Introducing some classes of stable systems without any smooth Lyapunov functions

For nonsmooth Filippov systems, the stability of the system is assumed to be proved by nonsmooth Lyapunov functions, such as piecewise smooth Lyapunov functions. This extension was based on the Filippov solution and Clarke generalized gradient. However, it is difficult to estimate the gradient of a non-smooth Lyapunov function. In some cases, the nonsmooth system can be divided into continuous and discontinuous components. If the Lebesgue measure of the discontinuous components is zero, the smooth Lyapunov function can be utilized to prove the stability of the system owing to the inner product of the gradient of the Lyapunov function of the discontinuous components being zero. In this paper, we apply the smooth Lyapunov function to prove the stability of the nonsmooth ratio-dependent predator-prey system. In contrast to the existing literature, in this paper, although the system is divided into continuous and discontinuous components, the inner product of the gradient of the Lyapunov function of the discontinuous part is not zero but negative. In the proof of stability, the negative value condition is stricter than the zero-value condition. This proof method only needs to construct a smooth Lyapunov function, which is simpler than a non-smooth Lyapunov function or other methods.

On a class of nonlinear stochastic dynamical systems: Analysis of the transient behaviour

Two-dimensional nonlinear dynamic system model of interspecific interaction and numerical simulation research on it

ABSTRACTBased on Radial Basis Function neural networks, a stable adaptive control scheme is proposed for a class of unknown nonlinear dynamic systems. It is shown that the scheme is stable and convergent in the sense that all the signal in the loop remain bounded, and in the ideal case, output tracking error will converge to zero and in the non-ideal case, the error will converge to a residue which is proportional to the modelling error bound. Simulation results are also provided to demonstrate performance of the scheme.

Electrochemical sensors have been broadly applied in various areas for diagnostics and controls. However, many electrochemical sensors are often cross-sensitive to one or more interfering compounds, which can make it rather challenging in attaining accurate concentration of the target compound. The sensor cross-sensitivity can cause instability issues in feedback control systems and inaccurate diagnostics. In many cases, a state estimation problem with cross-sensitive output is equivalent to a state estimation problem with limited number of state candidates that can lead to the same output. In this study, an innovative estimation error-based observer is proposed for state estimation for a general class of 1st-order nonlinear dynamic systems with cross-sensitive output. By constructing observers for each potential candidate and analyzing the respective estimation errors, the true state (or mode) of the system can be determined based on the cross-sensitive output measurement. The proposed algorithm was applied to estimate the ammonia coverage ratio of a Diesel selective catalytic reduction system with ammonia cross-sensitive <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\text{NO}_{\mathrm{x}}$</tex> sensor output. Simulation results show that the true ammonia coverage ratio can be accurately estimated with this method at both low and high ammonia coverage ratios. This method can be applied to a broad class of nonlinear dynamic systems with cross-sensitive outputs for state estimation.

The stability of nonlinear dynamical system of relative rotation is studied. Firstly, the dynamics equation of relative rotation autonomous nonlinear dynamical system with commonly damped force and forced excitation is deduced. Secondly, the stability of relative rotation nonlinear dynamical system is studied. For the nonlinear dynamical system, it is proved that the closed orbit bifurcation can occur under some conditions. Finally, The approximate solution of the equation under forced excitation is obtained by the method of multiple scales.

Nonlinear system analysis and design have been attracting increasing adherents in recent decades. Different normal form techniques, approximation methods, and perturbation procedures have been used to analyze these systems. In recent years, a novel approach, called modal series method, has been introduced and used to study the behavior of autonomous nonlinear dynamic systems of the form x = g(x). In this paper, we extend this approach to a wide class of nonautonomous nonlinear systems of the form x = g(x,i). Using two textbook examples, we show the capability of this extension to find approximate solutions for nonautonomous nonlinear systems.

Our thermodynamic approach to the study and design of robust optimal control processes in nonlinear (in general global unstable) dynamic systems used soft computing based on genetic algorithms with a fitness function as minimum entropy production. Control objects were nonlinear dynamic systems involving essentially nonlinear stochastic differential equations. An algorithm was developed for calculating entropy production rate in control object motion and in control systems. Part 1 discusses relation of the Lyapunov function (measure of stability) and the entropy production rate (physical measure of controllability). This relation was used to describe the following qualitative properties and important relations: dynamic stability motion (Lyapunov function), Lyapunov exponent and Kolmogorov-Sinai entropy, physical entropy production rates, and symmetries group representation in essentially nonlinear systems as coupled oscillator models. Results of computer simulation are presented for entropy-like dynamic behavior for typical benchmarks of dynamic systems such as Van der Pol, Duffing, and Holmes-Rand, and coupled oscillators. Parts 2 and 3 discuss the application of this approach to simulation of dynamic entropy-like behavior and optimal benchmark control as a 2-link manipulator in a robot for service use and nonlinear systems under stochastic excitation.

We develop a direct Lyapunov method for the almost sure open-loop stabilizability and asymptotic stabilizability of controlled degenerate diffusion processes. The infinitesimal decrease condition for a Lyapunov function is a new form of Hamilton--Jacobi--Bellman partial differential inequality of second order. We give local and global versions of the first and second Lyapunov theorems, assuming the existence of a lower semicontinuous Lyapunov function satisfying such an inequality in the viscosity sense. An explicit formula for a stabilizing feedback is provided for affine systems with smooth Lyapunov function. Several examples illustrate the theory.

The number of equilibrium points of perturbed nonlinear positive dynamical systems

In this thesis, we present a dynamical systems approach to adaptive controllers for locomotion control. The approach is based on a rigorous mathematical framework using nonlinear dynamical systems and is inspired by theories of self-organization. Nonlinear dynamical systems such as coupled oscillators are an interesting approach for the on-line generation of trajectories for robots with many degrees of freedom (e.g. legged locomotion). However, designing a nonlinear dynamical system to satisfy a given specification and goal is not an easy task, and, hitherto no methodology exists to approach this problem in a unified way. Nature presents us with satisfactory solutions for the coordination of many degrees of freedom. One central feature observed in biological subjects is the ability of the neural systems to exploit natural dynamics of the body to achieve efficient locomotion. In order to be able to exploit the body properties, adaptive mechanisms must be at work. Recent work has pointed out the importance of the mechanical system for efficient locomotion. Even more interestingly, such well suited mechanical systems do not need complicated control. Yet, in robotics, in most approaches, adaptive mechanisms are either missing or they are not based on a rigorous framework, i.e. they are based on heuristics and ad-hoc approaches. Over the last three decades there has been enormous progress in describing movement coordination with the help of Synergetic approaches. This has led to the formulation of a theoretical framework: the theory of dynamic patterns. This framework is mathematically rigorous and at the same time fully operational. However, it does not provide any guidelines for synthetic approaches as needed for the engineering of robots with many degrees of freedom, nor does it directly help to explain adaptive systems. We will show how we can extend the theoretical framework to build adaptive systems. For this purpose, we propose the use of multi-scale dynamical systems. The basic idea behind multi-scale dynamical systems is that a given dynamical system gets extended by additional slow dynamics of its parameters, i.e. some of the parameters become state variables. The advantages of the framework of multi-scale dynamical systems for adaptive controllers are 1) fully dynamic description, 2) no separation of learning algorithm and learning substrate, 3) no separation of learning trials or time windows, 4) mathematically rigorous, 5) low dimensional systems. However, in order to fully exploit the framework important questions have to be solved. Most importantly, methodologies for designing the feedback loops have to be found and important theoretical questions about stability and convergence properties of the devised systems have to be answered. In order to tackle this challenge, we first introduce an engineering view on designing nonlinear dynamical systems and especially oscillators. We will highlight the important differences and freedom that this engineering view introduces as opposed to a modeling one. We then apply this approach by first proposing a very simple adaptive toy-system, consisting of a dynamical system coupled to a spring-mass system. Due to its spring-mass dynamics, this system contains clear natural dynamics in the form of resonant frequencies. We propose a prototype adaptive multi-scale system, the adaptive frequency oscillator, which is able to adapt its intrinsic frequency to the resonant frequency of the body dynamics. After a small sidetrack to show that we can use adaptive frequency oscillators also for other applications than for adaptive controllers, namely for frequency analysis, we then come back to further investigation of the adaptive controller. We apply the same controller concept to a simple spring-mass hopper system. The spring-mass system consists of a body with two legs attached by rotational joints. The legs contain spring-damper elements. Finally, we present results of the implementation of the controller on a real robot, the experimental robot PUPPY II. This robot is a under-actuated robot with spring dynamics in the knee joints. It will be shown, that due to the appropriate simplification and concentration on relevant features in the toy-system the controller concepts works without a fundamental change on all systems from the toy system up to the real robot.

Modeling of Microblogging Social Networks: Dynamical System vs. Random Dynamical System

The usual invariance results for asymptotic stability for continuous autonomous finite-dimensional dynamical systems involve a positive definite Lyapunov function whose time derivative along the system motions is negative semidefinite (along with certain invariance conditions). This is equivalent to requiring that along the system motions, the Lyapunov function is nonincreasing at all times with increasing time. In this paper we establish an invariance result for asymptotic stability for continuous and discontinuous nonautonomous finite-dimensional dynamical systems, which requires a positive definite Lyapunov function which when evaluated along the system motions is nonincreasing only on certain unbounded discrete time sets E with increasing time. This allows that between the time instants determined by E, the Lyapunov function may increase (i.e., over some finite time intervals, the system may exhibit unstable behavior). We also show that the usual invariance theorem for asymptotic stability reduces to the invariance theorem for continuous dynamical systems presented herein. In addition, we establish a variant to the above result involving estimates of the asymptotic behavior of the system's motions. We apply our results to four examples. One of these involves the stabilization of conservative mechanical systems using energy dissipation intermittently.

This paper presents how to apply a newly developed general theory for the global transversality and tangency of flows in n-dimensional nonlinear dynamical systems to a 2-D nonlinear dynamical system (i.e. a periodically forced, damped Duffing oscillator). The global tangency and transversality of the periodic and chaotic motions to the separatrix for such a nonlinear system are discussed to help us understand the complexity of chaos in nonlinear dynamical systems. This paper presents the concept that the global transversality and tangency to the separatrix are independent of the Melnikov function (or the energy increment). Chaos in nonlinear dynamical systems makes the exact energy increment quantity to be chaotic no matter if the nonlinear dynamical systems have separatrices or not. The simple zero of the Melnikov function cannot be used to simply determine the existence of chaos in nonlinear dynamical systems. Through this paper, the expectation is that, from now on, one can use the alternative aspect to look into the complexity of chaos in nonlinear dynamical systems. Therefore, in this paper, the analytical conditions for global transversality and tangency of 2-D nonlinear dynamical systems are presented. The first integral quantity increment (i.e. the energy increment) for a certain time interval is achieved for periodic flows and chaos in the 2-D nonlinear dynamical systems. Under the perturbation assumptions and convergent conditions, the Melnikov function is recovered from the first integral quantity increment. A periodically forced, damped Duffing oscillator with a separatrix is investigated as a sampled problem. The corresponding analytical conditions for the global transversality and tangency to the separatrix are obtained and verified by numerical simulations. The switching planes and the corresponding local and global mappings are defined on the separatrix. The mapping structures are developed for local and global periodic flows passing through the separatrix. The mapping structures of global chaos in the damped Duffing oscillator are also discussed. Bifurcation scenarios of the damped Duffing oscillator are presented through the traditional Poincaré mapping section and the switching planes. The first integral quantity increment (i.e. L-function) is presented to observe the periodicity of flows. In addition, the global tangency of periodic flows in such an oscillator is measured by the G-function and G(1)-function, and is verified by numerical simulations. The first integral quantity increment of periodic flows is zero for their complete periodic cycles. Numerical simulations of chaos in such a Duffing oscillator are carried out through the Poincaré mapping sections. The conservative energy distribution, G-function and L-function along the displacement of Poincaré mapping points are presented to observe the complexity of chaos. The first integral quantity increment (i.e. L-function) of chaotic flows at the Poincaré mapping points is nonzero and chaotic. The switching planes of chaos are presented on the separatrix for a better understanding of the global transversality to the separatrix. The switching point distribution on the separatrix is presented and the switching G-function on the separatrix is given to show the global transversality of chaos on the separatrix. The analytical conditions are obtained from the new theory rather than the Melnikov method. The new conditions for the global transversality and tangency are more accurate and independent of the small parameters.

Elements of mathematical phenomenology of self-organization nonlinear dynamical systems: Synergetics and fractional calculus approach