Lyapunov Function Candidate Research Articles

Model following adaptive control for nodes in complex dynamical network via the state observer of links

Geometrically, a complex dynamical network (CDN) can be regarded as the interconnected system composed of the node subsystem (NS) and the link subsystem (LS) coupled with each other. Guided by this idea, in order to achieve the goal of each node following asymptotically its own reference target in a CDN, this paper investigates the model following adaptive control (MFAC) problem of NS via the dynamics of links, which implies that the LS plays the important dynamic auxiliary role in the MFAC realization of nodes. Meanwhile, we focus on the condition that the links state information is unavailable, due to sensor practical application and measurement cost constraints. To obviate this restriction, we construct the asymptotical state observer for the LS. Next, to achieve the control goal of this paper, an appropriate Lyapunov candidate function is selected, by which the MFAC scheme for NS is synthesized based on the state observer of LS. Finally, the simulation example is performed to demonstrate the theoretical results in this paper.

Robust Control Design of Under-Actuated Nonlinear Systems: Quadcopter Unmanned Aerial Vehicles with Integral Backstepping Integral Terminal Fractional-Order Sliding Mode

In this paper, a novel robust finite-time control scheme is specifically designed for a class of under-actuated nonlinear systems. The proposed scheme integrates a reaching phase-free integral backstepping method with an integral terminal fractional-order sliding mode to ensure finite-time stability at the desired equilibria. The core of the algorithm is built around proportional-integral-based nonlinear virtual control laws that are systematically designed in a backstepping manner. A fractional-order integral terminal sliding mode is introduced in the final step of the design, enhancing the robustness of the overall system. The robust nonlinear control algorithm developed in this study guarantees zero steady-state errors at each step while also providing robustness against matched uncertain disturbances. The stability of the control scheme at each step is rigorously proven using the Lyapunov candidate function to ensure theoretical soundness. To demonstrate the practicality and benefits of the proposed control strategy, simulation results are provided for two systems: a cart–pendulum system and quadcopter UAV. These simulations illustrate the effectiveness of the proposed control scheme in real-world scenarios. Additionally, the results are compared with those from the standard literature to highlight the superior performance and appealing nature of the proposed approach for underactuated nonlinear systems. This comparison underscores the advantages of the proposed method in terms of achieving robust and stable control in complex systems.

Distributed State Observer for Systems with Multiple Sensors under Time-Delay Information Exchange.

The issues of state estimations based on distributed observers for linear time-invariant (LTI) systems with multiple sensors are discussed in this paper. We deal with the scenario when the information exchange has known time delays, and aim at designing a distributed observer for each subsystem such that each distributed observer can estimate the system state asymptotically by rejecting the time delay. To begin with, by rewriting the target system in a connecting form, a subsystem which is affected by the time-delay states of other nodes is established. And then, for this subsystem, a distributed observer with time delay is constructed. Moreover, an equivalent state transformation is made for the observer error dynamic system based on the observable canonic decomposition theorem. Further, in order to ensure that the distributed observer error dynamic system is asymptotically stable even if there exists a time delay, a linear matrix inequality (LMI) which is relative to the Laplace matrix is elaborately set up, and a special Lyapunov function candidate based on the LMI is considered. Next, based on the Lyapunov function and Lyapunov stability theory, we prove that the error dynamic system of the distributed observer is asymptotically stable, and the observer gain is determined by a feasible solution of the LMI. Finally, a simulation example is given to illustrate the effectiveness of the proposed method.

Analysis of a fractional-order model for dengue transmission dynamics with quarantine and vaccination measures

A comprehensive mathematical model is proposed to study two strains of dengue virus with saturated incidence rates and quarantine measures. Imperfect dengue vaccination is also assumed in the model. Existence, uniqueness and stability of the proposed model are proved using the results from fixed point and degree theory. Additionally, well constructed Lyapunov function candidates are also applied to prove the global stability of infection-free equilibria. It is also demonstrated that the model is generalized Ulam-Hyers stable under some appropriate conditions. The model is fitted to the real data of dengue epidemic taken from the city of Espirito Santo in Brazil. For the approximate solution of the model, a non-standard finite difference(NSFD) approach is applied. Sensitivity analysis is also carried out to show the influence of different parameters involved in the model. The behaviour of the NSFD is also assessed under different denominator functions and it is observed that the choice of the denominator function could influence the solution trajectories. Different scenario analysis are also assessed when the reproduction number is below or above one. Furthermore, simulations are also presented to assess the epidemiological impact of dengue vaccination and quarantine measures for infected individuals.

Pinning Group Consensus of Multi-agent Systems Under DoS Attacks

In this paper, group consensus is investigated for a class of nonlinear multi-agent systems suffered from the DoS attacks. Firstly, a first-order nonlinear multi-agent system is constructed, which is divided into M subsystems and each subsystem has an unique leader. Then a protocol is proposed and a Lyapunov function candidate is chosen. By means of the stability theory, a sufficient criterion, which involves the duration of DoS attacks, coupling strength and control gain, is obtained for achieving group consensus in first-order system. That is, the nodes in each subsystem can track the leader of that group. Furthermore, the result is extended to nonlinear second-order multi-agent systems and the controller is also improved to obtain sufficient conditions for group consensus. Additionally, the lower bounds of the coupling strength and average interval of DoS attacks can be determined from the obtained sufficient conditions. Finally, several numerical simulations are presented to explain the effectiveness of the proposed controllers and the derived theoretical results.

Observer-based bipartite consensus tracking of nonlinear fractional-order multi-agent systems with pull-based dynamic event-triggered mechanism

This paper investigates the observer-based bipartite consensus tracking of nonlinear fractional-order multi-agent systems (FOMASs) by employing a pull-based dynamic event-triggered mechanism (DETM). First, considering that the relevant state information of each agent is not always measurable, a class of distributed observers is considered for each agent to estimate its state information. Then, a pull-based DETM for FOMASs is proposed to avoid continuous controller updates, in which the dynamic threshold is modulated according to the preset conditions. The pull-based DETM constructed in this paper enables each agent to update the controller only based on its own trigger instants. Furthermore, an observer-based dynamic event-triggered control protocol is designed to guarantee the bipartite consensus tracking of FOMASs. Correspondingly, sufficient conditions are obtained by using graph theory and choosing suitable Lyapunov candidate functions. Moreover, the Zeno behavior is precluded. Finally, two simulation examples are presented to illustrate the theoretical results efficiently.

Stochastic stabilization based on kinetic-potential energy shaping for stochastic mechanical port-Hamiltonian systems

This paper extends a kinetic-potential energy shaping method to stochastic mechanical port-Hamiltonian systems. The kinetic-potential energy shaping brings a new class of Lyapunov function candidates involving a cross term of the position and the momentum without solving partial differential equations for deterministic port-Hamiltonian systems. However, the conventional kinetic-potential energy shaping does not necessarily work for stochastic port-Hamiltonian systems due to energy increase by stochastic noise. Therefore, we first provide a modification properly compensated by using stochastic generalized canonical transformations. Then, two stochastic stability results are presented. The first result shows a necessary condition for stochastic asymptotic stability for an equilibrium state. The other one shows a necessary condition for stochastic bounded stability for a target state, which is not necessarily an equilibrium point.

Modified Universal Integral Regulator for Quadrotor UAV in the Presence of External Disturbances

Universal Integral Regulator (UIR) is a widely used control law for control problems, it is based on sliding mode control with the inclusion of a Conditional Integrator (CI). In the original approach of this control law, the main gain is considered a constant. The purpose of this paper is to solve a quadrotor tracking problem using a new nonlinear control law based on the quadrotor tracking problem using a new nonlinear control law based on the UIR, considering the main gain dependent on the tracking error (variable gain), the control law is called in the present work as Modified Universal Integral Regulator (MUIR). It is expected that the MUIR can improve the transient response and reduce the control demand when compared to previous approaches of similar controllers. The adopted Newton-Euler quadrotor model and the controller design are treated separately in two subsystems, attitude and position control loops. The stability of MUIR is demonstrated by the use of a candidate Lyapunov function. Finally, in order to validate the robustness and choice of the proposed controllers, several numerical simulations were developed in the presence of external disturbances. Less error and control activity during transient response were observed when compared to the original Universal Integral Regulator controller.

Novel Adaptive Control for Flexible-Joint Robots With Unknown Measurement Sensitivity

For the existing tracking control schemes of flexible-joint robots, precise sensor measurement is an implicit premise. However, idealized sensors are difficult to achieve due to manufacturing technology or other external factors. To this end, this paper further investigates the tracking control problem for flexible-joint robots with unknown measurement sensitivity. Specifically, for such multi-input multi-output Euler-Lagrange systems with completely unknown system dynamics, a novel measurement values-based adaptive control method is proposed by fusing sensitivity information and system variables into Lyapunov function candidates, where the restriction on system states in other the approximation lemma-based results is removed, since unknown nonlinearities are scaled by the structural characteristics of system variables. Above all, even if there are measurement errors, satisfactory tracking performance can be obtained by adjusting the design parameters, which is proved by rigorous theoretical analysis. Finally, hardware experiments further verify the effectiveness of the proposed method. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This work is motivated by the trajectory tracking control problem for flexible-joint robots under imprecise sensor measurements. Due to manufacturing technology limitations and component aging, there is inevitably a deviation between the measured values of sensors and real values, and this problem may become more prominent as the working environment of flexible-joint robots tends to become more complex. To our knowledge, most of the existing solutions for flexible-joint robots are developed based on precise sensor measurements, and they may fail to achieve satisfactory performance when real state information is not available. Moreover, the prior knowledge about model parameters and measurement sensitivity is difficult or impossible to exactly obtain in practice, which seriously hinders the further application of control methods that are dependent on system dynamics. To this end, this paper proposes a novel tracking control scheme based on measurement information for flexible-joint robots with unknown measurement sensitivity, where the dependence on model information is eliminated with the elaborately constructed Lyapunov function candidates, and the real tracking error is still adjusted to an acceptable range even if there are measurement errors. Preliminary experiments on a flexible-joint robot developed by Quanser company demonstrate the feasibility and effectiveness of the proposed method. In future studies, designing an effective scheme to achieve direct preset tracking control is the focus of the work.

Adaptive finite‐time stabilizing control of fractional‐order nonlinear systems with unmodeled dynamics via sampled‐data output‐feedback

AbstractThis article realizes an adaptive finite‐time sampled‐data output‐feedback stabilization for a class of fractional‐order nonlinear systems with unmodeled dynamics and unavailable states. K‐filters are constructed to estimate unavailable states, a dynamic signal is introduced to handle unmodeled dynamics and neural networks were used to approximate uncertain nonlinearities existed in stabilizer construction. With the help of backstepping technique, an adaptive sampled‐data output‐feedback stabilizer is exported, and such stabilizer with allowable design parameters and sampling period can render the corresponding closed‐loop system reaches practically finite‐time stable, which can be demonstrated by means of selected Lyapunov function candidates. In the end, two simulations with a numerical and an engineering examples are presented to verify the effectiveness of the proposed scheme.

Fully distributed adaptive asymptotical consensus tracking control of uncertain high‐order nonlinear multi‐agent systems with event‐triggered communication

AbstractIn this article, the distributed adaptive asymptotical consensus tracking control is considered for a class of high‐order nonlinear multi‐agent systems with matched unknown parameters under the event‐triggered communication mechanism. The controllers, adaptive laws and triggering conditions are presented under directed topology with a spanning tree, relieving communication burden among the connected agents since continuous monitoring of in‐neighbors is no longer needed. To handle the cross‐coupling term associated with asymmetric Laplacian matrix and unknown parameters of all agents when the candidate Lyapunov function is defined in terms of local consensus errors, adaptive immersion and invariance control technique is adopted. Adaptive estimators are also designed to estimate the constants related to global information, with which the proposed control scheme is fully distributed. It is shown that all closed‐loop system signals are globally uniformly bounded and all agents can track the reference signal asymptotically while ruling out Zeno behavior though a subset of the agents cannot have direct access to the reference signal information. A numerical example is illustrated to show the effectiveness of the proposed control protocol.

Synthesis of ℋ∞ Control for Descriptor Hybrid Systems with Actuator Saturation

This paper addresses a mode-dependent state-feedback H∞ control for stochastic descriptor hybrid systems, considering both the absence and presence of actuator saturation. Firstly, the necessary and sufficient conditions for the stochastic admissibility criterion with H∞ performance γ of the closed-loop system are proposed. Given the proposed non-convex condition, the author reformulates it into linear matrix inequalities (LMIs). Then, to extend the result to the systems with actuator saturation, the actuator-saturated control input is expressed as a linear combination of a given state-feedback control input and a virtual control input that always remains under the saturation level. To verify this expression, the set invariant condition is also suggested by using the singular mode-dependent Lyapunov function candidate. Therefore, the conditions for the existence of both the mode-dependent state-feedback H∞ control and the ellipsoidal shape invariant sets are successfully derived in terms of LMIs. Two numerical examples demonstrate the effectiveness of the proposed method by solving optimization problems subject to the proposed LMIs that minimize H∞ performance γ and maximize the invariant set, respectively.

Discrete Communication and Control Updating in Adaptive Dynamic Event-Triggered Consensus

This paper studies the consensus control problem faced with three essential demands, namely, discrete control updating for each agent, discrete-time communications among neighboring agents, and the fully distributed fashion of the controller implementation without requiring any global information of the network topology. Note that the existing related results only meet one or two demands at most, which are essentially not applicable to this problem. In this paper, a novel dynamic triggering condition is first presented, which is independent of the relative information among neighboring agents, providing potential opportunities to conquer the difficulties caused by the interaction of the above-mentioned three demands. Based on the proposed triggering condition, a framework on fully distributed adaptive consensus protocol with discrete communication and control updating is established. The thorough analysis of the adaptive gain is made by introducing a virtual adaptive gain to generate the Lyapunov function candidate, which significantly facilitates the consensus verification. Finally, numerical examples and comparisons are provided to verify the effectiveness of the proposed adaptive dynamic event-triggered protocols.

Stability Analysis and Controller Synthesis Using Single-Hidden-Layer ReLU Neural Networks

This paper presents algorithms to solve analysis and controller synthesis problems for dynamical systems modeled as a recurrent single-hidden-layer rectified linear unit neural network (ReLU NN), or equivalently, a piecewise affine dynamical system. Such models are interesting since they may arise through the use of modern machine learning methods for system identification, or as closed-loop solutions in certain classes of model predictive control (MPC) problems. A key idea in the proposed approach is to use piecewise affine Lyapunov functions parametrized as ReLU neural networks, and similarly parametrized controllers. This compatible representation between the Lyapunov function and the dynamics simplifies the automation of analysis of and controller synthesis for learned models. Our method of verifying a candidate Lyapunov function is faster than methods based on mixed integer programming. We ‘learn’ controllers and Lyapunov functions using both weight updates and network architecture search, without gradients. We demonstrate the proposed algorithm on examples involving learned models, explicit MPC controllers, and constrained controller synthesis.

Stability Analysis of Nonlinear Dynamic Systems Using the Lyapunov Method Approach

A new approach to the stability analysis of homogeneous nonlinear systems is proposed, based on the concept of candidate Lyapunovfunctions, where the focus is not on the positive definiteness of the candidate Lyapunovfunctions, but on the negative definiteness of their derivatives. After having ensured the negative definiteness of the derivative function, on the basis of the sign assignment of the primitive function, the stability of the equilibrium is analyzed, where the necessary and sufficient conditions are declared at the same time. The selection of the tendency of the Lyapunov candidate function is primarily performed in the form of a linear combination of some simple functions. The unknown coefficients in the structure of the candidate function are computed based on the negative definiteness of the derivative function. Then, using these coefficients in Lyapunov function, sign of primitive function in state space is argued. Thus, the stability/instability of the equilibrium point can be deduced from the triple sign settings of the candidate function. Furthermore, in the process of negative definiteness of the derivative function, the coefficients are obtained using two independent methods. The proposed theoretical results are supported and their effectiveness is demonstrated by numerical simulations.

Prescribed Time Fault-Tolerant Affine Formation Control for Multi-Agent Systems with Double-Integrator Dynamics

There is an increasing interest in the affine formation control of multi-agent systems, because it can change the centroid, orientation and scale of the formation by controlling only a few leaders. In this paper, the fault-tolerant affine formation control problem is addressed for double-integrator multi-agent systems with partial loss of efficiency and bias faults. Firstly, in order to track the leaders with dynamically changing accelerations, an acceleration observer with prescribed time convergence is proposed, which can estimate the ideal acceleration for each follower. Then, based on the acceleration observer, a fault-tolerant control algorithm is given. A new Lyapunov function candidate is constructed, based on which a sufficient condition to achieve the control objective is derived. Theoretical analysis shows that the formation tracking error can converge to zero within a prescribed time, and remain in a small neighborhood of zero after that time. Finally, numerical simulations are given to show the effectiveness of the proposed algorithm and compare it with existing results.

Adaptive Hamiltonian-Based Energy Control with Built-In Integrator for PEMFC Hybrid Power Conversion Architecture

In this article, an Adaptive Hamiltonian-Based Energy Control (AHBEC) with a built-in integrator is introduced for the Proton-Exchange Membrane Fuel Cell (PEMFC) hybrid power-conversion system. The presented additional built-in integrator is based on the Lyapunov and Hamiltonian functions. Unlike previous works, the control strategy proposed in this article aims to regulate the output current of the PEMFC stack, i.e., to provide sufficient power support to the load, and the integrator is built into the control loop in order to eliminate the steady-state current error caused by external disturbances and model parameter uncertainties. The large-signal stability of the whole system is demonstrated by selecting a suitable Lyapunov-candidate function. An experimental setup is constructed in the laboratory to verify the proposed control strategy. The experimental results demonstrate the robustness and effectiveness of the designed energy-control approach.

Adaptive control of a VTOL uncrewed aerial vehicle for high-performance aerobatic flight

We present two adaptive control methods conceived to enable a vertical take-off and landing (VTOL) uncrewed aerial vehicle (UAV) to perform a class of aerobatic maneuvers in the presence of aerodynamic-coefficient and torque-latency variations induced by changes of the local flow fields during high-speed aerobatic flight. First, we introduce a linear time-varying (LTV) dynamical model, which is assumed to have unknown time-dependent parameters, to describe the aerodynamic effects acting on the actuation dynamics of the system. Then, we present the design of a Lyapunov-based adaptive control scheme aimed to compensate for undesired behavior of the LTV actuator dynamics, according to which we derive the control and adaptation laws from a single Lyapunov candidate function. Next, we propose a modular adaptive control scheme to address the same problem, but in which the adaptation law is specified separately from that of the controller. We use modern nonlinear theory to deduce and analyze the conditions that guarantee the global asymptotic stability of both adaptive control strategies. In order to exemplify the controller synthesis procedures, we implemented both adaptive control methods on a quadrotor UAV to perform three different types of aerobatic-flight maneuvers—namely, triple flips, Pugachev’s cobras, and mixed flips. The obtained experimental results provide compelling evidence of the effectiveness of the two proposed methods to compensate for the undesired effects induced by aerodynamic-coefficient and torque-latency variations. Furthermore, the experimental data demonstrate that both adaptive schemes significantly improve the flight performance of the quadrotor UAV during the execution of aerobatic maneuvers, compared to those achieved with a controller that disregards the LTV actuator dynamics induced by high-speed aerodynamic effects. The suitability of the time-varying approach used to model the influence of high-speed local flow fields on the dynamics of the controlled UAV was indirectly validated through data obtained during the aerobatic-flight experiments presented here.

Fault estimation for cyber–physical systems with intermittent measurement transmissions via a hybrid observer approach

This paper investigates the problem of fault estimation for cyber–physical systems (CPSs) with intermittent measurement transmissions. A novel hybrid observer-based fault estimation scheme is proposed to generate estimations of the system state and the fault. Specifically, the correction terms of the designed observer are updated while new output measurements are available. Moreover, the estimation performance of the designed observer is analyzed within a framework of hybrid systems, and sufficient conditions are established to guarantee exponential input-to-state stability with a prescribed convergence rate based on the elaborately constructed Lyapunov function candidate. In contrast to the existing estimation schemes, the requirements of continuous or periodical communication are eliminated, and therefore the proposed one is more applicable for the modern industrial systems integrated with data communication networks. Finally, the application to an aircraft model is presented to illustrate the effectiveness of the developed methods.

Double-layer event-triggered policy of passivity-based output regulation under deception attacks for switched systems

The problem of secure output regulation (SOR) is solved from the passivity viewpoint for the networked switched system (NSS) permitting severely unstable dynamics in the presence of deception attacks (DAs). Dual Double-layer event-triggered policies (DLETPs) are developed to govern the appropriate triggering in a layered manner in conjunction with parameters of passivity, DAs, and switching characteristics. A tolerant switching strategy (TSS) is proposed to permit destabilizing switching outnumbers stabilizing switching and capture the passivity parameter. In the joint design of DLETPs and TSS in SOR for NSSs, the direct quantificational connections are showcased between the system’s stability, candidate Lyapunov function, event-triggering subject to DAs, and switching behaviors through the system’s passivity degree. Simulation results of speed regulation for an aero-engine model provide the effectiveness of the presented control methodology.