Fractional-order PD control at Hopf bifurcations in a fractional-order congestion control system

In this paper, we address the problem of the bifurcation control of a delayed fractional-order dual model of congestion control algorithms. A fractional-order proportional–derivative (PD) feedback controller is designed to control the bifurcation generated by the delayed fractional-order congestion control model. By choosing the communication delay as the bifurcation parameter, the issues of the stability and bifurcations for the controlled fractional-order model are studied. Applying the stability theorem of fractional-order systems, we obtain some conditions for the stability of the equilibrium and the Hopf bifurcation. Additionally, the critical value of time delay is figured out, where a Hopf bifurcation occurs and a family of oscillations bifurcate from the equilibrium. It is also shown that the onset of the bifurcation can be postponed or advanced by selecting proper control parameters in the fractional-order PD controller. Finally, numerical simulations are given to validate the main results and the effectiveness of the control strategy.

     

Adaptive neural network-based saturated control of robotic exoskeletons

In this paper, novel adaptive neural network (NN) controllers with input saturation are presented for n-link robotic exoskeletons. The controllers consist of a state feedback controller and an output feedback controller. Through utilizing auxiliary dynamics, the controllers provide a new framework for input saturated control of these robotic systems which can feature the global stability for state feedback control. To compensate for the unknown dynamics of the system, adaptive schemes based on NNs are exploited. Furthermore, adaptive robust terms are utilized to deal with unknown external disturbances. Stability studies show that the closed-loop system is globally uniformly ultimately bounded (UUB) with the state feedback controller, where the global property of the NN-based controller is achieved exploiting a smooth switching function and a robust control term. Also, the system is semi-globally UUB with the output feedback controller. Effectiveness of the controllers is validated by simulations and experimental tests.     

Generalized synchronization of the fractional-order chaos in weighted complex dynamical networks with nonidentical nodes

A fractional-order weighted complex network consists of a number of nodes, which are the fractional-order chaotic systems, and weighted connections between the nodes. In this paper, we investigate generalized chaotic synchronization of the general fractional-order weighted complex dynamical networks with nonidentical nodes. The well-studied integer-order complex networks are the special cases of the fractional-order ones. Based on the stability theory of linear fraction-order systems, the nonlinear controllers are designed to make the fractional-order complex dynamical networks with distinct nodes asymptotically synchronize onto any smooth goal dynamics. Numerical simulations are provided to verify the theoretical results. It is worth noting that the synchronization effect sensitively depends on both the fractional order ?? and the feedback gain k _i . Moreover, generalized synchronization of the fractional-order weighted networks can still be achieved effectively with the existence of noise perturbation.     

Chaotification for switched linear systems with controllers

This paper shows that two or more switched linear systems can generate chaotic dynamical behaviors by an appropriate switching rule as they at least consist of a controllable system and an unstable system with the expanding property. According to the results in the reference （Xie, L. L., Zhou, Y., and Zhao, Y. Criterion of chaos for switched linear systems with antrollers. International Journal of Bifurcation and Chaos, 20（12）, 4105-4109 （2010））, a nonlinear feedback gain is needed to generate chaotic dy- namics. A linear feedback control is usually used to approximate the nonlinear one for simulation. In order to obtain the exact control, as a main result of this paper, the con- troller is constructed by Russell＇s result, and a block diagram is included to interpret the realization of the controller. Numerical simulations are given to illustrate the generated chaotic dynamical behavior of the switched linear systems with some parameters and show the effects of the constructed controller.     

Barrier Lyapunov functions-based fixed-time stabilization of nonholonomic systems with unmatched uncertainties and time-varying output constraints

The fixed-time stabilization problem is addressed in this paper for a kind of nonholonomic systems in chained form with unmatched uncertainties and time-varying output constraints. A novel tan-type barrier Lyapunov function is introduced to deal with time-varying output constraints. Under the unified framework of the considered system with and without output constraints, a state feedback controller is designed with the aid of adding a power integrator technique and switching control strategy. It is shown that the suggested controller ensures the states of closed-loop system to zero in a given fixed time without disobeying the constraints. Finally, simulation results are given to confirm the efficacy of the presented control scheme.

     

Lyapunov-based fractional-order controller design to synchronize a class of fractional-order chaotic systems

In this paper, a novel adaptive fractional-order feedback controller is first developed by extending an adaptive integer-order feedback controller. Then a simple but practical method to synchronize almost all familiar fractional-order chaotic systems has been put forward. Through rigorous theoretical proof by means of the Lyapunov stability theorem and Barbalat lemma, sufficient conditions are derived to guarantee chaos synchronization. A wide range of fractional-order chaotic systems, including the commensurate system and incommensurate case, autonomous system, and nonautonomous case, is just the novelty of this technique. The feasibility and validity of presented scheme have been illustrated by numerical simulations of the fractional-order Chen system, fractional-order hyperchaotic Lü system, and fractional-order Duffing system.     

Robust synchronization of two different fractional-order chaotic systems with unknown parameters using adaptive sliding mode approach

In this paper, an adaptive sliding mode control method is introduced to ensure robust synchronization of two different fractional-order chaotic systems with fully unknown parameters and external disturbances. For this purpose, a fractional integral sliding surface is defined and an adaptive sliding mode controller is designed. In this method, no knowledge of the bounds of parameters and perturbation is required in advance and the parameters are updated through an adaptive control process. The proposed scheme is global and theoretically rigorous. Two examples are given to illustrate effectiveness of the scheme, in which the synchronizations between fractional-order chaotic Chen system and fractional-order chaotic Rössler system, between fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system, respectively, are successfully achieved. Corresponding numerical simulations are also given to verify the analytical results.

     

Emergence of diverse dynamical responses in a fractional-order slow–fast pest–predator model

To explore the impact of pest-control strategy on integrated pest management, a three-dimensional (3D) fractional- order slow–fast prey–predator model is introduced in this article. The prey community (assumed as pest) represents fast dynamics and two predators exhibit slow dynamical variables in the three-species interacting prey–predator model. In addition, common enemies of that pest are assumed as predators of two different species. Pest community causes serious damage to the economy. Fractional-order systems can better describe the real scenarios than classical-order dynamical systems, as they show previous history-dependent properties. We establish the ability of a fractional-order model with Caputo’s fractional derivative to capture the dynamics of this prey–predator system and analyze its qualitative properties. To investigate the importance of fractional-order dynamics on the behavior of the pest, we perform the local stability analysis of possible equilibrium points, using certain assumptions for different sets of parameters and reveal that the fractional-order exponent has an impact on the stability and the existence of Hopf bifurcations in the prey–predator model. Next, we discuss the existence, uniqueness and boundedness of the fractional-order system. We also observe diverse oscillatory behavior of different amplitude modulations including mixed mode oscillations (MMOs) for the fractional-order prey–predator model. Higher amplitude pest periods are interspersed with the outbreaks of small pest concentration. With the decrease of fractional-order exponent, small pest concentration increases with decaying long pest periods. We further notice that the reduced-order model is biologically significant and sensitive to the fractional-order exponent. Additionally, the dynamics captures adaptation that occurs over multiple timescales and we find consistent differences in the characteristics of the model for various fractional exponents.

     

LMI-based stabilization of a class of fractional-order chaotic systems

Based on the theory of stabilization of fractional-order LTI interval systems, a simple controller for stabilization of a class of fractional-order chaotic systems is proposed in this paper. We consider the structure of the chaotic systems as fractional-order LTI interval systems due to the limited amplitude of chaotic trajectories. We introduce a simple feedback controller for the interval system and then, based on a recently established theorem for stabilization of interval systems, we reach to a linear matrix inequality (LMI) problem. Solving the LMI yields an appropriate decoupling feedback control law which suffices to bring the chaotic trajectories to the origin. Several illustrative examples are given which show the effectiveness of the method.     

Decentralized direct adaptive output feedback fuzzy H ∞ tracking design of large-scale nonaffine nonlinear systems

A novel H _∞ tracking-based decentralized direct adaptive output feedback fuzzy controller is developed for a class of interconnected nonaffine uncertain nonlinear systems in this paper. By virtue of the proper filtering of the observation error dynamics to assure its strictly positive realness, the observer-based decentralized direct adaptive fuzzy control (DAFC) scheme is presented for a class of large-scale nonaffine nonlinear systems by the combination of H _∞ tracking technique, implicit function theorem, a state observer and a fuzzy inference system. The output feedback and adaptation mechanisms for each subsystem depend upon local measurements not only to achieve asymptotical tracking of a reference trajectory but to guarantee arbitrary small attenuation level of the mismatched errors and external disturbances on the tracking error. Simulation results confirm the effectiveness of the proposed decentralized output feedback scheme.     

Stabilization using fractional-order PI and PID controllers

This paper presents a solution to the problem of stabilizing a given fractional dynamic system using fractional-order PIλ and PI^λD^μ controllers. It is based on plotting the global stability region in the (k _p, k _i)-plane for the PI^λ controller and in the (k _p , k _i , k _d)-space for the PIλDμ controller. Analytical expressions are derived for the purpose of describing the stability domain boundaries which are described by real root boundary, infinite root boundary and complex root boundary. Thus, the complete set of stabilizing parameters of the fractional-order controller is obtained. The algorithm has a simple and reliable result which is illustrated by several examples, and hence is practically useful in the analysis and design of fractional-order control systems.     

Fractional-order complex T system: bifurcations,chaos control,and synchronization

In this paper, the fractional-order complex T system is proposed. The dynamics of the system including symmetry, the stability of equilibrium points, bifurcations with variation of system parameters, and derivative orders are investigated. Period-doubling and tangent bifurcations with appropriate derivative orders and system parameters are observed. Besides, the control problem of the system is examined by using the feedback control technique. Furthermore, based on the stability theory of fractional-order systems, the scheme of function projective synchronization for the fractional-order complex T system is presented. The function projective synchronization for the system is realized by designing an appropriate synchronization controller. Numerical simulations are used to demonstrate the effectiveness and feasibility of the proposed scheme.     

Robust tracking control method based on composite nonlinear feedback technique for linear systems with time-varying uncertain parameters and disturbances

In this paper, the composite nonlinear feedback control method is considered for robust tracking and model following of uncertain linear systems. The control law guarantees that the tracking error decreases asymptotically to zero in the presence of time varying uncertain parameters and disturbances. For performance improvement of the dynamical system, the proposed robust tracking controller consists of linear and nonlinear feedback parts without any switching element. The linear feedback law is designed to allow the closed loop system have a small damping ratio and a quick response while the nonlinear feedback law increases the damping ratio of the system as the system output approaches the output of the reference model. A new collection of different nonlinear functions used in the control law are offered to improve the reference tracking performance of the system. The proposed robust tracking controller improves the transient performance and steady state accuracy simultaneously. Finally, the simulations are provided to verify the theoretical results.     

Observer-based adaptive fuzzy finite-time prescribed performance tracking control for strict-feedback systems with input dead-zone and saturation

This paper considers the problems of finite-time prescribed performance tracking control for a class of strict-feedback nonlinear systems with input dead-zone and saturation simultaneously. The unknown nonlinear functions are approximated by fuzzy logic systems and the unmeasurable states are estimated by designing a fuzzy state observer. In addition, a non-affine smooth function is used to approximate the non-smooth input dead-zone and saturated nonlinearity, and it is varied to the affine form via the mean value theorem. An adaptive fuzzy output feedback controller is developed by backstepping control method and Nussbaum gain method. It guarantees that the tracking error falls within a pre-set boundary at finite time and all the signals of the closed-loop system are bounded. The simulation results illustrate the feasibility of the design scheme.

     

Fuzzy observer-based command filtered tracking control for uncertain strict-feedback nonlinear systems with sensor faults and event-triggered technology

For the trajectory tracking problem of nth-order uncertain nonlinear systems with sensor faults, a fuzzy controller based on command filtered and event-triggered technology is designed to improve the tracking error of the system. Concurrently, a fault-tolerant control scheme is introduced to effectively solve the problem of sudden output sensor failure. Additionally, the proposed controller can also greatly avoid complexity explosion problem of derivations of virtual control laws, which makes the design of the controller simpler. Furthermore, an effective observer is designed to solve the problem of system state immeasurability. Therefore, the proposed control scheme makes the design of the controller more convenient and flexible. According to Lyapunov stability theory, it is proved that all closed-loop signals are uniformly and ultimately bounded. Finally, two simulation examples of second-order nonlinear system and single-link robot show the effectiveness of the proposed scheme.

     

Output feedback \varvec{\mathcal {H}}_{{\varvec{\infty }}} control of stochastic nonlinear time-delay systems with state and disturbance-dependent noises

This paper is concerned with the output feedback \(\mathcal {H}_\infty \) control problem for a class of stochastic nonlinear systems with time-varying state delays; the system dynamics is governed by the stochastic time-delay It \(\hat{o}\) -type differential equation with state and disturbance contaminated by white noises. The design of the output feedback \(\mathcal {H}_\infty \) control is based on the stochastic dissipative theory. By establishing the stochastic dissipation of the closed-loop system, the delay-dependent and delay-independent approaches are proposed for designing the output feedback \(\mathcal {H}_\infty \) controller. It is shown that the output feedback \(\mathcal {H}_\infty \) control problem for the stochastic nonlinear time-delay systems can be solved by two delay-involved Hamilton–Jacobi inequalities. A numerical example is provided to illustrate the effectiveness of the proposed methods.     

PLC based fractional-order PID temperature control in pipeline: design procedure and experimental evaluation

Meccanica - Fractional-order control system design can be used for systems with non-local dynamics involving long-term memory effects. However, implementation of a fractional-order controller in...     

Observer design and output feedback stabilization for nonlinear multivariable systems with diffusion PDE-governed sensor dynamics

This paper addresses the problems of observer design and output feedback stabilization for a class of nonlinear multivariable systems, where the nonlinear system dynamics are described by ordinary differential equations (ODEs), and the sensor dynamics are governed by diffusion partial differential equations (PDEs). Based on the Luenberger observer theory, a Luenberger-type PDE-ODE cascaded observer is derived to estimate the state variables of the system. Then, an observer-based output feedback stabilizing controller is developed. The exponential stability of both the observer error system and closed-loop control system is proven via the Lyapunov direct method. Finally, numerical examples are provided to illustrate the effectiveness of the proposed design methods.     

Fuzzy adaptive high-gain-based observer backstepping control for SISO nonlinear systems with dynamical uncertainties

In this paper, a fuzzy adaptive output feedback control approach is developed for a class of SISO strict-feedback nonlinear systems with unmeasured states, unmodeled dynamics, and dynamical disturbances. In the backstepping recursive design, fuzzy logic systems are used to approximate the unknown nonlinear functions, a fuzzy adaptive high-gain observer is designed to estimate the unmeasured states; a dynamic signal is incorporated into the control scheme to dominate the dynamic uncertainties. Using the states estimates and combining the backstepping design technique, a fuzzy adaptive output feedback control is constructed recursively. It is proved that the proposed fuzzy adaptive output feedback control scheme can guarantee the all signals in the closed-loop system are semiglobally uniformly ultimately bounded (SUUB), and the observer and tracking error converges to a small neighborhood of the origin. The effectiveness of the proposed approach is illustrated via an example.