0
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

School of Aerospace Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332-0150

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

## Abstract

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.

<>
Your Session has timed out. Please sign back in to continue.

## References

L'Afflitto, A. , Haddad, W. M. , and Bakolas, E. , 2016, “ Partial-State Stabilization and Optimal Feedback Control,” Int. J. Robust Nonlinear Control, 26(5), pp. 1026–1050.
Bernstein, D. S. , 1993, “ Nonquadratic Cost and Nonlinear Feedback Control,” Int. J. Robust Nonlinear Control, 3(3), pp. 211–229.
Haddad, W. M. , and Chellaboina, V. , 2008, Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach, Princeton University Press, Princeton, NJ.
Lum, K.-Y. , Bernstein, D. S. , and Coppola, V. T. , 1995, “ Global Stabilization of the Spinning Top With Mass Imbalance,” Dyn. Stab. Syst., 10(4), pp. 339–365.
Vorotnikov, V. I. , 1998, Partial Stability and Control, Birkhäuser, Boston, MA.
Chellaboina, V. , and Haddad, W. M. , 2002, “ A Unification Between Partial Stability and Stability Theory for Time-Varying Systems,” IEEE Control Syst., 22(6), pp. 66–75.
Molinari, B. , 1973, “ The Stable Regulator Problem and Its Inverse,” IEEE Trans. Autom. Control, 18(5), pp. 454–459.
Moylan, P. J. , and Anderson, B. , 1973, “ Nonlinear Regulator Theory and an Inverse Optimal Control Problem,” IEEE Trans. Autom. Control, 18(5), pp. 460–465.
Jacobson, D. H. , 1977, Extensions of Linear-Quadratic Control Optimization and Matrix Theory, Academic Press, New York.
Jacobson, D. H. , Martin, D. H. , Pachter, M. , and Geveci, T. , 1980, Extensions of Linear-Quadratic Control Theory, Springer-Verlag, Berlin.
Freeman, R. A. , and Kokotović, P. V. , 1996, “ Inverse Optimality in Robust Stabilization,” SIAM J. Control Optim., 34(4), pp. 1365–1391.
Sepulchre, R. , Jankovic, M. , and Kokotovic, P. , 1997, Constructive Nonlinear Control, Springer, London.
Deng, H. , and Krstić, M. , 1997, “ Stochastic Nonlinear Stabilization—Part II: Inverse Optimality,” Syst. Control Lett., 32(3), pp. 151–159.
Speyer, J. , 1976, “ A Nonlinear Control Law for a Stochastic Infinite Time Problem,” IEEE Trans. Autom. Control, 21(4), pp. 560–564.
Bass, R. , and Webber, R. , 1966, “ Optimal Nonlinear Feedback Control Derived From Quartic and Higher-Order Performance Criteria,” IEEE Trans. Autom. Control, 11(3), pp. 448–454.
Rajpurohit, T. , and Haddad, W. M. , 2016, “ Partial-State Stabilization and Optimal Feedback Control for Stochastic Dynamical Systems,” American Control Conference (ACC), Boston, MA, July 6–8, pp. 6562–6567.
Kushner, H. J. , 1967, Stochastic Stability and Control, Academic Press, New York.
Khasminskii, R. Z. , 2012, Stochastic Stability of Differential Equations, Springer-Verlag, Berlin.
Kushner, H. J. , 1971, Introduction to Stochastic Control, Holt, Rinehart and Winston, New York.
Arnold, L. , 1974, Stochastic Differential Equations: Theory and Applications, Wiley Interscience, New York.
Sharov, V. , 1978, “ Stability and Stabilization of Stochastic Systems Vis-a-Vis Some of the Variables,” Avtom. Telemekh., 11(1), pp. 63–71 (in Russian).
Øksendal, B. , 1995, Stochastic Differential Equations: An Introduction With Applications, Springer-Verlag, Berlin.
Yamada, T. , and Watanabe, S. , 1971, “ On the Uniqueness of Solutions of Stochastic Differential Equations,” J. Math. Kyoto Univ., 11(1), pp. 155–167.
Watanabe, S. , and Yamada, T. , 1971, “ On the Uniqueness of Solutions of Stochastic Differential Equations II,” J. Math. Kyoto Univ., 11(3), pp. 553–563.
Meyn, S. P. , and Tweedie, R. L. , 1993, Markov Chains and Stochastic Stability, Springer-Verlag, London.
Folland, G. B. , 1999, Real Analysis: Modern Techniques and Their Applications, Wiley Interscience, New York.
Mao, X. , 1999, “ Stochastic Versions of the LaSalle Theorem,” J. Differ. Equations, 153(1), pp. 175–195.
Apostol, T. M. , 1957, Mathematical Analysis, Addison-Wesley, Reading, MA.
Arapostathis, A. , Borkar, V. S. , and Ghosh, M. K. , 2012, Ergodic Control of Diffusion Processes, Cambridge University Press, Cambridge, UK.
Curtis, H. D. , 2014, Orbital Mechanics for Engineering Students, Elsevier, Oxford, UK.
Junkins, J. , and Schaub, H. , 2009, Analytical Mechanics of Space Systems, AIAA Education Series, Reston, VA.
Culick, F. E. C. , 1976, “ Nonlinear Behavior of Acoustic Waves in Combustion Chambers—I,” Acta Astronaut., 3(9–10), pp. 715–734.
Paparizos, L. G. , and Culick, F. E. C. , 1989, “ The Two-Mode Approximation to Nonlinear Acoustics in Combustion Chambers—I: Exact Solution for Second Order Acoustics,” Combust. Sci. Technol., 65(1–3), pp. 39–65.
Yang, V. , Kim, S. I. , and Culick, F. E. C. , 1987, “ Third-Order Nonlinear Acoustic Waves and Triggering of Pressure Oscillations in Combustion Chambers—Part I: Longitudinal Modes,” AIAA Paper No. 87-1873.

## Figures

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)

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)

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)

Fig. 4

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

## Discussions

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections