Disturbance Observer Based Optimal Attitude Control of NSV Using $\theta -D$ Method

Abstract

In this paper, a disturbance observer based optimal attitude control scheme using $\theta -D$ method is presented for the near space vehicle (NSV). Firstly, $\theta -D$ method is used to design the optimal controller for the nominal system without considering the disturbance. Secondly, nonlinear disturbance observer (NDO) technique is applied to estimate the disturbance and the estimation result can be used as the disturbance compensation term. Then, the composite controller consisting of optimal controller and disturbance compensation term is proposed. The closed-loop system signals are proved to be uniformly ultimately bounded (UUB) using Lyapunov method. Finally, simulation results show the effectiveness of proposed control scheme.

Appendix

Proof:

Choose a Lyapunov candidate function

$$\begin{aligned} \begin{aligned} L(t)=\frac{1}{2}x^T\sum \limits _{i = 0}^\infty {{{\hat{T}}_i}} x+\frac{1}{2}\tilde{d}^T\tilde{d} \end{aligned} \end{aligned}$$

(18)

Taking the time derivative of (13), we have

$$\begin{aligned} \begin{aligned} \dot{L}(t)=&\left[ {x^T\sum \limits _{i = 0}^\infty {{{\hat{T}}_i}}+\frac{1}{2}x^T\sum \limits _{i = 0}^\infty {\frac{\partial \hat{T}_i}{\partial x}x}}\right] [f(x)+g(x)u+d]+\tilde{d}^T (\dot{d}-L\tilde{d})\\ \end{aligned} \end{aligned}$$

(19)

Since $u=-R^{-1}g^T(x)\sum \limits _{i = 0}^\infty {{{\hat{T}}_i}}x-g^{-1}(x)\hat{d}$, then we have

$$\begin{aligned} \begin{aligned}&\dot{L}(t)=-\frac{1}{2}u^TRu-\frac{1}{2}\left[ Q+\sum \limits _{i = 0}^\infty {D_i\theta ^i}\right] x+x^T\left[ \frac{1}{2}\sum \limits _{i = 0}^\infty {\frac{{{\partial \hat{T}}_i}}{\partial x}}x\right] \\&{}\left[ A_0+A(x)-g(x)R^{-1}g^T(x)\sum \limits _{i = 0}^\infty \hat{T}_i\right] x+x^T\left[ \frac{1}{2}\sum \limits _{i = 0}^\infty {\frac{{{\partial \hat{T}}_i}}{\partial x}}x\right] \tilde{d}+\tilde{d}^T (\dot{d}-L\tilde{d})\\&{}\le -\frac{1}{2}\bar{\lambda }\Vert x\Vert ^2-\lambda _{min}(L)\Vert \tilde{d}\Vert ^2+\frac{1}{2}\Vert \tilde{d}\Vert ^2+\frac{1}{2}\dot{d}^2+\Vert x\Vert ^2\left| \left| \frac{1}{2}\sum \limits _{i = 0}^\infty {\frac{{{\partial \hat{T}}_i}}{\partial x}}x\right| \right| \\&{}\left| \left| A_0+A(x)-g(x)R^{-1}g^T(x)\sum \limits _{i = 0}^\infty {{{\hat{T}}_i}}\right| \right| +\left| \left| \frac{1}{2}\sum \limits _{i = 0}^\infty {\frac{{{\partial \hat{T}}_i(x)}}{\partial x}}x\right| \right| (\frac{1}{2}\Vert x\Vert ^2+\frac{1}{2}\Vert \tilde{d}\Vert ^2) \end{aligned} \end{aligned}$$

(20)

where $\bar{\lambda }=\lambda _{min}[Q+\sum \limits _{i = 0}^\infty {D_i\theta ^i}]$.

Using the relationship $\frac{\partial \hat{T}_i}{\partial x}=\epsilon _i\frac{\partial \bar{T}_i}{\partial x}$, we have [6]

$$\begin{aligned} \begin{aligned}&\dot{L}(t)\le -\left[ {\frac{1}{2}\bar{\lambda }-\epsilon _i\left| \left| \frac{1}{2}\sum \limits _{i = 0}^\infty {\frac{{{\partial \bar{T}}_i}}{\partial x}}x\right| \right| } \right. \left. {\left( \left| \left| A_0+A(x)-g(x)R^{-1}g^T(x)\sum \limits _{i = 0}^\infty {{\hat{T}_i}}\right| \right| +1\right) } \right] \Vert x\Vert ^2\\&{}-\left[ \lambda _{min}(L)-\frac{1}{2}-\epsilon _i\left| \left| \frac{1}{2}\sum \limits _{i = 0}^\infty {\frac{{{\partial \bar{T}}_i}}{\partial x}}x\right| \right| \right] \Vert \tilde{d}\Vert ^2+\frac{1}{2}\bar{d}_M^2\ \end{aligned} \end{aligned}$$

(21)

Note that as long as x lies in a compact set with A(x) is bounded; g(x) is bounded as shown in Assumption 1; $\sum \limits _{i = 0}^\infty {{\hat{T}_i(x)}}$ is converge and positive in Lemma 1, one can always choose a set of $\epsilon _i$ such that

$$\begin{aligned} \begin{aligned} M=&\left[ {\frac{1}{2}\bar{\lambda }-\epsilon _i\left| \left| \frac{1}{2}\sum \limits _{i = 0}^\infty {\frac{{{\partial \bar{T}}_i}}{\partial x}}x\right| \right| } \right. \left. {\left( \left| \left| A_0+A(x)-g(x)R^{-1}g^T(x)\sum \limits _{i = 0}^\infty {{\hat{T}_i}}\right| \right| +1\right) } \right] \ge 0\\ N=&\left[ \lambda _{min}(L)-\frac{1}{2}-\epsilon _i\left| \left| \frac{1}{2}\sum \limits _{i = 0}^\infty {\frac{{{\partial \bar{T}}_i}}{\partial x}}x\right| \right| \right] \ge 0; B_d=\frac{1}{2}\bar{d}_M^2 \end{aligned} \end{aligned}$$

(22)

then we have

$$\begin{aligned} \begin{aligned}&\dot{L}(t)\le -M\Vert x\Vert ^2 -N\Vert \tilde{d}\Vert ^2+B_d \end{aligned} \end{aligned}$$

(23)

If $\Vert x\Vert >\sqrt{\frac{B_d}{M}}$ or $\Vert \tilde{d}\Vert >\sqrt{\frac{B_d}{N}}$, we have $\dot{L}(t)< 0$, then the disturbance estimation error $\tilde{d}$ and system states x are proved to be UUB.