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Research Papers

Partial-State Stabilization and Optimal Feedback Control for Stochastic Dynamical Systems

[+] Author and Article Information
Tanmay Rajpurohit

School of Aerospace Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332-0150
e-mail: tanmay.rajpurohit@gatech.edu

Wassim M. Haddad

School of Aerospace Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332-0150
e-mail: wm.haddad@aerospace.gatech.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received November 30, 2015; final manuscript received March 6, 2017; published online June 5, 2017. Assoc. Editor: Suman Chakravorty.

J. Dyn. Sys., Meas., Control 139(9), 091001 (Jun 05, 2017) (18 pages) Paper No: DS-15-1602; doi: 10.1115/1.4036033 History: Received November 30, 2015; Revised March 06, 2017

In this paper, we develop a unified framework to address the problem of optimal nonlinear analysis and feedback control for partial stability and partial-state stabilization of stochastic dynamical systems. Partial asymptotic stability in probability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that is positive definite and decrescent with respect to part of the system state which can clearly be seen to be the solution to the steady-state form of the stochastic Hamilton–Jacobi–Bellman equation, and hence, guaranteeing both partial stability in probability and optimality. The overall framework provides the foundation for extending optimal linear-quadratic stochastic controller synthesis to nonlinear-nonquadratic optimal partial-state stochastic stabilization. Connections to optimal linear and nonlinear regulation for linear and nonlinear time-varying stochastic systems with quadratic and nonlinear-nonquadratic cost functionals are also provided. Finally, we also develop optimal feedback controllers for affine stochastic nonlinear systems using an inverse optimality framework tailored to the partial-state stochastic stabilization problem and use this result to address polynomial and multilinear forms in the performance criterion.

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References

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Figures

Grahic Jump Location
Fig. 1

A sample average along with the sample standard deviation of the closed-loop system trajectories versus time: ω1(t) in blue, ω2(t) in red, and ω3(t) in green (See color figure online)

Grahic Jump Location
Fig. 2

A sample average along with the sample standard deviation of the control signal versus time: u1(t) in blue and u2(t) in red (See color figure online)

Grahic Jump Location
Fig. 3

A sample average along with the sample standard deviation of the closed-loop system trajectories versus time: q1(t) in blue, q2(t) in red, and q3(t) in green (See color figure online)

Grahic Jump Location
Fig. 4

A sample average along with the sample standard deviation of the control signal versus time (See color figure online)

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