Optimal control on special Euclidean group via natural gradient algorithm



Considering the optimal control problem about the control system of the special Euclidean group whose output only depends on its input is meaningful in practical applications. The optimal control considered here is described as the output matrix is as close as possible to the target matrix by adjusting the system input. The geodesic distance is adopted as the measure of the difference between the output matrix and the target matrix, and the trajectory of the control input obtained in the process is achieved. Furthermore, some numerical simulations are shown to illustrate our outcomes based on the natural gradient descent algorithm for optimizing the control system of the special Euclidean group.


本文借助于自然梯度算法研究特殊欧几里德群的最优控制问题。这一控制系统的输出仅仅与控制输入有关。具体来说, 文中考虑的特殊欧几里德群上的最优控制问题为: 通过调节系统的输入, 使得系统的输出矩阵尽可能的接近目标矩阵, 输出矩阵与目标矩阵的差异用相应矩阵流形的测地距离来描述, 同时, 在控制过程中, 可以得到系统输入的控制轨线。在文章的最后, 利用数值模拟进一步说明文中利用自然梯度算法来解决特殊欧几里德群的最优控制问题的可行性和有效性。

This is a preview of subscription content, access via your institution.


  1. 1

    Bloch A M, Crouch P E, Marsden J E. Optimal control and geodesics on quadratic matrix Lie groups. Found Comput Math, 2008, 8: 469–500

    MathSciNet  Article  MATH  Google Scholar 

  2. 2

    Cont A, Dubnov S, Assayag G. On the information geometry of audio streams with applications to similarity computing. IEEE Trans Audio Speech Lang Process, 2011, 19: 837–846

    Article  Google Scholar 

  3. 3

    Barbaresco F, Roussigny H. Innovative tools for Radar signal processing based on Cartan’s geometry of SPD matrices and information geometry. In: Proceedings of IEEE International Radar Conference, Rome, 2008. 1–6

    Google Scholar 

  4. 4

    Grenander U, Miller M I, Srivastava A. Hilbert-Schmidt lower bounds for estimators on matrix Lie groups for ATR. IEEE Trans Patt Anal Mach Intell, 1998, 20: 790–802

    Article  Google Scholar 

  5. 5

    Zefran M, Kumar V, Croke C. On the generation of smooth three-dimensional rigid body motions. Departmental Papers (MEAM), 1998. 244

    Google Scholar 

  6. 6

    Amari S, Douglas S C. Why natural gradient? In: Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, Seattle, 1998. 2: 1213–1216

    Google Scholar 

  7. 7

    Amari S. Natural gradient works efficiently in learning. Neural Comput, 1998, 10: 251–276

    Article  Google Scholar 

  8. 8

    Duan X M, Sun H F, Peng L Y, et al. A natural gradient descent algorithm for the solution of discrete algebraic Lyapunov equations based on the geodesic distance. Appl Math Comput, 2013, 219: 9899–9905

    MathSciNet  MATH  Google Scholar 

  9. 9

    Zhang Z N, Sun H F, Zhong F W. Natural gradient-projection algorithm for distribution control. Optim Contr Appl Met, 2009. 30: 495–504

    MathSciNet  Article  Google Scholar 

  10. 10

    Zhang Z N, Sun H F, Peng L Y. Natural gradient algorithm for stochastic distribution systems with output feedback. Differ Geom Appl, 2013, 31: 682–690

    MathSciNet  Article  MATH  Google Scholar 

  11. 11

    Duan X M, Sun H F, Peng L Y. Riemannian means on special Euclidean group and unipotent matrices group. Sci World J, 2013, 2013: 292787

    Article  Google Scholar 

  12. 12

    Curtis M L. Matrix Groups. New York: Springer-Verlag, 1979

    Book  MATH  Google Scholar 

  13. 13

    Zhang X D. Matrix Analysis and Application. Beijing: Springer, 2004

    Google Scholar 

  14. 14

    Jost J. Riemannian Geometry and Geometric Analysis. 3rd ed. Berlin: Springer, 2002

    Book  MATH  Google Scholar 

  15. 15

    Moakher M. A differential geometric approch to the geometric mean of symmetric positive-definite matrices. SIAM J Matrix Anal Appl, 2005, 26: 735–747

    MathSciNet  Article  MATH  Google Scholar 

  16. 16

    Wang H. Control of conditional output probability density functions for general nonlinear and non-Gaussian dynamic stochastic systems. IEE Proc-Control Theory Appl, 2003, 150: 55–60

    Article  Google Scholar 

  17. 17

    Wang H. Minimum entropy control of non-Gaussian dynamic stochastic systems. IEEE Trans Automat Control, 2002, 47: 398–403

    MathSciNet  Article  Google Scholar 

  18. 18

    Wang H. Robust control of the output probability density functions for multivariable stochastic systems with guaranteed stability. IEEE Trans Automat Control, 1999, 44: 2103–2107

    MathSciNet  Article  MATH  Google Scholar 

  19. 19

    Bastian M R, Gunther J H, Moon T K. A simplified natural gradient learning algorithm. Adv Artif Neural Syst, 2011, 2011: 407497

    Google Scholar 

  20. 20

    Das N, Dash P K, Routray A. A constrained sequential algorithm for source separation in a non-stationary environment using natural gradient. In: Proceedings of 2011 IEEE Recent Advances in Intelligent Computational Systems (RAICS), Trivandrum, 2011. 729–734

    Chapter  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Huafei Sun.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Li, C., Zhang, E., Jiu, L. et al. Optimal control on special Euclidean group via natural gradient algorithm. Sci. China Inf. Sci. 59, 112203 (2016). https://doi.org/10.1007/s11432-015-0096-3

Download citation


  • system control
  • natural gradient
  • special Euclidean group
  • geodesic distance
  • numerical simulation


  • 系统控制
  • 自然梯度
  • 特殊欧几里德群
  • 测地距离
  • 数值模拟