Global inverse optimal stabilization of stochastic nonholonomic systems

Abstract

Optimality has not been addressed in existing works on control of (stochastic) nonholonomic systems. This paper presents a design of optimal controllers with respect to a meaningful cost function to globally asymptotically stabilize (in probability) nonholonomic systems affine in stochastic disturbances. The design is based on the Lyapunov direct method, the backstepping technique, and the inverse optimal control design. A class of Lyapunov functions, which are not required to be as nonlinearly strong as quadratic or quartic, is proposed for the control design. Thus, these Lyapunov functions can be applied to design of controllers for underactuated (stochastic) mechanical systems, which are usually required Lyapunov functions of a nonlinearly weak form. The proposed control design is illustrated on a kinematic cart, of which wheel velocities are perturbed by stochastic noise.

Robust adaptive tracking control of underactuated ODINs under stochastic sea loads

“…As such, these Lyapunov functions can be chosen as class K ∞ functions of the Lyapunov functions used in this paper. These types of Lyapunov functions have been proposed for stabilization of stochastic nonholonomic systems in [48]. whereV 4 = W 4 (X e ) + Nτ El (τ min El ,τ max El ,τ El ,τ El ) + N δ El (δ min El , δ max El , δ El ,δ El ) + N τ Er (τ min Er , τ max Er , τ Er ,τ Er ) + N δ Er (δ min Er , δ max Er , δ Er ,δ Er ).…”

Three-dimensional Chained Nonholonomic Systems Stabilization Control via Dynamic Feedback

“…By the equation (4), (6), (7), (15), 12 ( , ) uu will not converge to zero except fixed-point stabilization. …”

“…Based on such typical nonholonomic wheeled mobile robot system to study the point stabilization and trajectory tracking control problem, when the control task switching between point stabilization and trajectory tracking, require the system to switch to the corresponding controller, and this kind of switch is likely to make the system produce shock, the components damage [11][12][13]. In order to avoid this kind of switch, scholars put forward the unified controller, the controller of processing a variety of control tasks at the same time.…”

Boundary control of transverse motion of flexible marine risers under stochastic loads

“…In the modal control approach, a lumped parameter system (LPS) governed by a set of ordinary differential equations (ODEs), described in terms of modal coordinates, is first obtained from the DPS by the Galerkin or assumed modes method. Then, classical control design methods, see Anderson and Moore (1990); Krstic et al (1995); Khasminskii (1980); Deng et al (2001); Mao (2007); Do (2015), can be applied to the LPS. This approach can only control a certain number of modes of a DPS and suffers from a spill-over problem, see Meirovitch (1997).…”

“…In the boundary control approach, the DPS is directly considered and controllers are implemented at the boundaries to control all the modes. Using the Lyapunov direct method in Khalil (2002), various boundary controllers of a proportional-derivative type have been proposed for flexible beam-like systems such as pipes and risers, see Stanway and Burrows (1981); Fard and Sagatun (2001); ; Fung and Tseng (1999); Pan (2008a, 2009); Do (2011); Ge et al (2010); He et al (2011); Nguyen et al (2013); Bohm et al (2014); Do (2017Do ( , 2016. There is serious problem with all of the above works: the environmental loads on the riser were assumed to be deterministic.…”