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A continuous nonlinear full-state time-delayed feedback control scheme is designed within the framework of geometric mechanics to stabilize the rigid body attitude motion, which is subject to an unknown constant time delay in feedback measurement. The attitude kinematics is globally described on the matrix Lie group S O ( 3 ) of rigid body rotations. A Morse–Lyapunov–Krasovskii functional is utilized to guarantee the asymptotic stability of the system, which yields the control gain matrices via linear matrix inequality stability conditions. In light of this result, a delayed feedback control scheme is also designed for the planar rotational motion to examine the almost global asymptotic stability of the system in the presence of an unknown time delay in feedback measurement. Simulations are performed for the proposed control schemes based on the discretized models of the controlled systems and the performance of the proposed controllers are evaluated by employing large maneuvers.

The delayed feedback stabilization of rigid spacecraft attitude dynamics in the presence of an unknown time-varying delay in the measurement is addressed. The attitude representation is parameterized using minimal attitude coordinates. The time-varying delay and its derivative are assumed to be bounded. By employing a linear state feedback controller via a Lyapunov–Krasovskii functional, a general delay-dependent stability condition is characterized for the closed-loop parameterized system in terms of a linear matrix inequality (LMI) whose solution gives the suitable controller gains. An estimate of the region of attraction of the controlled system is also obtained, inside which the asymptotic stability of parameterized system is guaranteed.

The delayed feedback control of rigid body attitude motion is addressed where there is an unknown time delay in the feedback measurement. The attitude motion is described on the tangent bundle TSO(3) to globally and uniquely represent the orientation of the rigid body, while a continuous nonlinear delayed feedback control law is proposed for local asymptotic stabilization on TSO(3). For this purpose, we introduce a notion based on the Lyapunov technique, which combines the Morse-Lyapunov method with Lyapunov-Krasovskii technique to yield a suitable Morse-Lyapunov-Krasovskii (M-L–K) functional, from which the stability conditions and proper control gain matrices are obtained in terms of linear matrix inequalities (LMIs). Simulations illustrate the performance of the proposed control scheme.

In this paper, a numerical approach is developed for the stability analysis of linear stochastic delay differential equations (SDDEs) in the parameter space based on the Chebyshev Spectral Continuous Time Approximation (CSCTA) technique. The CSCTA method is used to approximate an infinite-dimensional linear SDDE as a set of linear stochastic differential equations (SDEs). The mean and mean-square stability concepts are employed for the stochastic stability analysis of the resulting SDE. For this purpose, a set of linear deterministic differential equations for both the first and second moments are obtained using the Ito differential rule. Two examples are provided: a first order SDDE with multiplicative stochastic excitation and a second order SDDE with both additive and multiplicative stochastic excitation. In both examples the stability charts obtained from the proposed approach match those obtained using the stochastic semi-discretization method as described by Elbeyli et al. (Commun Nonlinear Sci Numer Simul 10(1):85–94, 2005). In the first example the stability results obtained from both numerical approaches are found to be less conservative than the Lyapunov-based stability region obtained by Samiei et al. (Int J Dyn Control 1(1):64–80, 2013).

In this paper, we obtain an analytical Lyapunov- based stability conditions for scalar linear and nonlinear stochastic systems with discrete time-delay. The Lyapunov–Krasovskii and Lyapunov–Razumikhin methods are applied with techniques from stochastic calculus to obtain the regions of mean square asymptotic stability in the parameter space. Both delay-independent and delay-dependent stability conditions are analyzed corresponding to both additive and multiplicative stochastic Brownian motion excitation in the Ito form. It is also shown that the derived sufficient conditions are less conservative in comparison with other numerical LMI-based Lyapunov approaches. A range of different stability charts are obtained based on the derived Lyapunov-based stability criteria, which are also compared with numerical first and second moment stability boundaries computed using the stochastic semidiscretization method. A Lipschitz condition is used to treat nonlinear functions of the current and delayed states.

In this paper, we design a geometric stochastic feedback controller to almost globally asymptotically stabilize the rigid body attitude in probability. The attitude motion is represented on the tangent bundle of SO(3) and is subject to an stochastic input torque with an unknown multiplicative diffusion coefficient. In addition, we assume that the variance parameter of the stochastic input torque is unknown. We, first, interpret the attitude dynamics in the Ito sense and the Frobenius norm of the unknown diffusion coefficient is approximated by an unknown bounded scalar parameter. An adaptive backstepping method and a suitable Morse-Lyapunov (M-L) function candidate is then employed to obtain a nonlinear continuous stochastic feedback control law. The almost global asymptotic stability of system is guaranteed in probability and the control gain matrix is obtained through solving LMI feasibility problem. A simulation example is performed to demonstrate the effectiveness of the proposed control scheme on TSO(3).

Discrete-time estimation of rigid body attitude and angular velocity without any knowledge of the attitude dynamics model, is treated using the discrete Lagrange-d'Alembert principle. Using body-fixed sensor measurements of direction vectors and angular velocity, a Lagrangian is obtained as the difference between a kinetic energy-like term that is quadratic in the angular velocity estimation error and an artificial potential obtained from Wahba's function. An additional dissipation term that depends on the angular velocity estimation error is introduced, and the discrete Lagrange-d'Alembert principle is applied to the Lagrangian with this dissipation. An explicit first order and a symmetric second-order version of this discrete-time filtering scheme are also presented, with a discussion of their advantages. A numerical simulation comparing the performances of the second-order estimator and the first-order estimator is carried out.

This paper investigates robust asymptotic stabilization of rigid body attitude dynamics evolving on the tangent bundle of SO(3) using geometric stochastic feedback control, where the system is subject to a stochastic input torque. To start with, the attitude dynamics is interpreted in the Ito sense. However, due to evolution of the kinematic differential equation of the system on SO(3), analyzing the stochastic system on TSO(3) is non-trivial. To address this challenging problem of robust asymptotic stabilization of attitude dynamics, the back-stepping method along with a suitable Morse-Lyapunov (M-L) function candidate with constant control gain parameters are used to obtain a nonlinear stochastic feedback control law. The control gain matrix and the M-L function control gain can be obtained by solving a feasible LMI, which can guarantee the robust asymptotic stability of the rigid body on TSO(3). Numerical simulations are performed to demonstrate and validate the effectiveness of the proposed controller in the state space of rigid body attitude motion in TSO(3).

In this paper, a descriptor dynamic model for a two-cooperative rigid-flexible link manipulators system which holds a rigid object in a horizontal workspace is derived. When a system of multi manipulators grasps an object, they form a closed chain that its equations consist of some algebraic constraints as well as dynamic equations. These algebraic constraints can cause impulsive behaviors in the system response and must be considered in modeling and control. Moreover, some of these algebraic constraints are dependent to each other and must be neglected in order to reach to the minimal form of the model. In this paper, a dynamic model for a rigid-flexible link manipulator which is derived in many literatures based on Lagrange equation and assumed mode method is considered. Then, the Minimal constraints equations of the system are derived. The closed chain motion of manipulators and constraints are linearized around the rigid-body motion, and a linearized dynamic descriptor model of the system which is the absolute description of the whole system is obtained. Finally, by using descriptor system theory, the behavior of the system is investigated. To evaluate the proposed method, a simulation was performed on a two-cooperative flexible-link manipulator system.

A novel method to solve the stabilization problem for a class of nonlinear affine single-input systems using neural networks is proposed in this paper. The controller is based on feedback linearization where the control signal is directly estimated by a nonlinear in parameter neural network (NLPNN). A modified Back Propagation (BP) algorithm with e-modification is used to update the weights of the network. The stability of overall closed-loop system is shown using Lyapunov method. To evaluate the performance of the proposed controller, a set of simulations is performed on a Rossler chaotic system.