2.1 Problem statement

Consider the following nonlinear system subject to unmodeled dynamics, coupling uncertainties and disturbances:

(1) \begin{align}\begin{array}{*{20}{l}}{{{\dot x}_1}\left( t \right)} {} = { {x_2}\left( t \right) + {d_1}\left( t \right)}\\[5pt] {{{\dot x}_2}\left( t \right)} {} = { f\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right) + {{\Delta }}f\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right) + {{\Lambda }}\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,u\left( t \right)\!,\zeta \left( t \right)} \right) + B\left( {u\left( t \right) + {d_2}\left( t \right)} \right)}\\[5pt] {\dot \zeta \left( t \right)} {} = { {f_\zeta }\left( {\zeta \left( t \right)\!,{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,u\left( t \right)} \right)\!,}\end{array}\end{align}

where ${x_1}\left( t \right)\!,{x_2}\left( t \right)$ denote the states, $u\left( t \right)$ is the input signal of the controlled system. $f\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right)$ is the smooth nonlinear function and ${{\Delta }}f\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right)$ denotes the model uncertain item. $B$ is a known matrix with proper dimensions and its smallest eigenvalue satisfies ${\lambda _{B,min}} \geqslant 1$ . ${d_1}\left( t \right)$ and ${d_2}\left( t \right)$ denote the mismatched and matched disturbances, respectively. With a reasonable analysis, it is assumed that the coupling uncertain term ${{\Lambda }}\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,u\left( t \right)\!,\zeta \left( t \right)} \right)$ is simultaneously affected by the system states ${x_1}\left( t \right)\!,{x_2}\left( t \right)$ , the input signal $u\left( t \right)$ , and the unmodeled dynamics $\zeta \left( t \right)$ . What’s more, the dynamic characteristics of $\zeta \left( t \right)$ can be expressed as $\dot \zeta \left( t \right) = {f_\zeta }\left( {\zeta \left( t \right)\!,{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,u\left( t \right)} \right)$ . In this paper, the time-varying asymmetric full state constraints are considered, that is, ${\underline{b}_1}\left( t \right) \lt {x_1}\left( t \right) \lt {\bar b_1}\left( t \right)$ and ${\underline{b}_2}\left( t \right) \lt {x_2}\left( t \right) \lt {\bar b_2}\left( t \right)$ , where all the predefined constraints are bounded. Table 1 shows the main variables used in this work, and more details in this table will be introduced in the following sections.

Table 1. The main variables in this paper

Our control objective is summarised as follows. Given the nonlinear system Equation (1) with unmodeled dynamics, matched disturbance and mismatched disturbances, the control objective is to propose a full-state-constrained intelligent adaptive controller that can force the system output to track the desired trajectory ${x_d}\left( t \right)$ as far as possible, and all the states satisfy the full state constraints.

2.2 Assumptions and lemmas

To achieve the control objective, the following related assumptions and lemmas are required.

Assumption 2. The coupling uncertainty satisfies the following inequality

(2) \begin{align}{{\Lambda }}\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,u\left( t \right)\!,\zeta \left( t \right)} \right) \le {\varphi _1}\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right) + {\varphi _2}\left( {\zeta \left( t \right)} \right) + {p_u}u\left( t \right)\end{align}

where ${\varphi _1}\left( \cdot \right)$ and ${\varphi _2}\left( \cdot \right)$ are both unknown non-negative smooth functions, ${p_u} \in \left( { - 1,1} \right)$ is a constant.

Assumption 3. The unmodeled dynamics are exponentially input-to-state practically stable(exp-ISpS), specifically, there has a Lyapunov function ${V_\zeta }\left( {\zeta \left( t \right)} \right)$ so that

(3) \begin{align}\begin{array}{*{20}{l}}{{\alpha _1}\left( {\zeta \left( t \right)} \right) \le {V_\zeta }\left( {\zeta \left( t \right)} \right) \le {\alpha _2}\left( {\zeta \left( t \right)} \right)}\\[9pt] {\dfrac{{\partial {V_\zeta }\left( {\zeta \left( t \right)} \right)}}{{\partial \zeta \left( t \right)}}\kappa \left( {\zeta \left( t \right)\!,{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,u\left( t \right)} \right) \le - {\gamma _1}{V_\zeta }\left( {\zeta \left( t \right)} \right) + \rho \left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right) + {\gamma _2},}\end{array}\end{align}

where ${\alpha _1}\left( \cdot \right)\!,{\alpha _2}\left( \cdot \right)$ belong to class ${K_{{\infty _{}}}}$ functions, $\rho \left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right) = {x_1}^T\left( t \right){x_1}\left( t \right) + {x_2}^T\left( t \right){x_2}\left( t \right)$ , and ${\gamma _1},{\gamma _2}$ are positive constants.

Lemma 1. [Reference Polycarpou and Ioannou23] Given any constant satisfying $\varepsilon \gt 0$ and any variable $z \in \mathbb{R}$ , the following inequality holds

(4) \begin{align}0 \le \left| z \right| - z{\rm{tanh}}\left( {\frac{z}{\varepsilon }} \right) \le \kappa \varepsilon \end{align}

where $\kappa $ is a constant satisfying $\kappa = {{\rm{e}}^{ - \left( {\kappa + 1} \right)}}$ , i.e., $\kappa = 0.2785$ .

Lemma 2. [Reference Wang, Yuan and Yang24] For any constant $\varepsilon \gt 0$ and vector $\xi \in {\mathbb{R}^n}$ , we have

(5) \begin{align}\left\| \xi \right\| \lt \frac{{{\xi ^T}\xi }}{{\sqrt {{\xi ^T}\xi + {\varepsilon ^2}} }} + \varepsilon \end{align}

Lemma 3. [Reference Wang, Yuan and Yang24] For any positive constant $\varepsilon \gt 0$ take into account the set ${{{\Omega }}_\varepsilon }$ defined by ${\Omega _\varepsilon }\;:\!=\; \left\{ {x|\left\| x \right\| \le 0.2554\varepsilon } \right\}$ . Next, for any $x \notin {{{\Omega }}_\varepsilon }$ , the following inequality is satisfied

(6) \begin{align}1 - 16{\rm{tan}}{{\rm{h}}^2}\left( {\frac{x}{\varepsilon }} \right) \le 0\end{align}

2.3 BIAS-RBFNN

Owing to the capacity of online learning and universal approximation of smooth nonlinear functions, adaptive neural network (ANN) control algorithms have been widely utilised to compensate for the effects caused by unmodeled dynamics or disturbances [Reference Liu, Zeng, Tong, Chen and Liu26–Reference Ni and Shi29]. Among these neural network control algorithms, RBFNN has become the most commonly used one because of its outstanding capacity for function approximation and fast calculation speed. However, to improve the approximation accuracy, the normal RBFNN-based control methods require more hidden nodes, which will increase the computational cost exponentially. In addition, the failure of the approximation may occur when the input of normal RBFNN deviates from the ‘approximation domain’. In order to solve the problems mentioned above, we will introduce an adaptive BIAS-RBFNN based control scheme, the conclusion that the anti-disturbance ability of the BIAS-RBFNN based scheme is stronger than the normal RBFNN based scheme has been proved by [Reference Liu, Li, Ge, Ji, Ouyang and Tee30].

According to Ref. [Reference Dai, Wang and Luo31], for any bounded continuous function $F\left( Z \right)$ which is defined on a closed set ${{{\Omega }}_Z}$ , the structure of RBFNN can be represented as follows

(7) \begin{align}F\left( Z \right) = {{{\Theta }}^T}{{\Phi }}\left( Z \right) + {\varepsilon _{{\Theta }}}\left( Z \right)\!,\forall Z \in {{{\Omega }}_Z}\end{align}

where $Z \in {\mathbb{R}^n}$ is the input vector, ${{\Theta }}$ denotes the optimal weight. ${\varepsilon _{{\Theta }}}$ is the optimal approximate error which can be reduced to any tiny value by increasing the number of nodes of the neural network. According to relevant references, the Gaussian functions are often used as the activation functions of RBFNN, which can be described as

(8) \begin{align}{S_j}\left( Z \right) = {\rm{exp}}\left[ { - \frac{{{{\left( {Z - {\mu _j}} \right)}^T}\left( {Z - {\mu _j}} \right)}}{{{\sigma ^2}}}} \right],j = 1,2, \ldots, m\end{align}

where ${\mu _j} = {\left[ {{\mu _{j1}},{\mu _{j2}}, \ldots, {\mu _{jn}}} \right]^T}$ represents the position of the hidden node, $\sigma $ denotes the width of Gaussian functions, $m$ is the number of hidden nodes in each channel.

The activation functions for the adaptive BIAS-RBFNN based control scheme taken in this paper are shown below

(9) \begin{align}{{{\Phi }}_{lj}}\left( Z \right) = {S_j}\left( Z \right) + {b_l}\end{align}

where ${b_l}$ represents the bias of each activation function, which is added to each activation function. According to Ref. [Reference Cybenko32], ${{{\Phi }}_{lj}}\left( Z \right)$ defined in Equation (9) meets the condition of the Tauber-Wiener function, which also has the capability to represent any continuous functions. Considering the length of this paper, we directly give the following lemma:

Based on Lemma 2.3, we can conclude that the BIAS-RBFNN-based control scheme discussed in this section can obtain more accurate function approximation results, and its anti-disturbance ability is stronger.

3.1 Nonlinear error transformation

To satisfy the time-varying full state constraints, a practical approach is to constrain the tracking errors of the system states. In general, most existing works can only ensure that the tracking errors ultimately converge, while ignoring the dynamic performance of the system, which may lead to inevitable oscillations and overshoots. Therefore, this work will propose a simple and effective nonlinear error transformation technique to enable the system states to smoothly track the desired signals without exceeding their constraint range.

By defining the desired output signal ${x_d}\left( t \right)$ , the tracking error of ${x_1}\left( t \right)$ and ${x_2}\left( t \right)$ can be expressed as

(10) \begin{align}\begin{array}{*{20}{l}}{{e_1}\left( t \right)} {} = { {x_1}\left( t \right) - {x_d}\left( t \right)}\\[5pt] {{e_2}\left( t \right)} {} = { {x_2}\left( t \right) - {x_{2c}}\left( t \right)}\end{array}\end{align}

where ${x_{2c}}\left( t \right)$ is the virtual control signal of ${e_1}\left( t \right)$ subsystem.

From the previous analysis, the constraints of the tracking errors can be expressed as

(11) \begin{align}\begin{array}{*{20}{l}}{{\underline{e}_1}\left( t \right)} {} = { {\underline{b}_1}\left( t \right) - {x_d}\left( t \right)}\\[5pt] {{{\bar e}_1}\left( t \right)} {} = { {{\bar b}_1}\left( t \right) - {x_d}\left( t \right)}\\[5pt] {{\underline{e}_2}\left( t \right)} {} = { {\underline{b}_2}\left( t \right) - {x_{2c}}\left( t \right)}\\[5pt] {{{\bar e}_2}\left( t \right)} {} = { {{\bar b}_2}\left( t \right) - {x_{2c}}\left( t \right)}\end{array}\end{align}

It is clear that if the error constraints ${\underline{e}_1}\left( t \right) \lt {e_1}\left( t \right) \lt {\bar e_1}\left( t \right)$ and ${\underline{e}_2}\left( t \right) \lt {e_2}\left( t \right) \lt {\bar e_2}\left( t \right)$ can be well ensured, the system states ${x_1}\left( t \right)$ and ${x_2}\left( t \right)$ can also be guaranteed to never violate the predefined constraints $\left( {{\underline{b}_1}\left( t \right)\!,{{\bar b}_1}\left( t \right)} \right)$ and $\left( {{\underline{b}_2}\left( t \right)\!,{{\bar b}_2}\left( t \right)} \right)$ , respectively. To guarantee the constraints of the tracking errors, the nonlinear error transformation technique based on a one-one mapping is utilised. Specifically, we define

(12) \begin{align}\begin{array}{*{20}{l}}{{s_1}\left( t \right)} {} = { {\rm{tan}}\left( {\dfrac{\pi }{2} \times \dfrac{{2{e_1}\left( t \right) - {{\bar e}_1}\left( t \right) - {\underline{e}_1}\left( t \right)}}{{{{\bar e}_1}\left( t \right) - {\underline{e}_1}\left( t \right)}}} \right)}\\[9pt] {{s_2}\left( t \right)} {} = { {\rm{tan}}\left( {\dfrac{\pi }{2} \times \dfrac{{2{e_2}\left( t \right) - {{\bar e}_2}\left( t \right) - {\underline{e}_2}\left( t \right)}}{{{{\bar e}_2}\left( t \right) - {\underline{e}_2}\left( t \right)}}} \right)}\end{array}\end{align}

Then, we can get that

(13) \begin{align}\begin{array}{*{20}{l}}{{e_1}\left( t \right)} {} = { \dfrac{{{{\bar e}_1}\left( t \right) - {\underline{e}_1}\left( t \right)}}{\pi }{\rm{arctan}}\left( {{s_1}\left( t \right)} \right) + \dfrac{{{{\bar e}_1}\left( t \right) + {\underline{e}_1}\left( t \right)}}{2}}\\[9pt] {{e_2}\left( t \right)} {} = { \dfrac{{{{\bar e}_2}\left( t \right) - {\underline{e}_2}\left( t \right)}}{\pi }{\rm{arctan}}\left( {{s_2}\left( t \right)} \right) + \dfrac{{{{\bar e}_2}\left( t \right) + {\underline{e}_2}\left( t \right)}}{2}}\end{array}\end{align}

By further analysis, we can conclude that

(14) \begin{align}\begin{array}{*{20}{l}}{\left\{ {\begin{array}{*{20}{l}}{{\rm{li}}{{\rm{m}}_{{s_1} \to - \infty }}{e_1}\left( t \right) = {\underline{e}_1}\left( t \right)}\\[5pt] {{\rm{li}}{{\rm{m}}_{{s_1} \to + \infty }}{e_1}\left( t \right) = {{\bar e}_1}\left( t \right)}\end{array}} \right.}\end{array},\begin{array}{*{20}{l}}{\left\{ {\begin{array}{*{20}{l}}{{\rm{li}}{{\rm{m}}_{{s_2} \to - \infty }}{e_2}\left( t \right) = {\underline{e}_2}\left( t \right)}\\[5pt] {{\rm{li}}{{\rm{m}}_{{s_2} \to + \infty }}{e_2}\left( t \right) = {{\bar e}_2}\left( t \right)}\end{array}} \right.}\end{array}\end{align}

Therefore, we can draw a conclusion that if the transformed states ${s_1}\left( t \right)$ and ${s_2}\left( t \right)$ are bounded, the tracking errors ${e_1}\left( t \right)$ and ${e_2}\left( t \right)$ will consequently be forced to stay within their constraint regions. To achieve this goal, the control problem can be converted to designing a proper adaptive control law to force the transformed states to be bounded.

3.2 Full-state-constrained intelligent adaptive controller design

In this section, a full-state-constrained BIAS-RBFNN-based intelligent adaptive controller will be constructed and the structure of the proposed control algorithm is demonstrated in Fig. 1.

Figure 1. The structure of the proposed full-state-constrained intelligent adaptive control algorithm.

Taking the derivative of ${s_1}\left( t \right)$ yields

(15) \begin{align}{\dot s_1}\left( t \right) = \mu _{{e_1}}^{{s_1}}\left( {{x_2}\left( t \right) + {d_1}\left( t \right) - {{\dot x}_d}\left( t \right)} \right) + L_{{e_1}}^{{s_1}},\end{align}

where $\mu _{{e_1}}^{{s_1}} = \frac{{\partial {s_1}\left( t \right)}}{{\partial {e_1}\left( t \right)}}$ , $L_{{e_1}}^{{s_1}} = \frac{{\partial {s_1}\left( t \right)}}{{\partial {{\bar e}_1}\left( t \right)}}{\dot{\bar{e}}_1}\left( t \right) + \frac{{\partial {s_1}\left( t \right)}}{{\partial {\underline{e}_1}\left( t \right)}}{\underline{\dot{e}}_1}\left( t \right)$ .

By defining ${s_0}\left( t \right) = \mathop \smallint \nolimits_0^t {s_1}\left( u \right){{\;\;}}du$ , the inner loop virtual signal ${x_{2c}}\left( t \right)$ can be designed as

(16) \begin{align}{x_{2c}}\left( t \right) = {\left( {\mu _{{e_1}}^{{s_1}}} \right)^{ - 1}}\left( { - {k_0}{s_0}\left( t \right) - {k_1}{s_1}\left( t \right) - L_{{e_1}}^{{s_1}}} \right) + {\dot x_d}\left( t \right) - {\hat{d}_1}\left( t \right)\end{align}

where ${k_0}$ and ${k_1}$ are positive constant feedback gains, ${\hat{d}_1}\left( t \right)$ denotes the estimated value of mismatched disturbance ${d_1}\left( t \right)$ . In this paper, a disturbance observer is used to compensate for the unmatched disturbance ${d_1}\left( t \right)$ and the design process is shown below.

(17) \begin{align}\begin{array}{*{20}{l}}{{{\hat{d}}_1}\left( t \right)} {} = { {\alpha _{{d_1}}}\left( t \right) + L{x_1}\left( t \right)}\\[5pt] {{{\dot \alpha }_{{d_1}}}\left( t \right)} {} = { - L{{\hat{d}}_1}\left( t \right) - L{x_2}\left( t \right)}\\[5pt] {{{\dot{\hat{d}}}_1}\left( t \right)} {} = { - L{{\hat{d}}_1}\left( t \right) + L\left( {{{\dot x}_1}\left( t \right) - {x_2}\left( t \right)} \right) = - L{{\tilde d}_1}\left( t \right)}\end{array}\end{align}

where ${\tilde d_1}\left( t \right) = {\hat{d}_1}\left( t \right) - {d_1}\left( t \right)$ , the gain of the disturbance observer is represented by $L$ , and ${\alpha _{{d_1}}}\left( t \right)$ is the internal state.

Substituting the virtual signal ${x_{2c}}\left( t \right)$ designed in (16) into ${\dot s_1}\left( t \right)$ yields that

(18) \begin{align}{\dot s_1}\left( t \right) = - {k_0}{s_0}\left( t \right) - {k_1}{s_1}\left( t \right) - \mu _{{e_1}}^{{s_1}}{\tilde d_1}\left( t \right) + \mu _{{e_1}}^{{s_1}}{e_2}\left( t \right)\end{align}

Taking the time derivative of ${s_2}\left( t \right)$ yields

(19) \begin{align}\begin{array}{*{20}{l}}{{{\dot s}_2}\left( t \right)} {} = { \mu _{{e_2}}^{{s_2}}\left( {{{\dot x}_2}\left( t \right) - {{\dot x}_{2c}}\left( t \right)} \right) + L_{{e_2}}^{{s_2}}}\\[5pt] {} {} = { \mu _{{e_2}}^{{s_2}}\left( {\begin{array}{*{20}{l}}{f\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right) + {{\Delta }}f\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right) + {{\Lambda }}\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,u\left( t \right)\!,\zeta \left( t \right)} \right)}\\[5pt] { + B\left( {u\left( t \right) + {d_2}\left( t \right)} \right) - {{\dot x}_{2c}}\left( t \right)}\end{array}} \right) + L_{{e_2}}^{{s_2}}}\end{array},\end{align}

where $\mu _{{e_2}}^{{s_2}} = \frac{{\partial {s_2}\left( t \right)}}{{\partial {e_2}\left( t \right)}}$ , $L_{{e_2}}^{{s_2}} = \frac{{\partial {s_2}\left( t \right)}}{{\partial {{\bar e}_2}\left( t \right)}}{\dot{\bar{e}}_2}\left( t \right) + \frac{{\partial {s_2}\left( t \right)}}{{\partial {\underline{e}_2}\left( t \right)}}{\underline{\dot{e}}_2}\left( t \right)$ .

Then, introduce the dynamic signal $r\left( t \right)$ as shown below

(20) \begin{align}\dot r\left( t \right) = - {\gamma _0}r\left( t \right) + \rho \left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right)\!,r\left( 0 \right) = {r_0}\end{align}

where ${\gamma _0} \in \left( {0,{{\;\;}}{\gamma _1}} \right)$ .

According to Assumption 2.2, we can get that

(21) \begin{align}s_2^T\Lambda \left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,u\left( t \right)\!,\zeta \left( t \right)} \right) \le \left\| {s_2^T\left( t \right)} \right\|\left( {{\varphi _1}\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right) + {\varphi _2}\left( {\zeta \left( t \right)} \right) + {p_u}u\left( t \right)} \right)\end{align}

Thanks to Lemma 2.2 and Assumption 3, Equation (21) AUcan be rewritten as

(22) \begin{align} \begin{array}{l} \left\| {s_2^T\left( t \right)} \right\|{\varphi _1}\left( {{x_1}\left( t \right),{x_2}\left( t \right)} \right) \le {s_2^T}\left( t \right){{\bar \varphi }_1}\left( {{s_2}\left( t \right),{x_1}\left( t \right),{x_2}\left( t \right)} \right) + {\varepsilon _1}\\[5pt] \left\| {s_2^T\left( t \right)} \right\|{\varphi _2}\left( {\zeta \left( t \right)} \right) \le \left\| {s_2^T\left( t \right)} \right\|{\varphi _2}^\circ \alpha _1^{ - 1}\left( {2r\left( t \right)} \right) + \left\| {s_2^T\left( t \right)} \right\|{\varphi _2}^\circ \alpha _1^{ - 1}\left( {2{\varepsilon _r}} \right) \end{array}\end{align}

where ${\varepsilon _1} \gt 0$ is an arbitrary constant,

(23) \begin{align}{\bar \varphi _1}\left( {{s_2}\left( t \right)\!,{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right) = \frac{{{\varphi _1}\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right)s_2^T{\varphi _1}\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right)}}{{\sqrt {{{[s_2^T{\varphi _1}\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right)]}^2} + \varepsilon _1^2} }}\end{align}

Further, using Young’s inequality, we can get that

(24) \begin{align}\left\| {s_2^T\left( t \right)} \right\|{\varphi _2}\left( {\zeta \left( t \right)} \right) \le s_2^T{\bar \varphi _2}\left( {{s_2}\left( t \right)\!,{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,r\left( t \right)} \right) + {\varepsilon _2} + \frac{1}{4}s_2^T\left( t \right){s_2}\left( t \right) + {\varepsilon _3}\end{align}

where ${\varepsilon _2} \gt 0$ is an arbitrary constant,

(25) \begin{align}\begin{array}{*{20}{l}}{{{\bar \varphi }_2}\left( {{s_2}\left( t \right)\!,{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,r\left( t \right)} \right) = \dfrac{{{\varphi _2}^ \circ \alpha _1^{ - 1}\left( {2r\left( t \right)} \right)s_2^T\left( t \right){\varphi _2}^ \circ \alpha _1^{ - 1}\left( {2r\left( t \right)} \right)}}{{\sqrt {{{\left[ {s_2^{T}\left( t \right){\varphi _2}^ \circ \alpha _1^{ - 1}\left( {2r\left( t \right)} \right)} \right]}^2} + \varepsilon _2^2} }}}\\[5pt] {{\varepsilon _3} = {{\left[ {{\varphi _2}^ \circ \alpha _1^{ - 1}\left( {2{\varepsilon _r}} \right)} \right]}^2}}\end{array}\end{align}

For the set-valued mapping of the outer-loop tracking error, BIAS-RBFNN is introduced to approximate the unknown nonlinear term

(26) \begin{align}\begin{array}{*{20}{l}}{{{{\Theta }}^T}{{{\Phi }}_b}\left( {{s_2}\left( t \right)\!,{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,r\left( t \right)} \right) + {\varepsilon _{{\Theta }}}} {} = { {{\bar \varphi }_1}\left( {{s_2}\left( t \right)\!,{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right) + {{\bar \varphi }_2}\left( {{s_2}\left( t \right)\!,{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,r\left( t \right)} \right)}\\[5pt] {} {}{ + {{\Delta }}f\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right){x_2}\left( t \right) + {s_2}{{\left( t \right)}^{ - 1}}{s_1}\left( t \right)\left( {{e_2}\left( t \right) - {s_2}\left( t \right)} \right)}\end{array}\end{align}

where ${\varepsilon _{{\Theta }}}$ is the RBFNN estimation error and ${{{\Phi }}_b}\left( {{s_2}\left( t \right)\!,{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,r\left( t \right)} \right)$ is the activation function with local errors.

In order to compensate and suppress the influence of the matched disturbance ${d_2}\left( t \right)$ and neural network estimation error ${\varepsilon _{{\Theta }}}$ on the controlled system. Firstly, the total disturbance $D\left( t \right)$ can be constructed in the following form:

(27) \begin{align}D\left( t \right) = B{d_2}\left( t \right) + {\varepsilon _{{\Theta }}}\end{align}

Based on Assumption 4, the following inequality holds

(28) \begin{align}\left| {D\left( t \right)} \right| = \left| {B{d_2}\left( t \right) + {\varepsilon _{{\Theta }}}} \right| \le {\varepsilon _v}\end{align}

According to Lemma 1, it’s easy to get that

(29) \begin{align}s_2^T\left( t \right)D\left( t \right) \le \left| {{s_2}\left( t \right)} \right|{\varepsilon _v} \le {\varepsilon _v}s_2^T\left( t \right){\rm{tanh}}\left( {\dfrac{{{s_2}\left( t \right)}}{\alpha }} \right) + \kappa \alpha {\varepsilon _v}\end{align}

In addition, the following inequality can be established by designing the control signal $u\left( t \right)$ to satisfy $s_2^T\left( t \right)u\left( t \right) \le 0$ , hence we have

(30) \begin{align}\begin{array}{*{20}{l}} s_2^T\left( t \right)u\left( t \right) + \left\| {{p_u}s_2^T\left( t \right)u\left( t \right)} \right\| \\[5pt] \le s_2^T\left( t \right)u\left( t \right) - \left\| {{p_u}} \right\|s_2^T\left( t \right)u\left( t \right)\\[5pt] \le \left( {1 - \left\| {{p_u}} \right\|} \right)s_2^T\left( t \right)u\left( t \right)\end{array}\end{align}

Based on the above analysis, the nominal outer loop controller can be designed as

(31) \begin{align}{u_c}\left( t \right) = {B^{ - 1}}\left( {\begin{array}{*{20}{l}}{ - {k_2}{s_2}\left( t \right) - f\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right) - {{{{\hat \Theta }}}^T}{{{\Phi }}_b}\left( {{s_2}\left( t \right)\!,{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,r\left( t \right)} \right) - \dfrac{1}{4}{s_2}\left( t \right)}\\[10pt] { - {\varphi _\rho }\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,{s_2}\left( t \right)} \right) - {{\hat \varepsilon }_v}{\rm{tanh}}\left( {\dfrac{{{s_2}\left( t \right)}}{\alpha }} \right) + {{\dot x}_{2c}}\left( t \right) - {{\left( {\mu _{{e_2}}^{{s_2}}} \right)}^{ - 1}}L_{{e_2}}^{{s_2}} - \mu _{{e_1}}^{{s_1}}{s_1}\left( t \right)}\end{array}} \right)\end{align}

The meaning of ${\varphi _\rho }\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,{s_2}\left( t \right)} \right)$ will be explained in the following section. Further, by defining $\vartheta = 1/{\rm{in}}{{\rm{f}}_{t \geqslant 0}}\left[ {B - I \cdot \left\| {{p_u}} \right\|} \right]$ , the actual loop controller can be designed as

(32) \begin{align}u\left( t \right) = - \frac{{\hat \vartheta B{u_c}\left( t \right)\left[ {s_2^T\left( t \right)\hat \vartheta B{u_c}\left( t \right)} \right]}}{{\sqrt {{{\left[ {s_2^T\left( t \right)\hat \vartheta B{u_c}\left( t \right)} \right]}^2} + \varepsilon _u^2} }}\end{align}

where ${\hat \varepsilon _v},{{\hat \Theta }},\hat \vartheta $ are the estimated values of ${\varepsilon _v},{{\Theta }},\vartheta $ , respectively.

Then the adaptive law of ${\hat \varepsilon _v},{{\hat \Theta }},\hat \vartheta $ are proposed as

(33) \begin{align}\begin{array}{*{20}{l}}{{{\dot{\hat{\varepsilon}} }_v} = s_2^T\left( t \right){\rm{tanh}}\left( {\dfrac{{{s_2}\left( t \right)}}{\alpha }} \right) - {\lambda _\varepsilon }{{\hat \varepsilon }_v}}\\[8pt] {{{\dot{\hat{\Theta}} }} = {{{\Gamma }}_{{\Theta }}}{{\Phi }}\left( {{s_2}\left( t \right)\!,{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,r\left( t \right)} \right)s_2^T\left( t \right) - {{{\Gamma }}_{{\Theta }}}{\lambda _{{\Theta }}}{{\hat \Theta }}}\\[3pt] {\dot{\hat{\vartheta}} = - {{{\Gamma }}_\vartheta }s_2^T\left( t \right)B{u_c}\left( t \right) - {{{\Gamma }}_\vartheta }{\lambda _\vartheta }\hat \vartheta }\end{array}\end{align}

where ${{{\Gamma }}_{{\Theta }}},{{{\Gamma }}_\vartheta },{\lambda _\varepsilon },{\lambda _{{\Theta }}},{\lambda _\vartheta }$ are all positive design parameters.

To illustrate the effectiveness and stability of the proposed controller designed by the FIAC scheme for a class of uncertain nonlinear systems with unmodeled dynamics, coupling uncertainties, and mismatched disturbances, the stability analysis is given in the following subsection.

3.3 Stability analysis

Based on the above analysis, the differential equations of the controlled nonlinear system can be rewritten as

(34) \begin{align}\begin{array}{l}{{\dot s}_0}\left( t \right) = {s_1}\left( t \right)\\[2pt] {{\dot s}_1}\left( t \right) = - {k_0}{s_0}\left( t \right) - {k_1}{s_1}\left( t \right) - \mu _{{e_1}}^{{s_1}}{{\tilde d}_1}\left( t \right) + \mu _{{e_1}}^{{s_1}}{e_2}\left( t \right)\\[3pt] {{\dot s}_2}\left( t \right) = \mu _{{e_2}}^{{s_2}}\left( {\begin{array}{*{20}{l}}{f\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right) + {{\Delta }}f\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right) + {{\Lambda }}\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,u\left( t \right)\!,\zeta \left( t \right)} \right)}\\[3pt] { + B\left( {u\left( t \right) + {d_2}\left( t \right)} \right) - {{\dot x}_{2c}}\left( t \right)}\end{array}} \right) + L_{{e_2}}^{{s_2}}\\[3pt] \dot \zeta \left( t \right) = {f_\zeta }\left( {\zeta \left( t \right)\!,{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,u\left( t \right)} \right)\end{array}.\end{align}

The purpose of this paper is to design a suitable controller to ensure the stability of the differential equations in Equation (34). The stability of the closed-loop nonlinear system with the proposed control method can be revealed by the following theorem.

Proof. According to Ref. [Reference Jiang and Praly33], the dynamic signal $r\left( t \right)$ has the following properties

(35) \begin{align}\begin{array}{*{20}{l}}{r\left( t \right) \geqslant 0,\forall t \geqslant 0}\\[2pt] {{V_\zeta }\left( {\zeta \left( t \right)} \right) \le r\left( t \right) + {\varepsilon _r}}\end{array}\end{align}

where ${\varepsilon _r} = {V_\zeta }\left( {\zeta \left( 0 \right)} \right) + \frac{{{\gamma _2}}}{{{\gamma _1}}}$ .

By defining ${{\tilde \Theta }} = {{\hat \Theta }} - {{\Theta }},\tilde \vartheta = \hat \vartheta - \vartheta, {\tilde \varepsilon _v} = {\hat \varepsilon _v} - {\varepsilon _v}$ , the Lyapunov function can be selected as

(36) \begin{align} \begin{array}{l} V = {V_1} + {V_2},{V_1} = \dfrac{1}{2}s_0^T\left( t \right){s_0}\left( t \right) + \dfrac{1}{2}s_1^T\left( t \right){s_1}\left( t \right) + \dfrac{1}{2}\tilde d_1^T\left( t \right)\tilde d_1^{}\left( t \right)\\[2pt] {V_2} = \dfrac{1}{2}{\left( {\mu _{{e_2}}^{{s_2}}} \right)^{ - 1}}s_2^T\left( t \right){s_2}\left( t \right) + \dfrac{1}{2}Tr\left( {{{\tilde \Theta }^T}\Gamma _\Theta ^{ - 1}\tilde \Theta } \right) + \dfrac{1}{{2\vartheta {\Gamma _\vartheta }}}{{\tilde \vartheta }^2} + \dfrac{1}{2}{{\tilde \varepsilon }^T}_v{{\tilde \varepsilon }_v} + \dfrac{r}{{{\Gamma _r}}} \end{array}\end{align}

where ${{{\Gamma }}_r} \gt 0$ is a positive constant.

Combining the differential equations in Equation (34), the derivative of ${V_1}$ can be expressed as

(37) \begin{align}{\dot V_1} = s_0^T\left( t \right){s_1}\left( t \right) + s_1^T\left( t \right)\left( { - {k_0}{s_0}\left( t \right) - {k_1}{s_1}\left( t \right) - \mu _{{e_1}}^{{s_1}}{{\tilde d}_1}\left( t \right) + \mu _{{e_1}}^{{s_1}}{e_2}\left( t \right)} \right) + \tilde d_1^T\left( t \right){\dot{\tilde{d}}_1}\left( t \right)\end{align}

From the designed disturbance observer Equation (17), it’s easy to conclude that

(38) \begin{align}{\dot{\tilde{d}}_1}\left( t \right) = - L{\tilde d_1}\left( t \right) - {\dot{d}_1}\left( t \right)\end{align}

Then, by defining ${\overline s_1} = {\left[ {s_0^T\left( t \right),\;s_1^T\left( t \right)} \right]^T}$ and using the following inequalities

(39) \begin{align}\begin{array}{*{20}{l}}{ - \mu _{{e_1}}^{{s_1}}s_1^T\left( t \right){{\tilde d}_1}\left( t \right) \le \dfrac{{\mu _{{e_1}}^{{s_1}}}}{2}s_1^T\left( t \right){s_1}\left( t \right) + \dfrac{{\mu _{{e_1}}^{{s_1}}}}{2}\tilde d_1^T\left( t \right){{\tilde d}_1}\left( t \right)}\\[10pt] { - \tilde d_1^T\left( t \right){{\dot{d}}_1}\left( t \right) \le \dfrac{1}{2}\tilde d_1^T\left( t \right){{\tilde d}_1}\left( t \right) + \dfrac{1}{2}\dot{d}_1^T\left( t \right){{\dot{d}}_1}\left( t \right)}\end{array}\end{align}

the derivative of ${V_1}$ in Equation (37) can be developed as

(40) \begin{align}\begin{array}{l}{{\dot V}_1} \le - \bar s_1^T\left( t \right)Q{{\bar s}_1}\left( t \right) - \mu _{{e_1}}^{{s_1}}s_1^T\left( t \right){e_2}\left( t \right) - \tilde d_1^T\left( t \right)\left( {L{{\tilde d}_1}\left( t \right) + {{\dot{d}}_1}\left( t \right)} \right) + \dfrac{{\mu _{{e_1}}^{{s_1}}}}{2}\tilde d_1^T\left( t \right){{\tilde d}_1}\left( t \right)\\[10pt] \,\,\,\,\, \le - \bar s_1^T\left( t \right)Q{{\bar s}_1}\left( t \right) - \mu _{{e_1}}^{{s_1}}s_1^T\left( t \right){e_2}\left( t \right) + \left( {\dfrac{{\mu _{{e_1}}^{{s_1}} + 1}}{2} - L} \right)\tilde d_1^T\left( t \right){{\tilde d}_1}\left( t \right) + \dfrac{1}{2}{{\dot{d}}^{T}}_1\left( t \right) + {{\dot{d}}_1}\left( t \right)\end{array}\end{align}

Where,

(41) \begin{align} Q = \left[ {\begin{array}{*{20}{c}} 0&{ - {I_n}}\\ {{k_0}{I_n}}& (-\frac{\mu _{{e_1}}^{{s_1}}}{2}+{k_1}){I_n} \end{array}} \right]\end{align}

Similarly, combining the dynamic characteristics of the dynamic signal $r\left( t \right)$ defined by (20), the derivative of ${V_2}$ is

(42) \begin{align}{\dot V_2} = {\left( {\mu _{{e_2}}^{{s_2}}} \right)^{ - 1}}s_2^T\left( t \right){\dot s_2}\left( t \right) + Tr\left( {{{\tilde \Theta }^T} \Gamma _\Theta ^{ - 1} \dot{\hat {\Theta}}} \right) + \frac{1}{{\vartheta {\Gamma _\vartheta }}}\tilde \vartheta \dot{\hat \vartheta} + {\tilde \varepsilon ^T}_v{\kern 1pt} {\dot{\hat \varepsilon }}_v - \frac{{{\gamma _0}}}{{{\Gamma _r}}}r\left( t \right) + \frac{{\rho \left( {{x_1}\left( t \right),{x_2}\left( t \right)} \right)}}{{{\Gamma _r}}}\end{align}

Thanks to the differential Equation (34), we can easily obtain that

(43) \begin{align}{\left( {\mu _{{e_2}}^{{s_2}}} \right)^{ - 1}}s_2^T\left( t \right){\dot s_2}\left( t \right) = s_2^T\left( t \right)\left( {\begin{array}{*{20}{l}}{f\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right) + {{\Delta }}f\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right) + {{\Lambda }}\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,u\left( t \right)\!,\zeta \left( t \right)} \right)}\\[5pt] { + B\left( {u\left( t \right) + {d_2}\left( t \right)} \right) - {{\dot x}_{2c}}\left( t \right) + {{\left( {\mu _{{e_2}}^{{s_2}}} \right)}^{ - 1}}L_{{e_2}}^{{s_2}}}\end{array}} \right)\end{align}

Combined with Equations (21)–(25) and (30), one has

(44) \begin{align} s_2^T\left( t \right){{\Lambda }}\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,u\left( t \right)\!,\zeta \left( t \right)} \right) &\le e_2^T\left( t \right){{\bar \varphi }_1}\left( {{s_2}\left( t \right)\!,{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right) + s_2^T\left( t \right){{\bar \varphi }_2}\left( {{s_2}\left( t \right)\!,{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,r\left( t \right)} \right)\nonumber \\[5pt] &+ \frac{1}{4}s_2^T\left( t \right){s_2}\left( t \right) - \left\| {{p_u}} \right\|s_2^T\left( t \right)u\left( t \right) + \mathop \sum \limits_{i = 1}^3 {\varepsilon _i} .\end{align}

With the aid of Equation (26), it’s clear that

(45) \begin{align} & s_2^T\left( t \right){{\Lambda }}\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,u\left( t \right)\!,\zeta \left( t \right)} \right) + s_2^T\left( t \right){{\Delta }}f\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right)\nonumber \\[5pt] & \le s_2^T\left( t \right){\Theta ^T}{\Phi _b}\left( {{s_2}\left( t \right)\!,{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,r\left( t \right)} \right) + s_2^T\left( t \right){\varepsilon _\Theta } + \frac{1}{4}s_2^T\left( t \right){s_2}\left( t \right) - \left\| {{p_u}} \right\|s_2^T\left( t \right)u\left( t \right) + \sum\limits_{i = 1}^3 {{\varepsilon _i}}.\end{align}

Then, combined with Equation (29), the following inequality can be constructed

(46) \begin{align}{\left( {\mu _{{e_2}}^{{s_2}}} \right)^{ - 1}}s_2^T\left( t \right){{\dot s}_2}\left( t \right) &\le s_2^T\left( t \right)\left( \begin{array}{l} f\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right) + {\Theta ^T}{\Phi _b}\left( {{s_2}\left( t \right)\!,{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,r\left( t \right)} \right) + \frac{1}{4}{s_2}\left( t \right)\\[5pt] - \left\| {{p_u}} \right\|u\left( t \right) + Bu\left( t \right) + {\varepsilon _v}\tanh \left( {\frac{{{s_2}\left( t \right)}}{\alpha }} \right) - {{\dot x}_{2c}}\left( t \right) + {\left( {\mu _{{e_2}}^{{s_2}}} \right)^{ - 1}}L_{{e_2}}^{{s_2}} \end{array} \right)\nonumber \\[5pt] &+ \sum\limits_{i = 1}^3 {{\varepsilon _i}} + \kappa \alpha {\varepsilon _v}\end{align}

Substituting ${u_c}\left( t \right)$ into Equation (46) yields that

(47) \begin{align} &{{{\left( {\mu _{{e_2}}^{{s_2}}} \right)}^{ - 1}}s_2^T\left( t \right){{\dot s}_2}\left( t \right) \le s_2^T\left( t \right)\left( {\begin{array}{*{20}{l}}- {k_2}{s_2}\left( t \right) - {\varphi _\rho }\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,{s_2}\left( t \right)} \right) + \left( {B - I\left\| {{p_u}} \right\|} \right)u\left( t \right) - B{u_c}\left( t \right)\\[9pt] { - {{{{\tilde \Theta }}}^T}{{{\Phi }}_b}\left( {{s_2}\left( t \right)\!,{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,r\left( t \right)} \right) - {{\tilde \varepsilon }_v}{\rm{tanh}}\left( {\dfrac{{{s_2}\left( t \right)}}{\alpha }} \right) - \mu _{{e_1}}^{{s_1}}{s_1}\left( t \right)}\end{array}} \right)}\nonumber\\[5pt] &{ + \mathop \sum \limits_{i = 1}^3 {\varepsilon _i} + \kappa \alpha {\varepsilon _v}} \end{align}

Then, referring to Lemma 2.2, we can get that

(48) \begin{align}\begin{array}{l}s_2^T\left( t \right)\left( {B - I\left\| {{p_u}} \right\|} \right)u\left( t \right) - s_2^T\left( t \right)B{u_c}\left( t \right)\\[9pt] = - \dfrac{{\left( {B - I\left\| {{p_u}} \right\|} \right){{\left[ {s_2^T\left( t \right)\hat \vartheta Bu_c^{}\left( t \right)} \right]}^2}}}{{\sqrt {{{\left[ {s_2^T\left( t \right)\hat \vartheta Bu_c^{}\left( t \right)} \right]}^2} + \varepsilon _u^2} }} - s_2^T\left( t \right)B{u_c}\left( t \right)\\[9pt] \le - \dfrac{1}{\vartheta }\dfrac{{{{\left[ {s_2^T\left( t \right)\hat \vartheta B{u_c}\left( t \right)} \right]}^2}}}{{\sqrt {{{\left[ {s_2^T\left( t \right)\hat \vartheta B{u_c}\left( t \right)} \right]}^2} + \varepsilon _u^2} }} - s_2^T\left( t \right)B{u_c}\left( t \right)\\[9pt] \le \frac{1}{\vartheta }\left( { - \left\| {s_2^T\left( t \right)\hat \vartheta Bu_c^{}\left( t \right)} \right\| + {\varepsilon _u}} \right) - s_2^T\left( t \right)B{u_c}\left( t \right)\\[9pt] \le \dfrac{1}{\vartheta }\left( {\tilde \vartheta s_2^T\left( t \right)B{u_c}\left( t \right) + {\varepsilon _u}} \right)\end{array}\end{align}

For any vector $\xi \in {\mathbb{R}^n}$ , we define

(49) \begin{align}{\rm{Tanh}}\left( {\xi \left( t \right)} \right) = {\left[ {{\rm{tanh}}{\xi _1}\left( t \right)\!,{\rm{tanh}}{\xi _2}\left( t \right)\!, \cdots, {\rm{tanh}}{\xi _n}\left( t \right)} \right]^T}.\end{align}

Hence, the following formula is established

(50) \begin{align}\begin{array}{*{20}{l}}{\dfrac{{\rho \left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right)}}{{{{{\Gamma }}_r}}}} {} = { \dfrac{{\rho \left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right)}}{{{{{\Gamma }}_r}}}\left( {1 - 16{\rm{Tan}}{{\rm{h}}^T}\left( {\dfrac{{s_2^T\left( t \right)}}{{{\varepsilon _\rho }}}} \right){\rm{Tanh}}\left( {\dfrac{{s_2^T\left( t \right)}}{{{\varepsilon _\rho }}}} \right)} \right)}\\[9pt] {} {}{ + {\rm{\;\;}}s_2^T\left( t \right){\varphi _\rho }\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,{s_2}\left( t \right)} \right)}\end{array}\end{align}

where

(51) \begin{align}{\varphi _\rho }\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right),{s_2}\left( t \right)} \right) = \frac{{16{s_2}\left( t \right)\rho \left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right)}}{{{{{\Gamma }}_r}s_2^T\left( t \right){s_2}\left( t \right)}}{\rm{Tan}}{{\rm{h}}^T}\left( {\frac{{s_2^T\left( t \right)}}{{{\varepsilon _\rho }}}} \right){\rm{Tanh}}\left( {\frac{{s_2^T\left( t \right)}}{{{\varepsilon _\rho }}}} \right)\end{align}

noting that ${\varphi _\rho }\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,{s_2}\left( t \right)} \right)$ is a non-singular function vector for ${s_2}\left( t \right)$ .

Substituting formulas (47), (48), (50) and (51) into formula (42), and combining Equation (40) and Assumption 4, we can derive the result of the differentiation of $V$ as follows

(52) \begin{align}\begin{array}{l}\dot V = {{\dot V}_1} + {{\dot V}_2}\\[9pt] \le - \bar s_1^T\left( t \right)Q{{\bar s}_1}\left( t \right) - \left( {L - \dfrac{{\mu _{{e_1}}^{{s_1}} + 1}}{2}} \right)\tilde d_1^T\left( t \right){{\tilde d}_1}\left( t \right) + \dfrac{1}{2}\bar d_1^2 - {k_2}s_2^T\left( t \right){s_2}\left( t \right)\\[9pt] + \dfrac{{\rho \left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right)}}{{{{{\Gamma }}_r}}}\left( {1 - 16{\rm{Tan}}{{\rm{h}}^T}\left( {\dfrac{{s_2^T\left( t \right)}}{{{\varepsilon _\rho }}}} \right){\rm{Tanh}}\left( {\dfrac{{s_2^T\left( t \right)}}{{{\varepsilon _\rho }}}} \right)} \right) + \dfrac{{\tilde \vartheta }}{\vartheta }s_2^T\left( t \right)B{u_c}\left( t \right)\\[9pt] + \dfrac{{{\varepsilon _u}}}{\vartheta } - s_2^T\left( t \right){{{{\tilde \Theta }}}^T}{{{\Phi }}_b}\left( {{s_2}\left( t \right)\!,{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,r\left( t \right)} \right) - s_2^T\left( t \right){{\tilde \varepsilon }_v}{\rm{tanh}}\left( {\dfrac{{{s_2}\left( t \right)}}{\alpha }} \right)\\[9pt] + Tr\left( {{{\tilde \Theta }^T}\Gamma _\Theta ^{ - 1} \dot {\hat \Theta} } \right) + \dfrac{1}{{\vartheta {\Gamma _\vartheta }}}\tilde \vartheta \dot {\hat \vartheta} + {{\tilde \varepsilon }^T}_v\; {{\dot {\hat \varepsilon} }_v} - \dfrac{{{\gamma _0}}}{{{\Gamma _r}}}r\left( t \right) + \sum\limits_{i = 1}^3 {{\varepsilon _i}} + \kappa \alpha {\varepsilon _v}\end{array}.\end{align}

By using the adaptive laws Equation (33) and considering the following inequalities

(53) \begin{align}\begin{array}{*{20}{l}}{ - Tr\left( {{{{{\tilde \Theta }}}^T}{{\hat \Theta }}} \right)} {}{ \le - \dfrac{1}{2}Tr\left( {{{{{\tilde \Theta }}}^T}{{\tilde \Theta }}} \right) + \dfrac{1}{2}Tr\left( {{{{\Theta }}^T}{{\Theta }}} \right)}\\[9pt] - {{\tilde \varepsilon }^T}_v\; {{\hat \varepsilon }_v} \le - \dfrac{1}{2}{{\tilde \varepsilon }^2}_v + \dfrac{1}{2}\varepsilon _v^2\\[9pt] { - {{\tilde \vartheta }^T}\hat \vartheta } {}{ \le - \dfrac{1}{2}{{\tilde \vartheta }^2} + \dfrac{1}{2}{\vartheta ^2}}\end{array}\end{align}

Therefore, we can further obtain the following result

(54) \begin{align}\begin{array}{l}\dot V \le - \bar s_1^T\left( t \right)Q{{\bar s}_1}\left( t \right) - \left( {L - \dfrac{{\mu _{{e_1}}^{{s_1}} + 1}}{2}} \right)\tilde d_1^T\left( t \right){{\tilde d}_1}\left( t \right) - {k_2}s_2^T\left( t \right){s_2}\left( t \right) - \dfrac{{{\lambda _\vartheta }}}{{2\vartheta }}{{\tilde \vartheta }^2}\\[9pt] \,\,\,\,\,\, - \dfrac{{{\lambda _{{\Theta }}}}}{2}Tr\left( {{{{{\tilde \Theta }}}^T}{{\tilde \Theta }}} \right) - \dfrac{{{\lambda _\varepsilon }}}{2}{{\tilde \varepsilon }^2}_v + \dfrac{1}{2}\bar d_1^2 + \dfrac{{\rho \left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)} \right)}}{{{{{\Gamma }}_r}}}\left( {1 - 16{\rm{Tan}}{{\rm{h}}^T}\left( {\dfrac{{s_2^T\left( t \right)}}{{{\varepsilon _\rho }}}} \right){\rm{Tanh}}\left( {\dfrac{{s_2^T\left( t \right)}}{{{\varepsilon _\rho }}}} \right)} \right)\\[9pt] \,\,\,\,\,\, + \dfrac{{{\lambda _\vartheta }}}{2}\vartheta + \dfrac{{{\lambda _{{\Theta }}}}}{2}Tr\left( {{{{\Theta }}^T}{{\Theta }}} \right) + \dfrac{{{\lambda _\varepsilon }}}{2}\varepsilon _v^2 + \dfrac{{{\varepsilon _u}}}{\vartheta } - \dfrac{{{\gamma _0}}}{{{{{\Gamma }}_r}}}r\left( t \right) + \mathop \sum \limits_{i = 1}^3 {\varepsilon _i} + \kappa \alpha {\varepsilon _v}\end{array}\end{align}

Ultimately, we can get the following conclusions

(55) \begin{align}\dot V \le - \gamma V + {k_f} + \frac{{\rho \left( {{x_1}\left( t \right){,_2}\left( t \right)} \right)}}{{{{{\Gamma }}_r}}}\left( {1 - 16{\rm{Tan}}{{\rm{h}}^T}\left( {\frac{{s_2^T\left( t \right)}}{{{\varepsilon _\rho }}}} \right){\rm{Tanh}}\left( {\frac{{s_2^T\left( t \right)}}{{{\varepsilon _\rho }}}} \right)} \right)\end{align}

where

(56) \begin{align}\begin{array}{*{20}{l}}{} {}{\gamma = {\rm{min}}\left\{ {2{\lambda _{{\rm{min}}}}\left( Q \right)\!,{\rm{\;\;}}2L - \left( {\mu _{{e_1}}^{{s_1}} + 1} \right)\!,{\rm{\;\;}}2\mu _{{e_2}}^{{s_2}}{k_2},{\rm{\;\;}}{\lambda _\varepsilon },{\rm{\;\;}}{\lambda _{{\rm{min}}}}\left( {{{{\Gamma }}_{{\Theta }}}} \right){\lambda _{{\Theta }}},{\rm{\;\;}}{{{\Gamma }}_\vartheta }{\lambda _\vartheta }} \right\}}\\[5pt] {} {}{{k_f} = \dfrac{{{\lambda _{{\Theta }}}}}{2}Tr\left( {{{{\Theta }}^T}{{\Theta }}} \right) + \dfrac{{{\lambda _\varepsilon }}}{2}\varepsilon _v^2 + \dfrac{{{\lambda _\vartheta }}}{2}\vartheta + \dfrac{{{{\bar d}_1}^2}}{2} + \dfrac{{{\varepsilon _u}}}{\vartheta } + \mathop \sum \limits_{i = 1}^3 {\varepsilon _i} + \kappa \alpha {\varepsilon _v}}\end{array}\end{align}

Define the following compact set

(57) \begin{align}\begin{array}{*{20}{l}}{{{{\Omega }}_f} = \left\{ {x \in {\mathbb{R}^n}\left| {V\left( x \right) \le {\gamma _2}/{\gamma _1}} \right.} \right\}}\\[5pt] {\Omega _\rho } = \left\{ {x\left| {\left\| x \right\| \le 0.2554\varepsilon } \right.} \right\}\end{array}\end{align}

Based on Lemma 3, it is easy to know that if ${s_2}\left( t \right) \in {{{\Omega }}_f} \cap {{{\Omega }}_\rho }$ , the solution of the closed-loop control system $\left[ {{s_0}\left( t \right)\!,{s_1}\left( t \right)\!,{s_2}\left( t \right)\!,{{\tilde \varepsilon }_v}\left( t \right)\!,{{\tilde \Theta }}\left( t \right)\!,\tilde \vartheta \left( t \right)} \right]$ are naturally bounded. If ${s_2}\left( t \right) \notin {{{\Omega }}_f} \cap {{{\Omega }}_\rho }$ , $\dot V \lt 0$ can be proved and $V\left( t \right)$ gradually decreases, and the solution will eventually converge to the set ${{{\Omega }}_f} \cap {{{\Omega }}_\rho }$ . Furthermore, according to Lemma 4, we know that when $t \to \infty $ , ${e_1}\left( t \right) \to 0$ , that is, the system tracking error gradually converges to zero. The proof is completed.