Lyapunov Feedback Control Research Articles

Low-thrust rendezvous trajectory generation for multi-target active space debris removal using the RQ-Law

The problem of space debris must be addressed to avoid a cascading collision in the near future that would critically damage humanity’s space assets and dependent terrestrial infrastructure. One method of meeting this challenge is to use a dedicated spacecraft to deorbit a series of prioritized debris. To avoid a large fuel requirement, electric spacecraft propulsion can be used, for which a multiple-rendezvous low-thrust trajectory is of interest. In this study, we develop such a trajectory for a predetermined subset of the Iridium 33 debris using a two-step procedure. First, the target rendezvous order is determined by using relevant distance metrics to approximate the transfer cost between debris objects. Next, the RQ-Law, a recently developed Lyapunov feedback control law for generating low-thrust rendezvous trajectories, is used to generate a trajectory that can take a spacecraft to each of the studied debris objects in the determined order. This method can be used to rapidly determine low-thrust multi-target multi-revolution rendezvous trajectories in the preliminary design stage for missions such as space debris removal and satellite servicing without requiring an initial guess.

Observer-based feedback control of two-level open stochastic quantum system

The observer-based feedback control for the two-level bilinear open stochastic quantum system is proposed in this paper. The state of open stochastic quantum system (OSQS) is described in the Cartesian coordinate system. The proposed state observer is designed by using state-dependent differential Riccati equation (SDDRE) and constructed for optimally estimating the state of OSQS from measurement output of the system. The state of observer is continuously updated by the output data of continuous weak measurement (CWM). A Lyapunov Feedback control is designed based on estimated state of the observer for the state transfer of OSQS. An exponential Lyapunov function is chosen to ensure the stability of the system. The observer-based Lyapunov feedback control (OLFC) strategy is developed according to the stochastic Lyapunov stability theorem. The numerical simulation results verify the achievability of the proposed OLFC strategy in terms of state estimation and state transfer of OSQS. Numerical simulations demonstrate that the observer tracks the state of system asymptotically with minimum error of ± 3%. The proposed OLFC has the ability to move the state of OSQS from arbitrary initial state to the final target eigenstate with high fidelity ≥ 90%.

Preliminary Design of Low-Energy, Low-Thrust Transfers to Halo Orbits Using Feedback Control

Calculating low-thrust trajectories that connect geocentric orbits to Earth-moon halo orbits involves designing the many-revolution thrust arc required to depart Earth vicinity. Existing methods are based on solving the corresponding optimal control problem and hence are computationally expensive, involve providing hard-to-obtain initial guesses, and do not lend themselves well to quick parametric trade studies. In this paper, a stochastic optimization algorithm is coupled with a Lyapunov feedback control law to explore the solution space for designing Earth-to-halo low-energy, low-thrust trajectories. While the objective is to minimize the fuel consumed for the transfer, the time of flight is constrained so as to optimally distribute the coast time between the low-thrust escape spiral and the low-energy arc. Engines are assumed to have a constant specific impulse and provide constant thrust throughout the transfer. The results obtained from this method are then compared with other available reference solutions in order to quantify its effectiveness in designing such transfers. Favorable comparisons thus obtained demonstrate the utility of the proposed method in achieving near-optimal solutions, combined with a large improvement in computation time.

Nonlinear reduced dynamics modelling and simulation of two-wheeled self-balancing mobile robot: SEGWAY system

ABSTRACTSegway is a self-balancing motorized two-wheeled vehicle which is able to carry the human body. In the presented paper, a problem of nonholonomic constrained mechanical systems is treated. New methods in nonholonomic mechanics are applied to a problem of a two-wheeled self-balancing robots motion ‘SEGWAY’. This method of the geometrical theory of general nonholonomic constrained systems on fibred manifolds and their jet prolongations, based on so-called Chetaev-type constraint forces, was proposed and developed in the last decade by Krupková and others. The equations of motion of a two-wheeled self-balancing robot are highly nonlinear and rolling without slipping condition can only be expressed by nonholonomic constraint equations. In this paper, the geometrical theory is applied to the above mentioned mechanical problem using the above mentioned Krupková approach. Additionally, the results of numerical solutions of constrained equations of motion derived within the theory are in good agreement with results of (1) [Maddahi, A., Shamekhi, A. H., & Ghaffari, A. (2015). A Lyapunov controller for self-balancing two-wheeled vehicles. Robotica, 33(1), 225–239]. using Lyapunov's feedback control design technique. The existence, continuity, and uniqueness of the solution for the proposed control system are proved utilizing the Filippov's solution (2). And with fuzzy controller proposed by [Qian, Q., Wu, J., & Wang, Z. (2017). A novel configuration of two-wheeled self-balancing robot/Nova konfiguracija samobalansirajuceg robota na dva kotaca (Original scientific paper/Izvorni znastveni clanak). Tehnicki Vjesnik-Technical Gazette, 24(2), 459–465].

B-plane targeting method for orbit maneuver using low thrust

In this research, a new control law for the trajectory correction maneuver (TCM) is proposed for a spacecraft assumed to be actuated by a low thruster such as an electric propulsion system. During the cruise phase of an interplanetary mission, trajectory errors accumulate continuously owing to unpredictable perturbation forces. In previous research, TCM algorithms were developed under the assumption that the thruster is impulsive. Since the required delta velocity for the TCM is relatively small and there are numerous advancements currently underway in electric propulsion systems, this paper proposes a new control law for the TCM using a low thruster. In the proposed control law, the B-plane targeting method widely used in interplanetary missions is employed as a target parameter and the Lyapunov feedback control theory is also used. The efficiency of the new control law is demonstrated through numerical simulations.

Optimization of low-thrust transfers to libration point orbits

Purpose – The purpose of this paper is to develop a superconvergent trajectory optimization method for the design of low-thrust transfer trajectories from parking orbit to libration point orbits near the collinear libration points. Design/methodology/approach – The optimization method is developed by merging the concept of Lyapunov feedback control law with the manifold dynamics. First, the whole transfer trajectory is divided into two segments, raising segment and coast segment. Then, the trajectories of each segment are described in different coordinate frames, and designed with Lyapunov feedback control law and the manifold dynamics, respectively. The Poincare section is used as an effective tool to search the compatible patching point between these two segments on the manifold. Finally, the transfer trajectory is optimized using sequential quadratic programming. Findings – In the numerical simulation, the proposed optimization method does not have any convergence problem. It is a fast and effective me...

Optimal strategy for low-thrust spiral trajectories using Lyapunov-based guidance

It is difficult and time-consuming to search the optimal low-thrust spiral trajectories. This is because the analytical solutions for long powered arcs are not available, and hundreds or even thousands of orbital revolutions are involved in transfer trajectories. This paper examines a smart guidance scheme based on Lyapunov feedback control which overcomes the difficulties found in control gains steering. In this guidance scheme, the artificial neural network is adopted to implement gains steering and the evolutionary algorithm is used as the learning algorithm for artificial neural network. In this paper, Earth J2 perturbation and Earth-shadow eclipse effects are considered. Finally, a comparison with solutions given by the literature demonstrates the effectiveness of the proposed method. Numerical simulation results show that the time-varying gains guidance scheme can decrease the transfer time with respect to constant gains cases.

A Lyapunov controller for self-balancing two-wheeled vehicles

SUMMARYSegway is a self-balancing motorized two-wheeled vehicle which is able to carry the human body. The main issue in design of such a vehicle is to choose a stable control system capable of keeping the rider close to the upright position over smooth and non-smooth surfaces. This work extends the research previously performed by the authors for design of a controller, using the feedback linearization technique, to increase the stability of a two-wheeled vehicle carrying human. This paper investigates the design and validation of a controller for an inertial mobile vehicle using the Lyapunov's feedback control design technique. The system equations of motion are derived followed by finding the Lyapunov function required to design the controller. Owing to the discontinuity, originating from a sign function in the control law, the proposed control system is discontinuous. Therefore, the existence, continuity, and uniqueness of the solution are proven utilizing the Filippov's solution. Afterwards, the asymptotic stability of the control system is proven using the extensions of Lyapunov's stability theory to nonsmooth systems, and LaSalle's invariant set theorem. Finally, the effectiveness of the proposed control system is validated using simulation studies. Results confirm that the controller keeps the system stable while provides good position tracking responses.

Generalized Guidance Scheme for Low-Thrust Orbit Transfer

The authors present an orbital guidance scheme for the satellite with an electrical propulsion system using a Lyapunov feedback control. The construction of a Lyapunov candidate is based on orbital elements, which consist of angular momentum and eccentricity vectors. This approach performs orbit transfers between any two arbitrary elliptic or circular orbits without any singularity issues. These orbital elements uniquely describe a non degenerate Keplerian orbit. The authors improve the reliability of the existing Lyapunov orbital guidance scheme by considering the energy term. Additional improvement is achieved by adding the penalty function. Furthermore, it is shown that the final suggested approach is suitable for the satellite passing the earth’s shadow area.

Feedback control of Rabi oscillations in circuit QED

We consider the feedback stabilization of Rabi oscillations in a superconducting qubit which is coupled to a microwave readout cavity. The signal is readout by homodyne detection of the in-phase quadrature amplitude of the weak-measurement output. By multiplying the time-delayed Rabi reference, one can extract the signal, with maximum signal-to-noise ratio, from the noise current. We further track and stabilize the Rabi oscillations by using Lyapunov feedback control to properly adjust the input Rabi drives. Theoretical and simulation results illustrate the effectiveness of the proposed control law.

A Lyapunov controller for stable haptic manipulation of hydraulic actuators

SUMMARYA scheme for bilateral control of hydraulic actuators is developed and experimentally evaluated in this paper. The control laws are derived based on Lyapunov's feedback control design technique. Owing to the discontinuity originating from a sign function in the control laws, the control system is non‐smooth. First, the existence, continuation, and uniqueness of Filippov's solution to the system are proven. Next, the extensions of Lyapunov's stability theory to non‐smooth systems and LaSalle's invariant set theorems are employed to prove the asymptotic stability of the control system. Effectiveness of the proposed controller is verified by simulation and experimental studies. It is shown that beside stability, the system has good transparency in terms of position and force exchanges between the master and slave actuators. Copyright © 2011 John Wiley & Sons, Ltd.

Low-Thrust, Multi-Revolution Orbit Transfer under the Constraint of a Switch Function without Prior Information

In this paper, a low-thrust, multi-revolution orbit transfer under the constraint of a complex switch function is investigated. First, the effect of the switch function, especially that of a switch function with no prior information, is analyzed. Then, by utilizing the concept of the Lyapunov feedback control law, the semi-analytical expressions of suboptimal thrust angles are derived, and a near-optimal solution could approach the optimal solution by adjusting the five weights in the Lyapunov function using sequential quadratic programming (SQP). In the novel method, except for optimization of the weight factors in the Lyapunov function, no iteration is contained in the process of design and optimization, so the method is rapid and has good convergence. This method also overcomes the drawbacks about convergence of traditional calculus of variation (COV). It is an effective method for the design of low-thrust, multi-revolution orbit transfer with no prior information switch functions.

A low‐thrust guidance law based on Lyapunov feedback control and hybrid genetic algorithm

PurposeThis paper seeks to examine the development of the on‐board guidance law for multi‐revolutions orbit transfer spacecraft with low‐thrust propulsion systems.Design/methodology/approachIn the research, first, a set of equinoctial elements is utilized to avoid the singularities in dynamical equation of classical orbit elements. A thruster switch law is derived by analyzing the efficiency of the changing of each orbit elements. Second, by using the theory of Lyapunov feedback control, analytic expressions of thrust angles are derived. Finally, the weights of the Lyapunov function are adjusted by hybrid genetic algorithm to improve the performance of the guidance law.FindingsFirst, the dynamical equations of classical orbit elements are always singularity during the orbit transfer. By using modified equinoctial elements, these singularities could be avoided. Second, the trajectory is sensitive to the weights in Lyapunov function. With reasonable weights, the key parameters under the control of the guidance law presented in this paper are very close to that of optimal trajectory.Research limitations/implicationsIn further research, some dynamical weights methods will be used in the control law to improve the performance index, and approach the optimal solution.Practical implicationsThe guidance law presented in this paper could be easily used as an on‐board algorithm for the multi‐revolutions orbit transfer or stationkeeping. Furthermore, it could also be utilized as an initial design method for low‐thrust orbit transfer.Originality/valueProviding a low‐thrust guidance law by combining the concept of Lyapunov feedback control with hybrid genetic algorithm. This method has a super convergence and a low‐computational cost.

On design of continuous Lyapunov's feedback control

A common approach to Lyapunov's stability control is to design a controller such that a Lyapunov function can be derived for the control system to ensure stability. This procedure often leads to a discontinuous controller. When the controller is implemented, the discontinuous terms are replaced with continuous functions to avoid chattering of the control signal. Two associated problems have been overlooked during this procedure. One is that discontinuous control systems are non-smooth, which violates the fundamental assumptions of solution theories and the applicability of Lyapunov's stability theory is questionable. Another problem is that the replacement of discontinuous terms may weaken stability, which can be critical. In this paper, we discuss proper stability analysis of discontinuous control systems using the extended Lyapunov's second method based on Filippov's solution concept for non-smooth systems. We further propose to utilize the concept of Lyapunov exponents to quantitatively analyze the stability of continuous control systems obtained by replacing the discontinuous terms in the discontinuous controllers. An example involving the stabilization of a two-link non-fixed-base robotic manipulator is presented for demonstration. This research fills the gap in designing continuous Lyapunov's stability controllers regarding limited available Lyapunov functions.

General Orthogonal State Variables and Applications to Lyapunov Functions and Feedback Control Systems

General Orthogonal State Variables and Applications to Lyapunov Functions and Feedback Control Systems