Nonlinear Time-varying Systems Research Articles

Passive Fuzzy Controller Design for the Parameter-Dependent Polynomial Fuzzy Model

This paper discusses a passive control issue for Nonlinear Time-Varying (NTV) systems subject to stability and attenuation performance. Based on the modeling approaches of Takagi-Sugeno (T-S) fuzzy model and Linear Parameter-Varying (LPV) model, a Parameter-Dependent Polynomial Fuzzy (PDPF) model is constructed to represent NTV systems. According to the Parallel Distributed Compensation (PDC) concept, a parameter-dependent polynomial fuzzy controller is built to achieve robust stability and passivity of the PDPF model. Furthermore, the passive theory is applied to achieve performance, constraining the disturbance effect on the PDPF systems. To develop the stability criteria, by introducing a parameter-dependent polynomial Lyapunov function, one can derive some stability conditions, which belong to the term of Sum-Of-Squares (SOS) form. Based on the Lyapunov function, two stability criteria are proposed to design the corresponding PDPF controller, such that the NTV system is robustly stable and passive. Finally, two examples are applied to demonstrate the effectiveness of the proposed stability criterion.

Practical semiglobal uniform exponential stability of nonlinear nonautonomous systems

UDC 517.9 We solve the following twofold problem: In the first part, we deduce Lyapunov sufficient conditions for practical uniform exponential stability of nonlinear perturbed systems under different conditions for the perturbed term. The second part presents a converse Lyapunov theorem for the notion of semiglobal uniform exponential stability for parametrized nonlinear time-varying systems. We establish the possibility of application of a perturbed parametrized system, by using Lyapunov theory, to the investigation of the robustness properties that may provide practical semiglobal uniform exponential stability with respect to perturbations.

Event-triggered tracking control for nonlinear time-varying delay systems with full state constraints and unknown control coefficients

This paper addresses the problem of event-triggered control for a class of nonlinear systems with unknown control coefficients, time-varying delays, full state constraints and external disturbances, simultaneously. Firstly, based on integral Barrier-Lyapunov Functionals, an adaptive event-triggered controller is designed to account for the impacts of complete state constraints, unknown time-varying delays and external disturbances. Using fuzzy logic systems, the unknown functions of systems are then roughly estimated. The system redundancy can be significantly reduced by implementing the event-triggered control technique in the meanwhile. By introducing the separation technique and Lyapunov–Krasovskii functionals, it is shown that the strategy can ensure tracking performance greatly and all the closed-loop signals of the system are bounded. Compared with the existing results, the proposed design scheme is less conservative and has a wilder application range. Finally, the simulation results show the effectiveness of the proposed approach.

Non-uniform Trajectory Tracking AILC for Nonlinear Time-varying Parameter System with Multiple Unknown Control Directions

In this paper, a new iterative learning controller is presented for nonlinear time-varying parameter system to solve the tracking problem of different target trajectories. Based on Nussbaum function and the Lyapunov-like synthesis design the learning controller to handle system dynamics with multi-unknown control directions. Over a finite time interval, the unknown time-varying parameter is considered to be periodic, so it is expanded using a Fourier series expansion, and the remaining terms are treated with a canonical series. The controller designed in this paper can ensure that all signals of the closed-loop system are bounded in a finite time interval [0,T], and can complete non-uniform target tracking. Finally, a simulation example is given to verify the effectiveness of the designed controller.

Discrete sliding mode control method for particle swarm optimization-based brushless DC motor of electric vehicle

Brushless DC motor (BLDCM) is a multivariable nonlinear time-varying system, which is difficult to control. The discrete sliding mode control method for BLDCM of electric vehicle on the basis of particle swarm optimization (PSO) is studied to improve the application of BLDCM in electric vehicle. The mathematical model of BLDCM of electric vehicle is established using the state formula. Based on the mathematical model of BLDCM, through the analysis of electromagnetic torque control of BLDCM, it is clear that controlling the angle between rotor flux and stator flux can accurately control the electromagnetic torque of BLDCM. The adaptive discrete sliding mode controller (SMC) is set to control the electromagnetic torque of BLDCM of electric vehicle, and the PSO algorithm is adopted to obtain the optimal parameters of the adaptive discrete SMC to realize the discrete sliding mode control of BLDCM of electric vehicle. According to experimental results, the proposed method can achieve the accurate control of torque and speed of BLDCM of electric vehicle, and increase the application of BLDCM in electric vehicle.

Distributed fault estimation over sensor networks: A bit rate allocation scheme

This paper investigates the issue of fault estimation (FE) in a distributed way for nonlinear time-varying systems via a digital communication channel. In order to relieve the communication limitation, a bit rate allocation scheme is made using data transmitted from sensors to estimators. Next, distributed estimators are designed in terms of the obtained decoded measurement outputs such that an upper bound of estimation error covariance (EEC) is ensured by resorting to the well-known mathematical induction approach. Moreover, the desired gains of constructed estimators are calculated at each iteration by minimizing the upper bound. Subsequently, a sufficient condition is presented to check the uniform boundedness of the designed recursive approaches of the distributed FE. Moreover, the monotonicity with respect to the bit rate is discussed for the fixed estimator gains. An example of Chua's circuits is finally presented to check the applicability of the developed distributed strategy.

Policy Iteration for Optimal Control of Discrete-Time Time-Varying Nonlinear Systems

Aimed at infinite horizon optimal control problems of discrete time-varying nonlinear systems, in this paper, a new iterative adaptive dynamic programming algorithm, which is the discrete-time time-varying policy iteration (DTTV) algorithm, is developed. The iterative control law is designed to update the iterative value function which approximates the index function of optimal performance. The admissibility of the iterative control law is analyzed. The results show that the iterative value function is non-increasingly convergent to the Bellman-equation optimal solution. To implement the algorithm, neural networks are employed and a new implementation structure is established, which avoids solving the generalized Bellman equation in each iteration. Finally, the optimal control laws for torsional pendulum and inverted pendulum systems are obtained by using the DTTV policy iteration algorithm, where the mass and pendulum bar length are permitted to be time-varying parameters. The effectiveness of the developed method is illustrated by numerical results and comparisons.

Asymptotic stability for nonlinear time-varying systems on time scales

This paper is concerned with nonlinear time-varying systems on time scales, which can cover not only trivial continuous and discrete dynamical systems, but also some other nontrivial cases. Some less conservative sufficient conditions for asymptotic stability of nonlinear systems are derived. Compared with the traditional stability results, the negative definiteness of the time derivatives of Lyapunov functions is weakened, that is, the time derivatives of related Lyapunov functions in this paper are allowed to be indefinite. Two examples about trivial and nontrivial systems are included to illustrate the effectiveness of the results obtained.

Data-driven solutions to spacecraft relative attitude-position fault-tolerant control

In this paper, the previous work of the authors on Subspace Predictive Control (SPC) in the context of spacecraft roto-translational relative motion is pursued, and the introduced novel SPC-based approach to fault-tolerant control (FTC) of nonlinear time-variant systems is further investigated. Firstly, the effectiveness of the proposed SPC-based framework for adaptive control allocation is exploited. Subsequently, separating nonlinear control and control allocation blocks yields a new layered SPC-based control scheme. Furthermore, designating a dedicated controller for each of the coupled motions leads to a novel distributed SPC-based architecture. Accordingly, three complete model-free fault-tolerant controllers for coupled nonlinear time-variant plants are developed, which solely need the knowledge of the occurrence time of faults as prerequisite. An internal fault diagnosis capability is also introduced, which makes the framework completely self-sufficient. Finally, the proposed framework is verified by challenging simulated scenarios.

A generalization of practical stability of nonlinear impulsive systems

Abstract In this paper, we introduce a new type of stability for nonlinear impulsive systems of differential equations, namely practical h-stability. By using the Lyapunov stability theory, some sufficient conditions which guarantee practical h-stability are established. Our original results generalize well-known fundamental stability results, practical stability, practical exponential stability and practical asymptotic stability for nonlinear time-varying impulsive systems. Then two classes of nonlinear impulsive systems, namely perturbed and cascaded impulsive systems, are discussed. Furthermore, the problem of practical h-stabilization for certain classes of nonlinear impulsive systems is considered. Finally, two numerical examples are given to show the effectiveness of our theoretical results.

A Nonlinear Filter by Pseudo-Formal Linearization for Nonlinear Time-Varying Systems

This paper is concerned with a nonlinear filter for a general class of nonlinear time-varying systems, by making use of a pseudo-formal linearization method and Kalman filter approach. The given nonlinear time-varying dynamic and measurement equations are piecewisely linearized by the formal linearization method using Taylor expansion so that Kalman filter can be applied for each linearized time-varying equations. Then the resulting filters are smoothly united into a single nonlinear filter by an automatic choosing function. The efficiency of this filter is shown through numerical examples.

On Averaging Technique for Nonlinear Time-Varying Systems without Local Lipschitz Continuity

The majority of literature for averaging methods deal with a class of nonlinear time-varying (NLTV) dynamics and their time-invariant averaged systems with locally Lipschitz continuous (LLC) condition. In this work, a class of uniformly continuous NLTV systems and their uniformly continuous averaged systems are considered. The first result shows the closeness of solutions between the original NLTV system and its averaged system with a sufficiently small time-scale separation parameter ε on a subset of a time interval, in which both systems have well-defined solutions. If the averaged system is finite-time stable (FTS), the second result shows that the original NLTV system will uniformly converge to an arbitrarily small neighborhood of the origin on a finite-time interval with sufficiently small ε. Simulation results support the theoretical finding.

Discrete-time Flatness and Linearization along Trajectories

The paper studies the relation between a nonlinear time-varying flat discrete-time system and the corresponding linear time-varying systems which are obtained by a linearization along trajectories. It is motivated by the continuous-time case, where it is well-known that the linearization of flat systems along trajectories results in linear time-varying systems which are again flat. Since flatness implies controllability, this constitutes an important verifiable necessary condition for flatness. In the present contribution, it is shown that this is also true in the discrete-time case: We prove that the linearized system is again flat, and that a possible flat output is given by the linearization of a flat output of the nonlinear system. Analogously, the map that describes the parameterization of the system variables of the linear system by this flat output coincides with the linearization of the corresponding map of the nonlinear system. The results are illustrated by two examples.

On the Absence of Spurious Local Trajectories in Time-Varying Nonconvex Optimization

In this article, we study the landscape of an online nonconvex optimization problem, for which the input data vary over time and the solution is a trajectory rather than a single point. To understand the complexity of finding a global solution of this problem, we introduce the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">spurious (i.e., nonglobal) local trajectory</i> as a generalization to the notion of spurious local solution in nonconvex (time-invariant) optimization. We develop an ordinary differential equation (ODE) associated with a time-varying nonlinear dynamical system which, at limit, characterizes the spurious local solutions of the time-varying optimization problem. We prove that the absence of spurious local trajectory is closely related to the transient behavior of the developed system. In particular, we show that if the problem is time-varying, the data variation may force all of the ODE trajectories initialized at arbitrary local minima at the initial time to gradually converge to the global solution trajectory. We study the Jacobian of the dynamical system along a local minimum trajectory and show how its eigenvalues are manipulated by the natural data variation in the problem, which may consequently trigger escaping poor local minima over time.

Distributed Fusion Filtering for Nonlinear Time-Varying Systems Over Amplify-and-Forward Relay Networks: An $H_{\infty }$ Quantized Framework

This paper is concerned with the distributed fusion filtering problem for a class of nonlinear time-varying systems subject to quantization effects within a finite-horizon <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> framework. To improve the communication quality, the amplify-and-forward (AaF) relay mechanism, which accounts for phenomenon of missing measurements, is utilized to schedule the data transmissions from the sensors to the remote filters. The dynamic quantization, as a result of the inherent limit of network bandwidth, is further considered in the communication process from the filters to the fusion center. The main objective of this paper is to propose a distributed fusion scheme that ensures both local and fusion <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> performance indices over a finite horizon. A sufficient condition is first established for guaranteeing a prescribed performance constraint on the local filtering error dynamics, and then the corresponding filter gains are calculated by solving a set of recursive matrix inequalities. Subsequently, with the help of the acquired local state estimates, the desired parameters of the fusion filters are designed in terms of the solution to a convex optimization problem. Finally, the effectiveness of the obtained theoretical results is testified by a numerical example.

An Extended Lyapunov-Density Condition for Nonlinear Time-Varying Systems

Lyapunov densities have been receiving a considerable amount of attention in the last two decades as a method of stability analysis. This paper extends the recently presented Lyapunov density conditions for nonlinear time-varying systems with introducing additional freedom in the inequalities on Lyapunov densities. Then the proposed inequality condition with the additional freedom leads to a convex formulation of robustly stabilizing state-feedback synthesis of nonlinear systems that have time-varying uncertainties. A numerical example demonstrates the usefulness of the proposed inequalities for the robust stabilization problem.

An online self-adaptive RBF network algorithm based on the Levenberg-Marquardt algorithm

ABSTRACT Aiming at the problem that the Levenberg-Marquardt (LM) algorithm can not train online radial basis function (RBF) neural network and the deficiency in the RBF network structure design methods, this paper proposes an online self-adaptive algorithm for constructing RBF neural network (OSA-RBFNN) based on LM algorithm. Thus, the ideas of the sliding window method and online structure optimization methods are adopted to solve the proposed problems. On the one hand, the sliding window method enables the RBF network to be trained online by the LM algorithm making the RBF network more robust to the changes in the learning parameters and faster convergence compared with the other investigated algorithms. On the other hand, online structure optimization can adjust the structure of the RBF network based on the information of training errors and hidden nodes to track the non-linear time-varying systems, which helps to maintain a compact network and satisfactory generalization ability. Finally, verified by simulation analysis, it is demonstrated that OSA-RBFNN exhibits a compact RBF network.

Event-based decentralized adaptive finite-time tracking control of interconnected nonlinear time-varying systems

Event-based decentralized adaptive finite-time tracking control of interconnected nonlinear time-varying systems

Finite-time stability of nonlinear time-varying systems with saturated impulse inputs

Finite-time stability of nonlinear time-varying systems with saturated impulse inputs

Nonconvex, Fully Distributed Optimization Based CAV Platooning Control Under Nonlinear Vehicle Dynamics

CAV platooning technology has received considerable attention, driven by the next generation smart transportation systems. This paper considers nonlinear vehicle dynamics and develops fully distributed optimization based CAV platooning control schemes via the platoon centered MPC approach for a possibly heterogeneous CAV platoon. The nonlinear vehicle dynamics leads to major difficulties in distributed algorithm development and control analysis. Specifically, the underlying MPC optimization problem is nonconvex and densely coupled. Further, the closed loop dynamics becomes a time-varying nonlinear system with non-vanishing external perturbations, making stability analysis rather complicated. To overcome these difficulties, we formulate the underlying MPC optimization problem as a locally coupled, albeit nonconvex, optimization problem and develop a sequential convex programming based fully distributed scheme for a general MPC horizon. Such a scheme can be effectively implemented for real-time computing using operator splitting methods. To analyze the closed loop stability, we apply various tools from global implicit function theorems, stability of linear time-varying systems, and Lyapunov theory for input-to-state stability to show that the closed loop system is locally input-to-state stable uniformly in all small coefficients pertaining to the nonlinear dynamic effects. Numerical tests on a heterogeneous CAV platoon in a real traffic condition illustrate the effectiveness of the proposed method.