Nonlinear Nonautonomous Systems Research Articles

Approximation-Free State-Feedback Backstepping Controller for Uncertain Pure-Feedback Nonautonomous Nonlinear Systems Based on Time-Derivative Estimator

A novel backstepping controller for uncertain single-input single-output pure-feedback nonaffine and nonautonomous nonlinear systems based on the time-derivative estimator (TDE) is proposed. Using TDEs, time-derivatives of error signals used in virtual control terms are directly estimated in every backstepping design steps. As a result, the control law has a relatively simple form. In addition, convergence of tracking error to a small neighborhood of origin is guaranteed regardless of unstructured uncertainties or unmatched disturbances in the controlled system. It does not require separate adaptive schemes or universal approximators such as neural networks or fuzzy logic systems adaptively tuned online to cope with system uncertainties. Simulation results demonstrated the simplicity and good performance of the proposed approximation-free controller.

Quasi-periodic solutions for a nonlinear non-autonomous Hamiltonian system

Quasi-periodic solutions for a nonlinear non-autonomous Hamiltonian system

Distributed cooperative adaptive tracking control for heterogeneous systems with hybrid nonlinear dynamics

The cooperative leader-following tracking for a group of heterogeneous mechanical systems with nonlinear hybrid order dynamics is studied. The controlled systems are considered to be composed of followers (agents) with hybrid first- and second-order time-varying dynamics. The leader is an unknown nonautonomous nonlinear system and can only give the state information of position and velocity to its neighboring followers. The followers are linked by the directed graph with fixed communication topology. And, not all of them have the information path to the leader directly. The directed information topology graph is required to have at least one spanning tree for position and velocity, respectively. Distributed cooperative adaptive control protocols are developed for all followers with first- or second-order dynamics to achieve the ultimate synchronization to the leader. The control protocols are designed based on the neural networks and the adaptive estimation algorithm for unknown time-varying functions and control coefficients. The convergence and boundedness of the synchronization error is proved by the Lyapunov theory. The simulation example verifies the correctness of the developed distributed control protocols.

On finite-time stability of nonautonomous nonlinear systems

ABSTRACT This paper is concerned with the finite-time stability (FTS) of nonautonomous nonlinear systems. Firstly, a new definition of FTS for nonautonomous nonlinear systems is introduced. Then some sufficient conditions on FTS for nonautonomous nonlinear systems are established based on Lyapunov function. Furthermore, the conditions are shown to be necessary and sufficient for linear time-varying systems. Finally, several numerical examples are given to illustrate our results.

Chaos in a non-autonomous nonlinear system describing asymmetric water wheels

We derive a water wheel model from first principles under the assumption of an asymmetric water wheel for which the water inflow rate is in general unsteady (modeled by an arbitrary function of time). Our model allows one to recover the asymmetric water wheel with steady flow rate, as well as the symmetric water wheel, as special cases. Under physically reasonable assumptions, we then reduce the underlying model into a non-autonomous nonlinear system. In order to determine parameter regimes giving chaotic dynamics in this non-autonomous nonlinear system, we consider an application of competitive modes analysis. In order to apply this method to a non-autonomous system, we are required to generalize the competitive modes analysis so that it is applicable to non-autonomous systems. The non-autonomous nonlinear water wheel model is shown to satisfy competitive modes conditions for chaos in certain parameter regimes, and we employ the obtained parameter regimes to construct the chaotic attractors. As anticipated, the asymmetric unsteady water wheel exhibits more disorder than does the asymmetric steady water wheel, which in turn is less regular than the symmetric steady state water wheel. Our results suggest that chaos should be fairly ubiquitous in the asymmetric water wheel model with unsteady inflow of water.

Detecting multi-pulse chaotic dynamics of high-dimensional non-autonomous nonlinear system for circular mesh antenna

This paper investigates the global bifurcations and multi-pulse jumping chaotic dynamics of circular mesh antenna. An equivalent continuum circular cylindrical shell is employed to represent the circular mesh antenna. Based on the four-dimension non-autonomous nonlinear governing equations of motion for the equivalent continuum circular cylindrical shell derived by Zhang et al. (2016, 2017), the improved extended Melnikov theory of the non-autonomous nonlinear system is utilized to conduct a theoretical analysis of the multi-pulse jumping chaotic motions for the equivalent continuum circular cylindrical shell. The thermal excitation and damping coefficient are considered as the controlling parameters to analyze their effect on the nonlinear vibrations and bifurcations of the equivalent continuum circular cylindrical shell. Numerical simulations are also introduced to further verify the existence of the multi-pulse jumping chaotic motions for the equivalent continuum circular cylindrical shell. The results obtained from the numerical simulations are compared to those obtained from the Melnikov theoretical prediction.

Search for Periodic Solutions of Highly Nonlinear Dynamical Systems

Numerical-analytical methods for finding periodic solutions of highly nonlinear autonomous and nonautonomous systems of ordinary differential equations are considered. Algorithms for finding initial conditions corresponding to a periodic solution are proposed. The stability of the found periodic solutions is analyzed using corresponding variational systems. The possibility of studying the evolution of periodic solutions in a strange attractor zone and on its boundaries is discussed, and interactive software implementations of the proposed algorithms are described. Numerical examples are given.

Distributed ESO based cooperative tracking control for high-order nonlinear multiagent systems with lumped disturbance and application in multi flight simulators systems

Based on extended state observer, a novel and practical design method is developed to solve the distributed cooperative tracking problem of higher-order nonlinear multiagent systems with lumped disturbance in a fixed communication topology directed graph. The proposed method is designed to guarantee all the follower nodes ultimately and uniformly converge to the leader node with bounded residual errors. The leader node, modeled as a higher-order non-autonomous nonlinear system, acts as a command generator giving commands only to a small portion of the networked follower nodes. Extended state observer is used to estimate the local states and lumped disturbance of each follower node. Moreover, each distributed controller can work independently only requiring the relative states and/or the estimated relative states information between itself and its neighbors. Finally an engineering application of multi flight simulators systems is demonstrated to test and verify the effectiveness of the proposed algorithm.

Optimal transport over nonlinear systems via infinitesimal generators on graphs

We present a set-oriented graph-based computational framework for continuous-time optimal transport over nonlinear dynamical systems. We recover provably optimal control laws for steering a given initial distribution in phase space to a final distribution in prescribed finite time for the case of non-autonomous nonlinear control-affine systems, while minimizing a quadratic control cost. The resulting control law can be used to obtain approximate feedback laws for individual agents in a swarm control application. Using infinitesimal generators, the optimal control problem is reduced to a modified Monge-Kantorovich optimal transport problem, resulting in a convex Benamou-Brenier type fluid dynamics formulation on a graph. The well-posedness of this problem is shown to be a consequence of the graph being strongly-connected, which in turn is shown to result from controllability of the underlying dynamical system. Using our computational framework, we study optimal transport of distributions where the underlying dynamical systems are chaotic, and non-holonomic. The solutions to the optimal transport problem elucidate the role played by invariant manifolds, lobe-dynamics and almost-invariant sets in efficient transport of distributions in finite time. Our work connects set-oriented operator-theoretic methods in dynamical systems with optimal mass transportation theory, and opens up new directions in design of efficient feedback control strategies for nonlinear multi-agent and swarm systems operating in nonlinear ambient flow fields.

Identification of Nonlinear Systems with Additive External Action

The problem of identification of nonautonomous nonlinear systems of ordinary differential equations is considered. An algorithm for reconstructing a system under harmonic external excitation is proposed. This algorithm is optimal from the point of view of computing time. An example of the solution of the system identification problem for the generalized Duffing system in the regime of deterministic chaos is considered.

Dynamics of a vibration system, taking into account the hereditary type friction and the mobility of a vibration limiter

The paper investigates the dynamics of some vibration systems taking into account the hereditary-type dry friction and mobility of a vibration limiter. The interaction of the vibration limiter and the vibration system occurs either according to Newton’s hypothesis (an absolutely rigid limiter) or softly (the limiter is mobile). A general mathematical model (MM) of systems has been developed, which is highly non-linear nonautonomous system with variable structure. A numerical-analytical approach using the mathematical apparatus of the point mapping method is implemented to study the dynamics of the MM. The peculiarity in the research approach is that the point mapping is not formed by the classical method (mappings of the Poincaré surface into itself), but by the duration of relative rest of the vibration system, which greatly facilitated the process of point mapping and its detailed study. The presence of floating boundaries of slip-motion plates required the creation of an original approach in constructing point mapping and interpreting the results obtained. The structure of the phase portrait of the MM as a function of the characteristics of the sliding and state friction forces, as well as on the type and position of the limiter was studied using the developed research methodology and the created software product. Based on the character of changes in bifurcation diagrams, it was possible to determine the main laws of changes in the motion regimes (occurrence of random complexity via the period possible transfer to chaos doubling process) when changing the parameters of the vibration system (amplitude and frequency of periodic action, the form of the functional dependence describing the change in the coefficient of friction of relative rest. Analytical results with and without a vibration limiter are compared.

The Decomposition Method in Issues of Researching Electric-Power Systems with Dry Friction

Electrical engines are widespread in all areas of industry and technology. Electrical actuators are widely used as an actuating motor regulating the interconnected controls of different machines. Increased demands for precision in interconnected controls make it very important to consider dry friction in the executive mechanism. The consideration of dry friction in a mathematical model of an automatic system considerably complicates its analysis. In all known practical cases of research into such issues, the research is carried out by means of computing or analytically, but using a considerably simplified model of the dry-friction law. This simplified presentation does not allow one to uncover and understand the true (or apparent) reasons for loss of stability in a system accompanied by friction self-oscillations of different kinds (periodical, chaotic etc.). The decomposition method of parameter space presented in the current work allows investigation of nonautonomous nonlinear multivariable systems via basic subsystems (both linear and nonlinear) that have a lower-order state space and are amenable to strict analysis.

Identification of minimal parameters for optimal suppression of chaos in dissipative driven systems

Taming chaos arising from dissipative non-autonomous nonlinear systems by applying additional harmonic excitations is a reliable and widely used procedure nowadays. But the suppressory effectiveness of generic non-harmonic periodic excitations continues to be a significant challenge both to our theoretical understanding and in practical applications. Here we show how the effectiveness of generic suppressory excitations is optimally enhanced when the impulse transmitted by them (time integral over two consecutive zeros) is judiciously controlled in a not obvious way. Specifically, the effective amplitude of the suppressory excitation is minimal when the impulse transmitted is maximum. Also, by lowering the impulse transmitted one obtains larger regularization areas in the initial phase difference-amplitude control plane, the price to be paid being the requirement of larger amplitudes. These two remarkable features, which constitute our definition of optimum control, are demonstrated experimentally by means of an analog version of a paradigmatic model, and confirmed numerically by simulations of such a damped driven system including the presence of noise. Our theoretical analysis shows that the controlling effect of varying the impulse is due to a subsequent variation of the energy transmitted by the suppressory excitation.

Controllability of non-autonomous nonlinear differential system with non-instantaneous impulses

In this paper, we applied the Rothe’s fixed point theorem to study the controllability of non-autonomous nonlinear differential system with non-instantaneous impulses in the space $$\mathbb {R}^{n}$$ . Also, we established the sufficient conditions for the controllability of the integro-differential equation as well as nonlocal problem. Finally, we have given an example to illustrate the application of these proposed results.

Adaptive control of nonlinear fractional-order systems using T–S fuzzy method

Owing to the superior capability of fractional differential equations in modeling and characterizing accurate dynamical properties of many high technology real world systems, the design and control of fractional-order systems have captured lots of attention in recent decades. In this paper, an adaptive intelligent fuzzy approach to controlling and stabilization of nonlinear non-autonomous fractional-order systems is proposed. Since dynamic equations of applied fractional-order systems usually contain various parameters and nonlinear terms, the Takagi–Sugeno (T–S) fuzzy models with if-then rules are adopted to describe the system dynamics. Also, as the nonlinear system parameters are assumed to be unknown, adaptive laws are derived to estimate such fluctuations. Simple adaptive linear-like control rules are developed based on the T–S fuzzy control theory. The stability of the resulting closed loop system is guaranteed by Lyapunov’s stability theory. Two illustrative numerical examples are presented to emphasize the correct performance and applicability of the proposed adaptive fuzzy control methodology. It is worth to notice that the proposed controller works well for stabilization of a wide class of either autonomous nonlinear uncertain fractional-order systems or non-autonomous complex systems with unknown parameters.

Stochastic sensitivity analysis of nonautonomous nonlinear systems subjected to Poisson white noise

The stochastic sensitivity function (SSF) method is extended to estimate the stationary probability distribution around periodic attractors of nonautonomous nonlinear dynamical systems subjected to Poisson white noise in this paper. After deriving the stochastic sensitivity functions of period-N cycle of mapping systems based on the characteristic of Poisson process, non-autonomous dynamical systems around periodic attractors are converted to mapping systems by constructing a stroboscopic map, and then the stochastic sensitivity functions of periodic attractors of nonautonomous nonlinear systems can be obtained by adopting the results of mapping systems. It is found that the stochastic sensitivity functions depend on the product of noise intensity and the arrival rate of Poisson processes. To illustrate the validity of the proposed method, a Henon map driven by Poisson processes and a Mathieu–Duffing oscillator under Poisson white noise are studied.

Aeroelastic Stability Estimation of Control Surfaces with Freeplay Nonlinearity

This work discusses the quantitative stability evaluation of aeroelastic problems with freeplay in the control surfaces. Stability estimation of linear time invariant and linear time periodic systems rely on eigenanalysis of state transition matrices and implies simplifications on the problems governed by nonlinear non-autonomous equations. Lyapunov Characteristic Exponents directly provide quantitative information on the stability of nonlinear non-autonomous dynamical systems. Stability estimation using Lyapunov Characteristic Exponents does not require a special reference solution and is consistent with the eigensolution of linear time invariant and Floquet-Lyapunov analysis of linear time periodic systems. Thus, they represent a natural generalization of conventional stability analysis. The Discrete QR method is used to practically estimate the Lyapunov Characteristic Exponents. The method is applied to a three-dimensional aeroelastic problem with freeplay introduced to the control surface.

Asymptotic Stability of Nonzero Solutions of Discontinuous Systems of Impulsive Differential Equations

Discontinuous systems of nonlinear non-autonomous differential equations with impulsive effects are the main object of investigation in the paper. These systems consist of two basic parts: (i) A set of non-linear nonautonomous systems of ordinary differential equations that define the continuous parts of the solutions. The right-hand sides of the systems are elements of the set of functions f = { f1, f2, ...} ; (ii) The conditions which consistently determine “the switching moments”. The structural change (discontinuity) of the right-hand side and impulsive perturbations take place at the moments of switching. In these moments, the trajectory meets the “switching sets”. They are parts of the hyperplanes, situated in the phase space of the system considered. Sufficient conditions are found so that the nonzero solutions of the studied discontinuous system with impulsive effects are asymptotically stable.

Periodic solutions to nonlinear nonautonomous system of differential equations

We prove a theorem on the existence of nonzero periodic solution to a system of differential equations by the method of fixed point of nonlinear operator defined on a topological product of two compact sets.

A robust adaptive observer for a class of singular nonlinear uncertain systems

ABSTRACTThis paper proposes a robust adaptive observer for a class of singular nonlinear non-autonomous uncertain systems with unstructured unknown system and derivative matrices, and unknown bounded nonlinearities. Unlike many existing observers, no strong assumption such as Lipschitz condition is imposed on the recommended system. An augmented system is constructed, and the unknown bounds are calculated online using adaptive bounding technique. Considering the continuous nonlinear gain removes the chattering which may appear in practical applications such as analysis of electrical circuits and estimation of interaction force in beating heart robotic-assisted surgery. Moreover, a simple yet precise structure is attained which is easy to implement in many systems with significant uncertainties. The existence conditions of the standard form observer are obtained in terms of linear matrix inequality and the constrained generalised Sylvester's equations, and global stability is ensured. Finally, simulation results are obtained to evaluate the performance of the proposed estimator and demonstrate the effectiveness of the developed scheme.