Robust adaptive dynamic surface control for hypersonic vehicles

Abstract

An adaptive dynamic surface control (DSC) scheme is proposed for the multi-input multi-output attitude control of near-space hypersonic vehicles (NHV). The proposed control strategy can improve the control performance of NHV despite uncertainties and external disturbances. The proposed controller combines dynamic surface control and radial basis function neural network (RBFNN) and is designed to control the longitudinal dynamics of NHV. The DSC technique is used to handle the problem of “explosion of complexity” inherent to the conventional backstepping method. RBFNN is used to approximate the unknown nonlinear function, and a robustness component is introduced in the controller to cancel the influence of compound disturbance and improve robustness and adaptation of the system. Simulation results show that the proposed strategy possesses good robustness and fast response.

Appendix

The detailed expressions of the vectors ${{{W}}_1}$ and matrices ${{{\varPi }}_2}$, $\varOmega _2$ are:

$$\begin{aligned} {{W}}_1^T&= \frac{1}{M} \left[ \begin{array}{c} {T_V}\cos \alpha - {D_V}\\ - Mg\cos \gamma \\ - T\sin \alpha - {D_\alpha }\\ {T_\lambda }\cos \alpha \\ {T_h}\cos \alpha - {D_h} - M{g_h}\sin \gamma \end{array} \right] \end{aligned}$$

(A-1)

$$\begin{aligned} {{\varOmega }_2}&= \left[ {\begin{array}{ccccc} {{w}}_{21}&{{w}}_{22}&{{w}}_{23}&{{w}}_{24}&{{w}}_{25} \end{array}}\right] \end{aligned}$$

(A-2)

where

$$\begin{aligned} {{{w}}_{21}}&= \left[ \begin{array}{c} {T_{VV}}\cos \alpha - {D_{VV}}\\ 0\\ - {T_V}\sin \alpha - {D_{V\alpha }}\\ {T_{V\lambda }}\cos \alpha \\ - {D_{Vh}} \end{array} \right] \end{aligned}$$

(A-3)

$$\begin{aligned} {{{w}}_{22}}&= {\left[ {0\begin{array}{*{20}{c}} {} \end{array}Mg\sin \gamma \begin{array}{*{20}{c}} {} \end{array}0\begin{array}{*{20}{c}} {} \end{array}0\begin{array}{*{20}{c}} {} \end{array} - {g_h}\cos \gamma } \right] ^T} \end{aligned}$$

(A-4)

$$\begin{aligned} {{{w}}_{23}}&= \left[ {\begin{array}{*{20}{c}} { - {T_V}\sin \alpha - {D_{\alpha V}}}\\ 0\\ { - T\cos \alpha - {D_{\alpha \alpha }}}\\ { - {T_\lambda }\sin \alpha }\\ { - {T_h}\sin \alpha - {D_{\alpha h}}} \end{array}} \right] \end{aligned}$$

(A-5)

$$\begin{aligned} {{{w}}_{24}}&= {\left[ {{T_{\lambda V}}\cos \alpha \begin{array}{*{20}{c}} {} \end{array}0\begin{array}{*{20}{c}} {} \end{array} - {T_\lambda }\sin \alpha \begin{array}{*{20}{c}} {} \end{array}{T_{\lambda \lambda }}\cos \alpha \begin{array}{*{20}{c}} {} \end{array}0} \right] ^T} \end{aligned}$$

(A-6)

$$\begin{aligned} {{{w}}_{25}}&= \left[ {\begin{array}{*{20}{c}} {{T_{hV}}\cos \alpha - {D_{hV}}}\\ { - M{g_h}\cos \gamma }\\ { - {T_h}\sin \alpha - {D_{h\alpha }}}\\ {{T_{h\lambda }}\cos \alpha }\\ {{T_{hh}}\cos \alpha - {D_{hh}} - M{g_{hh}}\sin \gamma } \end{array}} \right] \end{aligned}$$

(A-7)

$$\begin{aligned} {{{{\pi }}_1}}&= {\left[ {{\pi _{11}}\begin{array}{*{20}{c}} {} \end{array}{\pi _{12}}\begin{array}{*{20}{c}} {} \end{array}{\pi _{13}}\begin{array}{*{20}{c}} {} \end{array}{\pi _{14}}\begin{array}{*{20}{c}} {} \end{array}{\pi _{15}}} \right] ^T} \end{aligned}$$

(A-8)

where

$$\begin{aligned} {\pi _{11}}&= \frac{{\left( {{L_V} + {T_V}\sin \alpha } \right) V - \left( {L + T\sin \alpha } \right) }}{{M{V^2}}} + \frac{{g\cos \gamma }}{{{V^2}}} \end{aligned}$$

(A-9)

$$\begin{aligned} {\pi _{12}}&= \frac{{g\sin \gamma }}{V} \end{aligned}$$

(A-10)

$$\begin{aligned} {\pi _{13}}&= \frac{{{L_\alpha } + T\cos \alpha }}{{MV}} \end{aligned}$$

(A-11)

$$\begin{aligned} {\pi _{14}}&= \frac{{{T_\lambda }\sin \alpha }}{{MV}} \end{aligned}$$

(A-12)

$$\begin{aligned} {\pi _{15}}&= \frac{{{L_h} + {T_h}\sin \alpha }}{{MV}} - \frac{{{g_h}\cos \gamma }}{V} \end{aligned}$$

(A-13)

$$\begin{aligned} {{{\varPi }}_2}&= \left[ {{{\pi }_{21}}\begin{array}{*{20}{c}} {} \end{array}{{\pi }_{22}}\begin{array}{*{20}{c}} {} \end{array}{{\pi }_{23}}\begin{array}{*{20}{c}} {} \end{array}{{\pi }_{24}}\begin{array}{*{20}{c}} {} \end{array}{{\pi }_{25}}} \right] \end{aligned}$$

(A-14)

where

$$\begin{aligned}&{{\pi }_{21}} \nonumber \\&= \left[ \begin{array}{c} \frac{{\left( {{L_{VV}} + {T_{VV}}\sin \alpha } \right) }}{{MV}} - \frac{{2\left( {{L_V} + {T_V}\sin \alpha } \right) }}{{M{V^2}}} + \frac{{2\left( {L + T\sin \alpha } \right) }}{{M{V^3}}} - \frac{{2g\cos \gamma }}{{{V^3}}}\\ - \frac{{g\sin \gamma }}{{{V^2}}}\\ \frac{{{L_{V\alpha }} + {T_V}\cos \alpha }}{{MV}} - \frac{{{L_\alpha } + T\cos \alpha }}{{M{V^2}}}\\ \frac{{{T_{V\lambda }}\sin \alpha }}{{MV}} - \frac{{{T_\lambda }\sin \alpha }}{{M{V^2}}}\\ \frac{{{L_{Vh}} + {T_{Vh}}\sin \alpha }}{{MV}} - \frac{{{L_h} + {T_h}\sin \alpha }}{{M{V^2}}} + \frac{{{g_h}\cos \gamma }}{{{V^2}}} \end{array} \right] \end{aligned}$$

(A-15)

$$\begin{aligned}&{{\pi }_{22}} = {\left[ { - \frac{{g\sin \gamma }}{{{V^2}}}\begin{array}{*{20}{c}} {} \end{array}\frac{{g\cos \gamma }}{V}\begin{array}{*{20}{c}} {} \end{array}0\begin{array}{*{20}{c}} {} \end{array}0\begin{array}{*{20}{c}} {} \end{array}\frac{{{g_h}\sin \gamma }}{V}} \right] ^T} \end{aligned}$$

(A-16)

$$\begin{aligned}&{{\pi }_{23}} = \left[ \begin{array}{c} \frac{{{L_{\alpha V}} + {T_V}\cos \alpha }}{{MV}} - \frac{{{L_\alpha } + T\cos \alpha }}{{M{V^2}}}\\ 0\\ \frac{{{L_{\alpha \alpha }} - T\sin \alpha }}{{MV}}\\ \frac{{{T_\lambda }\cos \alpha }}{{MV}}\\ \frac{{{L_{\alpha h}} + {T_h}\cos \alpha }}{{MV}} \end{array} \right] \end{aligned}$$

(A-17)

$$\begin{aligned}&{{\pi }_{24}} = \left[ \begin{array}{c} \frac{{{T_{\lambda V}}\sin \alpha }}{{MV}} - \frac{{{T_\lambda }\sin \alpha }}{{M{V^2}}}\\ 0\\ \frac{{{T_\lambda }\cos \alpha }}{{MV}}\\ \frac{{{T_{\lambda \lambda }}\sin \alpha }}{{MV}}\\ \frac{{{T_{\lambda h}}\sin \alpha }}{{MV}} \end{array} \right] \end{aligned}$$

(A-18)

$$\begin{aligned}&{{\pi }_{25}}= \left[ \begin{array}{c} \frac{{{L_{hV}} + {T_{hV}}\sin \alpha }}{{MV}} - \frac{{{L_h} + {T_h}\sin \alpha }}{{M{V^2}}} + \frac{{{g_h}\cos \gamma }}{{{V^2}}}\\ \frac{{{g_h}\sin \gamma }}{V}\\ \frac{{{L_{h\alpha }} + {T_{h\alpha }}\sin \alpha + T\cos \alpha }}{{MV}}\\ \frac{{{T_{h\lambda }}\sin \alpha }}{{MV}}\\ \frac{{{L_{hh}} + {T_{hh}}\sin \alpha }}{{MV}} - \frac{{{g_{hh}}\cos \gamma }}{V} \end{array} \right] \end{aligned}$$

(A-19)