Dynamic Modeling and Vibration Control for a Nonlinear Three-Dimensional Flexible Manipulator

Abstract

In the previous chapters, modeling and vibration control of the flexible mechanical systems are restricted to one dimensional space, and only transverse deformation is taken into account. However, flexible systems may move in a three-dimensional (3D) space in practical applications. The control performance will be affected if the coupling effects between motions in three directions are ignored. In spatial and industrial environment, flexible manipulators have been widely used due to their advantages such as light weight, fast motion and low energy consumption [3, 7]. For dynamic analysis, the flexible manipulator system is regarded as a distributed parameter system (DPS) which is mathematically represented by partial differential equations (PDEs) and ordinary differential equations (ODEs) [2, 5, 8], however, these works are only considered in one dimensional space. To improve accuracy and reliability analysis, modeling and control of the flexible manipulator system in a 3D space is necessary. Therefore, several works have been done in dynamics modeling and control design when the coupling effect are taken into account.

Appendices

Appendix 1: Proof of the Theorem

Consider the Lyapunov candidate function as

$$\begin{aligned} V={{V}_{1}}+{{V}_{2}}+{{V}_{3}}+{{V}_{4}}+{{V}_{o}}\end{aligned}$$

(9.43)

where

$$\begin{aligned} {{V}_{1}}&=\frac{1}{2}\beta {{k}_{2}}{{\tilde{\varvec{\omega }}}^{T}}{{\mathbf {I}}_{h}}\tilde{\varvec{\omega }}+\frac{1}{2}\beta {{k}_{2}}{{\tilde{\mathbf {q}}}^{T}}{{\mathbf {k}}_{p}}\tilde{\mathbf {q}} +\frac{1}{2}\beta {{k}_{2}}\rho \int _{\varOmega }{{{\left[ \phi \frac{d\tilde{\mathbf {r}}}{dt}\right] }^{T}}\left[ \phi \frac{d\tilde{\mathbf {r}}}{dt}\right] dx}\end{aligned}$$

(9.44)

$$\begin{aligned} {{V}_{2}}&=\frac{1}{2}\beta {{k}_{2}}EI\int _{\varOmega }{\left[ {{\left( {{y}_{xx}}\right) }^{2}}+{{\left( {{z}_{xx}}\right) }^{2}}\right] dx} +\frac{1}{2}\beta {{k}_{2}}T\int _{\varOmega }{\left[ {{\left( {{y}_{x}}\right) }^{2}}+{{\left( {{z}_{x}}\right) }^{2}}\right] dx}\end{aligned}$$

(9.45)

$$\begin{aligned} {{V}_{3}}&=\frac{1}{2}{{m}}\mathbf {u}_{0}^{T}{{\mathbf {u}}_{0}}\end{aligned}$$

(9.46)

$$\begin{aligned} {{V}_{4}}&=\alpha \rho \int _{\varOmega }{x{{\left[ \phi \frac{d\tilde{\mathbf {r}}}{dt}\right] }^{T}}\left[ \phi \mathbf {{d}}_{x}\right] dx}+\alpha {{\tilde{\mathbf {q}}}^{T}}{{\mathbf {I}}_{h}}\tilde{\varvec{\omega }}+\alpha \rho \int _{\varOmega }{x{{\left[ {{\phi }_{1}}\frac{d\tilde{\mathbf {r}}}{dt}\right] }^{T}}\left[ {{\phi }_{2}}\tilde{\mathbf {q}}\right] dx}\end{aligned}$$

(9.47)

$$\begin{aligned} {{V}_{o}}&=\frac{1}{2}\tilde{\mathbf {d}}_{\tau }^{T}\tilde{\mathbf {d}}_{\tau }^{{}}+\frac{1}{2}\tilde{\mathbf {d}}_{u}^{T}\tilde{\mathbf {d}}_{u}^{{}}\end{aligned}$$

(9.48)

where the estimate errors are $\tilde{\mathbf {d}}_{\tau }^{{}}=\mathbf {d }_{\tau }^{{}}-\hat{\mathbf {d}}_{\tau }^{{}}$ and $\tilde{\mathbf {d}}_{u}=\mathbf {{d}}_{u}-\hat{\mathbf {d}}_{u}$, and $\tilde{\mathbf {r}}=\mathbf {r}$, ${d\tilde{\mathbf {r}}}/{dt}\;=\mathbf {{d}}_{t}+\tilde{\varvec{\omega }}\times \tilde{\mathbf {r}}$, and ${{\phi }_{1}}=\left[ \begin{matrix} 0 &{} 1 &{} 0\\ 0 &{} 0 &{} -1 \end{matrix} \right]$, ${{\phi }_{2}}=\left[ \begin{matrix} 0 &{} 0 &{} 1\\ 0 &{} 1 &{} 0 \end{matrix} \right]$.

To motivate the followings, we first focus our attention on Eq. (9.47) and we can obtain

$$\begin{aligned} \left| {{V}_{4}}\right|&\le \alpha \rho L\int _{\varOmega }{{{\left[ \phi \frac{d\tilde{\mathbf {r}}}{dt}\right] }^{T}}\left[ \phi \frac{d\tilde{\mathbf {r}}}{dt}\right] dx}+\frac{1}{2}\alpha \rho L\int _{\varOmega }{{{\left[ \phi \mathbf {{d}}_{x}\right] }^{T}}\left[ \phi \mathbf {{d}}_{x}\right] dx}\nonumber \\&+\frac{1}{2}\alpha \rho {{L}^{2}}{{\left[ {{\phi }_{2}}\tilde{\mathbf {q}}\right] }^{T}}\left[ {{\phi }_{2}}\tilde{\mathbf {q}}\right] +\frac{1}{2}\alpha {{{\tilde{\mathbf {q}}}}^{T}}{{\mathbf {I}}_{\mathbf {h}}}\tilde{\mathbf {q}}+\frac{1}{2}\alpha {{{\tilde{\varvec{\omega }}}}^{T}}{{\mathbf {I}}_{\mathbf {h}}}\tilde{\varvec{\omega }}\nonumber \\&\le {{\alpha }_{1}}\left( {{V}_{1}}+{{V}_{2}}\right) \end{aligned}$$

(9.49)

where ${{\alpha }_{1}}=\max \left[ \frac{2\alpha \rho L}{{{k}_{2}}\beta \rho },\frac{\alpha \rho L}{{{k}_{2}}\beta T},\frac{\alpha {{I}_{h1}}}{{{k}_{2}}\beta {{k}_{p1}}},\frac{\alpha \rho {{L}^{2}}+\alpha {{I}_{hi}}}{{{k}_{2}}\beta {{k}_{pi}}},\frac{\alpha }{{{k}_{2}}\beta }\right] ,i=2,3$.

Then, we have

$$\begin{aligned} -{{\alpha }_{1}}\left( {{V}_{1}}+{{V}_{2}} \right) \le {{V}_{4}}\le {{\alpha }_{1}}\left( {{V}_{1}}+{{V}_{2}} \right) \end{aligned}$$

Choosing $\beta$, ${{k}_{2}}$ and ${{\mathbf {k}}_{p}}$ to satisfy the inequality $0<{{\alpha }_{1}}<1$ yields ${{\alpha }_{2}}=1-{{\alpha }_{1}}>0$, ${{\alpha }_{3}}=1\text {+}{{\alpha }_{1}}>1$.

Then we further have

$$\begin{aligned} 0\le {{\alpha }_{2}}\left( {{V}_{1}}+{{V}_{2}} \right) \le {{V}_{1}}+{{V}_{2}}+{{V}_{4}}\le {{\alpha }_{3}}\left( {{V}_{1}}+{{V}_{2}} \right) \end{aligned}$$

Given the Lyapunov function in (9.43), we obtain

$$\begin{aligned} 0\le {{\alpha }_{2}}\left( {{V}_{1}}+{{V}_{2}}+{{V}_{3}}+{{V}_{o}}\right) \le V\le {{\alpha }_{3}}\left( {{V}_{1}}+{{V}_{2}}+{{V}_{3}}+{{V}_{o}}\right) \end{aligned}$$

(9.50)

Differentiating (9.43) with respect to time, we have

$$\begin{aligned} \dot{V}={{\dot{V}}_{1}}+{{\dot{V}}_{2}}+{{\dot{V}}_{3}}+{{\dot{V}}_{4}}+{{\dot{V}}_{o}}\end{aligned}$$

(9.51)

The term ${{\dot{V}}_{1}}$ is rewritten as

$$\begin{aligned} {{\dot{V}}_{1}}={{\dot{V}}_{11}}+{{\dot{V}}_{12}}+{{\dot{V}}_{13}}\end{aligned}$$

(9.52)

where

$$\begin{aligned} {{\dot{V}}_{11}}={{k}_{2}}\beta {{\tilde{\varvec{\omega }}}^{T}}{{\mathbf {I}}_{h}}{\dot{\tilde{\varvec{\omega }}}}+{{k}_{2}}\beta {{\tilde{\varvec{\omega }}}^{T}}\left( \tilde{\varvec{\omega }}\times \left( {{\mathbf {I}}_{h}}\tilde{\varvec{\omega }}\right) \right) \end{aligned}$$

(9.53)

$$\begin{aligned} {{\dot{V}}_{12}}={{k}_{2}}\beta \rho \int _{\varOmega }{{{\left[ \phi \frac{d\tilde{\mathbf {r}}}{dt}\right] }^{T}}\left[ \phi \frac{{{d}^{2}}\tilde{\mathbf {r}}}{d{{t}^{2}}}\right] dx}\end{aligned}$$

(9.54)

$$\begin{aligned} {{\dot{V}}_{13}}={{k}_{2}}\beta {{\tilde{\mathbf {q}}}^{T}}{{\mathbf {k}}_{p}}{\dot{\tilde{\mathbf{q}}}}\end{aligned}$$

(9.55)

Considering (9.15) and (9.30), we have

$$\begin{aligned} {{\dot{V}}_{11}}={{k}_{2}}\beta {{\tilde{\varvec{\omega }}}^{T}}\left( \varvec{\tau }+{{\mathbf {d}}_{\tau }}-M-N\right) \end{aligned}$$

(9.56)

and

$$\begin{aligned} {{{\dot{V}}}_{12}}&={{k}_{2}}\beta \rho \int _{\varOmega }{{{\left[ \phi \frac{d\tilde{\mathbf {r}}}{dt}\right] }^{T}}\left[ \phi \frac{{{d}^{2}}\tilde{\mathbf {r}}}{d{{t}^{2}}}\right] dx}\nonumber \\&={{k}_{2}}\beta \rho \int _{\varOmega }{{{\left[ \phi \mathbf {{d}}_{t}\right] }^{T}}\left[ \phi \left( \tilde{\varpi }+\mathbf {{d}}_{tt}\right) \right] dx}\nonumber +{{k}_{2}}\beta \rho \int _{\varOmega }{{{\left[ \phi \left( \tilde{\varvec{\omega }}\times \tilde{\mathbf {r}}\right) \right] }^{T}}\left[ \phi \left( \tilde{\varpi }+\mathbf {{d}}_{tt}\right) \right] dx}\nonumber \\&={{k}_{2}}\beta \rho \int _{\varOmega }{{{\left[ \phi \mathbf {{d}}_{t}\right] }^{T}}\left[ \phi \left( \tilde{\varpi }+\mathbf {{d}}_{tt}\right) \right] dx} +{{k}_{2}}\beta \rho \int _{\varOmega }{{{{\tilde{\varvec{\omega }}}}^{T}}\left( \tilde{\mathbf {r}}\times \left[ {{\phi }^{T}}\phi \left( \tilde{\varpi }+\mathbf {{d}}_{tt}\right) \right] \right) dx}\end{aligned}$$

(9.57)

Since

$$\begin{aligned} \rho \int _{\varOmega }{{{\left[ \phi \mathbf {{d}}_{t}\right] }^{T}}\left[ \phi \left( \tilde{\varpi }+\mathbf {{d}}_{tt}\right) \right] dx}&=-\int _{\varOmega }{{y}_{t}\left( EI{{y}_{xxxx}}-T{y}_{xx}\right) dx}\nonumber \\&-\int _{\varOmega }{{z}_{t}\left( EI{{z}_{xxxx}}-T{z}_{xx}\right) dx}\end{aligned}$$

(9.58)

and

$$\begin{aligned} \int _{\varOmega }{\rho \tilde{\mathbf {r}}\times \left[ {{\phi }^{T}}\phi \left( \tilde{\varpi }+\mathbf {{d}}_{tt}\right) \right] dx}&=T\tilde{\mathbf {r}}\times {{\mathbf {{d}}}_{x}(L,t)}-EI\tilde{\mathbf {r}}\times {{\mathbf {{d}}}_{xxx}(L,t)}+M+N \end{aligned}$$

(9.59)

we have

$$\begin{aligned} {{{\dot{V}}}_{12}}&={{k}_{2}}\beta {{{\tilde{\varvec{\omega }}}}^{T}}\left( M+N-EI\tilde{\mathbf {r}}\times {{{\mathbf {{d}}}}_{xxx}(L,t)}+T\tilde{\mathbf {r}}\times {{{\mathbf {{d}}}}_{x}(L,t)}\right) \nonumber \\&-{{k}_{2}}\beta \int _{\varOmega }{{z}_{t}\left( EI{{z}_{xxxx}}-T{z}_{xx}\right) dx} -{{k}_{2}}\beta \int _{\varOmega }{{y}_{t}\left( EI{{y}_{xxxx}}-T{y}_{xx}\right) dx}\end{aligned}$$

(9.60)

Substituting (9.28) into (9.55)

$$\begin{aligned} {{{\dot{V}}}_{13}}&={{k}_{2}}\beta {{{\tilde{\mathbf {q}}}}^{T}}{{\mathbf {k}}_{p}}{\dot{\tilde{\mathbf{q}}}}={{k}_{2}}\beta {{{\tilde{\mathbf {q}}}}^{T}}{{\mathbf {k}}_{p}}\mathbf {G}\tilde{\varvec{\omega }}\nonumber \\&={{k}_{2}}\beta {{{\tilde{\varvec{\omega }}}}^{T}}\left( {{\mathbf {G}}^{T}}{{\mathbf {k}}_{p}}\tilde{\mathbf {q}}\right) \end{aligned}$$

(9.61)

Then, we have

$$\begin{aligned} {{{\dot{V}}}_{1}}&={{{\dot{V}}}_{11}}+{{{\dot{V}}}_{12}}+{{{\dot{V}}}_{13}}\nonumber \\&={{k}_{2}}\beta {{{\tilde{\varvec{\omega }}}}^{T}}\left( \varvec{\tau }+{{\mathbf {d}}_{\tau }}-EI\tilde{\mathbf {r}}\times {{{\mathbf {{d}}}}_{xxx}(L,t)}+T\tilde{\mathbf {r}}\times {{{\mathbf {{d}}}}_{x}(L,t)}+{{\mathbf {G}}^{T}}{{\mathbf {k}}_{p}}\tilde{\mathbf {q}}\right) \nonumber \\&-{{k}_{2}}\beta \int _{\varOmega }{{y}_{t}\left( EI{{y}_{xxxx}}-T{y}_{xx}\right) dx} -{{k}_{2}}\beta \int _{\varOmega }{{z}_{t}\left( EI{{z}_{xxxx}}-T{z}_{xx}\right) dx}\end{aligned}$$

(9.62)

Considering the term ${{\dot{V}}_{2}}$, we have

$$\begin{aligned} {{\dot{V}}_{2}}={{k}_{2}}\beta EI\int _{\varOmega }{\left[ {y}_{xx}{{y}}_{xxt}+{z}_{xx}{{z}}_{xxt}\right] dx}+{{k}_{2}}\beta T\int _{\varOmega }{\left[ {y}_{x}{{y}}_{xt}+{z}_{x}{{z}}_{xt}\right] dx}\end{aligned}$$

(9.63)

Integrating (9.63) by parts with the boundary conditions, we obtain

$$\begin{aligned} {{{\dot{V}}}_{2}}&={{k}_{2}}\beta EI{y}_{xx}\left. { {{y}}_{xt}}\right| _{0}^{L}-{{k}_{2}}\beta EI{y}_{xxx}\left. {{y}_{t}}\right| _{0}^{L}+{{k}_{2}}\beta EI\int _{\varOmega }{{{y}_{xxxx}}{y}_{t}dx}\nonumber \\&+{{k}_{2}}\beta EI{z}_{xx}\left. {{{z}}_{x}}\right| _{0}^{L}-{{k}_{1}}\beta EI{z}_{xxx}\left. {{z}_{t}}\right| _{0}^{L}+{{k}_{2}}\beta EI\int _{\varOmega }{{{z}_{xxxx}}{z}_{t}dx}\nonumber \\&+{{k}_{2}}\beta T{y}_{x}\left. {{y}_{t}}\right| _{0}^{L}-{{k}_{2}}\beta T\int _{\varOmega }{{y}_{xx}{y}_{t}dx} +{{k}_{1}}\beta T{z}_{x}\left. {{z}_{t}}\right| _{0}^{L}-{{k}_{2}}\beta T\int _{\varOmega }{{z}_{xx}{z}_{t}dx}\nonumber \\&={{k}_{2}}\beta EI\int _{\varOmega }{{{y}_{xxxx}} {y}_{t}dx}+{{k}_{2}}\beta EI\int _{\varOmega }{{{z}_{xxxx}}{z}_{t}dx} -{{k}_{2}}\beta T\int _{\varOmega }{{y}_{xx}{y}_{t}dx} \nonumber \\&-{{k}_{2}}\beta T\int _{\varOmega }{{z}_{xx}{z}_{t}dx}-{{k}_{2}}\beta EI{{{{y}}}_{xxx}(L,t)}{{{{y}}}_{t}(L,t)}\nonumber \\&-{{k}_{2}}\beta EI{{{{z}}}_{xxx}(L,t)}{{{{z}}}_{t}(L,t)} +{{k}_{2}}\beta T{{{{y}}}_{x}(L,t)}{{{{y}}}_{t}(L,t)}+{{k}_{2}}\beta T{{{{z}_{x}}}(L,t)}{{{{z}}}_{t}(L,t)}\end{aligned}$$

(9.64)

Then we have

$$\begin{aligned} {{{\dot{V}}}_{1}}+{{{\dot{V}}}_{2}}&={{k}_{2}}\beta {{{\tilde{\varvec{\omega }}}}^{T}}\left( \varvec{\tau }+{{\mathbf {d}}_{\tau }}-EI\tilde{\mathbf {r}}\times {{{\mathbf {{d}}}}_{xxx}(L,t)}+T\tilde{\mathbf {r}}\times {{{\mathbf {{d}}}}_{x}(L,t)}\right) \nonumber \\&+{{k}_{2}}\beta {{{\tilde{\varvec{\omega }}}}^{T}}\left( {{\mathbf {G}}^{T}}{{\mathbf {k}}_{p}}\tilde{\mathbf {q}}\right) -{{k}_{2}}\beta EI{{{{y}}}_{xxx}(L,t)}{{{{y}_{t}}}(L,t)}-{{k}_{2}}\beta EI{{{{z}}}_{xxx}(L,t)}{{{{z}_{t}}}(L,t)}\nonumber \\&+{{k}_{2}}\beta T{{{{y}_{x}}}(L,t)}{{{{y}_{t}}}(L,t)}+{{k}_{2}}\beta T{{{{z}_{x}}}(L,t)}{{{{z}_{t}}}(L,t)}\nonumber \\&={{k}_{2}}\beta {{{\tilde{\varvec{\omega }}}}^{T}}\left( \varvec{\tau }+{{\mathbf {d}}_{\tau }}+{{\mathbf {G}}^{T}}{{\mathbf {k}}_{p}}\tilde{\mathbf {q}}\right) -{{k}_{2}}\beta EI{{{\mathbf {{d}}_{t}}}(L,t)}^{T}{{{\mathbf {{d}}}_{xxx}}(L,t)}\nonumber \\&-{{k}_{2}}\beta {{{\tilde{\varvec{\omega }}}}^{T}}\left( EI\tilde{\mathbf {r}}\times {{{\mathbf {{d}}}_{xxx}}(L,t)}-T\tilde{\mathbf {r}}\times {{{\mathbf {{d}}}_{x}}(L,t)}\right) +{{k}_{2}}\beta T{{{\mathbf {{d}}}_{t}}(L,t)}^{T}{{{\mathbf {{d}}}_{x}}(L,t)}\end{aligned}$$

(9.65)

By calculating, we get

$$\begin{aligned}&{{k}_{2}}T\beta {{{\tilde{\varvec{\omega }}}}^{T}}\left( \tilde{\mathbf {r}}\times {{{\mathbf {{d}}}_{x}}(L,t)}\right) -{{k}_{2}}EI\beta {{{\tilde{\varvec{\omega }}}}^{T}}\left( \tilde{\mathbf {r}}\times {{{\mathbf {{d}}}_{xxx}}(L,t)}\right) \nonumber \\&-{{k}_{2}}\beta EI{{{\mathbf {{d}}}}_{t}^{T}}{{{\mathbf {{d}}}_{xxx}}(L,t)} +{{k}_{2}}\beta T{{{\mathbf {{d}}}}_{t}^{T}}{{{\mathbf {{d}}}_{x}}(L,t)}\nonumber \\&=-{{k}_{2}}EI\beta {{\left[ \tilde{\varvec{\omega }}\times \tilde{\mathbf {r}}\right] }^{T}}{{{\mathbf {{d}}}_{xxx}}(L,t)}+{{k}_{2}}T\beta {{\left[ \tilde{\varvec{\omega }}\times \tilde{\mathbf {r}}\right] }^{T}}{{{\mathbf {{d}}}_{x}}(L,t)}\nonumber \\&-{{k}_{2}}\beta EI{{{\mathbf {{d}}_{t}}}^{T}}{{{\mathbf {{d}}}_{xxx}}(L,t)} +{{k}_{2}}\beta T{{{\mathbf {{d}}_{t}}}^{T}}{{{\mathbf {{d}}}_{x}}(L,t)}\nonumber \\&=-{{k}_{2}}\beta EI{{\left[ \frac{d{{{\tilde{\mathbf {r}}}}(L,t)}}{dt}\right] }^{T}}{{{\mathbf {{d}}}_{xxx}}(L,t)}+{{k}_{2}}\beta T{{\left[ \frac{d{{{\tilde{\mathbf {r}}}}(L,t)}}{dt}\right] }^{T}}{{{\mathbf {{d}}}_{x}}(L,t)}\end{aligned}$$

(9.66)

Then (9.65) is rewritten as

$$\begin{aligned} {{\dot{V}}_{1}}+{{\dot{V}}_{2}}&={{k}_{2}}\beta {{\tilde{\varvec{\omega }}}^{T}}\left( \varvec{\tau }+\mathbf {d}_{\tau }+{{\mathbf {G}}^{T}}{{\mathbf {k}}_{p}}\tilde{\mathbf {q}}\right) \nonumber \\&-{{k}_{2}}\beta EI{{\left[ \ \frac{d{{{\tilde{\mathbf {r}}}}(L,t)}}{dt}\ \right] }^{T}}\mathbf {d}_{xxx}{{(L,t)}}+{{k}_{2}}\beta T{{\left[ \ \frac{d{{{\tilde{\mathbf {r}}}}(L,t)}}{dt}\ \right] }^{T}}{\mathbf {d}_{x}}{{(L,t)}\ }\end{aligned}$$

(9.67)

Combining (9.36) and substituting (9.32) into (9.67) yields

$$\begin{aligned} {{{\dot{V}}}_{1}}+{{{\dot{V}}}_{2}}&=-{{k}_{2}}\beta {{{\tilde{\varvec{\omega }}}}^{T}}{{\mathbf {k}}_{d}}\tilde{\varvec{\omega }}-{{k}_{f}}{{k}_{2}}\beta {{{\tilde{\varvec{\omega }}}}^{T}}{{\mathbf {G}}^{T}}\mathbf {G}\tilde{\varvec{\omega }}\nonumber \\&-\frac{1}{2}\beta EIk_{1}^{2}{{\left[ {{{\mathbf {{d}}}_{x}}(L,t)}\right] }^{T}}{{\phi }^{T}}\phi {{{\mathbf {{d}}}_{x}}(L,t)}-{{k}_{2}}\beta {{{\tilde{\varvec{\omega }}}}^{T}}{{\mathbf {k}}_{q}}\tilde{\mathbf {q}}\nonumber \\&-\frac{1}{2}\beta EI{{\left[ \frac{d{{{\tilde{\mathbf {r}}}}(L,t)}}{dt}\right] }^{T}}{{\phi }^{T}}\phi \frac{d{{{\tilde{\mathbf {r}}}}(L,t)}}{dt}+\frac{1}{2}\beta EI\mathbf {u}_{0}^{T}{{\mathbf {u}}_{0}}\nonumber \\&+\beta EI{{k}_{1}}{{k}_{2}} {{\left[ {{{\mathbf {{d}}}_{x}}(L,t)}\right] }^{T}}{{\phi }^{T}}\phi {{{\mathbf {{d}}}_{xxx}}(L,t)}+{{k}_{2}}\beta {{{\tilde{\varvec{\omega }}}}^{T}}{{{\tilde{\mathbf {d}}}}_{\tau }}\nonumber \\&+\left( {{k}_{2}}\beta T-\beta EI{{k}_{1}}\right) {{\left[ \frac{d{{{\tilde{\mathbf {r}}}}(L,t)}}{dt}\right] }^{T}}{{\phi }^{T}}\phi {{{\mathbf {{d}}}_{x}}(L,t)}\nonumber \\&-\frac{1}{2}\beta EIk_{2}^{2}{{\left[ {{{\mathbf {{d}}}_{xxx}}(L,t)}\right] }^{T}}{{\phi }^{T}}\phi {{{\mathbf {{d}}}_{xxx}}(L,t)}\end{aligned}$$

(9.68)

According to Lemma 2.4, we have

$$\begin{aligned}&{{{\dot{V}}}_{1}}+{{{\dot{V}}}_{2}}\le -{{k}_{2}}\beta {{{\tilde{\varvec{\omega }}}}^{T}}{{\mathbf {k}}_{d}}\tilde{\varvec{\omega }}-{{k}_{f}}{{k}_{2}}\beta {{{\tilde{\varvec{\omega }}}}^{T}}{{\mathbf {G}}^{T}}\mathbf {G}\tilde{\varvec{\omega }}\nonumber \\&+\frac{1}{2}\beta EI\mathbf {u}_{0}^{T}{{\mathbf {u}}_{0}}+\frac{1}{2{{\sigma }_{3}}}{{k}_{2}}\beta {{{\tilde{\varvec{\omega }}}}^{T}}\tilde{\varvec{\omega }}+\frac{1}{2}{{\sigma }_{3}}{{k}_{2}}\beta {{{\tilde{\mathbf {d}}}}_{\tau }}^{T}{{{\tilde{\mathbf {d}}}}_{\tau }}\nonumber \\&+\frac{1}{2{{\sigma }_{2}}}\sqrt{{{k}_{2}}}\beta {{{\tilde{\varvec{\omega }}}}^{T}}{{\mathbf {k}}_{q}}\tilde{\varvec{\omega }}+\frac{1}{2}{{\sigma }_{2}}\sqrt{{{k}_{2}}}\beta {{{\tilde{\mathbf {q}}}}^{T}}{{\mathbf {k}}_{q}}\tilde{\mathbf {q}}\nonumber \\&-\left( \frac{1}{2}\beta EI-\frac{1}{2{{\sigma }_{0}}}\left| {{k}_{2}}\beta T-{{k}_{1}}\beta EI\right| \right) {{\left[ \frac{d{{{\tilde{\mathbf {r}}}}(L,t)}}{dt}\right] }^{T}}{{\phi }^{T}}\phi \frac{d{{{\tilde{\mathbf {r}}}}(L,t)}}{dt}\nonumber \\ {}&-\left( \frac{1}{2}\beta EIk_{1}^{2}-\frac{1}{2}{{\sigma }_{0}}\left| {{k}_{2}}\beta T-{{k}_{1}}\beta EI\right| \right) {{\left[ {{{\mathbf {{d}}}_{x}}(L,t)}\right] }^{T}}{{\phi }^{T}}\phi {{{\mathbf {{d}}}_{x}}(L,t)}\nonumber \\ {}&-\frac{1}{2}\beta EIk_{2}^{2}{{\left[ {{{\mathbf {{d}}}_{xxx}}(L,t)}\right] }^{T}}{{\phi }^{T}}\phi {{{\mathbf {{d}}}_{xxx}}(L,t)}+\beta EI{{k}_{1}}{{k}_{2}}{{\left[ {{{\mathbf {{d}}}_{x}}(L,t)}\right] }^{T}}{{\phi }^{T}}\phi {{{\mathbf {{d}}}_{xxx}}(L,t)}\end{aligned}$$

(9.69)

Since

$$\begin{aligned}&\frac{d{{\mathbf {u}}_{0}}}{dt}=\phi \left[ 2\tilde{\varvec{\omega }}\times {{{\mathbf {{d}}}_{t}}(L,t)}+{\dot{\tilde{\varvec{\omega }}}}\times {{{\tilde{\mathbf {r}}}}(L,t)}\mathbf {+} \tilde{\varvec{\omega }}\times \left( \tilde{\varvec{\omega }}\times {{{\tilde{\mathbf {r}}}}(L,t)}\right) + {{{\mathbf {{d}}}_{tt}}(L,t)}\right] \nonumber \\&+\phi \left[ {{k}_{1}} {{{\mathbf {{{d}}}}_{xt}}(L,t)}+{{k}_{1}}\tilde{\varvec{\omega }}\times {{{\mathbf {{d}}}_{x}}(L,t)}-{{k}_{2}} {{{\mathbf {{{d}}}}_{xxxt}}(L,t)}-{{k}_{2}}\tilde{\varvec{\omega }}\times {{{\mathbf {{d}}}_{xxx}}(L,t)}\right] \nonumber \\&=\phi \left[ {{{\tilde{\varpi }}}(L,t)}+ {{{\mathbf {{d}}}_{tt}}(L,t)}+{{k}_{1}} {{{\mathbf {{{d}}}}_{xt}}(L,t)}+{{k}_{1}}\tilde{\varvec{\omega }}\times {{{\mathbf {{d}}}_{x}}(L,t)}\right] \nonumber \\&-\phi \left[ {{k}_{2}} {{{\mathbf {{{d}}}}_{xxxt}}(L,t)}+{{k}_{2}}\tilde{\varvec{\omega }}\times {{{\mathbf {{d}}}_{xxx}}(L,t)}\right] \end{aligned}$$

(9.70)

we obtain

$$\begin{aligned} {{{\dot{V}}}_{3}}&={{m}}\mathbf {u}_{0}^{T}\frac{d{{\mathbf {u}}_{0}}}{dt} =\mathbf {u}_{0}^{T}\left( \mathbf {u}+{{\mathbf {d}}_{u}}+EI\phi {{{\mathbf {{d}}}_{xxx}}(L,t)}-T\phi {{{\mathbf {{d}}}_{x}}(L,t)}\right) \nonumber \\&+{{m}}\mathbf {u}_{0}^{T}\phi \left( {{k}_{1}} {{{\mathbf {{d}}}_{xt}}(L,t)}+{{k}_{1}}\tilde{\varvec{\omega }}\times {{{\mathbf {{d}}}_{x}}(L,t)}-{{k}_{2}} {{{\mathbf {{d}}}_{xxxt}}(L,t)}-{{k}_{2}}\tilde{\varvec{\omega }}\times {{{\mathbf {{d}}}_{xxx}}(L,t)}\right) \end{aligned}$$

(9.71)

Substituting (9.33) into (9.71), we have

$$\begin{aligned} {{{\dot{V}}}_{3}}&=-\mathbf {u}_{0}^{T}{{\mathbf {k}}_{u}}{{\mathbf {u}}_{0}}+\mathbf {u}_{0}^{T}{{{\tilde{\mathbf {d}}}}_{u}}\nonumber \\&\le -\mathbf {u}_{0}^{T}{{\mathbf {k}}_{u}}{{\mathbf {u}}_{0}}+\frac{1}{2}{{\sigma }_{6}}\mathbf {u}_{0}^{T}{{\mathbf {u}}_{0}}+\frac{1}{2{{\sigma }_{6}}}\tilde{\mathbf {d}}_{u}^{T}{{{\tilde{\mathbf {d}}}}_{u}}\end{aligned}$$

(9.72)

To go on, the term ${{\dot{V}}_{4}}$ can be written as

$$\begin{aligned} {{\dot{V}}_{4}}={{B}_{1}}+{{B}_{2}}+{{B}_{3}}+{{B}_{4}}+{{B}_{5}}+{{B}_{6}}\end{aligned}$$

(9.73)

where

$$\begin{aligned} {{B}_{1}}=\alpha \rho \int _{\varOmega }{x{{\phi }^{T}}{{\left[ \tilde{\varpi }+\mathbf {{d}}_{tt}\right] }^{T}}\left[ \phi \mathbf {{d}}_{x}\right] dx}\end{aligned}$$

(9.74)

$$\begin{aligned} {{B}_{2}}=\alpha \rho \int _{\varOmega }{x{{\left[ \phi \frac{d\tilde{\mathbf {r}}}{dt}\right] }^{T}}\left[ \phi \frac{d\mathbf {{d}}_{x}}{dt}\right] dx}\end{aligned}$$

(9.75)

$$\begin{aligned} {{B}_{3}}=\alpha \rho \int _{\varOmega }{x\phi _{1}^{T}{{\left[ \tilde{\varpi }\text {+} \mathbf {{d}}_{tt}\right] }^{T}}\left[ {{\phi }_{2}}\tilde{\mathbf {q}}\right] dx}\end{aligned}$$

(9.76)

$$\begin{aligned} {{B}_{4}}=\alpha \rho \int _{\varOmega }{x{{\left[ {{\phi }_{1}}\frac{d\tilde{\mathbf {r}}}{dt}\right] }^{T}}\left[ {{\phi }_{2}} {\dot{\tilde{\mathbf{q}}}}\right] dx}\end{aligned}$$

(9.77)

$$\begin{aligned} {{B}_{5}}=\alpha {{\tilde{\varvec{\omega }}}^{T}}{{\mathbf {I}}_{\mathbf {h}}}{\dot{\tilde{\mathbf{q}}}}\end{aligned}$$

(9.78)

$$\begin{aligned} {{B}_{6}}=\alpha {{\tilde{\mathbf {q}}}^{T}}\left( {{\mathbf {I}}_{\mathbf {h}}}{\dot{\tilde{\varvec{\omega }}}}+\tilde{\varvec{\omega }}\times \left( {{\mathbf {I}}_{\mathbf {h}}}\tilde{\varvec{\omega }}\right) \right) \end{aligned}$$

(9.79)

Using boundary conditions and integrating by parts, we obtain

$$\begin{aligned} {{{\dot{V}}}_{4}}&={{B}_{1}}+{{B}_{2}}+{{B}_{3}}+{{B}_{4}}+{{B}_{5}}+{{B}_{6}}\nonumber \\&\le -\alpha EIL{{{{y}_{x}}}(L,t)} {{{{y}_{xxx}}}(L,t)}-\alpha EIL {{{{z}_{x}}}(L,t)} {{{{z}_{xxx}}}(L,t)}+\frac{1}{2}\alpha TL{{\left( {{{{z}_{x}}}(L,t)}\right) }^{2}}+\frac{\alpha }{2{{\sigma }_{5}}}EIL{{\left( {{{{y}_{xxx}}}(L,t)}\right) }^{2}}\nonumber \\ {}&+\frac{\alpha }{2{{\sigma }_{5}}}EIL{{\left( {{{{z}_{xxx}}}(L,t)}\right) }^{2}} +\frac{\alpha }{2{{\sigma }_{7}}}TL{{\left( {{{{y}_{x}}}(L,t)}\right) }^{2}}+\frac{1}{2}\alpha TL{{\left( {{{{y}_{x}}}(L,t)}\right) }^{2}}\nonumber \\ {}&+\frac{\alpha }{2{{\sigma }_{7}}}TL{{\left( {{{{z}_{x}}}(L,t)}\right) }^{2}} -\frac{3}{2}\alpha EI\int _{\varOmega }{{{\left( {{y}_{xx}}\right) }^{2}}dx}-\frac{3}{2}\alpha EI\int _{\varOmega }{{{\left( {{z}_{xx}}\right) }^{2}}dx}\nonumber \\ {}&-\frac{1}{2}\alpha T\int _{\varOmega }{{{\left( {{y}_{x}}\right) }^{2}}dx}-\frac{1}{2}\alpha T\int _{\varOmega }{{{\left( {{z}_{x}}\right) }^{2}}dx}-\left( \frac{1}{2}\alpha \rho -{{\sigma }_{4}}\frac{\alpha \rho L}{2}\right) \int _{\varOmega }{{{\left[ \phi \frac{d\tilde{\mathbf {r}}}{dt}\right] }^{T}}\left[ \phi \frac{d\tilde{\mathbf {r}}}{dt}\right] dx}\nonumber \\&+\frac{\alpha }{2}{{\sigma }_{8}}\rho L\int _{\varOmega }{{{\left[ \phi \frac{d\tilde{\mathbf {r}}}{dt}\right] }^{T}}\left[ \phi \frac{d\tilde{\mathbf {r}}}{dt}\right] dx} +\frac{1}{2}\alpha L{{\left[ \phi \frac{d{{{\tilde{\mathbf {r}}}}(L,t)}}{dt}\right] }^{T}}\left[ \phi \frac{d{{{\tilde{\mathbf {r}}}}(L,t)}}{dt}\right] +\frac{1}{2}{{\sigma }_{9}}\alpha {{{\tilde{\varvec{\omega }}}}^{T}}{{\mathbf {I}}_{\mathbf {h}}}\tilde{\varvec{\omega }}\nonumber \\&+\frac{\alpha }{2{{\sigma }_{10}}}{{{\tilde{\varvec{\omega }}}}^{T}}{{\mathbf {k}}_{d}}\tilde{\varvec{\omega }}+\frac{\alpha \rho }{2{{\sigma }_{4}}}{{L}^{2}}\left( \tilde{\omega }_{2}^{2}+\tilde{\omega }_{3}^{2}\right) \mathbf {+}\frac{\alpha }{2{{\sigma }_{11}}}{{{\tilde{\mathbf {q}}}}^{T}}\tilde{\mathbf {q}} -\alpha {{{\tilde{\mathbf {q}}}}^{T}}{{\mathbf {k}}_{q}}\tilde{\mathbf {q}}-\frac{1}{2}\alpha \tilde{\eta }{{{\tilde{\mathbf {q}}}}^{T}}{{\mathbf {k}}_{p}}\tilde{\mathbf {q}+}\frac{\alpha }{2}{{\sigma }_{10}}{{{\tilde{\mathbf {q}}}}^{T}}{{\mathbf {k}}_{d}}\tilde{\mathbf {q}}\nonumber \\&+\frac{\alpha }{4}{{k}_{f}}{{{\tilde{\eta }}}^{2}}{{{\tilde{\mathbf {q}}}}^{T}}\tilde{\mathbf {q}+}\frac{\alpha }{2}{{\sigma }_{5}}EIL\tilde{q}_{3}^{2}+\frac{\alpha }{2}{{\sigma }_{7}}TL\tilde{q}_{3}^{2} +\frac{\alpha }{2}{{\sigma }_{7}}TL\tilde{q}_{2}^{2}+\frac{\alpha }{2}{{\sigma }_{5}}EIL\tilde{q}_{2}^{2}\mathbf {+}\frac{\alpha }{2}{{k}_{f}}{{\left[ \mathbf {G}\tilde{\varvec{\omega }}\right] }^{T}}\mathbf {G}\tilde{\varvec{\omega }}\nonumber \\&+\frac{1}{2{{\sigma }_{9}}}\alpha {{\left[ G\tilde{\varvec{\omega }}\right] }^{T}}{{\mathbf {I}}_{\mathbf {h}}}\left[ G\tilde{\varvec{\omega }}\right] \mathbf {+}\frac{\alpha }{2}{{\sigma }_{11}}{{{\tilde{\mathbf {d}}}}_{\tau }}^{T}{{{\tilde{\mathbf {d}}}}_{\tau }} +\frac{\alpha }{2{{\sigma }_{8}}}\rho L\left( {{\left[ G\tilde{\varvec{\omega }}\right] }^{T}}\phi _{2}^{T}{{\phi }_{2}}\left[ G\tilde{\varvec{\omega }}\right] \right) \end{aligned}$$

(9.80)

The last term of (9.51) is

$$\begin{aligned} {{{\dot{V}}}_{o}}&=\tilde{\mathbf {d}}_{\tau }^{T}{\dot{\tilde{\mathbf{d}}}}_{\tau }^{{}}+\tilde{\mathbf {d}}_{u}^{T}{\dot{\tilde{\mathbf{d}}}}_{u}^{{}} =-\tilde{\mathbf {d}}_{\tau }^{T}{\dot{\hat{\mathbf {d}}}}_{\tau }^{{}}-\tilde{\mathbf {d}}_{u}^{T}{\dot{\hat{\mathbf {d}}}}_{u}^{{}} +\tilde{\mathbf {d}}_{\tau }^{T}\dot{\mathbf {{d}}}_{\tau }^{{}}+\tilde{\mathbf {d}}_{u}^{T}{\dot{\mathbf {d}}}_{u}^{{}} \nonumber \\&\le -\tilde{\mathbf {d}}_{\tau }^{T}\left( {{{\dot{\mathbf {z}}}}_{\mathbf {1}}}+{{K}_{1}}\left( {{\mathbf {I}}_{\mathbf {h}}}\dot{\varvec{\omega }}+\omega \times \left( {{\mathbf {I}}_{\mathbf {h}}}\varvec{\omega }\right) \right) \right) -\tilde{\mathbf {d}}_{u}^{T}\left( {{{\dot{\mathbf {z}}}}_{\mathbf {2}}}+{{K}_{2}}{{m}}\phi \left[ \frac{{{d}^{2}}{{\mathbf {r}}_{L}}}{d{{t}^{2}}}\right] \right) +{{\sigma }_{12}}\tilde{\mathbf {d }}_{\tau }^{T}\tilde{\mathbf {d }}_{\tau }^{{}}\nonumber \\&+{{\sigma }_{13}}\tilde{\mathbf {d}}_{u}^{T}\tilde{\mathbf {d}}_{u}^{{}}+\frac{1}{{{\sigma }_{12}}}\dot{\mathbf {d }}_{\tau }^{T}\dot{\mathbf {d }}_{\tau }^{{}}+\frac{1}{{{\sigma }_{13}}}\dot{\mathbf {d}}_{u}^{T}\dot{\mathbf {d}}_{u}^{{}}\nonumber \\&=-\tilde{\mathbf {d}}_{\tau }^{T}\left( {{K}_{1}}-{{\sigma }_{12}}{{\mathbf {I}}_{3}} \right) \tilde{\mathbf {d}}_{\tau }^{{}}-\tilde{\mathbf {d}}_{u}^{T}\left( {{K}_{2}}-{{\sigma }_{13}}{{\mathbf {I}}_{3}} \right) \tilde{\mathbf {d}}_{u}^{{}} +\frac{1}{{{\sigma }_{12}}}\dot{\mathbf {d }}_{\tau }^{T}\dot{\mathbf {d }}_{\tau }^{{}}+\frac{1}{{{\sigma }_{13}}}\dot{\mathbf {d}}_{u}^{T}\dot{\mathbf {d}}_{u}^{{}} \end{aligned}$$

(9.81)

Applying (9.69), (9.72), (9.80) and (9.81), we have

$$\begin{aligned} \dot{V}&\le -\left( \frac{1}{2}\beta EIk_{2}^{2}-\frac{\alpha }{2{{\sigma }_{5}}}EIL-\frac{1}{2{{\sigma }_{1}}}\left| {{k}_{1}}{{k}_{2}}\beta EI-\alpha EIL\right| \right) {{\left[ \phi {{{\mathbf {{d}}}_{xxx}}(L,t)}\right] }^{T}}\left[ \phi {{{\mathbf {{d}}}_{xxx}}(L,t)}\right] \nonumber \\&-\left( \frac{1}{2}\beta EIk_{1}^{2}+\frac{\alpha EI}{L}-\frac{1}{2}\alpha TL-\frac{1}{2}{{\sigma }_{0}}\left| {{k}_{2}}\beta T-{{k}_{1}}\beta EI\right| -\frac{\alpha }{2{{\sigma }_{7}}}TL\right) {{\left[ \phi {{{\mathbf {{d}}}_{x}}(L,t)}\right] }^{T}}\left[ \phi {{{\mathbf {{d}}}_{x}}(L,t)}\right] \nonumber \\&-\left( -\frac{1}{2}{{\sigma }_{1}}\left| {{k}_{1}}{{k}_{2}}\beta EI-\alpha EIL\right| \right) {{\left[ \phi {{{\mathbf {{d}}}_{x}}(L,t)}\right] }^{T}}\left[ \phi {{{\mathbf {{d}}}_{x}}(L,t)}\right] -\frac{1}{2}\alpha EI\int _{\varOmega }{{{\left( {{y}_{xx}}\right) }^{2}}dx}-\frac{1}{2}\alpha EI\int _{\varOmega }{{{\left( {{z}_{xx}}\right) }^{2}}dx}\nonumber \\&-\frac{1}{2}\alpha T\int _{\varOmega }{{{\left( {{y}_{x}}\right) }^{2}}dx}-\frac{1}{2}\alpha \rho \left( 1-{{\sigma }_{4}}L-{{\sigma }_{8}}L\right) \int _{\varOmega }{{{\left[ \phi \frac{d\tilde{\mathbf {r}}}{dt}\right] }^{T}}\left[ \phi \frac{d\tilde{\mathbf {r}}}{dt}\right] dx}\nonumber \\&-\frac{1}{2}\left( \beta EI-\alpha L-\frac{1}{2{{\sigma }_{0}}}\left| {{k}_{2}}\beta T-{{k}_{1}}\beta EI\right| \right) {{\left[ \frac{d{{{\tilde{\mathbf {r}}}}(L,t)}}{dt}\right] }^{T}}{{\phi }^{T}}\phi \frac{d{{{\tilde{\mathbf {r}}}}(L,t)}}{dt}-\frac{1}{2}\alpha T\int _{\varOmega }{{{\left( {{z}_{x}}\right) }^{2}}dx}\nonumber \\&-{{{\tilde{\mathbf {q}}}}^{T}}\left( \alpha {{\mathbf {k}}_{q}}+\frac{1}{2}\alpha \tilde{\eta }{{\mathbf {k}}_{p}}-\frac{\alpha }{2}{{\sigma }_{10}}{{\mathbf {k}}_{d}}\right) \tilde{\mathbf {q}}+{{{\tilde{\mathbf {q}}}}^{T}}\left( \frac{\alpha }{4}{{k}_{f}}{{{\tilde{\eta }}}^{2}}{{\mathbf {I}}_{3}}+\frac{1}{2}{{\sigma }_{2}}\sqrt{{{k}_{2}}}\beta {{\mathbf {k}}_{q}}\mathbf {+}\frac{\alpha }{2{{\sigma }_{11}}}{{\mathbf {I}}_{3}}\right) \tilde{\mathbf {q}}\nonumber \\&+{{{\tilde{\mathbf {q}}}}^{T}}\left( \frac{\alpha }{2}{{\sigma }_{5}}EIL\phi ^{T}\phi +\frac{\alpha }{2}{{\sigma }_{7}}TL\phi ^{T}\phi \right) \tilde{\mathbf {q}}-\frac{1}{2}{{{\tilde{\varvec{\omega }}}}^{T}}\left( 2{{k}_{2}}\beta {{\mathbf {k}}_{d}}+\frac{1}{{{\sigma }_{3}}}{{k}_{2}}\beta {{\mathbf {I}}_{3}}-\frac{\alpha \rho }{{{\sigma }_{4}}}{{L}^{2}}\phi ^{T}\phi \right) \tilde{\varvec{\omega }}\nonumber \\&+\frac{1}{2}{{{\tilde{\varvec{\omega }}}}^{T}}\left( \frac{\alpha }{{{\sigma }_{10}}}{{\mathbf {k}}_{d}}+2{{\sigma }_{9}}\alpha {{\mathbf {I}}_{\mathbf {h}}}+\frac{1}{{{\sigma }_{2}}}\sqrt{{{k}_{2}}}\beta {{\mathbf {k}}_{q}}\right) \tilde{\varvec{\omega }}-{{\left[ G\tilde{\varvec{\omega }}\right] }^{T}}\left( {{k}_{2}}{{k}_{f}}\beta {{\mathbf {I}}_{3}}-\frac{\alpha }{2{{\sigma }_{8}}}\rho L\phi _{2}^{T}{{\phi }_{2}}\right) \left[ G\tilde{\varvec{\omega }}\right] \nonumber \\&-{{\left[ G\tilde{\varvec{\omega }}\right] }^{T}}\left( -\frac{1}{2{{\sigma }_{9}}}\alpha {{\mathbf {I}}_{\mathbf {h}}}-\frac{\alpha }{2}{{k}_{f}}{{\mathbf {I}}_{3}}\right) \left[ G\tilde{\varvec{\omega }}\right] -\mathbf {u}_{0}^{T}\left( {{\mathbf {k}}_{u}}-\frac{1}{2}\beta EI{{\mathbf {I}}_{2}}-\frac{1}{2}{{\sigma }_{6}}{{\mathbf {I}}_{2}}\right) {{\mathbf {u}}_{0}}\nonumber \\&-\tilde{\mathbf {d}}_{\tau }^{T}\left( {{K}_{1}}-\frac{\alpha }{2}{{\sigma }_{11}}{{\mathbf {I}}_{3}}-\frac{1}{2}{{\sigma }_{3}}{{k}_{2}}\beta {{\mathbf {I}}_{3}}-{{\sigma }_{12}}{{\mathbf {I}}_{3}}\right) \tilde{\mathbf {d}}_{\tau }-\tilde{\mathbf {d}}_{u}^{T}\left( {{K}_{2}}-\frac{1}{2{{\sigma }_{6}}}{{\mathbf {I}}_{2}}-{{\sigma }_{13}}{{\mathbf {I}}_{2}}\right) \tilde{\mathbf {d}}_{u}+\varepsilon \end{aligned}$$

(9.82)

where $\varepsilon =\frac{1}{{{\sigma }_{12}}}(\bar{d}_{\tau x}^{2}+\bar{d}_{\tau y}^{2}+\bar{d}_{\tau z}^{2})\text {+}\frac{1}{{{\sigma }_{13}}}(\bar{d}_{uy}^{2}+\bar{d}_{uz}^{2})$.

We design parameters ${{k}_{1}}$ ${{k}_{2}}$, ${{\mathbf {k}}_{p}}$, ${{\mathbf {k}}_{d}}$, ${{\mathbf {k}}_{q}}$, ${{\mathbf {k}}_{u}}$, ${{K}_{1}}$, ${{K}_{2}}$, $\alpha$, $\beta$ and ${{\sigma }_{n}}\left( n=0...13\right)$ to satisfy the following conditions:

$$\begin{aligned}&\beta EIk_{1}^{2}+\frac{2\alpha EI}{L}-{{\sigma }_{0}}\left| {{k}_{2}}\beta T-{{k}_{1}}\beta EI\right| -\alpha TL\nonumber \\&-\frac{\alpha }{{{\sigma }_{7}}}TL-{{\sigma }_{1}}\left| {{k}_{1}}{{k}_{2}}\beta EI-\alpha EIL\right| \ge 0 \end{aligned}$$

(9.83)

$$\begin{aligned} \beta EIk_{2}^{2}-\frac{\alpha }{{{\sigma }_{5}}}EIL-\frac{1}{{{\sigma }_{1}}}\left| {{k}_{1}}{{k}_{2}}\beta EI-\alpha EIL\right| \ge 0 \end{aligned}$$

(9.84)

$$\begin{aligned} \beta EI-\frac{1}{{{\sigma }_{0}}}\left| {{k}_{2}}\beta T-{{k}_{1}}\beta EI\right| -\alpha L\ge 0 \end{aligned}$$

(9.85)

$$\begin{aligned} {{k}_{2}}{{k}_{f}}\beta -\frac{1}{2{{\sigma }_{9}}}\alpha {{I}_{h1}}-\frac{\alpha }{2}{{k}_{f}}\ge 0 \end{aligned}$$

(9.86)

$$\begin{aligned} {{k}_{2}}{{k}_{f}}\beta -\frac{\alpha }{2{{\sigma }_{8}}}\rho L-\frac{1}{2{{\sigma }_{9}}}\alpha {{I}_{hi}}-\frac{\alpha }{2}{{k}_{f}}\ge 0\end{aligned}$$

(9.87)

$$\begin{aligned}&{{\delta }_{1}}=1-{{\sigma }_{8}}L-{{\sigma }_{4}}L>0\end{aligned}$$

(9.88)

$$\begin{aligned}&{{\delta }_{2}}=2\alpha {{k}_{q1}}+\alpha {{{\tilde{\eta }}}_{0}}{{k}_{p1}}-\alpha {{\sigma }_{10}}{{k}_{d1}}-\frac{1}{2}\alpha {{k}_{f}}{{{\tilde{\eta }}}^{2}}\nonumber \\&\text { }-{{\sigma }_{2}}\sqrt{{{k}_{2}}}\beta {{k}_{q1}}-\frac{\alpha }{2{{\sigma }_{11}}}>0\end{aligned}$$

(9.89)

$$\begin{aligned}&{{\delta }_{i+1}}=2\alpha {{k}_{qi}}+\alpha {{{\tilde{\eta }}}_{0}}{{k}_{pi}}-\alpha {{\sigma }_{10}}{{k}_{di}}-\frac{1}{2}\alpha {{k}_{f}}{{{\tilde{\eta }}}^{2}}\nonumber \\&\text { }-{{\sigma }_{2}}\sqrt{{{k}_{2}}}\beta {{k}_{qi}}-\alpha {{\sigma }_{7}}TL-\alpha {{\sigma }_{5}}EIL-\frac{\alpha }{2{{\sigma }_{11}}}>0\end{aligned}$$

(9.90)

$$\begin{aligned}&{{\delta }_{5}}=2{{k}_{2}}\beta {{k}_{d1}}+\frac{1}{{{\sigma }_{3}}}{{k}_{2}}\beta -\frac{\alpha }{{{\sigma }_{10}}}{{k}_{d1}}\nonumber \\&-2{{\sigma }_{9}}\alpha {{I}_{h1}}-\frac{1}{{{\sigma }_{2}}}\sqrt{{{k}_{2}}}\beta {{k}_{q1}}>0\end{aligned}$$

(9.91)

$$\begin{aligned}&{{\delta }_{i+4}}=2{{k}_{2}}\beta {{k}_{di}}+\frac{1}{{{\sigma }_{3}}}{{k}_{2}}\beta -\frac{\alpha \rho }{{{\sigma }_{4}}}{{L}^{2}}-\frac{\alpha }{{{\sigma }_{10}}}{{k}_{di}}\nonumber \\&\text { }-2{{\sigma }_{9}}\alpha {{I}_{hi}}-\frac{1}{{{\sigma }_{2}}}\sqrt{{{k}_{2}}}\beta {{k}_{qi}}>0\end{aligned}$$

(9.92)

$$\begin{aligned}&{{\delta }_{j+7}}={{k}_{uj}}-\frac{1}{2}\beta T-\frac{1}{2}{{\sigma }_{6}}>0\end{aligned}$$

(9.93)

$$\begin{aligned}&{{\delta }_{j+9}}={{K}_{2j}}-\frac{1}{2{{\sigma }_{6}}}-{{\sigma }_{13}}>0\end{aligned}$$

(9.94)

$$\begin{aligned}&{{\delta }_{m+11}}={{K}_{1m}}-\frac{\alpha }{2}{{\sigma }_{11}}-\frac{1}{2}{{\sigma }_{3}}{{k}_{2}}\beta -{{\sigma }_{12}}>0 \end{aligned}$$

(9.95)

where $m=1,2,3$, $j=1,2,$ $i=2,3$.

We can obtain

$$\begin{aligned} \dot{V}\le -{{\lambda }_{1}}\left( {{V}_{1}}+{{V}_{2}}+{{V}_{3}}+{{V}_{o}}\right) + \varepsilon \end{aligned}$$

(9.96)

where ${{\lambda }_{1}}=\min \left( \frac{\alpha {{\delta }_{1}}}{\beta {{k}_{1}}},\frac{2{{\delta }_{i+1}}}{\beta {{k}_{1}}{{k}_{pi}}},\frac{{{\delta }_{i+4}}}{\beta {{k}_{1}}{{I}_{hi}}},\frac{\alpha }{\beta {{k}_{1}}},\frac{2{{\delta }_{j+7}}}{{{m}}},2{{\delta }_{j+9}},2{{\delta }_{m+11}}\right)$

Combining (9.50) and (9.96), we have

$$\begin{aligned} \dot{V}\le -\lambda V + \varepsilon \end{aligned}$$

(9.97)

where $\lambda ={{{\lambda }_{1}}}/{{{\alpha }_{3}}}>0$.

Then, multiply Eq. (9.97), by ${{e}^{\lambda t}}$, we obtain

$$\begin{aligned} \frac{d}{dt}\left( \left( V(t){{e}^{\lambda t}} \right) \right) \le \varepsilon {{e}^{\lambda t}}\end{aligned}$$

(9.98)

Integration of the above inequalities, we obtain

$$\begin{aligned} V(t)\le V(0){{e}^{-\lambda t}}+\frac{\varepsilon }{\lambda }\left( 1-{{e}^{-\lambda t}} \right) \le V(0){{e}^{-\lambda t}}+\frac{\varepsilon }{\lambda } \end{aligned}$$

(9.99)

We can conclude that V(t) is bounded from (9.99).

According to Lemma 2.5, we have

$$\begin{aligned} \frac{\beta {{k}_{2}}T}{2L}{{y}^{2}}(x,t)\le \frac{\beta {{k}_{2}}T}{2}\int _{0}^{L}{{{({y}_{x})}^{2}}dx}\le {{V}_{2}}(t)\le \frac{V(t)}{{{\alpha }_{2}}} \end{aligned}$$

(9.100)

$$\begin{aligned} \frac{\beta {{k}_{2}}T}{2L}{{z}^{2}}(x,t)\le \frac{\beta {{k}_{2}}T}{2}\int _{0}^{L}{{{({z}_{x})}^{2}}dx}\le {{V}_{2}}(t)\le \frac{V(t)}{{{\alpha }_{2}}} \end{aligned}$$

(9.101)

Therefore we obtain y(x, t) and z(x, t) are uniformly bounded as follows

$$\begin{aligned} \left| y(x,t) \right| \le \sqrt{\frac{2L}{\beta {{k}_{2}}T{{\alpha }_{2}}}V(t)}\le \sqrt{\frac{2L}{\beta {{k}_{2}}T{{\alpha }_{2}}}\left( V(0){{e}^{-\lambda t}}+\frac{\varepsilon }{\lambda } \right) } \end{aligned}$$

(9.102)

$$\begin{aligned} \left| z(x,t) \right| \le \sqrt{\frac{2L}{\beta {{k}_{2}}T{{\alpha }_{2}}}V(t)}\le \sqrt{\frac{2L}{\beta {{k}_{2}}T{{\alpha }_{2}}}\left( V(0){{e}^{-\lambda t}}+\frac{\varepsilon }{\lambda } \right) } \end{aligned}$$

(9.103)

$\forall (x,t)\in [0,L]\times [0,\infty )$.

Similarly, we have

$$\begin{aligned} \left| {{{\tilde{q}}}_{i}} \right| \le \sqrt{\frac{2V(t)}{\beta {{k}_{2}}{{\alpha }_{2}}{{\lambda }_{\min }}({{\mathbf {k}}_{p}})}}\le \sqrt{\frac{2}{\beta {{k}_{2}}{{\alpha }_{2}}{{\lambda }_{\min }}({{\mathbf {k}}_{p}})}\left( V(0){{e}^{-\lambda t}}+\frac{\varepsilon }{\lambda } \right) }\text { }\left( i=1,2,3 \right) \end{aligned}$$

(9.104)

It follows that,

$$\begin{aligned} \underset{t\rightarrow \infty }{\mathop {\lim }}\,\left| y(x,t) \right| \le \sqrt{\frac{2L\varepsilon }{\beta {{k}_{2}}T{{\alpha }_{2}}\lambda }} \end{aligned}$$

(9.105)

$$\begin{aligned} \underset{t\rightarrow \infty }{\mathop {\lim }}\,\left| z(x,t) \right| \le \sqrt{\frac{2L\varepsilon }{\beta {{k}_{2}}T{{\alpha }_{2}}\lambda }} \end{aligned}$$

(9.106)

$\forall (x,t)\in [0,L]\times [0,\infty )\pounds \urcorner$ and

$$\begin{aligned} \underset{t\rightarrow \infty }{\mathop {\lim }}\,\left| {{{\tilde{q}}}_{i}} \right| \le \sqrt{\frac{2\varepsilon }{\beta {{k}_{2}}{{\alpha }_{2}}{{\lambda }_{\min }}({{\mathbf {k}}_{p}})\lambda }}\left( i=1,2,3 \right) \end{aligned}$$

(9.107)

$\forall t\in [0,\infty )$. Therefore, y(x, t), z(x, t) and ${{\tilde{q}}_{i}}\left( i=1,2,3 \right)$ are uniformly ultimate bounded.

Appendix 2: Simulation Program