Parabolic differential equations and lyapunov like functions.
Nonlinear parabolic differential equation stability, boundedness and uniqueness of solutions obtained by Liapunov direct method
LIAPUNOV FUNCTION↗
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Dynamics and Adaptive Control for Stability Recovery of Damaged Aircraft
This paper presents a recent study of a damaged generic transport model as part of a NASA research project to investigate adaptive control methods for stability recovery of damaged aircraft operating in off-nominal flight conditions under damage and or failures. Aerodynamic modeling of damage effects is performed using an aerodynamic code to assess changes in the stability and control derivatives of a generic transport aircraft. Certain types of damage such as damage to one of the wings or horizontal stabilizers can cause the aircraft to become asymmetric, thus resulting in a coupling between the longitudinal and lateral motions. Flight dynamics for a general asymmetric aircraft is derived to account for changes in the center of gravity that can compromise the stability of the damaged aircraft. An iterative trim analysis for the translational motion is developed to refine the trim procedure by accounting for the effects of the control surface deflection. A hybrid direct-indirect neural network, adaptive flight control is proposed as an adaptive law for stabilizing the rotational motion of the damaged aircraft. The indirect adaptation is designed to estimate the plant dynamics of the damaged aircraft in conjunction with the direct adaptation that computes the control augmentation. Two approaches are presented 1) an adaptive law derived from the Lyapunov stability theory to ensure that the signals are bounded, and 2) a recursive least-square method for parameter identification. A hardware-in-the-loop simulation is conducted and demonstrates the effectiveness of the direct neural network adaptive flight control in the stability recovery of the damaged aircraft. A preliminary simulation of the hybrid adaptive flight control has been performed and initial data have shown the effectiveness of the proposed hybrid approach. Future work will include further investigations and high-fidelity simulations of the proposed hybrid adaptive Bight control approach.
Nguyen, Nhan↗
Automated path planning of the Payload Inspection and Processing System
The Payload Changeout Room Inspection and Processing System (PIPS) is a highly redundant manipulator intended for performing tasks in the crowded and sensitive environment of the Space Shuttle Orbiter payload bay. Its dexterity will be exploited to maneuver the end effector in a workspace populated with obstacles. A method is described by which the end effector of a highly redundant manipulator is directed toward a target via a Lyapunov stability function. A cost function is constructed which represents the distance from the manipulator links to obstacles. Obstacles are avoided by causing the vector of joint parameters to move orthogonally to the gradient of the workspace cost function. A C language program implements the algorithm to generate a joint history. The resulting motion is graphically displayed using the Interactive Graphical Robot Instruction Program (IGRIP) produced by Deneb Robotics. The graphical simulation has the potential to be a useful tool in path planning for the PIPS in the Shuttle Payload Bay environment.
Byers, Robert M.↗
Hybrid Adaptive Flight Control with Model Inversion Adaptation
This study investigates a hybrid adaptive flight control method as a design possibility for a flight control system that can enable an effective adaptation strategy to deal with off-nominal flight conditions. The hybrid adaptive control blends both direct and indirect adaptive control in a model inversion flight control architecture. The blending of both direct and indirect adaptive control provides a much more flexible and effective adaptive flight control architecture than that with either direct or indirect adaptive control alone. The indirect adaptive control is used to update the model inversion controller by an on-line parameter estimation of uncertain plant dynamics based on two methods. The first parameter estimation method is an indirect adaptive law based on the Lyapunov theory, and the second method is a recursive least-squares indirect adaptive law. The model inversion controller is therefore made to adapt to changes in the plant dynamics due to uncertainty. As a result, the modeling error is reduced that directly leads to a decrease in the tracking error. In conjunction with the indirect adaptive control that updates the model inversion controller, a direct adaptive control is implemented as an augmented command to further reduce any residual tracking error that is not entirely eliminated by the indirect adaptive control.
Nguyen, Nhan↗
Attitude control of spacecraft using neural networks
This paper investigates the use of radial basis function neural networks for adaptive attitude control and momentum management of spacecraft. In the first part of the paper, neural networks are trained to learn from a family of open-loop optimal controls parameterized by the initial states and times-to-go. The trained is then used for closed-loop control. In the second part of the paper, neural networks are used for direct adaptive control in the presence of unmodeled effects and parameter uncertainty. The control and learning laws are derived using the method of Lyapunov.
Vadali, Srinivas R.↗
Analogy Between the Collatz Conjecture and Sliding Mode Control
The Collatz Conjecture is a famous mathematics problem that is simple to state and understand but remains unsolved. In this paper, the problem is recast as a discrete-time nonlinear system. and viewed in a new light from the perspective of nonlinear systems and feedback control theory. In particular, connections are made between the Collatz sequence of numbers and the behavior of closed-loop dynamical systems designed using a feedback control method called sliding mode control. Trajectories of such systems are characterized by a reaching phase and a sliding phase, the latter of which exhibits exponential convergence. As sliding mode control design is rooted in Lyapunov stability theory, the analogy suggests a new direction for proving or disproving the Collatz Conjecture. Although several possible approaches are discussed, no formal proof is given in this paper.
Collatz conjecture↗
Spacecraft Modeling, Attitude Determination, and Control: Quaternion-Based Approach
Spacecraft attitude determination and control is an important part of a spacecraft to achieve its designed mission. As of today, many spacecrafts have been successfully launched, and most of them have performed well as they were designed. Many research papers have been published to address the attitude determination and control design problems. Several text books are available for students to learn the technology and for engineers to use as references. The most popular spacecraft models for attitude determination algorithms and control design methods are the Euler angle models and the quaternion models . The Euler angle models have been proved very efficient as the linearized models are controllable, and all standard linear control system design methods are directly applicable. The drawbacks related to the Euler angle methods are (a) the designs based on linearized models may not globally stabilize the original nonlinear spacecraft, i.e., the design may not work when the attitude of the spacecraft is far away from the point where the linearization is performed; (b) the models depend on the rotational sequences, this can be error prone if several teams work on the same project and they use different rotational sequences; (c) for any rotational sequence, there is a singular point where the model is not applicable; and (d) since most attitude determination methods use quaternion to represent the spacecraft attitude, there is a need to transform quaternion into Euler angles. On the other hand, for quaternion models, people have found controllers that can globally stabilize nonlinear spacecraft systems; the models do not depend on rotational sequences and they have no singular point; and the quaternion is provided by attitude determination system and ready to use. The main problem with the quaternion model based control system design is that the linearized quaternion model is not controllable. Therefore, most published design methods heavily rely on Lyapunov functions for the nonlinear spacecraft system. But there is no systematic way to obtain a desired Lyapunov functions. Moreover, the Lyapunov function based designs focus on the closed-loop system stability but pay little attention to the closed-loop system performance. In a series of papers, the author proposed some reduced quaternion models which lead to some controllable linearized spacecraft models. Therefore, all standard linear system theory can be directly applied to analyze and design the spacecraft control systems. We showed that, in some cases, the designed control system is not only optimal for the linearized system, but also globally stabilize the original nonlinear system . Clearly, the reduced quaternion models do not depend on rotational sequences. Due to the special structure of the linearized spacecraft model, some most important design methods, such as LQR design and robust pole assignment design are very simple, enjoy the analytical solutions for some problems, have direct connection to the performance measures, such as settling time, rising time , and percentage of overshoot . All these features are attractive for high quality control system designs. The idea mentioned above is then extended to more spacecraft control problems using specific actuators such as magnetic torque bars and control momentum gyroscopes. These types of actuators may not provide exactly desired torques. Most existing methods use different conversions to get approximate solutions, meaning that these actuators may generate a torque close to but not equal to the desired one. Using the reduced quaternion models that incorporate the actuators into the system model, the control inputs are not torques but the operational parameters. The main benefit of this idea is that the control actions are not approximate but accurate. As all actuators have their operational limit, design with input constraints are also considered in this book by using recently developed interior-point optimization techniques. This book grows up from my research on the spacecraft attitude determination and control design methods in more than a decade which is focused on using reduced quaternion models because of their merits stated above. The book provides all necessary background materials on orbital dynamics, rotations and quaternion, frequently used reference frames, transformations between reference frames, space environment and disturbance torques, ephemeris astronomical vector calculations and measurement instruments, spacecraft control actuators and their models, so that the readers will get a global picture and can apply all these information into the spacecraft system modeling, attitude determination, and spacecraft control system designs, which is the main purpose of this book. This book is different from existing books in that we focus on quaternion based spacecraft control system designs and we consider only attitude control system design related problems, from spacecraft modeling, to attitude determination and estimation, to control system design method selection, to control algorithm development, and to the simulation of the control system designs. Moreover, this book addresses different attitude control tasks in the spacecraft life cycle, including spacecraft maneuver, orbit raising, attitude control, and rendezvous. Finally, this book emphasizes the state space design methods rather than the classical frequency design methods.
Attitude determination and control↗
Joint-space Lyapunov-based direct adaptive control of a kinematically redundant telerobot manipulator
This paper presents the design of a joint-space adaptive control scheme for controlling the slave arm motion of a dual-arm telerobot system developed at Goddard Space Flight Center (GSFC) to study telerobotic operations in space. Each slave arm of the dual-arm system is a kinematically redundant manipulator with seven degrees of freedom (DOF). Using the concept of model reference adaptive control (MRAC) and Liupunov direct method, we derive an adaptation algorithm that adjusts the PD controller gains of the control scheme. The development of the adaptive control scheme assumes that the slave arm motion is non-compliant and slowly varying. The implementation of the derived control scheme does not require the computation of manipulator dynamics which makes the control scheme sufficiently fast for real-time applications. Computer simulation study performed for the 7-DOF slave arm shows that the developed control scheme can efficiently adapt to sudden change in payload while tracking various test trajectories such as ramp or sinusoids with negligible position errors.
Nguyen, Charles C.↗
Harmonic oscillators and resonance series generated by a periodic unstable classical orbit
The presence of an unstable periodic classical orbit allows one to introduce the decay time as a purely classical magnitude: inverse of the Lyapunov index which characterizes the orbit instability. The Uncertainty Relation gives the corresponding resonance width which is proportional to the Planck constant. The more elaborate analysis is based on the parabolic equation method where the problem is effectively reduced to the multidimensional harmonic oscillator with the time-dependent frequency. The resonances form series in the complex energy plane which is equidistant in the direction perpendicular to the real axis. The applications of the general approach to various problems in atomic physics are briefly exposed.
Kazansky, A. K.↗
Design and implementation of fuzzy logic controllers
The main objectives of our research are to present a self-contained overview of fuzzy sets and fuzzy logic, develop a methodology for control system design using fuzzy logic controllers, and to design and implement a fuzzy logic controller for a real system. We first present the fundamental concepts of fuzzy sets and fuzzy logic. Fuzzy sets and basic fuzzy operations are defined. In addition, for control systems, it is important to understand the concepts of linguistic values, term sets, fuzzy rule base, inference methods, and defuzzification methods. Second, we introduce a four-step fuzzy logic control system design procedure. The design procedure is illustrated via four examples, showing the capabilities and robustness of fuzzy logic control systems. This is followed by a tuning procedure that we developed from our design experience. Third, we present two Lyapunov based techniques for stability analysis. Finally, we present our design and implementation of a fuzzy logic controller for a linear actuator to be used to control the direction of the Free Flight Rotorcraft Research Vehicle at LaRC.
Abihana, Osama A.↗
Adaptive Fuzzy Control of a Direct Drive Motor: Experimental Aspects
This paper presents a state feedback adaptive control method for position and velocity control of a direct drive motor. The proposed control scheme allows for integrating heuristic knowledge with mathematical knowledge of a system. It performs well even when mathematical model of the system is poorly understood. The controller consists of an adaptive fuzzy controller and a supervisory controller. The supervisory controller requires only knowledge of the upper bound and lower bound of the system parameters. The fuzzy controller is based on fuzzy basis functions and states of the system. The adaptation law is derived based on the Lyapunov function which ensures that the state of the system asymptotically approaches zero. The proposed controller is applied to a direct drive motor with payload and parameter uncertainty, and the effectiveness is experimentally verified. The real-time performance is compared with simulation results.
Medina, E.↗
Adaptive Fuzzy Control of a Direct Drive Motor
This paper presents a state feedback adaptive control method for position and velocity control of a direct drive motor. The proposed control scheme allows for integrating heuristic knowledge with mathematical knowledge of a system. It performs well even when mathematical model of the system is poorly understood. The controller consists of an adaptive fuzzy controller and a supervisory controller. The supervisory controller requires only knowledge of the upper bound and lower bound of the system parameters. The fuzzy controller is based on fuzzy basis functions and states of the system. The adaptation law is derived based on the Lyapunov function which ensures that the state of the system asymptotically approaches zero. The proposed controller is applied to a direct drive motor with payload and parameter uncertainty, and the effectiveness is verified by simulation results.
Medina, E.↗
Solar flux forecasting using mutual information with an optimal delay
Solar flux F(sub 10.7) directly affects the atmospheric density, thereby changing the lifetime and prediction of satellite orbits. For this reason, accurate forecasting of F(sub 10.7) is crucial for orbit determination of spacecraft. Our attempts to model and forecast F(sub 10.7) uncovered highly entangled dynamics. We concluded that the general lack of predictability in solar activity arises from its nonlinear nature. Nonlinear dynamics allow us to predict F(sub 10.7) more accurately than is possible using stochastic methods for time scales shorter than a characteristic horizon, and with about the same accuracy as using stochastic techniques when the forecasted data exceed this horizon. The forecast horizon is a function of two dynamical invariants: the attractor dimension and the Lyapunov exponent. In recent years, estimation of the attractor dimension reconstructed from a time series has become an important tool in data analysis. In calculating the invariants of the system, the first necessary step is the reconstruction of the attractor for the system from the time-delayed values of the time series. The choice of the time delay is critical for this reconstruction. For an infinite amount of noise-free data, the time delay can, in principle, be chosen almost arbitrarily. However, the quality of the phase portraits produced using the time-delay technique is determined by the value chosen for the delay time. Fraser and Swinney have shown that a good choice for this time delay is the one suggested by Shaw, which uses the first local minimum of the mutual information rather than the autocorrelation function to determine the time delay. This paper presents a refinement of this criterion and applies the refined technique to solar flux data to produce a forecast of the solar activity.
Ashrafi, S.↗
Low-thrust Orbit Transfers Using Candidate Lyapunov Functions with a Mechanism for Coasting
We consider low-thrust orbit transfers around a central body, where specified changes are sought in orbit elements except true anomaly. The desired changes in the remaining five elements can be arbitrarily large. Candidate Lyapunov functions are created based on analytic expressions for maximum rates of change of the orbit elements and the desired changes in the elements. These functions may be thought of as proximity quotients because they provide R measure of the proximity to the target orbit. The direction of thrust needed for steepest descent to the target orbit is also available analytically. The thrust is shutoff if the effectivity of the thrust at the current location on the osculating orbit is below some threshhold value. Thus, the equations of motion can be numerically integrated to obtain quickly and simply a transfer to the target orbit. A series of transfers can be easily computed to assess the trade-off between propellant mass and flight time. Preliminary comparisons to optimal solutions show that the method, while sub-optimal, performs well.
Candidate Lyapunoc functions↗