Vibration suppression and angle tracking of a fire-rescue ladder

Abstract

This paper mainly considers vibration suppression and angle tracking of a fire-rescue ladder system. The dynamical model is regarded as a segmented Euler–Bernoulli beam with gravity and tip mass, described by a set of motion equations and boundary conditions. Based on the nonlinear Euler–Bernoulli beam model, two active boundary controllers are proposed to achieve the control objectives. The elastic deflection and the angular error in the closed-loop system are proven to converge exponentially to a small neighborhood of zero. Numerical simulations based on finite difference method verify the effectiveness and the ascendancy of active boundary controllers.

Appendices

Appendices

Proof of step 1

Proof

Using the Young’s inequality and (10)–(12) to $V_{III}(t)$, we have

$$\begin{aligned} |V_{III}(t)|\le & {} \alpha \sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-} \rho _n|w(z,t)||{\dot{P}}(z,t)|\hbox {d}z\nonumber \\&+\xi \sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}\rho _nz|{w}'(z,t)||{\dot{P}}(z,t)|\hbox {d}z\nonumber \\&+\zeta \sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}\rho _n(L-z)|{w}'(z,t)||{\dot{P}}(z,t)|\hbox {d}z\nonumber \\\le & {} \sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}L\rho _{\max }[2L\alpha {\delta _1}+(\zeta +\xi ){\delta _2}]\nonumber \\&\times [w''(z,t)]^2\hbox {d}z +\sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}\Big [\frac{(\zeta +\xi )\rho _n}{2\delta _2}\nonumber \\&+\frac{\alpha \rho _n}{2\delta _1}\Big ][{\dot{P}}(z,t)]^2\hbox {d}z\nonumber \\\le & {} {\mu _1}V_1(t), \end{aligned}$$

(A.1)

where $\delta _1$ and $\delta _2$ are positive constants, $\rho _{\max }$ is the maximum value of $\rho _n,~n\in \{1,2,\ldots ,S\}$ and

$$\begin{aligned} {\mu _1}= & {} \max \Big \{\frac{2L\rho _{\max }[2L\alpha {\delta _1}+(\zeta +\xi ){\delta _2}]}{{\beta }EI_{\min }},\nonumber \\&\quad \frac{(\zeta +\xi )}{\beta \delta _2}+\frac{\alpha }{\beta \delta _1}\Big \} \end{aligned}$$

(A.2)

where $EI_{\min }$ is the minimum value of $EI_n,~n\in \{1,2,$$\ldots ,S\}.$ Therefore, we obtain

$$\begin{aligned} -{\mu _1}V_1(t){\le }V_3(t){\le }{\mu _1}V_1(t) \end{aligned}$$

(A.3)

When ${\mu _1}$ satisfies $0<{\mu _1}<1$, we have

$$\begin{aligned} 0\le \lambda _1(V_1(t)+V_2(t)){\le }V(t)\le \lambda _2(V_{I}(t)+V_{II}(t)). \end{aligned}$$

(A.4)

where $\lambda _1=1-\mu _1$ and $\lambda _2=1+\mu _1$.

Proof of step 2

Proof

Taking the derivative of (15) with respect to time, we have

$$\begin{aligned} {\dot{V}}(t) = {\dot{V}}_{I}(t) + {\dot{V}}_{II}(t) + {\dot{V}}_{III}(t). \end{aligned}$$

(B.1)

Using the motion equation (2), the first term of (B.1) can be written as

$$\begin{aligned} {\dot{V}}_{I}(t)= & {} -{\beta }\sum _{n=1}^{S}\int _{z_{n-1}^+}^{z_n^-} \rho _ng{\dot{P}}(z,t)\cos \theta (t)\hbox {d}z\nonumber \\&-{\beta }{EI_S}{\dot{P}}(L,t)P'''(L,t)-{\beta }EI_1P''(0,t){\dot{\theta }}(t)\nonumber \\&-{\beta }\sum _{n=1}^{S}\int _{z_{n-1}^+}^{z_n^-}c[{\dot{P}}(z,t)]^2\hbox {d}z. \end{aligned}$$

(B.2)

We define

$$\begin{aligned} B_1(t)= & {} -{\beta }\sum _{n=1}^{S}\int _{z_{n-1}^+}^{z_n^-}\rho _ng{\dot{P}}(z,t)\cos \theta (t)\hbox {d}z, \end{aligned}$$

(B.3)

$$\begin{aligned} B_2(t)= & {} -{\beta }EI_1P''(0,t){\dot{\theta }}(t). \end{aligned}$$

(B.4)

Using the Young’s inequality, we have

$$\begin{aligned} |B_1|\le & {} {\beta }\sum _{n=1}^{S}\int _{z_{n-1}^+}^{z_n^-}\rho _ng\frac{{\delta }_3}{2}[{\dot{P}}(z,t)]^2\hbox {d}z\nonumber \\&+{\beta }\sum _{n=1}^{S}\int _{z_{n-1}^+}^{z_n^-}\rho _ng\frac{1}{2\delta _3}[\cos \theta (t)]^2\hbox {d}z\nonumber \\\le & {} {\beta }\sum _{n=1}^{S}\int _{z_{n-1}^+}^{z_n^-}\rho _ng\frac{\delta _3}{2}[{\dot{P}}(z,t)]^2\hbox {d}z+\frac{{\beta }gL\rho _{\max }}{2\delta _3}\nonumber \\ \end{aligned}$$

(B.5)

$$\begin{aligned} |B_2|\le & {} {\beta }EI_1\frac{\delta _4}{2}[P''(0,t)]^2+{\beta }EI_1\frac{1}{2\delta _4}[{\dot{\theta }}(t)]^2,\nonumber \\ \end{aligned}$$

(B.6)

where $\delta _3$ and $\delta _4$ are positive constants. As a result, we obtain

$$\begin{aligned} {\dot{V}}_{I}(t)\le & {} {\beta }\sum _{n=1}^{S}\int _{z_{n-1}^+}^{z_n^-}\rho _ng\frac{\delta _3}{2} [{\dot{P}}(z,t)]^2\hbox {d}z +{\beta }EI_1\frac{1}{2\delta _4}[{\dot{\theta }}]^2\nonumber \\&+{\beta }EI_1\frac{\delta _4}{2}[P''(0,t)]^2 \nonumber \\&-\,{\beta }\sum _{n=1}^{S}\int _{z_{n-1}^+}^{z_n^-}c[{\dot{P}}(z,t)]^2\hbox {d}z\nonumber \\&-{\beta }{EI_S}{\dot{P}}(L,t)P'''(L,t)+\frac{{\beta }gL\rho _{\max }}{2\delta _3}. \end{aligned}$$

(B.7)

Using the motion equation (3), the second term of (B.1) can be written as

$$\begin{aligned} {\dot{V}}_{II}(t)= & {} {\gamma }\sum _{n=1}^{S}\int _{z_{n-1}^+}^{z_n^-}\rho _ngw(z,t)[{\dot{\theta }}(t)+\theta _e(t)]\sin \theta (t)\hbox {d}z\nonumber \\&+ {\gamma }[{\dot{\theta }}(t)+\theta _e(t)][M(t)+EI_1P''(0,t)\nonumber \\&+\, m_cgw(L,t)\sin \theta (t)+J_A{\dot{\theta }}(t)]\nonumber \\&+\,\beta \alpha ^2{m_c}u_a(t)\dot{u}_a(t) + k\theta _e(t){\dot{\theta }}(t). \end{aligned}$$

(B.8)

Using perfect square formula, we have

$$\begin{aligned} k\theta _e(t){\dot{\theta }}(t) = \frac{k}{2}[\theta _e(t)+{\dot{\theta }}(t)]^2-\frac{k}{2} [\theta _e(t)]^2-\frac{k}{2}[{\dot{\theta }}(t)]^2 \end{aligned}$$

(B.9)

Using the definition of $u_a(t)$ and the boundary condition (9), we have

$$\begin{aligned}&\nonumber \beta \alpha {m_c}u_a(t)\dot{u}_a(t) = \beta \alpha {m_c}u_a(t)[\ddot{P}(L,t)+\frac{\alpha -{\xi }L}{\beta }\dot{w}(L,t)]\\&\quad \nonumber = \beta \alpha {u_a}(t)[N(t)+EI_SP'''(L,t)\\&\quad -m_cg\cos \theta (t)+m_c\frac{\alpha -{\xi }L}{\beta }\dot{w}(L,t)].\end{aligned}$$

(B.10)

Defining $B_3={\gamma }\sum _{n=1}^{S}\int _{z_{n-1}^+}^{z_n^-}\rho _ngw(z,t)[{\dot{\theta }}(t)+\theta _e(t)] \times \sin \theta (t)\hbox {d}z$ and using the Young’s inequality and (10)–(12), we have

$$\begin{aligned} |B_3|\le & {} {\gamma }\sum _{n=1}^{S}\int _{z_{n-1}^+}^{z_n^-}\rho _ng|w(z,t)||[{\dot{\theta }}(t)+\theta _e(t)]|\hbox {d}z\nonumber \\\le & {} {\gamma }\sum _{n=1}^{S}\int _{z_{n-1}^+}^{z_n^-}\rho _ng\frac{\delta _5}{2}[w(z,t)]^2\hbox {d}z\nonumber \\&+{\gamma }\sum _{n=1}^{S}\int _{z_{n-1}^+}^{z_n^-}\rho _ng\frac{1}{2\delta _5}[{\dot{\theta }}(t)+\theta _e(t)]^2\hbox {d}z\nonumber \\\le & {} {\gamma }\sum _{n=1}^{S}\int _{z_{n-1}^+}^{z_n^-}2L^2\rho _ng{\delta _5}[w''(z,t)]^2\hbox {d}z\nonumber \\&+\frac{{\gamma }g\rho _{\max }L}{2\delta _5}[{\dot{\theta }}(t)+\theta _e(t)]^2, \end{aligned}$$

(B.11)

where $\delta _5$ is positive constant. Substituting (B.9)–(B.11) and the boundary control laws (13) and (14) to (B.8), we further have

$$\begin{aligned}&{\dot{V}}_{II}(t)\nonumber \\&\quad \le {\gamma }\sum _{n=1}^{S}\int _{z_{n-1}^+}^{z_n^-}2L^2\rho _ng{\delta _5}[w''(z,t)]^2\hbox {d}z -\frac{k}{2}[\theta _e(t)]^2\nonumber \\&\qquad -\,\frac{k}{2}[{\dot{\theta }}(t)]^2 -k_u[u_a(t)]^2 \nonumber \\&\qquad +\,\sqrt{\beta }u_a(t)EI_SP'''(L,t)\nonumber \\&\qquad -\,\Big ({k_{\theta }}-\frac{{\gamma }g\rho _{max}L}{2\delta _5} \Big )[{\dot{\theta }}(t)+\theta _e(t)]^2. \end{aligned}$$

(B.12)

The third term of (B.1) can be written as

$$\begin{aligned} {\dot{V}}_{III}(t)= & {} B_4+B_5+B_6+B_7+B_8+B_9, \end{aligned}$$

(B.13)

where

$$\begin{aligned} B_4= & {} \alpha \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}\rho _n{\dot{w}}(z,t){\dot{P}}(z,t)\hbox {d}z\\ B_5= & {} \alpha \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}\rho _nw(z,t){\ddot{P}}(z,t)\hbox {d}z\\ B_6= & {} \zeta \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}\rho _n(L-z){\dot{w}}'(z,t){\dot{P}}(z,t)\hbox {d}z\\ B_7= & {} \zeta \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}\rho _n(L-z)w'(z,t){\ddot{P}}(z,t)\hbox {d}z\\ B_8= & {} -\xi \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}\rho _nz{\dot{w}}'(z,t){\dot{P}}(z,t)\hbox {d}z\\ B_9= & {} -\xi \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}\rho _nzw'(z,t){\ddot{P}}(z,t)\hbox {d}z \end{aligned}$$

Using the definition of P(z, t), $B_4$ can be written as

$$\begin{aligned} B_4= & {} \alpha \sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}\rho _n[{\dot{P}}(z,t)]^2\hbox {d}z\nonumber \\&-\,\alpha \sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}\rho _nz{\dot{P}}(z,t){\dot{\theta }}(t)\hbox {d}z. \end{aligned}$$

(B.14)

Using the Young’s inequality, the seconde term of $B_4$ satisfies following inequality.

$$\begin{aligned}&|\alpha \sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}{\rho _n}z{\dot{P}}(z,t){\dot{\theta }}(t)\hbox {d}z|\nonumber \\&\quad \le \sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}\frac{\delta _{6}\alpha \rho _n}{2}z[{\dot{P}}(z,t)]^2\hbox {d}z\nonumber \\&\qquad + \frac{\alpha }{2\delta _{6}}\sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}{\rho _n}z[{\dot{\theta }}(t)]^2\hbox {d}z\nonumber \\&\quad \le \sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}\frac{\delta _{6}\alpha \rho _nL}{2}[{\dot{P}}(z,t)]^2\hbox {d}z \nonumber \\&\qquad \frac{\alpha {\rho _{\max }}L^2}{4\delta _{6}}[{\dot{\theta }}(t)]^2, \end{aligned}$$

(B.15)

where $\delta _{6}$ is a positive constant. So that $B_4$ satisfies

$$\begin{aligned} B_4\le & {} \alpha \sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}\rho _n(1+\frac{\delta _{6}L}{2})[{\dot{P}}(z,t)]^2\hbox {d}z\nonumber \\&+ \,\frac{\alpha {\rho _{\max }}L^2}{4\delta _{6}}[{\dot{\theta }}(t)]^2. \end{aligned}$$

(B.16)

Using the motion equation (2), $B_5$ can be written as

$$\begin{aligned} B_5= & {} -\alpha \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}{\rho _n}gw(z,t)\cos \theta (t)\hbox {d}z -\alpha {EI_S}w(L,t)\nonumber \\&\times P'''(L,t) -\alpha \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}{EI_n}[w''(z,t)]^2\hbox {d}z\nonumber \\&-\,{\alpha }c\sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}w(z,t){\dot{P}}(z,t)\hbox {d}z \end{aligned}$$

(B.17)

Using the Young’s inequality, (10) and (11), the first term of $B_5$ satisfies following inequality.

$$\begin{aligned}&|\alpha \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}{\rho _n}gw(z,t)\cos \theta (t)\hbox {d}z|\nonumber \\&\quad \le \alpha \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}{\rho _n}g\frac{\delta _7}{2}[w(z,t)]^2\hbox {d}z +\frac{\alpha }{2\delta _7}\sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}{\rho _n}g\hbox {d}z\nonumber \\&\quad \le 2L^2{\delta _7}g\alpha \sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}\rho _{\max }[w''(z,t)]^2\hbox {d}z\nonumber \\&\qquad +\frac{\alpha \rho _{\max }Lg}{2\delta _7}, \end{aligned}$$

(B.18)

where $\delta _{7}$ is a positive constant. Using the Young’s inequality and (10), the forth term of $B_5$ satisfies following inequality.

$$\begin{aligned}&|{\alpha }c\sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}w(z,t){\dot{P}}(z,t)\hbox {d}z|\nonumber \\&\quad \le \alpha \sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}2cL^2{\delta _8}[w''(z,t)]^2\hbox {d}z\nonumber \\&\qquad +\frac{c\alpha }{2\delta _8}\sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}[{\dot{P}}(z,t)]^2\hbox {d}z, \end{aligned}$$

(B.19)

where $\delta _{8}$ is a positive constant. So that $B_5$ satisfies

$$\begin{aligned} B_5\le & {} - \alpha {EI_S}w(L,t)P'''(L,t)+\frac{\alpha \rho _{\max }Lg}{2\delta _7}\nonumber \\&+ \,\frac{c\alpha }{2\delta _8}\sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}[{\dot{P}}(z,t)]^2\hbox {d}z - \alpha \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}({EI_n}\nonumber \\&-\,2L^2{\delta _7}g\rho _{\max }-2L^2c{\delta _8})[w''(z,t)]^2\hbox {d}z\nonumber \\ \end{aligned}$$

(B.20)

Using the definition of P(z, t) and integrating by part, $B_6$ can be written as

$$\begin{aligned} B_6= & {} \zeta \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}\rho _n(L-z) {\dot{P}}'(z,t){\dot{P}}(z,t)\hbox {d}z\nonumber \\&-\zeta \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}\rho _n(L-z){\dot{\theta }}(t){\dot{P}}(t)\hbox {d}z \nonumber \\= & {} \zeta \sum _{n=1}^{S-1}(\rho _n-\rho _{n+1})(L-{z_n^-}) [{\dot{P}}({z_n^-},t)]^2\nonumber \\&-\zeta \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}\rho _n(L-z) {\dot{\theta }}(t){\dot{P}}(z,t)\hbox {d}z\nonumber \\&+\frac{\zeta }{2}\sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}\rho _n[{\dot{P}}(z,t)]^2\hbox {d}z \end{aligned}$$

(B.21)

Using the Young’s inequality, the term $\zeta \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}\rho _n \times (L-z){\dot{\theta }}(t){\dot{P}}(z,t)\hbox {d}z$ in $B_6$ satisfies following inequality:

$$\begin{aligned}&|\zeta \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}\rho _n(L-z){\dot{\theta }}(t) {\dot{P}}(z,t)\hbox {d}z|\nonumber \\&\quad \le \frac{L\zeta \delta _9}{2}\sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-} \rho _n[{\dot{P}}(z,t)]^2\hbox {d}z \nonumber \\&\qquad +\,\frac{\zeta \rho _{\max }L^2}{2\delta _9}[{\dot{\theta }}(t)]^2\hbox {d}z \end{aligned}$$

(B.22)

where $\delta _{9}$ is a positive constant. So that $B_6$ satisfies

$$\begin{aligned} B_6\le & {} \zeta \sum _{n=1}^{S-1}(\rho _n-\rho _{n+1})(L-{z_n^-})[{\dot{P}}({z_n^-},t)]^2 \nonumber \\&\quad +\,\frac{\zeta \rho _{\max }L^2}{2\delta _9} \times [{\dot{\theta }}(t)]^2\hbox {d}z \nonumber \\&\quad +\,\frac{\zeta +L\zeta \delta _9}{2}\sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-} \rho _n[{\dot{P}}(z,t)]^2\hbox {d}z \end{aligned}$$

(B.23)

Using the motion equation (2), $B_7$ can be written as

$$\begin{aligned} B_7= & {} -\frac{\zeta }{2}L{EI_1}[w''(0,t)]^2 -\zeta \sum _{n=1}^{S-1}({\rho _n}-\rho _{n+1})g(L\nonumber \\&-\,{z_n^-})w({z_n^-},t)\cos \theta (t) -\zeta \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}w(z,t){\rho _n}g\nonumber \\&\times \cos \theta (t)\hbox {d}z +\frac{3\zeta }{2}\sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}{EI_n}[w''(z,t)]^2\nonumber \\&-\,\zeta \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}(L-z)w'(z,t)c{\dot{P}}(z,t)\hbox {d}z \end{aligned}$$

(B.24)

Considering ${\rho _n}>\rho _{n+1}$ and using the Young’s inequality and (12), we have the first term of $B_7$ satisfies

$$\begin{aligned}&|\zeta \sum _{n=1}^{S-1}({\rho _n}-\rho _{n+1})g(L-{z_n^-})w({z_n^-},t)\cos \theta (t)|\nonumber \\&\quad \le \zeta \rho _{\max }gL\sum _{n=1}^{S-1}|w({z_n^-},t)\cos \theta (t)|\nonumber \\&\quad \le \frac{\delta _{10}8gL^2(S-1)\zeta \rho _{\max }}{2}\sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}[w''(z,t)]^2\hbox {d}z\nonumber \\&\qquad + \frac{gL(S-1)\zeta \rho _{\max }}{2\delta _{10}} \end{aligned}$$

(B.25)

where $\delta _{10}$ is a positive constant. Similar with (B.18), the second term in $B_7$ satisfies

$$\begin{aligned}&|\zeta \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}w(z,t){\rho _n}g\cos \theta (t)\hbox {d}z|\nonumber \\&\quad \le 2L^2{\delta _7}g\varsigma \sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}\rho _{\max }[w''(z,t)]^2\hbox {d}z \nonumber \\&\qquad +\,\frac{\zeta \rho _{\max }Lg}{2\delta _7} \end{aligned}$$

(B.26)

Using the Young’s inequality and (11), the third term of $B_7$ satisfies following inequality.

$$\begin{aligned}&|\zeta \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}(L-z)w'(z,t)c{\dot{P}}(z,t)\hbox {d}z|\nonumber \\&\quad \le cL^2{\delta _{11}}\zeta \sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}[w''(z,t)]^2\hbox {d}z\nonumber \\&\qquad +\frac{cL\zeta }{2\delta _{11}}\sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}[{\dot{P}}(z,t)]^2\hbox {d}z \end{aligned}$$

(B.27)

where $\delta _{11}$ is a positive constant. So that $B_7$ satisfies

$$\begin{aligned} B_7\le & {} \zeta \sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-} \Big (2L^2{\delta _7}g\rho _{\max }+\frac{8(S-1)\rho _{\max }\delta _{10}gL^2}{2}\nonumber \\&+\frac{3}{2}{EI_n}+cL^2{\delta _{11}}\Big )[w''(z,t)]^2\hbox {d}z +\frac{\zeta \rho _{\max }Lg}{2\delta _7}\nonumber \\&+ \frac{(S-1)\zeta \rho _{\max }gL}{2\delta _{10}} +\frac{cL\zeta }{2\delta _{11}}\sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-} [{\dot{P}}(z,t)]^2\hbox {d}z\nonumber \\&-\frac{\zeta }{2}L{EI_1}[w''(0,t)]^2 \end{aligned}$$

(B.28)

Using the definition of P(z, t) and integrating by part, $B_8$ can be written as

$$\begin{aligned} B_8= & {} -\xi \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}\rho _nz{\dot{P}}'(z,t){\dot{P}}(z,t)\hbox {d}z\nonumber \\&+\xi \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}\rho _nz{\dot{\theta }}(t){\dot{P}}(t)\hbox {d}z \nonumber \\= & {} -\frac{\xi }{2}\sum _{n=1}^{S-1}(\rho _n-\rho _{n+1}){z_n^-}[{\dot{P}}({z_n^-},t)]^2\nonumber \\&-\,\frac{\xi }{2}\sum _{n=1}^S\rho _SL[{\dot{P}}(L,t)]^2 \nonumber \\&+\,\frac{\xi }{2}\sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}\rho _n[{\dot{P}}(z,t)]^2\hbox {d}z\nonumber \\&+\,\xi \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}\rho _nz{\dot{\theta }}(t){\dot{P}}(z,t)\hbox {d}z \end{aligned}$$

(B.29)

Using the Young’s inequality, the forth term of $B_8$ satisfies following inequality.

$$\begin{aligned}&|\xi \sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}{\rho _n}z {\dot{P}}(z,t){\dot{\theta }}(t)\hbox {d}z|\nonumber \\&\quad \le \sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}\frac{\delta _{12}\xi \rho _nL}{2} [{\dot{P}}(z,t)]^2\hbox {d}z \nonumber \\&\qquad +\, \frac{\xi {\rho _{\max }}L^2}{4\delta _{12}}[{\dot{\theta }}(t)]^2 \end{aligned}$$

(B.30)

where $\delta _{12}$ is a positive constant. So that $B_8$ satisfies

$$\begin{aligned} B_8\le & {} \frac{\xi }{2}\sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}\rho _n (1+\delta _{12}L)[{\dot{P}}(z,t)]^2\hbox {d}z \nonumber \\&-\,\frac{\xi }{2}\sum _{n=1}^S\rho _SL [{\dot{P}}(L,t)]^2 -\frac{\xi }{2}\sum _{n=1}^{S-1}(\rho _n-\rho _{n+1}){z_n^-}\nonumber \\&\times [{\dot{P}}({z_n^-},t)]^2 +\, \frac{\xi {\rho _{\max }}L^2}{4\delta _{12}}[{\dot{\theta }}(t)]^2 \end{aligned}$$

(B.31)

Using the motion equation (2) and integrating by part, $B_9$ can be written as

$$\begin{aligned} B_9= & {} \xi \sum _{n=1}^{S-1}(\rho _n-\rho _{n+1}){z_{n}^-}w({z_n^-},t)\cos \theta (t) \nonumber \\&-\,\sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}\xi \times {\rho _n}gw(z,t)\cos \theta (t)\hbox {d}z \nonumber \\&+\,\xi \rho _SLw(L,t)\cos \theta (t)\nonumber \\&+\,\xi {EI_S}Lw(L,t)P'''(L,t)\nonumber \\&+\,\xi \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}c zw'(z,t) {\dot{P}}(z,t)\hbox {d}z\nonumber \\&+\frac{3\xi }{2}\sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}{EI_n}[w''(z,t)]^2\hbox {d}z\nonumber \\ \end{aligned}$$

(B.32)

Similar with (B.25), (B.18) and (B.27), the first, third and sixth terms of $B_9$ satisfy respectively following inequalities:

$$\begin{aligned}&|\xi \sum _{n=1}^{S-1}(\rho _n-\rho _{n+1}){z_{n}^-}w({z_n^-},t) \cos \theta (t)|\nonumber \\&\quad \le \frac{8(S-1)\delta _{10}\xi \rho _{\max }gL^2}{2} \sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}[w''(z,t)]^2\hbox {d}z\nonumber \\&\qquad + \frac{(S-1)\xi \rho _{\max }gL}{2\delta _{10}} \end{aligned}$$

(B.33)

$$\begin{aligned}&|\xi \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}{\rho _n}gw(z,t)\cos \theta (t)\hbox {d}z|\nonumber \\&\quad \le 2L^2{\delta _7}g\xi \sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}\rho _{\max }[w''(z,t)]^2\hbox {d}z +\frac{\xi \rho _{\max }Lg}{2\delta _7}\nonumber \\ \end{aligned}$$

(B.34)

$$\begin{aligned}&|\xi \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}zw'(z,t)c{\dot{P}}(z,t)\hbox {d}z|\nonumber \\&\quad \le cL^2{\delta _{11}}\xi \sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}[w''(z,t)]^2\hbox {d}z\nonumber \\&\qquad +\frac{cL\xi }{2\delta _{11}}\sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}[{\dot{P}}(z,t)]^2\hbox {d}z \end{aligned}$$

(B.35)

Using the Young’s inequality and (12), the second term of $B_9$ satisfies following inequality:

$$\begin{aligned}&|\xi \rho _SLw(L,t)\cos \theta (t)|\nonumber \\\le & {} \frac{8\delta _{13}\xi \rho _{S}gL^2}{2}\sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}[w''(z,t)]^2\hbox {d}z + \frac{\xi \rho _{S}gL}{2\delta _{13}}\nonumber \\ \end{aligned}$$

(B.36)

So that $B_9$ satisfies

$$\begin{aligned} B_9\le & {} \xi \sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}\Big (\frac{8(S-1)\delta _{10}\rho _{\max }gL^2}{2} +\frac{8\delta _{13}\rho _{S}gL^2}{2}\nonumber \\&+\,\frac{3{EI_n}}{2}+2L^2{\delta _7}g\rho _{\max }+cL^2{\delta _{11}}\Big )[w''(z,t)]^2\hbox {d}z\nonumber \\&+\,\xi {EI_S}Lw(L,t)P'''(L,t) + \frac{\xi \rho _{S}gL}{2\delta _{13}} +\frac{\xi \rho _{\max }Lg}{2\delta _7}\nonumber \\&+\,\frac{cL\xi }{2\delta _{11}}\sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-}[{\dot{P}}(z,t)]^2\hbox {d}z + \frac{(S-1)\xi \rho _{\max }gL}{2\delta _{10}}\nonumber \\ \end{aligned}$$

(B.37)

Using (B.16), (B.20), (B.23), (B.28), (B.31) and (B.37) for (B.13), we obtain

$$\begin{aligned}&{\dot{V}}_{III}(t) \nonumber \\&\quad \le -\sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}\Big [\alpha {EI_n} -2L^2(\alpha +\zeta +\xi ){\delta _7}g\rho _{\max }\nonumber \\&\quad -\frac{3(\zeta +\xi )}{2}{EI_n} -\frac{8(S-1)(\zeta +\xi )\delta _{10}\rho _{\max }gL^2}{2}\nonumber \\&\quad -2L^2{\alpha }c{\delta _8} -\frac{8\xi \delta _{13}\rho _{S}gL^2}{2} -cL^2(\zeta +\xi ){\delta _{11}}\Big ]\nonumber \\&\quad \times [w''(z,t)]^2\hbox {d}z - (\alpha -{\xi }L){EI_S}w(L,t)P'''(L,t)\nonumber \\&\quad +\,\sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-} \Big [\frac{c\alpha }{2\delta _8} +\frac{cL(\zeta +\xi )}{2\delta _{11}} +\alpha \rho _n(1+\frac{\delta _{6}L}{2})\nonumber \\&\quad +\,\frac{\xi }{2}\rho _n(1+\delta _{12}L) +\frac{\zeta }{2}\rho _n+\frac{L\zeta \delta _9}{2}\rho _n\Big ] [{\dot{P}}(z,t)]^2\hbox {d}z\nonumber \\&\quad -\,\frac{\xi }{2}\sum _{n=1}^S\rho _SL[{\dot{P}}(L,t)]^2 + \Big (\frac{\xi {\rho _{\max }}L^2}{4\delta _{12}}+\frac{\zeta \rho _{\max }L^2}{2\delta _9}\nonumber \\&\quad +\, \frac{\alpha {\rho _{\max }}L^2}{4\delta _{6}}\Big ) [{\dot{\theta }}(t)]^2 -\frac{\zeta }{2}L{EI_1}[w''(0,t)]^2 \nonumber \\&\quad -\sum _{n=1}^{S-1}(\rho _n-\rho _{n+1})\Big [\frac{\xi }{2}{z_n^-}-\zeta (L-{z_n^-})\Big ][{\dot{P}}({z_n^-},t)]^2\nonumber \\&\quad +\frac{(\alpha +\zeta +\xi )\rho _{\max }Lg}{2\delta _7} + \frac{\xi \rho _{S}gL}{2\delta _{13}}\nonumber \\&\quad + \frac{(S-1)(\zeta +\xi )\rho _{\max }gL}{2\delta _{10}} \end{aligned}$$

(B.38)

Using (B.7), (B.12) and (B.38), we can obtain

$$\begin{aligned} {\dot{V}}(t)\le & {} -\sum _{n=1}^S\int _{z_{n-1}^+}^{z_n^-}\Big [\alpha {EI_n} -2L^2(\alpha +\zeta +\xi ){\delta _7}g\rho _{\max }\nonumber \\&-\,\frac{8(S-1)(\zeta +\xi )\delta _{10}\rho _{\max }gL^2}{2} -\frac{8\xi \delta _{13}\rho _{S}gL^2}{2}\nonumber \\&-\,\frac{3(\zeta +\xi )}{2}{EI_n} -2L^2{\alpha }c{\delta _8} -cL^2(\zeta +\xi ){\delta _{11}}\nonumber \\&-\,2L^2{\gamma }\rho _ng{\delta _5}\Big ][w''(z,t)]^2\hbox {d}z -\sum _{n=1}^{S}\int _{z_{n-1}^{+}}^{z_n^-} \Big [{\beta }c \nonumber \\&-\frac{c\alpha }{2\delta _8}-\alpha \rho _n(1+\frac{\delta _{6}L}{2}) -\frac{cL(\zeta +\xi )}{2\delta _{11}} -\frac{L\zeta \delta _9}{2}\rho _n\nonumber \\&-\frac{\zeta }{2}\rho _n -\frac{{\beta }\delta _3\rho _ng}{2} -\frac{\xi }{2}\rho _n(1+\delta _{12}L)\Big ] [{\dot{P}}(z,t)]^2\hbox {d}z\nonumber \\&-\sum _{n=1}^{S-1}(\rho _n-\rho _{n+1})\Big [\frac{\xi }{2}{z_n^-}-\zeta (L-{z_n^-})\Big ][{\dot{P}}({z_n^-},t)]^2\nonumber \\&-\, \Big (\frac{k}{2} -\frac{\xi {\rho _{\max }}L^2}{4\delta _{12}} -\frac{\zeta \rho _{\max }L^2}{2\delta _9} - \frac{\alpha {\rho _{\max }}L^2}{4\delta _{6}}\nonumber \\&-\frac{{\beta }EI_1}{2\delta _4}\Big )[{\dot{\theta }}(t)]^2 -\frac{\xi }{2}\sum _{n=1}^S\rho _SL[{\dot{P}}(L,t)]^2\nonumber \\&-\frac{EI_1}{2}({\zeta }L-{\beta }{\delta _4})[P''(0,t)]^2 -\Big ({k_{\theta }}-\frac{{\gamma }g\rho _{max}L}{2\delta _5}\Big )\nonumber \\&\times [{\dot{\theta }}(t)+\theta _e(t)]^2 -k_u[u_a(t)]^2 -\frac{k}{2}[\theta _e(t)]^2\nonumber \\&+ \frac{(S-1)(\zeta +\xi )\rho _{\max }gL}{2\delta _{10}} + \frac{\xi \rho _{S}gL}{2\delta _{13}}\nonumber \\&+\frac{(\alpha +\zeta +\xi )\rho _{\max }Lg}{2\delta _7} +\frac{{\beta }gL\rho _{\max }}{2\delta _3}\nonumber \\\le & {} -\lambda _3(V_{I}(t)+V_{II}(t))+\varepsilon , \end{aligned}$$

(B.39)

where

$$\begin{aligned} \varepsilon= & {} \frac{(S-1)(\zeta +\xi )\rho _{\max }gL}{2\delta _{10}} + \frac{\xi \rho _{S}gL}{2\delta _{13}}\nonumber \\&+\,\frac{(\alpha +\zeta +\xi )\rho _{\max }Lg}{2\delta _7} +\frac{{\beta }gL\rho _{\max }}{2\delta _3} \end{aligned}$$

(B.40)

and the positive constants $\alpha$, $\beta$, $\gamma$, $\zeta$, $\xi$, c, k, $k_u$, $k_{\theta }$, $\delta _1$-$\delta _{13}$ are chosen under the following conditions.

$$\begin{aligned} \beta&{>}&\max \Big \{\frac{2L\rho _{\max }(2L\alpha {\delta _1}{+}\zeta {\delta _2})}{EI_{\min }},\frac{\zeta }{\delta _2}{+}\frac{\alpha }{\delta _1}\Big \} \end{aligned}$$

(B.41)

$$\begin{aligned} \kappa _n= & {} \alpha {EI_n} -2L^2(\alpha +\zeta +\xi ){\delta _7}g\rho _{\max } -2L^2{\alpha }c{\delta _8}\nonumber \\&-\,\frac{8(S-1)(\zeta +\xi )\delta _{10}\rho _{\max }gL^2}{2} -2L^2{\gamma }\rho _ng{\delta _5}\nonumber \\&-\,\frac{3(\zeta +\xi )}{2}{EI_n} -\frac{8\xi \delta _{13}\rho _{S}gL^2}{2}\nonumber \\&-\,cL^2(\zeta +\xi ){\delta _{11}}>0,~n\in \{1,2,\ldots ,S\} \end{aligned}$$

(B.42)

$$\begin{aligned} \nu _n= & {} {\beta }c-\frac{c\alpha }{2\delta _8} -\frac{{\beta }\delta _3\rho _ng}{2} -\frac{cL(\zeta +\xi )}{2\delta _{11}} -\frac{L\zeta \delta _9}{2}\rho _n\nonumber \\&-\,\alpha \rho _n(1+\frac{\delta _{6}L}{2}) -\frac{\xi }{2}\rho _n(1+\delta _{12}L) -\frac{\zeta }{2}\rho _n>0,\nonumber \\&\quad n\in \{1,2,\ldots ,S\} \end{aligned}$$

(B.43)

$$\begin{aligned} \chi _1= & {} \min \Big \{\frac{2\nu _1}{\beta \rho _1},\ldots ,\frac{2\nu _S}{\beta \rho _S}\Big \}\nonumber \\= & {} -\frac{\delta _{6}{\alpha }L}{{\beta }} -g{\delta _3} -\frac{L\zeta \delta _9}{{\beta }} -\frac{2\alpha +{\zeta }+\xi (1+\delta _{12}L)}{\beta }\nonumber \\&-\,\frac{cL(\zeta +\xi )}{\beta \delta _{11}\rho _{\max }} -\frac{c\alpha }{\delta _8{\beta }\rho _{\max }} \end{aligned}$$

(B.44)

$$\begin{aligned} \chi _2= & {} \min \Big \{\frac{2\kappa _1}{{\beta }EI_1},\ldots ,\frac{2\kappa _S}{{\beta }EI_S}\Big \}\nonumber \\= & {} \frac{2\alpha }{\beta } -\frac{3(\zeta +\xi )}{\beta } -\frac{8(S-1)(\zeta +\xi )\delta _{10}\rho _{\max }gL^2}{{\beta }EI_{\max }}\nonumber \\&-\frac{4L^2{\delta _7}g(\alpha +\zeta +\xi )\rho _{\max }}{{\beta }EI_{\max }} -\frac{2cL^2{\delta _{11}}(\zeta +\xi )}{{\beta }EI_{\max }}\nonumber \\&-\frac{4cL^2{\delta _8}\alpha }{{\beta }EI_{\max }} -\frac{4L^2\rho _ng{\gamma }{\delta _5}}{{\beta }EI_{\max }} -\frac{8\xi \delta _{13}\rho _{S}gL^2}{{\beta }EI_{\max }} \end{aligned}$$

(B.45)

$$\begin{aligned} \sigma _1= & {} \frac{k}{2}-\frac{\rho _{\max }L^2}{2}(\frac{\zeta }{\delta _9}+\frac{2\xi }{\delta _{12}}+\frac{\alpha }{2\delta _{6}})-\frac{{\beta }EI_1}{2\delta _4}>0\nonumber \\ \end{aligned}$$

(B.46)

$$\begin{aligned} \sigma _2= & {} \frac{\xi }{2}{z_n^-}-\zeta (L-{z_n^-})>0\end{aligned}$$

(B.47)

$$\begin{aligned} \sigma _3= & {} {k_{\theta }}-\frac{{\gamma }g\rho _{\max }L}{2\delta _5}>0\end{aligned}$$

(B.48)

$$\begin{aligned} \sigma _4= & {} {\zeta }L-{\beta }{\delta _4}>0 \end{aligned}$$

(B.49)

where $EI_{\max }$ is the maximum value of $EI_n,~n\in \{1,2,$

$\ldots ,S\}$.

$$\begin{aligned} \lambda _3= & {} \min \Big \{\chi _1,\chi _2,\frac{2\sigma _3}{J_A\gamma },1,\frac{2k_u}{m_c}\Big \}>0. \end{aligned}$$

(B.50)

Combining (A.4) and (B.39), we finally obtain

$$\begin{aligned} {\dot{V}}(t)\le & {} -{\lambda }V(t)+\varepsilon , \end{aligned}$$

(B.51)

where $\lambda =\lambda _3/\lambda _2>0$.

Proof of step 3

Proof

Multiplying inequation (B.51) by $e^{{\lambda }t}$ yields

$$\begin{aligned} \frac{\partial }{{\partial }t}[V(t)e^{{\lambda }t}]\le & {} {\varepsilon }e^{{\lambda }t}. \end{aligned}$$

(C.1)

Integrating of the above inequality leads to

$$\begin{aligned} V(t)e^{{\lambda }t}-V(0)\le & {} \frac{\varepsilon }{\lambda }e^{{\lambda }t}-\frac{\varepsilon }{\lambda } \end{aligned}$$

(C.2)

then we can obtain

$$\begin{aligned} V(t) \le \Big [V(0)-\frac{\varepsilon }{\lambda }\Big ]e^{-{\lambda }t}+\frac{\varepsilon }{\lambda }{\le }V(0)e^{-{\lambda }t}+\frac{\varepsilon }{\lambda }{\in }{\mathscr {L}}_{\infty } \end{aligned}$$

(C.3)

which suggests V(t) is bounded. Using (16), (A.4) and (12), we obtain

$$\begin{aligned} \frac{{\beta }EI_n}{16L}[w(z,t)]^2\le & {} \sum _{n=1}^{S}\int _{z_{n-1}^+}^{z_n^-}\frac{{\beta }EI_n}{2}[w''(z,t)]^2\hbox {d}z\nonumber \\\le & {} V_{I}(t){\le }V_{I}(t)+V_{II}(t)\nonumber \\\le & {} \frac{1}{\lambda _1}V(t){\in }{\mathscr {L}}_{\infty } \end{aligned}$$

(C.4)

$\forall (z,t)\in [{z_{n-1}^+},{z_n^-}]\times [0,\infty ),n\in \{1,2,\ldots ,S\}.$ From (C.3) and (C.4), we have

$$\begin{aligned} |w(z,t)|\le & {} \sqrt{\frac{16L}{\lambda _1{\beta }EI_n}[V(0)e^{-{\lambda }t}+\frac{\varepsilon }{\lambda }]}\nonumber \\\le & {} \sqrt{\frac{16L}{\lambda _1{\beta }EI_n}[V(0)+\frac{\varepsilon }{\lambda }]} \end{aligned}$$

(C.5)

$\forall (z,t)\in [{z_{n-1}^+},{z_n^-}]\times [0,\infty ),n\in \{1,2,\ldots ,S\}.$ Further more, from (C.5), we have

$$\begin{aligned} \lim \limits _{t\rightarrow \infty }|w(z,t)|\le & {} \sqrt{\frac{16L\varepsilon }{\lambda \lambda _1{\beta }EI_n}} \end{aligned}$$

(C.6)

$\forall (z,t)\in [{z_{n-1}^+},{z_n^-}]\times [0,\infty ),n\in \{1,2,\ldots ,S\}.$ Using (17) and (A.4), we have

$$\begin{aligned} \frac{k}{2}[\theta _e(t)]^2 \le V_{II}(t){\le }V_{I}(t)+V_{II}(t)\le \frac{1}{\lambda _1}V(t){\in }{\mathscr {L}}_{\infty } \end{aligned}$$

(C.7)

From (C.3) and (C.7), we have

$$\begin{aligned} |\theta _e(t)| \le \sqrt{\frac{2}{k\lambda _1}[V(0)e^{-{\lambda }t}+\frac{\varepsilon }{\lambda }]} \le \sqrt{\frac{2}{k\lambda _1}[V(0)+\frac{\varepsilon }{\lambda }]},\nonumber \\~~\forall t\in [0,\infty )~~~~\nonumber \\ \end{aligned}$$

(C.8)

Further more, from (C.8), we have

$$\begin{aligned} \lim \limits _{t\rightarrow \infty }|\theta _e(t)|\le & {} \sqrt{\frac{2\varepsilon }{k\lambda \lambda _1}},~~\forall t\in [0,\infty ). \end{aligned}$$

(C.9)