Direct Lyapunov Approach Research Articles

Observer‐Based Robust Controller Design for Nonlinear Fractional‐Order Uncertain Systems via LMI

We discuss the observer‐based robust controller design problem for a class of nonlinear fractional‐order uncertain systems with admissible time‐variant uncertainty in the case of the fractional‐order satisfying 0 < α < 1. Based on direct Lyapunov approach, a sufficient condition for the robust asymptotic stability of the observer‐based nonlinear fractional‐order uncertain systems is presented. Employing Finsler’s Lemma, the systematic robust stabilization design algorithm is then proposed in terms of linear matrix inequalities (LMIs). The efficiency and advantage of the proposed algorithm are finally illustrated by two numerical simulations.

Neural network based adaptive fuzzy PID-type sliding mode attitude control for a reentry vehicle

This work investigates the attitude control of reentry vehicle under modeling inaccuracies and external disturbances. A robust adaptive fuzzy PID-type sliding mode control (AFPID-SMC) is designed with the utilization of radial basis function (RBF) neural network. In order to improve the transient performance and ensure small steady state tracking error, the gain parameters of PID-type sliding mode manifold are adjusted online by using adaptive fuzzy logic system (FLS). Additionally, the designed new adaptive law can ensure that the closed-loop system is asymptotically stable. Meanwhile, the problem of the actuator saturation, caused by integral term of sliding mode manifold, is avoided even under large initial tracking error. Furthermore, to eliminate the need of a priori knowledge of the disturbance upper bound, RBF neural network observer is used to estimate the disturbance information. The stability of the closed-loop system is proved via Lyapunov direct approach. Finally, the numerical simulations verify that the proposed controller is better than conventional PID-type SMC in terms of improving the transient performance and robustness.

The ellipsoidal invariant set of fractional order systems subject to actuator saturation: the convex combination form

The domain of attraction of a class of fractional order systems subject to saturating actuators is investigated in this paper. We show the domain of attraction is the convex hull of a set of ellipsoids. In this paper, the Lyapunov direct approach and fractional order inequality are applied to estimating the domain of attraction for fractional order systems subject to actuator saturation. We demonstrate that the convex hull of ellipsoids can be made invariant for saturating actuators if each ellipsoid with a bounded control of the saturating actuators is invariant. The estimation on the contractively invariant ellipsoid and construction of the continuous feedback law are derived in terms of linear matrix inequalities (LMIs). Two numerical examples illustrate the effectiveness of the developed method.

Time delay study for semi-active control of coupled adjacent structures using MR damper

The pounding phenomenon in adjacent structures happens in severing earthquakes that can cause great damages. Connecting neighboring structures with active and semi-active control devices is an effective method to avoid mutual colliding between neighboring buildings. One of the most important issues in control systems is applying online control force. There will be a time delay if the prose of producing control force does not perform on time. This paper proposed a time-delay compensation method in coupled structures control, with semi-active Magnetorheological (MR) damper. This method based on Newmark's integration is adopted to mitigate the time-delay effect. In this study, Lyapunov's direct approach is employed to compute demanded voltage for MR dampers. Using Lyapunov's direct algorithm guarantees the system stability to design a controller based on feedback. Because of the strong nonlinearity of MR dampers, the equation of motion of coupled structures becomes an involved equation, and it is impossible to solve it with the common time step methods. In present paper modified Newmark-Beta integration based on the instantaneous optimal control algorithm, used to solve the involved equation. In this method, the response of a coupled system estimated base on optimal control force. Two MDOF structures with different degrees of freedom are finally considered as a numeric example. The numerical results show, the Newmark compensation is an efficient method to decrease the negative effect of time delay in coupled systems; furthermore, instantaneous optimal control algorithm can estimate the response of structures suitable.

Quantized Control Under Round-Robin Communication Protocol

This paper analyzes the exponential stability of a discrete-time linear plant in feedback control over a communication network with $N$ sensor nodes, dynamic quantization, large communication delays, variable sampling intervals, and round-robin scheduling. The closed-loop system is modeled as a switched system with multiple-ordered time-varying delays and bounded disturbances. We propose a time-triggered zooming algorithm implemented at the sensors that preserves exponential stability of the closed-loop system. A direct Lyapunov approach is presented for initialization of the zoom variable. The proposed framework can be applied to the plants with polytopic type uncertainties. The effectiveness of the method is illustrated on cart-pendulum and quadruple-tank processes.

Disturbance Rejection for Fractional-Order Time-Delay Systems

This paper presents an equivalent-input-disturbance (EID-) based disturbance rejection method for fractional-order time-delay systems. First, a modified state observer is applied to reconstruct the state of the fractional-order time-delay plant. Then, a disturbance estimator is designed to actively compensate for the disturbances. Under such a construction of the system, by constructing a novel monochromatic Lyapunov function and using direct Lyapunov approach, the stability analysis and controller design algorithm are derived in terms of linear matrix inequality (LMI) technique. Finally, simulation results demonstrate the validity of the proposed method.

Synchronization for distributed parameter NNs with mixed delays via sampled-data control

A sampled-data synchronization scheme is developed for a class of distributed parameter neural networks with mixed time delays. The design approach of the desired sampled data controller is also presented. We propose a sampled-data controller design, in which the sampled-data (in time) measurements of the state are taken in a finite number of fixed sampling points in the spatial domain. It is supposed that the sampling intervals in time and in space are bounded. Based on the extended Wirtinger’s inequality, an appropriate Lyapunov–Krasovskii functional and some free-weighting matrices are used to establish simple and less conservative synchronization criteria, which can ensure that the master systems synchronize with the slave systems via direct Lyapunov approach in terms of linear matrix inequalities. Finally, numerical simulations are carried out to demonstrate the effectiveness of the proposed synchronization method.

Stabilization of Networked Control Systems With Sparse Observer-Controller Networks

In this technical note we provide a set of stability conditions for linear time-invariant networked control systems with arbitrary topology, using a Lyapunov direct approach. We then use these stability conditions to provide a novel low-complexity algorithm for the design of a sparse observer-based control network. We employ distributed observers by employing the output of other nodes to improve the stability of each observer dynamics. To avoid unbounded growth of controller and observer gains, we impose bounds on their norms. The effects of relaxation of these bounds is discussed when trying to find the complete decentralization conditions.

Control design of an unmanned hovercraft for agricultural applications

The efficient and precise application of agricultural materials such as fertilizer or herbicide can be greatly facilitated by autonomous operation. This is especially important under difficult conditions at remote sites. The purpose of this work is to develop an accurate nonlinear controller using a direct Lyapunov approach to ensure stability of an unmanned hovercraft prototype used for the execution of these agricultural tasks. Such a craft constitutes an underactuated system which has fewer actuators than degrees of freedom. The proposed closed loop system is simulated to demonstrate that a control law can stabilize both the actuated and unactuated degrees of freedom of the hovercraft. It is shown that the position and orientation of the hovercraft achieve high dynamic and steady performance. Keywords: controller, direct Lyapunov method, nonholonomic constraint, underactuated system, unmanned hovercraft DOI: 10.3965/j.ijabe.20150802.1468

Switching‐based asymptotic stabilisation of underactuated ships with non‐diagonal terms in their system matrices

Stabilisation is an issue of great practical importance to the ship industry. This study considers the global asymptotic stability controller for a class of underactuated surface vessel possessing non-diagonal inertia and damping matrices. For the synthesis of controller, the investigated underactuated model is first converted into two cascade connected subsystems using input and state transformations. For these subsystems, it is shown that the stabilisation problem of this study can be reduced to the one of the second subsystem. By combining the method of adding a power integrator and the direct Lyapunov approach, a finite time switching control strategy is developed guaranteeing global asymptotic convergence of the full states to origin. The simulation results testify to the effectiveness of the presented control approach.

Guaranteed interval of delay for a class of non-linear control systems

A new method is proposed in this article in order to determine a guaranteed interval of time delay in input signal for a class of non-linear control systems with state feedback. This method is based on the stability sufficient condition of a non-linear control system using the Lyapunov direct approach. For this purpose, the dynamic equations of the non-linear input-delayed system are reformulated as a non-linear system without delay, but with delay-dependent coefficients in the state-space description. The control strategy is considered a linear state feedback. Then, by introducing an appropriately related Lyapunov candidate, the stability sufficient condition of the closed-loop system is satisfied, leading to a guaranteed interval of delay according to the state feedback gain. The main results are gathered and proved as a lemma. The effectiveness of this proposed method is investigated by simulating the state feedback control of a non-linear synchronous generator with a delay in input.

Multilevel-Based Topology Design and Cell Patterning With Robotically Controlled Optical Tweezers

This paper presents the use of robotically controlled optical tweezers to move a group of biological cells into a desired region for required patterning. A multilevel-based topology is designed to present different cell patterning in the desired region. A potential function-based controller is developed to control the cells in forming the required patterning as well as achieving rotation and scaling of the desired patterning. A pattern regulatory control force is additionally designed and added to the cell patterning controller for particularly addressing the local minima problem that causes the cells to stop at the undesired positions. The stability of the controlled system is analyzed using a direct Lyapunov approach. Experiments are performed with a cell manipulation system equipped with holographic optical tweezers to demonstrate the effectiveness of the proposed approach.

Global Asymptotic Trajectory Tracking and Point Stabilization of Asymmetric Underactuated Ships with Non-Diagonal Inertia/Damping Matrices

In this work, we investigate the trajectory tracking and point stabilization problems of asymmetric underactuated surface ships with non-diagonal inertia and damping matrices. By combining the novel state and input transformations, the direct Lyapunov approach, and the nonlinear time-varying tools, the trajectory tracking controller is derived, guaranteeing global κ-exponential convergence of state trajectory to the reference one satisfying mild persistent exciting conditions. By properly designing the reference trajectory, the proposed tracking scheme is also generalized to achieve global uniform asymptotic point stabilization. Simulation examples are given to illustrate the effectiveness of the proposed control schemes.

Stabilisation of a class of 2-DOF underactuated mechanical systems via direct Lyapunov approach

This paper represents an alternative stabilisation procedure for a class of two degree-of-freedom underactuated mechanical systems based on a set of transformations and a Lyapunov function. After simplifying dynamic equations of the system via partial feedback linearisation and coordinate changes, the stability of the system is provided with Lyapunov’s direct method. Proposed control scheme is used on two different examples and asymptotic convergence for each system is proven by means of La Salle’s invariance principle. The designed controller is successfully illustrated through numerical simulations for each example.

Observers and initial state recovering for a class of hyperbolic systems via Lyapunov method

Recently the problem of estimating the initial state of some linear infinite-dimensional systems from measurements on a finite interval was solved by using the sequence of forward and backward observers Ramdani, Tucsnak, and Weiss (2010). In the present paper, we introduce a direct Lyapunov approach to the problem and extend the results to the class of semilinear systems governed by wave and beam equations with boundary measurements from a finite interval. We first design forward observers and derive Linear Matrix Inequalities (LMIs) for the exponential stability of the estimation errors. Further we obtain simple finite-dimensional conditions in terms of LMIs for an upper bound T∗ on the minimal time, that guarantees the convergence of the sequence of forward and backward observers on [0,T∗] for the initial state recovering. This T∗ represents also an upper bound on the observability time. For observation times bigger than T∗, these LMIs give upper bounds on the convergence rate of the iterative algorithm in the norm defined by the Lyapunov functions. In our approach, T∗ is found as the minimal dwelling time for the switched exponentially stable (forward and backward estimation error) systems with the different Lyapunov functions (Liberzon, 2003). The efficiency of the results is illustrated by numerical examples.

Fault tolerant autonomous rendezvous and docking architecture for spacecraft in presence of control actuator failures

PurposeThe purpose of this paper is to present novel robust fault tolerant control design architecture to detect and isolate spacecraft attitude control actuators and reconfigure to redundant backups to improve the practicality of actuator fault detection.Design/methodology/approachThe Robust Fault Tolerant Control is designed for spacecraft Autonomous Rendezvous and Docking (AR&D) using Lyapunov direct approach applied to non‐linear model. An extended Kalman observer is used to accurately estimate the state of the attitude control actuators. Actuators on all three axes (roll/pitch/yaw) sequentially fail one after another and the robust fault tolerant controller acts to reconfigure to redundant backups to stabilize the spacecrafts and complete the required maneuver.FindingsIn the simulations, the roll, pitch and yaw dynamics of the spacecraft are considered and the attitude control actuators failures are detected and isolated. Furthermore, by switching to redundant backups, the guarantee of overall stability performance is demonstrated.Research limitations/implicationsA real time actuator failure detection and reconfiguration process using robust fault tolerant control is applied for spacecraft AR&D maneuvers. Finding an appropriate Lyapunov function for the non‐linear dynamics is not easy and always challenging. Failure of actuators on all three axes at the same time is not considered. It is a very useful approach to solve self‐assembly problems in space, spacecraft proximity maneuvers as well as co‐operative control of planetary vehicles in presence of actuator failures.Originality/valueAn approach has been proposed to detect, isolate and reconfigure spacecraft actuator failures occurred in the spacecraft attitude control system. A Robust Fault Tolerant Control scheme has been developed for the nonlinear AR&D maneuver for two spacecrafts. Failures that affect the control performance characteristics are considered and overall performance is guaranteed even in presence of control actuator failures. The architecture is demonstrated through model‐based simulation.

Observers and initial state recovering for a wave equation: an LMI approach

Recently the problem of estimating the initial state of some linear infinite-dimensional systems from measurements on a finite interval was solved by using the sequence of forward and backward observers [14]. In the present paper, we introduce a direct Lyapunov approach to the problem and extend the results to the class of semilinear systems governed by 1-d wave equations with boundary measurements from a finite interval. We first design forward observers and derive Linear Matrix Inequalities (LMIs) for the exponential stability of the estimation errors. Further we find LMIs for an upper bound T* on the minimal time, that guarantees the convergence of the sequence of forward and backward observers on [0, T*] for the initial state recovering. For observation times bigger than T*, these LMIs give upper bounds on the convergence rate of the iterative algorithm in the norm defined by the Lyapunov functions. The efficiency of the results are illustrated by a numerical example.

Observer-based robust stabilisation of a class of non-linear fractional-order uncertain systems: an linear matrix inequalitie approach

This study focuses on the observer-based robust stabilisation problem of a class of non-linear fractional-order uncertain systems with admissible time-variant uncertainty and bounded perturbation on the observer gain. A sufficient condition for the robust asymptotic stability of the observer-based non-linear fractional-order uncertain systems is presented based on direct Lyapunov approach. Utilising the matrix’s singular value decomposition method (SVD), the systematic robust stabilisation design algorithm is then proposed. All the results are obtained in terms of linear matrix inequalities. A numerical example is provided to illustrate the efficiency and advantage of the proposed algorithm.

Sampled-data control of semilinear 1-d heat equations

AbstractA semilinear one-dimensional convection-diffusion equation with distributed control, coupled to the Dirichlet or to the mixed boundary conditions, is considered. A sampled-data controller design is developed, where the sampled-data (in time) measurements of the state are taken in a finite number of fixed sampling points in the spatial domain. Sufficient conditions for the exponential convergence of the state dynamics are derived in terms of Linear Matrix Inequalities (LMIs) depending on the bounds of the sampling intervals. The new method is based on the direct Lyapunov approach via Wirtinger's and Halanay's inequalities.

Adaptive fuzzy fault-tolerant attitude control of spacecraft

This paper investigates the attitude control of spacecraft in the presence of unknown mass moment of inertia matrix, external disturbances, actuator failures, and control input constraints. A robust adaptive controller is proposed with the utilization of fuzzy logic and backstepping techniques. The unit quaternion is employed to describe the attitude of spacecraft for global representation without singularities. The system uncertainty is estimated by introducing a fuzzy logic system. The adaptive mechanism has only two parameters to be adapted on-line because the adaptive law of the proposed controller is derived from the norm of the weight matrix. The stability of the closed-loop system is guaranteed by Lyapunov direct approach. Results of numerical simulations state that the proposed controller is successful in achieving high attitude performance in the presence of parametric uncertainties, external disturbances, actuator failures, and control input constraints.