Hybrid control of synchronization of fractional order nonlinear systems

Under the existence of model uncertainties and external disturbance, finite‐time projective synchronization between two identical complex and two identical real fractional‐order (FO) chaotic systems are achieved by employing FO sliding mode control approach. In this paper, to ensure the occurrence of synchronization and asymptotic stability of the proposed methods, a sliding surface is designed and the Lyapunov direct method is used. By using integer and FO derivatives of a Lyapunov function, three different FO real and complex control laws are derived. A hybrid controller based on a switching law is designed. Its behavior is more efficient that if the individual controllers were designed based on the minimization of an appropriate cost function. Numerical simulations are implemented for verifying the effectiveness of the methods.     

Fractional Order Modeling And Nonlinear Fractional Order Pi‐Type Control For PMLSM System

In this paper, firstly a fractional order (FO) model is proposed for the speed control of a permanent magnet linear synchronous motor (PMLSM) servo system. To identify the parameters of the FO model, a practical modeling algorithm is presented. The algorithm is based on a pattern search method and its effectiveness is verified by real experimental results. Second, a new fractional order proportional integral type controller, that is, (PI^μ)^λ or FO[FOPI], is introduced. Then a tuning methodology is presented for the FO[FOPI] controller. In this tuning method, the controller is designed to satisfy four design specifications: stability requirement, specified gain crossover frequency, specified phase margin, flat phase constraint, and minimum integral absolute error. Both set point tracking and load disturbance rejection cases are considered. The advantages of the tuning method are that it fully considers the stability requirement and avoids solving a complex nonlinear optimization problem. Simulations are conducted to verify the effectiveness of the proposed FO[FOPI] controller over classical FOPI and FO[PI] controllers.     

Tuning fractional order proportional integral controllers for fractional order systems

In this paper, two fractional order proportional integral controllers are proposed and designed for a class of fractional order systems. For fair comparison, the proposed fractional order proportional integral (FOPI), fractional order [proportional integral] (FO[PI]) and the traditional integer order PID (IOPID) controllers are all designed following the same set of the imposed tuning constraints, which can guarantee the desired control performance and the robustness of the designed controllers to the loop gain variations. This proposed design scheme offers a practical and systematic way of the controllers design for the considered class of fractional order plants. From the simulation and experimental results presented, both of the two designed fractional order controllers work efficiently, with improved performance comparing with the designed stabilizing integer order PID controller by the observation. Moreover, it is interesting to observe that the designed FO[PI] controller outperforms the designed FOPI controller following the proposed design schemes for the class of fractional order systems considered.     

Stabilization of Uncertain Multi‐Order Fractional Systems Based on the Extended State Observer

The extended state observer (ESO) based controller has been used successfully with integer‐order systems involving large uncertainties. In this paper, the robust control of uncertain multi‐order fractional‐order (FO) systems based on ESO is investigated. First, we transform the multi‐order FO system into an equivalent system in the form of a same‐order state‐space equation. Then, the ESO for the new system is established for estimating both the state and the total disturbance. Sufficient conditions for bounded‐input and bounded‐output stability are derived, and the asymptotic stability of the closed loop system is analyzed, based on whether the states are available or not. Finally, numerical simulations are presented to demonstrate the validity and feasibility of the proposed methodology.     

Stabilisability analysis of high-order unstable processes by fractional-order controllers

In this article, the sufficient stabilisability criteria for unstable time-delay processes by fractional-order controllers are investigated. The process is a high-order system with a single unstable pole, which also has a stable zero. The adopted approach to derive the sufficient conditions for stability is based on the well-known Nyquist stability criterion. Stabilisability is studied by applying fractional-order proportional integral and proportional derivative controllers and the results are given in terms of the maximum allowable value of time delay. Additionally, limitations on the parameters of the controllers which must be taken into account in the controller design are proposed.     

Robust Finite-time Synchronization of Non-identical Fractional-order Hyperchaotic Systems and Its Application in Secure Communication

This paper proposes a novel adaptive sliding mode control (SMC) method for synchronization of non-identical fractional-order (FO) chaotic and hyper-chaotic systems. Under the existence of system uncertainties and external disturbances, finite-time synchronization between two FO chaotic and hyperchaotic systems is achieved by introducing a novel adaptive sliding mode controller (ASMC). Here in this paper, a fractional sliding surface is proposed. A stability criterion for FO nonlinear dynamic systems is introduced. Sufficient conditions to guarantee stable synchronization are given in the sense of the Lyapunov stability theorem. To tackle the uncertainties and external disturbances, appropriate adaptation laws are introduced. Particle swarm optimization (PSO) is used for estimating the controller parameters. Finally, finite-time synchronization of the FO chaotic and hyper-chaotic systems is applied to secure communication.      

Stability and Stabilization of Fractional‐Order Systems with Different Derivative Orders: An LMI Approach

Stability and stabilization analysis of fractional‐order linear time‐invariant (FO‐LTI) systems with different derivative orders is studied in this paper. First, by using an appropriate linear matrix function, a single‐order equivalent system for the given different‐order system is introduced by which a new stability condition is obtained that is easier to check in practice than the conditions known up to now. Then the stabilization problem of fractional‐order linear systems with different fractional orders via a dynamic output feedback controller with a predetermined order is investigated, utilizing the proposed stability criterion. The proposed stability and stabilization theorems are applicable to FO‐LTI systems with different fractional orders in one or both of 0 <  α  < 1 and 1 ≤  α  < 2 intervals. Finally, some numerical examples are presented to confirm the obtained analytical results.     

基于模型依赖驻留时间的异步切换控制

研究在模型依赖平均驻留时间切换策略下切换线性系统的异步切换控制问题,同时考虑模型依赖的控制器滞后时间的约束问题.在实际情况下,信号传输和系统检测等原因会导致控制器的切换滞后于子系统.基于这类情况,首先将子系统运行的区间划分为子系统与控制器相匹配的区间和非匹配的区间,根据模型依赖的驻留时间策略设计出各子系统的控制器;然后,结合模型依赖的控制器滞后时间、系统参数和Lyapunov稳定条件推导出合适的驻留时间设计参数,且使得异步切换系统全局一致指数稳定;最后通过数值仿真验证了所提出方法的有效性.     

Output feedback controller design of Takagi–Sugeno fuzzy systems

This paper presents new systematic design methods of two types of output feedback controllers for Takagi–Sugeno (T–S) fuzzy systems, one of which is constructed with a fuzzy regulator and a fuzzy observer, while the other is an output direct feedback controller. In order to use the structural information in the rule base to decrease the conservatism of the stability analysis, the standard fuzzy partition (SFP) is employed to the premise variables of fuzzy systems. New stability conditions are obtained by relaxing the stability conditions derived in previous papers. The concept of parallel distributed compensation (PDC) is employed to design fuzzy regulators and fuzzy observers from the T–S fuzzy models. New stability analysis and design methods of output direct feedback controllers are also presented. The output feedback controllers design and simulation results for a nonlinear mass-spring-damper mechanical system show that these methods are effective.     

Conical tank level control using fractional order PID controllers: a simulated and experimental study

In this paper, simulated and experimental results on the conical tank level control are presented. PI/PID controllers of integer order (IO) as well as of fractional order (FO) are studied and compared. The tuning parameters are obtained first by using root locus (RL) and Ziegler and Nichols methods, for comparison purposes. Next, particle swarm optimization (PSO) is employed to determine the optimal controllers'' parameters using as fitness function the integral of the absolute value of tracking error (IAE). From the experimental results it is concluded that PI/FOPI are the controllers presenting the lowest IAE indexes, whereas PID/FOPID controllers present the lowest energy consumption by the control signal.     

Achievable closed-loop properties of systems under decentralizedcontrol: conditions involving the steady-state gain

The question of the existence of decentralized controllers for open-loop stable multivariable systems which provide particular closed-loop properties is investigated. In particular, we study the existence of decentralized controllers which provide integral action (Type I closed-loop performance) and also demonstrate one or more of: unconditional stability, integrity with respect to actuator and sensor failure, and decentralized unconditional stability. Necessary, sufficient, and, in some cases, necessary and sufficient conditions on the open-loop steady-state gain are derived such that there exists a controller which provides these desired closed-loop characteristics. These results provide the basis for a systematic approach to control structure selection for decentralized controller design     

Stability analysis of a class of nonlinear fractional‐order systems under control input saturation

This paper studies the stability of a class of nonlinear fractional‐order (FO) systems under input control saturation. Based on the Gronwall‐Bellman lemma and the sector‐bounded condition, sufficient conditions are provided to stabilize such systems by means of a state‐feedback controller. The performance of the proposed controller is tested with 2 FO chaotic systems, namely, the FO brushless direct‐current motor and the Chen systems. The results illustrate the good performance of the new controller under saturation effects.     

Analytical structures and analysis of the simplest fuzzy PI controllers

This paper deals with simplest fuzzy PI controllers which employ two fuzzy numbers on the universe of discourse (UOD) of each input variable, and three fuzzy numbers on the UOD of output variable. Analytical structures of such controllers are derived using triangular membership functions for fuzzification, different combinations of T-norms and T-conorms, different inference methods, and center of area (COA) method for defuzzification. Properties of these controllers are investigated. A comparative study is made on (i) the fuzzy PI controllers derived, and (ii) on the fuzzy PI controllers and their counterpart—conventional PI controller. Moreover, sufficient conditions for bounded-input bounded-output (BIBO) stability of fuzzy PI control systems are established using the well-known small gain theorem.     

时滞L PV 系统的稳定新判据及控制器设计

针对一类具有随参数变化状态时滞的线性参数变化系统，提出一种新的依赖于参数的Lyapunov稳定条件，该准则通过引入两个附加矩阵，解除了系统矩阵与依赖于参数的Lyapunov函数之闻的耦合，更易于系统的分析与综合，在此基础上设计了此类系统的状态反馈控制器，采用线性矩阵不等式技术，将控制器存在的充分条件转化为参数线性矩阵不等式的解存在条件，数值仿真验证了所提出算法的可行性。     

Linear fractional order controllers; A survey in the frequency domain

Today, there is a great tendency toward using fractional calculus to solve engineering problems. The control is one of the fields in which fractional calculus has attracted a lot of attention. On the one hand, fractional order dynamic models simulate characteristics of real dynamic systems better than integer order models. On the other hand, Fractional Order (FO) controllers outperform Integer Order (IO) controllers in many cases. FO-controllers have been studied in both time an frequency domain. The latter one is the fundamental tool for industry to design FO-controllers. The scope of this paper is to review research which has been carried out on FO-controllers in the frequency domain. In this review paper, the concept of fractional calculus and their applications in the control problems are introduced. In addition, basic definitions of the fractional order differentiation and integration are presented. Then, four common types of FO-controllers are briefly presented and after that their representative tuning methods are introduced. Furthermore, some useful continuous and discrete approximation methods of FO-controllers and their digital and analogue implementation methods are elaborated. Then, some Matlab toolboxes which facilitate utilizing FO calculus in the control field are presented. Finally, advantages and disadvantages of using FO calculus in the control area are discussed. To wrap up, this paper helps beginners to get started rapidly and learn how to select, tune, approximate, discretize, and implement FO-controllers in the frequency domain.     

多个线性时滞系统的关联稳定与协调控制

研究线性时滞系统的时滞无关关联稳定性和协调控制问题.利用双线性矩阵不等式(BMI),给出了两个系统关联稳定和协调镇定的充分条件,将关联和协调控制器的设计问题转化为具有BMI约束的非凸优化问题,并给出了求解这类问题的交替算法.最后,通过一个数值例子来说明本文结果的有效性.研究结果表明,不稳定的时滞系统可以通过设计关联或协调控制构成稳定的组合系统.     

An Approach to Design MIMO FO Controllers for Unstable Nonlinear Plants

This paper develops an approach to control unstable nonlinear multi-inputs multi-output (MIMO) square plants using MIMO fractional order (FO) controllers. The controller design uses the linear time invariant (LTI) state space representation of the nonlinear model of the plant and the diagonal closedloop transfer matrix (TM) function to ensure decoupling between inputs. Each element of the obtained MIMO controller could be either a transfer function (TF) or a gain. A TF is associated in turn with its corresponding FO TF. For example, a D (Derivative) TF is related to a FO TF of the form Dδ, δ = [0, 1]. Two applications were performed to validate the developed approach via experimentation: control of the angular positions of a manipulator, and control of the car and arm positions of a translational manipulator.      

Exponential stability and stabilization of uncertain linear time-varying systems using parameter dependent Lyapunov function

In this paper, the problem of exponential stability and stabilization for a class of uncertain linear time-varying systems is considered. The system matrix belongs to a polytope and the time-varying parameter as well as its time derivative are bounded. Based on a time-varying version of Lyapunov stability theorem, new sufficient conditions for the exponential stability and stabilization via parameter dependent state feedback controllers (i.e., a gain scheduling controllers) are given. Using parameter dependent Lyapunov function, the conditions are formulated in terms of two linear matrix inequalities without introducing extra useless decision variables and hence are simply verified. The results are illustrated by numerical examples.     

Delayed standard neural network models for control systems.

In order to conveniently analyze the stability of recurrent neural networks (RNNs) and successfully synthesize the controllers for nonlinear systems, similar to the nominal model in linear robust control theory, the novel neural network model, named delayed standard neural network model (DSNNM) is presented, which is the interconnection of a linear dynamic system and a bounded static delayed (or nondelayed) nonlinear operator. By combining a number of different Lyapunov functionals with S-procedure, some useful criteria of global asymptotic stability and global exponential stability for the continuous-time DSNNMs (CDSNNMs) and discrete-time DSNNMs (DDSNNMs) are derived, whose conditions are formulated as linear matrix inequalities (LMIs). Based on the stability analysis, some state-feedback control laws for the DSNNM with input and output are designed to stabilize the closed-loop systems. Most RNNs and neurocontrol nonlinear systems with (or without) time delays can be transformed into the DSNNMs to be stability-analyzed or stabilization-synthesized in a unified way. In this paper, the DSNNMs are applied to analyzing the stability of the continuous-time and discrete-time RNNs with or without time delays, and synthesizing the state-feedback controllers for the chaotic neural-network-system and discrete-time nonlinear system. It turns out that the DSNNM makes the stability conditions of the RNNs easily verified, and provides a new idea for the synthesis of the controllers for the nonlinear systems.     

Analytical structures and analysis of the simplest fuzzy PDcontrollers

This paper deals with simplest fuzzy PD controllers which employ only two fuzzy sets on the universe of discourse of each input variable, and three fuzzy sets on the universe of discourse of output variable. First, analytical structures of the simplest fuzzy PD controllers are derived via triangular membership functions for fuzzification, intersection T-norm, Lukasiewicz OR and Zadeh (1965) OR T-conorms, Mamdani's minimum, Larsen's product and drastic product inference methods, and center of area method for defuzzification. Properties of such fuzzy PD controllers are investigated. Based on these properties a comparative study is made on fuzzy controllers derived, and also on the fuzzy controllers and their counterpart-conventional linear PD controller. Finally, sufficient conditions for bounded-input bounded-output stability of fuzzy PD control systems are established using the well known small gain theorem.