Continuous Nonlinear Systems Research Articles

Hierarchical Decomposition of LTL Synthesis Problem for Nonlinear Control Systems

This paper deals with the control synthesis problem for a continuous nonlinear dynamical system under a Linear Temporal Logic (LTL) formula. The proposed solution is a top-down hierarchical decomposition of the control problem involving three abstraction layers of the problem, iteratively solved from the coarsest to the finest. The LTL planning is first solved on a small transition system only describing the regions of interest involved in the LTL formula. For each pair of consecutive regions of interest in the resulting accepting path satisfying the LTL formula, a discrete plan is then constructed in the partitioned workspace to connect these two regions while avoiding unsafe regions. Finally, an abstraction refinement approach is applied to synthesize a controller for the dynamical system to follow each discrete plan. The second main contribution, used in the third abstraction layer, is a new monotonicity-based method to over-approximate the finite-time reachable set of any continuously differentiable system. The proposed framework is demonstrated in simulation for a motion planning problem of a mobile robot modeled as a disturbed unicycle.

Dynamic Analysis of a Three-dimensional Non-linear Continuous System

Most physical phenomena are modeled as continuous or discrete dynamic systems of a second dimension or more, but because of the multiplicity of bifurcation parameters and the large dimension, researchers have big problems for the study of this type of systems. For this reason, this article proposes a new method that facilitates the qualitative study of continuous dynamic systems of three dimensions in general and chaotic systems in particular, which contains many parameters of bifurcations. This method is based on projection on the plane and on an appropriate bifurcation parameter.

A Special Kind of Sliding Mode Control for Nonlinear System With State Constraints

This paper presents a control strategy named auxiliary surfaces sliding mode control (AS-SMC) by using Positive Invariant Set (PIS) to control a class of continuous nonlinear systems with state constraints. The PIS can be regarded as a special kind of sliding surface, and its invariance ensures that the system state can satisfy the constraints in the convergence process. The stability analysis and a PIS theory proof are given. The control strategy is successfully tested in numerical simulations and even effectively applied to the coaxial unmanned helicopter flight control system on hardware in the loop (HIL) platform. The results verify the effectiveness of the proposed control strategy for the experimental set-up.

Adaptive Asymptotic Control for a Class of Uncertain Nonlinear Systems

This paper addresses the asymptotic tracking problem of adaptive neural control for a class of uncertain strict-feedback nonlinear systems. As a universal approximator, the neural network is widely utilized to solve the tracking control problem of unknown continuous nonlinear systems. Due to the existence of neural network approximation errors, previous neural network-based control approaches can only achieve the bounded tracking rather than the asymptotic tracking. This paper designs an asymptotic error eliminating term to achieve the adaptive neural asymptotic tracking. By utilizing the Lyapunov stability theory, all the variables of the resulting closed-loop system are proven to be semi-globally uniformly ultimately bounded, and the tracking error can converge to zero asymptotically by choosing design parameters appropriately. A simulation example is presented to show the effectiveness of the proposed control approach.

Adaptive fault tolerant control for time-varying delay system with actuator fault and mismatched disturbance

In this paper, the fault tolerant control (FTC) problem for a class of continuous Lipschitz nonlinear systems with actuator fault, mismatched disturbance, and time-varying delay is investigated. An adaptive integral sliding mode fault tolerant control (AISMFTC) technique is proposed. First, a composite observer is designed to estimate the states and disturbance simultaneously, by introducing a sliding mode observer, the actuator fault is reconstructed. Then, in accordance with the estimated information, an AISMFTC method is designed. In the AISMFTC scheme, two adaptive laws and compensators are contained to compensate the observer errors, deal with the effects of actuator fault and attenuate the disturbance. Furthermore, the H∞performance is considered for the robustness of the system. Finally, the effectiveness of the proposed method is illustrated by two examples.

Comparative Controlling of the Lorenz Chaotic System Using the SMC and APP Methods

The Lorenz chaotic system is based on a nonlinear behavior and this causes the system to be unstable. Therefore, two different controller models were developed and named as the adaptive pole placement and sliding mode control (SMC) methods for the establishment of continuous time nonlinear Lorenz chaotic system. In order to achieve this, an improved controller structure was developed first theoretically for both the controller methods and then tested practically using the numerical samples. During the establishment of adaptive pole placement method for the Lorenz chaotic system, various stages were applied. The nonlinear chaotic system was also linearized by means of Taylor Series expansion processes. In addition, the feedback matrix of the adaptive pole placement method was determined using linear Jacobian matrix. The chaotic system reached an equilibrium point by using both the SMC and adaptive pole placement methods; however the simulation results of the SMC had better success than adaptive pole placement control technique.

Nonlinear vibration isolation for fluid-conveying pipes using quasi-zero stiffness characteristics

Abstract Fluid-conveying pipes are always subjected to various excitations to cause unwanted vibrations. A quasi-zero stiffness system consisting of three linear springs is adopted as the nonlinear isolator to attenuate the transverse vibrations of fluid-conveying pipes induced by foundation excitations. A dynamic model of nonlinear forced vibration of the fluid-conveying pipe coupled with two nonlinear isolators is established for the nonlinear continuous system and validated by using two methods, Galerkin method and the finite difference method. The influence of the quasi-zero stiffness isolators on the vibration characteristics and vibration transmission of the pipe is investigated by analyzing the natural frequency, vibration mode, and nonlinear vibration response. The effects of flow speed of the fluid and the system parameters of the isolator are studied to evaluate the isolation performance. It is found that the quasi-zero stiffness isolator and fluid flow can shift several natural frequencies of vibration of the pipeline to the low-frequency region. When the linear stiffness of the vibration isolation is zero in the vertical direction, the first two modes of the bending vibration of the fluid-conveying pipe tend to become rigid mode. While achieving high-efficiency vibration isolation in the high-frequency region, the vibration in the low-frequency region is complicated. The flow speed of the fluid can deteriorate the performance of vibration isolation.

Filtering of T–S Fuzzy Systems With Nonuniform Sampling

The problem of asynchronous filtering of nonlinear systems is investigated in this paper. To facilitate analysis, a discrete-time Takagi-Sugeno fuzzy model with a fast and uniform period is introduced to approximate continuous nonlinear systems. A slow and nonuniform sampler between system and filter is proposed to overcome the contradictions between fast sampling period and limited bandwidth. The nonuniform sampling interval of sampler is subject to a Markov chain. To make full use of the partial information of sampling interval which is accessible to the filter, the asynchronous filter which is described by the hidden Markov model is introduced to reduce conservatism. By resorting to the augmentation approach and the Lyapunov functional method which depends on nonuniform sampling interval, some sufficient conditions for the existence of asynchronous filer are given to guarantee the stability and dissipativity of the augmented system. The gains of asynchronous filter are given via solving a set of linear matrix inequalities. Two examples are utilized to illustrate the validity of the developed filter design technique where the relationships between optimal dissipative performance indices and the mode synchronization rate are given.

Enhancement of Power System Transient Stability By Fractional Order Controlled STATCOM Tuned By PSO

This paper discusses the application of Fractional order PI controlled Static synchronous compensator for improvement rotor angle stability of inter connected power system. FACTS Controllers plays important role in enhancing the power system stability.Besides improving the stability margin of the power system it also aids the damping of inter area power oscillations. In the present work STATCOM is connected in multimachine power system .The dynamic response of the STATCOM is controlled by using fractional order controllers.The controller gains of the fractional controller are tuned by using PSO algorithm.It gives acceptable solutions to continuous non-linear systems with less computational effort. The performance of the proposed controller has been compared with integer order PI controllers at different locations of fault. In this paper a 3 machine 9 bus WSSC test power system is considered and simulated in MATLAB/SIMULINK.

Automated-Sampling-Based Stability Verification and DOA Estimation for Nonlinear Systems

This paper develops a new sampling-based method for stability verification of piecewise continuous nonlinear systems via Lyapunov functions. Depending on the nonlinear system dynamics, the candidate Lyapunov function and the set of states of interest, verifying stability requires solving complex, possibly non-convex or infeasible optimization problems. To avoid such problems, the proposed approach firstly distributes the verification of Lyapunov's inequality on a finite sampling of a bounded set of states of interest. Secondly, it extends the validity of Lyapunov's inequality to an infinite, bounded set of states by automatically exploiting local continuity properties. A sampling-based method for estimating the domain of attraction (DOA) via level sets of a validated Lyapunov function is also presented. Efficient state-space exploration is achieved using multi-resolution sampling and hyper-rectangles as basic sampling blocks. The operations that need to be performed for each sampling point in the state-space can be carried out in parallel, which improves scalability. The proposed methodology is illustrated for various benchmark examples from the literature.

Maximum Entropy Probability Density Principle in Probabilistic Investigations of Dynamic Systems.

In this study, we consider a method for investigating the stochastic response of a nonlinear dynamical system affected by a random seismic process. We present the solution of the probability density of a single/multiple-degree of freedom (SDOF/MDOF) system with several statically stable equilibrium states and with possible jumps of the snap-through type. The system is a Hamiltonian system with weak damping excited by a system of non-stationary Gaussian white noise. The solution based on the Gibbs principle of the maximum entropy of probability could potentially be implemented in various branches of engineering. The search for the extreme of the Gibbs entropy functional is formulated as a constrained optimization problem. The secondary constraints follow from the Fokker–Planck equation (FPE) for the system considered or from the system of ordinary differential equations for the stochastic moments of the response derived from the relevant FPE. In terms of the application type, this strategy is most suitable for SDOF/MDOF systems containing polynomial type nonlinearities. Thus, the solution links up with the customary formulation of the finite elements discretization for strongly nonlinear continuous systems.

Fault detection observer design for 2-D continuous nonlinear systems with finite frequency specifications

This research studies the problem of fault detection observer design for two-dimensional (2-D) continuous-time nonlinear systems in Takagi–Sugeno (T–S) form. Finite frequency (FF) specifications are used to design the observers, which makes observer designing different from previously proposed 2-D detection observers. Faults and disturbances are considered to be dominated in two different FF domain intervals. Fault sensitivity and disturbance robustness are measured by two FF performance indices, respectively. The aim of this paper is to design fault detection observers such that the residual error system has the sensitivity to faults and the desired robustness to disturbances. Sufficient conditions for the existence of a desired fault detection fuzzy observer are established in terms of linear matrix inequalities (LMIs). Simulation results indicate that faults can be detected effectively using the proposed method.

Event-Based Control for Networked T–S Fuzzy Systems via Auxiliary Random Series Approach

This paper presents an auxiliary random series approach to model the effect of network induced problems, such as data losses and transmission delay subject to event-based communication scheme for nonlinear continuous time systems. T-S fuzzy model is employed to describe the nonlinear systems. In order to save the bandwidth and energy, we introduce the event-triggered mechanism to reduce the number of data for transmission and computation. Thus, it is necessary to consider the influence of data losses, data disorder, and transmission delay since the transmitted data packets become more important. Consequently, it is very complicated to analyze the performance of such networked system and one of the most difficult part, in the authors' opinion, is to construct the mathematical model of closed-loop systems. In this paper, we present an auxiliary random series approach to describe the data transmitted in the system, and therefore, the closed-loop systems can be obtained. Associated with a tailor-made Lyapunov-Krasovskii functional, the stability analysis is processed and a fuzzy controller is designed. Asynchronous membership functions are considered to obtain more relaxed stability conditions. To clarify the effectiveness of the proposed method, a cart-damper-spring system is employed for simulation.

Sequent period-(2m − 1) motions to chaos in the van der Pol oscillator

In this paper, independent, symmetric, periodic motions in a van der Pol oscillator are predicted through a semi-analytical method. This semi-analytic method is based on the discretization of the corresponding continuous nonlinear system for an implicit mapping. Through the implicit mapping structures, stable and unstable periodic motions are obtained analytically. A sequence of periodic motions to chaos via 1(S) ◁ 3(S) ◁ ··· ◁ (2m − 1)(S) ◁ ··· is discovered. The stability and bifurcations of periodic motions are determined through eigenvalue analysis. The frequency–amplitude characteristics of periodic motions are discussed. Numerical simulations of the periodic motions are carried out for comparison of numerical and analytical results. Such a periodic motion sequence is for a better understanding of dynamics of the van der Pol oscillator.

Discretization of a Lyapunov Approach for Attraction Domain Estimation Through a BMI Optimization Problem

This paper proposes a Lyapunov method for estimating the asymptotic stability domain of polynomial discrete nonlinear systems. Based on an existing approach of attraction domain estimation for continuous nonlinear systems, we present an analogous technique focusing on discrete systems. The idea is to reformulate the problem as a bilinear matrix inequalities (BMI) optimization problem where the Lyapunov function is considered as a decision variable. The exploitation of polar coordinates and positive definite matrices properties defines the objective function and the constraint functions. One of the main advantages of this method is that the results are independent of the Lyapunov function. Furthermore, the obtained stability domain approaches the exact domain boundaries. An application example is illustrated to validate found results.

Event-triggered fuzzy filtering in networked control system for a class of non-linear system with time delays

ABSTRACTThis paper focuses on the design of H∞ filter and event-triggered scheme for a class of continuous non-linear networked control systems (NCSs) based on fuzzy system appeared with time delays. First, we consider the discrete event-triggered (ET) scheme to make efficient utilisation of bandwidth. Under this ET-scheme, sensor releases the data only when our sampled-data plant violates the specific event-triggered condition. Second, the T-S fuzzy system is used to model the non-linear NCSs. Another purpose of this paper is to design filters involving delays. Such filters have a more general form than the delay-free filters that have been mostly considered in the traditional studies. By using the time-delay system, co-design of event-triggered scheme and H∞ filter for the delayed NCSs is presented in a unified frame work. To choose the latest data packet and discard the dis-ordered packet logic, zero-order hold is inserted between the fuzzy filter and event generator. Then, by using the novel fuzzy Lyapunov–Krasovskii function approach with free-weighting matrix technique, H∞ filter design of event-triggered delay NCSs is proposed. Finally, to show the effective result of our co-design method, a tunnel-diode example is given.

Identification of Parameters of Nonlinear Dynamical Systems Simulated by Volterra Polynomials

We study the identification methods for the nonlinear dynamical systems described by Volterra series. One of the main problems in the dynamical system simulation is the problem of the choice of the parameters allowing the realization of a desired behavior of the system. If the structure of the model is identified in advance, then the solution to this problem closely resembles the identification problem of the system parameters. We also investigate the parameter identification of continuous and discrete nonlinear dynamical systems. The identification methods in the continuous case are based on application of the generalized Borel Theorem in combination with integral transformations. To investigate discrete systems, we use a discrete analog of the generalized Borel Theorem in conjunction with discrete transformations. Using model examples, we illustrate the application of the developed methods for simulation of systems with specified characteristics.

Adaptive NN finite-time tracking control of output constrained nonlinear system with input saturation

This paper considers the finite-time tracking control problem for the strict-feedback nonlinear continuous systems involving input saturation and output constraints. A sequence of desired and auxiliary virtual control signals and real control input is designed to derive a representation of the system estimation errors and stabilize the system. The proposed approach is further developed via a finite-time stability theory, barrier Lyapunov function, and neural network approximation scheme to achieve an expected performance of the considered system. According to the proposed scheme, we solve the finite-time tracking control problem of the nonlinear systems with input saturation. Then, a theorem is provided to address that all the signals and system states are bounded, and the system output is driven to track the reference signal in a finite time to a small neighborhood of zero and remains in the predefined compact sets. The effectiveness of the proposed scheme is confirmed via two simulation examples.

Functional observer-based fuzzy controller design for continuous nonlinear systems

ABSTRACTThis paper presents a novel method of synthesising controllers for nonlinear systems represented by Takagi--Sugeno fuzzy model using functional observer. The proposed method uses minimum order observer for estimating the control input of each rule as a function of states. Existence conditions of the functional observer-based controller are provided in the form of rank equality conditions. The stability conditions of the observer are obtained using quadratic Lyapunov function, and the conditions are presented in the form of linear matrix inequalities. Observer parameters can be obtained by solving these inequalities. Finally, systematic synthesis procedures for constructing the observer are presented. Examples with simulation output are given to demonstrate this method and to compare its performance with a conventional full-state observer-based controller.

Optimal Continuous Stochastic Control Systems with Incomplete Feedback: Approximate Synthesis

The sufficient e-optimality conditions for the control of the nonlinear continuous stochastic systems with incomplete feedback were formulated and proved. They enabled one to estimate the precision of approximate control as compared with the value-optimal performance functional. Relations were established to determine the e-optimal control, and a strategy was worked out to determine its use in minimizing the total mismatch of the resulting relations.