Hybrid active and passive control of a very large floating beam structure

Abstract

In this paper, we present a novel hybrid active and passive control method for very large floating structure (VLFS) to reduce the hydroelastic response, such that the resulting controlled VLFS can enhance its serviceability on the whole area. The floating beam structure is described as a distributed parameter system with partial differential equation (PDE). According to Lyapunov stability principle, a hybrid active and passive controller is designed to suppress the vibration of VLFS for the improvement in serviceability. In the active control design, two boundary controllers are developed to act on the upstream and downstream ends of VLFS, respectively. In passive control design, passive control components with high elastic rigidities are used to absorb the dynamic energy of VLFS from waves. Numerical simulations with comparison to the existing active control method are used to verify the effectiveness of the proposed control method. The parametric studies are given to examine the effects of various parameters to the vibration response of VLFS.

Appendices

Appendix 1

The kinetic energy $E_\mathrm{k}$ in the VLFS system can be described as

$$\begin{aligned} E_\mathrm{k} (t)=\frac{1}{2}\int _0^L {\rho \left( {\frac{\partial w(x,t)}{\partial t}} \right) ^{2}\hbox {d}x} \end{aligned}$$

(8)

where x and t represent the independent spatial and temporal variables, respectively. w(x, t) and $\dot{w}(x,t)$ are the position and velocity on the position x of VLFS system at the time t.

The bending and tension potential energy $E_\mathrm{p}$ in the VLFS system can be

$$\begin{aligned} E_\mathrm{p} (t)= & {} \frac{1}{2}\int _0^L {EI\left( x \right) \left( {\frac{\partial ^{2}w(x,t)}{\partial x^{2}}} \right) ^{2}\hbox {d}x}\nonumber \\&+\frac{1}{2}T\int _0^L {\left( {\frac{\partial w(x,t)}{\partial x}} \right) ^{2}\hbox {d}x} \end{aligned}$$

(9)

where EI and T are the bending rigidity and tension of the VLFS system, respectively.

The virtual work done by ocean wave uncertainties on the VLFS is given by

$$\begin{aligned} \delta W_f =\int _0^L {f(x,t)} \delta w(x,t) \mathrm {dx} \end{aligned}$$

(10)

where f(x, t) is the distributed load on the VLFS due to the hydrodynamic effects of the ocean waves.

The virtual work done by damping on the VLFS system is described by

$$\begin{aligned} \delta W_{d}= & {} -\int _0^L {c_\mathrm{d} \frac{\partial w(x,t)}{\partial t}} \delta w(x,t)\mathrm{dx} \nonumber \\&-\,c_\mathrm{d} \frac{\partial w(L,t)}{\partial t}\delta w(L,t)\nonumber \\&-\,c_\mathrm{d} \frac{\partial w(0,t)}{\partial t}\delta w(0,t) \end{aligned}$$

(11)

where $c_\mathrm{d}$ is the damping coefficient of the VLFS system.

The virtual work done by spring on the VLFS system is described by

$$\begin{aligned} \delta W_r= & {} -\int _0^L {k_\mathrm{c} w(x,t)} \delta w(x,t)\mathrm{dx} \nonumber \\&-\,k_\mathrm{c} w(L,t)\delta w(L,t)-k_\mathrm{c} w(0,t)\delta w(0,t) \end{aligned}$$

(12)

where $k_\mathrm{c}$ is the spring constant of the hydrostatic restoring force.

The virtual work done by the boundary control is written as

$$\begin{aligned} \delta W_m =u(L,t)\delta w(L,t)+v(0,t)\delta w(0,t) \end{aligned}$$

(13)

where u(L, t) is the control input at the position L of VLFS and v(0, t) is the control input at the position 0 of VLFS.

Thus, we have the total virtual work done on the VLFS as

$$\begin{aligned} \delta W=\delta W_f +\delta W_m +\delta W_d +\delta W_r \end{aligned}$$

(14)

Hamilton’s principle which is utilized to derive the governing equations of VLFS system is represented by

$$\begin{aligned} \int _{t_1 }^{t_2 } {\delta \left[ {E_\mathrm{k} (t)-E_\mathrm{p} (t)+W(t)} \right] \hbox {d}t} =0 \end{aligned}$$

(15)

where $t_1$ and $t_2$ are two time instants with $t_1<t<t_2$ and $\delta$ denotes the variational operator.

Then, we can obtain the governing equations of the system (1) with boundary conditions (1)–(3).

Appendix 2

Consider the system (1) and introduce the following appropriate integral Lyapunov functional candidates:

$$\begin{aligned} V(t)=V_1 (t)+V_2 (t)+V_3 (t) \end{aligned}$$

(16)

where

$$\begin{aligned} V_1 (t)= & {} \frac{\beta \rho }{2}\int _0^L {w_t^2 (x,t)} \hbox {d}x \nonumber \\&+\,\frac{\beta }{2}\int _0^L {EI(x)w_{xx}^2 (x,t)} \hbox {d}x \nonumber \\&+\,\frac{\beta T}{2}\int _0^L {w_x^2 (x,t)} \hbox {d}x \nonumber \\&+\,\gamma \rho \int _0^L {w(x,t)w_t (x,t)} \hbox {d}x\nonumber \\&+\,\frac{\beta k_c }{2}\int _0^L {w^{2}(x,t)} \hbox {d}x \end{aligned}$$

(17)

$$\begin{aligned} V_2 (t)= & {} \frac{c_\mathrm{d} }{2}\left[ {\alpha Lw_x (L,t)+\beta w_t (L,t)+\gamma w(L,t)} \right] ^{2} \nonumber \\&+\,\frac{c_\mathrm{d} }{2}\left[ {\beta w_t (0,t)+\gamma w(0,t)} \right] ^{2} \end{aligned}$$

(18)

$$\begin{aligned} V_3 (t)= & {} \alpha \rho \int _0^L {xw_t (x,t)} w_x (x,t)\hbox {d}x \end{aligned}$$

(19)

Define a new function as

$$\begin{aligned}&\Gamma (t)=\int _0^L \left( w_t^2 (x,t) +w_{xx}^2 (x,t)\right. \\&\left. \quad +\, w_x^2 (x,t)+w^2 (x,t)\right) \hbox {d}x \end{aligned}$$

Then, we have

$$\begin{aligned}&\left( {\frac{\beta }{2}\min \left( {\rho ,EI\left( x \right) ,T,k_\mathrm{c} } \right) -\frac{\gamma \rho }{2}} \right) \Gamma (t)\le V_1 (t)\nonumber \\&\quad \le \left( {\frac{\beta }{2}\max \left( {\rho ,EI\left( x \right) ,T,k_\mathrm{c} } \right) +\frac{\gamma \rho }{2}} \right) \Gamma (t) \end{aligned}$$

(20)

Appling Lemma 1 to Eq. (20) yields

$$\begin{aligned}&\left| {V_3 (t)} \right| \le \alpha \rho L\int _0^L \left( w_t^2 (x,t)+w_x^2 (x,t)\right) \hbox {d}x\nonumber \\&\quad \le s_1 \Gamma (t) \end{aligned}$$

(21)

where $s_1 =\alpha \rho L$.

Then, it can be obtained that

$$\begin{aligned} -s_1 \Gamma (t)\le V_3 (t)\le s_1 \Gamma (t) \end{aligned}$$

(22)

Combing Eqs. (20) and (22), we further obtain

$$\begin{aligned} s_2 \Gamma (t)\le V_1 (t)+V_3 (t)\le s_3 \Gamma (t) \end{aligned}$$

(23)

with

$$\begin{aligned} s_2= & {} \frac{\beta }{2}\min \left( {\rho ,EI\left( x \right) ,T,k_\mathrm{c} } \right) -\frac{\gamma \rho }{2}-s_1 \end{aligned}$$

(24)

$$\begin{aligned} s_3= & {} \frac{\beta }{2}\max \left( {\rho ,EI\left( x \right) ,T,k_\mathrm{c} } \right) +\frac{\gamma \rho }{2}+s_1 \end{aligned}$$

(25)

where $s_3$ is a positive constant, and $s_2$ will be positive due to the appropriate choice of parameters $\alpha$ and $\beta$.

Given the proposed Lyapunov functional (16), it can be obtained

$$\begin{aligned} s_4 \left( {\Gamma (t)+V_2 (t)} \right) \le V(t)\le s_5 \left( {\Gamma (t)+V_2 (t)} \right) \end{aligned}$$

(26)

where $s_4 =\min (s_2 ,1)$, $s_5 =\max (s_3 ,1)$ are positive constants.

Differentiating Eq. (16) V(t) with respect to time t yields

$$\begin{aligned} \dot{V}(t)=\dot{V}_1 (t)+\dot{V}_2 (t)+\dot{V}_3 (t) \end{aligned}$$

(27)

where

$$\begin{aligned} \dot{V}_1 (t)\le & {} -(\beta {c}_\mathrm{d} -\gamma \rho \nonumber \\&-\beta \sigma -\mu _1 \gamma c_\mathrm{d} )\int _0^L {w_t^2 (x,t)} \hbox {d}x \nonumber \\&+\left( {\frac{\beta }{\sigma }+\mu _2 \gamma } \right) \int _0^L {f^{2}(x,t)} \hbox {d}x\nonumber \\&-\left( {rk_\mathrm{c} -\frac{\gamma c_\mathrm{d} }{\mu _1 }-\frac{\gamma }{\mu _2 }} \right) \int _0^L {w^{2}(x,t)} \hbox {d}x \nonumber \\&-\, \gamma T\int _0^L {w_x^2 (x,t)} \hbox {d}x\nonumber \\&-\, \gamma \int _0^L {EI(x)w_{xx}^2 (x,t)} \hbox {d}x\nonumber \\&-\, \beta EI(L)w_{xxx} (L,t)w_t (L,t)) \nonumber \\&+\, \beta EI(0)w_{xxx} (0,t)w_t (0,t)\nonumber \\&+\beta Tw_x (L,t)w_t (L,t)\nonumber \\&-\, \beta Tw_x (0,t)w_t (0,t) \nonumber \\&+\, \gamma Tw(L,t)w_x (L,t)-\gamma Tw(0,t)w_x (0,t) \nonumber \\&-\, \gamma EI(L)w(L,t)w_{xxx} (L,t)\nonumber \\&+\, \gamma EI(0)w(0,t)w_{xxx} (0,t) \end{aligned}$$

(28)

Substituting boundary conditions (3) and (4) into $\dot{V}_2 (t)$, we obtain

$$\begin{aligned} \dot{V}_2 (t)= & {} \left[ {\beta w_t (0,t)+\gamma w(0,t)} \right] \nonumber \\&\quad \left[ {\begin{array}{l} \gamma c_\mathrm{d} w_t (0,t)+\beta v_t (t)-\beta Tw_{xt} (0,t) \\ +\, \beta \left( {EI(0)w_{xx} (0,t)} \right) _{xt} -\beta k_\mathrm{c} w_t (0,t) \\ \end{array}} \right] \nonumber \\&+ \left[ {\begin{array}{l} \alpha Lw_x (L,t) \\ +\, \beta w_t (L,t)+\gamma w(L,t) \\ \end{array}} \right] \nonumber \\&\quad \left[ {\begin{array}{l} c_\mathrm{d} \alpha Lw_{xt} (L,t)\\ +\, c_\mathrm{d} \gamma w_t (L,t)-\beta k_\mathrm{c} w_t (L,t) \\ +\, \beta \left( {EI(L)w_{xx} (L,t)} \right) _{xt} \\ +\, \beta u_t (t)-\beta Tw_{xt} (L,t) \\ \end{array}} \right] \end{aligned}$$

(29)

Substituting the system Eq. (1) into $\dot{V}_3 (t)$ and using the boundary conditions (2)–(4), we obtain the derivative of $V_3 (t)$ along VLFS system trajectory as

$$\begin{aligned}&\dot{V}_3 (t)\le -\left( {\frac{\alpha \rho }{2}-\frac{\alpha c_\mathrm{d} L}{\delta _1 }} \right) \int _0^L {w_t^2 (x,t)} \hbox {d}x\nonumber \\&\quad +\frac{\alpha L}{\delta _2 }\int _0^L {f^{2}(x,t)} \hbox {d}x \nonumber \\&\quad +\frac{\alpha k_\mathrm{c} }{2}\int _0^L {w^2 (x,t)} \hbox {d}x\nonumber \\&\quad -\frac{\alpha }{2}\int _0^L {\left( {4EI(x)-\left( {xEI(x)} \right) _x } \right) w_{xx}^2 (x,t)} \hbox {d}x\nonumber \\&\quad -\left( {\frac{\alpha T}{2}+\frac{\alpha \rho }{2}-\alpha \delta _1 c_\mathrm{d} L-\alpha \delta _2 L} \right) \int _0^L {w_x^2 (x,t)} \hbox {d}x \nonumber \\&\quad +\frac{\alpha TL}{2}w_x^2 (L,t)-\alpha EI(L)Lw_x (L,t)w_{xxx} (L,t) \nonumber \\&\quad -\frac{\alpha k_\mathrm{c} L}{2}w^{2}(L,t)+\frac{\alpha \rho L}{2}w_t^2 (L,t) \end{aligned}$$

(30)

Substituting the proposed law (5) and (6) into Eq. (29) as well as combining Eqs. (28)–(30), we have

$$\begin{aligned}&\dot{V}(t)\le -\left( {\beta c_\mathrm{d} +\frac{\alpha \rho }{2}-\gamma \rho -\beta \sigma -\mu _1 \gamma c_\mathrm{d} -\frac{\alpha c_\mathrm{d} L}{\delta _1 }} \right) \nonumber \\&\qquad \int _0^L {w_t^2 (x,t)} \hbox {d}x\nonumber \\&\quad +\left( {\frac{\beta }{\sigma }+\mu _2 \gamma +\frac{\alpha L}{\delta _2 }} \right) \int _0^L {f^{2}(x,t)} \hbox {d}x\nonumber \\&\quad -\left( {\gamma k_\mathrm{c} -\frac{\gamma c_\mathrm{d} }{\mu _1 }-\frac{\gamma }{\mu _2 }-\frac{\alpha k_\mathrm{c} }{2}} \right) \int _0^L {w^{2}(x,t)} \hbox {d}x \nonumber \\&\quad -\left( {\gamma T+\frac{\alpha T}{2}+\frac{\alpha \rho }{2}-\alpha \delta _1 c_\mathrm{d} L-\alpha \delta _2 L} \right) \int _0^L {w_x^2 (x,t)} \hbox {d}x \nonumber \\&\quad -\left( {2\alpha EI(x)-\frac{\alpha }{2}\left( {xEI(x)} \right) _x +\gamma EI\left( x \right) } \right) \int _0^L {w_{xx}^2 (x,t)} \hbox {d}x \nonumber \\&\quad -\, k\left[ {\alpha Lw_x (L,t)+\beta w_t (L,t)+\gamma w(L,t)} \right] ^{2} \nonumber \\&\quad -\, k_1 \left[ {\beta w_t (0,t)+\gamma w(0,t)} \right] ^{2}\nonumber \\&\quad -\left( {\beta g-\frac{\alpha \rho L}{2}-\frac{\alpha Lg}{\mu _3 }-\frac{\gamma g}{\mu _4 }} \right) w_t^2 (L,t) \nonumber \\&\quad -\left( {\frac{\alpha TL}{2}-\alpha Lg\mu _3 } \right) w_x^2 (L,t)\nonumber \\&\quad -\left( {\frac{\alpha k_\mathrm{c} L}{2}-\gamma g\mu _4 } \right) w_x^2 (L,t) \end{aligned}$$

(31)

where $K_\mathrm{D}$, g, r, $\beta$, $\sigma$, $\alpha$, $\delta _1$, $\delta _2$, $\delta _3$, $\delta _4$ and $\xi$ are chosen to satisfy the following inequalities:

$$\begin{aligned}&f_1 (\beta , \sigma , \alpha , \delta _i, \mu _i, \mathrm{g}, \gamma )=\beta c_\mathrm{d} +\frac{\alpha \rho }{2}\nonumber \\&\quad -\gamma \rho -\beta \sigma -\mu _1 \gamma c_\mathrm{d} -\frac{\alpha c_\mathrm{d} L}{\delta _1 }>0 \end{aligned}$$

(32)

$$\begin{aligned}&f_2 (\beta ,\sigma ,\alpha ,\delta _i ,\mu _i ,\mathrm{g},\gamma )=\gamma k_\mathrm{c} -\frac{\gamma c_\mathrm{d} }{\mu _1 }\nonumber \\&\quad -\frac{\gamma }{\mu _2 }-\frac{\alpha k_\mathrm{c} }{2}>0 \end{aligned}$$

(33)

$$\begin{aligned}&f_3 (\beta ,\sigma ,\alpha ,\delta _i ,\mu _i ,\mathrm{g},\gamma )=\gamma T+\frac{\alpha T}{2}+\frac{\alpha \rho }{2}\nonumber \\&\quad -\alpha \delta _1 c_\mathrm{d} L-\alpha \delta _2 L>0 \end{aligned}$$

(34)

$$\begin{aligned}&f_4 (\beta ,\sigma ,\alpha ,\delta _i ,\mu _i ,\mathrm{g},\gamma )=\beta g\nonumber \\&\quad -\frac{\alpha \rho L}{2}-\frac{\alpha Lg}{\mu _3 }-\frac{\gamma g}{\mu _4 }>0 \end{aligned}$$

(35)

$$\begin{aligned}&f_5 (\beta ,\sigma ,\alpha ,\delta _i ,\mu _i ,\mathrm{g},\gamma )=\frac{\alpha TL}{2}-\alpha Lg\mu _3 >0 \end{aligned}$$

(36)

$$\begin{aligned}&f_6 (\beta ,\sigma ,\alpha ,\delta _i ,\mu _i ,g,\gamma )=\frac{\alpha k_\mathrm{c} L}{2}-\gamma g\mu _4 >0 \end{aligned}$$

(37)

$$\begin{aligned}&f_7 (\beta ,\sigma ,\alpha ,\delta _i ,\mu _i ,\mathrm{g},\gamma )=2\alpha EI(x)\nonumber \\&\quad -\frac{\alpha }{2}\left( {xEI(x)} \right) _x +\gamma EI\left( x \right) >0 \end{aligned}$$

(38)

Then, we have

$$\begin{aligned}&\dot{V}(t)\le -\lambda _1 \Gamma (t)-\frac{2\min (k,k_1 )}{c_\mathrm{d} }V_2 +\varepsilon \le -\lambda _2\nonumber \\&\quad \left( {\Gamma (t) +V_2 } \right) +\varepsilon \end{aligned}$$

(39)

where $\varepsilon =\left( {\frac{\beta }{\sigma }+\mu _2 \gamma +\frac{\alpha L}{\delta _2 }} \right) \int _0^L {f^{2}(x,t)} \hbox {d}x$,$\lambda _1 =\min (f_1 ,f_2 ,f_3 )$ and $\lambda _2 =\min (\lambda _1 ,\frac{2\min (k,k_1 )}{T})$.

Furthermore, we have

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

(40)

where $\lambda ={\lambda _2 }/{s_5 }>0$.

By integrating the inequality (40), we obtain

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

(41)

It indicates that V(t) is bounded. We have

$$\begin{aligned}&\left( {\frac{\beta T}{2L}\hbox {+}\frac{\beta EI\left( x \right) }{2L^{3}}} \right) w^{2}(x,t)\le \frac{\beta T}{2}\int _0^L {w_x^2 (x,t)} \hbox {d}x\nonumber \\&\quad +\, \frac{\beta EI\left( x \right) }{2}\int _0^L {w_{xx}^2 (x,t)} \hbox {d}x\nonumber \\&\quad \le V_1 (t)\le \frac{1}{s_4 }V(t) \end{aligned}$$

(42)

Rearrangement of the term in the above inequality can yield

$$\begin{aligned} \left| {w(x,t)} \right|\le & {} \sqrt{\frac{1}{\frac{\beta T}{2L}\hbox {+}\frac{\beta EI\left( x \right) }{2L^{3}}}\left( {V(0)e^{-\lambda t}+\frac{\varepsilon }{\lambda }} \right) } \nonumber \\\le & {} \sqrt{\frac{1}{\frac{\beta T}{2L}\hbox {+}\frac{\beta EI\left( x \right) }{2L^{3}}}\left( {V(0)+\frac{\varepsilon }{\lambda }} \right) } \end{aligned}$$

(43)

where w(x, t) is uniformly bounded.