Full-state-constrained intelligent adaptive control for nonlinear systems with unmodeled dynamics and mismatched disturbances

Abstract

This paper develops a novel full-state-constrained intelligent adaptive control (FIAC) scheme for a class of uncertain nonlinear systems under full state constraints, unmodeled dynamics and external disturbances. The key point of the proposed scheme is to appropriately suppress and compensate for unmodeled dynamics that are coupled with other states of the system under the conditions of various disturbances and full state constraints. Firstly, to guarantee that the time-varying asymmetric full state constraints are obeyed, a simple and valid nonlinear error transformation method has been proposed, which can simplify the constrained control problem of the system states into a bounded control problem of the transformed states. Secondly, considering the coupling relationship between the unmodeled dynamics and other states of the controlled system such as system states and control inputs, a decoupling approach for coupling uncertainties is introduced. Thereafter, owing to the employed dynamic signal and bias radial basis function neural network (BIAS-RBFNN) improved on traditional RBFNN, the adverse effects of unmodeled dynamics on the controlled system can be suppressed appropriately. Furthermore, the matched and mismatched disturbances are reasonably estimated and circumvented by a mathematical inequality and a disturbance observer, respectively. Finally, numerical simulations are provided to demonstrate the effectiveness of the proposed FIAC strategy.

1.0 Introduction

As is known to all, it is unavoidable to encounter various constraints in actual systems. When these constraints are not satisfied, the performance and stability of the system will be affected to various degrees. In recent years, the research on state-constrained control, which mainly includes output-constrained control and full-state-constrained control, has become a hot issue studied by scholars in the field of control [1–6]. Full state constraints not only require system outputs to satisfy constraints but also have constraints on other states of the system. Therefore, in most practical applications, the full state constraints which can avoid excessive overshoot are more general than the output constraints [1]. Currently, the barrier Lyapunov functions (BLFs) have become the main tool to ensure that the full state constraints are obeyed [2, 7–9]. With the aid of BLF and the adaptive backstepping method, a novel adaptive tracking control algorithm for the strict-feedback nonlinear systems with full state constraints and dead zone was studied in [7]. In Ref. [8], based on the tan-type time-varying asymmetric BLF, an adaptive asymptotic tracking control method for the nonlinear systems with parametric uncertainty and time-varying asymmetric full state constraints was developed. Similarly, by introducing the time-varying asymmetric BLF in Ref. [9], all state variables were confined to predefined regions. However, considering practical engineering applications, the full-state-constrained control schemes mentioned above do not take into account the complex dynamic coupling uncertainties that may occur in the controlled nonlinear systems, which may lead to the inability to guarantee stability.

It is widely known that disturbances and uncertainties are inevitable in many practical control systems, such as flight control systems, robotics, vehicle systems and mechanical systems and so on [10–14]. However, various disturbances and uncertainties are usually nonlinear, unknown, and coupled with other variables, which makes their mathematical modeling and the design of effective suppression properties difficult. Therefore, the problem of compensation and suppression of disturbances and uncertainties has received extensive attention in the past few decades [14–21]. These come in a variety of forms, including sliding mode control (SMC) [14, 15], active disturbance rejection control (ADRC) [16–18], disturbance observer-based control (DOBC) [14, 19] and adaptive backstepping control (BC) [20, 21]. For a class of robotic systems with matched and mismatched disturbances, a high-order disturbance observer based sliding mode control method was designed in Ref. [14], which can successfully stabilise the robotic systems and suppress the matched and mismatched disturbances effectively. To realise the predictive state and the total uncertainty which represents the effects of the unknown nonlinear dynamics and external disturbances, an extended state observer was developed as a predictor in Ref. [16]. By using the backstepping technique, the authors in Ref. [20] designed an adaptive neural network controller for the uncertain nonlinear systems suffering from input delay and disturbances. In Ref. [21], an adaptive fuzzy backstepping dynamic surface controller was constructed for a class of strict-feedback uncertain nonlinear systems with unknown external disturbances. It is worth noting that the control methods mentioned above are all aimed at handling a single source of disturbances and uncertainties. There are usually multi-source disturbances and uncertainties acting on the actual physical systems, and the characteristic of the complex dynamic coupling in the multi-source uncertainties makes the design of the controller more challenging. Therefore, how to develop a controller to attenuate the effects of dynamic multi-source coupling uncertainties of the controlled system is still an open issue, which motivates this study again.

Motivated by the above discussions, a novel full-state-constrained intelligent adaptive control (FIAC) scheme is proposed for a class of nonlinear systems with unmodeled dynamics and mismatched disturbances in this study. The main contributions can be summarised as follows:

• • Different from most of the existing results on the control of uncertain nonlinear systems, this work takes into account the unmodeled dynamics that are coupled with the system states and control inputs and designs a decoupling method based on dynamic signal and improved RBFNN, namely BIAS-RBFNN, to estimate and circumvent the coupling uncertainties.

• • To improve the approximation accuracy and anti-disturbance ability of traditional RBFNN, the BIAS-RBFNN is introduced in this paper.

• • Compared to the existing state-constrained control schemes with multiple parameters and complex forms in Ref. [22], this work proposed a simple and effective nonlinear error transformation method to transform the constrained control problem of the system states into a bounded control problem of the transformed states, which greatly reduces the difficulty of controller design.

• • This paper provides a BIAS-RBFNN-based FIAC canonical form for the nonlinear systems, which can be applied to many other practical systems like robots, quadrotors, space unmanned systems, underwater unmanned systems, and so on.

This paper is organised as follows. In the next section, the full-state-constrained control problem with unmodeled dynamics and disturbances is formulated and some related assumptions and lemmas are introduced. Moreover, we give a detailed description of the novel BIAS-RBFNN. Then, the structure of the proposed full-state-constrained intelligent adaptive control scheme is built, followed by stability analysis. Finally, simulations are performed to verify the effectiveness of the proposed control method and a conclusion is presented.

2.0 Problem formulation and preliminaries

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 1. The system states ${x_1}\left( t \right)$ and ${x_2}\left( t \right)$ are both accessible. The desired signal ${x_d}\left( t \right)$ is bounded, smooth and twice differentiable.

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.

Assumption 4. For the unknown total disturbance $D\left( t \right)$ , there exists an unknown positive constant ${\varepsilon _v}$ such that $\left| {D\left( t \right)} \right| \le {\varepsilon _v}$ . What’s more, the rate of mismatched disturbance ${d_1}\left( t \right)$ is assumed to be bounded, that is $\left\| {{{\dot{d}}_1}\left( t \right)} \right\| \le {\bar d_1}$ .

Assumption 5. The initial values of the system states are all within the predefined constraints, i.e. ${\underline{b}_1}\left( 0 \right) \lt {x_1}\left( 0 \right) \lt {\bar b_1}\left( 0 \right)\!,{\underline{b}_2}\left( 0 \right) \lt {x_2}\left( 0 \right) \lt {\bar b_2}\left( 0 \right)$ .

Remark 1. Considering the possible coupling relationship between the unmodeled dynamics $\zeta \left( t \right)$ and other states in the nonlinear system, Assumption 2 briefly gives the common constraints for the coupling uncertainties $\Lambda \left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,u\left( t \right)\!,\zeta \left( t \right)} \right)$ . Assumption 3 provides some standard requirements of the unmodeled dynamics.

Lemma 1. [23] 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. [24] 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. [24] 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}

Lemma 4. [25] Let $f\;:\;\mathbb{R} \to \mathbb{R}$ represents any continuous differentiable function defined on $\left[ {0,\infty } \right)$ , and $\mathop {{\rm{lim}}}\limits_{t \to \infty } f\left( t \right)$ exists and possesses upper bound. If its derived function $\dot f\left( t \right)$ is uniformly continuous, then $\mathop {{\rm{lim}}}\limits_{t \to \infty } \dot f\left( t \right) = 0$ .

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 [26–29]. 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 [30].

According to Ref. [31], 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. [32], ${{{\Phi }}_{lj}}\left( Z \right)$ 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:

Lemma 5. [30] For the target functions with external disturbances, the RBFNNs with bias are more likely to achieve a higher approximation accuracy than the scheme of RBFNNs without bias.

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.

3.0 Controller design and stability analysis

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}

Remark 2. The main idea of nonlinear error transformation is that when the original constraint states contact the constraint boundary, the transformed states will tend toward infinity. Generally, some basic functions will approach infinity at their own singular points, which can help us achieve the above results, such as the independent variable of the tangent function approaching $\frac{\pi }{2}$ , the antilogarithm of the logarithmic function approaching 0, and so on.

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.

Theorem 1. Consider the nonlinear system shown in Equation (1), which is affected by unmodeled dynamics, coupled uncertainties, and various external disturbances . Assume that Assumption 1 $\sim $ 5 are satisfied. By using the nonlinear error transformation method Equation (12), the control laws Equations (31), (32), and the adaptive laws Equation(33), all signals have upper bounds and the tracking errors will converge to the compact sets. And all states will not violate their constraint boundaries.

Proof. According to Ref. [33], 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.

4.0 Simulation study

In this section, numerical simulations are conducted to demonstrate the validity of the proposed FIAC method. The simulations will be carried out in the following two aspects: a comparison of the proposed method with other methods and a comparison of the proposed method under different cases.

Based on the nonlinear system given by Equation (1), the specific parameters are given as

(58) \begin{align}\begin{array}{*{20}{l}} f\left( {{x_1}\left( t \right),{x_2}\left( t \right)} \right) = 0.2\sin \left( {{x_1}\left( t \right) + {x_2}\left( t \right)} \right),\;B = 2\\[5pt] \Delta f\left( {{x_1}\left( t \right),{x_2}\left( t \right)} \right) = 0.1\sin \left( {0.2{x_1}\left( t \right) + 0.1{x_2}\left( t \right)} \right)\end{array}\end{align}

The mismatched disturbance and matched disturbance are set as

(59) \begin{align}{d_1}\left( t \right) = 0.1\;{\rm{sin}}\left( {0.1{\rm{\;}}t} \right)\!,{\rm{\;\;}}{d_2}\left( t \right) = 0.2\;{\rm{sin}}\left( {0.2{\rm{\;}}t} \right)\end{align}

We choose the unmodeled dynamics as

(60) \begin{align}\dot \zeta \left( t \right) = - 12\zeta \left( t \right) + {x_1}{\left( t \right)^T}{x_1}\left( t \right) + {x_2}{\left( t \right)^T}{x_2}\left( t \right)\!,{\rm{\;\;}}\zeta \left( 0 \right) = 1\end{align}

Figure 2. The tracking performance of ${x_1}\left( t \right)$ under different control schemes.

Figure 3. The tracking performance of ${x_2}\left( t \right)$ under different control schemes.

Figure 4. The tracking performance of ${x_1}\left( t \right)$ of the proposed FIAC scheme under different cases.

Figure 5. The tracking performance of ${x_2}\left( t \right)$ of the proposed FIAC scheme under different cases.

Figure 6. The trajectories of the adaptive parameters of the proposed FIAC scheme.

Figure 7. The trajectories of the auxiliary signal $r\left( t \right)$ and the unmodeled dynamics $\zeta \left( t \right)$ .

The coupling uncertainty is affected by the system states, the input signal, and the unmodeled dynamics, which is supposed to be

(61) \begin{align}{{\Lambda }}\left( {{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,u\left( t \right)\!,\zeta \left( t \right)} \right) = - 2{\rm{sin}}\left( t \right){x_1}\left( t \right) + \zeta \left( {{x_1}\left( t \right) + {x_2}\left( t \right)} \right) - {\rm{sin}}\left( {\zeta \left( t \right)} \right)u\left( t \right)\end{align}

What’s more, the dynamic auxiliary signal $r\left( t \right)$ is designed as

(62) \begin{align}\dot r\left( t \right) = - 3r\left( t \right) + {x_1}{\left( t \right)^T}{x_1}\left( t \right) + {x_2}{\left( t \right)^T}{x_2}\left( t \right)\!,{\rm{\;\;}}r\left( 0 \right) = 0.5\end{align}

The time-varying asymmetric constraints of the states are set as

(63) \begin{align}\begin{array}{*{20}{l}}{} {}{{\underline{b}_1}\left( t \right) = - {e^{ - t}} - 1,{{\bar b}_1}\left( t \right) = 5{e^{ - t}} + 1}\\[5pt] {} {}{{\underline{b}_2}\left( t \right) = - 6{e^{ - t}} - 1,{{\bar b}_2}\left( t \right) = 2{e^{ - t}} + 1}\end{array}\end{align}

The input of the BIAS-RBFNN is selected as $Z\left( t \right) = {\left[ {{x_1}\left( t \right)\!,{x_2}\left( t \right)\!,{s_2}\left( t \right)\!,r\left( t \right)} \right]^T}$ and the position of the hidden nodes of each input channel is $\left[ { - 1,{\rm{\;\;}}0,{\rm{\;\;}}1} \right]$ , thus the total number of hidden nodes is ${3^4} = 81$ .

In this simulation, the initial conditions are set as follows: ${x_1}\left( 0 \right) = 2,{\rm{\;\;}}{x_2}\left( 0 \right) = 0.1,{\rm{\;\;}}{\hat \varepsilon _v}\left( 0 \right) = 0,{{\;\;\hat \Theta }}\left( 0 \right) = \left[ {0,{\rm{\;}}0,{\rm{\;}} \ldots, {\rm{\;}}0} \right]_{1 \times 81}^T,{\rm{\;\;}}\hat \vartheta \left( 0 \right) = 3$ . The control gains are designed as ${k_0} = 10,{\rm{\;\;}}{k_1} = 4,{\rm{\;\;}}{k_2} = 10$ . The adaptive parameters are given as ${\lambda _\varepsilon } = 0.3,{\rm{\;}}\lambda_{{\Theta }} = 0.1,{\rm{\;\;}}{\lambda _\vartheta } = 0.2$ and the gains of the adaptive laws are set as ${{{\Gamma }}_{{\Theta }}} = 10,{\rm{\;\;}}{{{\Gamma }}_\vartheta } = 1.2$ . The constants in the simulation are selected as ${\varepsilon _u} = 2,{\rm{\;\;}}{\varepsilon _\rho } = 0.1,{\rm{\;\;}}\alpha = 0.1,{\rm{\;\;}}{{{\Gamma }}_r} = 100$ . The gain of the disturbance observer is set as $L = 10$ . The time-varying desired signal is generated by ${x_d}\left( t \right) = 2{e^{ - t}}$ .

The simulation results are provided in Figs. 2–7. To verify the advantages of the proposed FIAC method, the full-state-constrained adaptive control (FAC) method without BIAS-RBFNN (BNN) and the FIAC method without fault-tolerant processing (FTP) are also taken into account for comparison, which is shown in Figs. 2 ${\rm{\;and\;}}$ 3. It can be seen from Figs. 2 ${\rm{\;and\;}}$ 3 that the proposed FIAC method can achieve excellent tracking performance and guarantee to never violate the predefined time-varying asymmetric full state constraints in simultaneous presence of the unmodeled dynamics and disturbances, which indicates the control objective has been realised successfully. It is obvious that although the tracking trajectories of the FAC method without BNN and the FIAC method without FTP converge gradually, the dynamic performance of the system is ignored, resulting in nonnegligible oscillation and overshoot, as well as the system states exceeding the constraint boundary. The varying trajectories of the adaptive parameters of the proposed FIAC method are shown in Fig. 6, which demonstrates the boundness of all the adaptive parameters. More specifically, it is shown in Fig. 7 that the purpose of suppressing unmodeled dynamics is achieved well by the dynamic auxiliary signal $r\left( t \right)$ .

To show the robustness of the proposed FIAC method, a numerical simulation is carried out under three different cases, which is shown in Figs. 4 and 5. Three different cases correspond to three groups of different disturbances and model uncertain items, the specific values to be set are shown in the Table 2.

In order to present a fair comparison, here we take the same initial conditions and the controller parameters which are mentioned above. As illustrated in Figs. 4 and 5, although the tracking trajectories will fluctuate with the increase of disturbances and uncertainty, the overall tracking effects still meet the requirements, which shows that the proposed FIAC method has strong anti-disturbance ability and robustness.

5.0 Conclusions

In this paper, we have proposed a novel FIAC scheme for a class of uncertain nonlinear systems subject to full state constraints, unmodeled dynamics and disturbances. Different from most of the existing control shemes that consider unmodeled dynamics of system, this paper also solves the problem of coupling relationship between unmodeled dynamics and system states, and proposes a decoupling algorithm for coupling uncertainties using an auxiliary dynamic signal and BIAS-RBFNN. Among them, BIAS-RBFNN is an improved version of traditional RBFNN, which has stronger robustness and higher estimation accuracy. Moreover, the nonlinear error transformation technology adopted in this paper is simple and effective, which ensures the states of the system do not exceed the predefined time-varying asymmetric constraints and reduces the difficulty of controller design to a certain extent. By utilising mathematical inequality and a disturbance observer, both matched and mismatched disturbances are reasonably estimated and circumvented. Last but not least, through two groups of comparative simulations, it can be clearly seen that the proposed FIAC scheme in this work achieves satisfactory tracking performance without violating predefined full state constraints, and all closed-loop signals are bounded.

Table 2. Three different cases of the simulation experiment