A general invariance principle for nonlinear time-varying systems with application to mobile robots

A general invariance principle, from the output-to-state point of view, is proposed for the dynamical analysis of nonlinear time-varying systems. This is achieved by the construction of a simple and intuitive criterion using integral inequality of the output function and modified detectability conditions. The proposed scheme can be viewed as an extension of the integral invariance principle (C.I. Byrnes and C.F. Martin, 1995) for time-invariant systems to time-varying systems. Such extension is nontrivial and can be used in various research areas such as adaptive control, tracking control and the control of driftless systems. An application to global tracking control of four-wheeled mobile robots is given to demonstrate the feasibility and validity of the proposed approach.

In this paper, a new approach to stability analysis of nonlinear dynamics of an underactuated autonomous underwater vehicle (AUV) is presented. AUV is a highly nonlinear robotic system whose dynamic model includes coupled terms due to the hydrodynamic damping factors. It is difficult to analyze the stability of a nonlinear dynamical system through Routh's stability approach because it contains nonlinear dynamic parameters owing to hydrodynamic damping coefficients. It is also difficult to analyze the stability of AUVs using Lyapunov's criterion and LaSalle's invariance principle. In this paper, we proposed the extended-Routh's stability approach to verify the stability of such nonlinear dynamic systems. This extended-Routh's stability approach is much easier as compared to the other existing methods. Numerical simulations are presented to demonstrate the efficacy of the proposed stability verification of the nonlinear dynamic systems, e.g., an AUV system dynamics.

An identification approach for linear and nonlinear time-variant structural systems via harmonic wavelets

MTN optimal control of MIMO non-affine nonlinear time-varying discrete systems for tracking only by output feedback

<p style='text-indent:20px;'>This paper demonstrates input-to-state stability (ISS) of the SIR model of infectious diseases with respect to the disease-free equilibrium and the endemic equilibrium. Lyapunov functions are constructed to verify that both equilibria are individually robust with respect to perturbation of newborn/immigration rate which determines the eventual state of populations in epidemics. The construction and analysis are geometric and global in the space of the populations. In addition to the establishment of ISS, this paper shows how explicitly the constructed level sets reflect the flow of trajectories. Essential obstacles and keys for the construction of Lyapunov functions are elucidated. The proposed Lyapunov functions which have strictly negative derivative allow us to not only establish ISS, but also get rid of the use of LaSalle's invariance principle and popular simplifying assumptions.

The stability and behavior of continuous-time nonlinear time-varying (NTV), nonlinear time-invariant (NTI) and linear time-varying (LTV) systems are generally very difficult to ascertain using current analytical tools. This paper presents sufficient conditions under which a NTV system with a bounded system matrix, A(t,X), is globally exponentially stable (GES). The continuous-time NTV system is modeled as an uncertain polytopic system even though A(t,X) may be reasonably known. It is shown that if all polytope-vertex matrices are Hurwitz, then the system is GES. Also, a method for determining the boundary estimate of a NTV system state-trajectory norm is developed. Presented is a unified stability analysis approach for a large class of NTV, NTI, and LTV systems. Even for large systems, application of the GES theorem presented is straight forward using readily available software and computational power. Examples are given that illustrate the application of the theorem offered.

This paper deals with the feedback stabilization of a hybrid PDE-ODE system which models an overhead crane with a flexible cable. The well-posedness of the closed-loop system is established, and asymptotic stabilization is proved, using LaSalle's Invariance Principle, for a class of nonlinear feedback laws. Estimates of the rate of decay are provided for a simplified model. Illustrative simulations are displayed.

A robust tracking control algorithm based mainly on the theory of variable structure systems and Lyapunov's direct method is developed in this paper, to treat the tracking control problem of a class of non-linear time-varying systems with uncertain dynamics. A nominal model is used instead of an exact one. together with a description of the bounds of plant uncertainty. The objective is to design a control law such that all the plants lying within a specified band of uncertainly will be driven to reach the goal. The modelling error, time-varying system dynamics, system uncertainties, and disturbances are all treated as general disturbances to the nominal system. This algorithm greatly eases the difficulties present in requiring a precise model of the physical system for the control purpose and for handling the non-linear time-varying system. The non-linear time-varying system is first linearized to a linear time-varying system (with time-varying coefficients completely defined) via the method of variable s...

The problem of asymptotic stabilization of nonlinear, "double integrator", open-loop stable systems via sign-indefinite damping injection is considered in this paper. A constructive procedure to reduce the problem to the solution of a set of partial differential equations is presented. Particular emphasis is given to mechanical systems, for which it is shown that the proposed approach obviates the usual detectability assumption needed to conclude asymptotic stability via LaSalle's invariance principle.

The global properties of a predator-prey model with nonlinear functional response and stage structure for the predator are studied using Lyapunov functions and LaSalle's invariance principle. It is found that, under hypotheses which ensure the uniform persistence of the system and the existence of a unique positive steady state, a feasible a priori lower bound condition on the abundance of the prey population ensures the global asymptotic stability of the positive steady state. A condition which leads to the extinction of the predators is indicated. We also obtain results on the existence and stability of periodic solutions. In particular, when (4.2) fails to hold and the unique positive steady state $E^*$ becomes unstable, the coexistence of prey and predator populations is ensured for initial populations not on the one-dimensional stable manifold of $E^*$, albeit with fluctuating population sizes.

Typhoid fever and malaria are two serious infectious diseases that are common in sub-Saharan Africa, and co-infection poses a serious threat to public health. Designing successful control measures requires an understanding of the dynamics of these illnesses. We created a deterministic compartmental model that divided the human population into seven groups: susceptible people, typhoid-only infected people, malaria-only infected people, co-infected people, recovered people, susceptible mosquitoes, and infected mosquitoes. We defined parameters for the global stability of the disease-free equilibria and determined the fundamental reproduction numbers for typhoid and malaria using the next-generation matrix technique. The global stability results were validated using Lyapunov functions and LaSalle's invariance principle. When both fundamental reproduction numbers are less than unity, the disease-free equilibrium is asymptotically stable worldwide.Numerical simulations highlight the threshold parameters that drive co-infection persistence and the combined impact of malaria–typhoid interventions. This study provides a theoretical basis for controlling malaria and typhoid co-infections through integrated interventions. The analytical thresholds derived can guide policymakers in optimizing combined control strategies in endemic regions such as Nigeria

If V is a Lyapunov function of an equation du/dt = u′ = Zu in a Banach space then asymptotic stability of an equilibrium point may be easily proved if it is known that sup(V′) &lt; 0 on sufficiently small spheres centered at the equilibrium point. In this paper weak asymptotic stability is proved for a bounded infinitesimal generator Z under a weaker assumption V′ ≤ 0 (which alone implies ordinary stability only) if some observability condition, involving Z and the Frechet derivative of V′, is satisfied. The proof is based on an extension of LaSalle′s invariance principle, which yields convergence in a weak topology and uses a strongly continuous Lyapunov function. The theory is illustrated with an example of an integro‐differential equation of interest in the theory of chemical processes. In this case strong asymptotic stability is deduced from the weak one and explicit sufficient conditions for stability are given. In the case of a normal infinitesimal generator Z in a Hilbert space, strong asymptotic stability is proved under the following assumptions Z*+Z is weakly negative definite and Ker Z = {0}. The proof is based on spectral theory.

Cholera is a common infectious disease caused by Vibrio cholerae, which has different infectivity. In this paper, we propose a cholera model with hyperinfectious and hypoinfectious vibrios, in which both human-to-human and environment-to-human transmissions are considered. By analyzing the characteristic equations, the local stability of disease-free and endemic equilibria is established. By using Lyapunov functionals and LaSalle's invariance principle, it is verified that the global threshold dynamics of the model can be completely determined by the basic reproduction number. Numerical simulations are carried out to illustrate the corresponding theoretical results and describe the cholera outbreak in Haiti. The study of optimal control helps us seek cost-effective solutions of time-dependent control strategies against cholera outbreaks, which shows that control strategies, such as vaccination and sanitation, should be taken at the very beginning of the outbreak and become less necessary after a certain period.

In this paper stabilization of infinite‐dimensional undamped second‐order systems is considered in the case where the input and output operators are collocated. The systems have an infinite number of poles and zeros on the imaginary axis. In the case where only position feedback is available, a parallel compensator is effective. The stabilizer is constructed by a P‐controller for the augmented system which consists of the controlled system and a parallel compensator. The asymptotic stability of the augmented system is proved by LaSalle's invariance principle under compactness of the resolvent.

A nonlinear controller is presented for the stabilization of the spherical inverted pendulum system. The control strategy is based on the Lyapunov approach in conjunction with LaSalle's invariance principle. The proposed controller is able to bring the pendulum to the unstable upright equilibrium point with the position of the movable base at the origin. The obtained closed-loop system has a very large domain of attraction, that can be as large as desired, for any initial position of the pendulum which lies above the horizontal plane.