Recursive terminal sliding mode control for hypersonic flight vehicle with sliding mode disturbance observer

Abstract

A recursive terminal sliding mode controller (RTSMC) based on sliding mode disturbance observer (SMDOB) is proposed for the longitudinal dynamics of a generic hypersonic flight vehicles (HFVs) in the presence of parametric uncertainties, measurement noises and external disturbances. First, a sliding mode tracking controller is presented by introducing recursive terminal sliding mode manifolds, in which each manifold will reach zero subsequently in finite time as well as the usual singularity problem will not occur. The RTSMC embraces advantages of both nonsingular terminal sliding mode control and high-order sliding mode control. Next, for the sake of enhancing the robustness of controller for uncertainties, a SMDOB is proposed to estimate and compensate the disturbances. Then, a composite controller that is composed of RTSMC and SMDOB is designed, and its stability is analyzed utilizing Lyapunov function method. Finally, numerical simulation is conducted for cruise flight condition of HFV. Simulation results show the expected control performance.

Appendices

Appendix 1

The detailed expressions of the vectors ${\varvec{\Omega }}_{1}, \varvec{\Pi }_{1}$ and matrices ${\varvec{\Omega }}_{2}, \varvec{\Pi }_{2}$ are as follows:

$$\begin{aligned}&{\varvec{\Omega }}_1 =\left[ {{\begin{array}{c} {\left( {\frac{\partial T}{\partial V}} \right) \cos \alpha -\frac{\partial D}{\partial V}} \\ {-\frac{m\mu \cos \gamma }{r^{2}}} \\ {-T\sin \alpha -\left( {\frac{\partial D}{\partial \alpha }} \right) } \\ {\left( {\frac{\partial T}{\partial \beta }} \right) \cos \alpha } \\ {\frac{2m\mu \sin \gamma }{r^{3}}} \\ \end{array} }} \right] ^\mathrm{T}\\&{\varvec{\Omega }}_2 =\left[ {{\begin{array}{ccccc} {\varvec{\omega }_{21} }&{} {\varvec{\omega }_{22} }&{} {\varvec{\omega }_{23} }&{} {\varvec{\omega }_{24} }&{} {\varvec{\omega }_{25} } \\ \end{array} }} \right] \end{aligned}$$

where

$$\begin{aligned}&\varvec{\omega }_{21} =\left[ {{\begin{array}{c} {\left( {\frac{\partial ^{2}T}{\partial V^{2}}} \right) \cos \alpha -\frac{\partial ^{2}D}{\partial V^{2}}} \\ 0 \\ {-\left( {\frac{\partial T}{\partial V}} \right) \sin \alpha -\frac{\partial ^{2}D}{\partial V\partial \alpha }} \\ {\left( {\frac{\partial ^{2}T}{\partial V\partial \beta }} \right) \cos \alpha } \\ 0 \\ \end{array} }} \right] , \varvec{\omega }_{22} =\left[ {{\begin{array}{c} 0 \\ {\frac{m\mu \sin \gamma }{r^{2}}} \\ 0 \\ 0 \\ {\frac{2m\mu \cos \gamma }{r^{3}}} \\ \end{array} }} \right] ,\\&\varvec{\omega }_{23} =\left[ {{\begin{array}{c} {-\left( {\frac{\partial T}{\partial V}} \right) \sin \alpha -\left( {\frac{\partial ^{2}D}{\partial V\partial \alpha }} \right) } \\ 0 \\ {-T\cos \alpha -\left( {\frac{\partial ^{2}D}{\partial \alpha ^{2}}} \right) } \\ {-\left( {\frac{\partial T}{\partial \beta }} \right) \sin \alpha } \\ 0 \\ \end{array} }} \right] \\&\varvec{\omega }_{24} =\left[ {{\begin{array}{c} {\frac{\partial ^{2}T}{\partial V\partial \beta }\cos \alpha } \\ 0 \\ {-\left( {\frac{\partial T}{\partial \beta }} \right) \sin \alpha } \\ 0 \\ 0 \\ \end{array} }} \right] , \varvec{\omega }_{25} =\left[ {{\begin{array}{c} 0 \\ {\frac{2m\mu \cos \gamma }{r^{3}}} \\ 0 \\ 0 \\ {-\frac{6m\mu \sin \gamma }{r^{4}}} \\ \end{array} }} \right] \\&\varvec{\Pi }_1 =\left[ {{\begin{array}{c} {\frac{{\partial L}/{\partial V+\left( {{\partial T}/{\partial V}} \right) \sin \alpha }}{mV}-\frac{L+T\sin \alpha }{mV^{2}}+\frac{\mu \cos \gamma }{V^{2}r^{2}}+\frac{\cos \gamma }{r}} \\ {\frac{\mu \sin \gamma }{Vr^{2}}-\frac{V\sin \gamma }{r}} \\ {\frac{{\partial L}/{\partial \alpha +T\cos \alpha }}{mV}} \\ {\frac{\left( {{\partial T}/{\partial \beta }} \right) \sin \alpha }{mV}} \\ {\frac{2\mu \cos \gamma }{Vr^{3}}-\frac{V\cos \gamma }{r^{2}}} \\ \end{array} }} \right] ^\mathrm{T}\\&\varvec{\Pi }_2 =\left[ {{\begin{array}{ccccc} {\varvec{\pi }_{21} }&{} {\varvec{\pi }_{22} }&{} {\varvec{\pi }_{23} }&{} {\varvec{\pi }_{24} }&{} {\varvec{\pi }_{25} } \\ \end{array} }} \right] \end{aligned}$$

$$\begin{aligned}&\varvec{\pi }_{21} =\left[ {{\begin{array}{c} {\frac{{\partial ^{2}L}/{\partial V^{2}+\left( {{\partial ^{2}T}/{\partial V^{2}}} \right) \sin \alpha }}{mV}-\frac{2\left[ {{\partial L}/{\partial V+\left( {{\partial T}/{\partial V}} \right) \sin \alpha }} \right] }{mV^{2}}} \\ +\frac{2\left( {L+T\sin \alpha } \right) }{mV^{3}}-\frac{2\mu \cos \gamma }{V^{3}r^{2}}{-\frac{\mu \sin \gamma }{V^{2}r^{2}}-\frac{\sin \gamma }{r}} \\ {\frac{\left( {{\partial ^{2}L}/{\partial \alpha \partial V}} \right) +\left( {{\partial T}/{\partial V}} \right) \cos \alpha }{mV}-\frac{{\partial L}/{\partial \alpha +T\cos \alpha }}{mV^{2}}} \\ {\frac{\left( {{\partial ^{2}T}/{\partial \beta \partial V}} \right) \sin \alpha }{mV}-\frac{\left( {{\partial T}/{\partial \beta }} \right) \sin \alpha }{mV^{2}}} \\ {-\frac{2\mu \cos \gamma }{V^{2}r^{3}}-\frac{\cos \gamma }{r^{2}}} \\ \end{array} }} \right] \\&\varvec{\pi }_{22} =\left[ {{\begin{array}{c} {-\frac{\mu \sin \gamma }{V^{2}r^{2}}-\frac{\sin \gamma }{r}} \\ {\frac{\mu \cos \gamma }{Vr^{2}}-\frac{V\cos \gamma }{r}} \\ 0 \\ 0 \\ {-\frac{2\mu \sin \gamma }{Vr^{3}}+\frac{V\sin \gamma }{r^{2}}} \\ \end{array} }} \right] ,\\&\varvec{\pi }_{23} =\left[ {{\begin{array}{c} {\frac{\left( {{\partial ^{2}L}/{\partial V\partial \alpha }} \right) +\left( {{\partial T}/{\partial V}} \right) \cos \alpha }{mV}-\frac{{\partial L}/{\partial \alpha +T\cos \alpha }}{mV^{2}}} \\ 0 \\ {\frac{{\partial ^{2}L}/{\partial \alpha ^{2}-T\sin \alpha }}{mV}} \\ {\frac{\left( {{\partial T}/{\partial \beta }} \right) \cos \alpha }{mV}} \\ 0 \\ \end{array} }} \right] \\&\varvec{\pi }_{24} =\left[ {{\begin{array}{c} {\frac{\left( {{\partial ^{2}T}/{\partial V\partial \beta }} \right) \sin \alpha }{mV}-\frac{\left( {{\partial T}/{\partial \beta }} \right) \sin \alpha }{mV^{2}}} \\ 0 \\ {\frac{\left( {{\partial T}/{\partial \beta }} \right) \cos \alpha }{mV}} \\ 0 \\ 0 \\ \end{array} }} \right] ,\\&\varvec{\pi }_{25} =\left[ {{\begin{array}{c} {-\frac{2\mu \cos \gamma }{V^{2}r^{3}}-\frac{\cos \gamma }{r^{2}}} \\ {-\frac{2\mu \sin \gamma }{Vr^{3}}+\frac{V\sin \gamma }{r^{2}}} \\ 0 \\ 0 \\ {-\frac{6\mu \cos \gamma }{Vr^{4}}+\frac{2V\cos \gamma }{r^{3}}} \\ \end{array} }} \right] \end{aligned}$$

Appendix 2

Without loss of generality, the angle of attack and flight path angle are assumed near this singularity to be

$$\begin{aligned} \alpha =\varepsilon ,\gamma =\frac{\pi }{2} \end{aligned}$$

(72)

where $\varepsilon$ is a small positive number.

Then,

$$\begin{aligned} b_{11}= & {} \left( {\frac{\rho V^{2}Sc_\beta \omega _n^2 }{2m}} \right) \cos \alpha =\left( {\frac{\rho V^{2}Sc_\beta \omega _n^2 }{2m}} \right) \cos \varepsilon \nonumber \\\approx & {} \frac{\rho V^{2}Sc_\beta \omega _n^2 }{2m} \end{aligned}$$

(73)

$$\begin{aligned} b_{12}= & {} -\left( {\frac{c_e \rho V^{2}S\bar{{c}}}{2mI_{yy} }} \right) \left( {T\sin \alpha +D_\alpha } \right) \nonumber \\\approx & {} -\left( {\frac{c_e \rho V^{2}S\bar{{c}}}{2mI_{yy} }} \right) \left( {T\varepsilon +\sigma } \right) \end{aligned}$$

(74)

where $\sigma =\frac{\partial \left( {\frac{1}{2}\rho V^{2}SC_D } \right) }{\partial \alpha }$ and which is a small real number.

$$\begin{aligned} b_{21}= & {} \left( {\frac{\rho V^{2}Sc_\beta \omega _n^2 }{2m}} \right) \sin (\alpha +\gamma )\nonumber \\= & {} \left( {\frac{\rho V^{2}Sc_\beta \omega _n^2 }{2m}} \right) \sin (\varepsilon \,+\,\frac{\pi }{2})\approx \frac{\rho V^{2}Sc_\beta \omega _n^2 }{2m} \end{aligned}$$

(75)

$$\begin{aligned} b_{22}= & {} \left( {\frac{c_e \rho V^{2}S\bar{{c}}}{2mI_{yy} }} \right) \left[ T\cos (\alpha +\gamma )\right. \nonumber \\&\left. +\,L_\alpha \cos \gamma -D_\alpha \sin \gamma \right] \nonumber \\\approx & {} \left( {\frac{c_e \rho V^{2}S\bar{{c}}}{2mI_{yy} }} \right) \left( {0+0-\sigma } \right) \nonumber \\= & {} -\sigma \left( {\frac{c_e \rho V^{2}S\bar{{c}}}{2mI_{yy} }} \right) \end{aligned}$$

(76)

In the above case, $b_{11}$ is equal to $b_{21}$, and the matrix $\mathbf{B}$ will be singular if $b_{12} =b_{22} =0$. Therefore, we give $b_{12}$ and $b_{22}$ some restrictions near the singularity, namely

$$\begin{aligned} b_{12} =\left\{ {{\begin{array}{l} {\delta _1^*,\left| {b_{12} } \right| \le \delta _1^*} \\ {b_{12} ,\left| {b_{12} } \right| >\delta _1^*} \\ \end{array} }} \right. \end{aligned}$$

(77)

$$\begin{aligned} b_{22} =\left\{ {{\begin{array}{l} {\delta _2^*,\left| {b_{22} } \right| \le \delta _2^*} \\ {b_{22} ,\left| {b_{22} } \right| >\delta _2^*} \\ \end{array} }} \right. \end{aligned}$$

(78)

moreover, $\delta _1^*\ne \delta _2^*, b_{12} \ne b_{22}$.

In that way, the matrix

$$\begin{aligned} \mathbf{B}=\left[ {{\begin{array}{ll} {b_{11} }&{}\quad {b_{12} } \\ {b_{21} }&{}\quad {b_{22} } \\ \end{array} }} \right] \ne \mathbf{0} \end{aligned}$$

(79)

And the controller in the paper can be used.