1 Introduction

Many actual systems can be described as high-order nonlinear systems, such as connected vehicles dynamics (e.g., a tractor with several trailers, etc.), synchronous motors, or aircraft wing rock dynamics [1,2,3,4,5]. Compared with strict-feedback and pure-feedback systems [6,7,8,9,10,11], high-order nonlinear systems are more general in the sense that positive-odd-integer powers appear in the dynamics. It is well documented in the literature that high-order nonlinear systems are intrinsically more challenging than strict-feedback and pure-feedback systems, as feedback linearization and backstepping methods fail to work [12]. To handle such problem, the adding-one-power-integrator method was successfully proposed in [13] by introducing iteratively one high power integrator instead of a linear one. Although a number of control problems (i.e., global robust stabilization [14], practical tracking control [15], and asymptotic tracking control [16]) have been solved, the system nonlinearities considered in the above-mentioned results are required to satisfy the growth constraints, i.e., \(|{\psi _{m}}(\cdot )|\le (|\chi _{1}|^{r_{1}}+\ldots +|\chi _{{m}}|^{r_{{m}}})\rho _{{m}}({\chi _{1}}, \ldots ,{\chi _{{m}}})\) with \(\rho _{{m}}(\cdot ) (1 \le m \le n-1)\) known nonnegative smooth functions. To remove this limitation, an intelligent tracking control method was firstly developed in [17] without involving any restrictive growth condition. The same strategy, together with the common Lyapunov function method, was extended to deal with switched high-order nonlinear systems in [18]. Most recently, a new error compensation method was further derived in [19] to improve the tracking accuracy. Despite those efforts above, these results rarely focus on the rate of convergence. To be specific, only the exponential convergence of tracking error is guaranteed in the aforementioned works, which reveals the convergence time tends to be infinite.

From a practical perspective, the rate of convergence is of great significance to the transient tracking performance [20, 21]. Recently, the finite-time tracking control for lower-/high-order nonlinear systems is investigated in [22,23,24], which makes the tracking error converge into the compact set within a finite time. Nevertheless, the convergence time achieved in [22,23,24] depends on the initial states of the system. This inevitably brings up a problem, that is, the convergence time cannot be accurately settled when the initial states of the system are unknown. To solve such problem, the fixed-time control [25,26,27,28] is proposed skillfully, by which the tracking error can converge into the compact set within a fixed time and its upper bound of convergence time is independent of system initial conditions. The authors in [29] established an adaptive fuzzy feedback fixed-time control mechanism for uncertain high-order nonlinear systems. However, the Lyapunov function in [25,26,27,28,29] will be augmented by additional estimation error variables when uncertainties exist in lower-/high-order model, leading to the result that the state tracking errors only can converge to a bounded compact set whose size cannot be set explicitly in advance (specified tracking accuracy) within fixed time. On the other hand, the above results neglect crucial aspects, such as state constraints, whose importance is explained hereafter.

Because state constraints generally exist in actual systems, ignoring these state constraints may deteriorate system performance and even endangers system safety [30, 31]. In order to prevent violations of state constraints, barrier Lyapunov function (BLF) is exploited to guarantee the non-violation of state constraints, while ensuring closed-loop stability [32]. Generally, the commonly seen forms include log-type BLF [33], tan-type BLF [34] and integral-type BLF [35]. Compared with common log-type BLF and integral-type BLF, the tan-type BLF can integrate constraint analysis into a general method, which can handle constrained/unconstrained situations [36,37,38]. Thus, the tan-type BLF has become a popular research topic. However, most of the existing results of tan-type BLF are concentrated on low-order nonlinear systems subject to full-state constraints and cannot be directly applied to high-order cases. On the other hand, the existing tan-type BLF is symmetric time-invariant, which cannot be used to handle asymmetric time-varying cases [34, 37, 38]. To our best knowledge, such issue has not been considered in the literature studies. In addition, the actuator faults are ubiquitous in engineering applications, which may cause the system instability and performance degradation [39,40,41,42,43]. Motivated by the above observations, the main innovative points of this work are listed as follows:

(1) Compared with existing fixed-time control designs [25,26,27,28,29], our proposed method can not only ensure fixed-time convergence, but also can make the tracking errors converge to a compact set whose size can be specified/predefined a priori.

(2) As  opposed  to  the  existing  symmetric  time-invariant tan-type BLFs [34, 37, 38], a new type of high-order tan-type BLF is proposed to guarantee that state variables are confined within some asymmetric time-varying sets all the time, and the initial conditions are inside of corresponding sets.

(3) A new BLF-based tuning function and a projection operator are delicately embedded into the control design to make the size of the convergence regions of state tracking errors adjustable in the framework of fixed-time stability. A variable-separable lemma is utilized to extract the virtual control signals and actuator faults in a “linear-like” manner.

2 Preliminaries and problem formulation

2.1 Problem statement

Consider the uncertain high-order nonlinear systems with the following form:

$$\begin{aligned} \left\{ \begin{aligned} {{{\dot{\xi }}}_i}&= {\varPsi _i}({{{\bar{\xi }}}_i}) + {\varGamma _i}({{{\bar{\xi }}}_i})\xi _{i + 1}^{{p_i}},\\ {{{\dot{\xi }}}_n}&= {\varPsi _n}(\xi ) + {\varGamma _n}(\xi ){u^{{p_n}}},\\ y&= {\xi _1},~1 \le i \le n - 1, \end{aligned} \right. \end{aligned}$$
(1)

where \(y\in {{\mathbb {R}}}\) is the system output; \(u \in {\mathbb {R}}\) is the control input (to be designed); \({{\bar{\xi }}_i} = {[{\xi _1},{\xi _2}, \ldots ,{\xi _i}]^T} \in {{\mathbb {R}}^i}\) and \(\xi = {[{\xi _1},{\xi _2}, \ldots ,{\xi _n}]^T} \in {{\mathbb {R}}^n}\) are the states. For \(i = 1, \ldots ,n\), \({\varPsi _i}( \cdot )\) and \({\varGamma _i}\left( \cdot \right) \) are unknown continuous functions, and \({p_i}\) are positive odd integers. In this paper, we consider the system (1) to be the time-varying full-state constraints, i.e., \(\xi _i(t)\) is required to remain in the set \({\underline{\varpi }}_{ci} (t)< \xi _i(t)< {\bar{\varpi }}_{ci} (t)\), where \({\underline{\varpi }}_{ci} (t)\) and \({\bar{\varpi }}_{ci} (t)\) such that \({\underline{\varpi }}_{ci} (t)< {\bar{\varpi }}_{ci} (t)\).

Remark 1

When \({p_{i}=1}\), dynamics (1) reduces to the models of works in [6,7,8,9, 21, 22, 26, 27, 32, 44, 45] which means (1) includes most of the existing nonlinear systems as special cases. Besides, the nonlinear functions \({\varPsi _i}(\cdot )\), here, are completely unknown, and they do not need to satisfy the growth constraints in [14,15,16], i.e., \(|{\varPsi _i}(\cdot )|\le (|\xi _{1}|^{p_{1}}+\ldots +|\xi _{i}|^{p_{i}})\rho _{i}({\xi _{1}}, \ldots ,{\xi _{i}})\) with \(\rho _{i}(\cdot ) (1 \le i \le n_i-1)\) known nonnegative smooth functions.

The actuator fault is taken into account, which contains no fault, partial fault, bias fault and stuck fault. In line with [38] and [43], suppose the actuator output when fault occurs is expressed as

$$\begin{aligned} \begin{aligned} u = {\mathcal {H}}{u_c} + {\eta _u}, \end{aligned}\end{aligned}$$
(2)

where \(0\le {\mathcal {H}}\le 1\) denotes the efficiency factor of the actuator and \({\eta _u}\) represents the unknown time-varying bias fault of the actuator. The actuator fault can represent the following fault patterns:

(1) \({\mathcal {H}}=1\) and \({\eta _u}=0\). In this case, the actuator is working normally, and free of failures and faults.

(2) \(0<{\mathcal {H}}<1\) and \({\eta _u}=0\). This indicates that the actuator is undergoing the failures/faults with partial loss of effectiveness (PLOE).

(3) \({\mathcal {H}}=0\). The case is said to be the failure with total loss of effectiveness (TLOE). This means that no matter the control \({u_c}\) takes any values, u is stuck at an unknown value.

Assumption 1

It is assumed that the efficiency factor \({\mathcal {H}}\) satisfies \(0< \mathcal {{\underline{H}}} \le {\mathcal {H}} < 1\) for the PLOE faults, where \( \mathcal {{\underline{H}}}\) is a known constant. In addition, the bias fault is assumed to be bounded by \({\eta _u} \le {\bar{\eta }}\) with \(\bar{\eta }\) being the known positive constant.

Remark 2

Please note that the actuator fault (2) in our paper has been extensively used in the literature [10, 39,40,41,42,43]. However, the common point in aforementioned works is to require the systems to be in low-order form, making the above methods fail due to the existing high powers, i.e., \(({\mathcal {H}}{u_c} + {\eta _u})^{{p_n}}\). A new design must be sought beyond the available schemes.

Without loss of generality, the following standard assumptions from the literature are made.

Assumption 2

For a continuously differentiable desir-ed trajectory \({y_r}\left( t \right) \), there exists a known positive function \({y_0}(t)\), i.e., \(\big | {{y_r}\left( t \right) } \big | \le {y_0}(t)\), \(\big | {{{\dot{y}}_r}\left( t \right) } \big | \le {y_0}(t)\), moreover \({y_0}\left( t \right) < {\bar{\varpi }}_{ci} (t)\).

Assumption 3

The signs of \({\varGamma _{i}( \cdot )}\) are known and there exist known positive constants \({{\underline{\varGamma }}_i} > 0\) and \({\bar{\varGamma }_i} > 0\), \(\left( {1 \le i \le n} \right) \) such that \({{\underline{\varGamma }}_{i}} \le {\varGamma _{i}( \cdot )} \le {\bar{\varGamma }_{i}}\).

Remark 3

Assumption 2 is standard and it is essential for the state constrained systems to select suitable bounded reference signal \(y_r(t)\) to be tracked. Thus, Assumption 2 is reasonable, and we can always provide a robust estimate about the upper bound of \(y_r(t)\). Assumption 3 is given to ensure the controllability of dynamics (1).

The  control  objective  is  to  design  a  fixed-time singularity-free fault-tolerant controller such that:

(1) The  asymmetric  and  time-varying  state  constraints under fault-tolerant control are not transgressed;

(2) The tracking errors can converge into the specified compact sets within fixed time.

The following lemmas are useful for deriving the main results.

Lemma 1

[5] For any \(\chi _1,\chi _2\in {\mathbb {R}}\) and positive odd integer b, there exist real-valued functions \({r_1}\left( { \cdot , \cdot } \right) \) and \({r_2}\left( { \cdot , \cdot } \right) \) such that

$$\begin{aligned} {\left( {{\chi _1} + {\chi _2}} \right) ^b} = {\ell }\left( {{\chi _1},{\chi _2}} \right) \chi _1^b + {\upsilon }\left( {{\chi _1},{\chi _2}} \right) \chi _2^b, \end{aligned}$$

where \({\ell }\left( {{\chi _1},{\chi _2}} \right) \in \big [{\underline{\ell }}, {{\bar{\ell }}} \big ]\) with \({{\underline{\ell }}}= 1-d\) and \({\bar{\ell }}=1+d\), where \(d = \sum \nolimits _{k = 1}^b {\frac{{b!}}{{k!\left( {b - k} \right) !}}} \frac{{b - k}}{b}{l^{\frac{b}{{b - k}}}}\) is an arbitrary constant taking value in \(\left( {0,1} \right) \) for some appropriately small constant l, \(\big | {{\upsilon }\left( {{\chi _1},{\chi _2}} \right) } \big | \le {{\bar{\upsilon }} (d)}=\sum \nolimits _{k = 1}^b {\frac{{b!}}{{k!\left( {b - k} \right) !}}} \frac{k}{b}{l^{ - \frac{b}{k}}}\) with \({\bar{\upsilon } (d)}\) being a positive constant.

Lemma 2

[17] Let \({\zeta _1} \in {\mathbb {R}}\), \({\zeta _2} \in {\mathbb {R}}\) and \({r_1}\) and \({r_2}\) be positive constants. For any \(\varpi > 0\), it holds that

$$\begin{aligned} {\big | {{\zeta _1}} \big |^{{r_1}}}{\big | {{\zeta _2}} \big |^{{r_2}}} \le \frac{{{r_1}\varpi {{\big | {{\zeta _1}} \big |}^{{r_1} + {r_2}}}}}{{{r_1} + {r_2}}} + \frac{{{r_2}{\varpi ^{ - {{{r_1}} \big / {{r_2}}}}}{{\big | {{\zeta _2}} \big |}^{{r_1} + {r_2}}}}}{{{r_1} + {r_2}}}. \end{aligned}$$

Lemma 3

[27] Consider the nonlinear system

$$\begin{aligned} \begin{aligned} \dot{x} = f\left( {x\left( t \right) } \right) ,x\left( 0 \right) = {x_0}, \end{aligned} \end{aligned}$$
(3)

if there exists a Lyapunov function \(V\left( x \right) \) for the system (3) such that

$$\begin{aligned} \dot{V}\left( x \right) \le - \alpha {V^p}\left( x \right) - \beta {V^q}\left( x \right) , \end{aligned}$$

where \(\alpha ,\beta > 0\); \(p > 1\); \(0< q < 1\), then the origin of the system (3) is fixed-time stable within

$$\begin{aligned} T \le {T_{\max }}: = \frac{1}{{\alpha \left( {p - 1} \right) }} + \frac{1}{{\beta \left( {1 - q} \right) }}. \end{aligned}$$

2.2 Radial basis function neural network

In theory, RBFNN can approximate any nonlinear function with arbitrary precision by choosing appropriate parameters [44], which is formulated as

$$\begin{aligned} \begin{aligned} \varPsi \left( \chi \right) = {w^T}\theta + \delta \left( \chi \right) , \end{aligned} \end{aligned}$$
(4)

where \(\delta \left( \chi \right) \) is the minimum approximation error of the RBFNN, the weight vector is expressed as

$$\begin{aligned} {w^ * } = \arg \mathop {\min }\limits _{w \subseteq {{\mathbb {R}}^n}} \left\{ {\varPsi \left( \chi \right) - {w^T}\theta } \right\} , \end{aligned}$$

\(\theta = {\left[ {{\theta _1}\left( \chi \right) , \ldots ,{\theta _n}\left( \chi \right) } \right] ^T}\) represents the basis function vector with

$$\begin{aligned} {\theta _i}\left( \chi \right) = \exp \left[ { - \frac{{{{\left( {\chi - {\iota _i}} \right) }^T}\left( {\chi - {\iota _i}} \right) }}{{\aleph _i^T{\aleph _i}}}} \right] ,i = 1, \cdots ,n, \end{aligned}$$

where n is the number of the RBFNN nodes, \({\iota _i} = {\left[ {{\iota _{i1}},{\iota _{i2}}, \ldots ,{\iota _{iq}}} \right] ^T}\) and \({\aleph _i}\) are the center and width of the Gaussian function, respectively. It is worth noting that, if the \(i\hbox {th}\) node is viewed to be activated, \({\theta _i}\left( \chi \right) \) has to be in the compact set \({\varOmega _\theta } = \left\{ {{\theta _i}\left( \chi \right) |1> {\theta _i}\left( \chi \right) \ge {\varepsilon _{\min }} > 0} \right\} \) with \({\varepsilon _{\min }}\) denoting the user-defined activation threshold. If the value of x is in the active region, i.e., \({\left\| {\chi - {\iota _i}} \right\| ^2} \le - \aleph _i^T{\aleph _i}\ln \left( {{\varepsilon _{\min }}} \right) \), the node i is called to be activated. Therefore, we can conclude that the smaller the value of \({\varepsilon _{\min }}\) is, the larger the size of active region is. However, the amount of computation will increase dramatically if the value of \({\varepsilon _{\min }}\) is chosen very small, which reveals that the importance of the selection of \({\varepsilon _{\min }}\).

3 Fixed-time adaptive controller design

A novel fixed-time singularity-free fault-tolerant control scheme is proposed for high-order nonlinear systems. A RBFNN \(w_i^T{\theta _i}\left( {{Z_i}} \right) \) with \({w_i} \in {R^i}\), \({\theta _i}\left( {{Z_i}} \right) \in {R^i}\) is utilized to handle the approximation of the unknown function \({{\bar{f}}_i}({Z_i})\) that will be specified latter. We use \({{\delta _i}\left( {{Z_i}} \right) }\) to denote the approximation error which is bounded by \({{\bar{\delta }}}\), i.e., \(\big | {{\delta _i}\left( {{Z_i}} \right) } \big | \le {\bar{\delta }}\). By lumping \(w_i\) and \({\theta _i}\left( {{Z_i}} \right) \) into the following vectors:

$$\begin{aligned} {\varXi _i}&= {\left[ {w_i^T,{\delta _i}\left( {{Z_i}} \right) } \right] ^T}, \end{aligned}$$
(5)
$$\begin{aligned} {\vartheta _i}\left( {{Z_i}} \right)&= {\left[ {\theta _i^T\left( {{Z_i}} \right) ,1} \right] ^T}, \end{aligned}$$
(6)

and we further have

$$\begin{aligned} {{\bar{\varPsi }}_i}\left( {{Z_i}} \right) = w_i^T{\theta _i}\left( {{Z_i}} \right) + {\delta _i}\left( {{Z_i}} \right) = \varXi _i^T{\vartheta _i}\left( {{Z_i}} \right) . \end{aligned}$$
(7)

To quantify the control objective, we define a constant \(q > 0\) as \(q = \mathop {\max }\limits _{1 \le i \le n} \left\{ {{p_i}} \right\} \) and consider the following change of coordinates:

$$\begin{aligned} \left\{ \begin{aligned} {s_1}&= {\xi _1} - {y_r},\\ {s_i}&= {\xi _i} - {\xi _{i,c}},\mathrm{{ }}i = 2,3, \ldots ,n, \end{aligned} \right. \end{aligned}$$
(8)

where \({\xi _{i,c}}\) represents the virtual control law which will be specified later. Before performing the controller design, define the following switch functions:

$$\begin{aligned}&{\varpi _i}\left( {{s_i}\left( t \right) } \right) :=\left\{ \begin{aligned}&{\varpi _{bi}}\left( {{s_i}\left( t \right) } \right) ,~\mathrm{{ if }}~{s_i}\left( t \right) > 0,\\&{\varpi _{ai}}\left( {{s_i}\left( t \right) } \right) ,~\mathrm{{ if }}~{s_i}\left( t \right) \le 0. \end{aligned} \right. \end{aligned}$$
(9)
$$\begin{aligned}&{\psi _i}\left( {{s_i}\left( t \right) } \right) :=\left\{ \begin{aligned}&1,~\mathrm{{ if }}~\big | {{s_i}\left( t \right) } \big | > {\beta _i},\\&0,~\mathrm{{ if }}~\big | {{s_i}\left( t \right) } \big | \le {\beta _i}. \end{aligned} \right. \end{aligned}$$
(10)
$$\begin{aligned}&\mathrm{{s}}{\mathrm{{g}}_i}\!\left( \! {{s_i}\left( t \right) } \!\right) \!\!\nonumber \\&\quad :=\!\!\!\left\{ \!\!\! \begin{aligned}&\quad \frac{{s_i^{q \!-\! {p_i} \!+\! 1}\!\left( t \right) }}{{\big | \! {s_i^{q\! - \!{p_i}\! +\! 1}\left( t \right) } \!\big |}},~~~~~~~~~~~~\mathrm{{ if }}\!~\big | {{s_i}\left( t \right) } \big | \!\!>\!\! {\beta _i},\\&\frac{{s_i^{q \!- \!{p_i}\! + \!1}\left( t \right) }}{{{{\left[ {\beta _i^2 \!\!-\!\! s_i^2\left( t \right) } \right] }^{2\left( {q\! -\! {p_i}\! +\! 1} \right) }} \!\!+\! \big |\! {s_i^{q \!-\! {p_i}\! + \!1}\left( t \right) }\! \big |}},~\!\mathrm{{if }}\!~\big | {{s_i}\left( t \right) } \big | \!\!\le \!\! {\beta _i}. \end{aligned} \right. \end{aligned}$$
(11)

where \({\varpi _i}\left( {{s_i}\left( t \right) } \right) \) is the asymmetric time-varying tracking error constraints, \(i \in \left\{ {1,2, \ldots ,n} \right\} \).

From the definitions of (10) and (11), one has

$$\begin{aligned} {\mathrm{s}}{\mathrm{g}}_{i}\!\left( {s_{i}}\left( t \right) \right) \!{\psi _{i}}\!\left( {s_{i}}\left( t \right) \right) \! \!=\!\!\left\{ \!\!\! \begin{aligned}&\frac{{s_{i}^{{q - {p_{i}} + 1}}\left( t \right) }}{{\big | {s_{i}^{{q - {p_{i}} + 1}}\left( t \right) } \big |}},~\!\mathrm{{ if }}~\big | {{s_{i}}\left( t \right) } \big | \!>\! {\beta _{i}},\\&~~~~~~~0,~~~~~~~~\mathrm{{ if }}~\big | {{s_{i}}\left( t \right) } \big | \!\le \!{\beta _{i}}, \end{aligned} \right. \nonumber \\ \end{aligned}$$
(12)

and

$$\begin{aligned} \begin{aligned} {\left[ {{\psi _i}\left( {{s_i}\left( t \right) } \right) } \right] ^m} = {\psi _i}\left( {{s_i}\left( t \right) } \right) , \end{aligned}\end{aligned}$$
(13)

where m represents positive integer.

Step  1: Using (1) and (8), differentiating \(s_1\) with respect to time yields

$$\begin{aligned} \begin{aligned} {\dot{s}_1} = {\varPsi _1}({\xi _1}) + {\varGamma _1}({\xi _1})\xi _2^{{p_1}} - {\dot{y}_r}. \end{aligned} \end{aligned}$$
(14)

Consider the high-order tan-type BLF as

$$\begin{aligned} \begin{aligned} {{V}_1} = \frac{{2\varpi _1^{q - {p_1} + 2}\left( t \right) }}{{\pi \left( {q - {p_1} + 2} \right) }}\tan \left( {\frac{{\pi \zeta _1^{q - {p_1} + 2}\left( t \right) }}{{2\varpi _1^{q - {p_1} + 2}\left( t \right) }}} \right) {\psi _1}. \end{aligned}\end{aligned}$$
(15)

where \(\zeta _1=|s_1|-{\beta _1}\). From (9), we know that \({\varpi _{b1}}\left( t \right) \) and \({\varpi _{a1}}\left( t \right) \) represents the lower and upper bound of tracking error, respectively, with \({\varpi _{b1}}\left( t \right)> {\beta _1} > 0\), \({\varpi _{a1}}\left( t \right)< - {\beta _1} < 0\).

Remark 4

If \(\varpi _1 (t) \) approaches to infinity, the term \( \frac{{2\varpi _1^{q - {p_1} + 2}\left( t \right) }}{{\pi \left( {q - {p_1} + 2} \right) }}\tan \left( {\frac{{\pi \zeta _1^{q - {p_1} + 2}\left( t \right) }}{{2\varpi _1^{q - {p_1} + 2}\left( t \right) }}} \right) \) will tend to \(\frac{\zeta _1^{q - {p_1} + 2}}{{{q} - {p_{1}} + 2}}\), and thus the proposed high-order tan-type BLF can deal with both the constrained case and the unconstrained case. The conventional BLF in [30, 31, 33] can only handle the constrained case due to the fact \(\mathop {\lim }_{\varpi _1 \rightarrow \infty } \frac{1}{{{q} - {p_{1}} + 2}}\times \log \bigg ( {\frac{{\varpi _1^{{q} - {p_{1}} + 2}}}{{\varpi _1^{{q} - {p_{1}} + 2} - \zeta _1^{{q} - {p_{1}} + 2}}}} \bigg )=0\).

Remark 5

It is well known that there have been many studies on the tan-type BLF constraint control problem [34, 37, 38], but they only consider the symmetric time-invariant scenarios. However, due to the influence of faults, asymmetric time-varying constraints are more common in actual systems, so the fault-tolerant control of asymmetric time-varying constraints is more practical. That is the reason why we construct asymmetric time-varying BLF in our paper.

Remark 6

In (15), the switch function \({\psi _1}\) is applied such that the bounds of steady-state tracking errors can be preset as \(\beta _1\). By utilizing a smooth switch \(\mathrm{{s}}{\mathrm{{g}}_1}\left( {{s_1}\left( t \right) } \right) \), the chattering phenomenon is effectively eliminated, and thus the transient performance of system is improved.

In view of (14) and (15), the time derivative of \({{{V}}_1}\) can be expressed as

$$\begin{aligned} {{\dot{{V}}}_1} =&\frac{{2\varpi _1^{q - {p_1} + 1}\left( t \right) {{{\dot{\varpi }}}_1}\left( t \right) }}{\pi }\tan \left( {\frac{{\pi \zeta _1^{q - {p_1} + 2}\left( t \right) }}{{2\varpi _1^{q - {p_1} + 2}\left( t \right) }}} \right) {\psi _1}\nonumber \\&+{l_1}{\left( {{{\bar{\varPsi }}}_1}({Z_1}) + {\varGamma _1}({\xi _1})\xi _2^{{p_1}}\right) \mathrm{{s}}{\mathrm{{g}}_1}\left( {{s_1}} \right) } {\psi _1}\nonumber \\&-{l_1}\frac{{{{{\dot{\varpi }}}_1}\left( t \right) }}{{{\varpi _1}\left( t \right) }}{\zeta _1}\left( t \right) {\psi _1}, \end{aligned}$$
(16)

where \({{\bar{\varPsi }}_1}({Z_1}) = {\varPsi _1}({\xi _1}) - {\dot{y}_r}\), \({l_1} = \zeta _1^{q - {p_1} + 1}\left( t \right) \varTheta _1^{ - 1}\), \({\varTheta _1} = {\cos ^2}\left( {\frac{{\pi \zeta _1^{q - {p_1} + 2}\left( t \right) }}{{2\varpi _1^{q - {p_1} + 2}\left( t \right) }}} \right) \) and \({Z_1} = {\left[ {{\xi _1},{{\dot{y}}_r}} \right] ^T}\). Noting (7), it can be known that (16) can be rewritten as

$$\begin{aligned} {{\dot{{V}}}_1}&=\frac{{2\varpi _1^{q - {p_1} + 1}\left( t \right) {{{\dot{\varpi }}}_1}\left( t \right) }}{\pi }\tan \left( {\frac{{\pi \zeta _1^{q - {p_1} + 2}\left( t \right) }}{{2\varpi _1^{q - {p_1} + 2}\left( t \right) }}} \right) {\psi _1}\nonumber \\&~~~+{l_1}{\left( \varXi _1^T{\vartheta _1}\left( {{Z_1}} \right) + {\varGamma _1}({\xi _1})\xi _2^{{p_1}}\right) \mathrm{{s}}{\mathrm{{g}}_1}\left( {{s_1}} \right) } {\psi _1}\nonumber \\&~~~ -{l_1}\frac{{{{{\dot{\varpi }}}_1}\left( t \right) }}{{{\varpi _1}\left( t \right) }}{\zeta _1}\left( t \right) {\psi _1}\nonumber \\&\le \frac{{2\varpi _1^{q - {p_1} + 1}\left( t \right) {{\dot{\varpi }}_i}\left( t \right) }}{\pi }\tan \left( {\frac{{\pi \zeta _1^{q - {p_1} + 2}\left( t \right) }}{{2\varpi _1^{q - {p_1} + 2}\left( t \right) }}} \right) {\psi _1}\nonumber \\&~~~+{l_1}{\left( \varPhi {\rho _1} + {\varGamma _1}({\xi _1})\xi _2^{{p_1}}\right) \mathrm{{s}}{\mathrm{{g}}_1}\left( {{s_1}} \right) } {\psi _1}\nonumber \\&~~~ -{l_1}\frac{{{{{\dot{\varpi }}}_1}\left( t \right) }}{{{\varpi _1}\left( t \right) }}{\zeta _1}\left( t \right) {\psi _1}, \end{aligned}$$
(17)

where \(\varPhi = \sqrt{\max \left\{ {w_i^T{w_i} + {\bar{\delta }}^2} \right\} } ,i = 1,2, \ldots ,n\), \({\rho _1} = {\mathrm{{sg}}_1}\left( {{s_1}} \right) \sqrt{\vartheta _1^T\left( {{Z_1}} \right) {\vartheta _1}\left( {{Z_1}} \right) + {\lambda _0}}\) with \({\lambda _0} > 0\) being the parameter to be designed. We are now in the position to handle the term \(\xi _2^{{p_1}}\) in (17) through Lemma 1 as

$$\begin{aligned} \begin{aligned} \xi _2^{{p_1}} = {\left( {{s_2} + {\xi _{2,c}}} \right) ^{{p_1}}} \le {{\bar{\upsilon }} _1}\big | {s_2^{{p_1}}} \big | + {\ell _1}\xi _{2,c}^{{p_1}}. \end{aligned}\end{aligned}$$
(18)

Substitute (18) into (17), and the derivative of \({{{V}}_1}\) is

$$\begin{aligned} {{\dot{{V}}}_1}&\le \frac{{2\varpi _1^{q - {p_1} + 1}\left( t \right) {{{\dot{\varpi }}}_1}\left( t \right) }}{\pi }\tan \left( {\frac{{\pi \zeta _1^{q - {p_1} + 2}\left( t \right) }}{{2\varpi _1^{q - {p_1} + 2}\left( t \right) }}} \right) {\psi _1}\nonumber \\&~~~ -\!{l_1}\!\frac{{{{{\dot{\varpi }}}_1}\left( t \right) }}{{{\varpi _1}\left( t \right) }}\!{\zeta _1}\!\left( t \right) \!{\psi _1}\!\!+\!\! \frac{1}{{{\varTheta _1}}}\!{{\bar{\varGamma }}_1}({\xi _1}){{{\bar{\upsilon }} }_1}\big |\! {\zeta _1^{q - {p_1} + 1}s_2^{{p_1}}} \!\big |\!{\psi _1}\nonumber \\&~~~+{l_1}\left( \varPhi {\rho _1} + {\varGamma _1}({\xi _1}){\ell _1}\xi _{2,c}^{{p_1}}\right) \mathrm{{s}}{\mathrm{{g}}_1}\left( {{s_1}} \right) {\psi _1}. \end{aligned}$$
(19)

According to Lemma 1, one has

(20)

Substitute (20) into (19) yields

$$\begin{aligned} {{{\dot{V}}}_1}&\le \frac{1}{{{\varTheta _1}}}{{\bar{\varGamma }}_1}({\xi _1}){{{\bar{\upsilon }} }_1}\left( \! {\phi _1^{\frac{{q - {p_1} + 1}}{q}}\zeta _1^{q + 1} + \phi _1^{ - \frac{{{p_1}}}{{q + 1}}}s_2^{q + 1}}\! \right) {\psi _1}\nonumber \\&~~~+\! \frac{{2\varpi _1^{q - {p_1} + 1}\!\left( t \right) {{\dot{\varpi }}_1}\!\left( t \right) }}{\pi }\!\tan \! \left( \! {\frac{{\pi \zeta _1^{q - {p_1} + 2}\left( t \right) }}{{2\varpi _1^{q - {p_1} + 2}\left( t \right) }}} \!\right) \!{\psi _1}\nonumber \\&~~~+\!{l_1}\left( \varPhi {\rho _1} + {\varGamma _1}({\xi _1}){\ell _1}\xi _{2,c}^{{p_1}}\right) \mathrm{{s}}{\mathrm{{g}}_1}\!\left( {{s_1}} \right) \!{\psi _1}\nonumber \\&~~~ -{l_1}\frac{{{{{\dot{\varpi }}}_1}\left( t \right) }}{{{\varpi _1}\left( t \right) }}{\zeta _1}\left( t \right) {\psi _1}. \end{aligned}$$
(21)

Design the singularity-free switching controller \({\xi _{2,c}}\) as

$$\begin{aligned} {\xi _{2,c}} =&- \left[ {\frac{1}{{{{\underline{\varGamma }}_1}({\xi _1})\underline{\ell }_1}}\left[ \frac{{{a_1}}}{{{l_1}}}{\tan ^p}\left( {\frac{{\pi \zeta _1^{q - {p_1} + 2}\left( t \right) }}{{2\varpi _1^{q - {p_1} + 2}\left( t \right) }}} \right) {\mathrm{{sg}}_1}\left( {{s_1}} \right) \right. } \right. \nonumber \\&+ \left( {{\beta _2}\! +\! 1} \right) \!{\mathrm{{sg}}_1}\left( {{s_1}} \right) \! +\! \frac{1}{{{l_1}{\varTheta _1}}}{{{\bar{\varGamma }}}_1}({\xi _1}){{\bar{\upsilon } }_1}\phi _1^{\frac{{q - {p_1} + 1}}{q}}\!\zeta _1^{q + 1}\nonumber \\&+ \frac{{2\varpi _1^{q - {p_1} + 1}\left( t \right) {{{\dot{\varpi }}}_1}\left( t \right) }}{{{l_1}\pi }}\tan \left( {\frac{{\pi \zeta _1^{q - {p_1} + 2}\left( t \right) }}{{2\varpi _1^{q - {p_1} + 2}\left( t \right) }}} \right) \nonumber \\&+ {\hbar _1}{\zeta _1}+ \frac{{{c_1}}}{{{l_1}}}{{\tan }^q}\left( {\frac{{\pi \zeta _1^{q - {p_1} + 2}\left( t \right) }}{{2\varpi _1^{q - {p_1} + 2}\left( t \right) }}} \right) {\mathrm{{sg}}_1}\left( {{s_1}} \right) \nonumber \\&\left. \left. +{{\hat{\varPhi }} {\rho _1}+ \frac{{{l_1}}}{4}{\mathrm{{sg}}_1}\left( {{s_1}} \right) + {\kappa ^2}{\rho _1}{\gamma _1} } \right] \! \right] ^{\frac{1}{{{p_1}}}}, \end{aligned}$$
(22)

where \({c_1}\), \({a_1}\), \(\kappa > 0\) are the parameters to be designed. The time-varying gain term \({\hbar _1}\) is chosen as

$$\begin{aligned} \begin{aligned} {\hbar _1} = \sqrt{{{\left( {\frac{{{{\dot{\varpi }}_{b1}}}}{{{\varpi _{b1}}}}} \right) }^2} + {{\left( {\frac{{{{{\dot{\varpi }}}_{a1}}}}{{{\varpi _{a1}}}}} \right) }^2} + {o_1}} \end{aligned}\end{aligned}$$
(23)

in which \({o_1} > 0\) is the parameter to be designed. The switched tuning function \({\gamma _1}\) is chosen as

$$\begin{aligned} {\gamma _1} \!\!= \!\!\left\{ \!\! \begin{aligned}&{\tan ^q}\!\!\left( \!\! {\frac{{\pi \zeta _1^{q - {p_1} + 2}\!\left( t \right) }}{{2\varpi _1^{q - {p_1} + 2}\!\left( t \right) }}}\!\!\right) \!{\rho _1}{\mathrm{{sg}}_1}\!\!\left( {{s_1}} \right) \!,~\!\mathrm{{ if }}~\big | {{s_1}\!\left( t \right) } \big |\! >\! {\beta _1},\\&~~~~~~~~~~~~~~~~0,~~~~~~~~~~~~~~~~~~~~\!\mathrm{{ if }}~\big | {{s_1}\!\left( t \right) } \big | \!\le \! {\beta _1}. \end{aligned} \right. \nonumber \\ \end{aligned}$$
(24)

Remark 7

In (22), when \(\zeta _1 \!\!\rightarrow \! 0\), it has \({\sin ^q}\!\!\left( \!\! {\frac{{\pi \zeta _1^{q - {p_1} + 2}\left( t \right) }}{{2\varpi _1^{q - {p_1} + 2}\left( t \right) }}} \!\!\right) \!\!\) \( \sim \!\! \left( \!\! {\frac{{\pi \zeta _1^{q - {p_1} + 2}\left( t \right) }}{{2\varpi _1^{q - {p_1} + 2}\left( t \right) }}}\! \!\right) ^q\). According to the L’Hospital’s rule, one has

$$\begin{aligned}&\mathop {\lim }\limits _{{\zeta _1} \rightarrow 0} \!\!\frac{{{{\tan }^q}\!\!\left( \!\! {\frac{{\pi \zeta _1^{q - {p_1} + 2}}}{{2\varpi _1^{q - {p_1} + 2}}}} \!\!\right) }}{{{l_1}}} \!\! \\&\quad =\!\! \mathop {\lim }\limits _{{\zeta _1} \rightarrow 0}\!\! \frac{{{{\sin }^q}\!\!\left( \!\! {\frac{{\pi \zeta _1^{q - {p_1} + 2}}}{{2\varpi _1^{q - {p_1} + 2}}}}\!\! \right) \!{{\cos }^{2 - q}}\!\!\left( \!\! {\frac{{\pi \zeta _1^{q - {p_1} + 2}}}{{2\varpi _1^{q - {p_1} + 2}}}}\!\! \right) }}{{\zeta _1^{q - {p_1} + 2}}}\\ \mathrm{{ }}&\quad \!\!=\!\! \mathop {\lim }\limits _{{\zeta _1} \rightarrow 0}\!\! \frac{{{{\left( {\frac{{\pi \zeta _1^{q - {p_1} + 2}}}{{2\varpi _1^{q - {p_1} + 2}}}} \right) }^q}}}{{\zeta _1^{q - {p_1} + 2}}} = 0, \end{aligned}$$

that is to say, there is no singularity problem if \({\varpi _1}\left( t \right) \ne 0\) in our proposed fixed-time controller. Furthermore, we note that the inequation \({\tan \left( {\frac{{\pi \zeta _1^{q - {p_1} + 2}}}{{2\varpi _1^{q - {p_1} + 2}}}} \right) > 0}\) holds since \({\zeta _1} < {\varpi _1}\); thus, it is feasible to design the tuning function (24).

In view of (23), it can be derived that

$$\begin{aligned} \begin{aligned} - {l_1}{\hbar _1}{\zeta _1}\!\! -\! {l_1}{\zeta _1}\!\left( t \right) \!\frac{{{{\dot{k}}_1}\left( t \right) }}{{{k_1}\left( t \right) }}\!\!=\!\! -\!\! \left( \!\! {{\hbar _1} \!+\! \frac{{{{\dot{k}}_1}\left( t \right) }}{{{k_1}\left( t \right) }}}\!\! \right) \!\!\frac{{\zeta _1^{q - {p_1} + 2}}}{\varTheta _1}\! < \!0. \end{aligned}\nonumber \\ \end{aligned}$$
(25)

Substituting (22) and (25) into (21) reaches

$$\begin{aligned} {{{\dot{V}}}_1}&\le - {c_1}{\tan ^q}\left( {\frac{{\pi \zeta _1^{q - {p_1} + 2}\left( t \right) }}{{2\varpi _1^{q - {p_1} + 2}\left( t \right) }}} \right) {\psi _1}- {l_1}\left( {{\beta _2} + 1} \right) {\psi _1}\nonumber \\&~~~ - {a_1}{\tan ^p}\left( {\frac{{\pi \zeta _1^{q - {p_1} + 2}\left( t \right) }}{{2\varpi _1^{q - {p_1} + 2}\left( t \right) }}} \right) \!{\psi _1}\!+\! {\tilde{\varPhi }} {\gamma _1} \!-\! {\kappa ^2}\gamma _1^2 \nonumber \\&~~~ + \frac{1}{{{\varTheta _1}}}{{\bar{\varGamma }}_1}({\xi _1}){{{\bar{\upsilon }} }_1}\phi _1^{ - \frac{{{p_1}}}{{q + 1}}}s_2^{q + 1}{\psi _1}- \frac{{l_1^2}}{4}{\psi _1}. \end{aligned}$$
(26)

Step  \(i~\left( {i \in \left\{ {2, \ldots ,n-1} \right\} } \right) \): The design process for step i follows recursively from step 1. From (1) and (8), the time derivative of variable \(s_i\) can be written as

$$\begin{aligned} \begin{aligned} {\dot{s}_i} = {{\bar{\varPsi }}_i}({Z_i}) + {\varGamma _i}({{\bar{\xi }}_i})\xi _{i + 1}^{{p_i}} - \frac{{\partial {\xi _{i,c}}}}{{\partial {{\hat{\varPhi }}} }}\dot{{\hat{\varPhi }}}, \end{aligned}\end{aligned}$$
(27)

where \({{\bar{\varPsi }}_i}({Z_i}) = {\varPsi _i}({{\bar{\xi }}_i}) - \sum \limits _{j = 1}^{i - 1} {\frac{{\partial {\xi _{j + 1,c}}}}{{\partial {\xi _j}}}{{{\dot{\xi }}}_j}} - \frac{{\partial {\xi _{i,c}}}}{{\partial {y_r}}}{\dot{y}_r}\) and \({Z_i} = {\left[ {{\xi _1},{\xi _2}, \ldots ,{\xi _i},{{\dot{y}}_r}} \right] ^T}\). Consider the high-order tan-type BLF as follows:

$$\begin{aligned} \begin{aligned} {{V}_i} \!=\! {{V}_{i - 1}}\mathrm{{ \!+\! }}\frac{{2\varpi _i^{q - {p_i} + 2}\left( t \right) }}{{\pi \left( {q - {p_i} + 2} \right) }}\!\tan \!\! \left( \! {\frac{{\pi \zeta _i^{q - {p_i} + 2}\left( t \right) }}{{2\varpi _i^{q - {p_i} + 2}\left( t \right) }}}\! \right) \!{\psi _i}. \end{aligned}\nonumber \\ \end{aligned}$$
(28)

where \(\zeta _i=|s_i|-{\beta _i}\). From (9), we know that \({\varpi _{bi}}\left( t \right) \) and \({\varpi _{ai}}\left( t \right) \) represent the lower and upper bound of tracking error, respectively, with \({\varpi _{bi}}\left( t \right)> {\beta _i} > 0\), \({\varpi _{ai}}\left( t \right)< - {\beta _i} < 0\). Let \({l_i} = \frac{{\zeta _i^{q - {p_i} + 1}\left( t \right) }}{{{\varTheta _i}}}\) and \({\varTheta _i} = {\cos ^2}\left( {\frac{{\pi \zeta _i^{q - {p_i} + 2}\left( t \right) }}{{2\varpi _i^{q - {p_i} + 2}\left( t \right) }}} \right) \), then the time derivative of \({{V}_i}\) is

$$\begin{aligned} {{{\dot{V}}}_i}&\le {{{\dot{V}}}_{i - 1}}+ {l_i} \left( \varPhi {\rho _i} + {\varGamma _i}({{{\bar{\xi }}}_i})\xi _{i + 1}^{{p_i}} - \frac{{\partial {\xi _{i,c}}}}{{\partial {\hat{\varPhi }}}}{\dot{\hat{\varPhi }}}\right) {\mathrm{{sg}}_i}\left( {{s_i}} \right) {\psi _i}\nonumber \\&~~~ +\frac{{2\varpi _i^{q - {p_i} + 1}\left( t \right) {{{\dot{\varpi }}}_i}\left( t \right) }}{\pi }\tan \left( {\frac{{\pi \zeta _i^{q - {p_i} + 2}\left( t \right) }}{{2\varpi _i^{q - {p_i} + 2}\left( t \right) }}} \right) {\psi _i}\nonumber \\&~~~-{l_i}\frac{{{{{\dot{\varpi }}}_i}\left( t \right) }}{{{\varpi _i}\left( t \right) }}{\zeta _i}\left( t \right) {\psi _i}, \end{aligned}$$
(29)

where \({\rho _i} = \mathrm{{s}}{\mathrm{{g}}_i}\left( {{s_i}} \right) \sqrt{\vartheta _i^T\left( {{Z_i}} \right) {\vartheta _i}\left( {{Z_i}} \right) + {\lambda _0}} \) with \({\lambda _0} > 0\) being the parameter to be designed. Along similar lines as (18)–(20), one has

$$\begin{aligned} {{{\dot{V}}}_i}&\le {{{\dot{V}}}_{i - 1}}\mathrm{{ + }}{l_i}\left( \varPhi {\rho _i} + {\varGamma _i}({{{\bar{\xi }}}_i}){\ell _i}\xi _{i + 1,c}^{{p_i}} - \frac{{\partial {\xi _{i,c}}}}{{\partial \hat{\varPhi }}}{\dot{{\hat{\varPhi }}}} \right) {\mathrm{{sg}}_i}\left( {{s_i}} \right) \!{\psi _i} \nonumber \\&~~~ +\! \frac{1}{{{\varTheta _i}}}{{{\bar{\varGamma }}}_i}({{{\bar{\xi }}}_i}){{\bar{\upsilon } }_i}\!\left( \! {\phi _i^{\frac{{q - {p_i} + 1}}{q}}\zeta _i^{q + 1} + \phi _i^{ - \frac{{{p_i}}}{{q + 1}}}s_{i + 1}^{q + 1}}\! \right) \!{\psi _i}\nonumber \\&~~~+\!\frac{{2\varpi _i^{q - {p_i} + 1}\left( t \right) {{{\dot{\varpi }}}_i}\left( t \right) }}{\pi }\!\tan \!\! \left( \! {\frac{{\pi \zeta _i^{q - {p_i} + 2}\left( t \right) }}{{2\varpi _i^{q - {p_i} + 2}\left( t \right) }}} \!\right) \!{\psi _i}\nonumber \\&~~~ -{l_i} \frac{{{{{\dot{\varpi }}}_i}\left( t \right) }}{{{\varpi _i}\left( t \right) }}{\zeta _i}\left( t \right) {\psi _i}. \end{aligned}$$
(30)

Design the singularity-free switching controller \({\xi _{i + 1,c}}\) as

$$\begin{aligned} {\xi _{i + 1,c}}\!\! =\!\!&-\! \!\!\left[ \! \frac{1}{{{{{{\underline{\varGamma }}}}_i}({{{\bar{\xi }}}_i})\underline{\ell }_i}}\!\Bigg [\!\frac{1}{{{l_i}{\psi _i}{\varTheta _{i\! -\! 1}}}}{{{\bar{\varGamma }}}_{i - 1}}\!({{{\bar{\xi }}}_{i - 1}}){{{\bar{\upsilon }} }_{i - 1}}\phi _{i - 1}^{ - \frac{{{p_{i - 1}}}}{{q + 1}}}\!\zeta _i^{q + 1}{\psi _{i - 1}} \right. \nonumber \\&+ \frac{{{c_i}}}{{{l_i}}}{{\tan }^q}\left( \! {\frac{{\pi \zeta _i^{q - {p_i} + 2}\left( t \right) }}{{2\varpi _i^{q - {p_i} + 2}\left( t \right) }}} \!\right) {\mathrm{{sg}}_i}\left( {{s_i}} \right) + \frac{1}{{{l_i}}}\beta _i^2{\psi _i}{\mathrm{{sg}}_i}\left( {{s_i}} \right) \nonumber \\&+ \frac{{{a_i}}}{{{l_i}}}{\tan ^p}\!\left( \! {\frac{{\pi \zeta _i^{q - {p_i} + 2}\left( t \right) }}{{2\varpi _i^{q - {p_i} + 2}\left( t \right) }}} \!\right) \!{\mathrm{{sg}}_i}\left( {{s_i}} \right) \!\!+\! {\kappa ^2}{\rho _i}\left( {{\gamma _{i - 1}}\! +\! {\gamma _i}} \right) \nonumber \\&+\!{\hat{\varPhi }} {\rho _i}\!+\! {\hbar _i}{\zeta _i}\!+ \! \frac{{{l_i}}}{4}{\mathrm{{sg}}_i}\left( {{s_i}} \right) \!+\! \frac{1}{{{l_i}{\varTheta _i}}}{{\bar{\varGamma }}_i}({{{\bar{\xi }}}_i}){{{\bar{\upsilon }} }_i}\phi _i^{\frac{{q - {p_i} + 1}}{q}}\zeta _i^{q + 1}\nonumber \\&+ \frac{{2\varpi _i^{q - {p_i} + 1}\left( t \right) {{{\dot{\varpi }}}_i}\left( t \right) }}{{{l_i}\pi }}\tan \left( {\frac{{\pi \zeta _i^{q - {p_i} + 2}\left( t \right) }}{{2\varpi _i^{q - {p_i} + 2}\left( t \right) }}} \right) {\psi _i} \nonumber \\&\left. \left. { + \left( {{\beta _{i + 1}} \!+\! 1} \right) \!{\mathrm{{sg}}_i}\left( {{s_i}} \right) }{ + \frac{{{l_i}}}{4}{\mathrm{{sg}}_i}\left( {{s_i}} \right) \!{\left( \! {\frac{{\partial {\xi _{i,c}}}}{{\partial {\hat{\varPhi }}}}} \!\right) ^2} } \right] \!\right] ^{\frac{1}{{{p_i}}}} \end{aligned}$$
(31)

here the time-varying gain term \({\hbar _i}\) is chosen as

$$\begin{aligned} \begin{aligned} {\hbar _i} = \sqrt{{{\left( {\frac{{{{\dot{\varpi }}_{bi}}}}{{{\varpi _{bi}}}}} \right) }^2} + {{\left( {\frac{{{{{\dot{\varpi }}}_{ai}}}}{{{\varpi _{ai}}}}} \right) }^2} + {o_i}} \end{aligned} \end{aligned}$$
(32)

in which \({o_i} > 0\) is the parameter to be designed. The switched tuning function \({\gamma _i}\) is chosen as

$$\begin{aligned} {\gamma _i} \!\!= \!\!\left\{ \!\!\! \begin{aligned}&{\tan ^q}\!\!\left( \!\! {\frac{{\pi \zeta _i^{q \!-\! {p_i}\! + \!2}\!\left( t \right) }}{{2\varpi _i^{q\! -\! {p_i}\! +\! 2}\!\left( t \right) }}}\! \!\right) \!\!{\rho _i}{\mathrm{{sg}}_i}\!\left( {{s_i}} \right) \! \!+\!\! {\gamma _{i \!-\! 1}},\mathrm{{ if }}\big | {{s_i}\left( t \right) } \big |\! \!>\!\! {\beta _i},\\ \mathrm{{ }}&~~~~~~~~~~~~~~~~~~{\gamma _{i - 1}},~~~~~~~~~~~~~~~~~\!\mathrm{{ if }}\!\!~\big | {{s_i}\left( t \right) } \big | \!\!\le \!\! {\beta _i}. \end{aligned} \right. \nonumber \\ \end{aligned}$$
(33)

Substituting (31) to (30) and using the fact that \(- {l_i}{\hbar _i}{\zeta _i} - {l_i}{\zeta _i}\left( t \right) \frac{{{{{\dot{\varpi }}}_i}\left( t \right) }}{{{\varpi _i}\left( t \right) }} = - \left( {{\hbar _i} + \frac{{{{{\dot{\varpi }}}_i}\left( t \right) }}{{{\varpi _i}\left( t \right) }}} \right) \frac{{\zeta _i^{q - {p_i} + 2}}}{{{\varTheta _i}}} < 0\), we can further have

$$\begin{aligned} {{{\dot{V}}}_i}&\le -\! \sum \limits _{m = 1}^i {c_m}{{\tan }^q}\left( {\frac{{\pi \zeta _m^{q - {p_m} + 2}\left( t \right) }}{{2\varpi _m^{q - {p_m} + 2}\left( t \right) }}} \right) {\psi _m}\!+\! {\tilde{\varPhi }} {\gamma _i}\!-\! \frac{{l_i^2}}{4}{\psi _i}\nonumber \\ \mathrm{{ }}&~~~ -\!\sum \limits _{m = 1}^i {a_m}{{\tan }^p}\left( {\frac{{\pi \zeta _m^{q - {p_m} + 2}\left( t \right) }}{{2\varpi _m^{q - {p_m} + 2}\left( t \right) }}} \right) {\psi _m}\!+\! {\wp _i}\!+ \!{\Im _i} \nonumber \\&~~~ + \!{l_i}\left( { -\! {\beta _{i + 1}} \!-\! 1} \right) {\psi _i}\!+ \! \frac{1}{{{\varTheta _i}}}{{{\bar{\varGamma }}}_i}({{\bar{\xi }}_i}){{{\bar{\upsilon }} }_i}\phi _i^{ - \frac{{{p_i}}}{{q + 1}}}s_{i + 1}^{q + 1}{\psi _i}, \end{aligned}$$
(34)

where

$$\begin{aligned} {\wp _i} =&- \frac{{l_{i - 1}^2}}{4}{\psi _{i - 1}} + {l_{i - 1}}\left( { - {\beta _i} - 1} \right) {\psi _{i - 1}} - \beta _i^2{\psi _i}\nonumber \\ {\Im _i} =&- {\kappa ^2}\gamma _i^2- \sum \limits _{m = 2}^i {{l_m}{\psi _m}{\mathrm{{sg}}_m}\left( {{s_m}} \right) \frac{{\partial {\xi _{m + 1,c}}}}{{\partial {\hat{\varPhi }} }}{\dot{{\hat{\varPhi }}} }} \nonumber \\&- \sum \limits _{m = 2}^i {\frac{{l_m^2}}{4}{\psi _m}{{\left( {\frac{{\partial {\xi _{m + 1,c}}}}{{\partial {\hat{\varPhi }} }}} \right) }^2}}. \end{aligned}$$
(35)

It should be worth noting that \({\wp _i} < 0\) is always satisfied. If \(0 \le {\beta _i} + 1\), the inequation \({l_{i - 1}}\left( { - {\beta _i} - 1} \right) {\psi _{i - 1}} \le 0\) holds, if \(0 > {\beta _i} + 1\), the inequation \({l_{i - 1}}\left( { - {\beta _i} - 1} \right) {\psi _{i - 1}} \le \frac{{l_{i - 1}^2}}{4}{\psi _{i - 1}} + {\left( { - {\beta _i} - 1} \right) ^2} \le \frac{{l_{i - 1}^2}}{4}{\psi _{i - 1}} + \beta _i^2{\psi _i}\) holds and thus \({\wp _i} < 0\).

Step n: Invoking (1) and (8), the dynamics of \({\dot{s}_n}\) are given by

$$\begin{aligned} \begin{aligned} {\dot{s}_n} = {{\bar{\varPsi }}_n}({Z_n}) + {\varGamma _n}(\xi ){\ell _n}{\mathcal {H}}^{{p_n}}{u_c^{{p_n}}}- \frac{{\partial {\xi _{i,c}}}}{{\partial {\hat{\varPhi }} }}{\dot{\hat{\varPhi }}}, \end{aligned}\end{aligned}$$
(36)

where \({{\bar{\varPsi }}_n}({Z_n}) \!=\! {\varPsi _n}(\xi ) \!- \!\sum \limits _{i = 1}^{n - 1}\! {\frac{{\partial {\xi _{i + 1,c}}}}{{\partial {\xi _i}}}{{{\dot{\xi }}}_i}} - \frac{{\partial {\xi _{n,c}}}}{{\partial {y_r}}}{\dot{y}_r}\!+ \! {\varGamma _n}(\xi ){\upsilon _n}{\eta _u^{{p_n}}} \) and \({Z_n} = {\left[ {{\xi _1},{\xi _2}, \ldots ,{\xi _n},{{\dot{y}}_r}} \right] ^T}\). Take the high-order tan-type BLF as

$$\begin{aligned} {{V}_n}&= \frac{{2\varpi _n^{q - {p_n} + 2}\left( t \right) }}{{\pi \left( {q - {p_n} + 2} \right) }}\tan \left( {\frac{{\pi \zeta _n^{q - {p_n} + 2}\left( t \right) }}{{2\varpi _n^{q - {p_n} + 2}\left( t \right) }}} \right) {\psi _n}\nonumber \\&~~~+ \frac{1}{{2\kappa }}{{\tilde{\varPhi }} ^2}+{{V}_{n - 1}}. \end{aligned}$$
(37)

where \(\zeta _n=|s_n|-{\beta _n}\). \(\kappa \) is the parameter to be designed. \({\hat{\varPhi }}\) is the estimate of \(\varPhi \) with \({\tilde{\varPhi }} = \varPhi - {\hat{\varPhi }}\) denoting the estimation error. Similarly to step i, define \({l_n} = \frac{{\zeta _n^{q - {p_n} + 1}\left( t \right) }}{{{\varTheta _n}}}\) and \({\varTheta _n} = {\cos ^2}\left( {\frac{{\pi \zeta _n^{q - {p_n} + 2}\left( t \right) }}{{2\varpi _n^{q - {p_n} + 2}\left( t \right) }}} \right) \), the time derivative of \({{V}_n}\) can be given by

$$\begin{aligned} {{{\dot{V}}}_n}&\le \! {{{\dot{V}}}_{n - 1}} \mathrm{{ + }}\frac{{2\varpi _n^{q - {p_n} + 1}\left( t \right) {{\dot{\varpi }}_n}\left( t \right) }}{\pi }\!\tan \!\left( \! {\frac{{\pi \zeta _n^{q - {p_n} + 2}\left( t \right) }}{{2\varpi _n^{q - {p_n} + 2}\left( t \right) }}}\! \right) \!{\psi _n}\nonumber \\&~~~+ {l_n}\left( \! {\varPhi {\rho _n} + {\varGamma _n}(\xi ){\ell _n}{\mathcal {H}}^{{p_n}}{u_c^{{p_n}}}\!-\!\frac{{\partial {\xi _{i,c}}}}{{\partial {\hat{\varPhi }}}}\dot{{\hat{\varPhi }}}}\! \right) \!{\mathrm{{sg}}_n}\!\left( {{s_n}} \right) {\psi _n}\nonumber \\&~~~- {l_n}\frac{{{{{\dot{\varpi }}}_n}\left( t \right) }}{{{\varpi _n}\left( t \right) }}{\zeta _n}\left( t \right) {\psi _n}- \frac{1}{\kappa }{\tilde{\varPhi }} \dot{{\hat{\varPhi }}}, \end{aligned}$$
(38)

where \({\rho _n} = \mathrm{{s}}{\mathrm{{g}}_n}\left( {{\zeta _n}} \right) \sqrt{\vartheta _n^T\left( {{Z_n}} \right) {\vartheta _n}\left( {{Z_n}} \right) + {\lambda _0}} \) with \({\lambda _0} > 0\) being the parameter to be designed. Design the singularity-free switching fault-tolerant controller \(u_c\) and adaptation law \({\hat{\varPhi }} \) as

$$\begin{aligned} u_c \!=\!&-\! \left[ {\frac{1}{{{{{\underline{\varGamma }}}_n}(\xi )}{\underline{\ell }}_n\mathcal {\underline{H}}^{{p_n}}}\left[ {\frac{{{c_n}}}{{{l_n}}}{{\tan }^q}\left( {\frac{{\pi \zeta _n^{q - {p_n} + 2}\left( t \right) }}{{2\varpi _n^{q - {p_n} + 2}\left( t \right) }}} \right) {\mathrm{{sg}}_n}\left( {{s_n}} \right) } \right. } \right. \nonumber \\&+{\hat{\varPhi }} {\rho _n} { \!+\! \frac{1}{{{l_n}{\psi _n}{\varTheta _{n - 1}}}}{{{\bar{\varGamma }}}_{n - 1}}\!({{{\bar{\xi }}}_{n - 1}}){{{\bar{\upsilon }} }_{n - 1}}\phi _{n - 1}^{ - \frac{{{p_{n - 1}}}}{{q + 1}}}\!\zeta _n^{q + 1}{\psi _{n \!-\! 1}}}\nonumber \\&+ \frac{{{a_n}}}{{{l_n}}}{\tan ^p}\!\left( \! {\frac{{\pi \zeta _n^{q - {p_n} + 2}\left( t \right) }}{{2\varpi _n^{q - {p_n} + 2}\left( t \right) }}}\! \right) \!\!{\mathrm{{sg}}_n}\left( {{s_n}} \right) \! +\!\! \frac{1}{{{l_n}}}\beta _n^2{\psi _n}{\mathrm{{sg}}_n}\!\left( {{s_n}} \right) \nonumber \\&+\! \frac{{2\varpi _n^{q - {p_n} + 1}\!\left( t \right) \!{{\dot{\varpi }}_n}\!\left( t \right) }}{{{l_n}\pi }}\!\tan \!\! \left( \! {\frac{{\pi \zeta _n^{q - {p_n} + 2}\left( t \right) }}{{2\varpi _n^{q - {p_n} + 2}\left( t \right) }}}\! \right) \!{\psi _n} \!+\! {\hbar _n}{\zeta _n} \nonumber \\&{\left. \left. +\! \frac{{{l_n}}}{4}{\mathrm{{sg}}_n}\!\left( {{s_n}} \right) \!\!{\left( {\frac{{\partial {\xi _{n,c}}}}{{\partial {\hat{\varPhi }}}}} \right) ^2}\!\!+\!\! {\kappa ^2}{\rho _n}\left( {{\gamma _{n - 1}}\! +\! {\gamma _n}} \right) \right] \!\right] ^{\frac{1}{{{p_n}}}}}\!\!, \end{aligned}$$
(39)
$$\begin{aligned} \dot{{\hat{\varPhi }}} = \mathrm{{Proj}}\left( {\kappa {\gamma _n}} \right) =\left\{ \begin{aligned}&\kappa {\gamma _n},~\mathrm{{ if }}~\big | {{s_n}\left( t \right) } \big | > {\beta _n},\\ \mathrm{{ }}&~0,~~~~\!\!\mathrm{{ if }}~\big | {{s_n}\left( t \right) } \big | \le {\beta _n}. \end{aligned} \right. \end{aligned}$$
(40)

where \({\hat{\varPhi }}(0) \in {\varOmega }\), and the time-varying gain term \({\hbar _n}\) is chosen as

$$\begin{aligned} \begin{aligned} {\hbar _n} = \sqrt{{{\left( {\frac{{{{\dot{\varpi }}_{bn}}}}{{{\varpi _{bn}}}}} \right) }^2} + {{\left( {\frac{{{{{\dot{\varpi }}}_{an}}}}{{{\varpi _{an}}}}} \right) }^2} + {o_n}} \end{aligned}\end{aligned}$$
(41)

in which \({o_n} > 0\) is the parameter to be designed. The switched tuning function is chosen as

$$\begin{aligned} {\gamma _n} \!\!= \!\!\left\{ \!\! \begin{aligned}&{\tan ^q}\!\left( \! {\frac{{\pi \zeta _n^{q\! -\! {p_n}\! +\! 2}\!\left( t \right) }}{{2\varpi _n^{q \!-\! {p_n}\! + \!2}\!\left( t \right) }}}\!\right) \!{\rho _n}{\mathrm{{sg}}_n}\!\left( \! {{s_n}} \!\right) \! +\! {\gamma _{n \!-\! 1}},~\!\!\mathrm{{ if }}\big | {{s_n}\left( t \right) } \big |\!\! >\!\! {\beta _n},\\&~~~~~~~~~~~~~~~~~~~{\gamma _{n - 1}},~~~~~~~~~~~~~~~~~~~\mathrm{{ if }}\big | {{s_n}\left( t \right) } \big | \!\!\le \!\! {\beta _n}. \end{aligned} \right. \nonumber \\ \end{aligned}$$
(42)

Using (34) and substituting (39) and (40) into (38) yields

$$\begin{aligned} {{{\dot{V}}}_n}&\le \! \frac{{{l_n}{\psi _n}}}{{{{\tan }^q}\left( {\frac{{\pi \zeta _n^{q - {p_n} + 2}\left( t \right) }}{{2\varpi _n^{q - {p_n} + 2}\left( t \right) }}} \right) }}\frac{{{\tilde{\varPhi }} }}{\kappa }\left[ {\kappa {\gamma _n}\! -\! \mathrm{{Proj}}\left( {\kappa {\gamma _n}} \right) } \right] \!+\! {\wp _n} \nonumber \\&~~~-\sum \limits _{i = 1}^n {a_i}{{\tan }^p}\left( {\frac{{\pi \zeta _i^{q - {p_i} + 2}\left( t \right) }}{{2\varpi _i^{q - {p_i} + 2}\left( t \right) }}} \right) {\psi _i}+ {\Im _n} \nonumber \\&~~~- \sum \limits _{i = 1}^n {c_i}{{\tan }^q}\left( {\frac{{\pi \zeta _i^{q - {p_i} + 2}\left( t \right) }}{{2\varpi _i^{q - {p_i} + 2}\left( t \right) }}} \right) {\psi _i}, \end{aligned}$$
(43)

where

$$\begin{aligned} {\wp _n}&= - \frac{{l_{n - 1}^2}}{4}{\psi _{n - 1}} + {l_{n - 1}}\left( { - {\beta _n} - 1} \right) {\psi _{n - 1}} - \beta _n^2{\psi _n},\nonumber \\ {\Im _n}&= - {\kappa ^2}\gamma _n^2 - \sum \limits _{i = 2}^n {\frac{{l_i^2}}{4}{\psi _i}{\mathrm{{sg}}_i}\left( {{s_i}} \right) {{\left( {\frac{{\partial {\xi _{i,c}}}}{{\partial {\hat{\varPhi }}}}} \right) }^2}} \nonumber \\&~~~ - \sum \limits _{i = 2}^n {{l_i}{\psi _i}{\mathrm{{sg}}_i}\left( {{s_i}} \right) \frac{{\partial {\xi _{i,c}}}}{{\partial {\hat{\varPhi }}}}\dot{{\hat{\varPhi }}} }. \end{aligned}$$
(44)

Similar to the analysis procedure in [44, 45], according to the property of projection operator (40), one has \({{{\tilde{\varPhi }} }}{\kappa ^{-1} }\left[ {\kappa {\gamma _n} - \mathrm{{Proj}}\left( {\kappa {\gamma _n}} \right) } \right] \le 0\) with \(\mathrm{{Proj}}\left( {\kappa {\gamma _n}} \right) \le {\kappa {\gamma _n}} \) (see Appendices E in [46]). The projection operator guarantees that the estimate \({{{\hat{\varPhi }} }}\) is always in a compact set \({\varOmega }\), i.e., \(\hat{\varPhi } \in {\varOmega }\). Along with the fact that \(\varPhi \in {\varOmega }\), the estimation error \({{\tilde{\varPhi }} }\) is ensured bounded. With the addition of the fact \(\frac{{{l_n}{\psi _n}}}{{{{\tan }^q}\left( {\frac{{\pi \zeta _n^{q - {p_n} + 2}\left( t \right) }}{{2\varpi _n^{q - {p_n} + 2}\left( t \right) }}} \right) }} \ge 0\). Incorporating the Young’s inequation, one arrives

$$\begin{aligned} ~~\!- \!\sum \limits _{i = 2}^n\! {{l_i}{\psi _i}{\mathrm{{sg}}_i}\!\left( {{s_i}} \right) \!\frac{{\partial {\xi _{i,c}}}}{{\partial \hat{\varPhi }}}\dot{{\hat{\varPhi }}} } \!\le \! \! -\! {\kappa ^2}\gamma _n^2\! -\!\! \sum \limits _{i = 2}^n \! {\frac{{l_i^2}}{4}{\psi _i}\!{{\left( \! {\frac{{\partial {\xi _{i,c}}}}{{\partial \hat{\varPhi }}}}\! \right) \!}^2}}\!, \end{aligned}$$
(45)

and therefore, the inequation \({\Im _n} \le 0\) always holds. Then, we can derive that

$$\begin{aligned} {{\dot{V}}_n} \le&-\sum \limits _{i = 1}^n {{c_i}{{\tan }^q}\left( {\frac{{\pi \zeta _i^{q - {p_i} + 2}\left( t \right) }}{{2\varpi _i^{q - {p_i}+ 2}\left( t \right) }}} \right) {\psi _i}} - \sum \limits _{i = 1}^n {a_i}\nonumber \\&\times {{\tan }^p}\left( {\frac{{\pi \zeta _i^{q - {p_i}+ 2}\left( t \right) }}{{2\varpi _i^{q - {p_i} + 2}\left( t \right) }}} \right) {\psi _i}. \end{aligned}$$
(46)

4 Stability analysis

At this point, the main result of this paper is given in Theorem 1.

Theorem 1

Consider the high-order nonlinear systems (1) subject to actuator faults (2), under Assumptions 1–3, the virtual controllers (22) and (31), the state-constrained fault-tolerant controller (39), and the adaption laws (40). By choosing appropriate design parameters, there exists a settling time independent of initial states such that tracking errors converge into the user-defined intervals and all signals of the closed-loop system are fixed-time stable and the time-varying state constraints will never be violated.

Proof

Consider the total high-order BLFs as

$$\begin{aligned} {\bar{V}}= \sum \limits _{i = 1}^n {\frac{{2\varpi _i^{q - {p_i} + 2}\left( t \right) }}{{\pi \left( {q - {p_i} + 2} \right) }}\tan \!\left( {\frac{{\pi \zeta _i^{q - {p_i} + 2}\left( t \right) }}{{2\varpi _i^{q - {p_i} + 2}\left( t \right) }}} \right) \!{\psi _i}}. \end{aligned}$$
(47)

Invoking (18), (19) and (21), it yields that

$$\begin{aligned} {\dot{{\bar{V}}}}&\le -\sum \limits _{i = 1}^n {c_i}{\tan ^q}\left( {\frac{{\pi \zeta _i^{q - {p_i} + 2}\left( t \right) }}{{2\varpi _i^{q - {p_i} + 2}\left( t \right) }}} \right) {\psi _i} + {{\tilde{\varPhi }}}{\gamma _n}\nonumber \\&~~~-\sum \limits _{i = 1}^n {a_i}{\tan ^p}\left( {\frac{{\pi \zeta _i^{q - {p_i} + 2}\left( t \right) }}{{2\varpi _i^{q - {p_i} + 2}\left( t \right) }}} \right) {\psi _i}. \end{aligned}$$
(48)

Substituting (45) into (48), we get

$$\begin{aligned} {{{\tilde{\varPhi }}}}{\gamma _n}&\! = \!{{{\tilde{\varPhi }}}}\sum \limits _{i = 1}^n \!{\tan ^q}\!\left( {\frac{{\pi \zeta _i^{q - {p_i} + 2}\left( t \right) }}{{2\varpi _i^{q - {p_i} + 2}\left( t \right) }}} \right) \!{\psi _i}\!\sqrt{\vartheta _i^T{\vartheta _i} + {\lambda _0}} \nonumber \\&\!\le \! {\omega _{{i_m}}}\!\sqrt{2n(2\! +\! {\lambda _0})} \psi _i^{1 - q}\!{\left( \! {\frac{{\pi \left( {q - {p_i} + 2} \right) }}{{2\varpi _i^{q - {p_i} + 2}\left( t \right) }}}\! \right) ^q}{\bar{V}}^q, \end{aligned}$$
(49)

where \({\omega _{{i_m}}}\) stands for the upper bound of estimation error \({{\tilde{\omega }} _i}\). Then, the time derivative of \({{{\bar{V}}}_i}\) becomes

$$\begin{aligned} {\dot{{\bar{V}}}_i}\begin{array}{*{20}{c}} \le \end{array} - {a_1}{V}_i^p\left( t \right) - {b_1}{V}_i^q\left( t \right) , \end{aligned}$$
(50)

where \({a_1} = \frac{{\pi \left( {q - {p_i} + 2} \right) {a_i}}}{{2\varpi _i^{q - {p_i} + 2}\left( t \right) }}\), \({b_1} = \mathop {\min }\limits _{i \in \left\{ {1,2, \ldots ,n} \right\} } \bigg \{ \frac{{\pi \left( {q - {p_i} + 2} \right) {c_i}}}{{2\varpi _i^{q - {p_i} + 2}\left( t \right) }} - {\omega _{{i_m}}}\sqrt{2n(2 + {\lambda _0})} \psi _i^{1 - q}{{\left( {\frac{{\pi \left( {q - {p_i} + 2} \right) }}{{2\varpi _i^{q - {p_i} + 2}\left( t \right) }}} \right) }^q} \bigg \}\). With the results obtained in (46), it leads to that the inequations \({{V}_n}\le 0\) hold; therefore, we can derive that \({\zeta _i}\) and \({\tilde{\varPhi }}\) are bounded. Due to that \(\zeta _i=|s_i|-{\beta _i}\) and the value of \({\beta _i}\) is user-defined, \({s_i}\) is bounded. Since \({\hat{\varPhi }} = \varPhi - {\tilde{\varPhi }} \) and \(\varPhi \) is a constant, we have \({\hat{\varPhi }}\) is bounded. Since \({\xi _i}\); \({V} \in {L_\infty }\). Therefore, all the signals of closed-loop system are bounded. In view of Lemma 2 and previous analysis, we can conclude that the tracking errors \(s_1\) will asymptotically converge to \(\beta \) within fixed time \(T=\frac{1}{a_1(p-1)}+\frac{2}{b_1(1-q)}\). The proof is thus completed. \(\square \)

Remark 8

Although some existing works such as [44] and [45] have dealt with the predefined accuracy issue, these works require the systems to be in low-order form and cannot preserve the fixed-time stability. Compared with the existing FTC strategies [39,40,41,42,43], our proposed method can not only achieve fixed-time convergence, but also guarantee the asymmetric time-varying state constraints. In other words, our proposed method includes existing approaches as special cases.

Remark 9

In Theorem 1, we can prove that the tracking error \(s_1\) asymptotically converges to a predefined interval \(\beta _1\) and similar analysis can be applied to prove that other state errors \(s_i\) also converge to prescribed interval \(\beta _i\). With this result, along with the adaptation law designed in (40), we have \({\lim _{t \rightarrow \infty }}\dot{{\hat{\varPhi }}} = 0\) and \({{\hat{\varPhi }} }\) ultimately converges to a constant. Such result is also demonstrated by the evolutions in Fig. 4b.

5 Simulation example

To verify the effectiveness of the proposed scheme in practical application, a poppet valve system which is one of the most commonly used components in hydraulic systems is considered as shown in Fig. 1 [1, pp. 54].

Fig. 1
figure 1

Poppet valve system

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A poppet valve is typically utilized to control the timing and quantity of gas or vapor flow into an engine, and its behavior can be modeled by the annular leakage equation. The input force F drives the poppet to move for regulating the volumetric flow rate \(Q_{\mathrm{{vol}}}=\lambda c^3\) of oil from the high-pressure to the low-pressure chamber, where \(\lambda =\frac{\pi r}{6 \mu L}\varDelta P\) is a lumped coefficient, \(c = \alpha y\) is the effective clearance of the annular passage with \(\alpha \) a constant and y the displacement of poppet, and where r, \(\mu \) and L are constants independent of the axial motion of poppet, and \(\varDelta P\) is the pressure drop between two chambers. The dynamics of oil volume V in upper chamber is given by

$$\begin{aligned} \begin{aligned} {\dot{V}}=Q_{\mathrm{{vol}}}-R(t), \end{aligned} \end{aligned}$$
(51)

where R is the lumped reduction rate of oil attributed to consumption and other leakages. Likewise, the equation of motion of the poppet is

$$\begin{aligned} \begin{aligned} m {\ddot{y}}=-k { \dot{y}}(t) +T(t)+F(t), \end{aligned}\end{aligned}$$
(52)

where m is the mass of the poppet, k is the viscous friction coefficient, T is the lumped elastic force, and F represents the input force. At this point, let us introduce the following notation substitutions:

$$\begin{aligned} \begin{aligned} \xi _{1}=V,~\xi _{2}=y,~\xi _{3}={ \dot{y}},~u=F, \end{aligned}\end{aligned}$$
(53)
Fig. 2
figure 2

a Evolution of \(y_1\) and \(y_r\), b evolution of \(\xi _2\) under two schemes, c evolution of \(\xi _3\) under two schemes

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Fig. 3
figure 3

a Profile of \(s_1\) under two schemes, b profile of \(s_2\) under two schemes, c profile of \(s_3\) under two schemes

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Then, the dynamic of systems (52) and (53) comes down to

$$\begin{aligned} {\dot{\xi }}_{1} = \varGamma _{1} \xi _{2}^3+\varPsi _{1},~ {\dot{\xi }}_{2} = \xi _{3},~ {\dot{\xi }}_{3} = \varGamma _{3} u +\varPsi _{3}, \end{aligned}$$
(54)

where \(\varGamma _{1}=\lambda \alpha ^3\), \(\varPsi _{1}=-R\), \(\varGamma _{3}=1/m\), \(\varPsi _{3}=\frac{1}{m}[T-k \xi _{3}]\) with \(m=7.5\mathrm{{kg}}\), \(k=2.5\mathrm{{N/m}}\), \(R=5\mathrm{{L/min}}\), \(\varDelta P=10\mathrm{{N/m^2}}\), \(T=5\mathrm{{N}}\), \(\mu =2.5\), \(L=5\), \(r=1.25\), \(\alpha =4.5\). The reference signal is \({y_r}\left( t \right) =\sin \left( t \right) + 0.5\cos \left( {2t} \right) \). While conducting the simulation, the initial state values are chosen as \({y_r}\left( 0 \right) = 0\), \({{\xi } _1}\left( 0 \right) = {\left[ { 0,0,0} \right] ^T}\) and \({{\hat{\varPhi }}}(0)=0.5 \), and the design parameters are chosen as \({c_1}=7.5\), \({c_2}=8.2\), \({c_3}=9\), \({a_1}=5.5\), \({a_2}=6\), \({a_3}=7\), \(\kappa =1\), \({o_1} =1.15\), \({o_2} =1.2\), \({o_3} =1.05\), \(p=1.1\), \(q=0.9\), \(\beta _1=0.03\). Consider the actuator fault \(u = {\mathcal {H}}{u_c} + {\eta _u}\) with \({\mathcal {H}}= \left\{ \!\! \begin{array}{l} ~1,~~t \le 10,\\ 0.6,~t > 10, \end{array} \right. \) and \({\eta _u}= \left\{ \!\! \begin{array}{l} ~0,~~t \le 10,\\ 1+0.5\mathrm{{sin}}(t),~t > 10, \end{array} \right. \). The system states are restrained in \({\underline{\varpi }}_{ci} (t)< \xi _i(t)< {\bar{\varpi }}_{ci} (t)\) with \({\underline{\varpi }}_{c1} (t)=0.1\mathrm{{cos}}(3t)-1.6\), \({\bar{\varpi }}_{c1} (t)=0.1\mathrm{{sin}}(3t)+1\), \({\underline{\varpi }}_{c2} (t)=0.2\mathrm{{cos}}(5t)-2\), \({\bar{\varpi }}_{c2} (t)=0.2\mathrm{{sin}}(5t)+2\), \({\underline{\varpi }}_{c3} (t)=2\mathrm{{sin}}(2t)-7.5\), \({\bar{\varpi }}_{c3} (t)=1.4\mathrm{{cos}}(2t)+31\), and the lower and upper bound of dynamics tracking error are given as \({\varpi _{b1}}\left( t \right) =0.1\mathrm{{cos}}(3t)-0.65\), \({\varpi _{a1}}\left( t \right) =0.1\mathrm{{sin}}(3t)+0.13\), \({\varpi _{b2}}\left( t \right) =0.2\mathrm{{cos}}(5t)-1.2\), \({\varpi _{a2}}\left( t \right) =0.2\mathrm{{sin}}(5t)+1.5\), \({\varpi _{b3}}\left( t \right) =2\mathrm{{sin}}(2t)-4.3\), \({\varpi _{a3}}\left( t \right) =1.4\mathrm{{cos}}(2t)+3.3\). The simulation results are shown in Figs. 2, 3 and 4, where the standard adaptive control scheme as in [17] is taken as a means of comparison. In order to expound the advantages of tracking performances, several performance indices, i.e., the integral time absolute error (ITAE) \(\left[ \int _0^T {t\big | {{{s_{1}}\left( t \right) } } \big |dt} \right] \), the root-mean-square error (RMSE) \(\left[ {\frac{1}{{T}}\int _0^T { {{s_{1}^2\left( t \right) } }dt} } \right] ^{\frac{1}{2}}\) and the mean absolute error (MAE) \(\left[ \! {\frac{1}{{T}}\int _0^T {\big | {{s_{1}\left( t \right) } } \big |dt} }\! \right] \) are introduced, and the calculation result is summarized in Table 1.

Fig. 4
figure 4

a The fault-tolerant controller \(u_c\) and actual control input u, b the adaptation law \({{\hat{\varPhi }}} \)

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Table 1 Performance indices under two schemes
Full size table

It can be perceived from Figs. 2a,  3a and Table 1 that the system output y tracks the desired trajectory \(y_r\) closely and the tracking error converges to a specified interval \(\beta _1=0.03\mathrm {ft/s}\) within fixed time, and the proposed control scheme possesses a better performance than the method in [17]. Figures 2b, c and 3b, c show the profiles of \(\xi _2\), \(\xi _3\), \(s_2\) and \(s_3\). The asymmetric and time-varying constraints are guaranteed via the proposed control scheme, while the system state constraint is violated with the control method in [17]. Besides, compared with the existing results, the fluctuation of tracking error is smaller by utilizing the proposed method. The boundness of the controller and adaptive parameters is illustrated in Fig. 4. As shown in Fig. 4, the proposed controller owns good robustness to the actuator faults.

6 Conclusions

An asymmetric time-varying barrier Lyapunov funct-ions-based nonsingular fixed-time adaptive FTC meth-od is constructed for high-order nonlinear systems in the presence of asymmetric time-varying state constraints and actuator faults. By constructing a new tuning function and a projection operator, the size of the convergence regions of state tracking errors in our case can be adjustable. The singularity and chatting problems are avoided by utilizing the switch function and smooth sign function. An interesting problem to be investigated in the future is how to solve the adaptive control problem for the high-order nonlinear systems in the presence of time-varying actuator fault. Besides, how to reduce the complexity of controller is still a very challenging problem [47,48,49], and this may be regarded as a possible future research topic.