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This study addresses the adaptive tracking control problem for a class of time-delay systems in strict-feedback form with unknown control gains and uncertain actuator non-linearity. The actuator non-linearity can be either backlash or dead zone, and the proposed approach does not require the knowledge of the bounds of non-linearity parameters. By applying an appropriate Lyapunov–Krasovskii functional and utilising the property of the well-defined trigonometric functions, the problems of time delay and controller singularity are avoided. The feasibility of using a static neural network to attenuate the effect of actuator non-linearity is proved with the aid of intermediate value theorem. Furthermore, it is proved that all closed-loop signals are bounded and the tracking error converges to a small residual set asymptotically. Two simulation examples are provided to demonstrate the effectiveness of the designed method.

This paper is concerned with the chaotic dynamics of a rotating pendulum system with bistable characteristics subjected to a viscous damping and a harmonic forcing. As a prototype of the single-degree-of-freedom system with bistable characteristics, this pendulum system exhibits a transition from smooth to discontinuous dynamics by changing a geometrical parameter. The dynamic behaviors of the unperturbed system with irrational nonlinearity bear significant similarities to the coupling of a simple pendulum and the smooth and discontinuous (SD) oscillator with the coexistence of the standard homoclinic orbits of Duffing type and pendulum type and the coexistence of the nonstandard homoclinic orbits of SD type and pendulum type in the smooth and discontinuous case, respectively. For the perturbed smooth system, we present an approximate technique to analytically obtain the lower bound line for horseshoes chaos arising from the homoclinic orbits of Duffing-type and Pendulum-type tangling, which overcomes the natural difficulties of solving the analytical expression of the homoclinic orbits and calculating the complicated Melnikov integrals. The chaotic thresholds of the perturbed discontinuous system are calculated by applying the numerical technique due to its non-smooth feature. Numerical simulations are carried out to certify the chaotic thresholds, which show the efficiency of the proposed techniques and demonstrate the predicated chaotic motions. Finally, different types of chaotic motions are illustrated via the cylindrical phase portraits. The contribution of this study is also helpful for exploring the dynamical behaviors of the complex nonlinear dynamical system containing the standard homoclinic or heteroclinic orbit in terms of the quantitative calculation.

In this paper, an adaptive fast terminal sliding mode control technique combined with a global sliding mode control scheme is investigated for the tracking problem of uncertain nonlinear third-order systems. The proposed robust tracking controller is formulated based on the Lyapunov stability theory and guarantees the existence of the sliding mode around the sliding surface in a finite time. Under the uncertainty and nonlinearity effects, the reaching phase is removed and the chattering phenomenon is eliminated. This scheme guarantees robustness against nonlinear functions, parameter uncertainties and external disturbances. The derivative of the state variable is replaced by a delay term in the form of an Euler approximation of the derivative function. Furthermore, the knowledge of upper bounds of the system uncertainties is not required, which is more flexible in the real implementations. Simulation results are presented to show the effectiveness of the suggested method.

This paper presents a robust tracking control strategy for servo systems with unknown backlash, employing adaptive parameter identification to address performance degradation caused by backlash nonlinearities. In high-precision positioning and rapid-response applications, backlash significantly compromises system performance. To address this challenge, a servo system model incorporating backlash nonlinearities is developed, and a novel adaptive inverse function is introduced for backlash compensation. The estimation error of unidentified parameters is indirectly obtained through the design of a state observer. Minimizing the estimation error facilitates the accurate identification of model parameters, encompassing those associated with backlash. Additionally, an adaptive law is designed to estimate the unknown upper bounds of disturbance dynamics. Then, a robust tracking controller is proposed, which dynamically adjusts control inputs in real time based on identified backlash parameters to counteract backlash-induced adverse effects. Theoretical analysis and simulation results demonstrate that the proposed strategy significantly improves tracking performance in servo systems with unknown backlash.

Time-variant nonlinear problems have always been a kind of complex research object in the field of control. The accuracy and efficiency of settling time-variant nonlinear inequality-equation (NIE) systems are often affected by the nonlinearity degree of the systems, and there are currently no complete algorithms to settle the time-variant NIE systems effectively. To settle this class of complex systems effectively, time-variant NIE systems are first equivalently transformed into a time-variant equation by introducing a nonnegative variable. Then, through the idea of zeroing neural network (ZNN) and the role of time-variant parameter-gain functions, a parameter-gain accelerated ZNN (PGAZNN) model is proposed to solve time-variant NIE systems. Theoretically, the stability of the proposed PGAZNN model is proved by strict mathematical analysis. In addition, the PGAZNN model can achieve fixed-time convergence, and the upper-bound of convergence time is estimated. Finally, numerical simulation example and symmetry trajectory tracking are given to verify the validity and correctness of the proposed PGAZNN model.

This paper proposes a new generalized adaptive gain sliding mode observer (GAGSMO) for estimating the unavailable states of a class of multi-input multi-output uncertain nonlinear systems. To further improve the estimation performance of conventional sliding mode observer, the observer gains of GAGSMO are designed for the first time as the generalized bounded positive functions of the available output errors and the upper bounds of disturbance terms. Due to the features of the designed observer gains, the GAGSMO has stronger robustness than the conventional sliding mode observer in the presence of system uncertainties and nonlinearities. The finite-time error convergence of GAGSMO is proved by the Lyapunov stability theorem in conjunction with the introduced mapping functions. Then, by catching sight of the inherent feature of sliding motion, a recursive mechanism based only on available estimation information is formulated to update the designed observer gains online in the sliding mode stage. With the recursive mechanism, the chattering level of GAGSMO is minimized, and the estimation accuracy of GAGSMO is further improved. The effectiveness and excellent performance of the proposed GAGSMO are illustrated with two numerical examples.

A Lawson-type exponential integrator combined with the Fourier pseudo-spectral method is provided for the nonlinear Double Sine-Gordon equation (DSGE), while the nonlinearity is characterized by β / ϵ with small parameter ϵ ∈ ( 0 , 1 ] and interaction parameter β ∈ ( 0 , + ∞ ) . In comparison to the Sine-Gordon equation, DSGE has many properties of solitons as well as its own unique new features. This is the first work to numerically simulate the physical phenomena arising from DSG kinks collisions. The improved uniform error bounds are proved by using the regularity compensation oscillatory (RCO) technique, which are O ( α 2 τ + h m ) up to the long time at T ϵ = T / α 2 , where α = ϵ for β ≥ 1 and α = ϵ / β for 0 < ϵ < β < 1 . Based on the uniform bounds, the error estimation for the discrete energy is derived. Furthermore, the improved error bounds are extended to two oscillatory DSGEs with O ( ϵ 2 ) and O ( ϵ 2 / β 2 ) wavelength in time. Numerical examples are provided to illustrate the accuracy and discrete energy property of the proposed method.

In this study, the design of an adaptive neural network-based fixed-time control system for a novel coaxial trans-domain hybrid aerial–underwater vehicle (HAUV) is investigated. A radial basis function neural network (RBFNN) approximation strategy-based adaptive fixed-time terminal sliding mode control (AFTSMC) scheme is proposed to solve the problems of the dynamic nonlinearity, model parameter perturbation, and multiple external disturbances of coaxial HAUV trans-media motion. A complete six-degrees-of-freedom model for a continuous water–air cross-domain model is first established based on the hyperbolic tangent transition function, and, subsequently, based on a basic framework of FTSMC, a fixed-time and fast-convergence controller is designed to track the target position and attitude signals. To reduce the dependence of the control scheme on precise model parameters, an RBFNN approximator is integrated into the sliding mode controller for the online model identification of the aggregate uncertainties of the coaxial HAUV, such as nonlinear unmodeled dynamics and external disturbances. At the same time, an adaptive technique is used to approximate the upper bound of the robust switching term gain in the controller, which further offsets the estimation error of the RBFNN and effectively attenuates the chattering effect. Based on Lyapunov stability theory, it is proven that the tracking error can converge in a fixed time. The effectiveness and superiority of the proposed control strategy are verified by several sets of simulation results obtained under typical working conditions.

We consider measures of nonlinearity (MoNs) of a polynomial curve in two-dimensions (2D), as previously studied in our Fusion 2010 and 2019 ICCAIS papers. Our previous work calculated curvature measures of nonlinearity (MoNs) using (i) extrinsic curvature, (ii) Bates and Watts parameter-effects curvature, and (iii) direct parameter-effects curvature. In this paper, we have introduced the computation and analysis of a number of new MoNs, including Beale’s MoN, Linssen’s MoN, Li’s MoN, and the MoN of Straka, Duník, and S̆imandl. Our results show that all of the MoNs studied follow the same type of variation as a function of the independent variable and the power of the polynomial. Secondly, theoretical analysis and numerical results show that the logarithm of the mean square error (MSE) is an affine function of the logarithm of the MoN for each type of MoN. This implies that, when the MoN increases, the MSE increases. We have presented an up-to-date review of various MoNs in the context of non-linear parameter estimation and non-linear filtering. The MoNs studied here can be used to compute MoN in non-linear filtering problems.

This paper investigates the optimized distributed fusion filtering (DFF) problem for a class of nonlinear discrete time-varying stochastic systems with randomly occurring uncertainty (ROU) and missing measurements (MMs). The stochastic nonlinearity is depicted in terms of statistical means. The phenomena of the ROU and MMs are considered during the modelling of state equation and measurement output respectively, which are characterized by Bernoulli distributed random variables. In order to deal with the effect of the parameter uncertainty, the method that the local estimation error covariances and cross-covariances from all estimators at every sample time are replaced by their upper bounds is adopted. Moreover, the minimum upper bounds for each filtering error covariance (FEC) are obtained by designing the corresponding filter gains. Based on the local filters, a new robust DFF algorithm is developed via the matrix-weighted fusion method. In addition, a sufficient condition concerning on the performance analysis of the developed algorithm is given, which can show that the boundedness of the upper bound for each FEC is guaranteed. Finally, a numerical example is provided to manifest the usefulness of the developed distributed fusion algorithm.