A near-optimal decoupling principle for nonlinear stochastic systems arising in robotic path planning and control

We consider nonlinear stochastic systems that arise in path planning and control of mobile robots. As is typical of almost all nonlinear stochastic systems, optimally solving the problem is intractable. Moreover, even if obtained it would require centralized control, while the path planning problem for mobile robots requires decentralized solutions. We provide a design approach which yields a tractable design that is quantifiably near-optimal. We exhibit a decoupling result under a small noise assumption consisting of the optimal open-loop design of nominal trajectory followed by decentralized feedback law to track this trajectory. As a corollary, we obtain a trajectory-optimized linear quadratic regulator design for stochastic nonlinear systems with Gaussian noise.

[1]  Pravin Varaiya,et al.  Stochastic Systems: Estimation, Identification, and Adaptive Control , 1986 .

[2]  David Q. Mayne,et al.  Model predictive control: Recent developments and future promise , 2014, Autom..

[3]  David Q. Mayne,et al.  Differential dynamic programming , 1972, The Mathematical Gazette.

[4]  H. Kushner.  Numerical Methods for Stochastic Control Problems in Continuous Time , 2000 .

[5]  J. Tsitsiklis,et al.  Robust, risk-sensitive, and data-driven control of markov decision processes , 2007 .

[6]  Dimitri P. Bertsekas,et al.  Dynamic programming and optimal control, 3rd Edition , 2005 .

[7]  A. D. Wentzell,et al.  Limit Theorems on Large Deviations for Markov Stochastic Processes , 1990 .

[8]  Yuval Tassa,et al.  Stochastic Differential Dynamic Programming , 2010, Proceedings of the 2010 American Control Conference.

[9]  H. Cruz-Suárez,et al.  Stochastic optimal control for small noise intensities: the discrete-time case , 2010 .

[10]  J. D. Perkins,et al.  Nonlinear Optimal Stochastic Control - Some Approximations when the Noise is Small , 1975, Optimization Techniques.

[11]  Karl Johan Åström,et al.  Control: A perspective , 2014, Autom..

[12]  Steven M. LaValle,et al.  Planning algorithms , 2006 .

[13]  John N. Tsitsiklis,et al.  The complexity of dynamic programming , 1989, J. Complex..

[14]  D. Q. Mayne,et al.  Robust and Stochastic MPC: Are We Going In The Right Direction? , 2015 .

[15]  W. Fleming.  Stochastic Control for Small Noise Intensities , 1971 .

[16]  E. Todorov,et al.  A generalized iterative LQG method for locally-optimal feedback control of constrained nonlinear stochastic systems , 2005, Proceedings of the 2005, American Control Conference, 2005..

[17]  Dimitri P. Bertsekas,et al.  Dynamic Programming and Optimal Control, Two Volume Set , 1995 .

[18]  Richard E. Kopp,et al.  On the Pontryagin Maximum Principle , 1962 .

[19]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[20]  An Approximation technique for small noise open-loop control problems† , 2007 .