Control Effectiveness Enhancement for the Hovering/Cruising Transition Control of a Ducted Fan UAV

Abstract

The ducted fan unmanned aerial vehicle (UAV) is capable of both hovering and high-speed cruising, while the transitional flight between hovering and cruising is one of the most challenging flight maneuverings. In this paper, we address the flight mode transition control of a ducted fan UAV with constrained control inputs. During our early flight tests, we have engaged a saturation problem on the control vanes that led to severe crashes. In order to maintain a sufficient control on the attitude suffering from input saturation, we propose a control effectiveness enhancement (CEE) algorithm, which compensates the deficiency on vane control effectiveness by utilizing the thrust-vectored property of the ducted fan UAV. In the meanwhile, to handle the variation of the complex aerodynamic effects acting on the vehicle, we adopt an adaptive full envelope flight control scheme to compensate all of the unmodeled nonlinear dynamics. A sufficient condition is also derived to ensure the stability of the closed-loop system, by which the aircraft is capable of tracking a given velocity trajectory with bounded tracking error. Finally, flight tests are conducted with comparative experiments. The result is satisfactory in accomplishing the desired transition course and resisting the risk of flight failure in the presence of input saturation, verifying the effectiveness of the proposed approach.

Appendix

In prior, we introduce some frequently used notations: ${{\lambda }_{\min }}(\centerdot )$ denotes the minimum eigenvalue of a matrix; ${{\lambda }_{\max }}(\centerdot )$ denotes the maximum eigenvalue of a matrix; ${{\left\| \centerdot \right\| }_{F}}$ denotes the Frobenius norm.

1.1 Proof of Theorem 1

The core function of the inner-loop design is to compensate the unmodeled nonlinear aerodynamic moment $\varvec{M}_a^B$ such that $\varvec{e}_{t1}$ converges to a small neighborhood around zero. This can be proved by reconstructing the closed-loop system into the form of a perturbed system and applying the Lyapunov stability theory.

1.1.1 The auxiliary system

Define the tracking errors ${{\varvec{\tilde{\omega }}}^{B}}={{\varvec{\hat{\omega }}}^{B}}-{{\varvec{\omega }}^{B}}$ and $\varvec{\tilde{M}}_{a}^{B}=\varvec{\hat{M}}_{a}^{B}-\varvec{M}_{a}^{B}$ where ${{\varvec{\hat{\omega }}}^{B}}$ and $\varvec{\hat{M}}_{a}^{B}$ are pre-defined by (18). From (16)(17)(18) we obtain the following error dynamics:

$$\begin{aligned} \left\{ \begin{array}{ll} {{{\varvec{\dot{\tilde{\omega }}}}}^{B}}=&{}\ {{\varvec{I}}^{-1}}\varvec{\tilde{M}}_{a}^{B}-{{\varvec{K}}_{1}}{{{\varvec{\tilde{\omega }}}}^{B}} \\ \varvec{\dot{\tilde{M}}}_{a}^{B}=&{}\ -{{\varvec{K}}_{\varvec{M}}}{{{\varvec{\tilde{\omega }}}}^{B}}-\varvec{\dot{M}}_{a}^{B}\\ {{{\varvec{\dot{e}}}}_{t1}}=&{}\ -{{\varvec{K}}_{uv}}{{{\varvec{{e}}}}_{t1}}-{{\varvec{I}}^{-1}}\varvec{K}_{uv}\varvec{\tilde{M}}_{a}^{B}+{{\varvec{I}}^{-1}}\varvec{\dot{M}}_{a}^{B} \end{array} \right. . \end{aligned}$$

(53)

Let ${{\varvec{z}}_{1}}=\begin{array}{ccc} [{{{\varvec{{e}}}}_{t1}}^{T} &{} {{({{{\varvec{\tilde{\omega }}}}^{B}})}^{T}} &{} {{(\varvec{\tilde{M}}_{a}^{B})}^{T}} \\ \end{array}{{]}^{T}}\in {{\mathbb {R}}^{9}}$. We can reconstruct (53) into a perturbed system:

$$\begin{aligned} {{\varvec{\dot{z}}}_{1}} = \varvec{H} {{\varvec{z}}_{1}} + \varvec{B} \varvec{\dot{M}}_{a}^{B}, \end{aligned}$$

(54)

where

$$\begin{aligned} \varvec{H} = \left[ \begin{array}{ccc} -{{\varvec{K}}_{uv}} &{} 0 &{} -{{\varvec{I}}^{-1}}{{\varvec{K}}_{uv}} \\ 0 &{} -{{\varvec{K}}_{1}} &{} {{\varvec{I}}^{-1}} \\ 0 &{} -{{\varvec{K}}_{\varvec{M}}} &{} 0 \\ \end{array} \right] , \quad \varvec{B} = \left[ \begin{array}{c} {{\varvec{I}}^{-1}} \\ 0 \\ -{{I}_{3\times 3}} \\ \end{array} \right] . \end{aligned}$$

(55)

Let ${{\varvec{T}}_{1}}\triangleq \text {diag}({{\varvec{K}}_{uv}},{{\varvec{K}}_{1}},\varvec{I} ^{-2}{{\varvec{K}}_{\varvec{M}}})$, solving the Lyapunov equation $\varvec{H}^T \varvec{P}_1+\varvec{P}_1 \varvec{H}=-\varvec{T}_1$ yields:Let ${{\varvec{T}}_{1}}\triangleq \text {diag}({{\varvec{K}}_{uv}},{{\varvec{K}}_{1}},\varvec{I} ^{-2}{{\varvec{K}}_{\varvec{M}}})$, solving the Lyapunov equation $\varvec{H}^T \varvec{P}_1+\varvec{P}_1 \varvec{H}=-\varvec{T}_1$ yields:

$$\begin{aligned}&{{\varvec{P}}_{1}}\triangleq \frac{1}{2}\left[ \begin{array}{ccc} I_{3\times 3} &{} 0 &{} \varvec{p}1 \\ 0 &{} I_{3\times 3} &{} \varvec{p}2 \\ \varvec{p}1 &{} \varvec{p}2 &{} \varvec{I} ^{-2}\varvec{p}3 \\ \end{array} \right] ,\nonumber \\&\left\{ \begin{array}{ll} \varvec{p}1\triangleq &{}\ {{\varvec{K}}_{uv}}{{\left[ \left( {{\varvec{K}}_{uv}}\varvec{+}{{\varvec{K}}_{\varvec{M}}} \right) \right] }^{-1}}\\ \varvec{p}2\triangleq &{}\ {{\left[ \left( {{\varvec{K}}_{1}}\varvec{+}{{\varvec{K}}_{\varvec{M}}} \right) \right] }^{-1}}\\ \varvec{p}3\triangleq &{}\ \left[ {{\varvec{K}}_{uv}}\varvec{+}{{\varvec{K}}_{\varvec{M}}}+{{\varvec{K}}_{uv}}^{2}\left( {{\varvec{K}}_{1}}\varvec{+}{{\varvec{K}}_{\varvec{M}}} \right) \right] \\ &{}\ \centerdot {{\left[ \left( {{\varvec{K}}_{1}}\varvec{+}{{\varvec{K}}_{\varvec{M}}} \right) \left( {{\varvec{K}}_{uv}}\varvec{+}{{\varvec{K}}_{\varvec{M}}} \right) {{\varvec{K}}_{\varvec{M}}} \right] }^{-1}}\\ \end{array} \right. . \end{aligned}$$

(56)

Consider the following Lyapunov candidate:

$$\begin{aligned} L_1={{\varvec{z}}_{1}}^{T}{{\varvec{P}}_{1}}{{\varvec{z}}_{1}}. \end{aligned}$$

(57)

From (56) and taking Assumption 1, it follows that:

$$\begin{aligned} \dot{L}_1= & {} \ -{{\varvec{z}}_{1}}^{T}{{\varvec{T}}_{1}}{{\varvec{z}}_{1}}+{{\varvec{z}}_{1}}^{T}{{\varvec{P}}_{1}}\varvec{B}\varvec{\dot{M}}_a^B\nonumber \\\le & {} \ -b_1\left\| {{\varvec{z}}_{1}} \right\| _{2}^{2}+a_1d_m\left\| {{\varvec{z}}_{1}} \right\| _{2}, \end{aligned}$$

(58)

where

$$\begin{aligned} \left\{ \begin{array}{ll} &{}b_1\triangleq \min \{{{\lambda }_{\min }}({{\varvec{K}}_{uv}}),{{\lambda }_{\min }}({{\varvec{K}}_{1}}),{{\lambda }_{\min }}({\varvec{I} ^{-2}{\varvec{K}}_{M}})\}\\ &{}a_1\triangleq \lambda _{\max } (\varvec{P}_1)\sqrt{\lambda _{\max } (\varvec{I}^{-1})+1}\\ \end{array}\right. . \end{aligned}$$

(59)

1.1.2 The closed-loop system

From (1)(13)(14)(16), we can establish the following perturbed system:

$$\begin{aligned} \left\{ \begin{array}{ll} {{{\varvec{\dot{e}}}}_{\varvec{\omega }}}=&{}\ -{{\varvec{K}}_{\varvec{\omega }}}{{\varvec{e}}_{\varvec{\omega }}}-{{{\varvec{{e}}}}_{t1}}\\ {{{\varvec{\dot{e}}}}_{\varvec{\eta }}}=&{}\ -{{\varvec{K}}_{\varvec{\eta }}}{{\varvec{e}}_{\varvec{\eta }}}+\varvec{Q}{{\varvec{e}}_{\varvec{\omega }}} \\ \end{array} \right. , \end{aligned}$$

(60)

which satisfies:

$$\begin{aligned}&\left\{ \begin{array}{ll} {{\left\| {{{\varvec{\dot{e}}}}_{\varvec{\eta }}} \right\| }_{2}}\le &{}\ -{{\lambda }_{min}}({{\varvec{K}}_{\varvec{\eta }}}){{\left\| {{\varvec{e}}_{\varvec{\eta }}} \right\| }_{2}}+{{\gamma }_{1}}{{\left\| {{\varvec{e}}_{\varvec{\omega }}} \right\| }_{2}} \\ {{\left\| {{{\varvec{\dot{e}}}}_{\varvec{\omega }}} \right\| }_{2}}\le &{}\ -{{\lambda }_{min}}({{\varvec{K}}_{\varvec{\omega }}}){{\left\| {{\varvec{e}}_{\varvec{\omega }}} \right\| }_{2}}+{{\left\| {{\varvec{z}}_{1}} \right\| }_{2}}\nonumber \\ \end{array}\right. ,\\&{{\gamma }_{1}}\triangleq \underset{-{{\varphi }_{0}}\le \varphi \le {{\varphi }_{0}}}{\mathop {\sup }}\,\left( {{\left\| \varvec{Q} \right\| }_{F}} \right) =\sqrt{1+2/{{\cos }^{2}}({{\varphi }_{0}})}. \end{aligned}$$

(61)

For positive parameters $c_1,c_2>0$, consider the following Lyapunov candidates:

$$\begin{aligned} {{L}_{2}}=\frac{1}{2}c_1{{\varvec{e}}_{\varvec{\eta }}}^{T}{{\varvec{e}}_{\varvec{\eta }}}+\frac{1}{2}{{\varvec{e}}_{\varvec{\omega }}}^{T}{{\varvec{e}}_{\varvec{\omega }}}+c_2{{L}_{1}}. \end{aligned}$$

(62)

Substituting (58)(61) into (62) yields:

$$\begin{aligned} \dot{L}_2\le & {} \ -{{\varvec{z}}_{2}}^{T}{{\Lambda }_{1}}{{\varvec{z}}_{2}}+c_2a_1 \left\| {{\varvec{z}}_{1}} \right\| _{2}d_{m}\nonumber \\\le & {} \ -{{b}_{2}}\left\| {{\varvec{z}}_{2}} \right\| _{2}^{2}+c_2a_1 \left\| {{\varvec{z}}_{2}} \right\| _{2}d_{m},\nonumber \\\le & {} \ -(1-\epsilon )b_2\left\| {{\varvec{z}}_{2}} \right\| _{2}^{2}, \qquad \forall \left\| {{\varvec{z}}_{2}} \right\| _{2} \ge (\sqrt{2}c_2a_1d_{m})/(\epsilon b_2), \end{aligned}$$

(63)

where

$$\begin{aligned} {{\varvec{z}}_{2}}\triangleq \left[ \begin{array}{c} {{\left\| {{\varvec{e}}_{\varvec{\eta }}} \right\| }_{2}} \\ {{\left\| {{\varvec{e}}_{\varvec{\omega }}} \right\| }_{2}} \\ {{\left\| {{\varvec{z}}_{{1}}} \right\| }_{2}} \\ \end{array} \right] ,\ {{\Lambda }_{1}}\triangleq \left[ \begin{array}{ccc} c_1{{\lambda }_{\min }}({{\varvec{K}}_{\varvec{\eta }}}) &{} -\frac{c_1{{\gamma }_{1}}}{2} &{} 0 \\ -\frac{c_1{{\gamma }_{1}}}{2} &{} {{\lambda }_{\min }}({{\varvec{K}}_{\varvec{\omega }}}) &{} -\frac{1}{2} \\ 0 &{} -\frac{1}{2} &{} {{c}_{2}}{{b}_{1}} \\ \end{array} \right] ,\ {{b}_{2}}\triangleq {{\lambda }_{\min }}({{\Lambda }_{1}}). \end{aligned}$$

(64)

${{\dot{L}}_{2}}$ is negative definite when ${{\Lambda }_{1}}$ is positive definite and $(1-\epsilon )>0$. This condition can be achieved by choosing:

$$\begin{aligned} \left\{ \begin{array}{ll} {{c}_{1}}<4{{\lambda }_{m}}({{\varvec{K}}_{\varvec{\eta }}}){{\lambda }_{m}}({{\varvec{K}}_{\varvec{\omega }}})/\gamma _{1}^{2}\\ {{c}_{2}}>\frac{{{\lambda }_{m}}({{\varvec{K}}_{\varvec{\eta }}})}{{{b}_{1}}\left[ 4{{\lambda }_{m}}({{\varvec{K}}_{\varvec{\eta }}}){{\lambda }_{m}}({{\varvec{K}}_{\varvec{\omega }}})-{{c}_{1}}\gamma _{1}^{2} \right] }\\ \end{array}\right. . \end{aligned}$$

(65)

And we can obtain the following:

$$\begin{aligned}&a_2\left\| {{\varvec{z}}_{2}} \right\| _{2}^{2} \le L_2 \le a_3\left\| {{\varvec{z}}_{2}} \right\| _{2}^{2},\nonumber \\&\left\{ \begin{array}{ll} &{} {{a}_{2}}\triangleq \frac{1}{2}\min \{1, c_1,c_2{{\lambda }_{m}}({{\varvec{P}}_{1}})\} \\ &{} {{a}_{3}}\triangleq \frac{1}{2}\max \{1, c_1,c_2a_1\}\\ \end{array}\right. . \end{aligned}$$

(66)

Let $\varepsilon _1=\sqrt{a_3/a_2},\ \varepsilon _2=(1-\epsilon )b_2/(2a_3)$, and $\mu =(\sqrt{2}c_2a_1d_{m1})/(\epsilon b_2)$. To achieve both $\mu <(r/\varepsilon _1)$ and $(1-\epsilon )>0$, we can choose:

$$\begin{aligned} (\sqrt{\frac{2a_3}{a_2}}a_1c_2d_{m})({rb_2})< \epsilon <1. \end{aligned}$$

(67)

Condition (67) can be easily achieved by some proper controller parameters $\varvec{K_{\eta }},{{\varvec{K}}_{\varvec{\omega }}},\varvec{K_M},\varvec{K}_1,\varvec{K}_{uv}$. For every initial condition that satisfies $\left\| \varvec{z}_2(t_0)\right\| _2\le (r/\varepsilon _1)$. According to Theorem 4.18 in [29], we have:

$$\begin{aligned}&\left\| \varvec{z}_2\right\| _2 \le \varepsilon _1 \left\| \varvec{z}_2(t_0)\right\| _2 e^{-\varepsilon _2(t-t_0)},\quad \forall t_0\le t\le t_0+T,\nonumber \\&\left\| \varvec{z}_2\right\| _2 \le r,\quad\quad\quad\quad\quad\quad\quad\quad\quad\; \forall t\ge t_0+T. \end{aligned}$$

(68)

This completes the proof.

1.2 Proof of Theorem 2

1.2.1 Property 1

First, we derive an important property which will be used later. Note that,

$$\begin{aligned} \varvec{R}_{B}^{I}({{\varvec{\eta }}_{d}})-\varvec{R}_{B}^{I}(\varvec{\eta })=\varvec{R}_{B}^{I}({{\varvec{\eta }}_{d}})\left[ {{{I}}_{3\times 3}}-{{\left( \varvec{R}_{B}^{I} \right) }^{T}}({{\varvec{e}}_{\varvec{\eta }}}) \right] . \end{aligned}$$

(69)

From (2) and ${{\varvec{e}}_{\varvec{\eta }}}={{\left[ \begin{array}{ccc} {{\varphi }_{e}} &{} {{\theta }_{e}} &{} {{\psi }_{e}} \\ \end{array} \right] }^{T}}$, we have:

$$\begin{aligned}&\ {{\left\| {{\varvec{I}}_{3\times 3}}-{{\left( \varvec{R}_{B}^{I} \right) }^{T}}({{\varvec{e}}_{\varvec{\eta }}}) \right\| }_{F}} \nonumber \\= & {} \ \sqrt{6-2\left( C_{{\psi }_{e}}C_{{\theta }_{e}}-S_{{\psi }_{e}}S_{{\varphi }_{e}}S_{{\theta }_{e}}+C_{{\psi }_{e}}C_{{\varphi }_{e}}+C_{{\varphi }_{e}}C_{{\theta }_{e}} \right) }. \end{aligned}$$

(70)

Applying $C_{{\alpha }_{1}}C_{{\alpha }_{2}}=[1-2{{S}^{2}}_({{\alpha }_{1}}/2)][1-2{{S}^{2}}_({{\alpha }_{2}}/2)]$, we can transform (70) into:

$$\begin{aligned}&\ \sqrt{6-2\left( C_{{\psi }_{e}}C_{{\theta }_{e}}-S_{{\psi }_{e}}S_{{\varphi }_{e}}S_{{\theta }_{e}}+C_{{\psi }_{e}}C_{{\varphi }_{e}}+C_{{\varphi }_{e}}C_{{\theta }_{e}} \right) } \nonumber \\\le & {} \ \sqrt{8{{S}^{2}}_{\frac{{\psi }_{e}}{2}}+8{{S}^{2}}_{\frac{{\theta }_{e}}{2}}+8{{S}^{2}}_{\frac{{\varphi }_{e}}{2}}+2S_{{\psi }_{e}}S_{{\varphi }_{e}}S_{{\theta }_{e}}} \nonumber \\\le & {} \ \sqrt{2{{\psi }_{e}}^{2}+2{{\theta }_{e}}^{2}+2{{\varphi }_{e}}^{2}+2\left| S{{\psi }_{e}}S{{\varphi }_{e}}S{{\theta }_{e}} \right| }. \end{aligned}$$

(71)

Since $0\le \left| S_{{\psi }_{e}}S_{{\varphi }_{e}}S_{{\theta }_{e}} \right| \le 1$, we have:

$$\begin{aligned} \left| S_{{\psi }_{e}}S_{{\varphi }_{e}}S_{{\theta }_{e}} \right| \le \frac{{{S}^{2}}_{{\psi }_{e}}+{{S}^{2}}_{{\varphi }_{e}}+{{S}^{2}}_{{\theta }_{e}}}{3}\le \frac{1}{3}\left( {{\psi }_{e}}^{2}+{{\varphi }_{e}}^{2}+{{\theta }_{e}}^{2} \right) . \end{aligned}$$

(72)

1.2.2 The closed-loop system

From (6), define

$$\begin{aligned} {{e}_{t2}}=A_{z}^{B}+{{u}_{Tn}}={{m}^{-1}}F_{az}^{B}-\left[ 1+{{k}_{Tv}}(\varvec{v}_{a}^{B}) \right] {{u}_{Ta}}. \end{aligned}$$

(73)

And from (13)(19)(20)(21)(73), we can establish the following perturbed system:

$$\begin{aligned} {{{\varvec{\dot{e}}}}_{\varvec{v}}}= & {} {{f}_{3}}({{\varvec{e}}_{\varvec{v}}},\varvec{e_\eta },e_{t2}) \nonumber \\= & {} -{{\varvec{K}}_{\varvec{v}}}{{\varvec{e}}_{\varvec{v}}}+\left[ \varvec{R}_{B}^{I}({{\varvec{\eta }}_{d}})-\varvec{R}_{B}^{I}(\varvec{\eta }) \right] \left[ \begin{array}{c} A_{x}^{B} \\ A_{y}^{B} \\ -{{u}_{Tn}} \\ \end{array} \right] -\varvec{R}_{B}^{I}(\varvec{\eta })\left[ \begin{array}{c} 0 \\ 0 \\ {{e}_{t2}} \\ \end{array} \right] . \end{aligned}$$

(74)

From (20)(21)(22)(74) and Property 1, we can obtain:

$$\begin{aligned}&\ {{\left\| {{f}_{3}}({{\varvec{e}}_{\varvec{v}}},\varvec{e_\eta },e_{t2}) -{{f}_{3}}({{\varvec{e}}_{\varvec{v}}},0,0) \right\| }_{2}} \nonumber \\\le & {} \ {{\left\| {{{I}}_{3\times 3}}-{{\left( \varvec{R}_{B}^{I} \right) }^{T}}({{\varvec{e}}_{\varvec{\eta }}}) \right\| }_{F}}{{\left\| {{\left[ \begin{array}{ccc} A_{x}^{B} &{} A_{y}^{B} &{} -{{u}_{Tn}} \\ \end{array} \right] }^{T}} \right\| }_{2}}+{{\left\| {{e}_{t2}} \right\| }_{2}} \nonumber \\\le & {} \ h_2{{\left\| {{\varvec{e}}_{\varvec{\eta }}} \right\| }_{2}}\left( {{\lambda }_{\max }}\left( {{\varvec{K}}_{\varvec{v}}} \right) {{\left\| {{\varvec{e}}_{\varvec{v}}} \right\| }_{2}}+{{h}_{1}} \right) +{{\left\| {{e}_{t2}} \right\| }_{2}}, \end{aligned}$$

(75)

where

$$\begin{aligned} {{d}_{1}}\triangleq {{\left\| \varvec{\dot{v}}_{d}^{I} \right\| }_{2}}+g, \quad h2\triangleq \sqrt{8/3} \end{aligned}$$

(76)

For positive parameters $c_3,c_4>0$, consider the following Lyapunov candidate:

$$\begin{aligned} {{L}_{3}}=\frac{1}{2}{{\varvec{e}}_{\varvec{v}}}^{T}{{\varvec{e}}_{\varvec{v}}}+c_3{{L}_{2}}+\frac{1}{2}c_4{{e}_{t2}}^{2}. \end{aligned}$$

(77)

The time derivative of $L_3$ is derived by:

$$\begin{aligned} {{{\dot{L}}}_{3}}= & {} \ \frac{\partial {{L}_{3}}}{\partial {{\varvec{e}}_{\varvec{v}}}}{{f}_{3}}({{\varvec{e}}_{\varvec{v}}},0,0)+\frac{\partial {{L}_{3}}}{\partial {{\varvec{e}}_{\varvec{v}}}}\left[ {{f}_{3}}({{\varvec{e}}_{\varvec{v}}},\varvec{e_\eta },e_{t2})-{{f}_{3}}({{\varvec{e}}_{\varvec{v}}},0,0) \right] \nonumber \\&\ +c_3{{{\dot{L}}}_{2}}+c_4{{e}_{t2}}{{{\dot{e}}}_{t2}}. \end{aligned}$$

(78)

In particular, substituting (24)(63)(73)(74)(75)(76) into (78) and considering that the velocity tracking error has an upper bound (${{\left\| {{\varvec{e}}_{\varvec{v}}} \right\| }_{2}}\le {{{e}}_{{vM} }}$), we can obtain:

$$\begin{aligned} {{{\dot{L}}}_{3}}\le & {} \ -{{\lambda }_{\min }}({{\varvec{K}}_{\varvec{v}}})\left\| {{\varvec{e}}_{\varvec{v}}} \right\| _{2}^{2}-c_3(1-\epsilon ){{b}_{2}}\left\| {{\varvec{z}}_{2}} \right\| _{2}^{2}-c_4{{K}_{uT}}{{e}_{t2}}^{2} \nonumber \\&\ +{{\left\| {{\varvec{e}}_{\varvec{v}}} \right\| }_{2}}{{e}_{t2}}+{{h}_{1}}{{h}_{2}}{{\left\| {{\varvec{e}}_{\varvec{\eta }}} \right\| }_{2}}{{\left\| {{\varvec{e}}_{\varvec{v}}} \right\| }_{2}} +{{h}_{2}}{{e}_{vM }}{{\lambda }_{\max }}\left( {{\varvec{K}}_{\varvec{v}}} \right) {{\left\| {{\varvec{e}}_{\varvec{\eta }}} \right\| }_{2}}{{\left\| {{\varvec{e}}_{\varvec{v}}} \right\| }_{2}} \nonumber \\\le & {} \ -{{\varvec{z}}_{3}}^{T}{{\Lambda }_{1}}{{\varvec{z}}_{3}}\nonumber \\\le & {} \ -b_3\left\| {{\varvec{z}}_{3}} \right\| _{2}^{2},\nonumber \\\forall&\ \left\| {{\varvec{z}}_{2}} \right\| _{2} \ge (\sqrt{2}c_2a_1d_{m1})/(\epsilon b_2),\quad 0<\epsilon <1, \end{aligned}$$

(79)

where

$$\begin{aligned} {{\varvec{z}}_{3}}\triangleq & {} \ \left[ \begin{array}{c} {{\left\| {{\varvec{e}}_{\varvec{v}}} \right\| }_{2}} \\ {{\left\| {{\varvec{z}}_{2}} \right\| }_{2}} \\ {{e}_{t2}} \\ \end{array} \right] ,\ {{\Lambda }_{2}}\triangleq \left[ \begin{array}{ccc} {{\lambda }_{\min }}({{\varvec{K}}_{\varvec{v}}}) &{} -\frac{{{h}_{3}}}{2} &{} -\frac{1}{2} \\ -\frac{{{h}_{3}}}{2} &{} {{c}_{3}}(1-\epsilon ){{b}_{2}} &{} 0 \\ -\frac{1}{2} &{} 0 &{} {{c}_{4}}{{K}_{uT}} \\ \end{array} \right] , \nonumber \\ {{h}_{3}}\triangleq & {} \ {{h}_{1}}{{h}_{2}}+{{h}_{2}}{{e}_{vM}}{{\lambda }_{\max }}\left( {{\varvec{K}}_{\varvec{v}}} \right) , \nonumber \\ {{b}_{3}}\triangleq & {} \ {{\lambda }_{\min }}({{\Lambda }_{2}}). \end{aligned}$$

(80)

${{\dot{L}}_{3}}$ is negative definite when ${{\Lambda }_{2}}$ is positive definite. This condition can be achieved by choosing:

$$\begin{aligned} \left\{ \begin{array}{ll} &{} {{c}_{3}}>\frac{h_{3}^{2}}{{4b_3}{{\lambda }_{\min }}({{\varvec{K}}_{\varvec{v}}})}\\ &{} {{c}_{4}}>\frac{{{c}_{3}}b_3}{{{K}_{uT}}\left( 4{{c}_{3}}b_3{{\lambda }_{\min }}({{\varvec{K}}_{\varvec{v}}})-h_{3}^{2} \right) }\\ \end{array}\right. . \end{aligned}$$

(81)

And we can obtain the following:

$$\begin{aligned}&a_4\left\| {{\varvec{z}}_{3}} \right\| _{2}^{2} \le L_4 \le a_5\left\| {{\varvec{z}}_{3}} \right\| _{2}^{2},\nonumber \\&\left\{ \begin{array}{ll} &{} {{a}_{4}}\triangleq \frac{1}{2}\min \{1, c_3a_2,c_4\} \\ &{} {{a}_{5}}\triangleq \frac{1}{2}\max \{1, c_3a_3,c_4\}\\ \end{array}\right. . \end{aligned}$$

(82)

Let $\varepsilon _3=\sqrt{a_5/a_4},\ \varepsilon _4=b_3/(2a_5)$. Analogously to that in the proof of Theorem 1, we can have:

$$\begin{aligned}&\left\| \varvec{z}_3\right\| _2 \le \varepsilon _3 \left\| \varvec{z}_3(t_0)\right\| _2 e^{-\varepsilon _4(t-t_0)},\ \forall t_0\le t\le t_0+T,\nonumber \\&\left\| \varvec{z}_3\right\| _2 \le r,\ \forall t\ge t_0+T. \end{aligned}$$

(83)

This completes the proof.