area mutually connected thermal power plant with generation rate constraint; group hunting search algorithm is adopted to explore the gain parameters of the controllers [13]. In [14], 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 [15], 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 [16], **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 **order** **systems** [25], 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 **some** **design** tools for simple **order** FOSs.

Show more
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

Show more
15 Read more

This paper deals with the output regulation of nonlinear **control** **systems** in **order** to guarantee desired performances in the presence of plant parameters variations. The proposed **control** law structures are based on the **fractional** **order** 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.

Show more
19 Read more

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.

Show more
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**.

Show more
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 [28] , 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.

Show more
21 Read more

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 **fractional** **order** **systems** 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 **fractional** **order** system,predictive functional **control** gives acceptable results.

Show more
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.

Show more
13 Read more

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.

Show more
15 Read more

15 Read more

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

Show more
19 Read more

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 [5] and [6].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 [4],[7] and [8] .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.

Show more
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 **fractional** **order** 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 [13] and literature [14] judge the stability of the system according to the Lyapunov equation; recently, Huang et al. [15] proposed a new method for judging the stability of **fractional**-**order** **systems**, 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 [16] and [17], the definition of hybrid synchronization is given respectively. Further, the literature [18] 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.

Show more
12 Read more

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 **control** **design** 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.

Show more
In this paper, a new approach to stability for **fractional** **order** **control** 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 **control** **systems**. The measured system and the controller are termed by a set of **fractional** **order** differential equations.

Show more
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.

Show more
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 [13]. 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 **order** **systems**. Moreover, region of the complex plane defined by (3) is not convex as shown in Figure 1. However, this problem has been solved in [4] in which the following result was proposed.

Show more
10 Read more

methods have been proposed to synchronize chaotic **systems** such as the sliding mode **control** method [2], 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.

17 Read more

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):

Show more
14 Read more

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.

15 Read more