The stable manifold approach for optimal swing up and stabilization of an inverted pendulum with input saturation

Fujimoto, Ryu; Sakamoto, Noboru

doi:10.3182/20110828-6-it-1002.01504

Cited by 20 publications

(22 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In recent decades, extensive investigations have been done in the area of nonlinear optimal control. Thus, various methods have been proposed to obtain an approximate solution to the HJB equation and/or approximation of the optimal control to obtain a feedback control for general nonlinear systems [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22].…”

Section: Literature Reviewmentioning

confidence: 99%

Infinite Horizon Nonlinear Quadratic Cost Regulator

Almubarak

Sadegh

Taylor

2019

2019 American Control Conference (ACC)

View full text Add to dashboard Cite

Section: Literature Reviewmentioning

confidence: 99%

Infinite Horizon Nonlinear Quadratic Cost Regulator

Almubarak

Sadegh

Taylor

2019

2019 American Control Conference (ACC)

View full text Add to dashboard Cite

“…The first results on this topic were obtained in [23,46]. Nevertheless, pendulum control problems still attract a lot of attention of researchers [2,4,13,19,21,28,33,35,37,45,47]. There is a large number of papers on various systems containing inverted pendulums, including a spherical one on a movable base [6,7,27].…”

Section: 3mentioning

confidence: 99%

Optimal spiral-like solutions near a singular extremal in a two-input control problem

Manita¹,

Ronzhina²

2021

DCDS-B

View full text Add to dashboard Cite

<p style='text-indent:20px;'>We study an optimal control problem affine in two-dimensional bounded control, in which there is a singular point of the second order. In the neighborhood of the singular point we find optimal spiral-like solutions that attain the singular point in finite time, wherein the corresponding optimal controls perform an infinite number of rotations along the circle <inline-formula><tex-math id="M1">\begin{document}$ S^{1} $\end{document}</tex-math></inline-formula>. The problem is related to the control of an inverted spherical pendulum in the neighborhood of the upper unstable equilibrium.</p>

show abstract

“…When the system is linear, however, and a quadratic cost functional is chosen, the HJB equation is reduced to the well-known Algebraic Riccati Equation (ARE) whose solution generates the prominent Linear Quadratic Regulator (LQR). Many different techniques have been proposed over the years to approximate the HJB equation's solution and/or approximate the associated optimal feedback control law for general nonlinear systems (Adurthi, Singla, and Majji, 2017;Al'Brekht, 1961;Almubarak, Sadegh, and Taylor, 2019;Beard, Saridis, and Wen, 1998;Fujimoto and Sakamoto, 2011;Garrard, 1972;Garrard and Jordan, 1977;Garrard, Enns, and Antony Snell, 1992;Kalise and Kunisch, 2018;Lawton and Beard, 1998;Lukes, 1969;Nishikawa, Sannomiya, and 1971;Oishi and Sakamoto, 2017;Sakamoto and van der Schaft, 2008;Tran, Suzuki, and Sakamoto, 2017;Wernli and Cook, 1975;Xin and Balakrishnan, 2005;Yoshida and Loparo, 1989).…”

Section: Introductionmentioning

confidence: 99%

“…The resulting algorithm has been successfully applied to control several nonlinear systems of academic interest such as an underactuated acrobot system (Horibe and Sakamoto, 2016), a constrained magnetic levitation system (Tran, Suzuki, and Sakamoto, 2017), and an inverted pendulum with saturated input (Fujimoto and Sakamoto, 2011). The main drawback of this method is that it requires a large amount of a-priori information to produce an approximate suboptimal solution.…”

Section: Introductionmentioning

confidence: 99%

Recursive Analytic Solution of Nonlinear Optimal Regulators

Sadegh,

Almubarak

2020

Preprint

View full text Add to dashboard Cite

The paper develops an optimal regulator for a general class of multi-input affine nonlinear systems minimizing a nonlinear cost functional with infinite horizon. The cost functional is general enough to enforce saturation limits on the control input if desired. An efficient algorithm utilizing tensor algebra is employed to compute the tensor coefficients of the Taylor series expansion of the value function (i.e., optimal cost-to-go). The tensor coefficients are found by solving a set of nonlinear matrix equations recursively generalizing the well-known linear quadratic solution. The resulting solution generates the optimal controller as a nonlinear function of the state vector up to a prescribed truncation order. Moreover, a complete convergence of the computed solution together with an estimation of its applicability domain are provided to further guide the user. The algorithm's computational complexity is shown to grow only polynomially with respect to the series order. Finally, several nonlinear examples including some with input saturation are presented to demonstrate the efficacy of the algorithm to generate high order Taylor series solution of the optimal controller.

show abstract

The stable manifold approach for optimal swing up and stabilization of an inverted pendulum with input saturation

Cited by 20 publications

References 8 publications

Infinite Horizon Nonlinear Quadratic Cost Regulator

Infinite Horizon Nonlinear Quadratic Cost Regulator

Optimal spiral-like solutions near a singular extremal in a two-input control problem

Recursive Analytic Solution of Nonlinear Optimal Regulators

Contact Info

Product

Resources

About