Lyapunov Redesign Technique Research Articles

Lyapunov Redesign-Based Optimal Consensus Control for Multi-Agent Systems With Uncertain Dynamics

This brief investigates an optimal consensus problem for a set of integrator systems with dynamic uncertainties. A hierarchical distributed control framework based on partial global information is given, which can reach the optimal solution of the team objective function. Each agent has an optimal control law based on the neighbors’ information and the gradients of the local cost functions, and a compensator derived with Lyapunov redesign technique to deal with the dynamic uncertainties. Then, to remove the constraint on initial conditions and the need for global knowledge of the state, the state differences between neighbors are involved to strengthen the optimal controller while local information feedback is included in the compensator. The resulted fully distributed algorithms can guarantee asymptotic convergence of the systems in the sense of input-to-state stability (ISS). The effectiveness of the method is illustrated by numerical examples.

Current profile and normalized beta control via feedback linearization and Lyapunov techniques

Simultaneous control of the current profile and normalized plasma beta is an essential control problem in the development of advanced tokamak scenarios. However, this control problem is especially challenging due to the nonlinear nature of the current, heat, and particle transport dynamics, as well as the difficulty to understand and accurately model such processes. In this work, a nonlinear, robust, model-based controller for the simultaneous regulation of the current profile and normalized beta has been designed using feedback linearization and Lyapunov redesign techniques. Feedback linearization avoids approximate linearization of the plasma dynamics, retaining the original physics content of the model. Moreover, the use of Lyapunov redesign techniques makes the controller robust against the uncertainties arising during the modeling process. The controller’s performance in the presence of unknown dynamics is tested in nonlinear, one-dimensional simulations using the Control Oriented Transport SIMulator (COTSIM) code, which employs plasma models that are significantly more complex than those employed for control synthesis.

Lyapunov Redesign for Input and State Delays Systems by Using Optimal Predictive Control and Ultimate Bound Approaches: Theory and Experiments

This article deals with the robust control design for linear systems with input and state delays, subject to nonvanishing matched disturbances. The controller is synthesized with the Lyapunov redesign technique in two stages: first, an optimal predictor compensates the input delay of the nominal system while a quadratic performance index is minimized. Then, the robust component guarantees that the solutions are ultimately bounded. The theoretical results are experimentally validated through a dehydration process, with the controller digital implementation on an industrial programmable automation controller cDAQ-9132.

Integrated Robust Control of the Global Toroidal Rotation and Total Plasma Energy in Tokamaks

Integrated-control solutions will play a significant role in future tokamaks, in which a variety of coupled control problems will need to be solved simultaneously by means of a limited number of actuators. In this article, the problem of simultaneously regulating the global toroidal rotation, Ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">φ</sub> , and total plasma energy, W, is tackled. These two 0-D variables, Ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">φ</sub> and W, depend on the ion toroidal rotation and electron temperature profiles, respectively. Both Ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">φ</sub> and W also depend on the electron density and safety factor profiles. The actuation methods considered in this article are co-current and counter-current neutral beam injection. A nonlinear, robust controller that makes use of Lyapunov redesign techniques is synthesized based on 0-D, control-oriented models of the Ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">φ</sub> and W dynamics. In addition, an actuator management scheme is designed to handle variations in the control priorities and availability of the neutral beam injectors. The actuator manager solves an optimization problem in real time in order to find the most appropriate course of action when unexpected changes occur. The integrated control architecture is tested for a DIII-D scenario by means of the 1-D code Control-Oriented Transport Simulator (COTSIM), which predicts the time evolution of the electron temperature, electron density, ion toroidal rotation, and safety factor profiles.

Modelling, design and control of non-isolated single-input multi-output Zeta-Buck-Boost converter

In this article, we focus on the analytical modeling of nonisolated single-input multiple-output (SIMO) dc–dc converters governed by linear differential algebraic equations (DAEs). The modeling challenge in nonisolated SIMO converters arises due to the switching among multiple DAEs governing the circuit. Averaged models of such converters, described by an ordinary differential equation, are derived using the notion of quasi-Weierstrass transformation. Using this technique, we derive the averaged model for a nonisolated Zeta–Buck–Boost (ZBB) converter. It is observed that the dynamics of the state variable corresponding to the algebraic constraint is uncontrollable and eliminating this state leads to a fourth-order completely controllable averaged model. This model is used to design a linear state-feedback controller for line regulation. In the case of bipolar SIMO converters, ZBB being an example, regulation of the two outputs can be achieved using any one of the two output integral states. Although the state-feedback controller successfully rejects the disturbance asymptotically, it is found that the deviations from the set point in the transient phase can be large. To see if smaller deviations can be achieved by appending the linear controller with a nonlinear term, we explore a nonlinear controller based on the Lyapunov redesign technique. Smaller deviations with complete disturbance rejection are possible if the disturbance is matched. When it is unmatched, small deviations are achieved at the expense of steady-state errors, whose estimate is explicitly found. The proposed model and control scheme are verified through a laboratory built hardware prototype.

Robust guaranteed cost control for a class of perturbed systems with multiple distributed time delays

In this paper, a robust guaranteed cost control is designed for linear systems with multiple pointwise and distributed state delays, when it is considered the presence of a perturbation that satisfies the matching condition. This result is carried out by combining a complete type Lyapunov-Krasovskii functional, a quadratic cost function, and the Lyapunov redesign technique. The obtained nonlinear state feedback is composed by three control laws: the first one stabilizes the nominal system (a new algorithm is presented), the second one is such that an upper bound of a cost function is guaranteed, and the last one compensates the perturbation effects, ensuring the robust stability of the closed loop system. In order to validate the proposal, numerical simulations are presented, and the controller is applied to regulate a real time dehydration process temperature, implemented on an industrial Programable Automatic Control NI cDAQ − 9132.

Flexible Structure Control of Unmatched Uncertain Nonlinear Systems via Passivity-based Sliding Mode Technique

This paper considers the problem of passivity-based sliding mode control of uncertain nonlinear systems. For this purpose, two cases are studied. In the first case, it is assumed that only the matched uncertainties exist in the system dynamics. In this case, the passivity-based control approach is used to design an appropriate sliding manifold with a flexible structure such that the asymptotic stability of the reduced-order model (i.e., the motion equations on the proposed sliding manifold) is guaranteed. In the second case, both matched and unmatched uncertainties are considered in the system dynamics and thus the reduced-order model has uncertain terms. In this case, the modified Lyapunov redesign technique is combined with the passivity-based control method for design of robust sliding manifold. The function that is basically used in the classical Lyapunov redesign method is the signum function, which is not differentiable. Since, for evaluating the derivative of the sliding manifold its equation must be differentiable, a new additional continuous control term is suggested. In addition, the robust practical stabilization of the motion equations is guaranteed without considering any limitation on the upper bound of the unmatched uncertainties. Finally, computer simulations are performed for a practical example (flexible spacecraft) and also a numerical example to demonstrate the ability of the proposed controller in robust stabilization of uncertain nonlinear dynamical systems.

Robust adaptive backstepping control for an uncertain nonlinear system with input constraint based on Lyapunov redesign

This paper presents the control problem for a class of n-order semi-strict nonlinear system subjects to unknown parameters, uncertainty, and input constraint. The controller is designed via combining backstepping control and Lyapunov redesign. Firstly, based on the Lyapunov function, a composite adaptive law is constructed to estimate the unknown parameters. To sequel, a robust term is designed to handle the matched and unmatched uncertainties on the basis of Lyapunov redesign technique. The “explosion of terms” problem that inherent in backstepping control is avoided by the robust second-order filters. Thirdly, an auxiliary signal provided by the auxiliary system is employed to handle the influence of the input constraint. It is proved that the closed-loop system is stable in a Lyapunov framework theory and the semi-global uniformly ultimate boundedness of all signals is achieved. Finally, numerical simulations are carried out to evaluate the performance of the proposed control strategy. Numerical example and the application of the hypersonic vehicle (HSV) tracking control are simulated to demonstrate the effectiveness of the proposed control scheme.

Nonlinear heading control of an autonomous underwater vehicle with internal actuators

Designs of hybrid autonomous underwater vehicles (AUVs), which use internal actuators instead of control surfaces to steer, have emerged recently in the ocean engineering community. This paper focuses on the heading autopilot design for a REMUS AUV by using an internal moving mass. A nonlinear dynamical system is first derived which describes the horizontal-plane motion of the vehicle coupled with an internal moving mass. It is shown that a displacement of the internal mass in the sway direction can affect the flow of the dynamical system in phase space. Using displacement as the system input, a LQR controller is designed to stabilize the heading angle of the vehicle by taking advantage of the position and inertia of the internal moving mass. The linear controller cannot deal with large perturbations due to the constraints on the maximum displacement and movement speed of the internal moving mass. Consequently, a tunnel thruster is added to the design in order to address the problem. A nonlinear full-state feedback law is derived based on backstepping and Lyapunov redesign technique to control the displacement of the internal mass and the force exerted by the tunnel thruster. Simulation results demonstrate the effectiveness of the proposed design.

코일 저항의 불확실성을 가지는 자기 부상 시스템의 강인 제어

Electromagnetic levitation system(EMLS) is one of the well known nonlinear systems due to its high degree of nonlinearities. Moreover, when there are uncertain parameters in EMLS, it is not easy to have an accurate control of EMLS. In this paper, we first apply a standard input-output feedback linearzing controller to EMLS and investigate the possible control error caused by uncertain coil resistor. Then, as a remedy, we design and apply a robust controller using Lyapunov redesign technique to deal with this uncertain coil resistor in the system. The validity of our robust controller is verified via system analysis and experimental results.

Guaranteed cost control using Lyapunov redesign for uncertain linear time delay systems

In this contribution, we combine the complete type functional approach with the guaranteed cost control strategy to design an optimal controller that stabilizes the system in closed loop and ensures an upper bound of the cost function. Then with the Lyapunov redesign technique, an additional control component is synthesized to robustify the design to a class of uncertainties that satisfy the matching condition.

Robust Nonlinear Observer And Observer‐Backstepping Control Design For Surface Ships

AbstractA robust passive non‐linear observer, utilizing the sliding mode concept and acceleration feedback (AFB) technique, is developed for ships. The main advantage of the proposed observer is that it is robust and that it takes the Coriolis‐centripetal matrix (C‐matrix) into account. The observer reconstructs velocities of ships and bias from slowly varying environmental disturbances. It also filters out the noise and wave frequency data from measurements to protect the actuators from wear and excessive fuel consumption. The sliding mode technique is introduced to improve robust performance against neglected disturbances, uncertainties, and unmodeled dynamics. The acceleration feedback technique and coordinate transformation are used for reshaping the inertia matrix and removing the C‐matrix from the mathematical model. Then, the observer design and stability analysis become simpler. An output feedback controller using observer backstepping and the Lyapunov redesign technique is derived, and the global stability of the observer and observer‐controller system is shown by Lyapunov stability theory. A set of simulations was carried out to verify the performance of the proposed observer and controller.

Robust global uniform asymptotic stabilization of underactuated surface vessels with unknown model parameters

SummaryThe full‐state stabilization scheme is proposed for the control of an underactuated surface vessel with unknown modeling parameters. By knowing only the upper/lower bounds of model parameters, the designed controller is the first one able to globally uniformly asymptotically stabilize all the states of the vessel to zero. The virtual surge velocity control law is first derived, which makes the Lyapunov function at the kinematic level non‐increasing, irrelevant to the yaw velocity, leaving a freedom for choosing the virtual yaw velocity control law to stabilize the other state variables. After finishing the design of virtual velocity law, the back‐stepping approach and the Lyapunov redesign technique are combined to obtain the actual force/torque control law despite parameter uncertainties. Simulation examples are given to illustrate the effectiveness of the proposed control law, showing that all the states and the control inputs are globally uniformly asymptotically convergent to zero under parameter uncertainties and are globally bounded under unknown external bounded disturbances. Copyright © 2013 John Wiley &amp; Sons, Ltd.

Robust Backstepping Control Based on a Lyapunov Redesign for Skid-Steered Wheeled Mobile Robots

This paper represents a robust backstepping tracking control based on a Lyapunov redesign for Skid-Steered Wheeled Mobile Robots (WMRs). We present kinematic and dynamic models that explicitly relate the perturbations to the skidding in order to improve the tracking performance during real running. A robust controller is synthesized in the backstepping approach and the Lyapunov redesign technique, which forces the error dynamics to stabilize to the reference trajectories. We design an additional feedback control - a Lyapunov redesign - such that the overall control stabilizes the actual system in the presence of uncertainty and perturbation with the knowledge of the Lyapunov function. Simulation results are provided to validate and analyse the performance and stability of the proposed controller.

Adaptive control scheme for grid-connected photovoltaic systems with unknown bounds

An adaptive control scheme for maximum power point tracking of a single-phase grid-connected photovoltaic system is presented. The difficulty on design a controller that may operate a photovoltaic system on its maximum power point (MPP) is that, this MPP depends on temperature and solar irradiance, ambient conditions that are time-varying and difficult to measure. A solution using an on-line sliding mode estimator is presented. It estimates three different parameters that depends on solar irradiance and temperature, eliminating the necessity of having any sensor for these environmental variables. It is capable of estimate time-varying parameters. A complete analysis was done taking into account the non-linearities showed by the closed-loop system. A variation of the Lyapunov redesign technique was used to find a controller that give globally asymptotically stable trajectories of the closed-loop system. An adaptive law was found to substitute a perturbation bound and also to eliminate possible chattering due to the discontinuous controller term. Computer simulations are presented to show the good performance of the controller.

Actuator Fault Reconstruction for Systems with Monotone Nonlinearities

AbstractThis paper considers observer‐based fault reconstruction for systems with monotone nonlinearities. The nonlinear term in the observer error dynamics satisfies a sector property. Using Lyapunov redesign techniques, a continuous nonlinear error feedback is designed according to the slope property of the nonlinear term to stabilize the error dynamics. Hence, the convergence of observer error is independent of the Lipschitz constant. If the observer error converges to zero asymptotically, then the continuous error feedback can be used to reconstruct the faults. As an extension, an adaptive scheme is developed for systems where the arguments of nonlinear functions are perturbed by unknown parameters. Finally, simulations of an electric driving system are presented to show the effectiveness of the schemes.

Adaptive control of uncertain nonlinear systems using mixed backstepping and Lyapunov redesign techniques

In this paper, a robust adaptive control law for a class of uncertain nonlinear systems is proposed. The proposed controller guarantees asymptotic output tracking of systems in the strict-feedback form with unknown static parameters, and matched and unmatched dynamic uncertainties. This controller takes advantages of a robust stability property of the Lyapunov redesign method and a systematic design procedure of the backstepping technique. In fact, the backstepping technique is employed to enrich the Lyapunov redesign method to compensate for not only matched – but also unmatched-uncertainties. On the other hand, using the Lyapunov redesign method in each step of the conventional backstepping technique makes backstepping robust. The suggested controller is designed through repeatedly utilizing the Lyapunov redesign method in each step of the backstepping technique. Simulation results reveal the efficiency of the Lyapunov redesign-based backstepping controller.

Robust simultaneous tracking and stabilization of wheeled mobile robots not satisfying nonholonomic constraint

A robust unified controller was proposed for wheeled mobile robots that do not satisfy the ideal rolling without slipping constraint. Practical trajectory tracking and posture stabilization were achieved in a unified framework. The design procedure was based on the transverse function method and Lyapunov redesign technique. The Lie group was also introduced in the design. The left-invariance property of the nominal model was firstly explored with respect to the standard group operation of the Lie group SE(2). Then, a bounded transverse function was constructed, by which a corresponding smooth embedded submanifold was defined. With the aid of the group operation, a smooth control law was designed, which fulfills practical tracking/stabilization of the nominal system. An additional component was finally constructed to robustify the nominal control law with respect to the slipping disturbance by using the Lyapunov redesign technique. The design procedure can be easily extended to the robot system suffered from general unknown but bounded disturbances. Simulations were provided to demonstrate the effectiveness of the robust unified controller.

Robust Control of Nonlinear Tape Transport Systems With and Without Tension Sensors

Web-winding systems, such as tape drives, are often modeled as linear and time-invariant, but at least two nonlinearities are common in these systems. First, the reel radii and moments of inertia change as web media spools from one reel to another. Second, friction can draw a thin layer of air between the layers of web media wrapped on the take-up reel, making the system’s spring and damping characteristics nonlinear by allowing a greater length of media to vibrate freely. In addition to these nonlinearities, there is often uncertainty in the motor parameters. In the first part of this paper, feedback linearization ideas motivate state feedback and changes of variables that transform the system into decoupled and intuitively meaningful tension and velocity loops. For the case where tension measurements are available, Lyapunov redesign techniques are then used to develop control laws that are robust with respect to these nonlinearities and uncertainties. The second part of this paper then develops an observer-based controller for the case where no tension measurements are available. Performance is established analytically for both the measurement-based and observer-based schemes. Simulations illustrate this performance.

Regulation of nonlinear systems using conditional integrators

This paper studies the regulation of nonlinear systems using conditional integrators. Previous work introduced the tool of conditional integrators that provide integral action inside a boundary layer while acting as stable systems outside, leading to improvement in transient response while achieving asymptotic regulation in the presence of unknown constant disturbances or parameter uncertainties. The approach, however, is restricted to a sliding mode control framework. This paper extends this tool to a fairly general class of state feedback control laws, with the stipulation that we know a Lyapunov function for the closed-loop system. Asymptotic regulation with improvement in transient response is done by using the Lyapunov redesign technique to implement the state feedback control as a saturated high-gain feedback and introducing a conditional integrator to provide integral action inside a boundary layer. Improvement in the transient response using conditional integrators is demonstrated with an experimental application to the pendubot. Copyright © 2005 John Wiley & Sons, Ltd.