Positive Odd Rational Numbers Research Articles

Adaptive Fuzzy Finite-Time Fault-Tolerant Consensus Tracking Control for High-Order Multiagent Systems With Directed Graphs.

This article investigates the distributed adaptive fuzzy finite-time fault-tolerant consensus tracking control for a class of unknown nonlinear high-order multiagent systems (MASs) with actuator faults and high powers (ratio of positive odd rational numbers). The fault models include both loss of effectiveness and bias fault. Compared with existing similar results, the MASs considered here are more general and complex, which include the special case when the powers are equal to 1. Besides, the functions in this article are completely unknown and do not need to satisfy any growth conditions. In the backstepping framework, an adaptive fuzzy fault-tolerant consensus tracking controller is designed via adding one power integrator technique and directed graph theory so that the controlled systems are semiglobal practical finite-time stability (SGPFTS). Finally, numerical simulation results further verify the effectiveness of the developed control scheme.

Event‐triggered finite‐time control for a class of switched nonlinear systems

AbstractThis work addresses the event‐triggered finite‐time control problem for a class of uncertain switched nonlinear systems with arbitrary switchings, whose powers are positive odd rational numbers. The key difference from the results of similar problems is that the systems considered in this article are more general, which contains a special case when the powers are equal to 1, and the powers have switching signals. It is well known that such nonlinear systems have challenges because of the uncontrollability in the linearization process and the backstepping technique that successfully developed for low‐order systems fail to work. To tackle this issue, combining the backstepping method, event‐triggered strategy with adding one power integrator technique, an event‐triggered control scheme is developed to make the controlled systems be globally finite‐time stable. Besides, the Zeno‐free behavior is proved to verify the feasibility of the proposed event‐triggered mechanism. Finally, simulation results are given to validate the effectiveness of the developed control strategy.

Global finite-time stabilization of switched high-order rational power nonlinear systems

This works is concerned with the finite-time optimal stabilization problem for a class of switched non-strict-feedback nonlinear systems whose powers are possibly different positive odd rational numbers in the sense the powers of each subsystem might differ from others. It is well known that high-order nonlinear systems are intrinsically challenging as feedback linearization and backstepping method successfully developed for low-order systems fail to work. To this purpose, the nested saturation homogeneous controller is constructively devised to achieve global finite-time stability. Furthermore, the corresponding design parameters are optimized by minimizing a well-defined cost function, and thus an optimal controller being independent of switching signals is obtained. Simulation results are eventually provided to validate the effectiveness of the proposed control scheme.

Adaptive finite‐time fault‐tolerant control for interconnected nonlinear systems

AbstractThis paper addresses the adaptive finite‐time output tracking control problem for nonlinear large‐scale systems whose power‐exponentials are positive odd rational numbers. The considered systems contain actuator loss‐of‐effectiveness and bias fault, which can be addressed by fault‐tolerant control technique. The intelligent approximator is presented to approximate the unknown nonlinear function. Meanwhile, the technique named adding a power integrator is used, which validly overcomes the problem of the system with power‐exponentials. A finite‐time fault‐tolerant controller is proposed based on the finite‐time Lyapunov stability theory and backstepping algorithm. Moreover, the tracking errors can converge to a small neighborhood of the origin in finite time. Finally, a simulation example is given to illustrate the effectiveness of the proposed adaptive finite‐time control strategies.

Distributed finite‐time output consensus tracking for a class of high‐order nonlinear multiagent systems with the powers of positive odd rational numbers and output constraints

SummaryThis article considers the distributed finite‐time output consensus tracking problem for a class of high‐order nonlinear multiagent systems with the powers of positive odd rational numbers and output constraints. First, since the finite‐time tracking design method for a single system with the powers of positive odd rational numbers cannot be extended to the finite‐time consensus tracking for multiagent systems, an especial configuration method of the powers is constructed. Second, a new exponential brake function (EBF) that purely depends on the outputs is proposed to address the asymmetric output constraints directly. By adjusting a related parameter, the large converted value can be avoided when the output approaches the constrained boundaries, which may reduce the control input over a large output scale. Third, given some reasonable assumptions, a distributed finite‐time output consensus tracking controller is proposed to guarantee that the tracking error converges to an adjustable small region around the origin in a finite time and that the output constraints of all subsystems are not violated. The result with output constraints can also be extended to the case without output constraints. Finally, simulation verifications also confirm the benefits and effectiveness of the proposed control methods.

Adaptive Neural Networks Finite-Time Optimal Control for a Class of Nonlinear Systems

This article addresses the finite-time optimal control problem for a class of nonlinear systems whose powers are positive odd rational numbers. First of all, a finite-time controller, which is capable of ensuring the semiglobal practical finite-time stability for the closed-loop systems, is developed using the adaptive neural networks (NNs) control method, adding one power integrator technique and backstepping scheme. Second, the corresponding design parameters are optimized, and the finite-time optimal control property is obtained by means of minimizing the well-defined and designed cost function. Finally, a numerical simulation example is given to further validate the feasibility and effectiveness of the proposed optimal control strategy.

Adaptive neural network finite-time tracking control for a class of high-order nonlinear multi-agent systems with powers of positive odd rational numbers and prescribed performance

This paper addresses the adaptive finite-time consensus tracking problem for high-order nonlinear multi-agent systems (MASs) with powers of positive odd rational numbers under prescribed performance. Since the virtual and actual control parts of the dynamics of each follower agent are power functions containing positive odd rational numbers, the method of adding a power integrator is used to overcome the controller design difficulties caused by the power functions. With the aid of a finite-time performance function (FTPF) and neural networks (NNs), a distributed adaptive finite-time consensus tracking controller with prescribed tracking performance is properly designed by the backstepping process. It is shown that the proposed control strategy can guarantee that the consensus tracking error converges to an arbitrarily small neighborhood around zero at any settling time, while all signals of the closed-loop system are semi-globally practical finite-time stable (SGPF-stable). Finally, a simulation example is presented to demonstrate the effectiveness of the proposed method.

Adaptive fuzzy finite‐time optimal control for switched nonlinear systems

SummaryThis article addresses the adaptive fuzzy finite‐time control problem for a class of switched nonlinear systems whose powers are positive odd rational numbers and vary with the switching signal. The fuzzy logic systems (FLSs) are used to approximate the unknown nonlinearities of the controlled systems, and then by combining backstepping control algorithm with adding a power integrator technique, an adaptive finite‐time controller is designed. By modifying and optimizing the relevant design parameters of the proposed adaptive finite‐time controller, a novel adaptive finite‐time optimal control approach is developed. The proposed two adaptive finite‐time optimal control schemes can guarantee semi‐globally practically finite‐time stability of the closed‐loop system. Moreover, the adaptive finite‐time optimal controller can also achieve optimized performance in relation to the cost functional. Finally, a simulation example is given to illustrate the effectiveness of the proposed two adaptive finite‐time control strategies.

Finite‐time optimal control for interconnected nonlinear systems

SummaryThis work addresses the finite‐time optimal control problem for a class of interconnected nonlinear systems with powers of positive odd rational numbers. A series of homogeneous controllers, which are capable of guaranteeing the local finite‐time stability of the closed‐loop systems, are first developed using the adding one power integrator method and backstepping technique. Then, the nested saturation controllers are further proposed to achieve global finite‐time stability. Furthermore, the corresponding design parameters are optimized, and thus, an optimal controller is obtained. A numerical simulation example is finally given to illustrate the effectiveness of the proposed control strategy.

Funnel Control of Uncertain High-Order Nonlinear Systems With Unknown Rational Powers

This article considers the funnel output tracking control for a class of high-order uncertain nonlinear systems with the powers of positive odd rational numbers, aiming to accomplish output tracking with prescribed accuracy when both the system nonlinearities and the powers of the system are unknown. For such a purpose, a robust funnel control algorithm, i.e., a continuous, static, and universal, state-feedback controller is explicitly constructed, which achieves the state errors evolving within the predesigned performance space. Benefits of the proposed funnel output tracking controller comparing with the current approaches lie in the fact that the exact knowledge of system nonlinearities, including generally required bounding functions, is not needed to be a priori. Moreover, all the powers in each high-order subsystem are permitted to be any unknown positive odd rational numbers as well. The efficacy of the developed algorithm is confirmed through two illustrative examples.

Adaptive quantised control for switched uncertain nonlinear systems with unknown control directions

This paper is concerned with the problem of global state-feedback regulation for a class of switched uncertain nonlinear systems with quantised input and unknown control directions under arbitrary switchings. A hysteretic quantiser is introduced to avoid actuator chattering. The powers of all subsystems are positive odd rational numbers and depend on the switching signals. Incorporating Nussbaum-type gain technique into adding a power integrator approach, a new adaptive controller is designed to guarantee that all the signals of the closed-loop system are bounded and the system state is globally asymptotically regulated to zero. An example is given to illustrate the validity of the proposed control scheme.

Robust finite‐time stabilization of a class of high‐order stochastic nonlinear systems subject to output constraint and disturbances

SummaryIn this paper, the robust stabilization problem is addressed for a class of high‐order stochastic nonlinear systems with output constraints and disturbances by using finite‐time control technique. One of the features of the considered stochastic systems is that the fractional powers are allowed to be any positive odd rational numbers, rather than grater than or equal to one. By constructing a novel tan‐type barrier Lyapunov function and using the adding a power integrator technique, the explicit steps on how to construct a backstepping‐like finite‐time controller have been developed to handle the robust stabilization and output constraint. Rigorous mathematical proof shows that the system states will finite‐time converge to a small region of the origin and the output constraint can be kept. Finally, a simulation example is given to illustrate the effectiveness of the proposed approach.

Adaptive finite‐time stabilization of stochastic nonlinear systems with the powers of positive odd rational numbers

SummaryIn this paper, we give some new results on finite‐time stabilization for stochastic nonlinear systems whose powers of integrators are all positive odd rational numbers (ie, numerators and denominators of the powers are all positive odd integers) but not necessary only high order or low order. By adopting new conditions on drift and diffusion terms, Lyapunov function, and controller design method, a state feedback controller is designed to guarantee stochastic nonlinear systems finite‐time stable in probability. Moreover, when unknown parameters exist in systems, we give the adaptive finite‐time state feedback controller. Numerical examples demonstrate the effectiveness of the proposed results.

Finite-time H∞ control for switched nonlinear systems

This paper studies the problem of finite-time control for switched nonlinear systems (SNSs), where the powers of chained integrators associated with individual subsystems can be different positive odd rational numbers from each other. First, the notion of finite-time for switched systems, as a performance index, is introduced. Contrary to the classical control, finite-time stability rather than asymptotic stability or practical stability needs to be satisfied in the finite-time control. Moreover, based on the method of multiple Lyapunov functions (MLFs) and the technique of adding a power integrator, a sufficient condition guaranteeing the solvability of the finite-time control problem for the system under consideration is derived via a designed switching law, where there is no subsystem whose corresponding control problem must be solvable. Finally, the effectiveness of the provided control strategy is demonstrated via a simulation example.

Adaptive Neural Network Finite-Time Control for Multi-Input and Multi-Output Nonlinear Systems With Positive Powers of Odd Rational Numbers.

This article investigates the adaptive neural network (NN) finite-time output tracking control problem for a class of multi-input and multi-output (MIMO) uncertain nonlinear systems whose powers are positive odd rational numbers. Such designs adopt NNs to approximate unknown continuous system functions, and a controller is constructed by combining backstepping design and adding a power integrator technique. By constructing new iterative Lyapunov functions and using finite-time stability theory, the closed-loop stability has been achieved, which further verifies that the entire system possesses semiglobal practical finite-time stability (SGPFS), and the tracking errors converge to a small neighborhood of the origin within finite time. Finally, a simulation example is given to elaborate the effectiveness and superiority of the developed.

Finite-time Stabilization with Output-constraints of A Class of High-order Nonlinear Systems

In this paper, the finite-time stabilization with output-constraint is investigated for a class of high-order nonlinear systems with the powers of positive odd rational numbers by constructing a Barrier Lyapunov function. First, sufficient conditions on characterizing the nonlinear functions of the considered systems are derived. Then, based on the technique of adding one power integrator, the global finite-time stabilizers of individual subsystems are systematically constructed to guarantee global finite-time stability with output constraints of the closed-loop nonlinear system. Finally, an example is provided to demonstrate the effectiveness of the proposed result.

Adaptive Finite-Time Stabilization of a Class of Uncertain Nonlinear Systems via Logic-Based Switchings

In this paper, global adaptive finite-time stabilization is investigated by logic-based switching control for a class of uncertain nonlinear systems with the powers of positive odd rational numbers. Parametric uncertainties entering the state equations nonlinearly can be fast time-varying or jumping at unknown time instants, and the control coefficient appearing in the control channel can be unknown. The bounds of the parametric uncertainties and the unknown control coefficient are not required to know a priori . Our proposed controller is a switching-type one, in which a nonlinear controller with two parameters to be tuned is first designed by adding a power integrator, and then a switching mechanism is proposed to tune the parameters online to finite-time stabilize the system. An example is provided to demonstrate the effectiveness of the proposed result.

Global finite-time stabilization of a class of switched nonlinear systems with the powers of positive odd rational numbers

In this paper, we will present new results on global finite-time stabilization for a class of switched strict-feedback nonlinear systems, whose subsystems have chained integrators with the powers of positive odd rational numbers (i.e., numerators and denominators of the powers are all positive odd integers but not necessarily relatively prime). All the powers in each equation of subsystems of the switched systems can be different. Based on the technique of adding a power integrator, the global finite-time stabilizers of individual subsystems are first systematically constructed to guarantee global finite-time stability of the closed-loop smooth switched system under arbitrary switchings, and then a co-design of stabilizers and a state-dependent switching law is proposed to achieve global finite-time stabilization of the closed-loop non-smooth switched systems. In the controller design, a common coordinate transformation of all subsystems is exploited to avoid using individual coordinate transformations for individual subsystems. We also give some sufficient conditions that enable our design by characterizing the powers of the chained integrators of the considered switched systems. Numerical examples are provided to demonstrate the effectiveness of the proposed results.

An expanded method to robustly practically output tracking control for uncertain nonlinear systems

This paper studies the global practical output tracking problem via state feedback for a class of uncertain nonlinear systems, whose chained integrator part has the power of positive odd rational numbers. Since the power is not restricted to be larger than or equal to one, the linearization of the system at the origin may fail. Nevertheless, it is proved in this paper that, under mild assumptions, globally practical output-tracking is achievable by state feedback controller.

Nonsmooth feedback stabilizer for strict-feedback nonlinear systems that may not be linearizable at the origin

We present a continuous feedback stabilizer for nonlinear systems in the strict-feedback form, whose chained integrator part has the power of positive odd rational numbers. Since the power is not restricted to be larger than or equal to one, the linearization of the system at the origin may fail. Nevertheless, we show that the closed loop system is globally asymptotically stable (GAS) with the proposed continuous (but, possibly not differentiable) feedback. We formulate a condition that enables our design by characterizing the powers of the given system. The condition also shows that our result is an extension of Qian and Lin [Non-lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization, Systems Control Lett. 42 (2001) 185–200] where the power of odd positive integers has been considered. New result on the global finite time stabilization problem is also presented.