To convert the original nonlinear set of differential equation into the linear one by using the equivalent linearization technique, the augmented state vector must be transformed to the centralized state vector
$$\begin{aligned} \mathbf {Y}^0(t)=\big [ \begin{array}{lllllllll} Y_1^0&Y_2^0&Y_3^0&Y_4^0&Y_5^0&Y_6^0&Y_7^0&Y_8^0&Y_9^0 \end{array}\big ] ^\mathrm{T}\end{aligned}$$
(18)
in accordance with the following expressions
$$\begin{aligned} Y_1^0(t)&= q_r(t)-\mu _{q_r}(t),\quad Y_2^0(t)=\dot{q}_r(t)-\mu _{\dot{q}_r}(t), \\ Y_3^0(t)&= z_d(t)-\mu _{z_d}(t),\quad Y_4^0(t)=\dot{z}_d(t)-\mu _{\dot{z}_d}(t), \\ Y_5^0(t)&= u_{M}(t)-\mu _{u_{M}}(t),\quad Y_6^0(t)=\dot{u}_{M}(t)-\mu _{\dot{u}_{M}}(t), \\ Y_7^0(t)&= v_0(t)-\mu _{v_0}(t),\quad Y_8^0(t)=\dot{v}_0(t)-\mu _{\dot{v}_0}(t), \\ Y_9^0(t)&= X(t)-\mu _{X}(t). \end{aligned}$$
(19)
The Equation (
15) can be then rewritten in the following form
$$\begin{aligned} d\mathbf {Y}^0(t)=\mathbf {c}^0\big (\mathbf {Y}^0(t),t\big )dt+\mathbf {b}(t)dW(t) \end{aligned}$$
(20)
where the centralized drift vector is given by
$$\begin{aligned} \mathbf {c}^0(\mathbf {Y}^0(t),t)=\mathbf {c}\big (\mathbf {Y}^0(t),t\big ) -\mathrm {E}\big [\mathbf {c}(\mathbf {Y}^0(t),t)\big ]. \end{aligned}$$
(21)
The diffusion vector is assumed to be independent of the state vector and defined as
$$\begin{aligned} \mathbf {b}(t)=\big [\begin{array}{llllllllll} 0&0&0&0&0&0&0&0&\alpha \sqrt{2\pi S_{0}} \end{array}\big ]^\mathrm{T} \end{aligned}$$
(22)
The elements of vector
\(\mathbf {c}^0(\mathbf {Y}^0(t),t)\) are obtained in following form
$$\begin{aligned} c_1^0(\mathbf {Y}^0(t))&= Y_2^0, \\ c_2^0(\mathbf {Y}^0(t))&= \frac{1}{m_r}\bigg \{-\tilde{c}_rY_2^0-\tilde{k}_r Y_1^0+\{k_d[Y_3^0-\varPhi _r(L)Y_1^0] \\&+\,c_d[Y_4^0-\varPhi _r(L)Y_2^0]\}\varPhi _r(L)+EA\frac{1}{L(\tau )}\varGamma _{rn}(Y_5^0Y_1^0 \\&-\,\mathrm {E}[Y_5^0Y_1^0]+Y_1^0\mu _{u_{M}}+Y_5^0\mu _{{q}_{r}}) \\&-\,EA\frac{1}{L(\tau )}\frac{\varPsi _L-1}{L}\varPhi _r(L)(Y_5^0Y_7^0-\mathrm {E}[Y_5^0Y_7^0] \\&+\,Y_7^0\mu _{u_{M}}+Y_5^0\mu _{v_0}) \\&+\,EA\frac{1}{2L}\kappa _{rr}\varGamma _{rn}\bigg [(Y_1^0)^3+3(Y_1^0)^2\mu _{{q}_{r}} \\&-\,3\mathrm {E}[(Y_1^0)^2]\mu _{{q}_{r}}+3Y_1^0\mu _{{q}_{r}}^2\bigg ] \\&-\,EA\frac{1}{2}\bigg (\frac{\varPsi _L-1}{L(\tau )}\bigg )^3\varPhi _r(L) \\&\times \bigg [(Y_7^0)^3+3(Y_7^0)^2\mu _{v_0}-3\mathrm {E}[(Y_7^0)^2]\mu _{v_0}+3Y_7^0\mu _{v_0}^2\bigg ] \\&+\,EA\frac{\varPsi _L-1}{L(\tau )}\varPhi _r(L)\bigg [\varGamma _{rn}-\frac{1}{2}\frac{1}{L}\kappa _{rr}\bigg ] \\&\times \bigg [Y_7^0(Y_1^0)^2+2Y_7^0Y_1^0\mu _{{q}_{r}}-2\mathrm {E}[Y_7^0Y_1^0]\mu _{{q}_{r}} \\&+\,Y_7^0\mu _{{q}_{r}}^2+(Y_1^0)^2\mu _{v_0}-\mathrm {E}[(Y_1^0)^2]\mu _{v_0}+2Y_1^0\mu _{{q}_{r}}\mu _{v_0}\bigg ] \\&+\,EA\bigg (\frac{\varPsi _L-1}{L(\tau )}\bigg )^2\bigg [\frac{1}{2}\varGamma _{rn}-\varPhi _r^2(L)\bigg ] \\&\times \bigg [(Y_7^0)^2Y_1^0+(Y_7^0)^2\mu _{{q}_{r}}\!\!-\mathrm {E}[(Y_7^0)^2]\mu _{{q}_{r}}+2Y_7^0Y_1^0\mu _{v_0} \\&-\, 2\mathrm {E}[Y_7^0(t)Y_1^0]\mu _{v_0}\!\!+2Y_7^0\mu _{v_0}\mu _{{q}_{r}}\!\!+Y_1^0\mu _{v_0}^2\bigg ]\!\!+\beta _r^{(0)}Y_7^0 \\&+\, \beta _r^{(1)}Y_8^0+\beta _r^{(2)}\big [Y_9^0-2\zeta _{f}\varOmega _0Y_8^0-\varOmega _0^2 Y_7^0\big ]\bigg \}, \\ c_3^0(\mathbf {Y}^0(t))&= Y_4^0, \\ c_4^0(\mathbf {Y}^0(t))&= \frac{1}{m_d}\bigg \{-k_d[Y_3^0-\varPhi _r(L)Y_1^0]-c_d[Y_4^0-\varPhi _r(L)Y_2^0], \\&+\,k_d\varPsi _L(Y_7^0)+c_d\varPsi _L(Y_8^0)\bigg \}, \\ c_5^0(\mathbf {Y}^0(t))&= Y_6^0, \\ c_6^0(\mathbf {Y}^0(t))&= \frac{EA}{M+m_d}\bigg \{-\frac{Y_5^0}{L(\tau )}-\frac{1}{2}\bigg [\frac{1}{L}\kappa _{rr}((Y_1^0)^2 \\&-\,\mathrm {E}[(Y_1^0)^2]+2Y_1^0\mu _{q_{r}})+2\frac{\varPsi _L-1}{L(\tau )}\varPhi _r(L)(Y_1^0Y_7^0 \\&-\,\mathrm {E}[Y_1^0Y_7^0]+Y_7^0\mu _{q_{r}}+Y_1^0\mu _{v_{0}}) \\&+\,\bigg (\frac{\varPsi _L-1}{L(\tau )}\bigg )^2\!\!\!((Y_7^0)^2-\mathrm {E}[(Y_7^0)^2]+2Y_7^0\mu _{v_{0}})\bigg ]\!\bigg \}, \\ c_7^0(\mathbf {Y}^0(t))&= Y_8^0, \\ c_8^0(\mathbf {Y}^0(t))&= Y_9^0-2\zeta _{f}\varOmega _0Y_8^0-\varOmega _0^2Y_7^0, \\ c_9^0(\mathbf {Y}^0(t))&= -\alpha Y_9^0. \end{aligned}$$
(23)
The original nonlinear system given by Eq. (
20) is replaced in further consideration by the linear system defined as
$$\begin{aligned} d\mathbf {Y}^0(t)=\mathbf {B}\mathbf {Y}^0(t)dt+\mathbf {b}(t)dW(t), \end{aligned}$$
(24)
where the centralized drift terms are adopted as a linear form of the state variables
$$\begin{aligned} c^0_{i,eq}\big (\mathbf {Y}^0(t)\big )=B_{im}Y^0_{m} \end{aligned}$$
(25)
and components
\(B_{im}\) are obtained as
$$\begin{aligned} B_{im}\kappa _{mj}=\mathrm {E}\big [Y^0_{j}c^0_{i}(\mathbf {Y}^0)\big ], \end{aligned}$$
(26)
which in the matrix form is as follows
$$\begin{aligned} \mathbf {B}\mathbf {\kappa }(t)=\mathrm {E}\big [\mathbf {c}^0\big (\mathbf {Y}^0(t)\big ) {\mathbf {Y}^0}^{\mathrm {T}}\big ]. \end{aligned}$$
(27)
Due to the assumption about the jointly Gaussian distribution of the state variables
\(\mathbf {Y}^0(t)\), the presented relationship for zero-mean Gaussian random vector
\(\mathbf {X}\) [
23] is taken into account
$$\begin{aligned} \mathrm {E}\big [\mathbf {X}f(\mathbf {X})\big ]=\mathrm {E}\big [\mathbf {X}\mathbf {X}^\mathrm {T}\big ]\mathrm {E}\big [\nabla f(\mathbf {X})\big ], \end{aligned}$$
(28)
where
\(f(\mathbf {X})\) denotes the non-linear function and
\(\nabla\) is defined as
\(\nabla =\bigg [\frac{\partial }{\partial X_1},\frac{\partial }{\partial X_2},\ldots , \frac{\partial }{\partial X_n}\bigg ]^\mathrm {T}\). Transposing both sides of Eq. (
27) and using Eq. (
28) yields
$$\begin{aligned} \mathbf {\kappa }(t)\mathbf {B}^\mathrm {T} =\mathbf {\kappa }(t)\mathrm {E}\big [\nabla {\mathbf {c}^0}^{\mathrm {T}} (\mathbf {Y}^0(t))\big ]. \end{aligned}$$
(29)
That defines the components of matrix
\(\mathbf {B}\) as
$$\begin{aligned} \mathbf {B}^\mathrm {T}=\mathrm {E}\big [\nabla {\mathbf {c}^0}^{\mathrm {T}}(\mathbf {Y}^0(t))\big ]. \end{aligned}$$
(30)
Using Eq. (
30) and the elements of centralized drift vector given by Eqs. (
23) the unknown matrix is obtained in the form
$$\begin{aligned} \mathbf {B}=\left[ \begin{array}{lllllllll} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ p^{(1)}_{r} & p^{(2)}_{r}& \frac{k_d\varPhi _r}{m_r}& \frac{c_d\varPhi _r}{m_r}& p^{(3)}_r & 0 & p^{(4)}_{r}& p^{(5)}_{r}& \frac{\beta _r^{(2)}}{m_r}\\ 0 & 0 & 0 & 1 & 0 & 0 & 0& 0 & 0 \\ \frac{k_d\varPhi _r}{m_d}& \frac{c_d\varPhi _r}{m_d} & -\frac{k_d}{m_d}& -\frac{c_d}{m_d}& 0& 0 & \frac{k_d\varPsi _L}{m_d}& \frac{c_d\varPsi _L}{m_d} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ p^{(6)}_{r} & 0 & 0 & 0 & p^{(7)}_{r}& 0 & p^{(8)}_{r}& 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -\varOmega _0^2 & -2\zeta _f\varOmega _0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\alpha \\ \end{array} \right] \end{aligned}$$
where
$$\begin{aligned} p^{(1)}_{r}&= \frac{1}{m_r}\bigg \{-\tilde{k}_r-k_d\varPhi _r^2+EA\frac{1}{L}\varGamma _{rn}\mu _{u_{M}} \\&+\, EA\frac{3}{2L}\kappa _{rr}\varGamma _{rn}\bigg [\mathrm {E}[(Y_1^0)^2]+\mu _{{q}_{r}}^2\bigg ] \\&+\, 2EA\frac{\varPsi _L-1}{L}\varPhi _r\bigg [\varGamma _{rn}-\frac{1}{2}\frac{1}{L}\kappa _{rr}\bigg ]\bigg [\mathrm {E}[Y_7^0Y_1^0]+\mu _{{q}_{r}}\mu _{v_0}\bigg ] \\&+\, EA\bigg (\frac{\varPsi _L-1}{L}\bigg )^2\bigg [\frac{1}{2}\varGamma _{rn}-\varPhi _r^2\bigg ]\bigg [\mathrm {E}[(Y_7^0)^2]+\mu _{v_0}^2\bigg ]\bigg \}, \\ p^{(2)}_{r}&= \frac{-\tilde{c}_r-c_d\varPhi _r^2}{m_r}, \\ p^{(3)}_r&=
\frac{EA}{m_rL}\big (\varGamma _{rn}\mu _{{q}_{r}}-\frac{\varPsi _L-1}{L}\varPhi _r\mu _{v_0}\big ), \\ p^{(4)}_{r}&= \frac{1}{m_r}\bigg \{EA\frac{\varPsi _L-1}{L}\bigg [-\frac{1}{L(\tau )}\varPhi _r\mu _{u_{M}} \\&- \,\frac{3}{2}\bigg (\frac{\varPsi _L-1}{L(\tau )}\bigg )^2\varPhi _r(L)\bigg (\mathrm {E}[(Y_7^0)^2]+\mu _{v_0}^2\bigg ) \\&+\, \varPhi _r(L)\bigg [\varGamma _{rn}-\frac{1}{2}\frac{1}{L}\kappa _{rr}\bigg ]\bigg (\mathrm {E}[(Y_1^0)^2]+\mu _{{q}_{r}}^2\bigg ) \\&+\, 2\bigg (\frac{\varPsi _L-1}{L}\bigg )\bigg [\frac{1}{2}\varGamma _{rn}- \varPhi _r^2(L)\bigg ]\bigg (\mathrm {E}[Y_7^0Y_1^0]+\mu _{v_0}\mu _{{q}_{r}}\bigg )\bigg ] \\&+\, \beta _r^{(0)}-\beta _r^{(2)}\varOmega _0^2\bigg \}, \\ p^{(5)}_{r}&= \frac{\beta _r^{(1)}-2\beta _r^{(2)}\zeta _{f}\varOmega _0}{m_r}, \\ p^{(6)}_{r}&= -\frac{EA}{M+m_d}\bigg [\frac{1}{L}\kappa _{rr}\mu _{q_{r}}+\frac{\varPsi _L-1}{L}\varPhi _r\mu _{v_{0}}\bigg ], \\ p^{(7)}_{r}&= -\frac{EA}{L(M+m_d)},\\ p^{(8)}_{r}&= -\frac{EA}{M+m_d}\bigg \{\frac{\varPsi _L-1}{L}\varPhi _r\mu _{q_{r}}+\bigg (\frac{\varPsi _L-1}{L}\bigg )^2\mu _{v_{0}}\bigg \}. \end{aligned}$$
(31)
To obtain the covariance matrix
\(\mathbf {R_{\mathbf {Y^0Y^0}}}=\mathrm {E}[\mathbf {Y^0}\mathbf {Y^0}^\mathrm {T}]\) and the differential equations governing the second-order statistical moments of the state vector
\(\mathbf {Y^0}(t)\), the following differential equation
$$\begin{aligned} \frac{d}{dt}\mathbf {R_{\mathbf {Y^0Y^0}}} =\mathbf {B}\mathbf {R_{\mathbf {Y^0Y^0}}}+\mathbf {R_{\mathbf {Y^0Y^0}}} \mathbf {B}^{\mathrm {T}}+\mathbf {d}\mathbf {d}^{\mathrm {T}} \end{aligned}$$
(32)
should be solved together with the differential equations for mean values defined as
$$\begin{aligned} \frac{d}{dt}\mathbf {\mu }(t)=\mathrm {E}\big [\mathbf {c}(\mathbf {Y}^0(t))\big ], \end{aligned}$$
(33)
where
$$\begin{aligned} \mathbf {\mu }(t)=\mathrm {E}[\mathbf {Y}(t)]. \end{aligned}$$
(34)
The higher-order joint statistical moments of the state vector
\(\mathbf {Y^0}(t)\) can be then easily obtained.