Use Of Lyapunov Stability Theory Research Articles

Anti-disturbance observer-based proportional-retarded control design for polytopic uncertain fractional-order systems

This study deals with the stabilisation problem of fractional-order time-delay systems with polytopic uncertainties and multiple disturbances with the use of Lyapunov stability theory. Notably, the multiple disturbances comprising both the known and unknown signals, are respectively characterised by the norm-bounded and exogenous system which signifies the harmonic signals with modelling perturbations. The control protocol is configured by the integration of proportional-retarded controller with gain perturbations and the output of an anti-disturbance observer. Precisely, the disturbance observer is taken into account for attenuating the influence of disturbance signals. Moreover, the implementation of a time-delay term in the feedback loop replicates the dynamic features of a derivative action that relies only on position measurements and its time differencing capabilities make it to be minimally sensitive against noise and provide a smoothing effect. By virtue of these circumstances, the asymptotic stability of the considered system is assured. With the aid of a suitable Lyapunov function, a set of robust order-dependent sufficient conditions is derived and then solved through the MATLAB LMI toolbox. As a result, the controller gain matrices can be computed and two numerical examples, including an electrical circuit model with simulation results are presented to certify the inherent potential of the theoretical outcomes.

Composite resilient state tracking and disturbance rejection control for periodic piecewise polynomial delay systems with multiform disturbances

In this study, we scrutinize the problems of resilient state tracking control (RSTC) and disturbance rejection (DR) for periodic piecewise polynomial delay systems in the presence of multiform disturbances and gain fluctuations. Specifically, we address both exosystem-induced disturbances and norm-bounded disturbances. Firstly, to tackle effectively the disturbances that are produced by exosystems, the disturbance observers are tailored, which estimates the disturbance in a precise manner. Secondly, the repercussions of norm-bounded disturbances can be lessened by accounting an extended dissipative performance. Moreover, by amalgamating the outcomes of the disturbance observer with the feedback-based tracking control, a DR-based RSTC is constructed. In addition, some parameter variations in the controller gain are taken into consideration in order to guarantee resiliency against unexpected flaws. Besides, the adequate conditions are derived in the form of linear matrix inequalities by making use of Lyapunov stability theory, new double integral inequality and extended Wirtinger’s inequality. Furthermore, the relation for acquiring the controller and observer gain matrices is formulated based on the established criteria. Finally, a numerical example with simulation depiction is presented to show the favour performance and capability of the designed controller.

Adaptive event-triggered control for networked interconnected systems with cyber-attacks

This paper investigates the secure adaptive event-triggered controller design for networked interconnected systems (NISs) with cyber-attacks. Firstly, an adaptive event-triggered mechanism (AETM) with dynamic threshold parameter is adopted to economize the limited network bandwidth which takes the abrupt data into consideration. A model of cyber-attacks is established for NISs with consideration of the malicious cyber-attacks. On account of the model of cyber-attacks and AETM, a mathematical model of NISs is established. For the built system, the sufficient condition for the asymptotic stability of the system is obtained by making use of Lyapunov stability theory and linear matrix inequality (LMI) technique, and the design algorithm of the controller is proposed. Finally, a simulation example is given to verify the validity of the theoretical results.

Robust control design for delayed periodic piecewise time-varying systems with actuator faults

This paper is mainly focused on the stabilization problem of uncertain delayed periodic piecewise time-varying systems inclusive of disturbances and faults in actuators. More specifically, the considered system is encompassed of periodic dynamics, which exhibits the nature of switched systems with fixed switching sequence and dwell time. The control protocol is configured in the form of both the present and past state information of the addressed system with passive performance. Moreover, the proposed control approach discloses the stabilization issue mainly by resolving the effect of faults in actuator components. Precisely, the desired periodic gain matrices of the developed controller are calculated by way of solving some matrix inequalities which are derived by making use of Lyapunov stability theory and matrix polynomial approach. As a result, the asymptotic stability of the considered system is ensured in conjunction with satisfied disturbance attenuation index. Conclusively, the simulation results of two numerical examples including mass-spring damping system are presented for validating the theoretical result.

Reflection-based technique for synthesis of structured controllers with guaranteed local convergence

This paper deals with stabilisation, and static output feedback synthesis, under structure constraints for linear time-invariant systems. The proposed approach relies on a prominent property of Hurwitz-stable matrices set and a Douglas-Rachford (DR) reflection method type for finding a point in the intersection of two closed sets. The method proposed in this paper emulates the genericity of the Linear Matrix Inequalities framework while keeping the feedback gain separated from any matrix with direct or indirect connection to the Lyapunov function. The link between the sequence of iterates generated by the proposed algorithm and a continuous dynamical system results in the use of Lyapunov stability theory for guaranteeing the local convergence. Several examples are given to prove the validity of the proposed method.

A Robust Guiding Torque Control Method for Automatic Steering Using LMI Algorithm

The existing path tracking methods usually neglect the effect of the drivers on the steering control. This paper proposes a robust steering control method of human-machine steering torque superposition based on linear matrix inequality (LMI) algorithm. First, the model for solving steering superposition torque introduces the steering system and steering resistance torque model in addition to the vehicle model, which increases the nonlinearity and uncertainty of system, and the human in torque superposition control also increases the external interferences. Therefore, this paper proposes a LMI robust control algorithm to reduce the external interference and the influence of uncertain factors on the system and improve the tracking performance of system, by use of Lyapunov stability theory and Schur complement property to convert the region pole assignment and robust control constraint conditions into LMI convex optimization problem. The next, the nonlinear vehicle dynamics solving model including Fiala tire model, steering column model is established; the nonlinear tire model is linearized by use of affine function, and the steering superposition control law is solved by use of LMI. Then, the union CarSim and Simulink simulation is conducted under different situations to verify the robustness and control performance of control system. Finally, through establishing the hardware-in-the-loop experiment table based on LabVIEW-RT, the effectiveness of control strategy is verified. The test results show that the method solves the model uncertainty and the robustness decrement problem resulting from human intervention, ensuring a good tracking performance, and a stable system at the same time.

Reliable Event-Triggered Asynchronous Extended Passive Control for Semi-Markov Jump Fuzzy Systems and Its Application

This paper is concerned with the reliable extended passive control problem for semi-Markov jump Takagi-Sugeno (T-S) fuzzy systems based on an event-triggered mechanism (ETM). An asynchronous approach is utilized to cope with the fuzzy-basis-dependent membership functions in networked fuzzy systems. For fear of the waste of communication resources, an ETM to screen the transmitted data is adopted for T-S fuzzy semiMarkov jump systems under consideration. Owing to the use of Lyapunov stability theory and some novel integral inequalities based on auxiliary functions, some sufficient conditions are obtained to guarantee that the resulting closed-loop system not only meets a prescribed mixed H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> and passive performance but also is stochastically stable. Then, by using a skillful matrix decoupling approach, the gains of the fuzzy controller may be expressed specifically. Lastly, the superiority and availability of the developed design method are extensively described and validated via its application to a truck control model.

Adaptive Trajectory Tracking Control of a High Altitude Unmanned Airship

Nonlinear dynamic model of a high-altitude unmanned airship, expressed by generalized coordinate, was built. A nonlinear compensation was introduced into the control loop to linearize and decouple the nonlinear system globally. In view of the imprecisely known inertia parameters of the airship, an adaptive law was proposed based on the feedback linearization to realize asymptotic tracking of any continuous time-varying desired trajectory from an arbitrary initial condition. The stability of the closed-loop control system was proved via the use of Lyapunov stability theory. Finally, numerical simulation results demonstrate the validity and effectiveness of the proposed adaptive control law.

Global analysis of a dynamical model for transmission of tuberculosis with a general contact rate

This paper deals with the global analysis of a dynamical model for the spread of tuberculosis with a general contact rate. The model exhibits the traditional threshold behavior. We prove that when the basic reproduction ratio is less than unity, then the disease-free equilibrium is globally asymptotically stable and when the basic reproduction ratio is great than unity, a unique endemic equilibrium exists and is globally asymptotically stable under certain conditions. The stability of equilibria is derived through the use of Lyapunov stability theory and LaSalle’s invariant set theorem. Numerical simulations are provided to illustrate the theoretical results.

Cryptanalysis of a chaotic communication scheme using adaptive observer

This paper addresses the cryptanalysis of a secure communication scheme recently proposed by Wu [Chaos 16, 043118 (2006)], where the information signal is modulated into a system parameter of a unified chaotic system. With the Kerckhoff principle, assuming that the structure of the cryptosystem is known, an adaptive observer can be designed to synchronize the targeted system, so that the transmitted information and the user-specific parameters are obtained. The success of adaptive synchronization is mathematically proved with the use of Lyapunov stability theory, based on the original assumption, i.e., the dynamical evolution of the information signal is available. A more practical case, but yet much more difficult, is also considered. As demonstrated with simulations, generalized synchronization is still possible, even if the derivative of the information signal is kept secret. Hence, the message can be coarsely estimated, making the security of the considered system questionable.

One approach to the control of uncertain dynamical systems

We present a controller design methodology for uncertain systems which is based on the constructive use of Lyapunov stability theory. The uncertainties, which are deterministic, are characterized by certain structural conditions and known as well as unknown bounds. As a consequence of the Lyapunov approach, the methodology is not restricted to linear or time-invariant systems. The robustness of these controllers in the presence of singular perturbations is considered. The situation in which the full state of the system is not available for measurement is also considered as are other generalizations.

On One Approach to the Control of Uncertain Systems

This paper is a personal history of one area of research with which I have been concerned over the past twenty years and which continues to attract my albeit waning attention, the control of dynamical systems based on uncertain models. Thus, it is a subjective account of various factors which I believe to have steered me to approach the problem via a constructive use of Lyapunov stability theory.