Stochastic stabilization and destabilization of ship maneuvering motion by multiplicative noise

Abstract

In 1982, Tsubokawa et al. (J SocNaval Arch Japan 1982: 101, 1982) examined ship maneuvering motion and revealed through numerical simulations that even a ship with unstable maneuvering characteristics could stabilize its course stability when wind or wave disturbance occurs. In this study, we assume that the facts confirmed by Tsubokawa et al. (J SocNaval Arch Japan 1982: 101, 1982) in their numerical calculations are attributed to the random variation of the coefficients inside the equations of motion. Moreover, in the control research field, certain contributions to stabilization and destabilization because of noise disturbance have been reported. Therefore, the existing works of Mao (Syst Control Lett 23: 2709, 1994), Arnold (Siam J Appl Mathemat 46: 427, 1986), and Kozin (SIAM J Appl Mathemat 21: 413, 1971) were explored and extended in this study. Moreover, we uncovered the mechanism of stabilization and destabilization of the ship maneuvering system.

Appendices

Appendix1 (Application of Mao’s method [2])

As indicated in previous sections, multiplicative noise stabilizes the system in this case. Therefore, in this section, the authors theoretically show the stability condition of Eq. 40.

We define x(t) as

$$\begin{aligned} x(t) \equiv \begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix} =\begin{bmatrix} r(t)\\ \textrm{d}r(t)/\textrm{d}t \end{bmatrix} \in \mathbb {R}^2. \end{aligned}$$

(89)

Then, we rewrite the SDE as follows:

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned} \textrm{d}x_1(t) =&x_2(t) \textrm{d}t\\ \textrm{d}x_2(t) =&( - \kappa x_2(t) - c_1 x_1(t) - c_3 x_1(t) ^ 3 ) \textrm{d}t \\&+ \Gamma x_1 \textrm{d}W(t), \end{aligned} \right. \\ \text {where} \quad \left\{ \begin{aligned} \kappa&\equiv \frac{T_{1} + T_{2}}{T_{1} T_{2}}\\ c_1&\equiv \frac{1 + K_0 K_{\textrm{D}}}{T_{1} T_{2}},\, c_3 \equiv \frac{\alpha _3}{T_{1} T_{2}}\\ \Gamma&\equiv \frac{\gamma }{T_{1} T_{2}}, \end{aligned} \right. \end{aligned} \end{aligned}$$

(90)

and W(t) is a one-dimensional standard Wiener process. The above equation can be written in the following vector form:

$$\begin{aligned} \begin{aligned} \textrm{d}x(t) = \mu (x(t), t) \textrm{d} t + \tilde{\Gamma } \, x(t) \textrm{d} W(t),\\ \text {where} \quad \left\{ \begin{aligned} \mu (x(t),t)&= \begin{bmatrix} x_2 \\ - \kappa x_2 - c_1 x_1 - c_3 x_1 ^ 3 \end{bmatrix}\\ \tilde{\Gamma }&= \begin{bmatrix} 0 &{} 0 \\ \Gamma &{} 0 \\ \end{bmatrix}, \end{aligned} \right. \end{aligned} \end{aligned}$$

(91)

.

Here the authors follow the approach of Mao [2]. Mao applied the It$\hat{\textrm{o}}$’s formula for $\log (\Vert x(t)\Vert ^2)$. By using $\log (\Vert x(t)\Vert ^2)$, the exponential stability of the system can be discussed. As Mao described in his paper [2], if the following condition:

$$\begin{aligned} \displaystyle \limsup _{t \rightarrow \infty } \frac{1}{t} \log (\Vert x(t)\Vert ^2) <0 \end{aligned}$$

(92)

is satisfied, then the system is almost surely exponentially stable. Then, he obtained the following equation in the integral form:

$$\begin{aligned} \begin{aligned} \begin{aligned}&\log \left( \Vert x(t)\Vert ^{2}\right) =\\&\log \left( \Vert x(t_0)\Vert ^{2}\right) \\&+ \int _{t_0}^{t} \Vert x(s)\Vert ^{-2} 2 x(s)^T \mu (x(s), s) \textrm{d} s \\&+ \frac{1}{2} \int _{t_0}^{t}\Vert x(s)\Vert ^{-4}\\&\cdot \left[ 2\Vert x(s)\Vert ^{2} \Vert \tilde{\Gamma } \, x(s)\Vert ^{2} - 4\left( x(s)^T \, \tilde{\Gamma } \, x(s)\right) ^{2}\right] \textrm{d} s\\&+ 2 \int _{t_0}^{t} \Vert x(s)\Vert ^2 \left( x(s)^T \, \tilde{\Gamma } \, x(s) \right) \textrm{d}W(s) \end{aligned} \end{aligned} \end{aligned}$$

(93)

or in differential form:

$$\begin{aligned} \begin{aligned}&\textrm{d} \log \left( \Vert x(t)\Vert ^{2}\right) =\\&\Vert x(t)\Vert ^{-2} 2 x^T(t) \mu (x(t), t) \textrm{d}t \\&+ \frac{1}{2} \Vert x(t)\Vert ^{-4}\\&\cdot \left[ 2\Vert x(t)\Vert ^{2} \Vert \tilde{\Gamma } \, x(t)\Vert ^{2}-4\left( x^T(t) \, \tilde{\Gamma } \, x(t)\right) ^{2}\right] \textrm{d}t \\&+ 2 \Vert x(t)\Vert ^{-2} \left( x^T(t) \, \tilde{\Gamma } \, x(t) \right) \textrm{d}W. \end{aligned} \end{aligned}$$

(94)

Then Mao [2] showed that the stability of the system can be evaluated with the use of Eqs. 93 and 94. This approach is applicable not only to the current system but also to other systems that appeared in our naval architecture research field.

Further, the authors analyze the system with $c_3 = 0$ as:

$$\begin{aligned} \begin{aligned} \textrm{d}x(t) = \mu (x(t), t) \textrm{d} t + \sigma (x(t),t) \textrm{d} W(t),\\ \text {where} \quad \left\{ \begin{aligned} \mu (x(t),t)&= \begin{bmatrix} x_2(t) \\ - \kappa x_2(t) - c_1 x_1(t) \end{bmatrix}\\ \sigma (x(t),t)&= \begin{bmatrix} 0 \\ \Gamma x_1(t) \\ \end{bmatrix}, \end{aligned} \right. \end{aligned} \end{aligned}$$

(95)

and W(t) is a one-dimensional standard Wiener process. As shown in the previous section, Mao [2] applied the It$\hat{\textrm{o}}$’s formula to $\log \left( \Vert x(t)\Vert ^2\right)$. However, considering the simple application of Mao’s original approach [2], it seems to be difficult to show the negativeness of $\log \left( \Vert x(t)\Vert ^{2}\right)$. Therefore, Hoshino [27] suggested to use the quadratic form function $F(x(t)) \equiv \log \left( x^T(t) \, P \, x(t) \right)$. If $x^T(t) \, P \, x(t)$ is positive definite and $F(x(t))<0$, then the system is almost surely exponentially stable. Here, P is a positive definite $2 \times 2$ matrix that has the following form:

$$\begin{aligned} P = \begin{bmatrix} p_1 &{} p_2 \\ p_3 &{} p_4 \\ \end{bmatrix} \end{aligned}$$

(96)

The corresponding components $p_i \, (i = 1,2,3,4)$ are determined by the following considerations. Now, we have:

$$\begin{aligned} x^T(t) P x(t) = p_1 x_1^2(t) + (p_2 + p_3) x_1(t) x_2(t) + p_4 x_2^2(t) \end{aligned}$$

(97)

Then, using $\log \left( x^T(t) P x(t) \right)$ instead of $\log \left( \Vert x(t)\Vert ^{2}\right)$, the stability can be judged. The discussion on this approach can be found in our literature [16].

Appendix2 (Application of Arnold’s method [3])

Arnold [3] demonstrated the condition of stability by obtaining the Lyapunov exponent. Now, the authors apply Arnold’s approach to the system Eq. 90 with $c_3=0$ is as follows:

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned}&\textrm{d}x_1(t) = x_2(t) \textrm{d}t\\&\textrm{d}x_2(t) = ( - \kappa x_2(t) - c_1 x_1(t) ) \textrm{d}t + \Gamma x_1(t) \textrm{d}W(t) \end{aligned} \right. \\ \end{aligned} \end{aligned}$$

(98)

where $\kappa$, $c_1$, and $\Gamma$ are the same as those in Eq. 90, and W(t) is a one-dimensional standard Wiener process. Arnold [3] tackled the stability condition for the system with linear damping and restoring terms, and the system resembles our target system. Furthermore, Arnold’s theory can be applied to the colored noise case. Let the external noise be $\xi (t)$, where $\xi (t)$ is an ergodic Markov process. The 2D system that Arnold tackled is as follows:

$$\begin{aligned} \frac{\textrm{d}^2 x_1(t)}{\textrm{d}t^2} + \kappa \frac{\textrm{d} x_1(t)}{\textrm{d} t} + \left[ c_1 + \bar{\Gamma } F\left( \xi \left( \frac{t}{\rho }\right) \right) \right] x_1(t) = 0 \end{aligned}$$

(99)

Hereafter, the authors show the results for Eq. 99, which extend the system of Eq. 98.

As is well known, the variable transformation

$$\begin{aligned} x_1(t) = y(t) \exp \left( -\frac{\kappa }{2} t\right) \end{aligned}$$

(100)

allows us to translate Eq. 99 into a system in which the damping term does not exist, which leads to:

$$\begin{aligned} \frac{\textrm{d}^2 y(t)}{\textrm{d}t^2} + \left[ c_1 - \frac{\kappa ^2}{4} + \bar{\Gamma } F\left( \xi \left( t \right) \right) \right] y(t) = 0 \end{aligned}$$

(101)

Now, the authors define a new variable and coefficient:

$$\begin{aligned} \tilde{F}(\xi (t)) = - F(\xi (t)), \quad \mathscr {C} = c_1 - \frac{\kappa ^2}{4}, \end{aligned}$$

(102)

then the system becomes

$$\begin{aligned} - \frac{\textrm{d}^2 y}{\textrm{d}t^2} + \bar{\Gamma } \tilde{F}(\xi (t)) y = \mathscr {C} y \end{aligned}$$

(103)

Arnold demonstrated the theoretical result of Lyapunov exponent $\lambda _{\textrm{L}}$ as follows:

$$\begin{aligned} \begin{aligned} \lambda _{\textrm{L}} = \left( - \frac{\kappa }{2}\right) \tilde{\lambda }_{\textrm{L}}\\ \text {where}\\ \begin{aligned} \tilde{\lambda }_{\textrm{L}} =&\sqrt{- \mathscr {C}} + \frac{\bar{\Gamma } ^ 2}{4 \mathscr {C}} \int _0^{\infty } \exp \left( -2 \sqrt{- \mathscr {C}} t \right) C(t) \textrm{d} t\\&+ o(\bar{\Gamma }^3) \end{aligned} \end{aligned} \end{aligned}$$

(104)

In this equation, $- \kappa / 2 < 0$ is constant, so hereafter the authors consider the sign of $\tilde{\lambda }_{\textrm{L}}$. Here, C(t) indicates the covariance of $F(\xi (t))$ defined as

$$\begin{aligned} C(t) \equiv \mathbb {E} [F(\xi (0)) \cdot F(\xi (t))]. \end{aligned}$$

(105)

For white noise, C(t) has the form of Dirac’s delta function, that is, $C(t) = \tilde{C} \delta _{\textrm{D}}(t)>0$. Here, $\delta _{\textrm{D}}(t)$ is Dirac’s delta function, and C is the positive constant. Therefore, we can explore additional manipulation of Eq. (104):

$$\begin{aligned} \begin{aligned} \tilde{\lambda }_{\textrm{L}} =&\sqrt{-\mathscr {C}} + \frac{ \tilde{C} \bar{\Gamma } ^ 2}{8 \mathscr {C}} + o(\bar{\Gamma }^3)\\ \end{aligned} \end{aligned}$$

(106)

Now, let us consider the negative condition of Eq. (106). The authors tackle the unstable system for the case $c_1 < 0$, and note that $\mathscr {C} < 0$. Therefore, the stability condition becomes:

$$\begin{aligned} {\bar{\Gamma }}^{2} > - \frac{8 (-\mathscr {C}) \sqrt{-\mathscr {C}}}{\tilde{C}} \end{aligned}$$

(107)

From the above-mentioned results, the authors notice that the present system can definitely be stabilized by adding some amount of multiplicative noise. This fact coincides with the behavior of our system. However, as shown in Figs. 4,5, additional increase in the intensity strength of the noise deteriorates system stability again. Therefore, the conclusions derived from the application of Arnold are inconsistent with these results. The source of this conflict is the assumption of Arnold’s theory [3]. Arnold applied perturbation theory in which $\bar{\Gamma }$ was sufficiently small. Therefore, with larger noise strength exceeding this assumption, the authors cannot judge system stability. Therefore, in order to explore not only stabilization but also destabilization with larger noise, the authors devise another approach.