Statistical control of control-affine nonlinear systems with nonquadratic cost functions: HJB and verification theorems

Won, Chang-Hee; Diersing, R.W.; Kang, Bryan

doi:10.1016/j.automatica.2010.06.027

Cited by 7 publications

(8 citation statements)

References 20 publications

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“…In this paper, we extend this result to n-th cumulants. This is not a simple extension of the results in (Won et al, 2010).…”

Section: Introductionmentioning

confidence: 48%

“…We proved that when n=2, the second cost cumulant HJB equation satisfies (1). The third and fourth cumulant cases were proved in (Won et al, 2010). Thus, we assume that this theorem holds for the second, third and (n-1)-th cumulant case.…”

Section: Problem Formulationmentioning

confidence: 97%

“…The study of cost control cumulant was initiated by (Sain, 1966). The authors extended the theory of cost cumulant control to third and fourth cumulants for a nonlinear system with nonquadratic cost and derived the corresponding HJB equations (Won et al, 2010). HJB equation was derive, but the solution was not determined.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Kang

Aduba

Won

2012

Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics

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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.

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“…In this paper, we extend this result to n-th cumulants. This is not a simple extension of the results in (Won et al, 2010).…”

Section: Introductionmentioning

confidence: 48%

Section: Problem Formulationmentioning

confidence: 97%

See 1 more Smart Citation

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

Kang

Aduba

Won

2012

Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics

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“…Statistical Nash differential game involves simultaneous optimization of the higher-order statistical properties of each individual player's cost function distribution through cumulants [38]. The optimization of cost function distribution through cost cumulant was initiated by…”

Section: Statistical Gamementioning

confidence: 99%

“…Won et al [38], extended the theory of cost cumulant to second, third and fourth cumulants for a nonlinear system with nonquadratic cost and derived the corresponding HJB equations. Statistical optimal control was applied to hybrid systems in [42].…”

Section: Statistical Gamementioning

confidence: 99%

<inline-formula> <tex-math notation="LaTeX">$N$</tex-math> </inline-formula>-Player Statistical Nash Game Control: <inline-formula> <tex-math notation="LaTeX">$m$</tex-math> </inline-formula>-th Cost Cumulant Optimization

Aduba

Won

2016

IEEE Trans. Automat. Contr.

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Game theory is the study of tactical interactions involving conflicts and cooperations among multiple decision makers called players with applications in diverse disciplines such as economics, biology, management, communication networks, electric power systems and control. This dissertation studies a statistical differential game problem where finite N players optimize their system performance by shaping the distribution of their cost function through cost cumulants. This research integrates game theory with statistical optimal control theory and considers a statistical Nash non-cooperative nonzero-sum game for a nonlinear dynamic system with nonquadratic cost functions. The objective of the statistical Nash game is to find the equilibrium solution where no player has the incentive to deviate once other players maintain their equilibrium strategy. The necessary condition for the existence of the Nash equilibrium solution is given for the m-th cumulant cost optimization using the Hamilton-Jacobi-Bellman (HJB) equations. In addition, the sufficient condition which is the verification theorem for the existence of Nash equilibrium solution is given for the m-th cumulant cost optimization using the Hamilton-Jacobi-Bellman (HJB) equations. However, solving the HJB equations even for relatively low dimensional game problem is not trivial, we propose to use neural network approximate method to find the solution of the HJB partial differential equations for the statistical game problem. Convergence proof of the neural network approximate method solution to exact solution is given. In addition, numerical examples are provided for the statistical game to demonstrate the applicability of the proposed theoretical developments. PREFACE "Far and away the best prize that life offers is the chance to work hard at work worth doing".

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Optimal control of markovian switching systems with applications to portfolio decisions under inflation

Fei

2015

Acta Mathematica Scientia

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Statistical control of control-affine nonlinear systems with nonquadratic cost functions: HJB and verification theorems

Cited by 7 publications

References 20 publications

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

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

<inline-formula> <tex-math notation="LaTeX">$N$</tex-math> </inline-formula>-Player Statistical Nash Game Control: <inline-formula> <tex-math notation="LaTeX">$m$</tex-math> </inline-formula>-th Cost Cumulant Optimization

Optimal control of markovian switching systems with applications to portfolio decisions under inflation

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