General Nonlinear Systems Research Articles

New trends in modeling and control of hybrid systems

New trends in modeling and control of hybrid systems

Decentralized Adaptive Command Filtered Neural Tracking Control of Large-Scale Nonlinear Systems: An Almost Fast Finite-Time Framework.

In this article, a decentralized adaptive finite-time tracking control scheme is proposed for a class of nonstrict feedback large-scale nonlinear interconnected systems with disturbances. First, a practical almost fast finite-time stability framework is established for a general nonlinear system, which is then applied to the design of the large-scale system under consideration. By fusing command filter technique and adaptive neural control and introducing two smooth functions, the "singular" and "explosion of complex" problems in the backstepping procedure are circumvented, while the obstacles caused by unknown interconnections are overcome. Moreover, according to the framework of practical almost fast finite-time stability, it is shown that all the closed-loop signals of the large-scale system are almost fast finite-time bounded, and the tracking errors can converge to arbitrarily small residual sets predefined in an almost fast finite time. Finally, a simulation example is presented to demonstrate the effectiveness of the proposed finite-time decentralized control scheme.

Asymptotic stabilization of general nonlinear fractional-order systems with multiple time delays

In this paper, two new control methods based on a Lyapunov-like function and a vector Lyapunov function separately were put forward to solve the asymptotic stabilization problem of general fractional-order nonlinear systems with multiple time delays. First, we deduced a new asymptotic stabilization control criterion based on a Lyapunov-like function after discussing the asymptotic stability criterion of general fractional-order nonlinear systems. Moreover, to address the problem of multiple time delays of the nonlinear system, we derived another asymptotic stabilization control criterion based on a vector Lyapunov function and an M-matrix via the new controller. Finally, the feasibility and effectiveness of the proposed controllers were verified by several common fractional-order nonlinear systems.

Dynamic reliability analysis of nonlinear structures using a Duffing-system-based equivalent nonlinear system method

To improve the analysis accuracy of dynamic reliability of a nonlinear system, an equivalent nonlinear system method is presented. In this method, general nonlinear systems are converted to equivalent Duffing nonlinear systems according to the minimum mean square error criterion, whose exact analytical solution of the random steady-state responses can be determined by a FokkerPlanckKolmogorov equation. Then the exact results of stochastic responses are used to analyze structural dynamic reliability. To use the equivalent nonlinear system method to analyze structural dynamic reliability is not only convenient for calculation but highly accurate. In addition, the presented equivalent nonlinear system has a parameter ε, which controls the degree of nonlinearity. Thus, it is easy to obtain the corresponding analysis results from converting the original system to equivalent nonlinear systems with different degrees of nonlinearity by changing the value of ε. Especially, when ε is zero, the equivalent nonlinear system will degenerate to the linear system, and leading to the analysis results of the equivalent linearization method. An example shows that the analysis results of the proposed equivalent nonlinear system method are reliable, and the calculation appears to be more accurate than that of the equivalent linearization method.

Finite-time stability analysis and stabilization by bounded linear time-varying feedback

This paper studies finite-time stability analysis and finite-time stabilization of linear systems by bounded linear time-varying feedback. On the one hand, (both local and global) finite-time stability of general nonlinear time-varying systems is investigated by the comparison principle and the notion of finite-time escaping functions. Some Lyapunov-like stability theorems are established. On the other hand, finite-time and prescribed-time stabilization of linear systems by bounded linear time-varying feedback are revisited based on the proposed finite-time stability theorems. Two classes of new time-varying high-gain functions are proposed to reduce the regulation time. Some connections of the proposed results to existing results on Lyapunov-inequality based finite-time stability analysis and nonlinear feedback based finite-time stabilization are revealed. The effectiveness of the proposed methods is illustrated by a numerical example.

A New Adaptive DS-Based Finite-Time Neural Tracking Control Scheme for Nonstrict-Feedback Nonlinear Systems

This article addresses the problem of adaptive finite-time neural tracking control for nonstrict-feedback nonlinear systems via dynamic surface (DS) technique. First, a new quasi-fast finite-time practical stability (QFPS) criterion is proposed for a class of general nonlinear systems. Then, the presented QFPS criterion is applied to design the desired adaptive finite-time neural tracking controller for a class of nonstrict-feedback nonlinear systems. The presented design scheme for the nonstrict-feedback nonlinear system has the following two features: 1) the “explosion of complexity” issue of the backstepping design is addressed by utilizing the DS technique and 2) the structural feature of Gaussian functions is applied to solve the design difficulties caused by the nonstrict-feedback form. It is proved that the designed controller for the nonstrict-feedback nonlinear system can make the resulting closed-loop system stabilizable in a quasi-fast finite time and the tracking error converges to a sufficiently small neighborhood of the origin. Finally, the simulation results are given to show the validity and practicability of the proposed design scheme.

Particle filtering for a class of cyber-physical systems under Round-Robin protocol subject to randomly occurring deception attacks

In this paper, the particle filtering problem is studied for a class of general nonlinear cyber-physical systems with non-Gaussian noises under Round-Robin protocol (RRP) subject to the randomly occurring deception attacks. In order to prevent the data from collisions and alleviate the communication overhead for the shared network with limited resources, the RRP is introduced in the sensor-to-filter channel to schedule the multiple sensors with a predefined transmission order. Under the RRP, only one sensor can be granted the access to the shared channel for measurement transmission at each time instant. A Bernoulli-distributed stochastic variable is utilized to describe the characteristic of random occurrence of deception attacks initiated by the adversaries. A RRP-based particle filtering algorithm is developed by establishing a modified likelihood function, where the statistical property of the randomly occurring deception attacks is exploited and the RRP-induced effect on the filter design is reflected. Finally, an illustrative example regarding the target tracking problem is provided to verify the feasibility and effectiveness of the developed particle filtering scheme.

Output feedback robust stabilisation for uncertain non‐linear systems with dead‐zone input

This study considers the output feedback robust stabilisation problem of non-linear systems with actuator non-symmetric dead zone and uncertain non-linearities including unmeasured states. Instead of constructing the dead-zone inverse, a robust control scheme is proposed. In particular, an input-driven observer is constructed, which uses the scaling gain to dominate non-linear terms. Based on non-separation principle and backstepping method, a time-varying and smooth output feedback controller is obtained to ensure that the resulting closed-loop systems are globally asymptotically stable. Also, the control scheme is generalised to more general non-linear systems in a lower triangular form. Finally, two examples are given to illustrate the effectiveness of the authors' results.

Pth moment exponential stability of general nonlinear discrete-time stochastic systems

pth moment exponential stability of general nonlinear discrete-time stochastic systems

A Unified Framework for Asymptotic Observer Design of Fuzzy Systems With Unmeasurable Premise Variables

This article develops a unified framework to design fuzzy-model-based observers of general nonlinear systems for both discrete-time and continuous-time cases. This observer problem is known as a challenging task due to the mismatch caused by the unmeasurable premise variables. To deal with this major challenge, we propose to rewrite the nonlinear system as a specific fuzzy model with two types of local nonlinearities: measurable and unmeasurable. Then, a differential mean value theorem for vector-valued functions is applied to local unmeasurable nonlinearities. This allows to represent the estimation error dynamics in a special polytopic form involving measurable membership functions and unknown but bounded time-varying parameters. Using Lyapunov-based arguments, design conditions in terms of linear matrix inequalities are derived to guarantee the asymptotic convergence of the estimation error. Three illustrative examples are given to demonstrate the interests of the new fuzzy observer framework in reducing: 1) the design conservatism and 2) the numerical complexity of the fuzzy observer structure for real-world applications.

Fractional-order adaptive fault-tolerant control for a class of general nonlinear systems

This paper proposes a series of fractional-order control methods (FOCMs) based on fractional calculus (FC) for a class of general nonlinear systems. In order to deal with the nonlinearities and uncertainties caused by both external and internal factors, the designed control schemes are adaptive, robust, fault-tolerant and do not involve detailed information of the system model. Besides, FC is combined to improve the control performance, especially in higher control accuracy, better anti-interference ability and stronger robustness. For a comprehensive consideration of the practical systems, three different actuator conditions are separately discussed, and the FOCMs are established aiming at these three different situations, respectively, and proved by theoretical analysis. The inverted pendulum system is adopted as simulation object, and the fractional-order schemes are verified and compared with integer-order controller and traditional PID controller. Simulation results make it clear that the proposed FOCMs are superior to other two schemes in control precision, robustness and anti-interference ability.

Robust Stabilization And Control Of Takagi-Sugeno Fuzzy Systems With Parameter Uncertainties And Disturbances Via State Feedback And Output Feedback

Most of the systems in the industry contain extreme non-linearity and uncertainties, which are hard to design and control utilizing general nonlinear systems. To conquer this sort of troubles, different plans have been produced in the most recent two decades, among which a popular methodology is Takagi-Sugeno fuzzy control. In this article, we present robust stabilization and control of Takagi-Sugeno (T-S) fuzzy systems with parameter uncertainties and disturbances. Initially, Takagi and Sugeno (TS) fuzzy model is used to represent a nonlinear system. Based on this T-S fuzzy model, fuzzy controller design schemes for state feedback and output feedback is also developed. Then, necessary conditions are derived for robust stabilization in the intelligence of Lyapunov asymptotic stability and are expressed in the arrangement of linear matrix inequalities (LMIs). The proposed system is implemented in the working platform of MATLAB and the simulation results are provided to illustrate the effectiveness of the proposed methods.

A multi-observer based estimation framework for nonlinear systems under sensor attacks

For general discrete-time nonlinear systems, we address the state estimation problem under sensor attacks. We provide a general estimation scheme, built around the idea of sensor redundancy and multi-observers, capable of reconstructing the system state in spite of sensor attacks and noise, for a large class of nonlinear plants and observers. This scheme has been proposed by others for linear systems and here we propose a unifying framework for a much larger class of nonlinear systems.

On universal stabilization property of Interconnection and Damping Assignment Control

It is well-known that the widely popular Interconnection and Damping Assignment Passivity-based Control (IDA-PBC) is “universally stabilizing”, in the sense that it generates all asymptotically stabilizing controllers for general nonlinear systems of the form ẋ=f(x,u). Unfortunately, the proof of this fact relies in the construction of a control law that is not globally defined. Although a standard regularization procedure may be used to enforce the required smoothness properties to the control signal, this procedure is quite technical and not constructive. To overcome this problem we provide an alternative construction that yields a continuous IDA-PBC law.

Extremum-seeking control for optimization of time-varying steady-state responses of nonlinear systems

Extremum-seeking control (ESC) is a useful tool for the steady-state performance optimization of plants for which limited knowledge about its dynamical behavior and disturbance situation is known. The case when the steady-state plant responses correspond to equilibrium solutions received a lot of attention. However, in many industrial applications plant performance is characterized by time-varying steady-state behavior. In those cases, no static parameter-to-steady-state performance map can be defined. In this work, we propose an ESC method that employs a so-called dynamic cost function to cope with time-varying steady-state responses of general nonlinear systems. We prove semi-global practical asymptotic stability of the closed-loop ESC scheme in the presence of bounded and time-varying external disturbances. Moreover, the working principle is illustrated by means of the real-time performance optimal tuning of a nonlinear control strategy for a motion control application.

Adaptive reduced order modeling for nonlinear dynamical systems through a new a posteriori error estimator: Application to uncertainty quantification

SummaryMultiquery problems such as uncertainty quantification (UQ), optimization of a dynamical system require solving a differential equation at multiple parameter values. Therefore, for large systems, the computational cost becomes prohibitive. This issue can be addressed by using a cheaper reduced order model (ROM) instead. However, the ROM entails error in the solution due to approximation in a lower dimensional subspace. Moreover, the ROM lacks robustness over a wide range of parameter values. To address these issues, first, an upper bound on the norm of the state transition matrix is derived. This bound, along with the residual in the governing equation, are then used to develop an error estimator for general nonlinear dynamical systems. Furthermore, this error estimator is used in conjunction with the modified greedy search algorithm proposed by Hossain and Ghosh (Int J Numer Methods Eng, 2018;116(12‐13): 741‐758) to adaptively construct a robust proper orthogonal decomposition‐based ROM. This adaptive ROM is subsequently deployed for UQ by invoking it in a statistical simulation. Two numerical studies: (i) viscous Burgers' equation and (ii) beam on nonlinear Winkler foundation, showed an improved accuracy of the error estimator compared to the current literature. A significant computational speed‐up in UQ is achieved.

Bursting dynamics in parametrically driven memristive Jerk system

Compared with general nonlinear systems, multi-time scale system has complex bursting dynamics and has received widespread attention. A memristor-based Jerk system with parametric excitation is proposed in this study. As the selected excitation frequency is far less than the natural frequency, implying the existence of an order gap between the excitation frequency and the natural one, the system can be considered as a classic fast-slow system with two timescales. In our system, when the slow-varying parameters periodically pass through the critical pitchfork bifurcation point periodically, a distinct time delay behavior can be observed. Complex bursting oscillations induced by the delayed pitchfork are revealed with different excitation amplitudes. By virtue of the fast-slow analysis method, the corresponding generation mechanisms are discussed by the transformed phase portraits, the time series, and the phase portraits. As the delay time interval induced by the pitchfork bifurcation is dependant not only on the excitation amplitude, but also on the excitation frequency, some excitation frequency related bursting patterns are also considered in our study. Finally, numerical simulations are provided to verify the validity of the study.

Robust Intelligent Control of SISO Nonlinear Systems Using Switching Mechanism.

In this article, a robust adaptive learning control strategy for uncertain single-input-single-output systems in strict-feedback form and controllability canonical form (CCF) is studied. For the strict-feedback system, the dynamic surface control is introduced while for the controllability canonical system, sliding-mode control is further constructed. The finite-time design is introduced for fast convergence. Under the switching mechanism, the intelligent design and the robust technique work together to obtain robust tracking performance. Once the states run out of the domain of intelligent control, the robust item will pull the states back while inside the neural working domain, the composite learning is developed to achieve higher approximation precision by building the prediction error for the weight update. The closed-loop system stability is analyzed via the Lyapunov approach. Especially for the CCF, the finite-time convergence is achieved while the system signals are globally uniformly ultimately bounded. Simulation studies on the general nonlinear systems and the flight dynamics show that the new design scheme obtains better tracking performance with higher precision and stronger robustness.

An accurate spectral collocation method for nonlinear systems of fractional differential equations and related integral equations with nonsmooth solutions

This paper aims to provide a rigorous analysis of exponential convergence of an adaptive spectral collocation method for a general nonlinear system of rational-order fractional initial value problems. The key idea of the proposed method is to adopt a smoothing transformation for the spectral collocation method to circumvent the curse of singularity at the beginning of time. As such, the singularity of the numerical approximation can be tailored to that of the singular solutions to a class of fractional initial value problems, leading to spectrally accurate approximation. Numerical examples are presented, which verify the theoretical predictions and demonstrate that the new formulation of the spectral method leads to better performance compared to other known numerical approaches with a relatively smaller number of degree of freedoms.

Identifying explicit expression of response probability density of nonlinear stochastic system: Information-theoretic method

For a general nonlinear stochastic system, the response probability density possesses crucial significance for response evaluation and reliability design. The existing literatures derive the response probability density by accurately or approximately solving the Fokker-Planck- Kolmogorov equation, by searching for the equivalent system according to some criterion and approximating the probability density of the original system by that of the equivalent one, or by numerically solving the original equation of motion and giving discrete information statistically. Herein, a novel procedure is established to identify the explicit expression of the stationary response probability density of a general nonlinear system directly from the discrete data of samples. The stationary probability density is expressed as an exponent form with the exponential power being linear combination of base functions elaborately selected. The Shannon information entropy is then arranged as a nonlinear function of the statistical moments of base functions which are evaluated by the discrete data of samples. The undetermined coefficients of base functions are finally determined by minimizing the Shannon information entropy. Three typical examples, i.e., Duffing oscillator, van der Pol system and a frictional system (as representatives of smooth systems, electric systems and non-smooth systems, respectively), are investigated to illustrate the accuracy of this procedure, the robustness to data noise, the insensitivity to sampling interval and strength of nonlinearity, and the low requirement on the amount of data.