Lyapunov-Krasovskii Functionals Research Articles

Hierarchical stability conditions for linear systems with interval time-varying delay

This paper proposes the hierarchical stability conditions for linear systems with an interval time-varying delay. In the matter of the delay, the upper and lower bounds are restricted, but there is no constraint on its derivative. Firstly, based on the state vectors and multiple integral state vectors involved in the delay-related generalized free-matrix-based integral inequalities (GFIIs), the hierarchical Lyapunov-Krasovskii functional (LKF) candidates are constructed. Then, the GFIIs are applied for the approximation of the integral quadratic terms, which will introduce the delay related nonlinear terms in the LKF differential. Next, the novel adjusted matrix-valued high and odd degree polynomial negative definite conditions (NDCs) are provided to acquire the hierarchical linear matrix inequality (LMI) conditions and cope with the nonlinear terms introduced by the GFIIs. Eventually, the advantages of the obtained stability conditions are checked through some numeric examples.

A new approach to stabilize diabetes systems with time-varying delays and disturbance rejection

In addition to meal intake, a diabetes system has a feedback behavior to balance the constituents of the chemicals, which causes instability. Therefore, a search for a feedback system that can stabilize the system with the introduced disturbances is a concern. In this article, a technique to stabilize a closed-loop system that considers the time-varying delay and rejecting disturbances is introduced. The closed-loop stability technique is designed and analyzed based on a selected suitable Lyapunov–Krasovskii functional (LKF) and satisfying the H∞ performance, and the integral part is upper bounded by a reciprocally convex inequality approach. The optimal closed-loop controller gain is obtained from the feasibility of the linear matrix inequality (LMI) by considering the minimum of the H∞ norm bound. To check the ability and performance of the proposed technique for the stability study, it is numerically applied to govern the blood glucose concentration of a 70 kg male patient co-experiencing both types (type 1 and 2) diabetes upon periodically glucose intake as external disturbances. The experiencing of the hybrid diabetes is achieved by varying the beta-cell functioning in insulin production. The proposed technique is able to stabilize and maintain the desired blood glucose concentration being inside a normal healthy range with both glucagon–glycogen conversion and insulin secretion delays.

Stability of port-Hamiltonian systems with mixed time delays subject to input saturation

In this paper, we investigate the stability of port-Hamiltonian systems with mixed time-varying delays as well as input saturation. Three types of time delays, including state delay, input delay, and output delay, are all assumed to be bounded. By introducing the output feedback control law and utilizing serval Lyapunov–Krasovskii functionals, we present three delay-dependent stability criteria in terms of the linear matrix inequality. Meanwhile, we use Wirtinger’s inequality, constraint conditions, and Lyapunov–Krasovskii functionals of triple and quadruple integral form to obtain less conservative results. Some numerical examples demonstrate and support our results.

H∞ dynamic observer design for a class of Lipschitz nonlinear discrete‐time systems with time varying delays

AbstractThis study explores the development of dynamic observer (HDO) for discrete‐time nonlinear systems (DTNLS) with time‐varying delay (TVD) and disturbances. The approach is to construct an augmented Lyapunov–Krasovskii function (LKF) with double summation terms, using the generalized reciprocally convex matrix inequality (GRCMI), as well as the Jensen‐based inequality (JBI) and the Wirtinger‐based inequality (WBI). These lead to less conservative time‐dependent conditions, represented as a set of linear matrix inequalities (LMIs) that can be efficiently solved using the LMI or YALMIP toolboxes. In addition, the proposed observer includes the widely used proportional observer (PO) and proportional integral observer (PIO) as specific cases. Two examples are presented to demonstrate the validity and effectiveness of the results.

Design and implementation for the state time-delay and input saturation compensator of gas turbine aero-engine control system

In order to obtain desired steady-state and transient performance of a gas turbine aero-engine during its operation, a usual way is to control the engine to operate in a manner close to its physical limits. The thermodynamic performance of a gas turbine aero-engine varies greatly within a wide working envelope, affected by factors such as engine performance degradation, actuator response lag and combustion time-delay. When the engine works in extreme conditions, for example rapid acceleration, some phenomena such as over-speed, over-temperature and over-pressure may occur. For this reason, various restrictions are often set in the engine control system to ensure safety during its operation, which are often realized by the form of saturation. In this paper, a compensation structure for gas turbine aero-engine state time-delay and input saturation is proposed. At first, the problems frequently occurred in the gas turbine aero-engine control system are investigated. And the gas turbine aero-engine control model is derived based on a thermodynamic non-linear model and its linearization model is established, together with its designed compensation structure. Differing from the existing technologies, a combination of dynamic system input saturation feedback and static state time-delay feedback is adopted in the proposed model. Then, the saturated nonlinearity and time-delay characteristic are introduced as convex combinations in the stability analysis. The local stability conditions of the closed-loop system are obtained by using Lyapunov–Krasovskii functionals and linear matrix inequalities (LMIs). Finally, hardware-in-loop (HIL) experiments are conducted to verify the feasibility of the proposed compensation method.

Membership-function-dependent [formula omitted] bumpless transfer control for switched interval type-2 fuzzy systems with time-delay

This paper addresses the H∞ bumpless transfer control for switched nonlinear delayed systems with exogenous perturbations via the interval type-2 fuzzy model. We firstly establish a novel characterization of bumpless transfer performance, which provides a framework to the bumpless transfer control of switched nonlinear systems. By using the multiple Lyapunov-Krasovskii functionals and the state-dependent switching strategy, sufficient conditions for asymptotically stability with H∞ bumpless transfer performance are obtained. More incisively, different from the existing results where the H∞ bumpless transfer control design conditions do not involve the information of membership functions, we adopt the piecewise-linear-membership-function technique to remove this limitation. Correspondingly, novel membership-function-dependent H∞ bumpless transfer criteria are derived, under which the state-dependent switching law and H∞ bumpless transfer controllers are jointly designed. Finally, a tunnel diode circuit model is exploited to verify the applicability and validity of the developed method.

State estimation and sliding-mode fault-tolerant control of a class of nonlinear systems with actuator faults under the event-triggered mechanism

The current work is an analytical study of the problem of state estimation and fault-tolerant control (FTC) based on dynamic event triggered mechanism (ETM) in the event of actuator fault for a class of nonlinear systems in a network environment. First, a state estimator module is designed in conjunction with the designed dynamic ETM, and an integral sliding surface (SS) is designed in the estimation space to obtain a variable time delay system after augmenting the system by combining the state estimator module and the SS. Second, the sufficient conditions that can make the augmented system stable under the ETM with H∞ performance index are obtained by the Lyapunov–Krasovskii function. The sufficient conditions are decoupled after the design of the sliding mode control (SMC) law that can stabilize the system, and the calculation of the state estimation gain matrix and the SMC gain matrix are obtained. Finally, the effectiveness of the method proposed in this study is verified with two simulation comparison experiments.

Sliding-mode fault-tolerant control of the six-rotor UAV with dead-zone-input under event-triggered mechanism

This paper investigates the problem of event-triggered mechanism(ETM)-based sliding-mode fault-tolerant control (FTC) for a six-rotor Unmanned Aerial Vehicle (UAV) with dead zone input (DZI) cases, considering potential actuator and sensor faults. Initially, a dynamic ETM is designed, followed by the development of a non-fragile observer utilizing this designed ETM. An integral sliding surface (SS) is then designed in the observation space, and the system is augmented and treated as a variable time delay system. Subsequently, sufficient conditions to ensure the stability of the augmented system with an H∞ performance index γ are obtained using the Lyapunov-Krasovskii function. Next, a sliding mode control (SMC) law is formulated to guide the sliding variables to the SS in finite time. Furthermore, sufficient conditions for ensuring system stability with an H∞ performance index γ are decoupled, and the calculation methods for the non-fragile observer gain matrix and the sliding mode gain matrix are obtained. Finally, to validate the effectiveness of the proposed method in this paper, simulation experiments are conducted.

Asymptotic Tracking Control for Uncertain Nonlinear Strict-Feedback Systems With Unknown Time-Varying Delays.

It is nontrivial to achieve asymptotic tracking control for uncertain nonlinear strict-feedback systems with unknown time-varying delays. This problem becomes even more challenging if the control direction is unknown. To address such problem, the Lyapunov-Krasovskii functional (LKF) is used to deal with the time delays, and the neural network (NN) is applied to compensate for the time-delay-free yet unknown terms arising from the derivative of LKF, and then an NN-based adaptive control scheme is constructed on the basis of backstepping technique, which enables the output tracking error to converge to zero asymptotically. Besides, with a milder condition on time delay functions, the notorious singularity issue commonly encountered in coping with time delay problems is subtly settled, which makes the proposed scheme simple in structure and inexpensive in computation. Moreover, all the signals in the closed-loop system are ensured to be semiglobally uniformly ultimately bounded, and the transient performance can be improved with proper choice of design parameters. Both the theoretical analysis and numerical simulation are carried out to validate the relevance of the proposed method.

Consensus Control of Multiagent Systems Under Switching Topology and Denial-of-Service Attacks

This article studies the consensus problem of multiagent systems (MASs) under denial-of-service (DoS) attacks and switching topology. Compared with the previous papers, under the switching topology, the impact of DoS attacks on the MAS is considered, and thus, the difficulty of consensus protocol design is further increased. First, a piecewise resilient consensus protocol is proposed to ensure that the MAS can achieve consensus under switching topology and DoS attacks. Then, by considering the different convergence rates caused by the switching topology in each sleep period of DoS attacks, a piecewise Lyapunov–Krasovskii function is established. The sufficient conditions on the global convergence of the system are obtained under switching topology and DoS attacks. In the meantime, the lower bound of average dwell time of the switching topology is determined under DoS attacks. The relationship between average dwell time and the parameters of DoS attacks is discussed. Finally, the simulation example is conducted to verify the effectiveness of the proposed method.

Stability Analysis of Time-Delay Switched System Based on Improved Lyapunov–Krasovskii Functionals

Stability Analysis of Time-Delay Switched System Based on Improved Lyapunov–Krasovskii Functionals

Improved results on reachable set estimation for singularly perturbed systems with time-varying delay

The problem of reachable set estimation for singularly perturbed systems with time-varying delays and bounded perturbation is studied in this paper. Firstly, a Lyapunov–Krasovskii function related to the perturbed parameter η is established. Secondly, introducing a method of integral inequality scaling and eliminated the influence of singularly perturbed parameter on the system within a certain range. The system state is therefore contained in a η-independent ellipsoid in accordance with a criterion imposed by linear matrix inequalities. Subsequently, examples are used to show that the findings are valid.

On the existence of diagonal Lyapunov–Krasovskii functionals for a class of nonlinear positive time-delay systems

This work investigates the absolute stability problem for positive Persidskii systems with delay. Right-hand sides of these systems are linear combinations of nonlinearities of a sector type. A special construction of diagonal Lyapunov–Krasovskii functionals is proposed and a criterion for the existence of such functionals guaranteeing the absolute stability is derived. Furthermore, sufficient conditions under which a common diagonal Lyapunov–Krasovskii functional can be constructed for a family of time-delay Persidskii systems are obtained. Four examples are presented demonstrating the efficiency of the developed approaches.

Disturbance observer‐based PID synchronization control of complex networks with probabilistic interval time‐varying delays and multiple disturbances

AbstractIn this article, the disturbance observer‐based proportional–integral–derivative (PID) control protocol is investigated for synchronization of complex networks with probabilistic interval time‐varying delays and multiple disturbances. For the sake of reducing conservatism, a method of probabilistic interval time‐varying delay is proposed by introducing two certain intervals with known probability distributions. Moreover, two different disturbances are taken into account. One of the disturbances is produced by an exogenous system which acts through the input channel, while the other is usual norm‐bounded disturbance. Simultaneously, a disturbance observer is designed to estimate and compensate the disturbance generated by the exogenous system. By constructing an appropriate Lyapunov–Krasovskii function, a sufficient criterion is obtained to ensure both the exponential synchronization and prescribed performance index. Finally, the simulation examples are employed to demonstrate the validity of the theoretical results.

A memory state-feedback controller via random pocket dropouts for multi-agent systems with external disturbance

This study explores the challenge of achieving consensus in multi-agent systems (MASs) when facing the random packet dropouts and disturbances. It employs memory state-feedback control (MSFC) in the context of undirected graphs and specified leader agents. The analysis focuses on mean square consensus, considering MASs within strongly connected networks or networks with undirected spanning trees. The MSFC approach is developed to ensure asymptotic consensus despite packet dropouts and also to reduce the impact of disturbances. Specifically, the consensus analysis leverages the Lyapunov-Krasovskii functional (LKF) framework, and the necessary conditions for implementing the proposed MSFC are established using linear matrix inequalities (LMIs). The system, augmented with an H ∞ attenuation level is guaranteed to achieve asymptotic mean-square stability according to the provided criteria. In conclusion, two examples are provided to illustrate the effectiveness and practicality of the proposed control mechanism.

A non-singular terminal sliding mode controller for a communication-based hybrid microgrid

The paper proposes a robust and effective networked control for a hybrid based islanded microgrid. Renewable energy sources with energy storage systems are linked in parallel using voltage source inverters to supply the loads in a microgrid. The master-slave control strategy is implemented. CAN and the ZigBee networks are selected to share data, where the master controller transmits the reference signals to the slave controllers in the microgrid via these two networks. As the control loop incorporates a communication network and energy storage systems, the impacts of external disturbances, parametric uncertainties, and time delay executed by the devices introduce challenges in control of voltage and current. Thereby, it is crucial to use a robust and stable control approach. Using the binary iteration algorithm, stability criteria in the form of linear matrix inequalities based on the Lyapunov-Krasovskii function are employed to derive the system's maximum allowable delay boundary. As the reliability of the communication network can be affected, a nonsingular terminal sliding mode controller is employed. The proposed robust control method was implemented in MATLAB/Simulink and the True-time 2.0 simulator and its performance were evaluated under different loads, uncertainties, disturbances, and communication impairments. Simulation results show that the desired performance i.e., maintaining the microgrid's stability with accurate voltage and current control even in unallowable delay limits is met.

A novel robust discrete-time integral sliding mode tracking control design for time-varying delay MIMO systems with unknown uncertainties

This paper proposes a novel robust tracking control scheme for discrete time linear uncertain Multiple-Input Multiple-Output (MIMO) systems subject to time-varying delay on the states. The considered system is affected by unknown but norm bounded uncertainties on parameters as well as matched disturbances on the states. The designed controller is based upon a proposed novel integral sliding surface and a new switching type of reaching law. Sufficient conditions based on Linear Matrix Inequalities (LMIs) and a suitable Lyapunov–Krasovskii Functional (LKF) are derived in order to guarantee the asymptotic stability of such system. The proposed controller ensures a good tracking performance despite the presence of the time varying delay and the matched/unmatched disturbances. Moreover and thanks to the proposed integral surface, the time reaching phase is eliminated and the chattering phenomenon is significantly reduced. The proposed controller is applied on an Autonomous Underwater Vehicle (AUV) to follow a prescribed desired trajectory. The simulation results illustrate the effectiveness of such controller.

Resilient and event-triggered control of singular Markov jump systems against cyber attacks

This paper is concerned with the H ∞ resilient and event-triggered control of singular Markov jump systems against deception attacks. The deception attacks are modelled as a random bounded signal which is governed by a Bernoulli distributed random variable. The event-triggered scheme is adopted to achieve a trade-off between system performance and network resources. Based on the technique of stochastic Lyapunov–Krasovskii functionals and linear matrix inequalities, efficient criteria are developed such that the closed-loop system is stochastically admissible with a certain H ∞ performance under deception attacks. Then, the co-design of resilient controller gains and event-triggered rules is provided in terms of a group of feasible LMIs. Finally, two examples are employed to verify the validity of our design.

Event‐triggered security control for fuzzy‐model‐based cyber‐physical systems under Denial‐of‐Service attacks and actuator faults

AbstractThis article addresses the issue of event‐triggered security control for nonlinear cyber‐physical systems, characterized by Denial‐of‐Service (DoS) attacks, time‐varying delays, and actuator faults. These complexities are captured using the Takagi–Sugeno fuzzy model. Significantly, this study focuses on DoS attacks targeting the controller‐to‐actuator channel, and subsequently introduces a model for actuator faults that considers the prolonged utilization or degradation of component parts. To optimize communication resource allocation, a model‐independent event‐triggered mechanism is proposed. Utilizing the Lyapunov–Krasovskii function approach, the objective of achieving stochastic stability is transformed into a linear matrix inequalities problem, thus deriving a set of sufficient conditions. The efficiency and validity of the proposed algorithms is demonstrated through simulation results.

Adaptive guaranteed cost control for nonlinear systems with unknown parameters and time delays based on fully actuated system approaches

This paper investigates the adaptive guaranteed cost stabilization (AGCS) problems for two classes of high-order nonlinear systems with unknown parameters (vector) and time delays. Firstly, based on the high-order fully actuated (HOFA) system approaches, the Lyapunov–Krasovskii functional (LKF) and the guaranteed cost control (GCC), a new AGCS strategy is proposed for HOFA nonlinear system with unknown parameter vector and time delays. Then, based on the above result, another AGCS controller for a class of strict-feedback systems (SFSs) with unknown parameters and time delays is obtained. Two designed controllers ensure that all of the states of two closed-loop systems are global boundedness, and preset arbitrarily the upper bound of cost functions (UBCFs) characterizing the output performance. More importantly, the UBCFs are independent of system initial values, unknown parameters (vector), and even time delays, which is difficult to achieve by using existing control methods. To do this, this paper introduces a local smooth nonlinear function (LSNF), and gives its corresponding lemma, which provide an important mathematical tool. Finally, three simulation examples, including an application in the electromechanical system, are given to prove the effectiveness and the practicability of our proposed control method.