Robust stochastic stabilization of attitude motion

Abstract

This study proposes robust stochastic stabilization of rigid body attitude motion within the framework of geometric mechanics, which can globally represent the attitude dynamics model. The system is subject to a stochastic input torque with an unknown variance parameter and an unknown nonlinear diffusion coefficient matrix. Our development starts with introducing a general notion of the stochastic stability in probability within the framework of geometric mechanics. Then, the Morse–Lyapunov (M–L) technique is employed to design a nonlinear continuous stochastic feedback control law. Finally, the asymptotic stability of the system is guaranteed in probability and the control gain parameters are obtained via solving a linear matrix inequality feasibility problem. An estimate of the region of attraction of the system is calculated to provide a better insight for tuning the control gain parameters. Two illustrative examples are performed based on the discretized model of the closed-loop system to demonstrate the effectiveness of the proposed control scheme.

Appendix

1.1 Appendix A: Proof of the Itô representation of attitude motion on $\mathrm {SO(3)}\times {\mathbb {R}}^3$

Let $\Omega _i(t),$$i=1,2,3$ be the solution of the ith element of SDE of Eq. (7b) in [0, t] such that

$$\begin{aligned} \Omega _i(t)= & {} \Omega _i(0)+\int _0^t{ {\mathfrak {F}}_i(s, \Omega (s),\tau (s))\mathrm {d}s}\nonumber \\&+\mathop {\underbrace{\int _0^t{{\mathfrak {G}}_i(s, \Omega (s))\mathrm {d}\beta (s)}}}\limits _{(S)}, \end{aligned}$$

(A.1)

where the second integral is interpreted as an Stratonovich stochastic integral and ${\mathfrak {F}}(t, \Omega (t),\tau (t))=-{\tilde{J}}^{-1}(t)\Omega (t)^\times J \Omega (t) +{\tilde{J}}^{-1}(t)\tau (t)$ and ${\mathfrak {G}}(t, \Omega (t))$ has been introduced after Eq. (8). The notations (S) or (I) are used to represent that the stochastic integral is interpreted in the Stratonovich and Itô, respectively. We aim to represents the Stratonovich integral of Eq. (A.1) in term of the Itô integral. Let $0=t_0<t_1<\cdots <t_n=t$ be a dissection of the interval [0, t] and $t_p, p=1,2,\ldots ,n$ represents an arbitrary time in this interval. In addition, we use the notation $\Omega _{i_p} \equiv \Omega _i(t_p)$ to show that the ith element of the angular velocity vector $\Omega$ is evaluated at the arbitrary time $t_p$, while $\Omega _p$ represents the angular velocity vector $\Omega$ is evaluated at $t_p$. The approach to transform the Stratonovich SDE of Eq. (7b) to its corresponding Itô form includes representing the Stratonovich stochastic integral in an equivalent Itô representation by replacing the Stratonovich integral in the solution of Eq. (A.1) by a Itô integral and some auxiliary terms. According to the Stratonovich integral definition [23], the second integral of Eq. (A.1) is interpreted as the stochastic limit of the Riemann integral as

$$\begin{aligned}&\mathop {\underbrace{\int _0^t{{\mathfrak {G}}_i(s,\Omega (s))\mathrm {d}\beta (s)}}}\limits _{(S)}\nonumber \\&\quad =\lim \limits _{n\rightarrow \infty }\sum \limits _{p=1}^{n}{\mathfrak {G}}_i(t_{p-1/2},\Omega _{p-1/2})(\beta _p-\beta _{p-1}). \end{aligned}$$

(A.2)

The Taylor series expansion of the term ${\mathfrak {G}}_i(t_{p-1/2},\Omega _{p-1/2})$ in Eq. (A.2) about $(t_{p-1},\Omega _{p-1})$ results in

$$\begin{aligned}&\mathop {\underbrace{\int _0^t{{\mathfrak {G}}_i(s,\Omega (s))\mathrm {d}\beta (s)}}}\limits _{(S)}=\lim \limits _{n\rightarrow \infty }\sum \limits _{p=1}^{n}\bigg [{\mathfrak {G}}_i(t_{p-1},\Omega _{p-1})\nonumber \\&\quad +\,\frac{\partial {\mathfrak {G}}_i(t_{p-1},\Omega _{p-1}) }{\partial t}(t_{p-1/2}-t_{p-1})\nonumber \\&\quad +\,\sum \limits _{k=1}^{3}\frac{\partial {\mathfrak {G}}_i(t_{p-1},\Omega _{p-1}) }{\partial \Omega _{k_p}}(\Omega _{k_{p-1/2}}-\Omega _{k_{p-1}}) +\cdots \bigg ]\nonumber \\&\quad \times (\beta _p-\beta _{p-1}). \end{aligned}$$

(A.3)

The second term on the right-hand side of Eq. (A.3) as well as higher order terms are omitted as they are of order higher than the increment $(t_p-t_{p-1})$ when $n \rightarrow \infty$. Considering the definition of the Itô stochastic integral, i.e., [23], Eq. (A.3) can be simplified as

$$\begin{aligned}&\mathop {\underbrace{\int _0^t{{\mathfrak {G}}_i(s,\Omega (s))\mathrm {d}\beta (s)}}}\limits _{(S)}= \mathop {\underbrace{\int _0^t{{\mathfrak {G}}_i(s,\Omega (s))\mathrm {d}\beta (s)}}}\limits _{(I)}\nonumber \\&\quad +\,\lim \limits _{n\rightarrow \infty }\sum \limits _{p=1}^{n}\bigg [\sum \limits _{k=1}^{3}\frac{\partial {\mathfrak {G}}_i(t_{p-1},\Omega _{p-1}) }{\partial \Omega _{k_p}}\nonumber \\&\quad \times \,(\Omega _{k_{p-1/2}}-\Omega _{k_{p-1}})\bigg ](\beta _p-\beta _{p-1}). \end{aligned}$$

(A.4)

In addition, the solution of Eq. (A.1) at the differential interval $[t_{p-1},t_{p-1/2}]$ is obtained as

$$\begin{aligned}&\Omega _{k_{p-1/2}}-\Omega _{k_{p-1}}= {\mathfrak {F}}_k(t_{p-1}, \Omega _{p-1},\tau _{p-1})(t_{p-1/2}-t_{p-1})\nonumber \\&\quad +\,{\mathfrak {G}}_k(t_{p-1},\Omega _{p-1})(\beta _{p-1/2}-\beta _{p-1}). \end{aligned}$$

(A.5)

Substituting Eq. (A.5) into Eq. (A.4) and dropping higher order terms yields

$$\begin{aligned}&\mathop {\underbrace{\int _0^t{{\mathfrak {G}}_i(s,\Omega (s))\mathrm {d}\beta (s)}}}\limits _{(S)}= \mathop {\underbrace{\int _0^t{{\mathfrak {G}}_i(s,\Omega (s))\mathrm {d}\beta (s)}}}\limits _{(I)} \nonumber \\&\quad +\,\lim \limits _{n\rightarrow \infty }\sum \limits _{p=1}^{n}\sum \limits _{k=1}^{3}\frac{\partial {\mathfrak {G}}_i(t_{p-1},\Omega _{p-1}) }{\partial \Omega _{k_p}}\nonumber \\&\quad {\mathfrak {G}}_k(t_{p-1},\Omega _{p-1})(\beta _{p-1/2}-\beta _{p-1})(\beta _p-\beta _{p-1}). \end{aligned}$$

(A.6)

Furthermore, the correlation function of the Brownian motion is determined as $R_\beta (t_s,t_l)=\sigma ^2\min (t_s,t_l)$, where $s,l\in \{1,2,\ldots ,n\}$ and $\min (\cdot ,\cdot )$ represents the minimum of two parameters, i.e., [21]. The correlation function is continuous at each interval $[t_s,t_l]$. Employing this fact as well as the independent increment property of the Brownian motion of $\beta (t)$ for the term $(\beta _{p-1/2}-\beta _{p-1})(\beta _p-\beta _{p-1})$ in Eq. (A.5) result in

$$\begin{aligned}&{\mathbb {E}}\{(\beta _{p-1/2}-\beta _{p-1})(\beta _p-\beta _{p-1})\}\nonumber \\&\quad ={\mathbb {E}}\{(\beta _{p-1/2}(\beta _p-\beta _{p-1})\}-{\mathbb {E}}\{\beta _{p-1}(\beta _p-\beta _{p-1})\}\nonumber \\&\quad ={\mathbb {E}}\{(\beta _{p-1/2}(\beta _p-\beta _{p-1})\}-0 \!=\!\sigma ^2(t_{p-1/2})\!-\!\sigma ^2(t_{p-1})\nonumber \\&\quad ={\sigma ^2}\left( \frac{1}{2}t_p+\frac{1}{2}t_{p-1}-t_{p-1}\right) \nonumber \\&\quad =\frac{\sigma ^2}{2}(t_p-t_{p-1})=\frac{\sigma ^2}{2}\mathrm {d}t. \end{aligned}$$

(A.7)

Substituting Eq. (A.7) into Eq. (A.6) and performing some algebra results in

$$\begin{aligned}&\mathop {\underbrace{\int _0^t{{\mathfrak {G}}_i(s,\Omega (s))\mathrm {d}\beta (s)}}}\limits _{(S)}= \mathop {\underbrace{\int _0^t{{\mathfrak {G}}_i(s,\Omega (s))\mathrm {d}\beta (s)}}}\limits _{(I)}\nonumber \\&\quad +\,\frac{\sigma ^2}{2}\int _0^t\sum \limits _{k=1}^{3}\frac{\partial {\mathfrak {G}}_i(s,\Omega (s)) }{\partial \Omega _{k}(s)} {\mathfrak {G}}_k(s,\Omega (s))\mathrm {d}s. \end{aligned}$$

(A.8)

Equation (A.8) represents the relationship between the Stratonovich and Itô integral, which was the main objective of this Appendix. Substituting Eq. (A.8) into the Startonovich stochastic integral of Eq. (A.1), the solution of the ith element of the Euler rotational equations of motion is obtained in terms of the Itô stochastic integral as

$$\begin{aligned}&\Omega _i(t)=\Omega _i(0)+\int _0^t\bigg [ {\mathfrak {F}}_i(s, \Omega (s),\tau (s))\nonumber \\&\quad +\,\frac{\sigma ^2}{2}\sum \limits _{k=1}^{3}\frac{\partial {\mathfrak {G}}_i(s,\Omega (s)) }{\partial \Omega _{k}(s)} {\mathfrak {G}}_k(s,\Omega (s))\bigg ]\mathrm {d}s\nonumber \\&\quad +\,\mathop {\underbrace{\int _0^t{{\mathfrak {G}}_i(s,\Omega (s))\mathrm {d}\beta (s)}}}\limits _{(I)}. \end{aligned}$$

(A.9)

Taking the differentiation of Eq. (A.9) with respect to t under the Leibniz’s integral rule and using the fact that $\mathrm {Pr}\{\beta (0)=0\}=1$ from the properties of the Brownian motion in Eq. (6) results in

$$\begin{aligned} \mathrm {d}\Omega _i(t)= & {} \bigg [{\mathfrak {F}}_i(t, \Omega (t),\tau (t))+\frac{\sigma ^2}{2}\sum \limits _{k=1}^{3}\frac{\partial {\mathfrak {G}}_i(t,\Omega (t)) }{\partial \Omega _{k}(t)}\bigg ]\mathrm {d}t\nonumber \\&+\,{\mathfrak {G}}_i(t,\Omega (t))\mathrm {d}\beta (t). \end{aligned}$$

(A.10)

Substituting ${\mathfrak {F}}_i(t, \Omega (t),\tau (t))$ defined in Eq. (A.1) into Eq. (A.10), the Itô SDE of Eq. (8b) is obtained.