Extended State Observer-Based Design of a Bilateral Dual-Kernel Fuzzy Control Algorithm

Abstract

For nonlinear problems in robotic systems, such as parametric uncertainties and external disturbances, this paper proposes a control method based on bilateral dual-kernel fuzzy control. To address the issue that joint angular velocities cannot be directly measured, an extended state observer (ESO) is introduced to simultaneously estimate the joint positions, velocities, and system nonlinearities, thereby achieving effective reconstruction of the system states. In terms of controller design, a dual-kernel function is adopted instead of the conventional single-kernel function. By exploiting its enhanced feature representation capability and fast response characteristics, the proposed approach improves the system dynamic response speed and reduces the settling time. For nonlinear residuals, the bilateral parallel control strategy further improves the approximation accuracy of the control system. Multiple dual-kernel fuzzy sub-controllers are integrated in a bilateral parallel manner, and the weighting parameters of both the fuzzy system and the bilateral structure are updated in real time based on the approximation error. This enables accurate approximation and compensation of the residuals estimated by the extended state observer. The stability of the closed-loop system is rigorously proved based on Lyapunov theory. Finally, simulations on the MATLAB R2022b platform and experiments on a robotic experimental platform are conducted to verify that the proposed bilateral dual-kernel fuzzy controller achieves significantly improved control accuracy for a two-degree-of-freedom robotic manipulator system compared with conventional controllers, thereby demonstrating the effectiveness and superiority of the proposed algorithm.

1. Introduction

In control systems, nonlinear issues such as model uncertainties, parametric uncertainties, and external disturbances pose significant challenges to control accuracy [1]. Traditional control methods that rely on precise system models often exhibit substantial performance degradation when confronted with complex uncertainties during the control process. To ensure stable operation of control systems in the presence of various uncertainties and external disturbances in complex nonlinear systems, a large number of control strategies, including adaptive control and robust sliding mode control, have been extensively developed.

To address model uncertainties, various control approaches have been developed for different application scenarios, including sliding mode controllers, adaptive controllers, and neural network-based control algorithms. The intuitionistic fuzzy adaptive sliding mode control system proposed in [2] models system uncertainties, thereby enhancing the stability and robustness of the control system. In [3], a data-driven sliding mode controller is developed to address nonlinearities arising from limited model information and unpredictable disturbances, enabling approximate compensation of system nonlinear dynamics. In [4], an integral sliding mode control (ISMC) approach based on preview repetitive control (PRC) is proposed to achieve effective control of continuous-time nonlinear systems in the presence of uncertainties, external disturbances, and norm-bounded nonlinearities. In [5], a reinforcement learning-based adaptive control approach is proposed for nonlinear systems with finite-time performance constraints, which enables the system states to satisfy prescribed performance specifications and achieve stable convergence within a predefined time without requiring an accurate model. In [6], an evolved fuzzy neural network is constructed to transform the nonlinear system model into an equivalent linear representation, thereby facilitating system control. In [7], an adaptive neural network control method is developed, in which neural networks continuously learn the unknown system dynamics, enabling accurate and robust control of nonlinear systems with unknown dynamics. A fuzzy system composed of a singleton fuzzifier, Gaussian membership functions, a product inference engine, and a center-average defuzzifier is a universal approximator [8], capable of approximating nonlinear functions with arbitrary accuracy. In particular, it exhibits strong environmental adaptability and disturbance rejection capability in parameter self-tuning fuzzy control schemes [9,10]. In [11], a fuzzy adaptive control strategy is proposed to ensure that the system state error can be rapidly regulated to zero, while effectively alleviating the chattering phenomenon. Phan et al. [12] proposed a position tracking control method for electro-hydrostatic actuators based on adaptive finite-time backstepping and state observation. By integrating finite-time convergence characteristics with a state observer, the proposed approach effectively enhanced the rapid tracking performance of the system under uncertain conditions. Li et al. [13] proposed an energy management strategy for hybrid power systems based on online extremum-seeking optimization. The proposed method achieved adaptive regulation of energy allocation through a real-time optimization mechanism while taking the influence of fuel cell degradation on system performance into consideration.

In practical operating environments of robotic systems, the direct measurement of joint angular velocities is often constrained by factors such as sensor accuracy, installation space, and environmental disturbances, making it difficult to obtain precise velocity signals [14]. The absence of accurate velocity information directly degrades the performance of trajectory tracking control, and conventional control algorithms that rely on full-state measurability [15,16] cannot be directly applied in such scenarios. To address the issue of unmeasurable states, various state observers have been developed and widely applied in robotic control systems. Among them, the extended state observer (ESO) [17,18], owing to its capability of simultaneously estimating system states and unknown disturbances, has become a commonly adopted approach for handling state unobservability in nonlinear systems. The ESO does not rely on an accurate system model and can achieve state reconstruction and disturbance estimation using only input–output information, making it well suited for robotic dynamic systems characterized by parametric uncertainties and complex disturbances [19,20,21,22,23]. Raheem et al. [24] proposed a method combining a nonlinear disturbance observer and super-twisting sliding mode control for knee joint assistive exoskeleton robots, which effectively improved the trajectory tracking accuracy of the system under uncertain disturbance conditions; Ismael et al. [25] proposed a robust control method based on LESO feedback linearization combined with an improved optimization algorithm for magnetic levitation systems, verifying the effectiveness of LESO in disturbance estimation of complex nonlinear systems. Alawad et al. [26] designed an ADRC control strategy based on a simplified ESO for wearable lower limb systems, realizing online estimation and compensation of system disturbances and uncertainties, and verifying its good robustness and tracking performance. Sliding mode observers [27,28] and Kalman filter-based observers [29,30] have also been widely applied in robotic state estimation. Sliding mode observers exhibit strong robustness and can effectively suppress the influence of external disturbances on the estimation results, whereas Kalman filter-based observers are well suited for state estimation in the presence of Gaussian noise. Through the analysis of the above-mentioned research, it can be seen that the existing methods generally adopt ESO/LESO for disturbance estimation and combine sliding mode control or the ADRC framework to achieve robust control, but there are still certain limitations in terms of complex nonlinear approximation ability and high-precision adaptive adjustment.

In summary, for nonlinear systems, existing control approaches such as adaptive control and sliding mode control have demonstrated satisfactory performance. However, certain limitations remain. Adaptive control [31,32] imposes significant computational requirements on control systems, while neural network-based methods [33,34,35] typically require large amounts of training data for parameter learning. Moreover, the chattering phenomenon induced by sliding mode control [36,37,38,39,40] may adversely affect control accuracy and reduce actuator lifespan. These limitations have promoted the increasingly important role of fuzzy control in controller design, among which the IF–THEN fuzzy inference mechanism has demonstrated particularly effective performance in modeling unknown nonlinear systems [41]. The aforementioned issues indicate that a single control mechanism is generally insufficient to simultaneously address model parametric uncertainties and external disturbances. Moreover, high-frequency oscillations that may arise during the parameter update process can lead to fluctuations in control signals, thereby degrading the overall control performance of the system. In addition, when dealing with partially unmeasurable parameters in nonlinear systems, it is necessary to incorporate observers into the control framework to ensure satisfactory system performance.

Reference [42] proposed a system modeling framework based on fuzzy chain structures from the perspective of complex system dynamics. The study demonstrated the effectiveness of fuzzy logical structures in characterizing nonlinear complex dynamic behaviors, thereby providing an important theoretical foundation for distributed fuzzy systems and the analysis of complex dynamical behaviors. Furthermore, Reference [43] proposed an adaptive prediction method integrating fuzzy logic with regression trees, which achieved effective prediction of disturbance behaviors in complex physical systems such as JET. This work demonstrated the application potential and scalability of fuzzy-based methods in practical complex engineering systems.

To further enhance the modeling and control capabilities of control systems for complex nonlinear dynamics, this paper considers a robotic manipulator as the research object. To address the issues of unmeasurable joint angular velocities and nonlinear disturbances, an extended state observer is integrated with a bilateral dual-kernel fuzzy control scheme, and an extended state observer-based bilateral dual-kernel fuzzy control algorithm is developed. First, the extended state observer is employed to estimate the unmeasurable joint angular velocities and system uncertainties. Subsequently, the bilateral dual-kernel fuzzy control scheme is utilized to accurately approximate and compensate for the observation residuals. Different types of kernel functions, such as Gaussian functions, bell-shaped functions, and PI-type functions, are introduced to construct dual-kernel fuzzy sub-controllers. These dual-kernel fuzzy sub-controllers provide a preliminary approximation of the observation residuals. By exploiting the different responses of sub-controllers with distinct initial parameters to the residual signals, a bilateral dual-kernel fuzzy controller is formed, thereby enhancing the approximation capability of the controller and further reducing the residual errors.

The main contributions of this paper are summarized as follows:

• Aiming at the problem that the joint angular velocity is unmeasurable during the actual operation of the robot, an extended state observer (ESO) is designed. By defining the unknown part and disturbance equation of the manipulator system, constructing augmented state variables and reasonably designing the observer gain parameters, this observer can realize the synchronous observation and estimation of joint position, velocity and the nonlinear part of the system.
• A dual-kernel fuzzy controller is designed. In the fuzzy controller, the dual-kernel function is used to replace the traditional single-kernel function to form the dual-kernel fuzzy controller. Compared with the traditional fuzzy controller, the dual-kernel fuzzy controller enhances the fast response ability to state signals, thereby improving the response speed of the controller and reducing the response time.
• A bilateral control strategy is proposed, wherein multiple unilateral dual-kernel fuzzy sub-controllers are integrated. The weighting parameters are continuously updated based on the approximation error, thereby improving the approximation accuracy of the control system in the presence of uncertainties.

The remainder of this paper is organized as follows. Section 2 presents the system structure and controller design. Section 3 provides the stability analysis of the proposed control scheme. Section 4 validates the effectiveness and superiority of the proposed controller through simulations conducted in MATLAB and experiments on a robotic platform. Finally, Section 5 concludes the paper and outlines directions for future research.

2. Controller Design

Consider the dynamic model of an n-DOF robotic manipulator system with external disturbances as follows:

$$M\left(q\right)\ddot{q}+C\left(q,\dot{q}\right)\dot{q}+G\left(q\right)=\tau +{\tau }_d$$

(1)

where q, $\dot{q}$, $\ddot{q}\in {\mathbb{R}}^n$ denote the joint position, velocity, and acceleration, respectively; $\tau \in {\mathbb{R}}^n$ represents the control input torque; $M\left(q\right)\in {\mathbb{R}}^{n\times n}$ is the symmetric positive definite inertia matrix; $C(q,\dot{q})\in {\mathbb{R}}^{n\times n}$ denotes the Coriolis and centrifugal force matrix; $G\left(q\right)\in {\mathbb{R}}^{n\times 1}$ is the gravity vector; and ${\tau }_d\in {\mathbb{R}}^n$ represents the external disturbance.

For the convenience of controller design, define $x_1=q$, $x_1={\left[x_{11},\cdots ,x_{1m}\right]}^{\top }\in {\mathbb{R}}^m$, $x_2=\dot{q}$, $x_2={\left[x_{21},\cdots ,x_{2m}\right]}^{\top }\in {\mathbb{R}}^m$. Then, the above equation can be transformed into the following form:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\hfill {\dot{x}}_1=x_2\hfill \\ \hfill {\dot{x}}_2=M^{-1}\left(x_1\right)\left(\tau +{\tau }_d-C\left(q,\dot{q}\right)-G\left(q\right)\right)\hfill \end{array}\right.\end{array}$$

(2)

Use $f\in {\mathbb{R}}^n$ to represent the unknown dynamics and disturbances in the system, defined as:

$$f=M^{-1}\left(\begin{array}{c}{\tau }_d-C\dot{q}-G\end{array}\right)$$

(3)

The robotic manipulator system can be expressed as:

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2\hfill \\ {\dot{x}}_2=M^{-1}\tau +f\hfill \end{array}\right.$$

(4)

Considering the tracking of a nonlinear reference system trajectory, the state-space model of the robotic manipulator can be obtained as follows:

$$\left\{\begin{array}{c}{\dot{x}}_{r1}={\dot{x}}_{r2}\hfill \\ {\dot{x}}_{r2}=-{\lambda }_1{x}_{r1}-{\lambda }_2{x}_{r2}+b_{r}r\hfill \end{array}\right.$$

(5)

where $x_r\in {\mathbb{R}}^{2n}$ represents the state of the reference system, ${\lambda }_1,{\lambda }_2,b_r\in \mathbb{R}$ are positive real constants, and $r\in {\mathbb{R}}^n$ denotes the external excitation signal.

Define the augmented state variable as $x_{re}={\left[x_r^{\top },r^{\top }\right]}^{\top }$, and the tracking error of the robotic manipulator system is given by:

$$e=x_r-x$$

(6)

For robotic manipulator systems, the joint angular velocity is typically not directly measurable. Therefore, the controller must be designed under conditions of unknown system nonlinearities and unmeasurable velocities. An effective control strategy and state observer are required to ensure that the tracking errors of joint positions and velocities remain within a small neighborhood around zero, while guaranteeing the stability of the closed-loop system. For an n-link robotic manipulator system, since certain system states cannot be directly measured, it is necessary to design a state observer to achieve effective control. According to Equations (3) and (4), the extended state is introduced as:

$$x_3=f(x,t)$$

(7)

Assumption: $\omega \left(t\right)$ is bounded, i.e., $\left|\omega \left(t\right)\right|\le \overline{w}$. Design the following extended state observer:

$$\left\{\begin{array}{c}{\dot{\widehat{x}}}_1={\widehat{x}}_2+k_1(x_1-{\widehat{x}}_1)\hfill \\ {\dot{\widehat{x}}}_2=\widehat{f}+M_0^{-1}\tau +k_2(x_1-{\widehat{x}}_1)\hfill \\ {\dot{\widehat{x}}}_3=k_3(x_1-{\widehat{x}}_1)\hfill \end{array}\right.$$

(8)

where ${\widehat{x}}_1$ and ${\widehat{x}}_2$ are the estimated values of the position and velocity state variables of the system, respectively, and ${\widehat{x}}_3=\widehat{f}$ denotes the estimate of the system uncertainties and external disturbances. $k_1,k_2,k_3>0$ represents the observer gain parameter. Since deviations may exist between the observer estimates and the true system states, the observation error of the robotic system is defined as follows:

$${\tilde{x}}_i=x_i-{\widehat{x}}_i\left(\begin{array}{c}i=1,2,3\end{array}\right)$$

(9)

According to Equations (9) and (10), the system state error is defined as follows:

$$\left\{\begin{array}{c}{\dot{\tilde{x}}}_1={\tilde{x}}_2-k_1{\tilde{x}}_1\hfill \\ {\dot{\tilde{x}}}_2={\tilde{x}}_3-k_2{\tilde{x}}_1\hfill \\ {\dot{\tilde{x}}}_3={\dot{x}}_3-k_3{\tilde{x}}_1\hfill \end{array}\right.$$

(10)

where, in the presence of $x_3=f\left(x\right)$ and under the assumption that $\dot{f}$ is bounded, the observation error system can be guaranteed to be exponentially stable by choosing an appropriate $k_3$. Perform a matrix transformation on Equation (11):

$$\dot{\tilde{X}}=A_0\tilde{X}+B_0\dot{f}$$

(11)

of which, $A_0=\left[\begin{array}{ccc}-l_1& 1& 0\\ -l_2& 0& 1\\ -l_3& 0& 0\end{array}\right]$, $B_0={\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^{\top }$.

The extended state observer is used to observe the uncertainties and external disturbances in the system. Since the extended state observer can not completely eliminate all uncertainties, there is a residual between the observed value ${\widehat{x}}_3$ and the actual value. The residual $\Delta \left(x\right)$ is defined as:

$$\Delta \left(x\right)=f(x,t)-{\widehat{x}}_3$$

(12)

Approximate the residuals using a bilateral dual-kernel fuzzy control system:

$$\Delta \left(x\right)=f_{res}+\epsilon \left(x\right)$$

(13)

Here, $f_{res}$ represents the approximation of the nonlinear residual—which includes system uncertainties and external disturbances—by the bilateral dual-kernel fuzzy controller, and $\epsilon \left(x\right)$ represents the nonlinear approximation error.

Definition of tracking error in a robotic system:

$$e_1=x_{r1}-x_1$$

(14)

$$e_2=x_{r2}-{\widehat{x}}_2$$

(15)

Since the joint angular velocity $x_2$ cannot be measured directly, the observer estimate is used in the control law, tracking error, and fuzzy approximation, while the true dynamics of the system are still determined by the actual states.

To facilitate the subsequent analysis, we construct a virtual controller of the following form:

$$v={\ddot{x}}_r-{\lambda }_1{e}_1-{\lambda }_2{e}_2$$

(16)

where ${\lambda }_1$ and ${\lambda }_2$ are the control gains of tracking error. The control input is designed as:

$$\begin{array}{cc}\hfill \mathrm{T}& =M({\ddot{x}}_r-{\lambda }_1{e}_1-{\lambda }_2{e}_2-{\widehat{x}}_3-f_{res})\hfill \\ \hfill & =M(\nu -{\widehat{x}}_3-f_{res})\hfill \end{array}$$

(17)

Establish a fuzzy control system:

Step 1: Define fuzzy sets $A_i^{l_i}(l_i=1,2,\cdots ,m_i)$ for the variable $x_i(i=1,2,\cdots ,n)$.

Step 2: Construct the fuzzy system $u_f$ using the following ${\displaystyle \prod _{i=1}^n}{m}_i$ fuzzy rules,

$$\begin{array}{c}\hfill {\begin{array}{cccc}IF& x_1& is& A\end{array}}_i^{l_i},\cdots \begin{array}{ccc}{x}_n& is& A_n^{l_n}\end{array},\begin{array}{cccc}TNEN& u_f& is& S^{l_1\cdots l_n}\end{array}\end{array}$$

(18)

where, $l_1=1,2,\cdots$, $m_i$, $i=1,2,\cdots ,n$.

The fuzzy controller is designed using a product inference engine, a singleton fuzzy set, and a center-average defuzzifier, i.e.,

$$u_f={\displaystyle \frac{{\sum }_{l_1=1}^{m_i}\cdots {\sum }_{l_1=1}^{m_i}{\overline{y}}_u^{l_1\cdots l_n}\left({\prod }_{i=1}^n{\mu }_{A_i}^{l_i}\left(x_i\right)\right)}{{\sum }_{l_1=1}^{m_i}\cdots {\sum }_{l_1=1}^{m_i}{\prod }_{i=1}^n{\mu }_{A_i}^{l_i}\left(x_i\right)}}$$

(19)

Let ${\overline{y}}_u^{l_1\cdots l_n}$ be the free parameter, placed in the set $\theta \in R^{\prod _{i=1}^n{m}_i}$, and the fuzzy controller is designed as:

$$u_f={\theta }^T\xi \left(x\right)$$

(20)

where $\theta$ is the parameter vector, and $\xi \left(x\right)={[{\xi }_1\left(x\right),\cdots ,{\xi }_n\left(x\right)]}^T$ is the regression vector.

$\xi \left(x\right)$ make the following definition:

$${\xi }_i\left(\overline{x}\right)={\displaystyle \frac{{\prod }_{i=1}^n{\mu }_{A_i}^{l_i}\left(x_i\right)}{{\sum }_{j=1}^m\left({\prod }_{i=1}^n{\mu }_{A_i}^{l_i}\left(x_i\right)\right)}}$$

(21)

Lemma 1 

([44]). For any given unknown nonlinear function $\beta \left(x\right)$ and any given constant $\epsilon >0$, there exists a fuzzy logic system (20) such that $sup\left|\beta \left(x\right)-u_f\right|<\epsilon$. Based on Lemma 1, the uncertainty term $f\left(x\right)$ of the nonlinear system can be approximated by (20).

Figure 1 shows the structure of the dual kernel fuzzy sub-controller. In the internal of the dual kernel fuzzy controller, the dual kernel function is used to replace the single kernel function, and the residual approximation error is obtained by calculating the fuzzy output of the dual kernel function and the system uncertainty residual.

In the fuzzy controller, Gaussian functions, bell-shaped functions, and PI-type functions are used as kernel functions to form a dual kernel structure. The Gaussian function, bell-shaped function, and PI-type function are given by Equations (22), (23) and (24), respectively.

$$G\left(x\right)=exp\left[{\displaystyle \frac{-{(x-m)}^2}{2{\sigma }^2}}\right]$$

(22)

$$P\left(x\right)={\displaystyle \frac{1}{1+k{(x-m)}^2}}$$

(23)

$$\pi \left(x\right)=c·\left({\displaystyle \frac{1}{1+e^{-(x-a)}}}-{\displaystyle \frac{1}{1+e^{-(x-b)}}}\right)+d$$

(24)

For the three fuzzy functions, the mean values of the fuzzy sets for all state variables are defined by selecting the partition points as [−45, 0, 45], [−25, 0, 25], and [−45, 0, 45], respectively.

Figure 2 is the structure diagram of the bilateral dual kernel fuzzy controller based on the state observer. The joint angular velocity is observed by the extended state observer, and the joint angular velocity tracking error $\dot{e}$ is obtained by the difference between the observed value ${\widehat{x}}_2$ and the reference joint angular velocity $x_{r2}$. The bilateral dual kernel output $f_{res}$ is compared with the observation value ${\widehat{x}}_3$ of the system nonlinearity in the extended state observer to obtain the nonlinear estimation residual $\Delta \left(x\right)$ of the system. The joint angle tracking error e, the joint angular velocity tracking error $\dot{e}$ and the system nonlinear estimation residual $\Delta \left(x\right)$ are input to the dual kernel fuzzy sub controller. Each dual kernel fuzzy sub-controller approximates the nonlinear residual of the system, and each approximation value is weighted and fused by bilateral weight parameters to form a bilateral dual kernel output. The output $f_{res}$ of the bilateral dual kernel fuzzy controller outputs the system controller together with the joint angle tracking error e and the joint angular velocity tracking error $\dot{e}$ to form the system control input.

According to Equation (20), the traditional single kernel fuzzy output, the following dual kernel fuzzy output is designed:

$$u_{fij}={\theta }_{ij}^{\top }{\xi }_{ij}$$

(25)

of which, $i=1,2,\cdots ,q$, represents the number of membership rules of fuzzy logic; j represents the number of membership functions of fuzzy logic, and $j=1,2$ represents that the controller adopts dual kernel fuzzy control.

$$u_{fi}=\sum u_{fij}$$

(26)

According to the bilateral learning strategy, the bilateral controller is composed of the outputs of the dual kernel fuzzy sub-controller

$$u_{fi}=\sum u_{fij}$$

(27)

where $U_f={\left[\begin{array}{cccc}{u}_{f1}& u_{f2}& \cdots & u_{fq}\end{array}\right]}^{\top }$ is the integration of the outputs of the dual kernel fuzzy sub controller, and $W_k={\left[\begin{array}{cccc}{w}_1& w_2& \cdots & w_k\end{array}\right]}^{\top }$ is the corresponding weight parameter. The bilateral fuzzy sub controllers form a bilateral structure, and the bilateral dual kernel fuzzy control output further approximates the estimated residual of the system uncertainty.

$$\Delta \left(x\right)={\displaystyle \sum _{k=1}^{N_p}}{\alpha }_k{\theta }_k^{\top }{\xi }_k\left(z\right)+\epsilon \left(x\right)$$

(28)

where $\epsilon \left(x\right)$ is the approximation error, and $\epsilon \left(x\right)$ is known to be bounded.

Fuzzy weight update rate:

$${\dot{\widehat{\theta }}}_k=-{\mathsf{\Gamma }}_k{\xi }_k\left(z\right)e_2$$

(29)

Bilateral weight update rate:

$${\dot{w}}_k=-{\gamma }_k{\dot{e}}^{\top }{\dot{\theta }}_k^{\top }{\xi }_k\left(z\right)$$

(30)

The output of dual kernel fuzzy sub-control needs to be weighted and fused to form a bilateral dual kernel fuzzy control output. In order to avoid the oscillation of the system state caused by the rapid change of the weight parameters of the system, the saturation conversion function is introduced to process the bilateral weight parameters.

Design saturation conversion function:

$$sat(z,l_d)={\displaystyle \frac{l_d}{1+e^{-z}}}-{\displaystyle \frac{l_d}{1+e^z}}$$

(31)

where $l_d$ represents the upper bound of the function, z is the output of the saturation conversion function, and the saturation conversion function is used to constrain the weight parameter update of the bilateral learner. Equation (30) is further rewritten as:

$${\dot{w}}_{ni}=sat(-{\gamma }_k{\dot{e}}^{\top }{\dot{\theta }}_k^{\top }{\xi }_k,l_d)$$

(32)

The pseudocode description of the control algorithm has been added in

Table 1. Pseudo-Code.

Input: Desired trajectory $q_d, {\dot{q}}_d$

Output: Control input u

1: Initialization:

Set initial states $q\left(0\right)$, $\dot{q}\left(0\right)$.

Set ESO states ${\dot{x}}_i\left(0\right)$.

Initialize fuzzy parameters ${\theta }_1$, ${\theta }_2$.

Set control gains and adaptation gains.

2: Loop for each sampling time t:

3: Measurement: obtain system output q.

4: Error computation:

$$\begin{gathered} e_1=q_d-q, \\ {e}_2={\dot{q}}_d-\dot{q}. \end{gathered}$$

5: ESO update (state and disturbance estimation):

$$\begin{gathered} {\dot{\widehat{x}}}_1={\widehat{x}}_2+k_1\left(x_1-{\widehat{x}}_1\right), \\ {\dot{\widehat{x}}}_2=\widehat{f}+M_0^{-1}\tau +k_2\left(x_1-{\widehat{x}}_1\right), \\ {\dot{\widehat{x}}}_3=k_3\left(x_1-{\widehat{x}}_1\right). \end{gathered}$$

Obtain estimates:

$$\begin{gathered} q_1={\dot{\widehat{x}}}_1, \\ {q}_2={\dot{\widehat{x}}}_2, \\ {q}_3={\dot{\widehat{x}}}_3. \end{gathered}$$

6: State reconstruction: $e_2={\dot{q}}_d-\dot{q}$.

7: Fuzzy system approximation (double-kernel structure):

$\widehat{f}={\theta }_1^{\top }{\xi }_1+{\theta }_2^{\top }{\xi }_2$

8: Bilateral fuzzy control law:

$$\begin{gathered} u_1=-{\lambda }_1{e}_1-{\lambda }_2{\widehat{e}}_2, \\ {u}_2=-\widehat{f}\left(x\right), \\ u=M\left({\ddot{x}}_r-{\lambda }_1{e}_1-{\lambda }_2{e}_2-{\widehat{x}}_3-f_{res}\right). \end{gathered}$$

9: Parameter adaptation law:

${\dot{\widehat{\theta }}}_k=-{\mathsf{\Gamma }}_k{\xi }_k\left(z\right)e_2$

10: Stability compensation update (ESO + fuzzy coupling): update gains if required based on Lyapunov design.

11: Apply control input: send u to mechanical system.

12: End loop

, which systematically presents the complete implementation process from state initialization, extended state observer update, fuzzy approximation calculation, control law generation and parameter adaptive update, thereby clearly demonstrating the engineering implementation steps and calculation logic of the proposed method.

3. Stability Analysis

Construct the following Lyapunov function:

$$V={\displaystyle \frac{1}{2}}{e}^{\top }e+{\displaystyle \frac{1}{2}}{\dot{e}}^{\top }\dot{e}+{\displaystyle \frac{1}{2}}{\tilde{X}}^{\top }P\tilde{X}+{\displaystyle \sum _{k=1}^{N_p}}{\displaystyle \frac{1}{2}}{\tilde{\theta }}_k^{\top }{\mathsf{\Gamma }}_k^{-1}{\tilde{\theta }}_k+{\displaystyle \sum _{k=1}^{N_p}}{\displaystyle \frac{1}{2{\gamma }_k}}{\overline{\alpha }}_k^2$$

(33)

The Lyapunov function is decomposed, and the extended state observer subsystem is analyzed as follows:

$$V_o={\displaystyle \frac{1}{2}}{\tilde{X}}^{T}P\tilde{X}$$

(34)

where $P>0$ satisfies the following conditions:

$$A_o^{T}P+P{A}_o=-Q,Q>0$$

(35)

Taking the derivative of Equation (34), we obtain:

$${\dot{V}}_o=-{\displaystyle \frac{1}{2}}{\tilde{X}}^{T}Q\tilde{X}+{\tilde{X}}^{T}P{B}_{o}w\left(t\right)$$

(36)

From Young’s inequality:

$$\begin{array}{cc}\hfill {\tilde{X}}^{T}P{B}_{0}w\left(t\right)& \le ∥\tilde{X}∥∥P{B}_0∥\left|w\right|\hfill \\ \hfill & \le c_1∥\tilde{X}∥\left|w\right|\hfill \\ \hfill & \le {\displaystyle \frac{\epsilon }{2}}{∥\tilde{X}∥}^2+{\displaystyle \frac{c_1^2}{2\epsilon }}{\left|w\right|}^2\hfill \end{array}$$

(37)

where $c_1=∥P{B}_0∥$. Substituting the above inequality into Equation (36), we can obtain:

$${\dot{V}}_o\le -{\displaystyle \frac{1}{2}}{\tilde{X}}^{T}Q\tilde{X}+{\displaystyle \frac{\epsilon }{2}}{∥\tilde{X}∥}^2+{\displaystyle \frac{c_1^2}{2\epsilon }}{\left|w\right|}^2$$

(38)

where $Q>0$,

$${\displaystyle \frac{1}{2}}\tilde{X}Q\tilde{X}\ge {\lambda }_{min}\left(Q\right){∥\tilde{X}∥}^2$$

(39)

Substituting (38) into the above expression yields:

$$\begin{array}{c}{\dot{V}}_0\le -{\displaystyle \frac{1}{2}}{\lambda }_{min}\left(Q\right){∥\tilde{X}∥}^2+{\displaystyle \frac{\epsilon }{2}}{∥\tilde{X}∥}^2+{\displaystyle \frac{c_1^2}{2\epsilon }}{\left|w\right|}^2\hfill \\ \le -\left({\displaystyle \frac{1}{2}}{\lambda }_{min}\left(Q\right)-{\displaystyle \frac{\epsilon }{2}}\right){∥\tilde{X}∥}^2+{\displaystyle \frac{c_1^2}{2\epsilon }}{\left|w\right|}^2\hfill \\ \le -{\lambda }_0{∥\tilde{X}∥}^2+c_0{w}^2\hfill \end{array}$$

(40)

From (40), it can be concluded that the estimation error of the state observer is uniformly bounded and ultimately converges to a bounded neighborhood, i.e., ${\tilde{x}}_i\to 0$. Consequently, ${\dot{e}}_1\to e_2$.

Define the Lyapunov function of the fuzzy parameter error term:

$$V_W={\displaystyle \sum _{i=1}^N}{\displaystyle \frac{1}{2{\gamma }_i}}{\tilde{W}}_i^{\top }{\tilde{W}}_i$$

(41)

By deriving the fuzzy parameter error term and substituting it into the fuzzy parameter update rate defined in Equation (29), the following can be obtained:

$${\dot{V}}_W={\displaystyle \sum _{i=1}^N}{\displaystyle \frac{1}{{\gamma }_i}}{\tilde{W}}_i^{\top }{\dot{W}}_i=-{\displaystyle \sum _{i=1}^N}{\tilde{W}}_i^{\top }{\mathsf{\Phi }}_i{e}_2$$

(42)

According to the fuzzy parameter error term above, the Lyapunov function of the bilateral weight parameter error term is defined:

$$V_{\alpha }={\displaystyle \frac{1}{2{\gamma }_{\alpha }}}{\tilde{\alpha }}^{\top }\tilde{\alpha }$$

(43)

By deriving the bilateral weight parameter error term and substituting it into the fuzzy parameter update rate defined in Equation (30), the following can be obtained:

$${\dot{V}}_{\alpha }={\displaystyle \frac{1}{{\gamma }_{\alpha }}}{\tilde{\alpha }}^{\top }\dot{\alpha }=-{\tilde{\alpha }}^{\top }{e}_2{\mathsf{\Phi }}^{\top }\widehat{W}$$

(44)

Define the tracking error Lyapunov function as follows:

$$V_1={\displaystyle \frac{1}{2}}{e}^{\top }e+{\displaystyle \frac{1}{2}}{\dot{e}}^{\top }\dot{e}$$

(45)

Differentiate (45) term by term:

$$\begin{array}{c}{\dot{V}}_1=e_1^{\top }{\dot{e}}_1+e_2^{\top }{\dot{e}}_2\hfill \\ =e_1^{\top }{e}_2+e_2^{\top }{\dot{e}}_2\hfill \\ =e_1^{\top }{e}_2+e_2^{\top }\left(-{\lambda }_1{e}_1-{\lambda }_2{e}_2+\epsilon \right)\hfill \\ =e_1^{\top }{e}_2-{\lambda }_1{e}_2^{\top }{e}_1-{\lambda }_2{e}_2^{\top }{e}_2+e_2^{\top }\epsilon \hfill \\ =-\left({\lambda }_1-1\right)e_1^{\top }{e}_2-{\lambda }_2{∥e_2∥}^2+e_2^{\top }\epsilon \hfill \end{array}$$

(46)

For the disturbance term $\epsilon$, Young’s inequality is employed:

$$e_2^{\top }\epsilon \le {\displaystyle \frac{1}{2}}{∥e_2∥}^2+{\displaystyle \frac{1}{2}}{∥\epsilon ∥}^2$$

(47)

Substituting (46) into the above expression yields:

$${\dot{V}}_1\le -\left({\lambda }_2-{\displaystyle \frac{1}{2}}\right){∥e_2∥}^2+{\displaystyle \frac{1}{2}}{∥\epsilon ∥}^2-\left({\lambda }_1-1\right)e_1^{\top }{e}_2$$

(48)

For the error term $e_1^{\top }{e}_2$, Young’s inequality is applied:

$$e_1^{\top }{e}_2\le {\displaystyle \frac{1}{2}}{∥e_1∥}^2+{\displaystyle \frac{1}{2}}{∥e_2∥}^2$$

(49)

Substituting (48) into the above expression yields:

$${\dot{V}}_1\le -c_1{∥e_1∥}^2-c_2{∥e_2∥}^2+{\displaystyle \frac{1}{2}}{∥\epsilon ∥}^2$$

(50)

From the above analysis, the extended state observer (ESO) is convergent, and the fuzzy approximation error is bounded. When the control gains satisfy ${\lambda }_1>1$ and ${\lambda }_2>{\displaystyle \frac{1}{2}}$, the system tracking error converges to zero.

Summarizing the above Lyapunov function derivations, it can be concluded that:

$$\dot{V}\le -{\alpha }_1{∥e_1∥}^2-{\alpha }_2{∥e_2∥}^2-{\alpha }_3{∥\tilde{x}∥}^2+{\displaystyle \frac{1}{2}}{∥\epsilon ∥}^2$$

(51)

Under the convergence condition of the extended state observer (ESO), the estimation error asymptotically approaches zero, and the estimated joint velocities gradually converge to their true values. By integrating the online approximation capability of the multilateral dual-kernel fuzzy system for system uncertainties, it can be proved that all signals in the closed-loop system are uniformly ultimately bounded. When the fuzzy approximation error tends to zero, the system tracking error asymptotically converges to the origin, thereby achieving asymptotic trajectory tracking control for the robotic manipulator system.

4. Experimental Validation

4.1. Simulation Validation

The extended state observer-based bilateral dual kernel fuzzy controller (ESO-BDKFC) proposed in this paper is compared with the control system based on linear extended state observer (LESO) and nonlinear extended state observer (NESO).

4.1.1. Parameter Setting

The dynamic performance of the manipulator is described by the following second-order nonlinear differential equations:

$$M\left(q\right)\ddot{q}+C(q,\dot{q})\dot{q}+G\left(q\right)=u+f$$

(52)

where

$$M=\left[\begin{array}{cc}{\displaystyle \frac{1}{3}}{m}_1{l}_1^2+{\displaystyle \frac{1}{3}}{m}_2\left(3l_1^2+l_2^2\right)+m_2{l}_1{l}_{2}cos\left(q_2\right)& {\displaystyle \frac{1}{3}}{m}_2{l}_2^2+{\displaystyle \frac{1}{2}}{m}_2{l}_1{l}_{2}cos\left(q_2\right)\\ {\displaystyle \frac{1}{3}}{m}_2{l}_2^2+{\displaystyle \frac{1}{2}}{m}_2{l}_1{l}_{2}cos\left(q_2\right)& {\displaystyle \frac{1}{3}}{m}_2{l}_2^2\end{array}\right]$$

(53)

$$C=\left[\begin{array}{cc}-{\displaystyle \frac{1}{2}}{m}_2{l}_1{l}_2{x}_{2}sin\left(q_2\right)& -{\displaystyle \frac{1}{2}}{m}_2{l}_1{l}_2({\dot{q}}_1+{\dot{q}}_2)sin\left(q_2\right)\\ {\displaystyle \frac{1}{2}}{m}_2{l}_1{l}_2{x}_{3}sin\left(q_2\right)& 0\end{array}\right]$$

(54)

$$G=\left[\begin{array}{c}{\displaystyle \frac{1}{2}}{m}_{2}g{l}_{2}cos(q_1+q_2)+{\displaystyle \frac{1}{2}}{m}_{1}g{l}_{1}cos\left(q_1\right)+m_{2}g{l}_{1}cos\left(q_1\right)\\ {\displaystyle \frac{1}{2}}{m}_{2}g{l}_{2}cos(q_1+q_2)\end{array}\right]$$

(55)

where $m_1$ and $m_2$ are the masses of links 1 and 2, respectively; $l_1$ and $l_2$ are the lengths of links 1 and 2, respectively; $q_1$ and $q_2$ are the joint positions of joints 1 and 2, respectively; ${\dot{q}}_1$ and ${\dot{q}}_2$ are the joint velocities of joints 1 and 2, respectively; and g is the gravitational acceleration. In the simulation, the physical parameters are set as $m_1=0.1\mathrm{kg}$, $m_2=0.1\mathrm{kg}$, $l_1=1\mathrm{m}$, $l_2=1\mathrm{m}$, and $g=9.8\mathrm{m}{\mathrm{s}}^{-2}$. Define the state as $x=[x_1,x_2,x_3,x_4]=[q_1,q_2,{\dot{q}}_1,{\dot{q}}_2]$.

Reference track:

$$\left\{\begin{array}{c}{q}_1={\displaystyle \frac{1}{2}}sin\left(5t\right)+{\displaystyle \frac{1}{2}}sin\left(2.5t\right),\hfill \\ q_2={\displaystyle \frac{1}{2}}sin\left(5t\right)+{\displaystyle \frac{1}{2}}sin\left(2.5t\right)\hfill \end{array}\right.$$

(56)

The adaptive rate of the controller proposed in this paper is shown in Equation (30), where the actuator saturation limits are set as $u_M=10$ and $u_m=-10$, respectively.

4.1.2. Simulation Results and Analysis

In order to compare the control effect of the controller from the data, the experimental performance index $\eta$ is constructed, which represents the average value of the accumulated absolute value of error, and the performance indicators of state 1 and state 2 are represented by ${\eta }_{x1}$ and ${\eta }_{x2}$, respectively. The following is the error index formula:

$$\eta ={\int }_0^T\left|e\right|dt$$

(57)

According to the data comparison in the comparison of error performance indicators of the three control methods in

Table 2. Comparison of error performance indexes of three control methods.

	LESO	NESO	ESO-BDKFC
${\eta }_{x11}$	0.001363	0.0005333	0.0001367
${\eta }_{x12}$	0.002156	0.002093	0.002032
${\eta }_{x21}$	0.00208	0.001217	0.0010015
${\eta }_{x22}$	0.008209	0.00769	0.007206

, ESO-BDKFC improves the performance of joint 1 by 60.9% and 74.4%, respectively, compared with LESO and NESO, and ESO-BDKFC improves the performance of LESO and Neso by 3% and 3%, respectively. In terms of the performance index of joint 2, ESO-BDKFC improved by 41.5% and 17.7% compared with LESO and NESO, respectively, and ESO-BDKFC improved by 6.32% and 6.29% compared with LESO and NESO, respectively.

Figure 3 and Figure 4 respectively show the position, velocity tracking curve and corresponding tracking error of joint 1. Figure 3 shows that ESO-BDKFC, LESO and NESO can track the position and speed of joint 1. Figure 4 shows that ESO-BDKFC is better than LESO and NESO in tracking the position and speed of joint 1. ESO-BDKFC has stable tracking performance, and the final tracking error can converge to near zero.

Figure 5 and Figure 6 are the trajectory tracking curves of joint 2 and the corresponding tracking errors, respectively. Figure 5 shows that the position and speed of joint 2 can track the upper reference signal in a very short time. Figure 6 shows that ESO-BDKFC and LESO are more stable and have obvious advantages in tracking performance than NESO in tracking the position of joint 2. For the velocity tracking error of joint 2, the final tracking error of ESO-BDKFC and LESO can converge to near zero, while the tracking error of the NESO comparison method is large and has a certain fluctuation. In the results shown in Figure 6, due to the coupling effect of Joint 1 on Joint 2, short-term fluctuations in tracking errors occur in Joint 2 during the commutation process. However, from the perspective of the overall dynamic response, the tracking errors of the three control methods can all converge stably and finally remain within the error range of [−0.05, 0.05], indicating that the proposed method still has good robustness and stability in the presence of coupling disturbances.

Figure 7 shows the control torque of joints 1 and 2. The graph shows that the control torque output of each control method is small and can be output within a certain range, ensuring the stability of the control torque output of the system.

4.2. Robot Platform Experimental Verification

The experimental environment selects the existing six-axis collaborative robot system controller and the corresponding control panel in the laboratory. The six-axis manipulator body is composed of 6 reconfigurable joint modules, connecting components, a base, and end components. In the robot system, the joint modules adopt 20,000-line incremental encoders for angle detection of the manipulator joint torque motors, and 17-bit absolute encoders for detecting the joint angles. The servo driver of the manipulator joint module uses a high-performance FPGA chip as the main processor, which can realize precise control of the current, speed, and position of the frameless torque motor used in the joint module.

As shown in Figure 8, the manipulator controller panel has a wealth of hardware peripheral interfaces such as microSD, USB, RS485, RS232 and GPIO, including a set of powerful monitoring software. The control system development platform used in the experiment supports the modular programming development of MATLAB/Simulink R2022b.

The robot control algorithm is simulated by MATLAB /Simulink. For the simulated control algorithm, the input and output interfaces are replaced by the input and output modules of Simulink of the controller. The Linux executable code and files can be automatically generated by compiling the whole module, and the corresponding control signals of the control algorithm can be generated after running on the controller. During the experiment, the control parameters were modified in real time by the monitoring software, and the joint torque motor and joint angle information of the system were collected by the incremental photoelectric encoder and the absolute angle encoder, respectively. The collected motion data were processed in MATLAB in offline mode, so as to realize the analysis of the position, speed, tracking error and other parameters of the hybrid robot.

In order to explore the tracking effect of the bilateral dual kernel fuzzy controller proposed in this paper in the robot system, the two-degree-of-freedom rotating joint manipulator in the robot system is used as the control object, and the tracking performance of the controller proposed in this chapter is verified by experiments on the robot platform. Compared with the setting of the comparison method in the simulation verification, the bilateral dual kernel fuzzy controller based on extended state observer (ESO-BDKFC) and the control system based on linear extended state observer (LESO) and nonlinear extended state observer (NLESO) proposed in this paper are used as experimental comparison methods.

4.2.1. Parameter Setting

In the bilateral dual kernel fuzzy controller based on state observer, the control parameters are selected as: $k_e={\left[230,150,12,5\right]}^{\top }$, $k_r={\left[-1,-1,-2,-2,1,1\right]}^{\top }$. The natural frequency of the second-order linear filter is $\omega =100$, the damping ratio is $\zeta =0.7$, the bilateral weight learning rate and the weight learning rate of the dual kernel fuzzy controller are ${\gamma }_1=\left[23,12\right]$ and ${\gamma }_2=\left[40,20\right]$, respectively, and the upper bound of the saturation conversion function is $\lambda =\left[20,20\right]$. The track of the system is:

$$\left\{\begin{array}{c}{q}_1={\displaystyle \frac{\pi }{6}}sin\left({\displaystyle \frac{t}{2}}+{\displaystyle \frac{\pi }{6}}\right),\hfill \\ q_2={\displaystyle \frac{1}{2}}sin\left({\displaystyle \frac{t}{4}}+{\displaystyle \frac{\pi }{6}}\right)\hfill \end{array}\right.$$

(58)

4.2.2. Experimental Results and Analysis

In order to compare the control effects of the four controllers from the data, the experimental performance index $\eta$ is constructed, which represents the average value of the accumulated absolute value of error, and the performance indexes of state 1 and state 2 are represented by ${\eta }_{x1}$ and ${\eta }_{x2}$, respectively. The following is the error index formula:

$$\eta ={\int }_0^T\left|e\right|dt$$

(59)

Table 3. Error performance index comparison in robot experimental platform.

	PID	LESO	NESO	ESO-BDKFC
${\eta }_{x1}$	0.49113	0.27782	0.27154	0.19716
${\eta }_{x2}$	0.76142	0.73529	0.55818	0.47008

shows the tracking error performance index of the four control methods for the state variables in the robot system. Among them, the performance of ESO-BDKFC is 29%, 27.4% and 59.9% higher than that of LESO, NESO and PID, respectively. In terms of the performance index of joint 2, ESO-BDKFC improved by 36.1%, 15.8% and 38.3% compared with LESO, NESO and PID, respectively.

Figure 9 shows the position tracking curves of joints 1 and 2 in the robot experimental platform under state observation. Figure 10 and Figure 11 are the position tracking errors of joints 1 and 2, respectively. Figure 9 shows that, compared with other comparison methods, the bilateral dual kernel fuzzy controller based on the state observer has a large initial tracking error for joint 1 at the initial time, but through learning, ESO-BDKFC can complete the convergence of tracking error in a short time and finally maintain in a small range.

Figure 10 shows that during the position tracking of joint 1, the bilateral dual kernel fuzzy controller based on the state observer can maintain a small initial error at the initial time and further reduce the tracking error through learning. In the whole tracking process, ESO-BDKFC shows better tracking characteristics than other comparative methods. Figure 11 shows that during the position tracking of joint 2, the bilateral dual kernel fuzzy controller based on the state observer can maintain a small initial error at the initial time and further reduce the tracking error through learning. In the whole tracking process, ESO-BDKFC shows better tracking characteristics than other comparative methods.

Figure 10 and Figure 11 show that since PID has no learning ability, its tracking error curve changes with time in a fixed waveform, while ESO-BDKFC, LESO and NESO can learn according to the system tracking error, so as to reduce the system tracking error. The initial tracking error of LESO for joints 1 and 2 is large and converges with time. The tracking effect of NESO is better than that of LESO; ESO-BDKFC has the minimum initial tracking error for joints 1 and 2, and can converge to near zero in a short time. The data change curves in Figure 10 and Figure 11 are consistent with the data in Table 3.

Figure 12 shows the control torque generated during the tracking process of joints 1 and 2 in the four control methods. Each control method can maintain the stable output of the control torque during the tracking process and has good output characteristics.

5. Discussion

Aiming at the problems of nonlinear disturbances and unmeasurable joint angular velocity existing in the robot system, this paper proposes a bilateral dual-core fuzzy control algorithm based on the Extended State Observer (ESO). Taking the robot system as the research object, a series of studies such as controller design, stability analysis, simulation and experimental verification have been carried out. The designed Extended State Observer (ESO) effectively solves the problem of unmeasurable joint angular velocity during the actual operation of the robot. The dual-core fuzzy controller significantly improves the response speed and nonlinear approximation ability of the controller, effectively shortens the system response time, and enhances the control real-time performance. The bilateral parallel control strategy further improves the approximation accuracy of the control system to nonlinear residuals. Through MATLAB simulation and robot experimental platform comprehensive verification, it is shown that the control effect of the bilateral dual-core fuzzy control algorithm based on ESO in the two-degree-of-freedom manipulator system is significantly better than that of PID, LESO and NESO, which proves the effectiveness and advancement of the method proposed in this paper.

Although the control method proposed in this paper has achieved good results, it still has the following shortcomings. The proposed fusion method of bilateral dual-core fuzzy control and extended state observer has good approximation ability and robustness in complex nonlinear systems, but its structure is relatively complex. When extended to high-dimensional, multi-joint or strong coupling systems, it may face problems of increased computational complexity and elevated structural design complexity. In terms of parameter tuning, since it involves multiple adjustment links such as fuzzy membership function parameters, dual-core structure weights, and ESO gains, the overall parameter tuning process has certain complexity and relies on prior experience or tuning strategies to a certain extent. In addition, the control performance of this method depends to a certain extent on the estimation accuracy of the ESO for system states and nonlinear disturbances; when the observation error increases, it may affect the control accuracy and convergence performance. This paper realizes the combination of the extended state observer and fuzzy control technology. In the future, we will explore the fusion research of the proposed algorithm with other advanced control algorithms, give full play to the advantages of each algorithm, reduce the algorithm complexity, and further improve the system’s control performance in scenarios of strong nonlinearity, large uncertainty and complex trajectory tracking.