Stackelberg game-based anti-disturbance control for unmanned surface vessels via integrative reinforcement learning

2.1. USVs dynamics

In this study, considering the unmodeled dynamics inherent to the unmanned vessel and the presence of external disturbances, its kinematics and dynamics are expressed as follows:

where $$ \aleph=[x,y,\psi]^{T} $$ represents the pose components. $$ \nu=\begin{bmatrix}u,\nu,r\end{bmatrix}^{T} $$ denotes the velocity vector, and $$ F(\nu) $$ represents the known dynamic characteristics of the unmanned vessel. $$ g $$ signifies the control input gain. $$ \pi_\tau $$ indicates the control input. $$ \mathcal{D}(\aleph,\nu,t) $$ accounts for time-varying external disturbances and the unmodeled dynamics of the system. Furthermore, the rotation matrix $$ \wp(\eta) $$ is given by:

Furthermore, considering the external disturbances experienced by the unmanned vessel, along with system uncertainties and unmodeled dynamics, the control input of the unmanned vessel is given as follows:

where $$ \pi_{\tau{c}} $$ is the disturbance-resistant controller to be designed, and $$ \pi_{\tau{d}} $$ represents the disturbance caused by unknown factors such as disturbances and the unmodeled dynamics of the unmanned vessel.

2.2. Control objectives

In this context, the attitude error vector \(e = \aleph - \aleph_d \) is introduced to assess the interaction between the command inputs to the unmanned vessel and the disturbances, which subsequently influences its trajectory performance. To effectively utilize the inherent structural characteristics of the unmanned vessel system (1), an intermediate virtual control variable is designed to streamline the controller design process while ensuring the convergence of the attitude error, as given below:

where $$ \Gamma_1 $$ is a positive definite control gain matrix. The difference between the actual velocity $$ v $$ of the unmanned vessel and the desired control law $$ v_d $$ plays a crucial role in determining the convergence of the pose error $$ e $$. To this end, the velocity error vector $$ {\Im _e} = v-v_d $$ is denoted as:

where $$ f({\Im _e})=F(v)+\mathcal{D}(\aleph,v,t)-\dot{v}_{d} $$ represents the unknown component in the dynamic system.

Control Objectives:

The control objective of this paper is to design an adaptive intelligent disturbance-resistant control scheme for unmanned vessels within the context of Stackelberg game theory. This scheme aims to achieve boundedness of the tracking error $$ e $$ and $$ \Im_e $$, as well as all signals within the closed-loop system, even in the presence of unknown dynamics and external disturbances.

To achieve the aforementioned control objectives, the following definitions are introduced before proceeding with the design of the controller in this paper:

Definition 1: (Stackelberg Game) A unique Stackelberg equilibrium control pair $$ \{\pi_{\tau{c}}^{*},\pi_{\tau{d}}^{*}(\pi_{\tau{c}}^{*})\} $$ must satisfy the following properties:

1. For the control output $$ \pi_{\tau c}$$, there exists a unique follower control output $$ \pi_{\tau d}^*\left(\pi_{\tau c}\right)$$ that minimizes the objective function \(J_1\). Moreover, for any $$ \pi_{\tau c}$$, it follows that $$ J_1\left(\pi_{\tau c}, \pi_{\tau d}^*\left(\pi_{\tau c}\right)\right) \leq J_1\left(\pi_{\tau c}, \pi_{\tau d}\left(\pi_{\tau c}\right)\right)$$.

2. For the optimal response of the follower $$ \pi_{\tau d}^*\left(\pi_{\tau c}\right)$$, there exists an optimal leader response $$ \pi_{\tau c}^*$$ such that $$ J_2\left(\pi_{\tau c}^*, \pi_{\tau d}^*\left(\pi_{\tau c}^*\right)\right) \leq J_2\left(\pi_{\tau c}, \pi_{\tau d}^*\left(\pi_{\tau c}\right)\right)$$ is satisfied.

3.1. Design framework for the disturbance-resistant controller based on Stackelberg game theory

In scenarios where the control inputs of unmanned vessels are subject to disturbances, it is imperative to identify the time-varying nature of these disturbances. To address this, we propose an intelligent optimal control strategy for disturbance rejection within the Stackelberg game framework, accounting for actuator input limitations. A non-cooperative game model is developed around the dynamic system of speed errors, wherein the disturbance signal $$ \left(\pi_{\tau d}\right)$$ is represented as the follower, and the disturbance rejection control strategy $$ \left(\pi_{\tau c}\right)$$ is represented as the leader in the Stackelberg game, as shown in Figure 1. This framework engenders a sequential decision-making process between the leader and follower, facilitating alternating iterative optimization, as outlined below:

Figure 1. Stackelberg game-based framework for anti-disturbance control of unmanned aerial vehicles.

1. The sequential decision-making begins with the initial output of the unmanned vessel's disturbance-resistant saturation controller $$ \pi_{\tau d}^0 $$.

2. Subsequently, the control output of the disturbances is regarded as the follower, interacting with the designed saturation controller to achieve the optimal control quantity aimed at maximizing the tracking error $$ e,{\Im _e} $$ and the control input $$ \pi_{\tau d}^* $$.

3. Conversely, the disturbance-resistant controller is regarded as the leader in this study, actively adjusting its strategy $$ \pi_{\tau c} $$ based on the follower's control output $$ \pi_{\tau d}^* $$, and selecting the optimal disturbance-resistant strategy $$ \pi_{\tau c}^* $$ to minimize the tracking error $$ e,{\Im _e} $$.

To establish a robust optimal disturbance rejection control strategy for unmanned vessels under alternating iterations between the leader and follower, the leader's decision-making process is contingent on the rational choices of the follower. This mutual dependence guarantees the simultaneous minimization of the cost functions for both parties. These cost functions are given as follows:

where $$ G,Q,R $$ represents positive definite matrices, and $$ \Pi(\pi_{\tau d})=\vartheta^{T}\dot{\hat{\lambda}}_{1} $$ denotes the control input influence associated with the follower.

It is noteworthy that the performance metric functions J₁ and J₂, as defined within the Stackelberg game framework, are characterized as follows:

1. J₁ encapsulates the robustness of the unmanned vessel system, particularly focusing on the degradation of tracking performance under maximum disturbance conditions.

2. J₂ governs the vessel's optimal tracking control performance, striving to minimize tracking error while minimizing control energy expenditure.

3. Through an alternating iterative optimization process of both J₁ (robustness) and J₂ (optimality), a Nash equilibrium is attained. In this equilibrium, the disturbance rejection robustness and optimal control (minimal energy consumption and tracking error) of the unmanned vessel cannot be further improved by independently altering either control strategy. Consequently, the system reaches a cooperative optimal state, thus fulfilling the design objectives of the proposed control strategy.

Furthermore, their corresponding value functions are expressed as follows:

The optimal value function for the follower is defined as

In this case, considering the velocity error dynamics (5), the Hamiltonian function for the follower is defined as

where $$ \nabla V_{1}=\partial V_{1}/\partial{\Im _e} $$ denotes the partial derivative of the Hamiltonian with respect to the variable $$ {\Im _e} $$. Furthermore, based on $$ \partial H_{1}/\partial\pi_{\tau d}=0 $$, the optimal control output $$ \pi_{\tau d}^* $$ for the follower is given by:

Furthermore, introducing the definition $$ \dot{\hat{\Lambda}}_{1}=\nabla\dot{V}_{1}=-\partial H_{1}/\partial{\Im _e} $$, the following co-state equations are obtained:

where $$ \nabla f=\partial f\left({\Im _e}\right)/\partial{\Im _e} $$. Thus, the follower adopts the optimal strategy (11) as its predetermined decision.

Subsequently, the leader's disturbance-resistant controller formulates its strategy by considering the follower's strategy (11) and the co-state constraints (12). Therefore, the constrained optimal control problem $$ \Pi(\tau_{d})=\vartheta^{T}\dot{\hat{\lambda}}_{1} $$ for the leader is expressed as:

where $$ \vartheta $$ is the designed Lagrange multiplier.

Thus, the Hamiltonian function for the leader can be expressed as follows:

where $$ \nabla V_{2}=\partial V_{2}/\partial{\Im _e} $$. By utilizing the necessary conditions for optimality, the leader's optimal disturbance-resistant saturation control strategy and co-state equations can be derived, with the specific steps outlined as follows:

To obtain the minimum cost function $$ V_{1}^{*},V_{2}^{*} $$ under the optimal control input $$ \pi_{\tau d}^{*},\pi_{\tau c}^{*} $$, the corresponding Hamilton-Jacobi (HJ) equation can be derived after introducing the definition $$ \nabla V_{1}^{*}=\partial V_{1}^{*}/\partial{\Im _e} $$, $$ \nabla V_{2}^{*}=\partial V_{2}^{*}/\partial{\Im _e} $$, as follows:

In a further step, by substituting \(\pi_{\tau d}^* \) (11) and \(\pi_{\tau c}^* \) (15) into (17), it can be inferred that:

Based on the above content, Theorem 1 is formulated as follows.

Theorem 1: Under the optimal disturbance influence strategy, the velocity error dynamics constrained by the cost function (6) can ensure stability by utilizing the optimal disturbance-resistant strategy designed in (15). Furthermore, considering the disturbance-resistant problem of the velocity error dynamics (5) with the cost function given by (7), the control for $$ \left\{\pi_{\tau c}^{*},\pi_{\tau d}^{*}(\pi_{\tau c}^{*})\right\} $$ achieves Stackelberg equilibrium if and only if the coupled HJ equations in (17) have a solution.

Proof: To prove the stability of the tracking error dynamics, $$ V_{2}^{*} $$ is selected as a candidate Lyapunov function. Its derivative with respect to (18) is calculated as follows:

Subsequently, based on $$ Q_{i}>0 $$ and $$ R_{i}>0 $$, the conclusion can be made through $$ \dot{V}_{2}^{*}<0 $$, indicating that under the optimal strategy, the tracking error system (5) can achieve asymptotic stability.

To demonstrate the Stackelberg equilibrium, it is noteworthy that the performance index (6) is reformulated as follows:

By combining expressions $$ \dot{V}_{1}^{*}=\left(\nabla V_{1}^{*}\right)^{T}\left[ f\left({\Im _e}\right)+g\pi_{\tau c}+g\pi_{\tau d} \right] $$ and $$ -2\left(\pi_{\tau d}^{*}\right)^{T}G\pi_{\tau d}=\left(\nabla V_{d}^{*}\right)^{T} g\pi_{\tau d}+\pi_{\tau c}R\pi_{\tau d}^{T} $$, and employing the complete square method, it yields:

Subsequently, starting from \(-\left(\pi_{\tau d}^{*}\right)^{T}G\pi_{\tau d}^{*} = \left(\pi_{\tau d}^{*}\right)^{T}G\pi_{\tau d}^{*} + \left(\nabla V_{d}^{*}\right)^{T} g\pi_{\tau d}^{*} + \pi_{\tau c}R\pi_{\tau d}^{*}\), and utilizing the coupled HJ equations for the follower as given in (18), this expression can be further simplified to:

By setting $$ \pi_{\tau d}=\pi_{\tau d}^{*} $$, the optimal value of the follower's cost function is $$ J_{1}\left({\Im _{e0}},\pi_{\tau d}^{*}\right)=V_{1}^{*}\left({\Im _{e0}}\right)-V_{1}^{*}\left({\Im _e}(\infty)\right) $$. According to condition $$ \dot{V}_{1}^{*}={\Im _e}^{T}Q{\Im _e}-\left(\pi_{\tau d}^{*}\right)^{T}G\pi_{\tau d}^{*}-\pi_{\tau c}^{T}R\pi_{\tau d}^{*} $$, it is able to conclude that when inequality $$ {\Im _e}^{T}Q{\Im _e}\leq-\pi_{\tau c}^{T}R\pi_{\tau d}^{*} $$ holds, one has:

Consequently, by applying a similar derivation process as outlined for the leader's cost function, we derive:

Therefore, combined with Figure 1, assuming that the control strategy of the follower \(\pi_{\tau d}\) and the leader's disturbance-rejection control strategy $$ \pi_{\tau c}$$ in the Stackelberg game satisfy the aforementioned conditions, such that neither the leader nor the follower can reduce their respective cost function values J₂ or J₂ by unilaterally adjusting their strategy, the proof of the Stackelberg game Nash equilibrium, as defined in Definition 1, is thus concluded.

3.2. Stackelberg game resolution via integral reinforcement learning techniques

As is well known, deriving an analytical solution to the coupled HJ Equation (17) is highly challenging due to the current nonlinear characteristics. To address this issue, this subsection employs an action-evaluation neural network algorithm based on integral reinforcement learning to solve the Stackelberg game in the disturbance-resistant context of the USVs system. This algorithm includes an auxiliary neural network capable of effectively handling the complexities arising from unknown dynamics. The discussion regarding the solution of the Stackelberg game will involve the follower, the unknown dynamics, and the leader.

To continue, an optimized evaluation neural network is developed to approximate the follower's optimal value function \(V_{1}^{*} \), enabling the representations of \(V_{1}^* \) and its gradient \(\nabla V_{1}^* \) as follows:

where $$ W_{1c}^{*} $$ represents the optimal weights of the follower's evaluation network, $$ \Phi_{_{1c}}=\Phi_{_{1c}}({\Im _e}) $$ denotes the activation function of the evaluation network, and $$ \zeta_{1c}=\zeta_{1c}({\Im _e}) $$ represents the estimation error. Furthermore, let $$ \nabla V_{1}^{*}=\partial V_{1}^{*} / \partial{\Im _e},\nabla\Phi_{1c}=\partial\Phi_{1c} / \partial{\Im _e} $$, $$ \nabla\zeta_{1c}=\partial\zeta_{1c}/\partial{\Im _e} $$.

Considering the unknown nature of the ideal weights $$ W_{1c}^{*} $$, a neural network approximation to approach $$ V_{1}^{*},\nabla V_{1}^{*} $$ is applied, resulting in:

where $$ \nabla\hat{V}_{1}=\partial\hat{V}_{1}/\partial z $$.

where $$ \hat{W}_\mathrm{la} $$ is the estimate of the optimal weights $$ W_{1}^{*} $$ of the optimal evaluation network.

To avoid utilizing the unknown dynamics \(f({\Im _e}) \) throughout the learning process, the Bellman error equation with arbitrary time integral $$ \lambda $$ is introduced as follows:

where $$ r_{1}\left({\Im _e},\hat{\pi}_{\tau d},\hat{\pi}_{\tau c}\right)=\hat{\pi}_{\tau d}^{T}R\hat{\pi}_{\tau d}^{T}-{\Im _e}^{T}Q{\Im _e}-\hat{\pi}_{\tau c}^{T}G\hat{\pi}_{\tau c} $$, $$ \hat{\pi}_{\tau c} $$ is the control strategy of the leader to be designed. Additionally, $$ \Delta\Phi_{\mathrm{lc}}=\Phi_{\mathrm{lc}}(t)-\Phi_{\mathrm{lc}}(t-\lambda) $$.

In addition, with the objective of minimizing the error \(E_{1c} = \frac{1}{2} e_{1c}^{T} J^{T} e_{1c} \), the optimal weights \(W_{1c}^{*} \) of the evaluation network are adaptively adjusted, leading to:

where $$ \Phi_{_{1c}}=\left(1+\Delta\Phi_{{1c}}^{T}\Delta\Phi_{{1c}}\right)^{2} $$. Additionally, $$ k_{_{1c}}>0 $$ represents the learning rate for adjusting the follower's evaluation network.

By substituting the follower's control output (27) into equations (12) and (16), it can be deduced that:

Additionally, to fully leverage the state information for disturbance-resistant control and to enhance the multifunctionality of the unmanned vessel's behavior, the optimal value function \(V_2 \) is decomposed as follows:

where $$ E_{V_{2}}^{*}=V_{2}^{*}-\Gamma_{{\Im _e}}{\Im _e}^{T}{\Im _e}-2\Gamma_{e}{\Im _e}^{T}\wp(\aleph)e-2\Gamma_{h}{\Im _e}^{T}f(z) $$, $$ \Gamma_{{\Im _e}},\Gamma_{s},\Gamma_{h} $$ is the positive definite controller gain. By taking the partial derivative with respect to \({\Im _e} \) and substituting (31) into (15), one obtains:

where $$ \nabla E_{V_{*}}^{*}=\partial E_{V_{*}}^{*}/\partial{\Im _e} $$.

Consequently, the evaluation network for designing the leader's (disturbance-resistant) control strategy is developed to approximate the value function \(E_{V_{2}}^{*} \) and its gradient \(\nabla E_{V_{2}}^{*} \) as follows:

Where \(W_{2c}^{*} \) represents the optimal weights of the leader's evaluation network, and \(\Phi_{2c}, \zeta_{2c} \) denote the activation function and estimation error of the leader's evaluation network, respectively.

Similarly, by estimating \(E_{V_{2}}^{*} \) and \(\nabla E_{V_{2}}^{*} \), one has:

where $$ \nabla\hat{E}_{V_{2}}=\partial\hat{E}_{V_{2}}/\partial{\Im _e} $$.

As a result of the preceding analysis, the leader's disturbance-resistant controller can be expressed as follows:

where $$ \hat{W}_{2a} $$ is the estimate of the optimal weights $$ {W}_{2c}^* $$ of the leader's evaluation network.

Moreover, akin to the formulation employed for the follower, the Bellman error corresponding to the leader can be expressed as:

where $$ r_{2}\left({\Im _e},\hat{\pi}_{\tau d},\hat{\pi}_{\tau c}\right)={\Im _e}^{T}Q{\Im _e}+\hat{\pi}_{\tau d}^{T}G\hat{\pi}_{\tau d}+\vartheta^{T}\dot{\hat{\Lambda}}_{1} $$ and $$ \Delta\Phi_{2c}=\Phi_{2c}\left(t\right)-\Phi_{2c}\left(t-\lambda\right) $$.

Based on this, the adaptive update law for the evaluation network weights that minimizes the objective function \(E_{2c}=\frac{1}{2}e_{2c}^{J^{T}}e_{2c}^{J}\) are designed as follows:

where $$ \Phi_{2c}=\left(1+\Delta\Phi_{2c}^{T}\Delta\Phi_{2c}\right)^{2} $$. Additionally, \(k_{2c}>0 \) represents the learning rate for the adaptive update law of the evaluation network weights.

Additionally, to maintain the stability of the policy updates, the weight update rule for the action network is reformulated as follows:

where, $$ D_{1c}=\nabla\Phi_{1c}gR^{-1}g^{T}\nabla\Phi_{1c}^{T},\quad D_{2c}=\nabla\Phi_{2c}gR^{-1}g^{T}\nabla\Phi_{2c}^{T} $$.

Building upon the preceding analysis, we can formulate the following Theorem 2:

Theorem 2: Consider an unmanned vessel system with partially unknown dynamics, subjected to the approximately optimal disturbance strategy (27) and update rules (29) and (38). The unmanned vessel system is designed with an approximately optimal disturbance-resistant control strategy (35), which includes update rules (37) and (38), ensuring ideal tracking of the trajectory under disturbances, while keeping all signals, as well as the tracking errors \(e \) and \(\Im_e \), bounded within the closed-loop system.

Proof: The following Lyapunov function is utilized in our analysis to assess system stability and performance:

where $$ \mathrm{L}_{1}=e^{T}e/2+{\Im _e}^{T}{\Im _e}/2 , \mathrm{L}_{2}=\tilde{W}_{2c}^{T}k_{2c}^{-1}\tilde{W}_{2c}/2+\tilde{W}_{2a}^{T}k_{2a}^{-1}\tilde{W}_{2a}/2 $$, $$ \mathrm{L}_{3}=V_{a}^{*}\left({\Im _e}\right)+\tilde{W}_{1c}^{T}k_{1c}^{-1}\tilde{W}_{1c}/2+\tilde{W}_{1a}^{T}k_{1a}^{-1}\tilde{W}_{1a}/2 $$.

Step 1: Taking the derivative of \(L_1 \) with respect to time, it obtains \(\dot{\mathrm{L}_{1}}=e^{T}\dot{e}+{\Im _e}^{T}\dot{{\Im _e}} \). Combining this with (4) and $$ e^{T}\dot{e}=-e^{T}\Gamma_{1}e+e^{T}v_{d}{\Im _e} $$, one derives \({\Im _e}^{T}\dot{{\Im _e}} \), which can be expressed as:

Step 2: By taking the time derivative of \(L_2\), we derive \(\dot{L}_{2} = -\tilde{W}_{2a}^{T}k_{2a}^{-1}\dot{\hat{W}}_{2a} - \tilde{W}_{2c}^{T}k_{2c}^{-1}\dot{\hat{W}}_{2c}\). Following this, the substitution of equations (37) and (38) into \(\dot{L}_2\) yields the subsequent result:

where $$ k_{2c}=\left\|\Phi_{2c}\right\|^{2}/\Gamma_{2c} $$. Meanwhile, let $$ p_{c}=\int_{t-\lambda}^{t}\biggl(r_{2c}\bigl({\Im _e}lta,\hat{\pi}_{\tau a},\hat{\pi}_{\tau c}\bigr)+\vartheta\dot{\hat{\pi}}_{\tau c}\biggr)ds $$. To further elaborate, combining with $$ \varepsilon_{l\dot{g}b}^{2c}={\Im _e}^{T}Q{\Im _e}+\left(\pi_{\tau c}^{*}\right)^{T}R\pi_{\tau c}^{*}+\vartheta\dot{\lambda}_{1}+\left(\nabla\Phi_{2c}^{'}W_{2c}^{*}+\nabla V_{c}^{1}\right)^{T}\left[f\left({\Im _e}\right)+g\pi_{\tau d}^{*}+g\pi_{\tau c}^{*}\right] $$, it yields:

where $$ \overline{p}_{c}=\int_{t-\lambda}^{t}\varepsilon_{hjb}^{c}-\left(W_{2c}^{*}\right)^{T}\nabla\Phi_{2c}\left[f\left({\Im _e}\right)+g\pi_{\tau c}^{*}+g\pi_{\tau d}^{*}\right] $$, $$ D_{2c}=\Delta\Phi_{2c}\left(\nabla V_{c}^{*}\right)^{T}\Psi^{T}gR^{-1}g^{T}\Psi\nabla\Phi_{2c}^{T} $$. To take it a step further, $$ \dot{L}_2 $$ can be further derived as:

where, $$ \prod_{1}=gG^{-1}g^{T}\nabla\Phi_{1c}^{T} ,\quad\Pi_{3}=\nabla\Phi_{1c}^{T}\prod_{1} $$.

Step 3: By performing a time derivative of \(L_3\), we obtain \(\dot{L}_{3} = -\tilde{W}_{1a}^{T}k_{1a}^{-1}\dot{\hat{W}}_{1a} - \tilde{W}_{1c}^{T}k_{1c}^{-1}\dot{\hat{W}}_{1c}\). The integration of this result, in conjunction with equations (29) and (38), leads to the following outcome:

where $$ \Gamma_{1a}=\left(1+\Delta\Phi_{1c}^T\Delta\Phi_{1c}\right)^2 $$ and $$ k_{_{1c}}=\left\|\Phi_{_{1c}}\right\|^{2}/\Gamma_{_{1a}} $$.

By integrating \(\varepsilon_{hjb}^{1a} = -{\Im _e}^{T}Q{\Im _e} + \left(\pi_{\tau d}^{*}\right)^{T}G\pi_{\tau d}^{*} + \hat{\pi}_{\tau c}^{T}R\pi_{\tau d}^{*} + \left(W_{1c}^{*}\right)^{T}\nabla\Phi_{1c}\left[f\left({\Im _e}\right) + g\hat{\pi}_{\tau c} + g\pi_{\tau d}^{*}\right] \) with \(p_{a} = \int_{t-\lambda}^{t} \Big(r_{1a}\big({\Im _e}, \hat{\pi}_{\tau d}, \hat{\pi}_{\tau c}\big)\Big)ds \), and applying a differentiation process analogous to that employed in deriving \(\dot{L}_2 \), we can subsequently obtain \(\dot{L}_3 \) as follows:

where $$ \Pi_{2}=g\Psi R^{-1}g^{T}\Psi\nabla\Phi_{2c}^{T},\Pi_{4}=\nabla\Phi_{2c}\Psi^{T}gR^{-1}g^{T}\Psi\nabla\Phi_{2c}^{T} $$.

Step 4: In light of the extensive analysis provided above, we are now positioned to derive:

where $$ \Psi_{1}=\frac{\overline{p}_{c}}{\Gamma_{2c}}\Delta\Phi_{2c}^{T}+k_{1c}\left(W_{1c}^{*}\right)^{T} $$, $$ \Psi_{2}=k_{2a}\left(W_{2c}^{*}\right)^{T}-\frac{\lambda}{4\Gamma_{2c}}\left(W_{2c}^{*}\right)^{T}\Delta\Phi_{2c}\left(W_{2c}^{*}\right)^{T}\Pi_{4} $$, $$ \Psi_{3}=\frac{\lambda}{2\Gamma_{1a}}D_{1c}+\frac{\overline{p}_{a}}{\Gamma_{1a}}\Delta\Phi_{1c}^{T}+k_{1c}\left(W_{1c}^{*}\right)^{T} $$, $$ \Psi_{4}=k_{1a}\left(W_{1c}^{*}\right)^{T}-\frac{\lambda}{4\Gamma_{1a}}\left(W_{1c}^{*}\right)^{T}\Delta\Phi_{1c}\left(W_{1c}^{*}\right)^{T}\Pi_{3i} $$, $$ \Psi_{5}=\frac{\lambda}{4\Gamma_{1a}}\Delta\Phi_{2c}\left(W_{2c}^{*}\right)^{T}\Pi_{4}-\frac{\lambda}{2\Gamma_{2c}}D_{2c} $$, $$ \Psi_{6}=\frac{\lambda}{4\Gamma_{1a}}\Delta\Phi_{2c}\left(W_{1c}\right)^{T}\Pi_{3}-\frac{\lambda}{2\Gamma_{1a}}D_{1c} $$.

In addition, leveraging the principles outlined in Young's inequality, the following result can be deduced:

In this way, it can be effectively simplified to:

where $$ a = \min \left\{ {2{\lambda _{\min }}\left( {{\Gamma _1}} \right),{\lambda _{\min }}\left( {{K_{\Im _e} }} \right),\frac{{k_{2c}^{}}}{{{\lambda _{\min }}\left( {\Gamma _{2c}^{ - 1}} \right)}},\frac{{k_{2a}^{}}}{{{\lambda _{\min }}\left( {\Gamma _{2a}^{ - 1}} \right)}},} \right. $$$$ \left. {\frac{{k_{1c}^{}}}{{{\lambda _{\min }}\left( {\Gamma _{1c}^{^{ - 1}}} \right)}},\frac{{k_{1a}^{}}}{{{\lambda _{\min }}\left( {\Gamma {{_{1a}^{ - 1}}^{}}} \right)}}} \right\} $$, $$ b=\frac{\left(\Psi_1\right)^2}{2k_{2c}}+\frac{\left(\Psi_2\right)^2}{2k_{2a}}+\frac{\left(\Psi_3\right)^2}{2k_{1c}}+\frac{\left(\Psi_4\right)^2}{2k_{1a}}+\frac{b_{\Im _e}^2}{2k_{\Im _e}} $$.

In addition, recognizing that:

Consequently, leveraging the principles established by the Lyapunov stability theorem ^[19], it can be inferred that the variables \(e, {\Im_e}, \tilde{W}_{2a}, \tilde{W}_{2c}, \tilde{W}_{1a}, \) and \(\tilde{W}_{1c} \) remain constrained within bounded limits throughout the operation of the closed-loop system.

Authors' contributions

Writing - original draft: Meng, Y.

Writing - review: Liu, C.

Writing - editing: Zhao, J.

Conceptualization: Huang, J.

Validation: Jing, G.

Availability of data and materials

Not applicable.

Financial support and sponsorship

This work was supported by the National Key R&D Program of China (2022YFB3902702); National Natural Science Foundation of China (62103250, 62273223, 62333011, and 62336005); Project of Science and Technology Commission of Shanghai Municipality, China (22JC1401401).

Conflicts of interest

Liu, C. is a Junior Editorial Board Member of the journal Intelligence & Robotics. He is not involved in any steps of editorial processing, notably including reviewer selection, manuscript handling, or decision-making. The other authors declare that there are no conflicts of interest.

Ethical approval and consent to participate

Not applicable.

Consent for publication

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Copyright

© The Author(s) 2025.