The paper deals with a graphicalapproach to investigation of robuststability for a feedback control loop with an uncertain fractionalordertime-delay plant and integer order or fractionalorder controller. Robuststability analysis is based on plotting the value sets for a suitable range of frequencies and subsequent verification of the zero exclusion condition fulfillment. The computational examples present the typical shapes of the value sets of a family of closed-loop characteristic quasipolynomials for a fractionalorder plant with uncertain gain, time constant, or time-delay term, respectively, and also for combined cases. Moreover, the practically oriented example focused on robuststability analysis of main irrigation canal pool controlled by either classical integer order PID or fractionalorder PI controller is included as well.
The robust controller design method suggested here is capable of ensuring closed loop stability for arbitrary order plants with additive uncertainty, which makes is applicable for wide range of plants. Future research can be done in the area of controller design for multi-area power system generation control, multivariable feed-back controlsystems, and robust performance for arbitrary order plants with additive uncertainty.
simpler procedure, and it should have no difficulties for prac- tical design engineers to accept this approach. Clearly, the MADB with the first-order finite difference approximation is comparable with the LMI method. Furthermore, we found good agreement between the third-order finite difference approximation and the fourth-order Pade approximation. The simulation based results for the MADB show that the estimated MADB through the proposed method sufficiently achieves the system stability. A simple controller design method has been developed by the authors based on the method presented in this paper. In the controller design method a stabilizing controller can be derived for a given network timedelay. In all the case studies or examples, only linear system examples are given. The method is lim- ited to linear systems only. The authors are now working on extending the methods to nonlinear systems, such as, multiconverter and inverter system and engine and electrical power generation systems [32, 33].
Up to now, controlling and synchronizing chaos in fractional-order financial systems has also been in- vestigated, for example, in [1, 10, 13, 14, 27, 32, 41, 43, 47, 46]. In , a sliding mode controller was designed to synchronize fractional-order financial systems in master-slave structure. In , a nec- essary condition was given to show the existence of 1-scroll, 2-scroll even multi-scroll chaotic at- tractors in fractional-order financial systems. Ac- tive control method was used in [27, 47]. An active controller with multiple conflicting objectives was constructed in . It should be highlighted that a key assumption in the above literatures is that the model of the financial systems should be known. However, most of real world systems are subjected to system uncertainties and external disturbances, especially in financial systems [7, 8, 27, 33, 34, 36, 44]. On the other hand, in financial systems, sys- tem uncertainties do exist because of limited siz- es of weather variables, political events, and other human factors. The existence of systems uncer- tainties and external disturbances could decrease the control performance, or even lead to instability of the system . It is meaningful to consider the control of financial systems with system uncer- tainties and external disturbances. Thanks to the works of Li et al. , the Lyapunov direct method (also called the Lyapunov second method) has been extended to fractional-order nonlinear systems. In this paper, a robust adaptive controller is proposed to solve the control problem of fractional-order fi- nancial chaotic systems with both system uncer- tainties and bounded external disturbances. The fractional-order Lyapunov approach is used to an- alyze the stability of the system. Specifically, the main contributions of this study include:
The problem of timedelay compensation in deadbeat control for power converters are considered but not solved systematically, by far. A linear phase-lead compensation solution is successfully employed in repetitive controlsystems to compensate the timedelay -. However, it is impractical to be adopted in the conventional deadbeat control frame due to its incausal lead-time item. A state estimator is adopted for compensation of computational delay . Also focused on this problem, another simple design method of two steps forward prediction approach is proposed in the frame of model predictive control , , . In these solutions, computational delay effects are effectively removed and control accuracy is prominently improved. However, as mentioned above, apart from computational delay, many other factors lead to the delay problem in practical systems. In these cases, the above mentioned approaches are not suitable and fail to achieve satisfying control performance. Therefore, a universal delay compensation approach for the deadbeat control schemes should be investigated in practical applications.
. (10) Remark. Important in the theory of nonlinear physical fractional-ordersystems is the use the dynamics caused by their observed. Continuous flow in phase space determines the behavior of the system can be studied using a discrete display the induced flow at the section of Poincare. It is a relativistic terms in the rights continuous flow and its discrete in timedelay.
The growth of practical applications in the use of fractional calculus has attracted the attention of many engineers and researchers. The advantages of using fractional operators have been shown in the literature [5, 7, 25, 27, 28, 33]. For instance, it has been shown that the behavior of viscoelastic materials can be correctly described by a fractional model with a small number of model parameters compared to using con- ventionals integer-order models with a large number of model parameters [9, 11, 16]. In addition, it has been shown that the fractional-order controllers are more sophisti- cated than regular integer-order controllers due to employing more control parameters .
The guaranteed cost control was investigated for parameter uncertain systems with timedelay in . Kim  studied the robuststability of time-delayed linear systems with uncertainties. The problem of delay-dependent robuststability was investigated for systems with time-varying structured un- certainties and time-varying delays . A robust controller was proposed in  for delay-dependent neutral systems with mixed delays and time-varying structured uncertainties. Sliding mode control scheme was presented for the robust stabilization of uncertain linear input-delaysystems with non- linear parametric perturbations in . In , the stability of systems in the presence of bounded uncertain time-varying delays in the feedback loop was analysed. Han  studied the absolute sta- bility for a class of nonlinear neutral systems using a discretized Lyapunov functional approach. In , an adaptive neural control scheme was proposed for a class of uncertain multi-input multi-output (MIMO) nonlinear state time-varying delaysystems in a triangular control structure with unknown nonlinear dead-zones and gain signs.
In this paper, a new approach to stability for fractionalordercontrol system is proposed. Here a dynamic system whose behavior can be modeled by means of differential equations involving fractional derivatives. Applying Laplace transforms to such equations, and assuming zero initial conditions, causes transfer functions with no integer powers of the Laplace transform variable s to appear. In recent time, the application of fractional derivatives has become quite apparent in modeling mechanical and electrical properties of real materials. Fractional integrals and derivatives have originated wide application in the controlsystems. The measured system and the controller are termed by a set of fractionalorder differential equations.
et al. , disturbance observer method was proposed to ensure that the teleoperation of motion controlsystems was robust to model uncertainties and time-delay. This involved the design of ad hoc flters, namely, a high-pass flter to handle high-frequency noises and a low-pass flter to compensate for the e“ect of delay and disturbances assumed to have low-frequency variations. In Bresch-Pietri et al. , an adaptive control scheme was proposed to completely reject a constant disturbance in the presence of unknown but bounded constant input delay and uncertain parameters. In Han et al. , a sliding mode control (SMC) was proposed in the presence of time-varying in- put delay and bounded matched disturbances to achieve ultimate boundedness of the closed loop system using a singular perturbation approach. The idea of adaptive robustcontrol (ARC) with rigorous stability analysis and its application to motion control problems was presented in Yao and Jiang  and references therein. However, their control law does not address delays in the input. Re- cently, in L´echapp´e et al. [2015b], a new prediction scheme with full state feedback has been proposed that is robust to external disturbances in the presence of input delay, then extended to output feedback in L´echapp´e et al. [2015a], and then to an unknown delay case in L´echapp´e et al. . But they do not address the e“ect of parametric uncertainties explicitly.
We propose a fractional-order controller to stabilize unstable fractional-order open-loop systems with interval uncertainty whereas one does not need to change the poles of the closed-loop system in the proposed method. For this, we will use the robuststability theory of Fractional-Order Linear Time Invariant FO-LTI systems. To determine the control parameters, one needs only a little knowledge about the plant and therefore, the proposed controller is a suitable choice in the control of interval nonlinear systems and especially in fractional-order chaotic systems. Finally numerical simulations are presented to show the eﬀectiveness of the proposed controller.
Tang et al.  propose a novel fractional sliding mode strat- egy with a dynamic essence in order to control several delay based chaotic systems in a master slave configuration. The control strategy is simulated on a multivariable delay chaotic robot by taking into consideration the chattering problem of the sliding mode control. Another innovative sliding mode strategy is detailed in  with applications upon a nonlin- ear robotic exoskeleton. Parameter uncertainties and external disturbances lead to a timedelay estimation based model-free fractionalorder nonsingular fast terminal sliding mode con- trol (MFF-TSM), The fractionalorder controller is designed to for tracking performance, fast speed of convergence, and chatter-free control inputs lacking singularities. Asymptotical stability is also investigated through the Lyapunov theorem. Cascade control structures for delayed systems are exempli- fied in . Both master and slave controllers are of frac- tional order. For the slave controller, a fractionalorder PD is chosen for the time delayed process, while for the master control, the fractionalorder SMC law is employed. Numeri- cal simulations validated the proposed approach in terms of time domain performance and stability of the closed loop system. Bode’s ideal transfer function and internal model control principles are used in  to analytically develop a fractionalorder controller combined with a modified Smith predictor targeting robustness to process’ uncertainties. The simulations prove the veracity of the method for systems with long time delays. Another control combination is done in  where a hierarchical structure has an event-based supervisor and a lower level fractionalorder PI controller applied to a wind turbine. The purpose of the supervisor is to analyze and determine the states of the process, while the fractionalorder controller’s main purpose is to ensure maximum power generation with peak performance and reliability.
Lyapunov stability is a general theory available for any differential equation. Over the past researches on time-delaysystems, many useful approaches are applied to guarantee the stability or stabilization of systems. The application of Lyapunov-Krasovskii functional theory has first started for system without neither uncertainties nor control  , some robuststability conditions based on LMI approach are given. Then, the guaranteed cost control problem for neutral timedelay system with feed-back control is investigated. Some papers are interested on stability and stabilization of this type of system where a linear–quadratic cost function is considered
Recently, a ﬁnite-timestability analysis of fractionaltime-delaysystems has ﬁrst been presented and reported in . But until now, only a few papers studied the stability of fractional neutral systems. Though the Lyapunov approach of nonlinear fractional neutral system was extended in , it is diﬃcult to use the Lyapunov method to study the stabil- ity of fractional neutral systems due to the complicated fractional derivatives. However, based on the algebraic approach, Zhang et al.  obtained some suﬃcient conditions for fractional neutral dynamical systems.
In this paper, a novel conformable fractionalorder (FO) sliding mode control technique is studied for a class of FO chaotic systems in the presence of uncertainties and disturbances. First, a novel FO nonlinear surface based on conformable FO calculus is proposed to design the FO sliding mode controller. Then, asymptotic stability of the controller is derived by means of the Lyapunov direct method via conformable FO operators. The stability analysis is performed in the sliding and reaching phase. In addition, the realization of reaching phase is guaranteed in finite time and the reaching time is calculated analytically. The proposed controlapproach has some superiorities. Reduction of the chattering phenomenon, high robustness against the uncertainty and external disturbance, and fast convergence speed are the main advantages of the proposed control scheme. Moreover, it has simple calculations because of using conformable FO operators in the control design. The numerical simulations verify the efficiency of the proposed controller.
fuzzy systems with interval time-varying delay. Most recently, a novel LKF is es- tablished via the delay-decomposition method, then by means of employing the reciprocally convex approach,  has achieved less conservative results than those in [5, 10, 14, 15, 16, 17, 18] for the uncertain T-S fuzzy systems with time-varying delay. However, when revisiting this problem, we find that the aforementioned works still leave plenty of room for improvement.
As an important class of hybrid systems, hybrid stochastic diﬀerential equations (SDEs) (also known as SDEs with Markovian switching) have been widely employed to model many practical systems that have variable structures subject to random abrupt changes, which may result from abrupt phenomena such as random failures and repairs of compo- nents, sudden environment changes, etc. One of the important issues in the study of hybrid SDEs is the automatic control, with subsequent emphasis being placed on the analysis of stability. A great number of signiﬁcant results on this subject have been reported in the literature; see, for instance, [–] and the references therein. In particular, we refer the reader to the book .
The fractional calculus dates from the 17th century , and it can be deﬁned as a classical mathematical notion and a generalization of the ordinary diﬀerentiation and integration not necessarily integer. The signiﬁcance of fractional-order representation is that it is more adequate to describe real world systems than those of integer-order models . Fractional- order calculus is focused on the whole time and space, but the integer-order calculus only concerned with local attribute at particular time and a certain position . Due to these advantages, fractional calculus is developing fast [4, 5], and its various applications are extensively used in many ﬁelds of science and engineering: in material engineering , chaos systems [7–9], economic systems , robotics , and in many more [12–16].
We investigate the stability of nonlinear nonautonomous discrete-timesystems with delaying ar- guments, whose linear part has slowly varying coeﬃcients, and the nonlinear part has linear majo- rants. Based on the “freezing” technique to discrete-timesystems, we derive explicit conditions for the absolute stability of the zero solution of such systems.
It is very easy to verify this local Lipschitz assumption. For example, the assumption is satisfied if f and g are continuously differentiable in x and y or they are differentiable in x and y with locally bounded derivatives. It is known that this classical assumption covers many hybrid SDDEs in the real world (see, e.g., books [23, 24] and the references therein). Of course, this assumption is not enough to guarantee the global solution (i.e., no explosion at a finite time). A standard additional condition for the existence and uniqueness of the global solution of the SDDE (4.1) would be the linear growth condition (see, e.g., [18, 23]). However, our aim here is to study the structured robuststability and boundedness of highly nonlinear SDDEs that do not satisfy the linear growth condition. We hence need to propose alternative assumptions.