Adaptive fuzzy finite-time output feedback fault-tolerant control for MIMO stochastic nonlinear systems with non-affine nonlinear faults

Abstract

This paper investigates adaptive fuzzy finite-time output feedback fault-tolerant tracking control problem for multiple-input multiple-output (MIMO) stochastic non-strict feedback nonlinear systems with non-affine nonlinear faults and states unmeasured. Firstly, high-gain fuzzy state observers are constructed to overcome the state unmeasured problem in the system. Secondly, the mean-value theorem and input compensation techniques are employed to decouple non-affine nonlinear faults, and the resulting unknown control coefficients problem is handled using Nussbaum functions. In addition, dynamic surface control techniques and error compensation mechanisms are considered in the controller design, which effectively decreases the computational complexity of the control scheme. By combining the adaptive backstepping technique with finite-time control, a novel adaptive fuzzy dynamic surface finite-time output feedback fault-tolerant tracking control algorithm is proposed. The designed controller guarantees that all signals in the stochastic systems are semi-global finite-time stable in probability (SGFSP) as well as the tracking errors can converge to a small neighborhood within the origin in finite-time. Finally, two simulation examples are introduced to verify the effectiveness of the suggested control strategy.

Introduction

In recent years, the problem of controller design for nonlinear systems received extensive attention and achieved fruitful results,^1–6 among which the adaptive backstepping technique gradually stands out due to its unique superiority in controller design. Recently, fuzzy logic systems (FLSs) and neural networks (NNs) attracted the attention of scholars for their powerful approximation ability of nonlinear unknown functions, and scholars combined them with adaptive backstepping techniques to design a series of new adaptive fuzzy or NN control strategies for nonlinear systems.^7–10 Among them, Xiao et al. ⁷ designed fuzzy adaptive tracking control for pure feedback systems with unknown dead zones and full state constraints. Shi et al. ⁸ proposed a dynamic surface-based fuzzy adaptive robust asymptotic tracking control scheme for nonlinear systems with unknown control directions. In literature, ⁹ Cui et al. designed fuzzy adaptive tracking control algorithms for MIMO switching nonlinear systems, and in literature, ¹⁰ Sui et al. investigated the problem of prescribed performance fuzzy adaptive finite-time output feedback control for MIMO nonlinear systems. However, the above control approaches do not apply to more general non-strict feedback nonlinear systems (NSFNSs). For this reason, in the literature, ¹¹ the variable separation technique was used to solve the adaptive control design problem for NSFNSs. It is worth noting that the nonlinear functions ¹¹ are restricted to monotonically increasing functions, which are certainly a strict condition. Subsequently, in literatures,^12,13 the authors cleverly removed the restriction of monotonically increasing nonlinear functions by using the properties of the FLS/NN, and designed a new fuzzy adaptive fixed-time tracking control strategy and a novel adaptive NN tracking control strategy for NSFNSs, respectively. On this basis, Li et al. ¹⁴ investigated the observer-based fuzzy adaptive finite-time dynamic surface tracking control problem for MIMO nonlinear systems with non-lower triangular structure. Nevertheless, the control objects studied^7–14 are all deterministic systems, which overlook the effects of stochastic disturbances.

Admittedly, many practical systems are usually accompanied by stochastic disturbances, which are the source of instability in control systems, and therefore, it is necessary to consider the influences of many stochastic disturbances such as external disturbances and modeling errors in controller design. For stochastic NSFNSs limited by other external disturbances, Pan et al. ¹⁵ developed an asymptotic tracking control scheme and implemented disturbance decay, which is the first application of the backstepping technique to the stochastic nonlinear systems. Wang et al. ¹⁶ designed event-triggered adaptive fuzzy tracking controller for stochastic pure feedback nonlinear systems, and the designed controller ensured that overall signals in the stochastic closed-loop systems are semi-globally uniformly and ultimately bounded in probability. In literatures,^17,18 adaptive NN tracking control schemes were proposed for stochastic NSFNSs and switched stochastic NSFNSs, respectively, and the designed control schemes achieve that output signal of the systems asymptotically track to the desired signal. Compared to the asymptotic tracking,^15–18 finite-time control received much attention because of its advantages of quicker response performance, higher robustness, and better control accuracy. Yin et al. ¹⁹ first introduced the concept of finite-time stability for stochastic nonlinear systems, and Min et al. ²⁰ introduced the related stochastic Lyapunov function theorem and defined SGFSP. In the literature, ²¹ Sui et al. proposed a new adaptive fuzzy control scheme for MIMO stochastic NSFNSs based on finite-time stability, and the designed controller ensures that overall signals in the systems considered are SGFSP. However, one point should be emphasized that the control elements of the control systems in the above control results are fault-free.

In many practical systems, control elements often occurred faults, which often lead to the instability of control systems. Therefore, how to design suitable control strategies for nonlinear systems with faults become important and many results have been achieved.^22–28 Among them, Liu et al. ²² first investigated adaptive neural finite-time FLC problem for switched nonlinear systems with bias and loss of effectiveness faults in the actuators. Zhang et al. ²³ employed the Nussbaum function to address the loss of effectiveness faults in the actuators and proposed an event-triggered adaptive output feedback control strategy for switched large-scale nonlinear systems. In literatures,^24–27 fuzzy adaptive fault-tolerant tracking control, adaptive NN output feedback fault-tolerant tracking control, fuzzy adaptive nonsingular fixed-time fault-tolerant tracking control, and finite-time adaptive NN fault-tolerant tracking control were studied for nonlinear systems, respectively, and the systems considered^24–27 were limited to lock-up and the loss of effectiveness faults in the actuators. Furthermore, Xiong et al. ²⁸ had designed the adaptive output feedback FTC strategy for nonlinear systems with sensor faults. However, the above FLC schemes are not effective for nonlinear systems with non-affine nonlinear faults. In the literature, ²⁹ Li et al. employed the Butterworth low-pass filter to deal with the non-affine nonlinear fault problem and designed an adaptive fuzzy output feedback FLC scheme for nonlinear systems. Subsequently, this treatment approach was widely utilized. Dong et al. ³⁰ proposed a prescribed performance fuzzy output feedback FLC strategy for multiagent systems with non-affine nonlinear faults, and Shao et al. ³¹ designed a fuzzy adaptive fixed-time FLC scheme for nonlinear interconnected systems with non-affine nonlinear faults. In literatures,^32,33 the fuzzy adaptive output-constrained FLC problem and the adaptive fuzzy optimal FLC problem were investigated separately for stochastic nonlinear systems with non-affine nonlinear faults. It is worth noting that none of the above FTC strategies^29–33 consider finite-time control. To the best of the authors’ knowledge, the adaptive fuzzy finite-time FTC problem for MIMO stochastic nonlinear systems with non-affine nonlinear faults and non-lower triangular structure is rarely investigated in existing results, which is one of the motivations of this paper. In addition, the treatment method that uses the Butterworth low-pass filter for non-affine nonlinear faults in Refs. [29–33] has certain limitations, therefore, how to eliminate the limitations in Refs. [29–33] and to design a new treatment method, is another motivation of this paper. Moreover, the consideration of MIMO stochastic NSFNSs makes controller design more challenging.

In addition, actual systems are often subject to special environmental and other factors, which cause the states of the control system to be unmeasured. Therefore, output feedback control needs to be taken into account in controller design. By constructing the state observer, Chen et al. ³⁴ designed an adaptive output feedback tracking control algorithm for NSFNSs. In the literature, ³⁵ Tong et al. constructed the fuzzy state observer to design an adaptive fuzzy output feedback tracking control strategy for NSFNSs with unmeasured states. The adaptive fuzzy finite-time output feedback controller design problem for MIMO nonlinear systems was addressed in Ref.[36]. Sun et al. ³⁷ proposed an adaptive NN output feedback control scheme for stochastic nonlinear systems with states unmeasured, and Li et al. ³⁸ presented a fuzzy adaptive output feedback control strategy for stochastic MIMO nonlinear systems with unknown control coefficients and unknown dead zones. In the literature, ³⁹ Zhang et al. constructed a high-gain fuzzy state observer as a special structure to overcome the problem that the system is not being processed by the fuzzy function, and designed a finite-time adaptive fuzzy output feedback FLC algorithm for NSFNSs with actuator faults. Furthermore, the problem of “complexity explosion” is also frequently encountered during the traditional controller design process. In the literature, ⁴⁰ the dynamic surface control (DSC) technology was used to address the problem, which introduces the first-order filter in the backstepping design process. However, more errors will be produced when the order of the system is increased. To tackle this problem, the corresponding error compensation system is constructed ⁴¹ to address the influence of the error generated with the DSC.

Based on the above discussion, this paper aims to design adaptive fuzzy dynamic surface finite-time FTC strategies for MIMO stochastic NSFNSs with non-affine nonlinear faults and state unmeasured. In the process of designing the controller, the FLS is used to deal with the unknown nonlinear functions in the system. On this basis, the high-gain fuzzy state observers are constructed. Combining the finite-time control, the backstepping technique, and Nussbaum functions, and employing DSC techniques and the error compensation mechanism, an observer-based adaptive fuzzy dynamic surface finite-time FTC scheme is proposed. In contrast to the existing results, the main characteristics and advantages of this paper are as follows:

1) Compared to stochastic nonlinear systems,^17–20 MIMO nonlinear systems,^9,10,16 and NSFNSs,^11–13,34 MIMO stochastic NSFNSs are considered, which means that the controller designs more challenging due to simultaneously considering non-strict structure and stochastic disturbance.

2) Distinct from the linear fault problem considered,^22–28 the system in this paper is subject to more complex non-affine nonlinear faults. Inspired by work in literature, ⁴² the mean-value theorem and the compensation techniques are employed to deal with the non-affine nonlinear fault problem, and the resulting unknown control coefficients are handled using Nussbaum functions. In contrast to^29–33,43 using Butterworth low-pass filters to overcome the above-mentioned issue, this paper directly decouples the non-affine nonlinear faults, which removes the limitation,^29–33,43 such as the high-frequency and low-pass properties, and the boundedness for functions.

3) Different from the adaptive fuzzy FTC approaches,^{23,25,26,28–33} the finite-time control theory is considered in the adaptive fuzzy FTC design algorithm of this paper, which the controller designed in this article not only has better tracking performance but also ensures that all signals in the stochastic systems are SGFSP. In addition, the DSC technique and error compensation mechanisms are incorporated into the process of designing the controller, which effectively eliminates the issue of “explosion of complexity,”^22–33 and optimizes the control algorithm.

Problem formulation and Preliminaries

Problem formulations

Consider the following MIMO stochastic NSFNSs with non-affine nonlinear faults

$\{\begin{array}{l}d{x}_{i,j}=\left(x_{i,j+1}+f_{i,j}\left(x\right)\right)dt+g_{i,j}\left(x_i\right)d{\omega }_i,i=1,2,\dots ,N;j=1,2,\dots ,n_i-1\\ d{x}_{i,n_i}=\left(u_i+f_{i,n_i}\left(x\right)+{\alpha }_i\left(t-T_0\right){\delta }_i\left(x_i,u_i\right)\right)dt+g_{i,n_i}\left(x_i\right)d{\omega }_i,\\ y_i=x_{i,1},\end{array}$

(1)

where $x={\left[x_1,\dots ,x_i,\dots ,x_N\right]}^T\in R^N$ with $x_i={\left[x_{i,1},\dots ,x_{i,n_i}\right]}^T\in R^{n_i}$ are the state vectors of the system, f _i (x) and g _i (x _i ) are the smooth nonlinear uncertain functions of the system. u _i ∈ R are the system input, and y _i ∈ R are the system output. ω _i is an r _i -dimensional independent Wiener process contenting the incremental covariance $E\left\{d{\omega }_i\cdot d{{\omega }_i}^T\right\}={\sigma }_i\left(t\right){\sigma }_i{\left(t\right)}^{T}dt$ , and σ _i (t) is an unknown incremental covariance matrix. δ _i (x _i , u _i ) ∈ R is the non-affine nonlinear fault function, and α _i (t − T₀) ∈ R is the time profile of the fault that occurs at some unknown time T₀, which is described as follows

${\alpha }_i\left(t-T_0\right)=\{\begin{array}{l}0,t<T_0,\\ 1-e^{-{\mathrm{\varrho }}_i\left(t-T_0\right)},t\le T_0,\end{array}$

(2)

where ϱ _i > 0 represents the unknown fault evolution rate. By employing the mean-value theorem in Ref. [51], the non-affine nonlinear fault function δ _i (x _i , u _i ) can be written as

${\delta }_i\left(x_i,u_i\right)={\delta }_i\left(x_i,u_{i_0}\right)+{\delta }_{u_{{\varsigma }_i}}\left(u_i-u_{i_0}\right),$

(3)

where ${\delta }_{u_{{\varsigma }_i}}=\left(\partial {\delta }_i\left(x_i,u_i\right)/\partial u_i\right){|}_{u_i=u_{{\varsigma }_i}}\in R$ with $u_{{\varsigma }_i}\in \left(u_{i_0},u_i\right)$ . By selecting $u_{i_0}=0$ , equation (3) can be written as ${\delta }_i\left(x_i,u_i\right)={\delta }_i\left(x_i,0\right)+{\delta }_{u_{{\varsigma }_i}}{u}_i$ . Defining $h_i\left(x_i\right)=\left({\delta }_{u_{{\varsigma }_i}}-b_i\right)u_i+{\delta }_i\left(x_i,0\right)$ , one gets

${\delta }_i\left(x_i,u_i\right)=b_i{u}_i+h_i\left(x_i\right).$

(4)

Substituting (4) into (1), one gets

$\{\begin{array}{l}d{x}_{i,j}=\left(x_{i,j+1}+f_{i,j}\left(x\right)\right)dt+g_{i,j}\left(x_i\right)d{\omega }_i,i=1,2,\dots ,N;j=1,2,\dots ,n_i-1\\ d{x}_{i,n_i}=\left({\overline{f}}_{i,n_i}\left(x\right)+\left({\alpha }_i\left(t-T_0\right)b_i+1\right)u_i\right)dt+g_{i,n_i}\left(x_i\right)d{\omega }_i,\\ y_i=x_{i,1},\end{array}$

(5)

where b _i are gain constants and it follows from the discussion in Ref. [46] that b _i ≠ − $\frac{1}{{\alpha }_i(t-T_0)}$ ∈ R, and ${\overline{f}}_{i,n_i}\left(x\right)=f_{i,n_i}\left(x\right)+{\alpha }_i\left(t-T_0\right)h_i\left(x_i\right)$ .

Remark 1. There exist many practical control systems that can be modeled as systems (1), such as the system of two inverted pendulums connected by a spring, ⁵² the two continuous stirred tank reactor system, ⁵³ and so on. Compared to stochastic nonlinear systems,^17–20 MIMO nonlinear systems,^9,10,16 and NSFNSs,^11–13,34 the MIMO stochastic NSFNSs that are more general structure and consistent with actual systems are considered, which means that the designed control scheme has wider applicability.

Remark 2. Distinct from the linear fault problem considered,^22–28 the system in this paper is subject to more general non-affine nonlinear faults, which fault functions containing system states x _i and the system inputs u _i , simultaneously. Therefore, the controllers^22–28 are not applicable. The mean-value theorem and compensation techniques are employed in this article to decouple the non-affine nonlinear faults, and the resulting unknown control coefficients are handled using Nussbaum functions. In contrast to works^29–33,43 using Butterworth low-pass filters to overcome the issue of the non-affine nonlinear faults, the treatment method of this paper removes the limitation,^29–33,43 such as the high-frequency and low-pass property, and the boundedness for functions.

The control objective in this study is to propose an adaptive fuzzy dynamic surface finite-time output feedback FTC algorithm for the MIMO stochastic NSFNSs (1) with non-affine nonlinear fault and unmeasurable states such that

(i) Overall signals within the closed-loop MIMO systems are SGFSP;

(ii) Output signals of the system y _i can track given signals y_i,d in the mean square sense within finite-time.

Preliminaries

Stochastic finite time stability

The following stochastic system is given

$dx=f\left(x\right)dt+g\left(x\right)d\omega ,$

(6)

where x and ω are designed in the system (1). f(x): R ⁿ → R ⁿ and g(x): R ⁿ → R^n×r are locally Lipschitz functions in x and f(0) = 0, g(0) = 0.

Definition 1. ⁴⁴ For any given $V\left(x\right)\in {\mathcal{C}}^2$ , the differential operator $\mathcal{L}$ is defined as follows

$\mathcal{L}V\left(x\right)=\frac{\partial V\left(x\right)}{\partial x}f+\frac{1}{2}Tr\left\{g^T\frac{{\partial }^{2}V}{\partial x^2}g\right\},$

(7)

where $Tr\{\cdot \}$ denotes the trace of the matrix.

Assumption 1. ⁴⁵ The disturbance covariance $g^T\sigma \left(t\right)\sigma {\left(t\right)}^{T}g=\overline{\sigma }{\overline{\sigma }}^T$ is bounded.

Definition 2. ⁴¹ For any initial values x(t₀) = x₀, there is a constant τ and a setting time T(τ, x₀) such that for t ≥ T(τ, x₀), one has E(‖x(t; x₀)‖) ≤ τ. Then, the system (1) is SGFSP.

Lemma 1 ⁴⁴ For the stochastic system (6), if exists the functions $V\left(x\right)\in {\mathcal{C}}^2$ and $k_1,k_2\in {\mathcal{K}}_{\infty }$ satisfy

$\begin{array}{l}{k}_1\left(\Vert x\Vert \right)\le V\left(x\right)\le k_2\left(\Vert x\Vert \right),\\ \mathcal{L}V\le -{\mu }_{1}V\left(x\right)-{\mu }_2{V}^{\beta }\left(x\right)+C,\end{array}$

(8)

where μ₁ > 0, μ₂ > 0, 0 < β < 1, 0 < τ < 1, and 0 < C < ∞ are constants. Then, the nonlinear system (6) is SGFSP, and the setting time T(x₀) satisfies

$E\left[T\left(x_0\right)\right]\le \frac{1}{\left(1-\beta \right){\mu }_1}\ln \left[\frac{{\mu }_{1}V{\left(x_0\right)}^{1-\beta }+{\mu }_2\tau }{{\mu }_1{\left(\frac{C}{\left(1-\tau \right)/{\mu }_2}\right)}^{\frac{1-\beta }{\beta }}+{\mu }_2\tau }\right].$

(9)

Lemma 2 ⁴⁶ For any real numbers $D_i\in R,i=1,2,\dots ,n,p\in (0,1]$ , there is

${\left({\displaystyle \sum }_{i=1}^n\left|D_i\right|\right)}^P\le {\displaystyle \sum }_{i=1}^n{\left|D_i\right|}^p\le n^{1-p}{\left({\displaystyle \sum }_{i=1}^n\left|D_i\right|\right)}^P.$

(10)

Lemma 3 ⁴⁷ For real variables ∀x, y, by defining the positive constants l, m, n, the subsequent inequality can be satisfied

${\left|x\right|}^m{\left|y\right|}^n\le \frac{m}{m+n}l{\left|x\right|}^{m+n}+\frac{n}{m+n}{l}^{-m/n}{\left|y\right|}^{m+n}.$

(11)

Lemma 4 (Young’s Inequality ⁴⁸ ). For ∀(ϱ, ψ) ∈ R², the following inequality holds

$\mathrm{\varrho }\psi \le \frac{p^o}{o}{\left|\mathrm{\varrho }\right|}^o+\frac{1}{ϵp^{ϵ}}{\left|\psi \right|}^{ϵ},$

(12)

where p > 0, o > 1, ϵ > 1, and (o − 1) (ϵ − 1) = 1.

Nussbaum-type function

Definition 3. ⁴⁹ A continuous function N(ξ) is referred to as the Nussbaum-type when it satisfies

$\begin{array}{l}\underset{s\to +\infty }{\text{liminf}}\frac{1}{s}{\int }_0^{s}N\left(\xi \right)d\xi =+\infty ,\\ \underset{s\to +\infty }{\text{liminf}}\frac{1}{s}{\int }_0^{s}N\left(\xi \right)d\xi =-\infty ,\end{array}$

(13)

there are many selections for N(ξ) that can satisfy the above properties. For example: ξ²cos(ξ), ξ²sin(ξ), exp(ξ²) cos(ξ²), $e^{{\xi }^2}\hspace{0.17em}\mathit{cos}\left(\left(\pi /2\right)\xi \right)$ and so on.

Lemma 5 ³⁹ Consider the smooth functions V(x) and ξ(t) defined on [0, t _f ) with V(t) ≥ 0, ∀t ∈ [0, t _f ), and if following inequality holds, the N(ξ) be a Nussbaum-gain function

$0\le V\left(t\right)\le a_1+e^{-a_{2}t}{\int }_0^t\left(\gamma N\left(\xi \right)+1\right)\dot{\xi }{e}^{a_2\tau }d\tau ,$

(14)

where a₁ is a suitable design constant, and a₂ is positive parameters, and γ ≠ 0 is the constant. Then, V(t), ξ(t) and ${\int }_0^t\left(\gamma N\left(\xi \right)+1\right)\dot{\xi }d\tau$ are bounded on [0, t _f ).

Fuzzy logic systems

Lemma 6 ⁵⁰ Given any continuous function $f\left(Z\right)$ defined over the compact set Ω. There is an FLS $W^{T}S\left(Z\right)$ , which satisfies

$\underset{Z\in \Omega }{\mathit{sup}}\left|f\left(Z\right)-W^{T}S\left(Z\right)\right|\le \epsilon ,$

(15)

where ɛ > 0, $W={\left[W_1,\dots ,W_N\right]}^T$ , $S\left(Z\right)={\left[p_1\left(Z\right),\dots ,p_N\left(Z\right)\right]}^T/{\displaystyle \sum }_{i=1}^N{p}_i\left(Z\right)$ represent a given positive constant, the ideal weight, and the fuzzy basis function, respectively, where N >1 is the number of fuzzy rules, and p _i , i = 1, …, N is selected as Gaussian functions.

Adaptive fuzzy finite time output feedback fault-tolerant control design

This section will design fuzzy state observers with high gain, and by using the adaptive backstepping technique, an adaptive fuzzy dynamic surface finite-time output feedback FTC strategy will be designed.

High gain fuzzy state observers design

According to Lemma 6, the functions $f_{i,j}\left(x\right),j=1,\dots ,n_i-1$ , ${\overline{f}}_{i,n_i}\left(x\right)$ can be approximated using the FLSs as

${\widehat{f}}_{i,j}\left(\widehat{x}|{\widehat{\theta }}_{i,j}\right)={\widehat{\theta }}_{i,j}^T{\phi }_{i,j}\left(\widehat{x}\right),{\widehat{\overline{f}}}_{i,n_i}\left(\widehat{x}|{\widehat{\theta }}_{i,n_i}\right)={\widehat{\theta }}_{i,n_i}^T\hspace{0.17em}{\phi }_{i,n_i}\left(\widehat{x}\right),$

(16)

and the optimal parameter θ _i and minimum approximation error ɛ _i are designed as

$\begin{array}{l}{\theta }_{i,j}=\mathit{arg}\underset{{\widehat{\theta }}_{i,j}\in {\Omega }_{i,j}}{\min }\left(\underset{x\in U_i}{\sup }\left|f_{i,j}\left(x\right)-{\widehat{f}}_{i,j}\left(\widehat{x}|{\widehat{\theta }}_{i,j}\right)\right|\right),j=1,\dots ,n_i-1\\ {\theta }_{i,n_i}=\mathit{arg}\underset{{\widehat{\theta }}_{i,n_i}\in {\Omega }_{i,n_i}}{\min }\left(\underset{x\in U_i}{\sup }\left|{\overline{f}}_{i,n_i}\left(x\right)-{\widehat{\overline{f}}}_{i,n_i}\left(\widehat{x}|{\widehat{\theta }}_{i,n_i}\right)\right|\right),\\ {\epsilon }_{i,j}=f_{i,j}\left(x\right)-{\widehat{f}}_{i,j}\left(\widehat{x}|{\theta }_{i,j}\right),j=1,\dots ,n_i-1\\ {\epsilon }_{i,n_i}={\overline{f}}_{i,n_i}\left(x\right)-{\widehat{\overline{f}}}_{i,n_i}\left(\widehat{x}|{\theta }_{i,n_i}\right),\end{array}$

(17)

where $\left|{\epsilon }_{i,j}\right|\le {{\epsilon }_{i,j}}^{*}$ , j = 1, …, n _i , ${{\epsilon }_{i,j}}^{*}>0$ are positive constants. U _i and Ω _i are compact sets.

Therefore, equation (5) can be rewritten as follows

$\{\begin{array}{l}d{x}_{i,j}=\left(A_i{x}_i+{\displaystyle \sum }_{j=1}^{n_i-1}{B}_{i,j}{f}_{i,j}\left(x\right)+B_{i,n_i}\left({\overline{f}}_{i,n_i}\left(x\right)+\left({\alpha }_i\left(t-T_0\right)a_i+1\right)u_i\right)\right)dt+g_{i}d{\omega }_i,\\ y_i=C_i{x}_i,\end{array}$

(18)

where A _i are matrix of order n*n with

$A_i=\left[\begin{array}{llll}0& & & \\ ⋮& & I_{n_i-1}& \\ 0& 0& \dots & 0\end{array}\right],$

(19)

and B_i,j are matrix of order n*1 with $B_{i,j}={\left[\underset{j}{\underbrace{0,\dots ,1}},\dots ,0\right]}^T$ . C _i = [1,…,0] ^T , and $g_i={\left[g_{i,1}\left(x_i\right),\dots ,g_{i,n_i}\left(x_i\right)\right]}^T$ .

For the system (18), the high-gain fuzzy state observer is designed as follows

$\{\begin{array}{l}{\dot{\widehat{x}}}_{i,j}={\widehat{x}}_{i,j+1}+{\widehat{f}}_{i,j}\left(\widehat{x}|{\widehat{\theta }}_{i,j}\right)+o_i^j{k}_{i,j}\left(x_{i,1}-{\widehat{x}}_{i,1}\right),j=1,\dots n_i-1\\ {\dot{\widehat{x}}}_{i,n_i}=\left({\alpha }_i\left(t-T_0\right)b_i+1\right)u_i+{\widehat{\overline{f}}}_{i,n_i}\left(\widehat{x}|{\widehat{\theta }}_{i,n_i}\right)+o_i^{n_i}{k}_{i,n_i}\left(x_{i,1}-{\widehat{x}}_{i,1}\right),\end{array}$

(20)

where o _i and k_i,j are design parameters satisfying o _i > 1 and k_i,j > 0, respectively. ${\widehat{x}}_{i,j}$ and ${\widehat{\theta }}_{i,j}$ represent the estimation of x_i,j and θ_i,j, respectively. Defining $e_i=x_i-{\widehat{x}}_i$ , i = 1, …, n _i , and e _i is the observer error vector. Combining (18) and (20), one has

$d{e}_i=\left(A_{o_i}{e}_i+{\displaystyle \sum }_{j=1}^{n_i}{B}_{i,j}{\stackrel{\sim }{\theta }}_{i,j}{\phi }_{i,j}\left(\widehat{x}\right)+{\epsilon }_i\right)dt+g_{i}d{\omega }_i.$

(21)

where

$A_{o_i}=\left[\begin{array}{cccc}-o_i^1{k}_{i,1}& & & \\ ⋮& & I_{n_i-1}& \\ -o_i^{n_i}{k}_{i,n_i}& 0& \dots & 0\end{array}\right],$

(22)

and ${\epsilon }_i={\left[{\epsilon }_{i,1},\dots {\epsilon }_{i,n_i}\right]}^T$ ${\stackrel{\sim }{\theta }}_{i,j}={\theta }_{i,j}-{\widehat{\theta }}_{i,j}$ , j = 1, …, n _i are parameter errors.

The coordinate transformations are defined as follows

${\xi }_i=O_i{e}_i,O_i=\text{diag}\left\{1,o^{-1}\dots ,o^{1-n_i}\right\}.$

(23)

Then, (21) can be written as

$d{\xi }_i=\left(o{A}_{i_1}{\xi }_i+O_i{\displaystyle \sum }_{j=1}^{n_i}{B}_{i,j}{\stackrel{\sim }{\theta }}_{i,j}{\phi }_{i,j}\left(\widehat{x}\right)+O_i{\epsilon }_i\right)dt+O_i{g}_{i}d{\omega }_i,$

(24)

where

$A_{i_1}=\left[\begin{array}{cccc}-k_{i,1}& & & \\ ⋮& & I_{n_i-1}& \\ -k_{i,n_i-1}& & & \\ -k_{n_i}& 0& \dots & 0\end{array}\right].$

(25)

Obviously, $A_{i_1}$ is a strict Hurwitz matrix. Thus, for any matrix $Q_i=Q_i^T>0$ , there is the matrix $P_i=P_i^T>0$ satisfying

$A_{i_1}^T{P}_i+P_i{A}_{i_1}=-Q_i.$

(26)

The Lyapunov function candidate is chosen as

$V_{\xi }={\displaystyle \sum }_{i=1}^N{V}_{{\xi }_i}={\displaystyle \sum }_{i=1}^N{\xi }_i^T{P}_i{\xi }_i.$

(27)

Combining (26) and (27), one has

$L{V}_{\xi }={\displaystyle \sum }_{i=1}^N\left(o{\xi }_i^T\left(A_{i_1}^T{P}_i+P_i{A}_{i_1}\right){\xi }_i+2{\xi }_i^T{P}_i{O}_i\left({\displaystyle \sum }_{j=1}^{n_i}{B}_{i,j}{\stackrel{\sim }{\theta }}_{i,j}{\phi }_{i,j}\left(\widehat{x}\right)+{\epsilon }_i\right)+Tr\left\{{\sigma }_i{O_i}^T{g_i}^T{P}_i{g}_i{O}_i{\sigma }_i^T\right\}\right).$

(28)

By employing Lemma 4 and $0<{\phi }_{i,j}^T(\cdot ){\phi }_{i,j}(\cdot )\le 1$ , one can be given that

$\begin{array}{l}2{\xi }_i^T{P}_i{O}_i{\displaystyle \sum }_{j=1}^{n_i}{B}_{i,j}{\stackrel{\sim }{\theta }}_{i,j}{\phi }_{i,j}\left(\widehat{x}\right)\le {\Vert P_i\Vert }^2{\displaystyle \sum }_{j=1}^{n_i}{{\stackrel{\sim }{\theta }}_{i,j}}^T{\stackrel{\sim }{\theta }}_{i,j}+n_i{\Vert {\xi }_i\Vert }^2,\\ 2{\xi }_i^T{P}_i{O}_i{\epsilon }_i\le {\Vert {\xi }_i\Vert }^2+{\Vert P_i\Vert }^2{{\epsilon }_i^{*}}^2,\\ \text{Tr}\left\{{\sigma }_i{O_i}^T{g_i}^T{P}_i{g}_i{O}_i{\sigma }_i^T\right\}\le \frac{1}{2}{\Vert P_i\Vert }^2+\frac{1}{2}{\left|{\overline{\sigma }}_i{{\overline{\sigma }}_i}^T\right|}^2,\end{array}$

(29)

where ${\epsilon }_i^{*}={\left[{\epsilon }_{i,1}^{*},\dots {\epsilon }_{i,n_i}^{*}\right]}^T$ . Substituting (29) into (28), one gets

$L{V}_{\xi }\le {\displaystyle \sum }_{i=1}^N\left(-{\lambda }_{i,0}{\Vert {\xi }_i\Vert }^2+{\displaystyle \sum }_{j=1}^{n_i}{\Vert P_i\Vert }^2{\stackrel{\sim }{\theta }}_{i,j}^T\hspace{0.17em}{\stackrel{\sim }{\theta }}_{i,j}+C_0\right),$

(30)

where λ_i,0 = λ _min (Q _i )o − n _i − 1, $C_0={\Vert P_i\Vert }^2\left(1/2+{{\epsilon }_i^{*}}^2\right)+1/2{\left|{\overline{\sigma }}_i{{\overline{\sigma }}_i}^T\right|}^2$ .

Adaptive fuzzy dynamic surface finite-time output-feedback fault-tolerant control design

This section will design the adaptive fuzzy dynamic surface finite-time output feedback FTC scheme by using the backstepping technique and the high-gain fuzzy state observer described before.

The tracking error variable is defined as

$\begin{array}{l}{z}_{i,1}=x_{i,1}-x_{i,d},\\ z_{i,j}={\widehat{x}}_{i,j}-x_{i,j,d},j=2,\dots ,n_i,\end{array}$

(31)

where x_i,d = y_i,d, and y_i,d are given output signals of the system. x_i,j,d are extra state variables, which are attained from the first-order filter on intermediate control function α_i,j. The following first-order filter can be defined as

${\iota }_{i,j}{\dot{x}}_{i,j,d}={\alpha }_{i,j-1}-x_{i,j,d},{\alpha }_{i,j-1}\left(0\right)=x_{i,j,d}\left(0\right),$

(32)

where ι_i,j, j = 2, …, n _i are design parameters.

Meantime, the following compensation function q_i,j, j = 1, …, n _i is depicted to disposal of the error caused by filter α_i,j−1 − x_i,j,d, which has an impact on the stability of the system

$\{\begin{array}{l}{\dot{q}}_{i,j}=-c_{i,j}{q}_{i,j}+\left(x_{i,j+1,d}-{\alpha }_{i,j}\right)+q_{i,j+1}-l_{i,j}sign\left(q_{i,j}\right),j=1,\dots n_i-1\\ {\dot{q}}_{i,n_i}=-c_{i,n_i}{q}_{i,n_i}-l_{i,n_i}sign\left(q_{i,n_i}\right),\end{array}$

(33)

where l_i,j, j = 1, …, n _i are design parameters, and the compensated tracking error signals are defined as follows

$v_{i,j}=z_{i,j}-q_{i,j},j=1,\dots ,n_i.$

(34)

The detailed process of designing the controllers is shown in the following

Step1: According to v_i,1 = z_i,1 − q_i,1 and z_i,1 = x_i,1 − x_i,d, one gets

$\begin{array}{l}d{v}_{i,1}=d{z}_{i,1}-d{q}_{i,1}=d{x}_{i,1}-d{x}_{i,d}-d{q}_{i,1}\\ =\left(x_{i,2}+f_{i,1}\left(x\right)-{\dot{x}}_{i,d}-{\dot{q}}_{i,1}\right)dt+g_{i,1}\left(x_i\right)d{\omega }_i\\ =\left(v_{i,2}+q_{i,2}+o{\xi }_{i,2}+x_{i,2,d}+{\theta }_{i,1}^T{\phi }_{i,1}\left(\widehat{x}\right)+{\epsilon }_{i,1}-{\dot{x}}_{i,d}-{\dot{q}}_{i,1}\right)dt+g_{i,1}\left(x_i\right)d{\omega }_i.\end{array}$

(35)

The Lyapunov function candidate is selected as

$V_1=V_{\xi }+{\displaystyle \sum }_{i=1}^N\left(\frac{1}{4}{v}_{i,1}^4+\frac{1}{2r_{i,1}}{{\stackrel{\sim }{\theta }}_{i,1}}^T{\stackrel{\sim }{\theta }}_{i,1}+\frac{1}{2{\overline{r}}_{i,1}}{{\stackrel{\sim }{\Phi }}_{i,1}}^2\right),$

(36)

where ${\stackrel{\sim }{\Phi }}_{i,1}={\Phi }_{i,1}-{\widehat{\Phi }}_{i,1}$ is the parameter error with ${\Phi }_{i,1}={\Vert {\theta }_{i,1}\Vert }^2$ . Then, the time derivative of V₁ is obtained as

$\begin{array}{l}L{V}_1\le L{V}_{\xi }+{\displaystyle \sum }_{i=1}^N({v_{i,1}}^3(v_{i,2}+q_{i,2}+o{\xi }_{i,2}+x_{i,2,d}+{\theta }_{i,1}^T{\phi }_{i,1}\left(\widehat{x}\right)+{\epsilon }_{i,1}-{\dot{x}}_{i,d}-{\dot{q}}_{i,1}-{{\theta }_{i,1}}^T{\phi }_{i,1}\left({\widehat{x}}_{i,1}\right)\\ +{{\stackrel{\sim }{\theta }}_{i,1}}^T{\phi }_{i,1}\left({\widehat{x}}_{i,1}\right))++{{\widehat{\theta }}_{i,1}}^T{\phi }_{i,1}\left({\widehat{x}}_{i,1}\right)\frac{3}{2}{v}_{i,1}^2{g}_{i,1}^T{\sigma }_{i,1}{{\sigma }_{i,1}}^T{g}_{i,1}-\frac{1}{r_{i,1}}{{\stackrel{\sim }{\theta }}_{i,1}}^T{\stackrel{.}{\widehat{\theta }}}_{i,1}-\frac{1}{{\overline{r}}_{i,1}}{\stackrel{\sim }{\Phi }}_{i,1}{\stackrel{.}{\widehat{\Phi }}}_{i,1}).\end{array}$

(37)

Substituting ${\dot{q}}_{i,1}$ into (37), one gets

$\begin{array}{l}L{V}_1\le L{V}_{\xi }+{\displaystyle \sum }_{i=1}^N({v_{i,1}}^3(v_{i,2}+c_{i,1}{q}_{i,1}+o{\xi }_{i,2}+{\alpha }_{i,1}+{\theta }_{i,1}^T{\phi }_{i,1}\left(\widehat{x}\right)+{\epsilon }_{i,1}-{\dot{x}}_{i,d}-{{\theta }_{i,1}}^T{\phi }_{i,1}\left({\widehat{x}}_{i,1}\right)\\ +{{\widehat{\theta }}_{i,1}}^T{\phi }_{i,1}\left({\widehat{x}}_{i,1}\right)+{{\stackrel{\sim }{\theta }}_{i,1}}^T{\phi }_{i,1}\left({\widehat{x}}_{i,1}\right)+l_{i,j}\text{sign}\left(q_{i,1}\right))+\frac{3}{2}{v}_{i,1}^2{g}_{i,1}^T{\sigma }_{i,1}{{\sigma }_{i,1}}^T{g}_{i,1}-\frac{1}{r_{i,1}}{{\stackrel{\sim }{\theta }}_{i,1}}^T{\stackrel{.}{\widehat{\theta }}}_{i,1}-\frac{1}{{\overline{r}}_{i,1}}{\stackrel{\sim }{\Phi }}_{i,1}{\stackrel{.}{\widehat{\Phi }}}_{i,1}).\end{array}$

(38)

By utilizing Lemma 4 and $0<{\phi }_{i,1}^T(\cdot ){\phi }_{i,1}(\cdot )\le 1$ , the following inequalities can be obtained

$\begin{array}{l}{v_{i,1}}^{3}o{\xi }_{i,2}\le \frac{1}{2}{v_{i,1}}^6+\frac{1}{2}{o}^2{\Vert {\xi }_i\Vert }^2,\\ {v_{i,1}}^3{v}_{i,2}\le \frac{3}{4}{v_{i,1}}^4+\frac{1}{4}{v_{i,2}}^4,\\ {v_{i,1}}^3{\epsilon }_{i,1}\le \frac{3}{4}{v_{i,1}}^4+\frac{1}{4}{{\epsilon }_{i,1}^{*}}^4,\\ {v_{i,1}}^3{l}_{i,1}sign\left(q_{i,1}\right)\le \frac{3}{4}{v_{i,1}}^4+\frac{1}{4}{l_{i,1}}^4,\\ {v_{i,1}}^3\left({{\theta }_{i,1}}^T{\phi }_{i,1}\left(\widehat{x}\right)-{{\theta }_{i,1}}^T{\phi }_{i,1}\left({\widehat{x}}_{i,1}\right)\right)\le {v_{i,1}}^6{\Phi }_{i,1}+1,\\ \frac{3}{2}{v}_{i,1}^2{g}_{i,1}^T{\sigma }_{i,1}{{\sigma }_{i,1}}^T{g}_{i,1}\le \frac{3}{4}{v_{i,1}}^4+\frac{3}{4}{\left|{\overline{\sigma }}_{i,1}{{\overline{\sigma }}_{i,1}}^T\right|}^2.\end{array}$

(39)

Substituting (39) into (38), one has

$\begin{array}{l}L{V}_1\le L{V}_{\xi }+{\displaystyle \sum }_{i=1}^N({v_{i,1}}^3(c_{i,1}{q}_{i,1}+{\alpha }_{i,1}+\frac{11}{4}{v}_{i,1}+\frac{1}{2}{v_{i,1}}^3-{\dot{x}}_{i,d}+{{\widehat{\theta }}_{i,1}}^T{\phi }_{i,1}\left({\widehat{x}}_{i,1}\right)+{{\stackrel{\sim }{\theta }}_{i,1}}^T{\phi }_{i,1}\left({\widehat{x}}_{i,1}\right)\\ +{v_{i,1}}^3{\widehat{\Phi }}_{i,1})+\frac{1}{4}{v_{i,1}}^4+\frac{1}{4}{v_{i,2}}^4+\frac{1}{2}{o}^2{\Vert {\xi }_i\Vert }^2+\frac{1}{4}{l_{i,1}}^4+\frac{1}{4}{{\epsilon }_{i,1}^{*}}^4+1+\frac{3}{4}{\left|{\overline{\sigma }}_{i,1}{{\overline{\sigma }}_{i,1}}^T\right|}^2\\ -\frac{1}{r_{i,1}}{{\stackrel{\sim }{\theta }}_{i,1}}^T{\stackrel{.}{\widehat{\theta }}}_{i,1}+{\stackrel{\sim }{\Phi }}_{i,1}\left({v_{i,1}}^6-\frac{1}{{\overline{r}}_{i,1}}{\stackrel{.}{\widehat{\Phi }}}_{i,1}\right)).\end{array}$

(40)

The virtual controller α_i,1 and adaptive law ${\widehat{\theta }}_{i,1}$ , ${\widehat{\Phi }}_{i,1}$ are constructed as follows

${\alpha }_{i,1}=-c_{i,1}{z}_{i,1}-\frac{11}{4}{v}_{i,1}-\frac{1}{2}{v_{i,1}}^3+{\dot{x}}_{i,d}-{{\widehat{\theta }}_{i,1}}^T{\phi }_{i,1}\left({\widehat{x}}_{i,1}\right)-{v_{i,1}}^3{\widehat{\Phi }}_{i,1}-s_{i,1}{v_{i,1}}^{4\beta -3},$

(41)

${\stackrel{.}{\widehat{\theta }}}_{i,1}=r_{i,1}{v_{i,1}}^3{\phi }_{i,1}\left({\widehat{x}}_{i,1}\right)-m_{i,1}{\widehat{\theta }}_{i,1},$

(42)

${\stackrel{.}{\widehat{\Phi }}}_{i,1}={\overline{r}}_{i,1}{v_{i,1}}^6-{\overline{m}}_{i,1}{\widehat{\Phi }}_{i,1}.$

(43)

where s_i,1, r_i,1, m_i,1, ${\overline{r}}_{i,1},\beta =\frac{2N-1}{2N+1}$ , β = 2N − 1/2N + 1 are positive design parameters, and N ≤2 is an integer. Substituting (41)–(43) into (40), one has

$\begin{array}{l}L{V}_1\le L{V}_{\xi }+{\displaystyle \sum }_{i=1}^N(-\left(c_{i,1}-\frac{1}{4}\right){v_{i,1}}^4+\frac{1}{4}{v_{i,2}}^4-s_{i,1}{v_{i,1}}^{4\beta }+\frac{1}{2}{o}^2{\Vert {\xi }_i\Vert }^2+\frac{1}{4}{l_{i,1}}^4\\ +\frac{1}{4}{{\epsilon }_{i,1}^{*}}^4+1+\frac{3}{4}{\left|{\overline{\sigma }}_{i,1}{\overline{\sigma }}_{i,1}^T\right|}^2+\frac{m_{i,1}}{r_{i,1}}{\stackrel{\sim }{\theta }}_{i,1}^T{\widehat{\theta }}_{i,1}+\frac{{\overline{m}}_{i,1}}{{\overline{r}}_{i,1}}{\stackrel{\sim }{\Phi }}_{i,1}{\widehat{\Phi }}_{i,1}).\end{array}$

(44)

Substituting LV _ξ into (44), we can get

$\begin{array}{l}L{V}_1\le {\displaystyle \sum }_{i=1}^N(-{\lambda }_{i,1}{\Vert {\xi }_i\Vert }^2+{\displaystyle \sum }_{j=1}^{n_i}{\Vert P_i\Vert }^2{{\stackrel{\sim }{\theta }}_{i,j}}^T{\stackrel{\sim }{\theta }}_{i,j}-\left(c_{i,1}-\frac{1}{4}\right){v_{i,1}}^4+\frac{1}{4}{v_{i,2}}^4\\ -s_{i,1}{v_{i,1}}^{4\beta }+\frac{m_{i,1}}{r_{i,1}}{\stackrel{\sim }{\theta }}_{i,1}^T{\widehat{\theta }}_{i,1}+\frac{{\overline{m}}_{i,1}}{{\overline{r}}_{i,1}}{\stackrel{\sim }{\Phi }}_{i,1}{\widehat{\Phi }}_{i,1}+C_1),\end{array}$

(45)

where λ_i,1 = λ_i,0 − 1/2o², and $C_1=C_0+1/4{l_{i,1}}^4+1/4{{\epsilon }_{i,1}^{*}}^4+1+3/4{\left|{\overline{\sigma }}_{i,1}{{\overline{\sigma }}_{i,1}}^T\right|}^2$ .

Step j(2 ≤ j ≤ n _i − 1): From v_i,j = z_i,j − q_i,j, and $z_{i,j}={\widehat{x}}_{i,j}-x_{i,j,d}$ , one can be deduced that

${\dot{v}}_{i,j}={\dot{z}}_{i,j}-{\dot{q}}_{i,j}={\dot{\widehat{x}}}_{i,j}-{\dot{x}}_{i,j,d}-{\dot{q}}_{i,j}.$

(46)

The Lyapunov function candidate is chosen as

$V_j=V_{j-1}+{\displaystyle \sum }_{i=1}^N\left(\frac{1}{4}{v}_{i,j}^4+\frac{1}{2r_{i,j}}{{\stackrel{\sim }{\theta }}_{i,j}}^T{\stackrel{\sim }{\theta }}_{i,j}+\frac{1}{2{\overline{r}}_{i,j}}{{\stackrel{\sim }{\Phi }}_{i,j}}^2\right),$

(47)

where ${\stackrel{\sim }{\Phi }}_{i,j}={\Phi }_{i,j}-{\widehat{\Phi }}_{i,j}$ is the parameter error with ${\Phi }_{i,j}={\Vert {\theta }_{i,j}\Vert }^2$ . Then, the time derivative of V _i is obtained as

$\begin{array}{l}L{V}_j\le L{V}_{j-1}+{\displaystyle \sum }_{i=1}^N\left({v_{i,j}}^3\left({\widehat{x}}_{i,j+1}+{\widehat{\theta }}_{i,j}^T{\phi }_{i,j}\left(\widehat{x}\right)+o_i^j{k}_{i,j}{e}_{i,1}-{\dot{x}}_{i,j,d}-{\dot{q}}_{i,j}\right)-\frac{1}{r_{i,j}}{\stackrel{\sim }{\theta }}_{i,j}^T{\stackrel{.}{\widehat{\theta }}}_{i,j}-\frac{1}{{\overline{r}}_{i,j}}{\stackrel{\sim }{\Phi }}_{i,j}{\stackrel{.}{\widehat{\Phi }}}_{i,j}\right)\\ \le L{V}_{j-1}+{\displaystyle \sum }_{i=1}^N({v_{i,j}}^3(v_{i,j+1}+{\dot{x}}_{i,j+1,d}+q_{i,j+1}+o_i^j{k}_{i,j}{e}_{i,1}+{\theta }_{i,j}^T{\phi }_{i,j}\left(\widehat{x}\right)-{\stackrel{\sim }{\theta }}_{i,j}^T{\phi }_{i,j}\left(\widehat{x}\right)\\ +{\widehat{\theta }}_{i,j}^T{\phi }_{i,j}\left({\widehat{\underset{\_}{x}}}_{i,j}\right)+{\stackrel{\sim }{\theta }}_{i,j}^T{\phi }_{i,j}\left({\widehat{\underset{\_}{x}}}_{i,j}\right)-{\theta }_{i,j}^T{\phi }_{i,j}\left({\widehat{\underset{\_}{x}}}_{i,j}\right)-{\dot{x}}_{i,j,d}-{\dot{q}}_{i,j})-\frac{1}{r_{i,j}}{\stackrel{\sim }{\theta }}_{i,j}^T{\stackrel{.}{\widehat{\theta }}}_{i,j}-\frac{1}{{\overline{r}}_{i,j}}{\stackrel{\sim }{\Phi }}_{i,j}{\stackrel{.}{\widehat{\Phi }}}_{i,j}),\end{array}$

(48)

where ${\widehat{\underset{\_}{x}}}_{i,j}={\left[{\widehat{x}}_{i,1},\dots ,{\widehat{x}}_{i,j}\right]}^T$ . Substituting ${\dot{q}}_{i,j}$ into (48), one has

$\begin{array}{l}L{V}_j\le L{V}_{j-1}+{\displaystyle \sum }_{i=1}^N({v_{i,j}}^3(c_{i,j}{q}_{i,j}+v_{i,j+1}+{\alpha }_{i,j}+o_i^j{k}_{i,j}{e}_{i,1}-{\dot{x}}_{i,j,d}+{\theta }_{i,j}^T{\phi }_{i,j}\left(\widehat{x}\right)-{\stackrel{\sim }{\theta }}_{i,j}^T{\phi }_{i,j}\left(\widehat{x}\right)\\ +{\widehat{\theta }}_{i,j}^T{\phi }_{i,j}\left({\widehat{\underset{\_}{x}}}_{i,j}\right)+{\stackrel{\sim }{\theta }}_{i,j}^T{\phi }_{i,j}\left({\widehat{\underset{\_}{x}}}_{i,j}\right)-{\theta }_{i,j}^T{\phi }_{i,j}\left({\widehat{\underset{\_}{x}}}_{i,j}\right)+l_{i,j}\text{sign}\left(q_{i,j}\right))-\frac{1}{r_{i,j}}{\stackrel{\sim }{\theta }}_{i,j}^T{\stackrel{.}{\widehat{\theta }}}_{i,j}-\frac{1}{{\overline{r}}_{i,j}}{\stackrel{\sim }{\Phi }}_{i,j}{\stackrel{.}{\widehat{\Phi }}}_{i,j}).\end{array}$

(49)

By utilizing Lemma 4 and $0<{\phi }_{i,j}^T(\cdot ){\phi }_{i,j}(\cdot )\le 1$ , the following inequalities can be obtained

$\begin{array}{l}{v_{i,j}}^3{v}_{i,j+1}\le \frac{3}{4}{v_{i,j}}^4+\frac{3}{4}{v_{i,j+1}}^4,\\ {v_{i,j}}^3{l}_{i,j}sign\left(q_{i,j}\right)\le \frac{3}{4}{v_{i,j}}^4+\frac{1}{4}{l_{i,j}}^4,\\ {v_{i,j}}^3\left({\theta }_{i,j}^T{\phi }_{i,j}\left(\widehat{x}\right)-{\theta }_{i,j}^T{\phi }_{i,j}\left({\widehat{\underset{¯}{x}}}_{i,j}\right)\right)\le {v_{i,j}}^6{\Phi }_{i,j}+1,\\ -{v_{i,j}}^3{\stackrel{\sim }{\theta }}_{i,j}^T{\phi }_{i,j}\left(\widehat{x}\right)\le \frac{1}{2}{v_{i,j}}^6+\frac{1}{2}{{\stackrel{\sim }{\theta }}_{i,j}}^T{\stackrel{\sim }{\theta }}_{i,j}.\end{array}$

(50)

Substituting (50) into (49), one can yield that

$\begin{array}{l}L{V}_j\le L{V}_{j-1}+{\displaystyle \sum }_{i=1}^N({v_{i,j}}^3(c_{i,j}{q}_{i,j}+{\alpha }_{i,j}+\frac{3}{2}{v}_{i,j}+\frac{1}{2}{v_{i,j}}^3+o_i^j{k}_{i,j}{e}_{i,1}-{\dot{x}}_{i,j,d}+{\widehat{\theta }}_{i,j}^T{\phi }_{i,j}\left({\widehat{\underset{\_}{x}}}_{i,j}\right)+{\stackrel{\sim }{\theta }}_{i,j}^T{\phi }_{i,j}\left({\widehat{\underset{\_}{x}}}_{i,j}\right)\\ +{v_{i,j}}^3{\widehat{\Phi }}_{i,j})+\frac{1}{4}{v_{i,j+1}}^4+\frac{1}{4}{l_{i,j}}^4+\frac{1}{2}{{\stackrel{\sim }{\theta }}_{i,j}}^T{\stackrel{\sim }{\theta }}_{i,j}+1-\frac{1}{r_{i,j}}{\stackrel{\sim }{\theta }}_{i,j}^T{\stackrel{.}{\widehat{\theta }}}_{i,j}+{\stackrel{\sim }{\Phi }}_{i,j}\left({v_{i,j}}^6-\frac{1}{{\overline{r}}_{i,j}}{\stackrel{.}{\widehat{\Phi }}}_{i,j}\right)).\end{array}$

(51)

The virtual controller α_i,j and adaptive law ${\widehat{\theta }}_{i,j}$ , ${\widehat{\Phi }}_{i,j}$ are constructed as follows

${\alpha }_{i,j}=-c_{i,j}{z}_{i,j}-\frac{3}{2}{v}_{i,j}-\frac{1}{2}{v_{i,j}}^3+{\dot{x}}_{i,j,d}-{\widehat{\theta }}_{i,j}^T{\phi }_{i,j}\left({\widehat{\underset{\_}{x}}}_{i,j}\right)-{v_{i,j}}^3{\widehat{\Phi }}_{i,j}-o_i^j{k}_{i,j}{e}_{i,1}-s_{i,j}{v_{i,j}}^{4\beta -3},$

(52)

${\stackrel{.}{\widehat{\theta }}}_{i,j}=r_{i,j}{v_{i,j}}^3{\phi }_{i,j}\left({\widehat{\underset{\_}{x}}}_{i,j}\right)-m_{i,j}{\widehat{\theta }}_{i,j},$

(53)

${\stackrel{.}{\widehat{\Phi }}}_{i,j}={\overline{r}}_{i,j}{v_{i,j}}^6-{\overline{m}}_{i,j}{\widehat{\Phi }}_{i,j}.$

(54)

where s_i,j, r_i,j, m_i,j, ${\overline{r}}_{i,j}$ , ${\overline{m}}_{i,j}$ are positive design parameters. Substituting (52)–(54) into (51), one has

$L{V}_j\le L{V}_{j-1}+{\displaystyle \sum }_{i=1}^N\left(-c_{i,j}{v_{i,j}}^4+\frac{1}{4}{v_{i,j+1}}^4-s_{i,j}{v_{i,j}}^{4\beta }+\frac{1}{4}{l_{i,j}}^4+\frac{1}{2}{\stackrel{\sim }{\theta }}_{i,j}^T\hspace{0.17em}{\stackrel{\sim }{\theta }}_{i,j}+1+\frac{m_{i,j}}{r_{i,j}}{\stackrel{\sim }{\theta }}_{i,j}^T{\widehat{\theta }}_{i,j}+\frac{{\overline{m}}_{i,j}}{{\overline{r}}_{i,j}}{\stackrel{\sim }{\Phi }}_{i,j}{\widehat{\Phi }}_{i,j}\right).$

(55)

Substituting LV_j−1 into (55), one gets

$\begin{array}{l}L{V}_j\le {\displaystyle \sum }_{i=1}^N(-{\lambda }_{i,1}{\Vert {\xi }_i\Vert }^2+{\displaystyle \sum }_{j=1}^{n_i}{\Vert P_i\Vert }^2{\stackrel{\sim }{\theta }}_{i,j}^T{\stackrel{\sim }{\theta }}_{i,j}-{\displaystyle \sum }_{k=1}^j\left(c_{i,k}-\frac{1}{4}\right){v_{i,k}}^4+\frac{1}{4}{v_{i,k+1}}^4-{\displaystyle \sum }_{k=1}^j{s}_{i,k}{v_{i,k}}^{4\beta }\\ +\frac{1}{2}{\displaystyle \sum }_{k=2}^j{\stackrel{\sim }{\theta }}_{i,k}^T{\stackrel{\sim }{\theta }}_{i,k}+{\displaystyle \sum }_{k=1}^j\frac{m_{i,k}}{r_{i,k}}{\stackrel{\sim }{\theta }}_{i,k}^T{\widehat{\theta }}_{i,k}+{\displaystyle \sum }_{k=1}^j\frac{{\overline{m}}_{i,k}}{{\overline{r}}_{i,k}}{\stackrel{\sim }{\Phi }}_{i,k}{\widehat{\Phi }}_{i,k}+C_i),\end{array}$

(56)

where $C_i=C_{i-1}+1/4{l_{i,j}}^4+1$ .

Stepn _i : Considering $v_{i,n_i}=z_{i,n_i}-q_{i,n_i}$ , and $z_{i,n_i}={\widehat{x}}_{i,n_i}-x_{i,n_i,d}$ , one can be deduced that

${\dot{v}}_{i,n_i}={\dot{z}}_{i,n_i}-{\dot{q}}_{i,n_i}={\dot{\widehat{x}}}_{i,n_i}-{\dot{x}}_{i,n_i,d}-{\dot{q}}_{i,n_i.}$

(57)

The Lyapunov function candidate is selected as

$V=V_{n_i-1}+{\displaystyle \sum }_{i=1}^N\left(\frac{1}{4}{v}_{i,n_i}^4+\frac{1}{2r_{i,n_i}}{\stackrel{\sim }{\theta }}_{i,n_i}^T\hspace{0.17em}{\stackrel{\sim }{\theta }}_{i,n_i}\right).$

(58)

Then, the time derivative of V is obtained as

$\begin{array}{l}LV=L{V}_{n_i-1}+{\displaystyle \sum }_{i=1}^N({v_{i,n_i}}^3\left(\left({\alpha }_i\left(t-T_0\right)a_i+1\right)u_i+{\widehat{\theta }}_{i,n_i}^T{\phi }_{i,n_i}\left(\widehat{x}\right)+o_i^{n_i}{k}_{i,n_i}{e}_{i,1}-{\dot{x}}_{i,n_i,d}-{\dot{q}}_{i,n_i}\right)\\ +\frac{1}{r_{i,n_i}}{\stackrel{\sim }{\theta }}_{i,n_i}^T\left(r_{i,n_i}{v_{i,n_i}}^3{\phi }_{i,n_i}\left(\widehat{x}\right)-{\stackrel{.}{\widehat{\theta }}}_{i,n_i}\right)-{v_{i,n_i}}^3{\stackrel{\sim }{\theta }}_{i,n_i}^T{\phi }_{i,n_i}\left(\widehat{x}\right)).\end{array}$

(59)

Letting γ _i = α _i (t − T₀)a _i + 1, and substituting ${\dot{q}}_{i,n_i}$ into (59), one has

$\begin{array}{l}LV=L{V}_{n_i-1}+{\displaystyle \sum }_{i=1}^N({v_{i,n_i}}^3(c_{i,n_i}{q}_{i,n_i}+{\gamma }_i{u}_i+{\widehat{\theta }}_{i,n_i}^T{\phi }_{i,n_i}\left(\widehat{x}\right)+o_i^{n_i}{k}_{i,n_i}{e}_{i,1}-{\dot{x}}_{i,n_i,d}\\ +l_{i,n_i}sign\left(q_{i,n_i}\right))+\frac{1}{r_{i,n_i}}{\stackrel{\sim }{\theta }}_{i,n_i}^T\left(r_{i,n_i}{v_{i,n_i}}^3{\phi }_{i,n_i}\left(\widehat{x}\right)-{\stackrel{.}{\widehat{\theta }}}_{i,n_i}\right)-{v_{i,n_i}}^3{\stackrel{\sim }{\theta }}_{i,n_i}^T{\phi }_{i,n_i}\left(\widehat{x}\right)).\end{array}$

(60)

By utilizing Lemma 4 and $0<{\phi }_{i,n_i}^T(\cdot ){\phi }_{i,n_i}(\cdot )\le 1$ , the following inequalities can be obtained

$\begin{array}{l}{v_{i,n_i}}^3{l}_{i,n_i}sign\left(q_{i,n_i}\right)\le \frac{3}{4}{v_{i,n_i}}^4+\frac{1}{4}{l_{i,n_i}}^4,\\ {v_{i,n_i}}^3{\stackrel{\sim }{\theta }}_{i,n_i}^T{\phi }_{i,n_i}\left(\widehat{x}\right)\le \frac{1}{2}{v_{i,n_i}}^6+\frac{1}{2}{\stackrel{\sim }{\theta }}_{i,n_i}^T\hspace{0.17em}{\stackrel{\sim }{\theta }}_{i,n_i}.\end{array}$

(61)

Substituting (61) into (60), one can yield that

$\begin{array}{l}LV\le L{V}_{n_i-1}+{\displaystyle \sum }_{i=1}^N({v_{i,n_i}}^3\left(c_{i,n_i}{q}_{i,n_i}+{\gamma }_i{u}_i+\frac{3}{4}{v}_{i,n_i}+\frac{1}{2}{v_{i,n_i}}^3+o_i^{n_i}{k}_{i,n_i}{e}_{i,1}-{\dot{x}}_{i,n_i,d}+{\widehat{\theta }}_{i,n_i}^T{\phi }_{i,n_i}\left(\widehat{x}\right)\right)\\ +\frac{1}{4}{l_{i,n_i}}^4+\frac{1}{2}{{\stackrel{\sim }{\theta }}_{i,n_i}}^T{\stackrel{\sim }{\theta }}_{i,n_i}+\frac{1}{r_{i,n_i}}{\stackrel{\sim }{\theta }}_{i,n_i}^T\left(r_{i,n_i}{v_{i,n_i}}^3{\phi }_{i,n_i}\left(\widehat{x}\right)-{\stackrel{.}{\widehat{\theta }}}_{i,n_i}\right)).\end{array}$

(62)

The controller $u_i={\alpha }_{i,n_i}$ and adaptive law ${\widehat{\theta }}_{i,n_i}$ are constructed as follows

${\alpha }_{i,n_i}=N\left({\zeta }_i\right){\overline{\alpha }}_{i,n_i},\stackrel{.}{{\zeta }_i}=v_{i,n_i}^3{\overline{\alpha }}_{i,n_i},$

(63)

${\overline{\alpha }}_{i,n_i}=c_{i,n_i}{z}_{i,n_i}+\frac{3}{4}{v}_{i,n_i}+\frac{1}{2}{v_{i,n_i}}^3-{\dot{x}}_{i,n_i,d}+{\widehat{\theta }}_{i,n_i}^T{\phi }_{i,n_i}\left(\widehat{x}\right)+o_i^{n_i}{k}_{i,n_i}{e}_{i,1}+s_{i,n_i}{v_{i,n_i}}^{4\beta -3},$

(64)

${\stackrel{.}{\widehat{\theta }}}_{i,n_i}=r_{i,n_i}{v_{i,n_i}}^3{\phi }_{i,n_i}\left(\widehat{x}\right)-m_{i,n_i}{\widehat{\theta }}_{i,n_i}.$

(65)

Substituting (63)–(65) into (62), one has

$LV\le L{V}_{n_i-1}+{\displaystyle \sum }_{i=1}^N\left(\left({\gamma }_{i}N\left({\zeta }_i\right)+1\right)\stackrel{.}{{\zeta }_i}-c_{i,n_i}{v_{i,n_i}}^4-s_{i,n_i}{v_{i,n_i}}^{4\beta }+\frac{1}{4}{l_{i,n_i}}^4+\frac{1}{2}{{\stackrel{\sim }{\theta }}_n}^T{\stackrel{\sim }{\theta }}_{i,n_i}+\frac{m_{i,n_i}}{r_{i,n_i}}{\stackrel{\sim }{\theta }}_{i,n_i}^T{\widehat{\theta }}_{i,n_i}\right).$

(66)

Substituting $L{V}_{n_i-1}$ into (66), one can be given that

$\begin{array}{l}LV\le {\displaystyle \sum }_{i=1}^N(-{\lambda }_{i,1}{\Vert {\xi }_i\Vert }^2+{\displaystyle \sum }_{j=1}^{n_i}{\Vert P_i\Vert }^2{{\stackrel{\sim }{\theta }}_{i,j}}^T{\stackrel{\sim }{\theta }}_{i,j}+\left({\gamma }_{i}N\left({\zeta }_i\right)+1\right)\stackrel{.}{{\zeta }_i}-{\displaystyle \sum }_{j=1}^{n_i}\left(c_{i,j}-\frac{1}{4}\right){v_{i,j}}^4\\ -{\displaystyle \sum }_{j=1}^{n_i}{s}_{i,j}{v_{i,j}}^{4\beta }+\frac{1}{2}{\displaystyle \sum }_{j=2}^{n_i}{{\stackrel{\sim }{\theta }}_{i,j}}^T{\stackrel{\sim }{\theta }}_{i,j}+{\displaystyle \sum }_{j=1}^{n_i}\frac{m_{i,j}}{r_{i,j}}{\stackrel{\sim }{\theta }}_{i,j}^T{\widehat{\theta }}_{i,j}+{\displaystyle \sum }_{j=1}^{n_i-1}\frac{{\overline{m}}_{i,j}}{{\overline{r}}_{i,j}}{\stackrel{\sim }{\Phi }}_{i,j}{\widehat{\Phi }}_{i,j}+C_n),\end{array}$

(67)

where $C_n=C_{n-1}+1/4{l_{i,n_i}}^4$ .

Remark 3. Compared to the existing control results, the controller design in this paper is different in two points. On the one hand, more general MIMO stochastic NSFNSs and non-affine nonlinear faults are considered, which makes the controller design more difficult and complex. On the other hand, this paper designs the controller based on finite time control and dynamic surface control technique, which not only makes the controller have better tracking performance but also reduces the computational complexity during the controller design. The detailed block diagram of the proposed control scheme in this paper is shown in Figure 1 below.

Figure 1.

The block diagram of adaptive fuzzy dynamic surface finite-time output feedback FTC algorithm.

Stability analysis

Theorem 1. Consider the MIMO stochastic NSFNSs (1) with non-affine nonlinear faults and unmeasured states under assumption 1. The fuzzy state observer with high gain is established by (20), the adaptive fuzzy dynamic surface finite-time output feedback FTC algorithm consists of virtual controllers (41), (52), and the actual controller (64), and the parameter adaptive laws (42), (43), (53), (54), and (65) can ensure that overall signals with the studied system (1) are SGFSP, and the tracking errors of the controller developed in this study can coverage to a small neighborhood around the origin within finite-time.

Proof: Combining the definition of θ, the following inequality holds

$\begin{array}{l}\frac{m_{i,j}}{r_{i,j}}{\stackrel{\sim }{\theta }}_{i,j}^T{\widehat{\theta }}_{i,j}\le -\frac{m_{i,j}}{2r_{i,j}}{\stackrel{\sim }{\theta }}_{i,j}^T{\stackrel{\sim }{\theta }}_{i,j}+\frac{m_{i,j}}{2r_{i,j}}{\theta }_{i,j}^T{\theta }_{i,j},\\ \frac{{\overline{m}}_{i,j}}{{\overline{r}}_{i,j}}{\stackrel{\sim }{\Phi }}_{i,j}{\widehat{\Phi }}_{i,j}\le -\frac{{\overline{m}}_{i,j}}{2{\overline{r}}_{i,j}}{\stackrel{\sim }{\Phi }}_{i,j}^2+\frac{{\overline{m}}_{i,j}}{2{\overline{r}}_{i,j}}{\Phi }_{i,j}^2.\end{array}$

(68)

According to Lemma 3, by choosing $x={\lambda }_{\mathit{min}}\left(Q_i\right)/2{\Vert {\xi }_i\Vert }^2,1/2r_{i,j}{\stackrel{\sim }{\theta }}_{i,j}^T{\stackrel{\sim }{\theta }}_{i,j},1/2{\overline{r}}_{i,j}{\stackrel{\sim }{\Phi }}_{i,j}^2$ , y = 1, m = β, n = 1 − β, l = 1/β, and Δ = (1 − β) (1/β)^β/β−1, the following inequality holds

$\begin{array}{l}{\left(\frac{1}{2r_{i,j}}{\stackrel{\sim }{\theta }}_{i,j}^T{\stackrel{\sim }{\theta }}_{i,j}\right)}^{\beta }\le \frac{1}{2r_{i,j}}{\stackrel{\sim }{\theta }}_{i,j}^T{\stackrel{\sim }{\theta }}_{i,j}+\mathrm{\Delta },\\ {\left(\frac{1}{2{\overline{r}}_{i,j}}{\stackrel{\sim }{\Phi }}_{i,j}^2\right)}^{\beta }\le \frac{1}{2{\overline{r}}_{i,j}}{\stackrel{\sim }{\Phi }}_{i,j}^2+\mathrm{\Delta },\\ {\left(\frac{{\lambda }_{\mathit{min}}\left(Q_i\right)}{2}{\Vert {\xi }_i\Vert }^2\right)}^{\beta }\le \frac{{\lambda }_{\mathit{min}}\left(Q_i\right)}{2}{\Vert {\xi }_i\Vert }^2+\mathrm{\Delta }.\end{array}$

(69)

Substituting (68) and (69) into (67), one has

$\begin{array}{l}LV\le {\displaystyle \sum }_{i=1}^N(-{\lambda }_{i,2}{\Vert {\xi }_i\Vert }^2-{\displaystyle \sum }_{i=1}^{n_i}\left(c_{i,j}-\frac{1}{4}\right){v_{i,j}}^4-{\displaystyle \sum }_{j=1}^{n_i}{s}_{i,j}{v_{i,j}}^{4\beta }-\frac{m_{i,1}-1}{2r_{i,1}}{\stackrel{\sim }{\theta }}_{i,1}^T{\stackrel{\sim }{\theta }}_{i,1}-{\displaystyle \sum }_{j=2}^{n_i}\left(\frac{m_{i,j}-1}{2r_{i,j}}-\frac{1}{2}\right){{\stackrel{\sim }{\theta }}_{i,j}}^T{\stackrel{\sim }{\theta }}_{i,j}\\ -{\displaystyle \sum }_{j=1}^{n_i-1}\frac{{\overline{m}}_{i,j}-1}{2{\overline{r}}_{i,j}}{\stackrel{\sim }{\Phi }}_{i,j}^2-{\left(\frac{{\lambda }_{\mathit{min}}\left(Q_i\right)}{2}{\Vert {\xi }_i\Vert }^2\right)}^{\beta }-{\left({\displaystyle \sum }_{j=1}^{n_i}\frac{1}{2r_{i,j}}{\stackrel{\sim }{\theta }}_{i,j}^T{\stackrel{\sim }{\theta }}_{i,j}\right)}^{\beta }-{\left({\displaystyle \sum }_{j=1}^{n_i-1}\frac{1}{2{\overline{r}}_{i,j}}{\stackrel{\sim }{\Phi }}_{i,j}^2\right)}^{\beta })+\left(\gamma N\left(\zeta \right)+1\right)\dot{\zeta }+C_{n+1},\end{array}$

(70)

where λ_i,2 = 1/2λ _min (Q _i ) − n _i − 3/2, $C_{n+1}={\displaystyle \sum }_{i=1}^N\left(C_n+{\displaystyle \sum }_{j=1}^{n_i}{m}_{i,j}/2r_{i,j}{\theta }_{i,j}^T{\theta }_{i,j}+{\displaystyle \sum }_{j=1}^{n_i-1}{\overline{m}}_{i,j}/2{\overline{r}}_{i,j}{\Phi }_{i,j}^2+2n_i\mathrm{\Delta }\right)$ , and $\left(\gamma N\left(\zeta \right)+1\right)\dot{\zeta }={\displaystyle \sum }_{i=1}^N\left({\gamma }_{i}N\left({\zeta }_i\right)+1){\zeta }_i\right)+1){\gamma }_{i}N\stackrel{.}{{\zeta }_i}$ .

According to the above analysis, and defining $a=\mathit{min}\{\left(1/2{\lambda }_{\mathit{min}}\left(Q_i\right)-n_i-3/2\right)/{\lambda }_{\mathit{max}}\left(P_i\right),4c_{i,j}-1,m_{i,1}-1,m_{i,k}-1-r_{i,k},\hspace{0.17em}{\overline{m}}_{i,l}-1,j=1,\dots ,n_i,k=2,\dots ,n_i,l=1,\dots ,n_i-1\}$ , $b=\mathit{min}\left\{s_{i,j}{4}^{\beta },1/2{\lambda }_{\mathit{max}}\left(P_i\right),1\right\}$ , the following inequality can be obtained

$LV\le -aV-b{V}^{\beta }+\left(\gamma N\left(\zeta \right)+1\right)\dot{\zeta }+C_{n+1}.$

(71)

Then, one gets

$LV\le -aV+\left(\gamma N\left(\zeta \right)+1\right)\dot{\zeta }+C_{n+1}.$

(72)

Multiplying both sides of (72) by e ^at at the same time and integrating it over [0,t] yields

$\begin{array}{l}V\le \frac{C_{n+1}}{a}+\left(V\left(0\right)-\frac{C_{n+1}}{a}\right)e^{-at}+e^{-at}{\int }_0^t\left(\gamma N\left(\zeta \right)+1\right)\dot{\zeta }{e}^{a\tau }d\tau \\ \le \frac{C_{n+1}}{a}+V\left(0\right)+e^{-at}{\int }_0^t\left(\gamma N\left(\zeta \right)+1\right)\dot{\zeta }{e}^{a\tau }d\tau .\end{array}$

(73)

From (73), and using Lemma 5, it follows that $\left(\gamma N\left(\zeta \right)+1\right)\dot{\zeta }$ is bounded on [0, t _f ), and suppose that $\left|\left(\gamma N\left(\zeta \right)+1\right)\dot{\zeta }\right|\le d$ (d is a positive constant). Let C = C_n+1 + d, one can be given that

$LV\le -aV+C.$

(74)

Therefore, equation (71) can be written as

$LV\le -aV-b{V}^{\beta }+C.$

(75)

From Lemma 1 and (75), it is easy to know that the system (1) is SGFSP. By choosing 0 < τ < 1, V will converge to the set $\psi =\left\{x|E\left[V\right]\le {\left\{C/\left(1-\tau \right)b\right\}}^{1/\beta }\right\}$ in finite-time T and

$E\left[T\right]\le \frac{1}{\left(1-\beta \right)a}\ln \left[\frac{aV{\left(x_0\right)}^{1-\beta }+b\tau }{a{\left(\frac{C}{\left(1-\tau \right)/{\mu }_2}\right)}^{\frac{1-\beta }{\beta }}+b\tau }\right].$

(76)

By defining ϵ = {C/(1 − τ)b}^1/β, the following inequality can be obtained

$\left[E\left(v_i^2\right)\right]\le {\left[E\left(v_i^4\right)\right]}^{\frac{1}{2}}\le {\left(4EV\right)}^{\frac{1}{2}}\le 2\sqrt{ϵ},\forall t\ge T,$

(77)

$\left[E\left({\stackrel{\sim }{{\theta }_i}}^2\right)\right]\le 2r\left(EV\right)\le 2rϵ,$

(78)

where r is the upper bound of r _i . That is, the closed-loop signals v _i and $\stackrel{\sim }{\theta }$ are bounded in sense of mean square value.

The following proves the above additional compensating function q_i,j can coverage in finite-time T₁. The Lyapunov function is chosen as $V_Q={\displaystyle \sum }_{i=1}^N{\displaystyle \sum }_{j=1}^{n_i}1/2q_{i,j}^2$ , where the first-order derivative is as follows

$L{V}_Q={\displaystyle \sum }_{i=1}^N\left({\displaystyle \sum }_{j=1}^{n_i-1}\left(q_{i,j}-c_{i,j}{q}_{i,j}+\left(x_{i,j+1,d}-{\alpha }_{i,j}\right)+q_{i,j+1}-l_{i,j}sign\left(q_{i,j}\right)\right)+q_{i,n_i}\left(-c_{i,n_i}{q}_{i,n_i}-l_{i,n_i}sign\left(q_{i,n_i}\right)\right)\right).$

(79)

Moreover, it can deduce that $\left(x_{i,j+1,d}-{\alpha }_{i,j}\right)\le {\eta }_{i,j}$ can be achieved in finite-time T₂ with a known constant η_i,j. According to Lemma 4, it can be obtained

$q_{i,j}{q}_{i,j+1}\le \frac{1}{2}{q}_{i,j}^2+\frac{1}{2}{q}_{i,j+1}^2.$

(80)

Substituting (80) into (79), one has

$L{V}_Q\le {\displaystyle \sum }_{i=1}^N\left(-\left(c_{i,1}-\frac{1}{2}\right)q_{i,1}^2-\left(c_{i,2}-1\right)q_{i,2}^2,\dots ,-\left(c_{i,n_i-1}-1\right)q_{i,n_i-1}^2-\left(c_{i,n_i}-\frac{1}{2}\right)q_{i,n_i}^2-{\displaystyle \sum }_{j=1}^{n_i}{q}_{i,j}\left(l_{i,j}-{\eta }_{i,j}\right)\right).$

(81)

By choosing the appropriate parameter to make l_i,j ≥ η_i,j hold, it can be obtained that

$L{V}_Q\le -Q_1{V}_Q-Q_2{V}_Q^{\frac{1}{2}},$

(82)

where $Q_1=\min \left\{\left(c_{i,1}-1/2\right),\left(c_{i,2}-1\right),\dots ,\left(c_{i,n_i-1}-1\right),\left(c_{i,n_i}-1/2\right)\right\}>0$ , $Q_2=\min \left\{\left|l_{i,j}-{\eta }_{i,j}\right|j=1,\dots ,n_i\right\}>0$ . Hence, the V _Q can converge to a small residual set of the origin in finite-time T₁.

From (77), (78), and (82), the $E\left({\left|y-y_d\right|}^2\right)=E\left({\left|z_1\right|}^2\right)\le 2\sqrt{ϵ}$ can be attained. Thus, the tracking error z₁ can converge to a small residual set of the origin in finite-time $\overline{T}=\max \left\{T,T_1+T_2\right\}$ .

The proof is completed.

Remark 4. In literatures, ^54,55 the event-triggered adaptive fuzzy finite-time control was investigated for nonholonomic multirobot systems and underactuated unmanned surface vehicles, respectively. In the literature, ⁵⁶ the prescribed performance adaptive fuzzy optimization control was studied for nonlinear vehicle platoon. Note that it is a meaningful and interesting topic to further consider the event-triggered adaptive fuzzy finite-time output feedback FLC problem and the prescribed performance adaptive fuzzy finite-time output feedback FLC problem for stochastic nonlinear systems subject to unmeasured states and non-affine nonlinear faults.

Simulation

In this section, two simulation examples are presented to verify the effectiveness of the proposed adaptive fuzzy dynamic surface finite-time output feedback FTC strategy.

Example 1: Consider the MIMO stochastic NSFNSs with non-affine nonlinear faults as follows

$\{\begin{array}{l}d{x}_{i,1}=\left(x_{i,2}+f_{i,1}\left(x\right)\right)dt+g_{i,1}\left(x_i\right)d{\omega }_i,\\ d{x}_{i,2}=\left(u_i+f_{i,2}\left(x\right)+{\alpha }_i\left(t-T_0\right){\delta }_i\left(x_i,u_i\right)\right)dt+g_{i,2}\left(x_i\right)d{\omega }_i,\\ y_i=x_{i,1},i=1,2,\end{array}$

(83)

where $f_{1,1}\left(x\right)=1-\hspace{0.17em}\mathit{cos}\left(x_{1,1}{x}_{1,2}{x}_{2,1}{x}_{2,2}\right)+x_{1,1},f_{1,2}\left(x\right)=x_{1,1}{x}_{2,1}{x}_{2,2}{e}^{x_{1,2}}+2x_{1,1}\hspace{0.17em}\mathit{cos}\left(x_{1,1}{x}_{1,2}\right),f_{2,1}\left(x\right)=2-2\hspace{0.17em}\mathit{cos}\left(x_{1,1}{x}_{1,2}{x}_{2,1}{x}_{2,2}\right)+3x_{1,1}{x}_{2,1}{x}_{2,2},f_{2,2}\left(x\right)=x_{2,1}{x}_{2,2}{e}^{x_{2,2}}+x_{1,1}{x}_{1,2}{x}_{2,2},g_{1,1}\left(x_1\right)=0.1x_{1,1}{x}_{1,2},g_{1,2}\left(x_1\right)=0.5\hspace{0.17em}\mathit{cos}\left(x_{1,1}{x}_{1,2}\right),g_{2,1}\left(x_2\right)=0.2x_{2,1}\hspace{0.17em}\sin \left(x_{2,1}{x}_{2,2}\right),$ and $g_{2,2}\left(x_2\right)=0.2\hspace{0.17em}\mathit{sin}\left(x_{2,1}{x}_{2,2}\right)$ . δ _i (x _i , u _i ) = 10(x_i,1x_i,2 + sin(u _i )) + 5,

${\alpha }_i\left(t-T_0\right)=\{\begin{array}{l}0,t<T_0,\\ 1-e^{-{\mathrm{\varrho }}_i\left(t-T_0\right)},t\ge T_0,\end{array}$

(84)

where ϱ₁ = ϱ₂ = 8, T₀ = 10, and the given reference signals are y_1,d = y_2,d = sin(t).

The fuzzy state observer is designed as

$\{\begin{array}{l}{\dot{\widehat{x}}}_{i,1}={\widehat{x}}_{i,2}+{\widehat{f}}_{i,1}\left(\widehat{x}|{\widehat{\theta }}_{i,1}\right)+o_i{k}_{i,1}\left(x_{i,1}-{\widehat{x}}_{i,1}\right),i=1,2\\ {\dot{\widehat{x}}}_{i,2}=(\left({\alpha }_i\left(t-T_0\right)b_i+1\right)u_i+{\widehat{\overline{f}}}_{i,2}\left(\widehat{x}|{\widehat{\theta }}_{i,2}\right)+o_i^2{k}_{i,2}\left(x_{i,1}-{\widehat{x}}_{i,1}\right).\end{array}$

(85)

In the simulation experiment, the compensating signal can be designed as follows

$\{\begin{array}{l}{\dot{q}}_{i,1}=-c_{i,1}{q}_{i,1}+\left(x_{i,2,d}-{\alpha }_{i,1}\right)+q_{i,2}-l_{i,1}sign\left(q_{i,1}\right),i=1,2,\\ {\dot{q}}_{i,2}=-c_{i,2}{q}_{i,2}-l_{i,2}sign\left(q_{i,2}\right).\end{array}$

(86)

In addition, the virtual controllers α _i and adaptive law θ _i , Φ₁ are designed as follows

${\alpha }_{i,1}=-c_{i,1}{z}_{i,1}-\frac{11}{4}{v}_{i,1}-\frac{1}{2}{v_{i,1}}^3+{\dot{x}}_{1,d}-{{\widehat{\theta }}_{i,1}}^T{\phi }_{i,1}\left({\widehat{x}}_{i,1}\right)-{v_{i,1}}^3{\widehat{\Phi }}_{i,1}-s_{i,1}{v_{i,1}}^{4\beta -3},{\alpha }_{i,2}=N\left({\zeta }_i\right){\overline{\alpha }}_{i,2},\stackrel{.}{{\zeta }_i}=v_{i,2}^3{\overline{\alpha }}_{i,2},{\overline{\alpha }}_{i,2}=c_{i,2}{z}_{i,2}+\frac{3}{4}{v}_{i,2}+\frac{1}{2}{v_{i,2}}^3-{\dot{x}}_{i,2,d}+{\widehat{\theta }}_{i,2}^T{\phi }_{i,2}\left(\widehat{x}\right)+o_i^2{k}_{i,2}{e}_{i,1}+s_{i,2}{v_{i,2}}^{4\beta -3},{\stackrel{.}{\widehat{\theta }}}_{i,1}=r_{i,1}{v_{i,1}}^3{\phi }_{i,1}\left({\widehat{x}}_{i,1}\right)-m_{i,1}{\widehat{\theta }}_{i,1},{\stackrel{.}{\widehat{\theta }}}_{i,2}=r_{i,2}{v_{i,2}}^3{\phi }_{i,2}\left(\widehat{x}\right)-m_{i,2}{\widehat{\theta }}_{i,2},{\stackrel{.}{\widehat{\Phi }}}_{i,1}={\overline{r}}_{i,1}{v_{i,1}}^6-{\overline{m}}_{i,1}{\widehat{\Phi }}_{i,1},i=1,2.$

(87)

In this simulation, the $e^{{\zeta }^2}\hspace{0.17em}\mathit{cos}\left(\left(\pi /2\right)\zeta \right)$ function is employed as Nussbaum function. The initial state is selected as x_1,2(0) = x_2,1(0) = 0.1 and other initial values are 0. Then, the design parameters are selected as b₁ = b₂ = 1, o₁ = o₂ = 5, k_1,1 = k_2,2 = 4, k_2,1 = 3, k_1,2 = 8, l_1,1 = l_1,2 = l_2,1 = l_2,2 = 0.01, c_1,1 = c_2,1 = 21, c_1,2 = 20, c_2,2 = 25, s_1,1 = s_1,2 = s_2,1 = s_2,2 = 0.5, β = 999/1001, and ι_1,2 = 21, ι_2,2 = 14, r_1,1 = r_1,2 = r_2,1 = 0.1, r_2,2 = 1, m_1,1 = m_1,2 = m_2,1 = m_2,2 = 0.2, ${\overline{r}}_{1,1}={\overline{r}}_{2,1}=0.1,{\overline{m}}_{1,1}={\overline{m}}_{2,1}=0.2$ . In addition, the fuzzy membership functions are selected as ${\mu }_{F_{i,j}^l}=\mathit{exp}\left(-{\left({\widehat{x}}_{i,j}-l\right)}^2/2\right),i=1,2,j=1,2,l=-3,-2,-1,0,1,2,3$ .

The simulation results are depicted in Figures 2–15. Figure 2 shows the system output x_1,1 and the given signal y_1,d. Figure 3 shows the system output x_2,1 and the given signal y_2,d. Figure 4 shows the tracking errors z_1,1 and z_2,1. Figure 5 shows the controllers u₁ and u₂. Figure 6 shows the system state x_1,1 and the system state estimate ${\widehat{x}}_{1,1}$ . Figure 7 shows the system state x_1,2 and the system state estimate ${\widehat{x}}_{1,2}$ . Figure 8 shows the system state x_2,1 and the system state estimate ${\widehat{x}}_{2,1}$ . Figure 9 shows the system state x_2,2 and the system state estimate ${\widehat{x}}_{2,2}$ . Figure 10 shows the ζ₁ and N(ζ₁). Figure 11 shows the ζ₂ and N(ζ₂). Figure 12 shows the adaptive parameters ${\widehat{\theta }}_{1,1}$ and ${\widehat{\theta }}_{1,2}$ . Figure 13 shows the adaptive parameters ${\widehat{\theta }}_{2,1}$ and ${\widehat{\theta }}_{2,2}$ . Figure 14 shows the system output x_1,1 and given signal y_1,d without the FTC technology in this paper. Figure 15 shows the system output x_2,1 and given signal y_2,d without the FTC technology in this paper.

Figure 2.

The system output x_1,1 and the given signal y_1,d (Example 1).

Figure 3.

The system output x_2,1 and the given signal y_2,d (Example 1).

Figure 4.

The tracking error z_1,1 and z_2,1 (Example 1).

Figure 5.

The controllers u₁ and u₂ (Example 1).

Figure 6.

The system state x_1,1 and the system state estimate ${\widehat{x}}_{1,1}$ (Example 1).

Figure 7.

The system state x_1,2 and the system state estimate ${\widehat{x}}_{1,2}$ (Example 1).

Figure 8.

The system state x_2,1 and the system state estimate ${\widehat{x}}_{2,1}$ (Example 1).

Figure 9.

The system state x_2,2 and the system state estimate ${\widehat{x}}_{2,2}$ (Example 1).

Figure 10.

The ζ₁ and N(ζ₁) (Example 1).

Figure 11.

The ζ₂ and N(ζ₂) (Example 1).

Figure 12.

The adaptive parameters ${\widehat{\theta }}_{1,1}$ and ${\widehat{\theta }}_{1,2}$ (Example 1).

Figure 13.

The adaptive parameters ${\widehat{\theta }}_{2,1}$ and ${\widehat{\theta }}_{2,2}$ (Example 1).

Figure 14.

The system output x_1,1 and given signal y_1,d without the FTC technology (Example 1).

Figure 15.

The system output x_2,1 and given signal y_2,d without the FTC technology (Example 1).

The results in Figures 2–13 show that the controller designed in this paper can ensure that the overall signals with the system (83) is SGFSP and the tracking errors of the study controller can converge in a small neighborhood around the origin within finite-time. It is obvious from Figures 14 and 15 that the control system without the FTC technique proposed is unstable.

Remark 5. As can be seen from Figures 2 and 3, the control scheme designed in this paper achieves excellent tracking performance. From Figures 4 and 5, it can be seen that at the 10th second, the system occurs faults that lead to larger tracking errors, and then the system signals quickly track the given signals based on the designed controller. Noteworthily, after 10 seconds, the tracking error of the system inevitably increases compared to before 10 seconds due to the occurrence of faults.

Remark 6. From Figures 6–9, it can be seen that the trajectories of the system states and the estimated value of the system states are roughly identical, which demonstrates the effectiveness and reasonableness of the proposed high-gain fuzzy state observer. The control performance and system stabilization can be ensured by adjusting the adaptive law. As can be seen from Figures 12 and 13, the adaptive law becomes gradually smaller with time due to the fact that the system is progressively stabilized. In addition, Figures 14 and 15 further verify the effectiveness of the adaptive fuzzy dynamic surface finite-time output feedback FTC proposed in this paper.

Example 2: Consider the system of two inverted pendulums connected by springs in Ref. [52] which is depicted in Figure 16, when the system is subjected to non-affine nonlinear faults, the actual system in Ref.[52] can be rewritten as follows

$\begin{array}{l}\{\begin{array}{l}d{x}_{1,1}=x_{1,2}dt,\\ d{x}_{i,2}=\left[\left(\frac{m_{1}gr}{J_1}-\frac{k{r}^2}{4J_1}\right)\mathit{sin}\left(x_{1,1}\right)+\frac{kr}{2J_1}\left(l-b\right)+\frac{a_1{u}_1}{J_1}+\frac{k{r}^2}{4J_1}\mathit{sin}\left(x_{2,1}\right)+{\alpha }_1\left(t-T_0\right){\delta }_1\left(x_1,u_1\right)\right]dt+g_{1,2}\left(x_1\right)d{\omega }_1,\\ y_1=x_{1,1},\end{array}\\ \{\begin{array}{l}d{x}_{2,1}=x_{2,2}dt,\\ d{x}_{i,2}=\left[\left(\frac{m_{2}gr}{J_2}-\frac{k{r}^2}{4J_2}\right)\mathit{sin}\left(x_{2,1}\right)+\frac{kr}{2J_2}\left(l-b\right)+\frac{a_2{u}_2}{J_2}+\frac{k{r}^2}{4J_2}\mathit{sin}\left(x_{1,2}\right)+{\alpha }_2\left(t-T_0\right){\delta }_2\left(x_2,u_2\right)\right]dt+g_{2,2}\left(x_2\right)d{\omega }_2,\\ y_2=x_{2,1},\end{array}\end{array}$

(88)

where m₁ = m₂ = 2 kg, J₁ = J₂ = 10 kg, g = 9.81 m/s², k = 10 N/m, r = 0.5 m, l = 0.5 m, b = 0.4 m, g_1,2(x₁) = sin(x_1,1x_1,2), and g_2,2(x₂) = sin(x_2,1x_2,2). δ _i (x _i , u _i ) = 10(x_i,1x_i,2 + sin(u _i ))

${\alpha }_i\left(t-T_0\right)=\{\begin{array}{l}0,t<T_0,\\ 1-e^{-{\mathrm{\varrho }}_i\left(t-T_0\right)},t\ge T_0,\end{array}$

(89)

where ϱ₁ = ϱ₂ = 8, T₀ = 10, and the given reference signals are y_1,d = sin(t), y_2,d = cos(t).

Figure 16.

Two inverted pendulums connected by springs.

The fuzzy state observer is designed as

$\{\begin{array}{l}{\dot{\widehat{x}}}_{i,1}={\widehat{x}}_{i,2}+o_i{k}_{i,1}\left(x_{i,1}-{\widehat{x}}_{i,1}\right),i=1,2\\ {\dot{\widehat{x}}}_{i,2}=\left(0.1+{\alpha }_i\left(t-T_0\right)b_i\right)u_i+{\widehat{\overline{f}}}_{i,2}\left(\widehat{x}|{\widehat{\theta }}_{i,2}\right)+o_i^2{k}_{i,2}\left(x_{i,1}-{\widehat{x}}_{i,1}\right).\end{array}$

(90)

It is noting that x_i,1 are linear differential equations. Then, the virtual control α_i,1 is described as ${\alpha }_{i,1}=-c_{i,1}{z}_{i,1}-11/4v_{i,1}-1/2{v_{i,1}}^3+{\dot{x}}_{1,d}-s_{i,1}{v_{i,1}}^{4\beta -3}$ and other controls similar to (87). The compensating signal is similar to (86). In this simulation, the $e^{{\zeta }^2}\hspace{0.17em}\mathit{cos}\left(\left(\pi /2\right)\zeta \right)$ function is employed as Nussbaum function. The initial state is selected as x_1,1(0) = x_1,2(0) = x_2,1(0) = x_2,2(0) = 0.1 and other initial values are 0. Then, the design parameters are selected as b₁ = b₂ = 1, o₁ = o₂ = 2, k_1,1 = k_2,1 = 2, k_1,2 = k_2,2 = 30, l_1,1 = l_1,2 = l_2,1 = l_2,2 = 0.01, c_1,1 = c_1,2 = 20, c_2,1 = 12, c_2,2 = 18, s_1,1 = s_1,2 = s_2,1 = s_2,2 = 0.5, β = 999/1001, and ι_1,2 = 12, ι_2,2 = 21, r_1,1 = r_1,2 = r_2,1 = r_2,2 = 0.1, m_1,1 = m_2,1 = 0.3, m_1,2 = 0.4, m_2,2 = 0.2, ${\overline{r}}_{1,1}={\overline{r}}_{2,1}=0.1,{\overline{m}}_{1,1}={\overline{m}}_{2,1}=0.2$ . In addition, the fuzzy membership functions are selected as ${\mu }_{F_{i,j}^l}=\mathit{exp}\left(-{\left({\widehat{x}}_{i,j}-l\right)}^2/2\right),i=1,2,j=1,2,l=-3,-2,-1,0,1,2,3$ .

The simulation results are depicted in Figures 17–30. Figure 17 shows the system output x_1,1 and the given signal y_1,d. Figure 18 shows the system output x_2,1 and the given signal y_2,d. Figure 19 shows the tracking errors z_1,1 and z_2,1. Figure 20 shows the controllers u₁ and u₂. Figure 21 shows the system state x_1,1 and the system state estimate ${\widehat{x}}_{1,1}$ . Figure 22 shows the system state x_1,2 and the system state estimate ${\widehat{x}}_{1,2}$ . Figure 23 shows the system state x_2,1 and the system state estimate ${\widehat{x}}_{2,1}$ . Figure 24 shows the system state x_2,2 and the system state estimate ${\widehat{x}}_{2,2}$ . Figure 25 shows the ζ₁ and N(ζ₁). Figure 26 shows the ζ₂ and N(ζ₂). Figure 27 shows the adaptive parameters ${\widehat{\theta }}_{1,1}$ and ${\widehat{\theta }}_{1,2}$ . Figure 28 shows the adaptive parameters ${\widehat{\theta }}_{2,1}$ and ${\widehat{\theta }}_{2,2}$ . Figures 29 and 30 show the system outputs x_1,1, x_2,1 and given signals y_1,d, y_2,d without the FTC technology in this paper.

Figure 17.

The system output x_1,1 and the given signal y_1,d (Example 2).

Figure 18.

The system output x_2,1 and the given signal y_2,d (Example 2).

Figure 19.

The tracking error z_1,1 and z_2,1 (Example 2).

Figure 20.

The controllers u₁ and u₂ (Example 2).

Figure 21.

The system state x_1,1 and the system state estimate ${\widehat{x}}_{1,1}$ (Example 2).

Figure 22.

The system state x_1,2 and the system state estimate ${\widehat{x}}_{1,2}$ (Example 2).

Figure 23.

The system state x_2,1 and the system state estimate ${\widehat{x}}_{2,1}$ (Example 2).

Figure 24.

The system state x_2,2 and the system state estimate ${\widehat{x}}_{2,2}$ (Example 2).

Figure 25.

The ζ₁ and N(ζ₁) (Example 2).

Figure 26.

The ζ₂ and N(ζ₂) (Example 2).

Figure 27.

The adaptive parameters ${\widehat{\theta }}_{1,1}$ and ${\widehat{\theta }}_{1,2}$ (Example 2).

Figure 28.

The adaptive parameters ${\widehat{\theta }}_{2,1}$ and ${\widehat{\theta }}_{2,2}$ (Example 2).

Figure 29.

The system output x_1,1 and given signal y_1,d without the FTC technology (Example 2).

Figure 30.

The system output x_2,1 and given signal y_2,d without the FTC technology (Example 2).

The results in Figures 17–28 show that the controller designed in this paper can ensure that the overall signals with the system (88) is SGFSP and the tracking errors can converge in a small neighborhood around the origin within finite-time. It is obvious from Figures 29 and 30 that the control system without the FTC algorithm proposed is unstable.

Remark 7. As can be seen from Figures 17–20, the proposed control algorithm has a fast response performance. Moreover, from Figure 19, it can be seen that at the 10th second, the system occurs faults, and therefore, the tracking errors become larger, and then the system signals quickly track to the given signals based on the designed controller.

Remark 8. As can be seen from Figures 21–24, the trajectories of system states and the estimated values of the system states are roughly identical, which validates the effectiveness and reasonableness of the constructed high-gain fuzzy state observer. From Figure 27, noteworthily, the adaptive law is the highest at the beginning, which means that the subsystem 1 is more unstable, after that, the adaptive law decreases. From Figure 28, it can be seen that the subsystem 2 is more unstable after the fault occurs. Therefore, the adaptive law of subsystem 2 starts to get bigger from 10 s with the peak at 18 s, after that, the adaptive law starts to decay. In addition, Figures 29 and 30 further verify the effectiveness of the adaptive fuzzy dynamic surface finite-time output feedback FTC proposed in this paper.

Conclusion

This paper considers the controller design challenge for MIMO stochastic NSFNSs with non-affine nonlinear faults and unmeasurable states. Employing input compensation techniques and mean-value theorems to deal with the non-affine nonlinear fault problem, which results in unknown nonlinear functions and unknown control coefficients being handled using FLSs and Nussbaum functions, respectively. In addition, fuzzy state observers with high gain have been established to address the unmeasurable states. By combining the adaptive backstepping technique, DSC approaches, error compensation mechanisms, and finite-time control theory, a novel adaptive fuzzy dynamic surface finite-time output feedback FTC algorithm is designed, which ensures that overall signals in the stochastic system are bounded, and the tracking errors of the controller developed in this paper can converge in a small neighborhood around the origin within finite-time. Finally, two simulation results have been presented to verify the effectiveness of the proposed control strategy. Extending the results of this research to the event-triggered adaptive fuzzy finite-time output feedback FLC and the prescribed performance adaptive fuzzy finite-time output feedback, FLC is one future area of interest.