Invariant Set Theorem Research Articles

Global Stability of a Host-Vector Model for Pine Wilt Disease with Nonlinear Incidence Rate

Based on classical epidemic models, this paper considers a deterministic epidemic model for the spread of the pine wilt disease which has vector mediated transmission. The analysis of the model shows that its dynamics are completely determined by the basic reproduction numberR0. Using a Lyapunov function and a LaSalle's invariant set theorem, we proved the global asymptotical stability of the disease-free equilibrium. We find that ifR0≤1, the disease free equilibrium is globally asymptotically stable, and the disease will be eliminated. IfR0>1, a unique endemic equilibrium exists and is shown to be globally asymptotically stable, under certain restrictions on the parameter values, using the geometric approach method for global stability, due to Li and Muldowney and the disease persists at the endemic equilibrium state if it initially exists.

Dynamic analysis of an SEIR model with distinct incidence for exposed and infectives.

An SEIR model with vaccination strategy that incorporates distinct incidence rates for the exposed and the infected populations is studied. By means of Lyapunov function and LaSalle's invariant set theorem, we proved the global asymptotical stable results of the disease-free equilibrium. The sufficient conditions for the global stability of the endemic equilibrium are obtained using the compound matrix theory. Furthermore, the method of direct numerical simulation of the system shows that there is a periodic solution, when the system has three equilibrium points.

Stability analysis and design of a class of MIMO fuzzy control systems

This paper presents a new stability analysis method dedicated to a class of fuzzy control systems FCSs controlling multi input-multi output MIMO nonlinear processes by means of Takagi-Sugeno-Kang fuzzy logic controllers FLCs. The stability analysis of the FCSs is carried out using LaSalle's global invariant set theorem by the separate stability analysis of each fuzzy rule in MIMO fuzzy control systems. Therefore the complexity of the stability analysis is reduced and the adding of new fuzzy rules can be conducted easily; this modification of FLC structure requires the fulfillment of only one of the conditions of the stability analysis theorem suggested in this paper. Another advantage of the stability analysis approach proposed in this paper is that the derivative of the Lyapunov function candidate must be only negative semi-definite in comparison with Lyapunov's stability theorem where it must be negative definite. The conservativeness of stability conditions is thus reduced, and this enables the convenient design of FLCs. The applicability and efficiency of the theoretical results are illustrated by numerical simulations for a representative MIMO process which deals with the level control in a three spherical tank system.

Iterative performance improvement of fuzzy control systems for three tank systems

This paper suggests the performance improvement of fuzzy control systems (FCSs) for three tank systems using iterative feedback tuning (IFT). The stable design of Takagi–Sugeno–Kang fuzzy controllers is guaranteed by means of a stability theorem based on LaSalle’s global invariant set theorem formulated for a class of multi input-multi output (MIMO) nonlinear processes. An IFT algorithm characterized by setting the step size to guarantee the FCS stability is proposed. The theoretical approaches are applied in a case study that deals with the IFT-based stable design of fuzzy controllers dedicated to the level control of a cylindrical three tank system as a representative MIMO system. A set of experimental results for a laboratory setup illustrates the performance improvement.

Dynamics Analysis of an SEIQS Model with a Nonlinear Incidence Rate

This paper considers an SEIQS model with nonlinear incidence rate. By means of Lyapunov function and LaSalle’s invariant set theorem, we proved the global asymptotical stable results of the disease-free equilibrium. It is then obtained the sufficent conditions for the global stability of the endemic equilibrium by the compound matrix theory. In addition, we also study the phenomena of limit cycle of the systems with the numerical simulations.

An SEIRS Model with a Nonlinear Incidence Rate

This paper considers an SEIRS model with nonlinear incidence rate. By means of Lyapunov function and LaSalle's invariant set theorem, we proved the global asymptotical stable results of the disease-free equilibrium. It is then obtained the sufficient conditions for the global stability of the endemic equilibrium by the compound matrix theory. In addition, we also study the phenomena of limit cycle of the systems with the numerical simulations.

Stable Nonlinear Control Allocation for Aircraft with Multiple Control Effectors

With respect to aircraft with redundant multiple control effectors, a nonlinear controller, which is composed of a virtual control law and a dynamic control allocation with position constraints of each effector, is designed. Based on Lyapunov stability theory and LaSalle invariant set theorem, asymptotic stabilities of upper control subsystem, dynamic control allocation subsystem and overall closed-loop system are proved respectively. Simulation results show the effectiveness of the proposed method.

Boundary stabilization of vibration of nonlocal micropolar elastic media

In this paper, we study the stabilization problem of vibration of linearized three-dimensional nonlocal micropolar elasticity. For this purpose, we need to demonstrate the well-posedness of the system of equations governing the vibration of three-dimensional nonlocal micropolar media for both forced (i.e. with boundary feedback) and unforced cases. We assume the non-homogeneous system of equations for the unforced (uncontrolled) case to establish the well-posedness. It should be pointed out that the well-posedness of the evolution equations in micropolar case has been studied by many authors; but, the well-posedness in the nonlocal micropolar is an open problem. Our tools in well-posedness analysis are the semigroup techniques. Afterwards, we pursue the stabilization problem and show that the vibration of the nonlocal micropolar elastic media will be eventually dissipated under boundary feedback actions consisting of stress and couple stress feedback laws. These control laws are simple, linear and can be easily implemented in practical applications. The stabilization proof is accomplished using Lyapunov stability and LaSalle’s invariant set theorems.

Tracking control and generalized projective synchronization of a class of hyperchaotic system with unknown parameter and disturbance

In this paper, the tracking control and generalized projective synchronization of a class of hyperchaotic system with unknown parameter and disturbance are investigated. Based on the LaSalle’s invariant set theorem, a robust adaptive controller is contrived to acquire tracking control and generalized projective synchronization and parameter identification simultaneously. It is proved theoretically that the proposed scheme can allow us to drive the hyperchaotic system to any desired reference signals, including hyperchaotic signals, chaotic signals, periodic orbits or fixed value by the given scaling factor. The presented simulation results further demonstrate that the proposed method is effective and robust.

A Lyapunov controller for stable haptic manipulation of hydraulic actuators

SUMMARYA scheme for bilateral control of hydraulic actuators is developed and experimentally evaluated in this paper. The control laws are derived based on Lyapunov's feedback control design technique. Owing to the discontinuity originating from a sign function in the control laws, the control system is non‐smooth. First, the existence, continuation, and uniqueness of Filippov's solution to the system are proven. Next, the extensions of Lyapunov's stability theory to non‐smooth systems and LaSalle's invariant set theorems are employed to prove the asymptotic stability of the control system. Effectiveness of the proposed controller is verified by simulation and experimental studies. It is shown that beside stability, the system has good transparency in terms of position and force exchanges between the master and slave actuators. Copyright © 2011 John Wiley & Sons, Ltd.

Adaptive chaotic control of permanent magnet synchronous motor

Based on the LaSalle's invariant set theorem, an adaptive controller is constructed to acquire the chaotic control for the permanent magnet synchronous motor. And then a extended adaptive controller is developed by introducing a control strength factor. In comparison with the previous schemes, the present control method is simple, flexible, and is applicable in practice because of reduced control cost. Simulation results are presented to show the effectiveness and the robust of the proposed method.

Stable design of Takagi-Sugeno fuzzy controllers for a laboratory three-tank system

This paper suggests a stable design approach dedicated to the Takagi-Sugeno fuzzy controllers for the water-level control of a laboratory three-tank system. The design approach is based on a simplified application of LaSalle’s global invariant set theorem. An original stability analysis theorem offers sufficient inequality-type stability conditions that enable the controller design and tuning. Real-time experimental results are included to validate the theoretical results.

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.

Fuzzy Control System Performance Enhancement by Iterative Learning Control

This paper suggests low-cost fuzzy control solutions that ensure the improvement of control system (CS) performance indices by merging the benefits of fuzzy control and iterative learning control (ILC). The solutions are expressed in terms of three fuzzy CS (FCS) structures that employ ILC algorithms and a unified design method focused on Takagi-Sugeno proportional-integral fuzzy controllers (PI-FCs). The PI-FCs are dedicated to a class of servo systems with linear/linearized controlled plants characterized by second-order dynamics and integral type. The invariant set theorem by Krasovskii and LaSalle with quadratic Lyapunov function candidates is applied to guarantee the convergence of the ILC algorithms and enable proper setting of the PI-FC parameters. The linear PI controller parameters tuned by the extended symmetrical optimum method are mapped onto the PI-FC ones by the modal equivalence principle. Real-time experimental results for a dc-based servo speed CS are included.

Passivity‐based control of a magnetically levitated flexible beam

AbstractThis paper solves the asymptotic stabilization problem for a magnetically levitated flexible beam using a nested‐loop passivity‐based controller design. Passivity analyses reveal that the system can be decomposed into two passive subsystems: a mechanical subsystem that consists of a flexible beam with both ends free and that defines a passive map from external forces to the velocity of the points on the flexible beam at which the external forces act; and an electrical subsystem that consists of a pair of electromagnets and that defines a strictly output‐passive map from voltages applied across the electromagnets to magnetic fluxes. The standard method for designing passivity‐based controllers leads to a nonlinear feed‐forward controller for the electrical subsystem, which enables the electrical subsystem to generate given desired magnetic forces, and an output feedback compensator for the mechanical subsystem, which computes the desired forces required to regulate the position and vibration of the beam. The asymptotic stability of each controller may be proven using Lyapunov's stability theory and LaSalle's invariant set theorem. Numerical simulations confirm the asymptotic stability of the equilibrium configuration of the closed‐loop system formed by the magnetically levitated flexible beam together with the proposed controllers. Copyright © 2008 John Wiley & Sons, Ltd.

Design and Experiments for a Class of Fuzzy Controlled Servo Systems

This paper proposes a new fuzzy control solution employing 2-DOF proportional-integral-fuzzy controllers dedicated to a class of servo systems. The controlled plants in these systems, widely used in mechatronics applications, can be characterized by second-order dynamics with integral type. The original design method suggested here starts with linear design results in terms of the extended symmetrical optimum method accompanied by an iterative feedback tuning (IFT) algorithm. Next, these results are transferred to the fuzzy case by the modal equivalence principle. The convergence of the IFT algorithm is guaranteed by the derivation of sufficient global asymptotic stability conditions based on Krasovskii-LaSalle's invariant set theorem with quadratic Lyapunov function candidate. Real-time experimental results corresponding to a low-cost fuzzy controlled servo system validate the theoretical approaches.

PASSIVATION-BASED INTEGRATED GUIDANCE AND CONTROL OF UNDERACTUATED MARINE CRAFT

A passivation approach is proposed for the guidance and control of under-actuated marine craft, which integrates trajectory planning and dynamic tasking in a single formulation. Trajectory and velocity requirements are specified through the design of a manifold in the phase space, and convergence proof is based on the invariant set theorem. Two design cases, path tracking and course keeping, are presented in detail.

On the Stability and Control of Nonlinear Dynamical Systems via Vector Lyapunov Functions

Vector Lyapunov theory has been developed to weaken the hypothesis of standard Lyapunov theory in order to enlarge the class of Lyapunov functions that can be used for analyzing system stability. In this paper, we extend the theory of vector Lyapunov functions by constructing a generalized comparison system whose vector field can be a function of the comparison system states as well as the nonlinear dynamical system states. Furthermore, we present a generalized convergence result which, in the case of a scalar comparison system, specializes to the classical Krasovskii-LaSalle invariant set theorem. In addition, we introduce the notion of a control vector Lyapunov function as a generalization of control Lyapunov functions, and show that asymptotic stabilizability of a nonlinear dynamical system is equivalent to the existence of a control vector Lyapunov function. Moreover, using control vector Lyapunov functions, we construct a universal decentralized feedback control law for a decentralized nonlinear dynamical system that possesses guaranteed gain and sector margins in each decentralized input channel. Furthermore, we establish connections between the recently developed notion of vector dissipativity and optimality of the proposed decentralized feedback control law. Finally, the proposed control framework is used to construct decentralized controllers for large-scale nonlinear systems with robustness guarantees against full modeling uncertainty.

Global stability of an SEI epidemic model with general contact rate☆

This paper consider an SEI epidemic model with general contact rate that incorporates constant recruitment and have infectious force in the latent period and infected period. By means of Lyapunov function and LaSalle’s invariant set theorem, we proved the global asymptotical stable results of the disease-free equilibrium and the epidemic equilibrium by using the Poincarè–Bendixson property.

Global stability of an SEI epidemic model with general contact rate

This paper consider an SEI epidemic model with general contact rate that incorporates constant recruitment and have infectious force in the latent period and infected period. By means of Lyapunov function and LaSalles invariant set theorem, we proved the global asymptotical stable results of the disease-free equilibrium and the epidemic equilibrium by using the Poincar�–Bendixson property. � 2004 Elsevier Ltd. All rights reserved.