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Modified deep deterministic policy gradient based on active disturbance rejection control for hypersonic vehicles

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Abstract

For the attitude control of hypersonic vehicles, a control scheme based on linear active disturbance rejection (LADRC) and modified deep deterministic policy gradient (MDDPG) is proposed. Firstly, LADRC is used to deal with uncertainty and nonlinear problems in the attitude control process. For the tedious manual parameter tuning process, MDDPG is used to optimize the control gains and bandwidth of LADRC. Secondly, a modified reward function and an early stop criterion are introduced in the MDDPG algorithm to improve the optimization performance. Then, another MDDPG is used as an auxiliary control, which is combined with LADRC for attitude control to improve the robustness and accuracy of the control. The proposed method considers the robustness and rapidity of the control. Finally, the effectiveness of the proposed method is proved by simulation. Compared with traditional LADRC, LADRC based on the Q-learning algorithm, LADRC based on the traditional DDPG algorithm, and LADRC based on MDDPG algorithm, the proposed method has a better control effect and can avoid a lot of manual parameter tuning.

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Funding

This work was supported by the National Natural Science Foundation of China under Grants 62203331.

Author information

Authors and Affiliations

  1. School of Electrical Engineering and Automation, Tianjin University of Technology, Tianjin, 300384, China

    Li Xu, Ji Yuehui, Song Yu, Liu Junjie & Gao Qiang

  2. Tianjin Key Laboratory for Control Theory & Applications in Complicated Industry Systems, Tianjin, 300384, China

    Li Xu, Ji Yuehui, Song Yu, Liu Junjie & Gao Qiang

Authors
  1. Li Xu
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  2. Ji Yuehui
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  3. Song Yu
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  4. Liu Junjie
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  5. Gao Qiang
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Correspondence to Liu Junjie.

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Appendix A

Appendix A

The expression for \({c_x}\) is as follows

$$\begin{aligned} {c_x} = {c_{{x_0}}} + {c_{x{\delta _e}}} + {c_{x{\delta _a}}} + {c_{x{\delta _r}}} \end{aligned}$$
(A1)

where

$$c_{{x_{0} }} = {\text{ }}\left\{ {\begin{array}{*{20}l} {8.7173 \times 10^{{ - 2}} + 3.179 \times 10^{{ - 3}} \cdot \alpha + \left( { - 3.307} \right) \times 10^{{ - 2}} \cdot M} \hfill & {} \hfill \\ { + \left( { - 1.25} \right) \times 10^{{ - 4}} \cdot M \cdot \alpha + 5.036 \times 10^{{ - 3}} \cdot M^{2} } \hfill & {} \hfill \\ { + \left( { - 1.1} \right) \times 10^{{ - 3}} \cdot \alpha ^{2} + 1.405 \times 10^{{ - 7}} \cdot M^{2} \cdot \alpha ^{2} } \hfill & {} \hfill \\ { + \left( { - 3.658} \right) \times 10^{{ - 4}} \cdot M^{3} + 3.175 \times 10^{{ - 4}} \cdot \alpha ^{3} } \hfill & {} \hfill \\ { + 1.274 \times 10^{{ - 5}} \cdot M^{4} + \left( { - 2.985} \right) \times 10^{{ - 5}} \cdot \alpha ^{4} } \hfill & {} \hfill \\ { + \left( { - 1.705} \right) \times 10^{{ - 7}} \cdot M^{5} + 9.766 \times 10^{{ - 7}} \cdot \alpha ^{5} ,\begin{array}{*{20}c} {} & {} \\ \end{array}} \hfill & {M>4.0} \hfill \\ \end{array} } \right.{\text{ }}$$
(A2)
$$c_{{x\delta _{e} }} = {\text{ }}\left\{ {\begin{array}{*{20}l} {4.5548 \times 10^{{ - 4}} + \left( {{\text{ - }}1.1436} \right) \times 10^{{ - 4}} \cdot M} \hfill & {} \hfill \\ { + 2.5411 \times 10^{{ - 5}} \cdot \alpha + \left( { - 3.6417} \right) \times 10^{{ - 5}} \cdot \delta _{e} } \hfill & {} \hfill \\ { + \left( { - 5.3015} \right) \times 10^{{ - 7}} \cdot M \cdot \alpha \cdot \delta _{e} } \hfill & {} \hfill \\ { + 3.014 \times 10^{{ - 6}} \cdot M^{2} + 3.2187 \times 10^{{ - 6}} \cdot \alpha ^{2} } \hfill & {} \hfill \\ { + 6.9629 \times 10^{{ - 6}} \cdot \delta _{e}^{2} } \hfill & {} \hfill \\ { + 2.1026 \times 10^{{ - 12}} \cdot M^{2} \cdot \alpha ^{2} \cdot \delta _{e}^{2} ,\begin{array}{*{20}c} {} & {} \\ \end{array} } \hfill & {M>4.0} \hfill \\ \end{array} } \right.{\text{ }}$$
(A3)
$$\begin{aligned} {c_{x{\delta _a}}}= & {} {c_{x{\delta _e}}} \end{aligned}$$
(A4)
$$c_{{x{\delta _r} }} = {\text{ }}\left\{ {\begin{array}{*{20}l} {7.50 \times 10^{{ - 4}} + \left( {{\text{ - }}2.29} \right) \times 10^{{ - 5}} \cdot \alpha + \left( { - 9.69} \right) \times 10^{{ - 5}} \cdot M} \hfill & {} \hfill \\ { + 8.76 \times 10^{{ - 7}} \cdot \alpha ^{2} + 2.70 \times 10^{{ - 6}} \cdot M^{2}, \begin{array}{*{20}c} {} & {} \\ \end{array}} \hfill & {M>4.0} \hfill \\ \end{array} } \right.$$
(A5)

The expression for \({c_y}\) is as follows

$$\begin{aligned} {c_y} = {c_{{y_0}}} + {c_{y{\delta _e}}} + {c_{y{\delta _a}}} \end{aligned}$$
(A6)

where

$$c_{{y_{0} }} = {\text{ }}\left\{ {\begin{array}{*{20}l} { - 8.19 \times 10^{{ - 2}} + 4.70 \times 10^{{ - 2}} \cdot M + 1.86 \times 10^{{ - 2}} \cdot \alpha } \hfill & {} \hfill \\ { + \left( { - 4.73} \right) \times 10^{{ - 4}} \cdot M \cdot \alpha + \left( { - 9.19} \right) \times 10^{{ - 3}} \cdot M^{2} } \hfill & {} \hfill \\ { + \left( { - 1.52} \right) \times 10^{{ - 4}} \cdot \alpha ^{2} + 7.74 \times 10^{{ - 4}} \cdot M^{3} } \hfill & {} \hfill \\ { + 4.08 \times 10^{{ - 6}} \cdot \alpha ^{3} + 5.99 \times 10^{{ - 7}} \cdot M^{2} \cdot \alpha ^{2} } \hfill & {} \hfill \\ { + \left( { - 2.93} \right) \times 10^{{ - 5}} \cdot M^{4} + \left( { - 3.91} \right) \times 10^{{ - 7}} \cdot \alpha ^{4} } \hfill & {} \hfill \\ { + 4.12 \times 10^{{ - 7}} \cdot M^{5} + 1.30 \times 10^{{ - 8}} \cdot \alpha ^{5} ,\begin{array}{*{20}c} {} & {} \\ \end{array} } \hfill & {M>4.0} \hfill \\ \end{array} } \right.{\text{ }}$$
(A7)
$$c_{{y\delta _{e} }} = {\text{ }}\left\{ {\begin{array}{*{20}l} { - 1.45 \times 10^{{ - 5}} + 7.10 \times 10^{{ - 6}} \cdot M + 1.01 \times 10^{{ - 4}} \cdot \alpha } \hfill & {} \hfill \\ { + \left( { - 4.14} \right) \times 10^{{ - 4}} \cdot \delta _{e} + \left( { - 3.51} \right) \times 10^{{ - 6}} \cdot \alpha \cdot \delta _{e} } \hfill & {} \hfill \\ { + 8.72 \times 10^{{ - 6}} \cdot M \cdot \delta _{e} + \left( {1.70} \right) \times 10^{{ - 7}} \cdot M \cdot \alpha \cdot \delta _{e} ,\begin{array}{*{20}c} {} & {} \\ \end{array} } \hfill & {M>4.0} \hfill \\ \end{array} } \right.$$
(A8)
$$\begin{aligned} {c_{y{\delta _a}}}= & {} {c_{y{\delta _e}}} \end{aligned}$$
(A9)

The expression of \({m_z}\) is as follows

$$\begin{aligned} {m_z} = {m_{{z_0}}} + {m_{z{\delta _e}}} + {m_{z{\delta _\alpha }}} + {m_{z{\delta _r}}} + {m_{zz}}\frac{{{\omega _z}{L_c}}}{{2V}} \end{aligned}$$
(A10)

where

$$m_{{z_{0} }} = {\text{ }}\left\{ {\begin{array}{*{20}l} { - 2.192 \times 10^{{ - 2}} + 7.739 \times 10^{{ - 3}} \cdot M + \left( { - 2.260} \right) \times 10^{{ - 3}} \cdot \alpha } \hfill & {} \hfill \\ { + 1.808 \times 10^{{ - 4}} \cdot M \cdot \alpha + 8.849 \times 10^{{ - 4}} \cdot M^{2} } \hfill & {} \hfill \\ { + 2.616 \times 10^{{ - 4}} \cdot \alpha ^{2} + \left( { - 2.880} \right) \times 10^{{ - 7}} \cdot M^{2} \cdot \alpha ^{2} } \hfill & {} \hfill \\ { + 4.617 \times 10^{{ - 5}} \cdot M^{3} + \left( { - 7.887} \right) \times 10^{{ - 5}} \cdot \alpha ^{3} } \hfill & {} \hfill \\ { + \left( { - 1.143} \right) \times 10^{{ - 6}} \cdot M^{4} + 8.288 \times 10^{{ - 6}} \cdot \alpha ^{4} } \hfill & {} \hfill \\ { + 1.082 \times 10^{{ - 8}} \cdot M^{5} + \left( { - 2.789} \right) \times 10^{{ - 7}} \cdot \alpha ^{5} ,\begin{array}{*{20}c} {} & {} \\ \end{array} } \hfill & {M>4.0} \hfill \\ \end{array} } \right.{\text{ }}$$
(A11)
$$m_{{z\delta _{e} }} = {\text{ }}\left\{ {\begin{array}{*{20}l} { - 5.67 \times 10^{{ - 5}} + \left( { - 1.51} \right) \times 10^{{ - 6}} \cdot M} \hfill & {} \hfill \\ { + \left( { - 6.59} \right) \times 10^{{ - 5}} \cdot \alpha + 2.89 \times 10^{{ - 4}} \cdot \delta _{e} } \hfill & {} \hfill \\ { + 4.48 \times 10^{{ - 6}} \cdot \alpha \cdot \delta _{e} + \left( { - 4.46} \right) \times 10^{{ - 6}} \cdot M \cdot \alpha } \hfill & {} \hfill \\ { + \left( { - 5.87} \right) \times 10^{{ - 6}} \cdot M \cdot \delta _{e} } \hfill & {} \hfill \\ { + 9.72 \times 10^{{ - 8}} \cdot M \cdot \alpha \cdot \delta _{e} ,\begin{array}{*{20}c} {} & {} \\ \end{array} } \hfill & {M>4.0} \hfill \\ \end{array} } \right.{\text{ }}$$
(A12)
$$\begin{aligned} {m_{z{\delta _a}}}= & {} {m_{z{\delta _e}}} \end{aligned}$$
(A13)
$$m_{{z\delta _{r} }} = \left\{ {\begin{array}{*{20}l} { - 2.79 \times 10^{{ - 5}} \cdot \alpha + \left( { - 5.89} \right) \times 10^{{ - 8}} \cdot \alpha ^{2} } \hfill & {} \hfill \\ { + 1.58 \times 10^{{ - 3}} \cdot M^{2} + 6.42 \times 10^{{ - 8}} \cdot \alpha ^{3} } \hfill & {} \hfill \\ { + ( - 6.69) \times 10^{{ - 4}} \cdot M^{3} + \left( { - 2.10} \right) \times 10^{{ - 8}} \cdot \alpha ^{4} } \hfill & {} \hfill \\ { + 1.05 \times 10^{{ - 4}} \cdot M^{4} + 3.14 \times 10^{{ - 9}} \cdot \alpha ^{5} } \hfill & {} \hfill \\ { + \left( { - 7.74} \right) \times 10^{{ - 6}} \cdot M^{5} + \left( { - 2.18} \right) \times 10^{{ - 10}} \cdot \alpha ^{6} } \hfill & {} \hfill \\ { + 2.70 \times 10^{{ - 7}} \cdot M^{6} + 5.74 \times 10^{{ - 12}} \cdot \alpha ^{7} } \hfill & {} \hfill \\ { - 3.58 \times 10^{{ - 9}} \cdot M^{7}, \begin{array}{*{20}c} {} & {} \\ \end{array} } \hfill & {M>4.0} \hfill \\ \end{array} } \right.$$
(A14)
$$m_{{zz}} = {\text{ }}\left\{ {\begin{array}{*{20}l} { - 1.36 + 0.386M + 7.85 \times 10^{{ - 4}} \cdot \alpha } \hfill & {} \hfill \\ { + 1.40 \times 10^{{ - 4}} \cdot M \cdot \alpha + \left( { - 5.42} \right) \times 10^{{ - 2}} \cdot M^{2} } \hfill & {} \hfill \\ { + 2.36 \times 10^{{ - 3}} \cdot \alpha ^{2} + \left( { - 1.95} \right) \times 10^{{ - 6}} \cdot M^{2} \cdot \alpha ^{2} } \hfill & {} \hfill \\ { + 3.80 \times 10^{{ - 3}} \cdot M^{3} + \left( { - 1.48} \right) \times 10^{{ - 3}} \cdot \alpha ^{3} } \hfill & {} \hfill \\ { + \left( { - 1.30} \right) \times 10^{{ - 4}} \cdot M^{4} + 1.69 \times 10^{{ - 4}} \cdot \alpha ^{4} } \hfill & {} \hfill \\ { + 1.71 \times 10^{{ - 6}} \cdot M^{5} + \left( { - 5.93} \right) \times 10^{{ - 6}} \cdot \alpha ^{5} ,\begin{array}{*{20}c} {} & {} \\ \end{array} } \hfill & {M>4.0} \hfill \\ \end{array} } \right.{\text{ }}$$
(A15)

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Xu, L., Yuehui, J., Yu, S. et al. Modified deep deterministic policy gradient based on active disturbance rejection control for hypersonic vehicles. Neural Comput & Applic 36, 4071–4081 (2024). https://doi.org/10.1007/s00521-023-09302-5

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