill \\ \hfill & -2c_2({\displaystyle \frac{1}{2c_2}}{\tilde{\delta }}_1^2)-2c_3({\displaystyle \frac{1}{2c_3}}{\tilde{\delta }}_2^2)+\Delta \hfill \end{array}$$

(37)

□

Ultimately, based on the aforementioned equations, we can conclude that ${\tilde{\delta }}_0$${\tilde{\delta }}_1$ and ${\tilde{\delta }}_2$ exponentially converge to a boundary related to Δ. Consequently, there exists a constant ${\overline{\delta }}_0\ge {\delta }_0,{\overline{\delta }}_0\ge {\widehat{\delta }}_0,{\overline{\delta }}_1\ge {\delta }_1,{\overline{\delta }}_1\ge {\widehat{\delta }}_1,{\overline{\delta }}_2\ge {\delta }_2,{\overline{\delta }}_2\ge {\widehat{\delta }}_2$ that meets the necessary conditions, ensuring that $\dot{V}\le 0$ holds true. This validates the stability of the system.

4. Simulation Analysis

To substantiate the efficacy of the proposed approach in mitigating the effects of backlash, a series of simulation experiments were meticulously conducted and analyzed. The system parameters were set as $K_i=5$ Wb, $m=5{\mathrm{kg}/\mathrm{m}}^2$, $B=4\mathrm{N}·\mathrm{m}/\mathrm{s}$. The backlash model parameters were set as follows: $b=1$, $n_1=1$, $s_r=0.5$, $s_l=-0.5$. The control parameters were set as $\lambda =2$, $k_w=4$, $k_1=15$, ${\mu }_1=0.8$, ${\mu }_2=0.4$, ${\mu }_3=0.7$, $c_1=0.3$, $c_2=2.8$, $c_3=0.45$, $c_4=0.0001$, $c_5=0.0002$, $c_6=0.0001$, $\Gamma =diag\left\{\begin{array}{cccc}0.001,& 0.4,& 4.2,& 3.1\end{array}\right\}$. We set the initial values of all other involved parameters to 0. The expression for $d\left(t\right)=f(x_1,x_2,t)/m$ is provided in [35], mainly including frictional nonlinearity and ripple torque, with the specific form as follows:

$$\begin{array}{c}\hfill d\left(t\right)=a_{1}sin\left({\omega }_1{x}_1\right)+a_{2}sin\left(3{\omega }_1{x}_1\right)+a_{3}sin\left(5{\omega }_1{x}_1\right)+\\ \hfill \xi x_2+\left[f_c+(f_s-f_c)e^{-{\left(x_2/{\omega }_s\right)}^2}+f_v{x}_2\right]sign\left(x_2\right)\end{array}$$

(38)

where $a_1,a_2,a_3$ represent constants and ${\omega }_1$ represents the position period; ${\omega }_s$ and $\xi$ denote the static angular velocity and equivalent viscous coefficient; $f_c,f_s$ and $f_v$ represent the coulomb friction, static friction, and viscous friction parameters, respectively. The parameters are set as $a_1=0.04$, $a_2=0.001$, $a_3=0.0001$, $\xi =0.002$, ${\omega }_s=0.0001$, $f_c=0.001$, $f_s=1.001$, $f_v=0.001$. To ensure fairness of comparison, the two groups of simulations use the same parameters.

Case 1: The given reference trajectory is $x_d=2sin\left(2t\right)$. As illustrated in Figure 1, the tracking trajectory exhibits a high degree of congruence with the reference trajectory. The tracking results demonstrate superior performance, with an overshoot of approximately 2.04% and steady-state error fluctuations within ±0.001. These metrics confirm the high tracking accuracy of the proposed method.

Figure 2 presents the control law, and it can be discerned from the figure that the output performance of the control law $Bl\left(u\right)$, following the treatment of the backlash, is marginally superior to that of u. To quantify the differences between the control input u and backlash-compensated input $Bl\left(u\right)$, two metrics are introduced: (1) Input Absolute Integral Value ($IAU=\int \left|u\left(t\right)\right|dt$) representing signal mean, and (2) Input Variance Integral Value ($ISDU=\int {(u\left(t\right)-u_{av})}^{2}dt$, where $u_{av}$ is the average value of $u\left(t\right)$) representing oscillation intensity. For Figure 2, u shows $IAU$ = 168.16 and $ISDU$ = 1268.7, while $Bl\left(u\right)$ yields IAU = 168.96 and ISDU = 1290.9. These results demonstrate that while the two inputs have similar average magnitudes (with u being marginally smaller), $Bl\left(u\right)$ exhibits $1.7\%$ greater oscillation ($ISDU$ difference of 22.2).

The online identification results of system parameters are depicted in Figure 3a. From the graph, we can observe that $v_1$, $v_2$, $v_3$, $v_4$ converges rapidly to the true values, demonstrating a commendable performance in parameter identification. Figure 3b depicts the lumped disturbance upper bound. It is evident from the figure that the upper limit of the disturbance is rapidly discernible.

The comparison between the estimated disturbance upper bound and the actual disturbance $d\left(t\right)$ is shown in Figure 4. The figure demonstrates that $d\left(t\right)$ fluctuates entirely within the estimated upper bound, confirming the effectiveness of both the designed bound and the adaptive parameter estimation.

Case 2: The second set of reference signals is provided as $x_d=sin\left(\pi t\right)+0.4cos\left(t\right)$ with all other parameters being the same as the first case.

The simulation results presented in Figure 5a elucidate the tracking trajectory. It is evident that, despite the absence of tracking in the initial few seconds, the system rapidly adapts and successfully follows the input signal. Figure 5b delineates the tracking error plot, which reveals that, apart from a more substantial error in the initial seconds, the subsequent error consistently remains within a narrow range, indicative of commendable tracking performance. Figure 6 exhibits the simulation effect of the control law, mirroring the first set, thereby indicating a stable control output. The same result holds for Figure 6, where u gives $IAU=198.15$ and $ISDU=1696.6$ versus $Bl\left(u\right)$’s $IAU=198.83$ and $ISDU=1718.8$, confirming the consistency across different reference signals. Figure 7a illustrates the parameter identification, with effects analogous to the first set, and Figure 7b also presents results congruent with the first case. The upper bound estimation of the disturbance is shown in Figure 8. The results indicate that the estimated upper bound effectively encompasses disturbance $d\left(t\right)$, ensuring system stability under the interference of lumped disturbance.

5. Experimental Results

To evaluate the effectiveness of the method proposed in this paper, experimental validation was conducted on an open-source multi-motor drive control integrated experimental platform. The experimental platform comprises a PMSM-DC motor towing system with gear transmission. The PMSM serves as the primary control object while the DC motor provides controllable load conditions. This configuration can effectively demonstrate the clearance compensation and disturbance rejection capabilities of the proposed algorithm. The equipment used included an experimental box, the motors, and a PC. The experimental equipment is shown in Figure 9, with the motor parameters listed in

Table 1. Basic motor parameters of the permanent magnet synchronous motor.

Name	Unit	Parameter
Rated output power	W	400
Polar coordinates	-	4
Rated voltage	VDC	48
Rated speed	RPM	3000
Maximum speed	RPM	4500
Rated torque	N.m	1.27
Rated current	A	12.5
Line back EMF constant	V/kr/min	6.8
Rotor moment of inertia	kg.m2	0.00003
Linear resistance	$\mathsf{\Omega }$	0.25
Linear inductance	mH	0.5
Weight	kg	1.6
Feedback component	-	Incremental, ABZ + UVW, 2500 PPR

and

Table 2. Parameters table of brushed DC motor.

Name	Unit	Parameter
Rated voltage	V	24
No-load speed	RPM	3300
No-load current	A	1.5
Rated torque	N.m	0.7
Rated speed	RPM	2800
Rated current	A	12
Rated power	W	200
Back electromotive force constant	V/KRPM	7.27
Moment of inertia	kg.m2	0.93
Motor length	mm	147
Motor weight	kg	3.2

.

C1: The adaptive parameter identification-based controller with unknown backlash compensation (APIBC), as proposed in this paper, is configured with control parameters set to $k_1=1.8$, $\lambda =0.5$, $k_w=5$, $c_1=0.2$, $c_2=0.001$, $c_3=0.2$, $c_4=1$, $c_5=1000$, $c_6=20$, ${\mu }_1=0.1$, ${\mu }_2=0.005$, ${\mu }_3=0.07$, $n_1=1$, $b=1$, $\Gamma =diag\left\{\begin{array}{cccc}0.03,& 0.03,& 0.0005,& 0.005\end{array}\right\}$. The tracking trajectory is defined as $x_d=\pi sin\left(t\right)$.

C2: A traditional PI controller is used for fair comparison, with parameters configured as $k_p=0.3$ and $k_i$ = 0.00025. Additionally, the reference trajectory is set to $x_d=\pi sin\left(t\right)$.

Figure 10a illustrates the tracking strategy diagram. The analysis of this figure clearly shows that both controllers are capable of closely aligning the actual position with the reference trajectory. Furthermore, it is evident that the controller proposed in this paper exhibits superior tracking performance.

Figure 10b illustrates the tracking error diagram. Analysis of the figure clearly demonstrates that the performance of the controller designed in this paper is significantly superior to that of the PI controller. The error of the proposed controller is within the range of $\pm 0.04$, whereas the PI controller exhibits an error within the range of $\pm 0.09$, effectively halving the error compared to the PI controller and indicating enhanced robustness.

Figure 11 illustrates the control law. Experimental results further validate the conclusions, with u showing $IAU$ = 4833.3 and $ISDU$ = 346,090 compared to $Bl\left(u\right)$’s $IAU$ = 4839.8 and $ISDU$ = 347,930—a $0.53\%$ increase in oscillation magnitude. The results of the online parameter identification for the system are presented in Figure 12a. Upon closer inspection, it becomes evident that the parameters, denoted as $v_1,v_2,v_3,v_4$, converge rapidly to their true values, thereby demonstrating remarkable parameter identification performance. Figure 12b illustrates the disturbances experienced by the system, providing a clear depiction of their upper bound. Importantly, the parameters swiftly converge to this upper bound, further reinforcing the system’s superior performance.

6. Conclusions

This paper addresses the nonlinearity of servo systems with unknown backlash by proposing a robust tracking control strategy based on adaptive parameter identification. This approach aims to mitigate the negative impact of backlash phenomena on the dynamic performance of the system, particularly in applications requiring high precision positioning and rapid response. We constructed a comprehensive model of the servo system that captures the nonlinear characteristics of backlash and introduced an adaptive inverse function to tackle the compensation issue, ensuring stable operation of the system under uncertainty. Additionally, we designed an adaptive control law to accurately estimate the unknown upper bounds of the disturbance dynamics and developed a robust tracking controller capable of dynamically adjusting control inputs. Through theoretical analysis and simulation validation, the results demonstrate that the proposed control strategy significantly enhances the tracking performance of servo systems with unknown backlash, providing innovative insights and important guidance for practical applications in servo system control.