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Prescribed performance control for automatic carrier landing with disturbance

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Abstract

This paper presents a novel automatic carrier landing controller which holds a constant angle of attack during final approach with prescribed performance in the presence of external disturbances and carrier deck motion. Based on the nonlinear model of the aircraft, backstepping technique is adopted as the main control frame. To improve the stability during final approach, a novel control structure which maintains a constant angle of attack is proposed. By using performance constrained guidance law, the proposed controller is capable of guaranteeing trajectory tracking errors within prescribed performance, which means that the tracking errors are confined within prescribed convergence rates and maximum overshoots. Moreover, considering the deck motion of the carrier and inherent phase lag of the aircraft, deck motion compensation is included. Furthermore, nonlinear disturbance observers are introduced to eliminate the affects of unknown disturbances, while command filters are employed as well, avoiding complicated computations for time derivatives of virtual controls. Finally, simulation results clarify and verify the proposed control scheme.

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grant 61503010, in part by the Aeronautical Science Foundation of China under Grant 2016ZA51001, and in part by the Fundamental Research Funds for the Central Universities under Grant YWF-17-BJ-Y-143 and YWF-18-BJ-Y-108.

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Authors and Affiliations

  1. School of Aeronautic Science and Engineering, Beihang University, Beijing, 100191, People’s Republic of China

    Zhiyuan Guan & Yunpeng Ma

  2. Seventh Research Division, School of Automation Science and Electrical Engineering, Beihang University, Beijing, 100191, People’s Republic of China

    Zewei Zheng

  3. Institute of Disaster Prevention, Sanhe, Hebei, People’s Republic of China

    Na Guo

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  1. Zhiyuan Guan
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  2. Yunpeng Ma
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Correspondence to Zewei Zheng.

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Appendices

Appendix A

$$\begin{aligned}&T = T_{\mathrm {max}} \delta _p,\rho = 1.225\hbox { kg/m}^3, Q = \frac{1}{2} \rho V^2 \\&\begin{bmatrix} Y\\ D\\ C \end{bmatrix} = \text {QS}\begin{bmatrix} C_Y\\ C_D\\ C_C \end{bmatrix} = \text {QS}\begin{bmatrix} C_Y^0 + C_Y^\alpha \alpha \\ C_D^0 + C_D^\alpha \alpha + C_D^{\alpha ^2} \alpha ^2\\ C_C^\beta \beta \end{bmatrix} \\&\begin{bmatrix} L\\ M\\ N \end{bmatrix} = \text {QS}\begin{bmatrix} l C_L\\ c C_M\\ l C_N \end{bmatrix} \\&\quad = \text {QS}\begin{bmatrix} l ( C_L^\beta \beta + C_L^{\delta _\alpha } \delta _\alpha + C_L^{\delta _r} \delta _r + C_L^p p + C_L^r r )\\ c (C_M^0 + C_M^{\delta _e} \delta _e + C_M^q q )\\ l ( C_N^\beta \beta + C_N^{\delta _\alpha } \delta _\alpha + C_N^{\delta _r} \delta _r + C_N^p p + C_N^r r ) \\ \end{bmatrix} \end{aligned}$$

where \(C_Y,C_D,C_C\) are the lift, drag, lateral forces coefficients, respectively, \(C_L,C_M,C_N\) are rolling, pitching, yawing moments coefficients, respectively. l denotes wing span, c denotes mean aerodynamic chord, \(\delta _a,\delta _e,\delta _r\) are the deflection angles of aileron, elevator, rudder; \(\delta _p\) is the degree of throttle (%). \(T_{\mathrm {max}}\) is the maximum thrust of the engine, T is the current thrust.

Appendix B

$$\begin{aligned}&\varvec{f}_1 = \varvec{f}_1(\varvec{x}_2 \mathrm {,} V_k) = \begin{bmatrix} V_k(\cos \gamma \sin \chi - \chi ) \\ -V_k(\sin \gamma - \gamma ) \end{bmatrix} \\ \\&\varvec{b}_1 = \varvec{b}_1(V_k) = \begin{bmatrix} V_k&0\\ 0&-V_k \end{bmatrix} \\ \\&\varvec{f}_2 = \varvec{f}_2(\varvec{x}_2 \mathrm {,} \varvec{x}_3 \mathrm {,} V_k) \\&\quad = \begin{bmatrix} \frac{1}{m V_k \cos \gamma } (T(-\cos (\alpha +\sigma )\sin \beta \cos \mu ) \\ + \text {QSC}_C^\beta \beta \cos \mu )\\ -\frac{1}{m V_k}(T(-\sin (\alpha + \sigma )\cos \mu \\ - \cos (\alpha + \sigma )\sin \beta \sin \mu )+\text {QSC}_c^\beta \beta \sin \mu \\ -\text {QSC}_Y^0 \cos \mu +mg\cos \gamma ) \end{bmatrix} \\ \\&\varvec{b}_2 = \varvec{b}_2(\varvec{x}_2 \mathrm {,} V_k \mathrm {,} Q) \\&\quad = \frac{1}{mV_k}\begin{bmatrix} \frac{T\sin (\alpha + \sigma ) + QS(C_Y^0 + C_Y^\alpha \alpha )}{\cos \gamma }&0\\ 0&\text {QSC}_Y^\alpha \\ \end{bmatrix} \\ \\&\varvec{f}_3 = \varvec{f}_3(\varvec{\dot{x}}_2 \mathrm {,} \varvec{x}_2 \mathrm {,} \varvec{x}_3) \\&\quad = \begin{bmatrix} \dot{\gamma } + \frac{-\dot{\gamma } \cos \mu - \dot{\chi }\sin \mu \cos \gamma }{\cos \beta } \\ -\dot{\gamma } \sin \mu + \dot{\chi } \cos \mu \cos \gamma \\ \frac{\dot{\gamma } \sin \beta \cos \mu + \dot{\chi }(\sin \gamma \cos \beta + \sin \beta \sin \mu \cos \gamma )}{\cos \beta } \end{bmatrix} \\&\varvec{b}_3= \varvec{b}_3(\varvec{x}_2 \mathrm {,} \varvec{x}_3) \\&\quad = \begin{bmatrix} -\cos \alpha \tan \beta&\frac{1}{\cos \beta }&-\sin \alpha \tan \beta \\ \sin \alpha&0&-\cos \alpha \\ \frac{\cos \alpha }{\cos \beta }&0&\frac{\sin \alpha }{\cos \beta } \end{bmatrix} \\ \\&\varvec{f}_4 = \varvec{f}_4(\varvec{x}_3 \mathrm {,} \varvec{x}_4 \mathrm {,} Q) \\&\varvec{f}_4(1) =\frac{1}{I_x I_z - I_{xz}^2}((I_y I_z - I_z^2 -I{xz}^2)rq \\&\qquad +\, (I_x I_{xz} - I_y I_{xz} - I_z I_{xz})pq + I_z \text {QSl}(C_L^\beta \beta \\&\qquad +\, C_L^p p + C_L^r r) + I_{xz}\text {QSl}(C_N^\beta \beta + C_N^p p + C_N^r r)) \\&\varvec{f}_4(2) = \frac{1}{I_y}((I_z - I_x)pr -I_{xz}p^2 + I_{xz}r^2 \\&\qquad ~~~~~+\, \text {QSC}(C_M^0 + C_M^q q)) \\&\varvec{f}_4(3) = \frac{1}{I_x I_z - I_{xz}^2}((I_x^2 - I_x I_y + I_{xz}^2)pq \\&\qquad -\, (I_x I_{xz} - I_y I_{xz} - I_z I_{xz})rq + I_{xz} \text {QSl}(C_L^\beta \beta \\&\qquad +\, C_L^p p + C_L^r r) + I_x\text {QSl}(C_N^\beta \beta + C_N^p p + C_N^r r)) \\&\varvec{b}_4 = \varvec{b}_4(Q) \\&\quad = \text {QS}\begin{bmatrix} l\frac{I_z C_L^{\delta _a} + I_{xz}C_N^{\delta _a}}{I_xI_z - I^2_{xz}}&0&l \frac{I_zC_L^{\delta _r} + I_{xz}C_N^{\delta _r}}{I_xI_z - I^2_{xz}} \\ 0&c\frac{C_M^{\delta _e}}{I_y}&0 \\ l\frac{I_{xz} C_L^{\delta _a} + I_{x}C_N^{\delta _a}}{I_xI_z - I^2_{xz}}&0&l\frac{I_{xz} C_L^{\delta _r} + I_{x}C_N^{\delta _r}}{I_xI_z - I^2_{xz}} \end{bmatrix} \\ \end{aligned}$$

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Guan, Z., Ma, Y., Zheng, Z. et al. Prescribed performance control for automatic carrier landing with disturbance. Nonlinear Dyn 94, 1335–1349 (2018). https://doi.org/10.1007/s11071-018-4427-3

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