# Performance Shaping through Cost Cumulants and Neural Networks-based Series Expansion

### Bei Kang, Chukwuemeka Aduba, Chang-Hee Won

#### Abstract

The performance shaping method is addressed as a statistical optimal control problem. In statistical control, we shape the distribution of the cost function by minimizing n-th order cost cumulants. The n-th cost cumulant, Hamilton-Jacobi-Bellman (HJB) equation is derived as the necessary condition for the optimality. The proposed method provides an approach to control a higher order cost cumulant for stochastic systems, and generalizes the traditional linear-quadratic-Gaussian and Risk-Sensitive control methods. This allows the cost performance shaping via the cost cumulants. Moreover, the solution of general n-th cost cumulant control is provided by numerically solving the HJB equations using neural network method. The results of this paper are demonstrated through a satellite attitude control example.

#### References

- Alberkht, E. G. (1961). On the Optimal Stabilization of Nonlinear Systems. PMM - Journal of Applied Mathematics and Mechanics, 25:1254-1266.
- Beard, R., Saridis, G., and Wen, J. (1997). Sufficient Conditions for the Convergence of Galerkin Approximations to the Hamilton-Jacobi Equation. Automatica, 33(12):2159-2177.
- Chen, T., Lewis, F. L., and Abu-Khalaf, M. (2007). A Neural Network Solution for Fixed-Final Time Optimal Control of Nonlinear Systems. Automatica, 43:482- 490.
- Fleming, W. H. and Rishel, R. W. (1975). Deterministic and Stochastic Optimal Control. Springer-Verlag, New York.
- Kang, B. and Won, C. (2010). Nonlinear Second Cost Cumulant Control using Hamilton-Jacobi-Bellman Equation and Neural Network Approximation. In Proc. of the American Control Conference, Baltimore, MD.
- Lim, A. E. and Zhou, X. Y. (2001). Risk-Sensitive Control with HARA Utility. IEEE Transactions on Automatic Control, 46(4):563-578.
- Sain, M. K. (1966). Control of Linear Systems According to the Minimal Variance Criterion-A New Approach to the Disturbance Problem. IEEE Transactions on Automatic Control, AC-11(1):118-122.
- Sain, M. K. (1967). Performance Moment Recursions, with Application to Equalizer Control Laws. In Proc. of Annual Allerton Conference on Circuit and System Theory, Monticello, IL.
- Smith, P. J. (1995). A Recursive Formulation of the Old Problem of Obtaining Moments from Cumulants and Vice Versa. The American Statistician, 49(2):217- 218.
- Stuart, A. and Ord, J. K. (1987). Kendall's Advanced Theory of Statistics:Distribution Theory. Oxford University Press, New York, 5th edition.
- Won, C., Diersing, R. W., and Kang, B. (2010). Statistical Control of Control-Affine Nonlinear Systems with Nonquadratic Cost Function:HJB and Verification Theorems. Automatica, 46(10):1636-1645.
- Won, C.-H. (1999). Comparative Study of Various Control Methods for Attitude Control of a LEO Satellite. Aerospace Science and Technology, 3(5):323-333.

#### Paper Citation

#### in Harvard Style

Kang B., Aduba C. and Won C. (2012). **Performance Shaping through Cost Cumulants and Neural Networks-based Series Expansion** . In *Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,* ISBN 978-989-8565-21-1, pages 196-201. DOI: 10.5220/0004032101960201

#### in Bibtex Style

@conference{icinco12,

author={Bei Kang and Chukwuemeka Aduba and Chang-Hee Won},

title={Performance Shaping through Cost Cumulants and Neural Networks-based Series Expansion},

booktitle={Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,},

year={2012},

pages={196-201},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0004032101960201},

isbn={978-989-8565-21-1},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,

TI - Performance Shaping through Cost Cumulants and Neural Networks-based Series Expansion

SN - 978-989-8565-21-1

AU - Kang B.

AU - Aduba C.

AU - Won C.

PY - 2012

SP - 196

EP - 201

DO - 10.5220/0004032101960201