area mutually connected thermal power plant with generation rate constraint; group hunting search algorithm is adopted to explore the gain parameters of the controllers . In , PI controller design is performed by using optimization for FOSs; first, controller parameters for a stable control are calculated by using the stability boundary locus method and then optimization is used to provide the best control. In , a new robust FOPID controller to stabilize a perturbed nonlinear chaotic system on one of its unstable ﬁxed points is proposed based on the PID actions using the bifurcation diagram. In , fractional-order discrete synchronization of a new fourth -order memristor chaotic oscillator and the dynamic properties of the fractional-order discrete system are investigated; a new method for synchronizing is proposed and validated. In spite of the existence of a great deal of publications about FOSs some of which have just been mentioned above, most of the present analysis and design techniques deal with sophisticated and rather special applications [17-24]. Although the step response characteristics such as rise time, settling time, delay time, overshoot and some others are well known by explicit formulas for simple integer ordersystems , such formulas are not available for FOSs. And a compact publication yielding the relations between the design parameters and the step response characteristics of even simple FOSs are not yet present. The purpose of this paper is to fulfill this vacancy and to supply somedesign tools for simple order FOSs.
The CRONE control methodology has been extended to MIMO systems [Nelson-Gruel et al., 2009]. Its main principle is to optimize the param- eters of a nominal and diagonal open loop trans- fer function matrix whose diagonal elements are deﬁned by (11). It can be used to control a beam and tank system (Fig. 14) that models an aircraft wing. This system exhibits extremely low-damped vibrations that depend on the level of ﬁlling of the tank (Fig. 15). About 200 sec was required to obtain damped vibrations. These vibrations are measured by two piezoelectric ceramics ( y l and y h ). Two other
This paper deals with the output regulation of nonlinear controlsystems in order to guarantee desired performances in the presence of plant parameters variations. The proposed control law structures are based on the fractionalorder PI (FOPI) and the CRONE control schemes. By introducing the multimodel approach in the closed-loop system, the presented design methodology of fractional PID control and the CRONE control guarantees desired transients. Then, the multimodel approach is used to analyze the closed-loop system properties and to get explicit expressions for evaluation of the controller parameters. The tuning of the controller parameters is based on a constrained optimization algorithm. Simulation examples are presented to show the effectiveness of the proposed method.
Abstract— Control of unstable systems with conventional PID controllers gives poor set-point tracking and disturbance rejection performance. The use of set-point weighted PID controllers (SWPID) to improve the control performance with respect to set-point tracking and disturbance rejection have been attempted. This is due to the fact that, SWPID will reduce proportional and derivative kicks in the control action. However, the control signal of SWPID controller is still inheriting the PID’s undesired oscillations in the control signal. This leads to faster degradation of actuators. In this work, a fractional-order low- pass filter is designed alongside SWPID controller for unstable systems. Incorporating such filter will help to reduce undesired oscillations. The result’s comparison shows that the performance of SWPID with fractional-order filter is better compared to its performance with an integer-order filter. This is true for all the three unstable systems considered.
Abstract. This paper focuses on the synchronization issues between a class of fractional-order and integer-order chaotic systems. A closed-loop control system is introduced following the linear feedback control and fractional-order stability theories to address the synchronization issues. Appropriate coefficients in this paper mentioned synchronization are adopted to guarantee the finite time asymptotical stability of resulting synchronization error due to the disturbances. The proposed control scheme is validated using simulations, and the results illustrate that the proposed controller can implement the synchronization between a class of fractional-order chaotic systems and integer-order chaotic systems, two variable structure fractional-order chaotic systems or two mismatched fractional-order chaotic systems.
In reality, the agents might be affected by the interaction among neighboring agents, but also by its own intrinsic nonlinear dynamic. So the MASs with intrinsic nonlinear dynamics are considered recently in [2,3,5,14,18] . Since the limited view field or nonuniform sensing ranges of sensors, one agent may be able sense another agent, but not vice versa. The com- munication topology among the agents, in general is directed. Taking into consideration these practical cases, in this paper, we consider the consensus problem of fractional-order double integrator MASs with intrinsic nonlinear dynamics and general directed topologies using only relative output information. Due to the well-known Leibniz rule for fractional derivatives is invalid  , how to construct a suitable Lyapunov function for analysing the stability of nonlinear fractional-order MASs is very challenging. The output feedback based consensus control of double integrator MASs in the presence of nonlinear fractional-order dynamics is even more challenging as the communication topology among the agents is not only directed but also local.
Abstract – The fractional calculus is the area of mathematics that handles derivatives and integrals of any arbitrary order (fractional or integer, real or complex order). Predictive Functional Control (PFC) is one of the most popular methods of model predictive control. The implementation of the predictive functional controller (PFC) on the fractionalordersystems has been presented in this paper. The effect of various approximations, sensitivity analysis, tuning of predictive functional controller parameters, the effect of delay and noise analysis of the fractional-order system has been considered. It has been shown that, in fractionalorder system,predictive functional control gives acceptable results.
Synchronization of chaos has widely spread as an important issue in nonlinear systems and is one of the most important branches on the problem of controlling of chaos. In this paper, among different chaotic systems the economy chaotic system has been selected. The main aim of this paper is the designing based on the active sliding mode control for the synchronization of fractional- order chaotic systems. The chaos in the economic series could have serious and very different consequences in common macro-economy models. In this paper, this article expressed the various positions of synchronization in economic system that include of changes in the coefficients of the system ,changes in the initial conditions of the system and different fractional-order synchronization on the economic system. . in which Synchronization is shown in some examples.
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 robust stability 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.
fractional-order adaptation laws are designed to update the controller parameters. By employing the fractional-order expansion of classical Lyapunov stability method, a robust controller is designed for fractional-order chaotic systems. The system states asymptotically converge to the origin and all signals in the closed-loop system remain bounded. A counterexample is constructed to show that the fractional-order
On the one hand, since a Lyapunov-type inequality has found many applications in the study of various properties of solutions of diﬀerential equations, such as oscillation the- ory, disconjugacy and eigenvalues problems, there have been many extensions and gener- alizations as well as improvements in this ﬁeld, e.g., to nonlinear second order equations, to delay diﬀerential equations, to higher order diﬀerential equations, to diﬀerence equa- tions and to diﬀerential and diﬀerence systems. We refer the readers to [–] (integer or- der). Fractional diﬀerential equations have gained considerable popularity and importance due to their numerous applications in many ﬁelds of science and engineering including physics, population dynamics, chemical technology, biotechnology, aerodynamics, elec- trodynamics of complex medium, polymer rheology, control of dynamical systems. With the rapid development of the theory of fractional diﬀerential equation, there are many
The turbine power is measured from the equation(2) is given to the outer loop controller that is FOESC which gives an approximated value of the optimal turbine speed with respect to Fig.2.It is then given to the inner loop nonlinear control. The non linear control used here is feedback linearization which performs FOC and avoids magnetic saturation of the IG. It is a closed loop drives the turbine speed to the optimal value found by the MPPT and drives the rotor flux to the reference flux value given manually. The conventional FOC control method with P&O method is shown in  and .The PI controller used causes high response time and high overshooting if error is unexpectedly very high. It is also difficult to design PI since unpredictable variations in the machine parameters, external load disturbance and non linear dynamics. The other methods used for FOC concept are Fuzzy logic, gain scheduled PI and relative gain array. The feedback linearization gives a faster response and desired response can be obtained by adjusting the feedback gains. The controller gives stator frequency and stator voltage given to modulation, the modulation and pulse generation for MC can be referred from , and  .The MC regulates the stator electrical frequency to control the turbine speed. The stator voltage amplitude can be maintained to regulate the rotor flux. The turbine speed variation does not affect rotor flux. Similarly the rotor flux reference can be varied even independently of reference optimal speed found by the MPPT. This is an improvement over FOC.
stability theory is crucial, and it is an important basis for judging whether a system can operate normally. It is also an important basis to prove chaotic synchronization. At present, the main chaotic synchronization criterion is based on the synchronization criterion of Lyapunov exponent and the synchronization criterion based on Lyapunov function. The literature [5-7] is based on the synchronization criterion of the Lyapunov function. However, in the fractionalorder system, due to the more complicated system, there are of course some methods that are different from the integer order stability judgment. The literature [8,9] separately calculates the range of the value of the coefficient matrix of the analysis system, and then judges the stability criterion of the fractional-order linear system; Hu Jianbing [10-12] is a long-term commitment to fractional nonlinear stability. Research; literature  and literature  judge the stability of the system according to the Lyapunov equation; recently, Huang et al.  proposed a new method for judging the stability of fractional-ordersystems, that is, constructing a suitable function first. Then analyze the positive and negative of its eigenvalues to determine whether the system is stable. The hybrid synchronization problem of chaotic systems has only appeared in recent years. Hybrid synchronization, that is, synchronization and anti-synchronization coexist, is actually a generalization of synchronization and projection synchronization. In the literature  and , the definition of hybrid synchronization is given respectively. Further, the literature  designed a simple linear hybrid synchronous controller, and proposed a synchronization criterion, but the conditions of the criterion is related to the state variables of the drive system.
In this paper a fractional calculus based control strategy for speed control of a DC motor with load changes is presented. The relevance of the paper to the research field consists in the simplicity of the approach, yet yielding a robust controller that can meet the performance specifications for significant load changes. The robustness of the fractional-order PI controller and its performances are compared against an integer-order PI controller. In order to evaluate the robustness of the controllers a change in the motor loading unit is considered for the conducted experiments. Due to the change in the brake unit, the gain and time constant of the system are also modified. The performances of both classical integer-order approach and fractional-order approach are analyzed through simulations and real-time experiments. The controldesign method and the application are kept simple, yet effective to illustrate basic time domain and frequency domain concepts. The experimental results revealed better performances of the fractional approach in comparison with the classical one.
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.
The classical mass-spring-damper system is a challenging system as each mass-spring construction introduces a peak in the frequency response of the system, resulting in resonance frequencies and high oscillations if damping is poor (like the case in this paper). Traditionally, these kind of systems are difficult to control by an integer-order proportional-integral- derivative (PID) controller as this controller has only one pair of zeros to compensate the system. Therefore, a controller of higher order would be more suitable to control poorly damped systems such as the mass-spring-damper. Advanced controllers such as fractional-order controllers may be better but also more complex as they can be approximated by high order integer- order transfer functions.
Remark 1 Throughout the article, triplet (A, B, C) is always supposed to be minimal. Testing if the eigenvalues of matrix A belong to a region of the left half plane defined by (3) with 1 < ν < 2 is a well-known problem in LMI control theory because it corresponds to a performance requirement on the damping ratio of the system. A solu- tion of this problem is provided by the LMI region framework . Extending this LMI condition to the case 0 < ν < 1 is far from trivial because the location of eigenva- lues in this region corresponds to unstable integer ordersystems. Moreover, region of the complex plane defined by (3) is not convex as shown in Figure 1. However, this problem has been solved in  in which the following result was proposed.
methods have been proposed to synchronize chaotic systems such as the sliding mode control method , active control method [3-6], linear and non linear feedback control method [7-8], adaptive control method [9-10], backstepping control [11-12] and impulse control method [13- 14]. Using these methods, numerous synchronization problems of well- known chaotic systems such as L¨u, R¨ossler, Lorenz, Chen, Genesio have been studied.
The dynamics model of an autonomous ground vehicle repre- sents the study of the relationship between the various forces action on a robot mechanism and their accelerations. This is mainly used for simulation study and analysis of vehicle’s design and a motion controller design for the vehicle. The description of the mechanism of the robot movement is given in terms of its component parts; bodies, joints and the para- meters that characterize them. In fact, several parameters are required to define the dynamic model of a given rigid body such inertia, centre of mass and applied forces. The energy- based Lagrangian approach can be used to derive the dynamic model of the autonomous vehicle which is represented in the following general form Fierro and Lewis (1997):
The rest of this paper is organized as follows. In Section , we give some necessary nota- tions, concepts, and lemmas. In Section , two suﬃcient conditions ensuring convergence results of the system () are presented. An interesting example is given in the ﬁnal section to demonstrate the application of our main results.