Showing papers on "Lyapunov function published in 2015"

Dynamic Triggering Mechanisms for Event-Triggered Control

Abstract: In this technical note, we present a new class of event triggering mechanisms for event-triggered control systems. This class is characterized by the introduction of an internal dynamic variable, which motivates the proposed name of dynamic event triggering mechanism. The stability of the resulting closed-loop system is proved and the influence of design parameters on the decay rate of the Lyapunov function is discussed. For linear systems, we establish a lower bound on the inter-execution time as a function of the parameters. The influence of these parameters on a quadratic integral performance index is also studied. Some simulation results are provided for illustration of the theoretical claims.

Nonlinear Robust Adaptive Tracking Control of a Quadrotor UAV Via Immersion and Invariance Methodology

Abstract: This paper presents a novel asymptotic tracking controller for an underactuated quadrotor unmanned aerial vehicle using the robust integral of the signum of the error (RISE) method and an immersion and invariance (I&I)-based adaptive control methodology The control system is decoupled into two parts: the inner loop for attitude control and the outer loop for position control The RISE approach is applied in the inner loop for disturbance rejection, whereas the I&I approach is chosen for the outer loop to compensate for the parametric uncertainties The asymptotic tracking of the time-varying 3-D position and the yaw motion reference trajectories is proven via the Lyapunov-based stability analysis and LaSalle's invariance theorem Real-time experiment results, which are performed on a hardware-in-the-loop simulation testbed, are presented to illustrate the performance of the proposed control scheme

Dynamic Surface Control Using Neural Networks for a Class of Uncertain Nonlinear Systems With Input Saturation

Abstract: In this paper, a dynamic surface control (DSC) scheme is proposed for a class of uncertain strict-feedback nonlinear systems in the presence of input saturation and unknown external disturbance. The radial basis function neural network (RBFNN) is employed to approximate the unknown system function. To efficiently tackle the unknown external disturbance, a nonlinear disturbance observer (NDO) is developed. The developed NDO can relax the known boundary requirement of the unknown disturbance and can guarantee the disturbance estimation error converge to a bounded compact set. Using NDO and RBFNN, the DSC scheme is developed for uncertain nonlinear systems based on a backstepping method. Using a DSC technique, the problem of explosion of complexity inherent in the conventional backstepping method is avoided, which is specially important for designs using neural network approximations. Under the proposed DSC scheme, the ultimately bounded convergence of all closed-loop signals is guaranteed via Lyapunov analysis. Simulation results are given to show the effectiveness of the proposed DSC design using NDO and RBFNN.

Non-singular fixed-time terminal sliding mode control of non-linear systems

Abstract: This study addresses a fixed-time terminal sliding-mode control methodology for a class of second-order non-linear systems in the presence of matched uncertainties and perturbations. A newly defined non-singular terminal sliding surface is constructed and a guaranteed closed-loop convergence time independent of initial states is derived based on the phase plane analysis and Lyapunov tools. The simulation results of a single inverted pendulum in the end are included to show the effectiveness of the proposed methodology.

Adaptive Neural Output Feedback Control of Uncertain Nonlinear Systems With Unknown Hysteresis Using Disturbance Observer

Abstract: In this paper, an adaptive neural output feedback control scheme is proposed for uncertain nonlinear systems that are subject to unknown hysteresis, external disturbances, and unmeasured states. To deal with the unknown nonlinear function term in the uncertain nonlinear system, the approximation capability of the radial basis function neural network (RBFNN) is employed. Using the approximation output of the RBFNN, the state observer and the nonlinear disturbance observer (NDO) are developed to estimate unmeasured states and unknown compounded disturbances, respectively. Based on the RBFNN, the developed NDO, and the state observer, the adaptive neural output feedback control is proposed for uncertain nonlinear systems using the backstepping technique. The first-order sliding-mode differentiator is employed to avoid the tedious analytic computation and the problem of “explosion of complexity” in the conventional backstepping method. The stability of the whole closed-loop system is rigorously proved via the Lyapunov analysis method, and the satisfactory tracking performance is guaranteed under the integrated effect of unknown hysteresis, unmeasured states, and unknown external disturbances. Simulation results of an example are presented to illustrate the effectiveness of the proposed adaptive neural output feedback control scheme for uncertain nonlinear systems.

A Survey of Linear Parameter-Varying Control Applications Validated by Experiments or High-Fidelity Simulations

Abstract: This paper provides a survey of results in linear parameter-varying (LPV) control that have been validated by experiments and/or high-fidelity simulations. The LPV controller synthesis techniques employed in the references of this survey are briefly reviewed and compared. The methods are classified into polytopic, linear fractional transformation, and gridding-based techniques and it is reviewed how in each of these approaches, synthesis can be carried out as a convex optimization problem via a finite number of linear matrix inequalities (LMIs) for both parameter-independent and parameter-dependent Lyapunov functions. The literature is categorized with regard to the application, the complexity induced by the controlled system’s dynamic and scheduling orders, as well as the synthesis method. Exemplary cases dealing with specific control design problems are presented in more detail to point control engineers to possible approaches that have been successfully applied. Furthermore, key publications in LPV control are related to application achievements on a timeline.

Reinforcement-Learning-Based Robust Controller Design for Continuous-Time Uncertain Nonlinear Systems Subject to Input Constraints

Abstract: The design of stabilizing controller for uncertain nonlinear systems with control constraints is a challenging problem The constrained-input coupled with the inability to identify accurately the uncertainties motivates the design of stabilizing controller based on reinforcement-learning (RL) methods In this paper, a novel RL-based robust adaptive control algorithm is developed for a class of continuous-time uncertain nonlinear systems subject to input constraints The robust control problem is converted to the constrained optimal control problem with appropriately selecting value functions for the nominal system Distinct from typical action-critic dual networks employed in RL, only one critic neural network (NN) is constructed to derive the approximate optimal control Meanwhile, unlike initial stabilizing control often indispensable in RL, there is no special requirement imposed on the initial control By utilizing Lyapunov’s direct method, the closed-loop optimal control system and the estimated weights of the critic NN are proved to be uniformly ultimately bounded In addition, the derived approximate optimal control is verified to guarantee the uncertain nonlinear system to be stable in the sense of uniform ultimate boundedness Two simulation examples are provided to illustrate the effectiveness and applicability of the present approach

Adaptive Control of Hydraulic Actuators With LuGre Model-Based Friction Compensation

Abstract: This paper concerns high-accuracy tracking control for hydraulic actuators with nonlinear friction compensation Typically, LuGre model-based friction compensation has been widely employed in sundry industrial servomechanisms However, due to the piecewise continuous property, it is difficult to be integrated with backstepping design, which needs the time derivation of the employed friction model Hence, nonlinear model-based hydraulic control rarely sets foot in friction compensation with nondifferentiable friction models, such as LuGre model, Stribeck effects, although they can give excellent friction description and prediction In this paper, a novel continuously differentiable nonlinear friction model is first derived by modifying the traditional piecewise continuous LuGre model, then an adaptive backstepping controller is proposed for precise tracking control of hydraulic systems to handle parametric uncertainties along with nonlinear friction compensation In the formulated nonlinear hydraulic system model, friction parameters, servovalve null shift, and orifice-type internal leakage are all uniformly considered in the proposed controller The controller theoretically guarantees asymptotic tracking performance in the presence of parametric uncertainties, and the robustness against unconsidered dynamics, as well as external disturbances, is also ensured via Lyapunov analysis The effectiveness of the proposed controller is demonstrated via comparative experimental results

Adaptive NN Tracking Control of Uncertain Nonlinear Discrete-Time Systems With Nonaffine Dead-Zone Input

Abstract: In the paper, an adaptive tracking control design is studied for a class of nonlinear discrete-time systems with dead-zone input. The considered systems are of the nonaffine pure-feedback form and the dead-zone input appears nonlinearly in the systems. The contributions of the paper are that: 1) it is for the first time to investigate the control problem for this class of discrete-time systems with dead-zone; 2) there are major difficulties for stabilizing such systems and in order to overcome the difficulties, the systems are transformed into an n -step-ahead predictor but nonaffine function is still existent; and 3) an adaptive compensative term is constructed to compensate for the parameters of the dead-zone. The neural networks are used to approximate the unknown functions in the transformed systems. Based on the Lyapunov theory, it is proven that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded and the tracking error converges to a small neighborhood of zero. Two simulation examples are provided to verify the effectiveness of the control approach in the paper.

Pinning Synchronization of Directed Networks With Switching Topologies: A Multiple Lyapunov Functions Approach

Abstract: This paper studies the global pinning synchronization problem for a class of complex networks with switching directed topologies. The common assumption in the existing related literature that each possible network topology contains a directed spanning tree is removed in this paper. Using tools from $M$ -matrix theory and stability analysis of the switched nonlinear systems, a new kind of network topology-dependent multiple Lyapunov functions is proposed for analyzing the synchronization behavior of the whole network. It is theoretically shown that the global pinning synchronization in switched complex networks can be ensured if some nodes are appropriately pinned and the coupling is carefully selected. Interesting issues of how many and which nodes should be pinned for possibly realizing global synchronization are further addressed. Finally, some numerical simulations on coupled neural networks are provided to verify the theoretical results.

Neural Network Control of a Rehabilitation Robot by State and Output Feedback

Abstract: In this paper, neural network control is presented for a rehabilitation robot with unknown system dynamics. To deal with the system uncertainties and improve the system robustness, adaptive neural networks are used to approximate the unknown model of the robot and adapt interactions between the robot and the patient. Both full state feedback control and output feedback control are considered in this paper. With the proposed control, uniform ultimate boundedness of the closed loop system is achieved in the context of Lyapunov's stability theory and its associated techniques. The state of the system is proven to converge to a small neighborhood of zero by appropriately choosing design parameters. Extensive simulations for a rehabilitation robot with constraints are carried out to illustrate the effectiveness of the proposed control.

Control barrier function based quadratic programs with application to bipedal robotic walking

Abstract: This paper presents a methodology for the development of control barrier functions (CBFs) through a backstepping inspired approach. Given a set defined as the superlevel set of a function, h, the main result is a constructive means for generating control barrier functions that guarantee forward invariance of this set. In particular, if the function defining the set has relative degree n, an iterative methodology utilizing higher order derivatives of h provably results in a control barrier function that can be explicitly derived. To demonstrate these formal results, they are applied in the context of bipedal robotic walking. Physical constraints, e.g., joint limits, are represented by control barrier functions and unified with control objectives expressed through control Lyapunov functions (CLFs) via quadratic program (QP) based controllers. The end result is the generation of stable walking satisfying physical realizability constraints for a model of the bipedal robot AMBER2.

Event-Triggered State Estimation for Complex Networks With Mixed Time Delays via Sampled Data Information: The Continuous-Time Case

Abstract: In this paper, the event-triggered state estimation problem is investigated for a class of complex networks with mixed time delays using sampled data information. A novel state estimator is presented to estimate the network states. A new event-triggered transmission scheme is proposed to reduce unnecessary network traffic between the sensors and the estimator, where the sampled data is transmitted to the estimator only when the so-called “event-triggered condition” is satisfied. The purpose of the problem addressed is to design an estimator for the complex network such that the estimation error is ultimately bounded in mean square. By utilizing Lyapunov theory combined with the stochastic analysis approach, sufficient conditions are established to guarantee the ultimate boundedness of the estimation error in mean square. Then, the desired estimator gain matrices are obtained via solving a convex problem. Finally, a numerical example is given to illustrate the effectiveness of the results.

Review on computational methods for Lyapunov functions

Abstract: Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ different methods such as series expan- sion, linear programming, linear matrix inequalities, collocation methods, al- gebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function.

Top Tension Control of a Flexible Marine Riser by Using Integral-Barrier Lyapunov Function

Abstract: This paper presents a boundary controller for a flexible marine riser to suppress the riser's vibration with a top tension constraint. The flexible marine riser is described by a distributed parameter system with a partial differential equation and four ordinary differential equations. The boundary controller is designed at the top boundary of the riser based on an integral-barrier Lyapunov function to suppress the riser's tension at top. Adaptive control is designed when the system parametric uncertainty exists. With the proposed robust adaptive boundary control, uniformed boundedness under the ocean disturbance can be achieved. Stability analysis of the closed-loop system is given using the Lyapunov stability theory. Simulation results illustrate the effectiveness of the proposed boundary controller with top tension constraint.

Reinforcement Learning Design-Based Adaptive Tracking Control With Less Learning Parameters for Nonlinear Discrete-Time MIMO Systems

Abstract: Based on the neural network (NN) approximator, an online reinforcement learning algorithm is proposed for a class of affine multiple input and multiple output (MIMO) nonlinear discrete-time systems with unknown functions and disturbances. In the design procedure, two networks are provided where one is an action network to generate an optimal control signal and the other is a critic network to approximate the cost function. An optimal control signal and adaptation laws can be generated based on two NNs. In the previous approaches, the weights of critic and action networks are updated based on the gradient descent rule and the estimations of optimal weight vectors are directly adjusted in the design. Consequently, compared with the existing results, the main contributions of this paper are: 1) only two parameters are needed to be adjusted, and thus the number of the adaptation laws is smaller than the previous results and 2) the updating parameters do not depend on the number of the subsystems for MIMO systems and the tuning rules are replaced by adjusting the norms on optimal weight vectors in both action and critic networks. It is proven that the tracking errors, the adaptation laws, and the control inputs are uniformly bounded using Lyapunov analysis method. The simulation examples are employed to illustrate the effectiveness of the proposed algorithm.

Cooperative Control of Multi-Agent Systems With Unknown State-Dependent Controlling Effects

Abstract: This paper investigates the cooperative control problem of uncertain high-order nonlinear multi-agent systems on directed graph with a fixed topology. Each follower is assumed to have an unknown controlling effect which depends on its own state. By the Nussbaum-type gain technique and the function approximation capability of neural networks, a distributed adaptive neural networks-based controller is designed for each follower in the graph such that all followers can asymptotically synchronize the leader with tracking errors being semi-globally uniform ultimate bounded. Analysis of stability and parameter convergence of the proposed algorithm are conducted based on algebraic graph theory and Lyapunov theory. Finally, a example is provided to validate the theoretical results.