Hierarchical Composite Anti-Disturbance Control for Robotic Systems Using Robust Disturbance Observer

Abstract

A new anti-disturbance control strategy is presented for a class of nonlinear robotic systems with multiple disturbances. This strategy is named hierarchical composite anti-disturbance control (HCADC). Two types of disturbances are studied. One is generated by an exogenous system with uncertainty and the other is described by an uncertain vector with the bounded H ₂ norm. The hierarchical control strategy is established which includes a disturbance observer based controller (DOBC) and an H _∞ controller, where DOBC is used to reject the first type of disturbance and H _∞ controller is used to attenuate the second. Stability analysis for both the error estimation systems and the composite closed-loop system is provided. Simulations for a two-link manipulator system show that the desired disturbance attenuation and rejection performances can be guaranteed.

Appendices

Appendix A: Proof of the Lemma 11.1

Denoting a Lyapunov candidate

$$ \begin{array}{*{20}c} {\prod (x,t) = x^T P_x + \frac{1}{{\lambda ^2 }}\int_0^t {\left[ {\left\| {U\dot x} \right\|^2 - \left\| {f(\dot x,\tau )} \right\|^2 } \right]d\tau } } \\ { + \frac{1}{{\lambda _1^2 }}\int_0^t {\left[ {\left\| {U_1 x} \right\|^2 - \left\| {f_1 (x,\tau )} \right\|^2 } \right]d\tau } } \\ \end{array} $$

(11.12)

then we have

$$ \begin{array}{l} \begin{array}{*{20}c} {\dot \prod (x,t) = \dot x^T P_x + x^T P\dot x + \frac{1}{{\lambda ^2 }}\left[ {\left\| {U\dot x} \right\|^2 - \left\| {f(\dot x,\tau )} \right\|^2 } \right]} \\ { + \frac{1}{{\lambda _1^2 }}\left[ {\left\| {U\dot x} \right\|^2 - \left\| {f(\dot x,\tau )} \right\|^2 } \right]} \\ \end{array} \\ = x^T \left[ {PA + A^T P + \frac{1}{{\lambda ^2 }}A^T U^T UA + \frac{1}{{\lambda _1^2 }}U_1^T U_1 } \right]x \\ + x^T \left[ {\lambda PF + \frac{1}{\lambda }A^T U^T UF\quad \lambda _1 PF_1 + \frac{{\lambda _1 }}{{\lambda ^2 }}A^T U^T UF_1 } \right]q \\ + q^T \left[ {\lambda PF + \frac{1}{\lambda }A^T U^T UF\quad \lambda _1 PF_1 + \frac{{\lambda _1 }}{{\lambda ^2 }}A^T U^T UF_1 } \right]^T x \\ + q^T \left[ {\begin{array}{*{20}c} {F^T U^T UF - 1} & {\frac{{\lambda _1 }}{\lambda }F^T U^T F_1 } \\ {\frac{{\lambda _1 }}{\lambda }F_1^T U^T UF} & {\frac{{\lambda _1^2 }}{{\lambda ^2 }}F_1^T U^T UF_1 - 1} \\ \end{array}} \right]q \\ = \left[ {\begin{array}{*{20}c} x \\ q \\ \end{array}} \right]^T \left[ {\begin{array}{*{20}c} {PA + A^T P + C_H^T C_H } & {PB_H + C_H^T D_H } \\ {B_H^T P + D_H^T C_H } & {D_H^T D_H - 1} \\ \end{array}} \right]\left[ {\begin{array}{*{20}c} x \\ q \\ \end{array}} \right] \\ \end{array} $$

(11.13)

where B _H , C _H , D _H is denoted by (11.7), and $q^{T}=[-(\frac{1}{\lambda}f)^{T}\ (\frac{1}{\lambda_{1}}f_{1})^{T}]$. Denote

$$M_1=\left[\begin{array}{c@{\quad}c}PA+A^TP+C_H^TC_H&PB_H+C_H^TD_H\\[4pt]B_H^TP+D_H^TC_H&D_H^TD_H-I\end{array}\right].$$

Based on Schur complement, it can be seen that M ₁<0⇔M ₂<0 where

$$ M_2 = \left[ {\begin{array}{*{20}c} {PA + A^T P} & {PB_H } & {C_H^T } \\ {B_H^T P} & { - I} & {D_H^T } \\ {C_H } & {D_H } & { - 1} \\ \end{array}} \right]. $$

Thus $\dot{\Pi}(x,t)<0\Leftrightarrow M_{2}<0$, i.e. if M ₂<0 holds, system (11.1) is asymptotic stable.

Appendix B: Proof of the Lemma 11.2

For system (11.5), select a Lyapunov candidate as (11.12), and denote the following auxiliary function (known as the storage function)

$$J:=\int^{t}_0\left[z^Tz-\gamma^2w^Tw+\dot{\Pi}(x,t)\right]dt.$$

It can be verified that

$$ \begin{array}{l} z^T z - \gamma ^2 w^T w + \dot \prod (x,t) \\ = x^T C^T C_x - \gamma ^2 w^T w + \dot x^T Px + x^T P\dot x + \frac{1}{{\lambda ^2 }}\left[ {\left\| {U\dot x} \right\|^2 - \left\| {f(\dot x,\tau )} \right\|^2 } \right] \\ + \frac{1}{{\lambda _1^2 }}\left[ {\left\| {U\dot x} \right\|^2 - \left\| {f(\dot x,\tau )} \right\|^2 } \right] \\ = x^T \left[ {PA + A^T P + \frac{1}{{\lambda ^2 }}A^T U^T UA + \frac{1}{{\lambda _1^2 }}U_1^T U_1 + C^T C} \right]x \\ + w^T \left[ { + \frac{1}{{\lambda ^2 }}B^T U^T UB - \gamma ^2 I} \right]w \\ + q^T \left[ {\begin{array}{*{20}c} {F^T U^T F - 1} & {\frac{{\lambda _1 }}{\lambda }F^T U^T UF_1 } \\ {\frac{{\lambda _1 }}{\lambda }F_1^T U^T UF} & {\frac{{\lambda _1^2 }}{{\lambda ^2 }}F_1^T U^T UF_1 - 1} \\ \end{array}} \right] \\ + q^T \left[ {\begin{array}{*{20}c} {\frac{1}{\lambda }F^T U^T F_1 } \\ {\frac{\lambda }{\lambda }F_1^T U^T UB} \\ \end{array}} \right]w + w^T \left[ {\begin{array}{*{20}c} {\frac{1}{\lambda }F^T U^T UB} \\ {\frac{{\lambda _1 }}{{\lambda ^2 }}F_1^T U^T UB} \\ \end{array}} \right]^T q \\ + x^T \left[ {\lambda PF + \frac{1}{\lambda }A^T U^T UF\quad \lambda _1 PF_1 + \frac{{\lambda _1 }}{{\lambda ^2 }}A^T U^T UF_1 } \right]q \\ + q^T \left[ {\lambda PF + \frac{1}{\lambda }A^T U^T UF\quad \lambda _1 PF_1 + \frac{{\lambda _1 }}{{\lambda ^2 }}A^T U^T UF_1 } \right]^T x \\ + x^T \left[ {PB + \frac{1}{{\lambda ^2 }}A^T U^T UB} \right]w + w^T \left[ {PB + \frac{1}{{\lambda ^2 }}A^T U^T UB} \right]^T x \\ = \left[ {\begin{array}{*{20}c} x \\ q \\ w \\ \end{array}} \right]^T \left[ {\begin{array}{*{20}c} {PA + A^T P + C_H^T C_H + C^T C} & {PB + \frac{1}{{\lambda ^2 }}A^T U^T UB} & {PB + C_H^T E_H } \\ {B_H^T P + D_H^T C_H } & {D_H^T D_H - 1} & {D_H^T E_H } \\ {B^T P + E_H^T C_H } & {E_H^T D_H } & {E_H^T E_H - \gamma ^2 I} \\ \end{array}} \right] \\ \times \left[ {\begin{array}{*{20}c} x \\ q \\ w \\ \end{array}} \right]. \\ \end{array} $$

(11.14)

Denote M ₃ as

$$ M_3 = \left[ {\begin{array}{*{20}c} {PA + A^T P + C_H^T C_H + C^T C} & {PB + C_H^T D_H } & {PB + C_H^T E_H } \\ {B_H^T P + D_H^T C_H } & {D_H^T D_H - 1} & {D_H^T E_H } \\ {B^T P + E_H^T C_H } & {E_H^T D_H } & {E_H^T E_H - \gamma ^2 I} \\ \end{array}} \right] $$

(11.15)

where B _H , C _H , D _H is also denoted by (11.7), and

$$ \begin{array}{*{20}c} {E_H = \left[ {\begin{array}{*{20}c} { - \frac{1}{\lambda }UB} \\ 0 \\ \end{array}} \right],} & {q^T = \left[ {\begin{array}{*{20}c} { - \left( {\frac{1}{\lambda }f} \right)^T } & {\left( {\frac{1}{{\lambda _1 }}f_1 } \right)^T } \\ \end{array}} \right]} \\ \end{array}. $$

Based on Schur complement, it can be seen that M ₃<0⇔M ₄<0 where

$$ M_4 = \left[ {\begin{array}{*{20}c} {PA + A^T P} & {PB_H } & {PB} & {C_H^T } & {C^T } \\ {B_H^T P} & { - I} & 0 & {D_H^T } & 0 \\ {B^T P} & 0 & { - \gamma ^2 I} & {E_H^T } & 0 \\ {C_H } & {D_H } & {E_H } & { - I} & 0 \\ C & 0 & 0 & 0 & { - I} \\ \end{array}} \right]. $$

Exchanging of rows and columns yields that M ₄<0⇔M ₅<0, where

$$ M_5 = \left[ {\begin{array}{*{20}c} {PA + A^T P} & {PB_H } & {C_H^T } & {PB} & {C^T } \\ {B_H^T P} & { - I} & {D_H^T } & 0 & 0 \\ {C_H } & {D_H } & { - I} & {E_H } & 0 \\ {B^T P} & 0 & {E_H^T } & { - \gamma ^2 I} & 0 \\ C & 0 & 0 & 0 & { - I} \\ \end{array}} \right]. $$

Since M ₅<0 implies that M ₂<0, system (11.5) is asymptotically stable in the absence of the disturbance d(t) with zero initial condition. Furthermore, since $J=\int^{\infty}_{0}[z^{T}z-\gamma^{2}w^{T}w]dt+\Pi(x,t)$, then J<0 holds when M ₅<0 holds, which implies that ‖z‖₂<γ‖d‖₂.