2.1. Problem statement

The airframe structure and corresponding coordinate systems of the tilt tri-rotor UAV used in this paper are shown in Figure 1.

Figure 1.

Tilt tri-rotor UAV structure.

The dynamics of tilt tri-rotor UAV described by Newton Euler equation is given as [Reference Wang and Xian27]

(1) \begin{align}\begin{array}{l}\tau = J\dot \varOmega + \varOmega \times J\varOmega + {d_\tau }\\[5pt] \varOmega = R \cdot \dot {Z}\end{array}\end{align}

where, ${Z} = {\left[ {\phi \;\;\theta \;\;\psi } \right]^T} \in {\mathbb{R}^3}$ is the attitude vector, $\phi, \;\theta, \;\psi $ represent roll angle, pitch angle and yaw angle, respectively. $\varOmega = {\left[ {{\varOmega _1}\;\;{\varOmega _2}\;\;{\varOmega _3}} \right]^T} \in {\mathbb{R}^3}$ expressed in body coordinate frame denotes the angular velocity vector with respect to the inertial coordinate system. $\tau \in {\mathbb{R}^3}$ is the control torque of three attitude channels. $J = {\rm{diag}}\!\left\{ {{{\left[ {{J_\phi }\;\;{J_\theta }\;\;{J_\psi }\;} \right]}^T}} \right\} \in {\mathbb{R}^{3 \times 3}}$ represents the moment of inertia matrix of UAV. ${d_\tau } \in {\mathbb{R}^3}$ is the unknown time-varying disturbance torque. $R \in {\mathbb{R}^{3 \times 3}}$ denotes rotation matrix from inertial coordinate system to body coordinate system which is given as

(2) \begin{align}R = \left[ {\begin{array}{c@{\quad}c@{\quad}c}1 & {}0 & {}{ -\!\sin \theta }\\[5pt] 0 & {}{\cos \phi } & {}{\cos \theta \sin \phi }\\[5pt] 0 & {}{ -\!\sin \phi } {}& {\cos \theta \cos \phi }\end{array}} \right]\end{align}

For a class of tilt tri-rotor UAVs subject to coupling unmodeled dynamics, actuator faults and external disturbances, the dynamics can be described as follows

(3) \begin{align}\begin{array}{l}\tau = {K_u}u + {\tau _f}\\[5pt] \tau + \varLambda \!\left( {Z,\;\varOmega, \;\zeta } \right) = J\dot \varOmega + \varOmega \times J\varOmega + {d_\tau }\\[5pt] \varOmega = R \cdot \dot {Z}\end{array}\end{align}

where ${\tau _f} \in {\mathbb{R}^3}$ is the unknown constant additive fault of control torques, ${K_u} = {\rm{diag}}\!\left\{ {{{\left[ {{K_u}_\phi \;{K_u}_\theta \;\;{K_u}_\psi \;} \right]}^T}} \right\} \in {\mathbb{R}^{3 \times 3}}$ represents the actuator fault factor matrix with ${K_u}_\phi, \;{K_u}_\theta, \;{K_u}_\psi $ being the constants belong to $(0,1]$ . $\varLambda \!\left( {Z,\;\varOmega, \;\zeta } \right)$ denotes the dynamic uncertainties which is affected by the system states and the unmodeled dynamics $\zeta $ simultaneously. What’s more, the dynamic of $\zeta $ are modeled as $\dot \zeta = {f_\zeta }\!\left( {\zeta, Z,\;\varOmega } \right)$ in this work.

Defining

(4) \begin{align}\begin{array}{l}{x_1} = Z,\;{x_2} = \Omega, \;x = {\left[ {x_1^T,x_2^T} \right]^T}\\[5pt] f\!\left( x \right) = - {J^{ - 1}}\!\left( {{x_2} \times J{x_2}} \right),\;d = {\tau _f} - {d_\tau },\;\\[5pt] \Delta u = \left( {{K_u} - I} \right)u,\;g\!\left( x \right) = {J^{ - 1}}\end{array}\end{align}

The dynamics of tilt tri-rotor UAVs can be rewritten as follows

(5) \begin{align}\begin{array}{l}{{\dot x}_1} = {R^{ - 1}}{x_2}\\[5pt] {{\dot x}_2} = f\!\left( x \right) + g\!\left( x \right)\left[ {u + \Delta u + d + \Lambda \!\left( {x,\zeta } \right)} \right]\\[5pt] \dot \zeta = {f_\zeta }\!\left( {\zeta, x} \right)\end{array}\end{align}

2.2. Assumptions and lemmas

Before presenting the main results, some useful assumptions and lemmas are listed as follows:

Assumption 1. The coupling dynamic uncertainties $\Lambda \!\left( {x,\zeta } \right)$ satisfy

(6) \begin{align}\Lambda \!\left( {x,\zeta } \right) \le {\varphi _1}\!\left( x \right) + {\varphi _2}\!\left( \zeta \right)\end{align}

where ${\varphi _1}\!\left( \cdot \right)$ and ${\varphi _2}\!\left( \cdot \right)$ are two unknown nonnegative smooth functions.

Assumption 2. The unmodeled dynamics $\zeta $ is set as exponentially input-to-state practically stable (exp-ISpS), that is, there has a Lyapunov function ${V_\zeta }\!\left( \zeta \right)$ such that

(7) \begin{align}\begin{array}{l}{\alpha _1}\!\left( \zeta \right) \le {V_\zeta }\!\left( \zeta \right) \le {\alpha _2}\!\left( \zeta \right)\\[9pt] \dfrac{{\partial {V_\zeta }\!\left( \zeta \right)}}{{\partial \zeta }}\kappa \!\left( {\zeta, x} \right) \le - {\gamma _1}{V_\zeta }\!\left( \zeta \right) + \rho \!\left( x \right) + {\gamma _2}\end{array}\end{align}

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

Lemma 1. [Reference Ning, Zhang and Wang28] For any constant $\epsilon \gt 0$ and vector $\xi \in {\mathbb{R}^n}$ , we have

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

Lemma 2. [Reference Zhu, Xia and Fu29] Considering the nonlinear system $\dot x = f\!\left( x \right)$ and supposing that there exists continuous differentiable function $V\!\left( x \right)$ , scalars $c \gt 0$ , $0 \lt p \lt 1$ and $0 \lt \varepsilon \lt \infty $ , then if the following inequality

(9) \begin{align}\dot V\!\left( x \right) \le - c{V^p}\!\left( x \right) + \varepsilon \end{align}

is valid, then the nonlinear system $\dot x = f\!\left( x \right)$ is practical finite-time stable (PFS).

Lemma 3. [Reference Wang, Yuan and Yang30] For any positive constant $\delta \gt 0$ and the set ${\Omega _\delta }$ defined by ${\Omega _\delta }\;:\!=\; \left\{ {x|\left\| x \right\| \lt 0.2554\delta } \right\}$ , the following inequality is satisfied if $x \notin {\Omega _\delta }$

(10) \begin{align}1 - 16\;{\tanh ^2}\!\left( {\frac{x}{\delta }} \right) \le 0\end{align}

3.1. Control structure

In this section, a novel NN-based robust adaptive super-twisting sliding mode fault-tolerant control scheme will be designed to solve the attitude tracking problem of the tilt tri-rotor UAV subject to unmodeled dynamics, actuator faults and external disturbances. Figure 2 shows the structure of the proposed controller.

Figure 2.

The structure of the proposed controller.

3.2. Robust adaptive super-twisting sliding mode fault-tolerant controller design

In order to realise the attitude tracking control of tilt tri-rotor UAV, define the inner loop attitude tracking error as

(11) \begin{align}{e_1} = {x_1} - {y_d}\end{align}

where ${e_1} = {\left[ {{e_{1\phi }}\;\,{e_{1\theta }}\;\,{e_{1\psi }}} \right]^T} \in {\mathbb{R}^{3 \times 1}}$ and ${e_{1\phi }},\;{e_{1\theta }},\;{e_{1\psi }}$ are the attitude tracking errors of $\phi, \;\theta, \;\psi $ respectively, ${y_d} = {\left[ {{\phi _d}\!\left( t \right)\;\;{\theta _d}\!\left( t \right)\;\;{\psi _d}\!\left( t \right)} \right]^T} \in {R^{3 \times 1}}$ represents the command signal.

According to (5), we can get the differential of ${e_1}$ as

(12) \begin{align}{\dot e_1} = {R^{ - 1}}{x_2} - {\dot y_d}\end{align}

By defining ${e_0} = \int_0^t {{e_1}\!\left( s \right)} \;{\kern 1pt} d{\kern 1pt} s$ and the outer loop attitude tracking error as ${e_2} = {x_2} - {x_{2c}}$ , the inner loop virtual control signal can be obtained as

(13) \begin{align}{x_{2c}} = R\!\left( { - {k_0}{e_0} - {k_1}{e_1} + {{\dot y}_d}} \right)\end{align}

where ${k_0},{k_1} \in {\mathbb{R}^{3 \times 3}}$ are positive definite diagonal gain matrices.

Substituting (13) into (12) has

(14) \begin{align}{\dot e_1} = - {k_0}{e_0} - {k_1}{e_1} + {R^{ - 1}}{e_2}\end{align}

According to (5), the differential of ${e_2}$ can be computed as

(15) \begin{align}{\dot e_2} = f\!\left( x \right) + g\!\left( x \right)\!\left[ {u + \Delta u + d + \Lambda \!\left( {x,\zeta } \right)} \right] - {\dot x_{2c}}\end{align}

Then, the dynamic signal $r$ is introduced, and its dynamics are described as

(16) \begin{align}\dot r = - {\gamma _0}r + \rho \!\left( x \right),\;\;r\!\left( 0 \right) = {r_0}\end{align}

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

According to Assumption 1, we have

(17) \begin{align}e_2^Tg\!\left( x \right)\Lambda \!\left( {x,\zeta } \right) \le \!\left\| {e_2^Tg} \right\|\left( {{\varphi _1}\!\left( x \right) + {\varphi _2}\!\left( \zeta \right)} \right)\end{align}

Based on Lemma 1 and (17), the following inequalities hold

(18) \begin{align}\begin{array}{l}\left\| {e_2^Tg\!\left( x \right)} \right\|{\varphi _1}\!\left( x \right) \le e_2^Tg\!\left( x \right){{\bar \varphi }_1}\!\left( {{e_2},x} \right) + {\varepsilon _1}\\[7pt] \left\| {e_2^Tg\!\left( x \right)} \right\|{\varphi _2}\!\left( \zeta \right) \le \left\| {e_2^Tg\!\left( x \right)} \right\|{\varphi _2}^\circ \alpha _1^{ - 1}\left( {2r} \right) + \left\| {e_2^Tg\!\left( x \right)} \right\|{\varphi _2}^\circ \alpha _1^{ - 1}\!\left( {2{\varepsilon _r}} \right)\end{array}\end{align}

where ${\varepsilon _1}$ is a positive constant,

(19) \begin{align}{\bar \varphi _1}\!\left( {{e_2},x} \right) = \frac{{{\varphi _1}\!\left( x \right)e_2^Tg\!\left( x \right){\varphi _1}\!\left( x \right)}}{{\sqrt {{{[e_2^Tg\!\left( x \right){\varphi _1}\!\left( x \right)\!]}^2} + \varepsilon _1^2} }}\end{align}

Further, using Young’s inequality, one has

(20) \begin{align}\left\| {e_2^Tg\!\left( x \right)} \right\|{\varphi _2}\!\left( \zeta \right) \le e_2^Tg\!\left( x \right){\bar \varphi _2}\!\left( {{e_2},x,r} \right) + {\varepsilon _2} + \frac{1}{4}e_2^Tg{\left( x \right)^T}g\!\left( x \right){e_2} + {\varepsilon _3}\end{align}

where ${\varepsilon _2} \gt 0$ is also a positive constant,

(21) \begin{align}\begin{array}{l}{{\bar \varphi }_2}\!\left( {{e_2},x,r} \right) = \dfrac{{{\varphi _2}^\circ \alpha _1^{ - 1}\!\left( {2r} \right)e_2^Tg\!\left( x \right){\varphi _2}^\circ \alpha _1^{ - 1}\!\left( {2r} \right)}}{{\sqrt {{{\!\left[ {e_2^Tg\!\left( x \right){\varphi _2}^\circ \alpha _1^{ - 1}\!\left( {2r} \right)} \right]}^2} + \varepsilon _2^2} }}\\[22pt] {\varepsilon _3} = {\left[ {{\varphi _2}^\circ \alpha _1^{ - 1}\!\left( {2{\varepsilon _r}} \right)} \right]^2}\end{array}\end{align}

With the ability of approximating an arbitrary unknown continuous function on desired accuracy, NNs have attracted extensive attention in the field of nonlinear control [Reference Liang, Liu, Zhang and Huang31–Reference Wang, Pan and Liang33]. Moreover, radial basis function neural network (RBFNN) is one of the most widely used methods in NN control [Reference Liu, Li, Ge, Ji, Ouyang and Tee34]. Therefore, employing RBFNNs as the general network framework to estimate and compensate the system uncertainties in the tilt tri-rotor UAV yields

(22) \begin{align}{\Theta ^T}\Phi \!\left( {{Z_N}} \right) + {\varepsilon _\Theta } = {\bar \varphi _1}\!\left( {{e_2},x} \right) + {\bar \varphi _2}\!\left( {{e_2},x,r} \right) + \Delta u\end{align}

where $\Theta $ denotes the unknown optimal weight, ${Z_N}$ is the input vector, ${\varepsilon _\Theta }$ denotes the optimal approximation error which satisfies $\left| {{\varepsilon _\Theta }} \right| \le {\bar \varepsilon _\Theta }$ with ${\bar \varepsilon _\Theta }$ being a small positive constant, $\Phi \!\left( {{Z_N}} \right)$ denotes the activation function which can be described as the following Gaussian function

(23) \begin{align}\Phi \!\left( {{Z_N}} \right) = \exp \!\left[ { - \frac{{{{\!\left( {{Z_N} - \mu } \right)}^T}\!\left( {{Z_N} - \mu } \right)}}{{{\sigma ^2}}}} \right]\end{align}

where $\mu $ denotes the position vector of the hidden nodes and $\sigma $ represents the width of Gaussian function. Furthermore, we define $\tilde * = \hat * - * $ with $\tilde * $ denoting the estimation error and $\hat * $ denoting the estimate of the corresponding unknown variable ^*, respectively.

Then, based on the above analysis, the robust adaptive super-twisting sliding mode fault-tolerant controller can be designed as

(24) \begin{align}\begin{array}{l}u = {g^{ - 1}}\!\left( x \right)\!\left[ { - v - {k_2}{e_2} - {R^{ - 1}}{e_1} - f\!\left( x \right) + {{\dot x}_{2c}}} \right] - {{\hat \Theta }^T}\Phi \!\left( Z \right) - \frac{1}{4}g\!\left( x \right){e_2}\\[5pt] v = {{\hat \vartheta }_1}{\chi _1}\!\left( {{e_2}} \right) + \int_0^t {{{\hat \vartheta }_2}{\chi _2}\!\left( {{e_2}\!\left( s \right)} \right)ds} \end{array}\end{align}

where ${k_2} \in {\mathbb{R}^{3 \times 3}}$ is positive definite diagonal gain matrix,

(25) \begin{align}\begin{array}{l}{\chi _1}\!\left( {{e_2}} \right) = {\upsilon _1}\dfrac{{{e_2}}}{{{{\left\| {{e_2}} \right\|}^{1/2}}}} + {\upsilon _2}{e_2}\\[15pt] {\chi _2}\!\left( {{e_2}} \right) = \dfrac{1}{2}\upsilon _1^2\dfrac{{{e_2}}}{{\left\| {{e_2}} \right\|}} + \dfrac{3}{2}{\upsilon _1}{\upsilon _2}\dfrac{{{e_2}}}{{{{\left\| {{e_2}} \right\|}^{1/2}}}} + \upsilon _2^2{e_2}\end{array}\end{align}

${\upsilon _1}$ , ${\upsilon _2}$ are arbitrary positive constants. ${\hat \vartheta _1}$ , ${\hat \vartheta _2}$ are adaptive gains with the following update laws

(26) \begin{align}\begin{array}{l}{{\dot{\hat{\vartheta}} }_1} = \dfrac{{2\kappa }}{{1 + 4\varepsilon _1^2}}{\chi _1}^T\!\left( {{e_2}} \right){\chi _2}\!\left( {{e_2}} \right) - \dfrac{{{\gamma _{{\vartheta _1}}}}}{{1 + 4\varepsilon _1^2}}{{\hat \vartheta }_1}\\[15pt] {{\dot{\hat{\vartheta}} }_2} = \kappa + 4{\varepsilon _1}^2 + 2{\varepsilon _1}{{\hat \vartheta }_1}\end{array}\end{align}

where ${\varepsilon _1}$ , $\kappa $ and ${\gamma _{{\vartheta _1}}}$ are positive constants.

And the adaptive law of $\hat \Theta $ is given as

(27) \begin{align}\dot{\hat{\Theta}} = {\Gamma _\Theta }\Phi \!\left( Z \right)e_2^Tg\!\left( x \right) - {\Gamma _\Theta }{\lambda _\Theta }\hat \Theta \end{align}

where ${\varGamma _\varTheta }$ and ${\lambda _\varTheta }$ are all positive design parameters.

3.3. Stability analysis

Proof. Based on the above analysis, the differential equations of the tilt tri-rotor UAV can be held

(28) \begin{align}\begin{array}{l}{{\dot e}_0} = {e_1}\\[5pt] {{\dot e}_1} = - {k_0}{e_0} - {k_1}{e_1} + {R^{ - 1}}{e_2}\\[5pt] {{\dot e}_2} = f\!\left( x \right) + g\!\left( x \right)\!\left[ {u + \Delta u + d + \Lambda \!\left( {x,\zeta } \right)} \right] - {{\dot x}_{2c}}\\[5pt] \dot \zeta = {f_\zeta }\!\left( {\zeta, x} \right)\end{array}\end{align}

As indicated in Ref. [Reference Jiang and Praly35], the dynamic signal $r$ satisfies the following properties

(29) \begin{align}\begin{array}{l}r\!\left( t \right) \geq 0,\forall t \geq 0\\[5pt] {V_\zeta }\!\left( \zeta \right) \le r\!\left( t \right) + {\varepsilon _r}\end{array}\end{align}

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

Substituting (24) into ${\dot e_2}$ and denoting ${d_1} = g\!\left( x \right)d$ , one has

(30) \begin{align}\begin{array}{c}{{\dot e}_2} = - {{\hat \vartheta }_1}{\chi _1}\!\left( {{e_2}} \right) + z + {d_1} - {k_2}{e_2} - {R^{ - 1}}{e_1} - g\!\left( x \right){{\hat \Theta }^T}\Phi \!\left( Z \right) - \dfrac{1}{4}g{\!\left( x \right)^T}g\!\left( x \right){e_2} + g\!\left( x \right)\!\left[ {\Delta u + \Lambda \!\left( {x,\zeta } \right)} \right]\\[10pt] \dot z = - {{\hat \vartheta }_2}{\chi _2}\!\left( {{e_2}} \right)\end{array}\end{align}

Associated with (25), there exists an unknown positive constant ${g_1}$ satisfying

(31) \begin{align}\left\| {{d_1}} \right\| \le {g_1}\left\| {{e_2}} \right\|\end{align}

In order to make the subsequent proof clearer, the stability proof process for the ${\dot e_2}$ part in (30) will adopt two different ways according to the respective characteristics of different state variables. Considering the first part $ - {\hat \vartheta _1}{\chi _1}\!\left( {{e_2}} \right) + z + {d_1}$ of ${\dot e_2}$ , and denoting $\eta = {\left[ {{\chi _1}^T\!\left( {{e_2}} \right)\quad \,{z^T}} \right]^T}$ and $\Gamma \!\left( {{e_2}} \right) = {\textstyle{1 \over 2}}{\upsilon _1}{\left\| {{e_2}} \right\|^{ - 1/2}} + {\upsilon _2}$ , it is easily obtained that ${\chi _2}\!\left( {{e_2}} \right) = \Gamma \!\left( {{e_2}} \right){\chi _1}\!\left( {{e_2}} \right)$ . Correspondingly, another part of ${\dot e_2}$ is the remaining items.

Then, taking the derivative of $\eta $ has

(32) \begin{align}\begin{array}{c}\dot \eta = \varGamma \!\left( {{e_2}} \right)\left[ {\begin{array}{*{20}{c}}{ - {{\hat \vartheta }_1}{\chi _1} + z + {d_1}}\\[5pt] { - {{\hat \vartheta }_2}{\chi _1}}\end{array}} \right]\\[15pt] = \varGamma \!\left( {{e_2}} \right)\!\left( {\hat A\eta + {d_\eta }} \right)\end{array}\end{align}

where

\begin{align*}\hat A = \left[ {\begin{array}{c@{\quad}c}{ - {{\hat \vartheta }_1}{I_n}} & {}{{I_n}}\\[5pt] { - {{\hat \vartheta }_2}{I_n}}& {}0\end{array}} \right],\;\;{d_\eta } = {\left[ {\begin{array}{c@{\quad}c}{{d_1}^T} {}& 0\end{array}} \right]^T}\end{align*}

${I_n}$ denotes the $n \times n$ identity matrix.

For simplicity, the total Lyapunov function is naturally divided into the following three Lyapunov sub-functions and the total stability proof process is naturally expanded through these three parts. The following Lyapunov candidate functions are selected as

(33) \begin{align}\begin{array}{c}{V_1} = \dfrac{1}{2}e_0^T{e_0} + \dfrac{1}{2}e_1^T{e_1}\\[10pt] {V_2} = {\eta ^T}P\eta + \dfrac{1}{2}{\left( {{{\hat \vartheta }_1} - {\vartheta _1}} \right)^2} + \dfrac{1}{2}{\left( {{{\hat \vartheta }_2} - {\vartheta _2}} \right)^2}\\[10pt] {V_3} = \dfrac{1}{2}e_2^T{e_2} + \dfrac{1}{2}Tr\!\left( {{{\tilde \Theta }^T}\Gamma _\Theta ^{ - 1}\tilde \Theta } \right) + \dfrac{r}{{{\Gamma _r}}}\end{array}\end{align}

where ${\Gamma _r} \gt 0$ is a positive constant, $P$ is a positive definite matrix satisfying

(34) \begin{align}P = \left[ {\begin{array}{c@{\quad}c}{(\kappa + 4{\varepsilon _1}^2){I_n}} & {}{ - 2{\varepsilon _1}{I_n}}\\[5pt] { - 2{\varepsilon _1}{I_n}}& {}{{I_n}}\end{array}} \right]\end{align}

${\vartheta _1}$ and ${\vartheta _2}$ are positive constants satisfying

(35) \begin{align}\begin{array}{l}{\vartheta _1} \gt \dfrac{{2{\varepsilon _1}\!\left( {\kappa + 4{\varepsilon _1}^2} \right) + \left( {\kappa + 4{\varepsilon _1}^2} \right)\upsilon _2^{ - 1}{g_1}}}{\kappa } + \dfrac{{{\varepsilon _1}{{\left( {\upsilon _2^{ - 1}{g_1}} \right)}^2}}}{{2\kappa }}\\[15pt] {\vartheta _2} = \kappa + 4{\varepsilon _1}^2 + 2{\varepsilon _1}{\vartheta _1}\end{array}\end{align}

Firstly, based on (28), calculating derivative of ${V_1}$ has

(36) \begin{align}{\dot V_1} = e_0^T{e_1} + e_1^T\!\left( { - {k_0}{e_0} - {k_1}{e_1} + {R^{ - 1}}{e_2}} \right)\end{align}

Then, by defining ${\bar e_1} = {\left[ {e_0^T,\;e_1^T} \right]^T}$ , equation (36) can be rewritten as

(37) \begin{align}{\dot V_1} = - \bar e_1^TQ{\bar e_1} + e_1^T{R^{ - 1}}{e_2}\end{align}

where

(38) \begin{align}Q = \left[ {\begin{array}{c@{\quad}c}0 & {}{ - {I_n}}\\[5pt] {{k_0}{I_n}} & {}{{k_1}{I_n}}\end{array}} \right]\end{align}

Secondly, the stability analysis process for ${V_2}$ is conducted and the derivative of ${V_2}$ is as follows

(39) \begin{align}\begin{array}{c}{{\dot V}_2} = {{\dot \eta }^T}P\eta + {\eta ^T}P\dot \eta + \left( {{{\hat \vartheta }_1} - {\vartheta _1}} \right){{\dot{\hat{\vartheta}} }_1} + \left( {{{\hat \vartheta }_2} - {\vartheta _2}} \right){{\dot{\hat{\vartheta}} }_2}\\[5pt] = \Gamma \!\left( {{s_2}} \right)\left[ {{\eta ^T}\!\left( {{{\hat A}^T}P + P\hat A} \right)\eta + 2{\eta ^T}P{d_\eta }} \right] + \left( {{{\hat \vartheta }_1} - {\vartheta _1}} \right){{\dot{\hat{\vartheta}} }_1} + \left( {{{\hat \vartheta }_2} - {\vartheta _2}} \right){{\dot{\hat{\vartheta}} }_2}\\[5pt] = \Gamma \!\left( {{s_2}} \right)\left[ {{\eta ^T}\left( {{A^T}P + PA} \right)\eta + 2{\eta ^T}P{d_\eta }} \right] + \Gamma \!\left( {{s_2}} \right){\eta ^T}\left[ {{{\left( {\hat A - A} \right)}^T}P + P\!\left( {\hat A - A} \right)} \right]\eta \\[5pt] + \left( {{{\hat \vartheta }_1} - {\vartheta _1}} \right){{\dot{\hat{\vartheta}} }_1} + \left( {{{\hat \vartheta }_2} - {\vartheta _2}} \right){{\dot{\hat{\vartheta}} }_2}\end{array}\end{align}

(40)

where

(41) \begin{align}A = \left[ {\begin{array}{c@{\quad}c}{ - {\vartheta _1}{I_n}} & {}{{I_n}}\\[5pt] { - {\vartheta _2}{I_n}} {}& 0\end{array}} \right]\end{align}

Substituting (34) into ${\dot V_2}$ yields

(42) \begin{align}\begin{array}{c}{{\dot V}_2} = - \Gamma \!\left( {{e_2}} \right)\left\{ \begin{array}{l}\!\left[ {2\!\left( {\kappa + 4{\varepsilon _1}^2} \right){\vartheta _1} - 4{\varepsilon _1}{\vartheta _2}} \right]{\chi _1}^T\!\left( {{e_2}} \right){\chi _1}\!\left( {{e_2}} \right) + 4{\varepsilon _1}{z^T}z\\[5pt] + 2\!\left[ {{\vartheta _2} - \left( {\kappa + 4{\varepsilon _1}^2} \right) - 2{\varepsilon _1}{\vartheta _1}} \right]{\chi _1}^T\!\left( {{e_2}} \right)z - 2\!\left( {\kappa + 4{\varepsilon _1}^2} \right){d_1}^T{\chi _1}\!\left( {{e_2}} \right) + 4{\varepsilon _1}{d_1}^Tz\end{array} \right\}\\[5pt] \; + \Gamma \!\left( {{e_2}} \right){\eta ^T}\!\left[ {{{\left( {\hat A - A} \right)}^T}P + P\!\left( {\hat A - A} \right)} \right]\eta + \left( {{{\hat \vartheta }_1} - {\vartheta _1}} \right){{\dot{\hat{\vartheta}} }_1} + \left( {{{\hat \vartheta }_2} - {\vartheta _2}} \right){{\dot{\hat{\vartheta}} }_2}\end{array}\end{align}

Based on Cauchy–Schwarz inequality [Reference Trinh36], equation (42) can be rewritten as

(43) \begin{align}\begin{array}{c}{{\dot V}_2} \le - \Gamma \!\left( {{s_2}} \right)\!\left\{ \begin{array}{c}\left[ {2\!\left( {\kappa + 4{\varepsilon _1}^2} \right){\vartheta _1} - 4{\varepsilon _1}{\vartheta _2}} \right]{\chi _1}^T\!\left( {{e_2}} \right){\chi _1}\!\left( {{e_2}} \right) + 2\!\left[ {{\vartheta _2} - \left( {\kappa + 4{\varepsilon _1}^2} \right) - 2{\varepsilon _1}{\vartheta _1}} \right]\chi _1^T\!\left( {{e_2}} \right)z\\[5pt] + 4{\varepsilon _1}{z^T}z - 2\!\left( {\kappa + 4{\varepsilon _1}^2} \right)\left\| {{d_1}} \right\|\left\| {{\chi _1}\!\left( {{e_2}} \right)} \right\| - 4{\varepsilon _1}\!\left\| {{d_1}} \right\|\left\| z \right\|\end{array} \right\}\\[5pt] + \Gamma \!\left( {{e_2}} \right){\eta ^T}\!\left[ {{{\left( {\hat A - A} \right)}^T}P + P\!\left( {\hat A - A} \right)} \right]\eta + \left( {{{\hat \vartheta }_1} - {\vartheta _1}} \right){{\dot{\hat{\vartheta}} }_1} + \left( {{{\hat \vartheta }_2} - {\vartheta _2}} \right){{\dot{\hat{\vartheta}} }_2}\end{array}\end{align}

According to (31) and (25), we have

(44) \begin{align}\left\| {{d_1}} \right\| \le {g_1}\left\| {{s_2}} \right\| \le \frac{{{g_1}}}{{{\upsilon _2}}}\left\| {{\chi _1}\!\left( {{s_2}} \right)} \right\|\end{align}

Therefore, combining (44) and ${\vartheta _2}$ in (35), there is the following one

(45) \begin{align}\begin{array}{c}{{\dot V}_2} \le - \Gamma \!\left( {{e_2}} \right)\left\{ \begin{array}{l}\left[ {2\!\left( {\kappa + 4{\varepsilon _1}^2} \right){\vartheta _1} - 4{\varepsilon _1}{\vartheta _2}} \right]{\left\| {{\chi _1}\!\left( {{e_2}} \right)} \right\|^2} + 4{\varepsilon _1}{\left\| z \right\|^2}\\[5pt] - 2\!\left( {\kappa + 4{\varepsilon _1}^2} \right)\upsilon _2^{ - 1}{g_1}{\left\| {{\chi _1}\!\left( {{e_2}} \right)} \right\|^2} - 4{\varepsilon _1}\upsilon _2^{ - 1}{g_1}\left\| z \right\|\left\| {{\chi _1}\!\left( {{e_2}} \right)} \right\|\end{array} \right\}\\[5pt] + \Gamma \!\left( {{e_2}} \right){\eta ^T}\left[ {{{\!\left( {\hat A - A} \right)}^T}P + P\!\left( {\hat A - A} \right)} \right]\eta + \left( {{{\hat \vartheta }_1} - {\vartheta _1}} \right){{\dot{\hat{\vartheta}} }_1} + \left( {{{\hat \vartheta }_2} - {\vartheta _2}} \right){{\dot{\hat{\vartheta}} }_2}\end{array}\end{align}

Denoting $\bar \eta = {\left[ {\begin{array}{*{20}{c}}{\left\| {{\chi _1}\!\left( {{e_2}} \right)} \right\|} {}{\left\| z \right\|}\end{array}} \right]^T}$ yields

(46) \begin{align}\begin{array}{c}{{\dot V}_2} \le - \Gamma \!\left( {{e_2}} \right){{\bar \eta }^T}\Pi \bar \eta + + \Gamma \!\left( {{e_2}} \right){\eta ^T}\left[ {{{\left( {\hat A - A} \right)}^T}P + P\!\left( {\hat A - A} \right)} \right]\eta \\[5pt] + \left( {{{\hat \vartheta }_1} - {\vartheta _1}} \right){{\dot{\hat{\vartheta}} }_1} + \left( {{{\hat \vartheta }_2} - {\vartheta _2}} \right){{\dot{\hat{\vartheta}} }_2}\end{array}\end{align}

where

\begin{align*}\Pi = \left[ {\begin{array}{c@{\quad}c}{2\kappa {\vartheta _1} - 4{\varepsilon _1}\!\left( {\kappa + 4{\varepsilon _1}^2} \right) - 2\!\left( {\kappa + 4{\varepsilon _1}^2} \right)\upsilon _2^{ - 1}{g_1}} & {}{ - 2{\varepsilon _1}\upsilon _2^{ - 1}{g_1}}\\[5pt] { - 2{\varepsilon _1}\upsilon _2^{ - 1}{g_1}}& {}{4{\varepsilon _1}}\end{array}} \right]\end{align*}

based on (35), the conclusion $\Pi \gt 0$ is valid.

Then, substituting ${\tilde \vartheta _1} = {\hat \vartheta _1} - {\vartheta _1}$ , ${\tilde \vartheta _2} = {\hat \vartheta _2} - {\vartheta _2}$ into (46) yields

(47) \begin{align}\begin{array}{c}{{\dot V}_2} \le - \Gamma \!\left( {{e_2}} \right){{\bar \eta }^T}\Pi \bar \eta \\[5pt] + \Gamma \!\left( {{e_2}} \right){\left[ {\begin{array}{*{20}{c}}{{\chi _1}\!\left( {{e_2}} \right)}\\[5pt] z\end{array}} \right]^T}\left[ {\begin{array}{c@{\quad}c}{\!\left( { - 2\!\left( {\kappa + 4{\varepsilon _1}^2} \right){{\tilde \vartheta }_1} + 4{\varepsilon _1}{{\tilde \vartheta }_2}} \right){I_n}}& {}{\!\left( {2{\varepsilon _1}{{\tilde \vartheta }_1} - {{\tilde \vartheta }_2}} \right){I_n}}\\[5pt] {\!\left( {2{\varepsilon _1}{{\tilde \vartheta }_1} - {{\tilde \vartheta }_2}} \right){I_n}} {}& 0\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{\chi _1}\!\left( {{e_2}} \right)}\\[5pt] z\end{array}} \right]\\[10pt] + \left( {{{\hat \vartheta }_1} - {\vartheta _1}} \right){{\dot{\hat{\vartheta}} }_1} + \left( {{{\hat \vartheta }_2} - {\vartheta _2}} \right){{\dot{\hat{\vartheta}} }_2}\end{array}\end{align}

Clearly, $\left\| {\bar \eta } \right\| = \left\| \eta \right\|$ is valid. In addition, the relationship ${\dot{\hat{\vartheta}} _2} = 2{\varepsilon _1}{\dot{\hat{\vartheta}} _1},{\tilde \vartheta _2} = 2{\varepsilon _1}{\tilde \vartheta _1}$ can be verified based on (26) and (35). Then by using the adaptive law (26), we can get the following inequality

(48) \begin{align} {{\dot V}_2} &\le - \left( {\frac{{{\upsilon _1}}}{{2{{\left\| {{e_2}} \right\|}^{1/2}}}} + {\upsilon _2}} \right){\lambda _{\min }}\!\left( \Pi \right){\left\| \eta \right\|^2} - 2\kappa \Gamma \!\left( {{e_2}} \right){\left\| {{\chi _1}\left( {{e_2}} \right)} \right\|^2}{{\tilde \vartheta }_1} + \left( {1 + 4{\varepsilon _1}^2} \right){{\dot{\hat{\vartheta}} }_1}{{\tilde \vartheta }_1}\nonumber\\[5pt] &\le - \left( {\frac{{{\upsilon _1}}}{{2{{\left\| {{e_2}} \right\|}^{1/2}}}} + {\upsilon _2}} \right){\lambda _{\min }}\!\left( \Pi \right){\left\| \eta \right\|^2} - {\gamma _{{\vartheta _1}}}{{\tilde \vartheta }_1}{{\hat \vartheta }_1}\nonumber\\[5pt] &\le - \left( {\frac{{{\upsilon _1}}}{{2{{\left\| {{e_2}} \right\|}^{1/2}}}} + {\upsilon _2}} \right)\frac{{{\lambda _{\min }}\!\left( \Pi \right)}}{{{\lambda _{\max }}\!\left( P \right)}}{\eta ^T}P\eta - \frac{{{\gamma _{{\vartheta _1}}}}}{2}{{\tilde \vartheta }_1}^2 + \frac{{{\gamma _{{\vartheta _1}}}}}{2}\vartheta _1^2\\[5pt] &\le - \left( {\frac{{{\upsilon _1}}}{{2{{\left\| {{e_2}} \right\|}^{1/2}}}} + {\upsilon _2}} \right)\frac{{{\lambda _{\min }}\left( \Pi \right)}}{{{\lambda _{\max }}\left( P \right)}}{\eta ^T}P\eta - \frac{{{\gamma _{{\vartheta _1}}}}}{2}\frac{2}{{1 + 4{\varepsilon _1}^2}}\left( {\frac{{1 + 4{\varepsilon _1}^2}}{2}{{\left( {{{\hat \vartheta }_1} - {\vartheta _1}} \right)}^2}} \right) + \frac{{{\gamma _{{\vartheta _1}}}}}{2}\vartheta _1^2\nonumber \end{align}

Considering ${\upsilon _1}{\left\| {{e_2}} \right\|^{1/2}} \le \left\| {{\chi _1}\!\left( {{e_2}} \right)} \right\|$ and

\begin{align*}{\left\| {{e_2}} \right\|^{1/2}} \le \frac{{{{\left( {{\eta ^T}P\eta } \right)}^{1/2}}}}{{{\upsilon _1}{\lambda _{\min }}^{1/2}\!\left( P \right)}}\end{align*}

the inequality (48) can be rewritten as

(49) \begin{align}{\dot V_2} \le - {\gamma _1}{\!\left( {{\eta ^T}P\eta } \right)^{1/2}} - {\gamma _2}{\eta ^T}P\eta - {\gamma _3}\left[ {\frac{1}{2}{{\left( {{{\hat \vartheta }_1} - {\vartheta _1}} \right)}^2} + \frac{1}{2}{{\left( {{{\hat \vartheta }_2} - {\vartheta _2}} \right)}^2}} \right] + \frac{{{\gamma _{{\vartheta _1}}}}}{2}\vartheta _1^2\end{align}

where

\begin{align*}{\gamma _1} = \frac{{{\upsilon _1}^2{\lambda _{\min }}\!\left( \Pi \right){\lambda _{\min }}^{1/2}\!\left( P \right)}}{{2{\lambda _{\max }}\!\left( P \right)}},\quad {\gamma _2} = \frac{{{\upsilon _2}{\lambda _{\min }}\!\left( \Pi \right)}}{{{\lambda _{\max }}\!\left( P \right)}},\;\;\;\;{\gamma _3} = \frac{{{\gamma _{{\vartheta _1}}}}}{{1 + 4\varepsilon _1^2}}\end{align*}

Then, it can be concluded that

(50) \begin{align}{\dot V_2} \le - {c_2}V_2^p + {C_2}\end{align}

where ${c_2} = \min \!\left( {{\gamma _1},{\gamma _2},{\gamma _3}} \right)$ , $p \in \left( {\frac{1}{2},1} \right)$ , ${C_2} = \frac{{{\gamma _{{\vartheta _1}}}}}{2}\vartheta _1^2$ . Then from Lemma 2, it can be proven that the system states are bounded as

(51) \begin{align}{\Omega _X} = \left\{ {X{\rm{|}}{V_2} \le {{\left( {\frac{{{C_2}}}{{{c_2}\left( {1 - {\theta _0}} \right)}}} \right)}^{\frac{1}{p}}}} \right\}\end{align}

after finite time

(52) \begin{align}T \le \frac{{{V_2}^{1 - p}\!\left( {X\!\left( {{t_0}} \right)} \right)}}{{{c_2}{\theta _0}\!\left( {1 - p} \right)}}\end{align}

where ${\theta _0} \in \left( {0,\;1} \right)$ and $X\!\left( {{t_0}} \right)$ represents the state vector of the closed-loop system at the initial time ${t_0}$ .

After that, the stability analysis of the third part of the total Lyapunov function will be carried out. Equally, combining the dynamics of dynamic signal $r\!\left( t \right)$ in (16), the derivative of ${V_3}$ can be obtained as

(53) \begin{align} {{\dot V}_3} &= e_2^T\left\{ { - {k_2}{e_2} - {R^{ - 1}}{e_1} - g{{\hat \varTheta }^T}\Phi \!\left( Z \right) - \frac{1}{4}{g^T}g{e_2} + g\left[ {\Delta u + \Lambda \!\left( {x,\zeta } \right)} \right]} \right\}\nonumber\\[5pt] & \quad + Tr\!\left( {{{\tilde \varTheta }^T}\varGamma _\varTheta ^{ - 1}\dot{\hat{\varTheta}} } \right) - \frac{{{\gamma _0}}}{{{\varGamma _r}}}r + \frac{{\rho \left( x \right)}}{{{\varGamma _r}}} \\[5pt] &= - {k_2}e_2^T{e_2} - {R^{ - 1}}e_2^T{e_1} - e_2^Tg{{\hat \varTheta }^T}\Phi \!\left( Z \right) - \frac{1}{4}e_2^T{g^T}g{e_2} + e_2^Tg\left[ {\Delta u + \Lambda \!\left( {x,\zeta } \right)} \right]\nonumber\\[5pt] &\quad + Tr\!\left( {{{\tilde \varTheta }^T}\varGamma _\varTheta ^{ - 1}\dot{\hat{\varTheta}} } \right) - \frac{{{\gamma _0}}}{{{\varGamma _r}}}r + \frac{{\rho \!\left( x \right)}}{{{\varGamma _r}}}\nonumber \end{align}

Combined with (17)-(21), we have

(54) \begin{align}e_2^Tg\varLambda \!\left( {x,\zeta } \right) \le e_2^Tg{\bar \varphi _1}\!\left( {{e_2},x} \right) + e_2^Tg{\bar \varphi _2}\!\left( {{e_2},x,r} \right) + \frac{1}{4}e_2^T{g^T}g{e_2} + \sum\limits_{i = 1}^3 {{\varepsilon _i}} \end{align}

In addition, based on (22), there is the following one

(55) \begin{align}e_2^Tg\left[ {\Delta u + \varLambda \!\left( {x,\zeta } \right)} \right] \le e_2^Tg{\varTheta ^T}\varPhi \!\left( Z \right) + e_2^Tg{\varepsilon _\varTheta } + \frac{1}{4}e_2^T{g^T}g{e_2} + \sum\limits_{i = 1}^3 {{\varepsilon _i}} \end{align}

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

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

Then, one has

(57) \begin{align}\frac{{\rho \!\left( x \right)}}{{{\Gamma _r}}} = \frac{{\rho \!\left( x \right)}}{{{\Gamma _r}}}\!\left( {1 - 16\;{{{\mathop{\rm Tanh}\nolimits} }^T}\!\left( {{\textstyle{{{e_2}} \over {{\varepsilon _\rho }}}}} \right){\mathop{\rm Tanh}\nolimits} \!\left( {{\textstyle{{{e_2}} \over {{\varepsilon _\rho }}}}} \right)} \right) + e_2^T{\varphi _\rho }\!\left( {x,{e_2}} \right)\end{align}

where

(58) \begin{align}{\varphi _\rho }\!\left( {x,{e_2}} \right) = \frac{{16{e_2}\rho \!\left( x \right)}}{{{\Gamma _r}e_2^T{e_2}}}{{\mathop{\rm Tanh}\nolimits} ^T}\!\left( {{\textstyle{{{e_2}} \over {{\varepsilon _\rho }}}}} \right){\mathop{\rm Tanh}\nolimits} \!\left( {{\textstyle{{{e_2}} \over {{\varepsilon _\rho }}}}} \right)\end{align}

note that ${\varphi _\rho }\!\left( {x,{e_2}} \right)$ is a non-singular function vector for ${e_2}$ .

Substituting (55), (57) and (58) into (53), we have

(59) \begin{align}\begin{array}{c}{{\dot V}_3} \le - {k_2}e_2^T{e_2} - {R^{ - 1}}e_2^T{e_1} - e_2^Tg{{\tilde \varTheta }^T}\Phi \!\left( Z \right) + e_2^Tg{\varepsilon _\varTheta } + \sum\limits_{i = 1}^3 {{\varepsilon _i}} \\[5pt] + Tr\!\left( {{{\tilde \varTheta }^T}\Gamma _\varTheta ^{ - 1}\dot{\hat{\varTheta}} } \right) + \dfrac{{\rho \!\left( x \right)}}{{{\Gamma _r}}}\!\left( {1 - 16\;{{{\mathop{\rm Tanh}\nolimits} }^T}\!\left( {{\textstyle{{{e_2}} \over {{\varepsilon _\rho }}}}} \right){\mathop{\rm Tanh}\nolimits} \!\left( {{\textstyle{{{e_2}} \over {{\varepsilon _\rho }}}}} \right)} \right) - \frac{{{\gamma _0}}}{{{\varGamma _r}}}r\end{array}\end{align}

By using the adaptive law of $\hat \Theta $ in (27), it’s easily obtained that

(60) \begin{align}{\dot V_3} \le - {k_2}e_2^T{e_2} - {R^{ - 1}}e_2^T{e_1} - {\lambda _\varTheta }Tr\!\left( {{{\tilde \varTheta }^T}\hat \varTheta } \right) + e_2^Tg{\varepsilon _\Theta } + \sum\limits_{i = 1}^3 {{\varepsilon _i}} - \frac{{{\gamma _0}}}{{{\varGamma _r}}}r + \frac{{\rho \!\left( x \right)}}{{{\varGamma _r}}}\!\left( {1 - 16\;{{{\mathop{\rm Tanh}\nolimits} }^T}\!\left( {{\textstyle{{{e_2}} \over {{\varepsilon _\rho }}}}} \right){\mathop{\rm Tanh}\nolimits} \!\left( {{\textstyle{{{e_2}} \over {{\varepsilon _\rho }}}}} \right)} \right)\end{align}

Considering that

(61) \begin{align}\begin{array}{l} - Tr\!\left( {{{\tilde \varTheta }^T}\hat \varTheta } \right) \le - \dfrac{1}{2}Tr\!\left( {{{\tilde \varTheta }^T}\tilde \varTheta } \right) + \dfrac{1}{2}Tr\!\left( {{\varTheta ^T}\varTheta } \right)\\[10pt] e_2^Tg\!\left( x \right){\varepsilon _\Theta } \le \dfrac{{g{{\!\left( x \right)}^T}g\!\left( x \right)}}{2}e_2^T{e_2} + \dfrac{{{{\bar \varepsilon }^2}_\Theta }}{2}\end{array}\end{align}

${\dot V_3}$ in (60) can be rewritten as

(62) \begin{align}\begin{array}{l}{{\dot V}_3} \le - \left( {{k_2} - \dfrac{{g{{\!\left( x \right)}^T}g\!\left( x \right)}}{2}} \right)e_2^T{e_2} - {R^{ - 1}}e_2^T{e_1} - \dfrac{{{\lambda _\varTheta }}}{2}Tr\!\left( {{{\tilde \varTheta }^T}\tilde \varTheta } \right) + \dfrac{{{\lambda _\varTheta }}}{2}Tr\!\left( {{\varTheta ^T}\varTheta } \right) + \dfrac{{{{\bar \varepsilon }^2}_\varTheta }}{2} + \sum\limits_{i = 1}^3 {{\varepsilon _i}} \\[10pt] - \dfrac{{{\gamma _0}}}{{{\varGamma _r}}}r + \dfrac{{\rho \left( x \right)}}{{{\varGamma _r}}}\!\left( {1 - 16\;{{{\mathop{\rm Tanh}\nolimits} }^T}\left( {{\textstyle{{{e_2}} \over {{\varepsilon _\rho }}}}} \right){\mathop{\rm Tanh}\nolimits} \left( {{\textstyle{{{e_2}} \over {{\varepsilon _\rho }}}}} \right)} \right)\end{array}\end{align}

Combining with ${\dot V_1}$ in (37) and ${\dot V_3}$ in (62), the following results can be obtained

(63) \begin{align}\begin{array}{l}{{\dot V}_1} + {{\dot V}_3}\\[10pt] \le - \bar e_1^TQ{{\bar e}_1} - \left( {{k_2} - \dfrac{{g{{\!\left( x \right)}^T}g\!\left( x \right)}}{2}} \right)e_2^T{e_2} - \dfrac{{{\lambda _\varTheta }}}{2}Tr\!\left( {{{\tilde \varTheta }^T}\tilde \varTheta } \right) + \dfrac{{{\lambda _\varTheta }}}{2}Tr\!\left( {{\varTheta ^T}\varTheta } \right) + \dfrac{{{{\bar \varepsilon }^2}_\varTheta }}{2} + \sum\limits_{i = 1}^3 {{\varepsilon _i}} \\[10pt] - \dfrac{{{\gamma _0}}}{{{\varGamma _r}}}r + \dfrac{{\rho \left( x \right)}}{{{\varGamma _r}}}\!\left( {1 - 16\;{{{\mathop{\rm Tanh}\nolimits} }^T}\!\left( {{\textstyle{{{e_2}} \over {{\varepsilon _\rho }}}}} \right){\mathop{\rm Tanh}\nolimits} \!\left( {{\textstyle{{{e_2}} \over {{\varepsilon _\rho }}}}} \right)} \right)\\[10pt] \le - \gamma \!\left( {{V_1} + {V_3}} \right) + {k_f} + \dfrac{{\rho \!\left( x \right)}}{{{\varGamma _r}}}\!\left( {1 - 16\;{{{\mathop{\rm Tanh}\nolimits} }^T}\!\left( {{\textstyle{{{e_2}} \over {{\varepsilon _\rho }}}}} \right){\mathop{\rm Tanh}\nolimits} \!\left( {{\textstyle{{{e_2}} \over {{\varepsilon _\rho }}}}} \right)} \right)\end{array}\end{align}

where

(64) \begin{align}\begin{array}{l}\gamma = \min \left\{ {2{\lambda _{\min }}\!\left( Q \right),\;2{k_2} - g{{\!\left( x \right)}^T}g\!\left( x \right),\;{\lambda _{\min }}\!\left( {{\varGamma _\varTheta }} \right){\lambda _\varTheta }} \right\}\\[10pt] {k_f} = \sum\limits_{i = 1}^3 {{\varepsilon _i}} + \dfrac{{{\lambda _\varTheta }}}{2}Tr\!\left( {{\varTheta ^T}\varTheta } \right) + \dfrac{{{{\bar \varepsilon }^2}_\varTheta }}{2}\end{array}\end{align}

Define the following compact set

(65) \begin{align}\begin{array}{l}{\varOmega _f} = \left\{ {x \in {\mathbb{R}^n}\left| {{V_1}\!\left( x \right) + {V_3}\!\left( x \right) \le \frac{{{\gamma _2}}}{{{\gamma _1}}}} \right.} \right\}\\[5pt] {\varOmega _\rho } = \left\{ {x\left| {\left\| x \right\| \lt 0.2554\varepsilon } \right.} \right\}\end{array}\end{align}

Based on Lemma 3, it is easy to know that if ${e_2}\!\left( t \right) \in {\varOmega _f} \cap {\varOmega _\rho }$ , the solution of the closed-loop control system $\left[ {{e_0},{e_1},{e_2},\tilde \varTheta } \right]$ is naturally bounded. If ${e_2}\!\left( t \right) \notin {\varOmega _f} \cap {\varOmega _\rho }$ , ${\dot V_1} + {\dot V_3} \lt 0$ can be proved and ${V_1} + {V_3}$ gradually decreases, and the solution will eventually converge to the set ${\varOmega _f} \cap {\varOmega _\rho }$ . Furthermore, with Barbalat’s lemma [Reference Lavretsky and Wise9] and the bounded ${e_0}$ , we conclude that when $t \to \infty $ , ${e_1} \to 0$ , that is, the system tracking errors gradually converge to zero.

In summary, by combining the above three parts of the proof process and results, we can draw the conclusion that the robust adaptive super-twisting sliding mode fault-tolerant controller designed in this paper can ensure the stability of the tilt tri-rotor UAV closed-loop system (5), and all the closed-loop signals are bounded as ${\Omega _\chi } = {\Omega _X} \cap {\Omega _f}$ . The proof is completed.

4.1. Comparisons with other methods

To demonstrate the advantages of the proposed RASTC method, the RASTC method without coupling uncertainties compensation (CUC) and traditional SMC method are taken into account for comparison, the simulation results are shown in Figs. 3–5.

From Figs. 3–5, we can find that the command signals can be accurately tracked by using the proposed RASTC method. However, both the RASTC without CUC method and the SMC method produce non-ideal tracking errors. In addition, the remarkable chattering phenomenon cannot be avoided when using the SMC controller. Obviously, the proposed control method can change timely to surmount the negative effects caused by the multiple uncertainties and the varying command signals. Therefore, it is readily to know that the tracking performances is satisfactory under the proposed RASTC controller. Thereafter, the simulation results of other parameters of the proposed method are as shown in Figs. 6–9, from which we can conclude that the boundedness of all the closed-loop signals is guaranteed.

To further quantitatively evaluate the tracking performance among the three controllers, integral squared errors (ISE) will be used as the performance index in this work. The quantitative comparison results are shown in Table 1, where ${\rm{ISE = }}\int_0^t {\!\left( {e_{1\phi }^2\!\left( t \right) + \;e_{1\theta }^2\!\left( t \right) + \;e_{1\psi }^2\!\left( t \right)} \right)} \;dt$ . Based on Table 1, it can be seen that the designed control scheme has the smallest ISE value, followed by the SMC method, and the RASTC without CUC method has the highest ISE value. According to Ref. [Reference Liu, Wang, Zheng, Dong and Sun37], a smaller ISE value not only indicates less overall fluctuation in tracking errors, but also reflects a faster convergence speed. Therefore, combining the performance index ISE with simulation results in Figs. 3–5, it can be seen that the proposed controller is able to provide a higher accuracy and a faster convergence speed for the TRUAVs subject to unmodeled dynamics, actuator faults and external disturbances.

4.2. Comparisons under different cases

Furthermore, for the purpose of verifying the robustness of the proposed method, the simulations under different cases which are three groups of disturbance torques ${d_\tau }$ , actuator faults parameters $\tau = {K_u}u + {\tau _f}$ and dynamic uncertainties $\Lambda \!\left( {Z,\;\varOmega, \;\zeta } \right)$ are set up. The specific parameter settings are shown in Table 2.

Table 1.

The performance comparisons of three controllers

Table 2.

Model parameter values of different cases $/({\rm{N}} \cdot {\rm{m)}}$

Figure 10.

The trajectories of the roll angle $\phi $ under different cases.

Figure 11.

The trajectories of the pitch angle $\theta $ under different cases.

Figure 12.

The trajectories of the yaw angle $\psi $ under different cases.

The corresponding results are given in Figures 10–12. It can be clearly seen that under Case 1, the system states including roll angle $\phi $ , pitch angle $\theta $ and yaw angle $\psi $ can track their command signals appropriately despite disturbances, actuator faults and dynamic uncertainties. As the disturbances, actuator faults and uncertainties gradually increase, the attitude angles of different channels oscillate to a small extent and then converge quickly, which satisfies the requirements of tracking control. To sum up, owing to the proposed RASTC scheme, superior tracking performance can be achieved, and the disturbances, actuator faults as well as dynamic uncertainties are sufficiently suppressed, demonstrating satisfactory stability and robustness.