分数阶控制系统

随着基于分数阶次的数学理论日益完善,分数阶控制系统也越来越广泛地被研究和讨论。为了完善分数阶控制系统的理论体系,给出了分数阶系统的总体综述。介绍了分数微积分的定义,给出了分数阶线性定常系统的传递函数和状态空间描述,简要介绍了分数阶控制系统的复频域分析和分数阶控制器及控制器的设计方法,并分析了几种设计方法的优缺点。分数系统的研究必然能找到合适的切入点广泛地进入现代控制领域,为真正实现工业控制自动化提供强有力的理论依据。     

一类分数阶非线性混沌系统的同步控制

在分数阶非线性系统同步控制的研究中,针对一类分数阶非线性混沌系统,研究了基于分数阶控制器的同步方法.利用状态反馈方法和分数阶微积分定义,设计了分数阶混沌系统同步控制器.进一步,根据分数阶非线性系统稳定性理论、Mittag-Leffler函数、Laplace变换以及Gronwall不等式,证明了同步控制器的有效性.最后,通过数值仿真,实现了初始值不同的两个分数阶非线性混沌系统同步.误差响应曲线表明研究的分数阶非线性系统同步响应速度快,控制精度高,验证了本文所设计的混沌同步控制方案的可行性.     

Fractional Order Modeling of Human Operator Behavior with Second Order Controlled Plant and Experiment Research

Modeling human operator's dynamics plays a very important role in the manual closed-loop control system, and it is an active research area for several decades. Based on the characteristics of human brain and behavior, a new kind of fractional order mathematical model for human operator in single-input single-output (SISO) systems is proposed. Compared with the traditional models based on the commonly used quasilinear transfer function method or the optimal control theory method, the proposed fractional order model has simpler structure with only few parameters, and each parameter has explicit physical meanings. The actual data and experiment results with the second-order controlled plant illustrate the effectiveness of the proposed method.      

Practical Application of Fractional Order Controllers to a Delay Thermal System

This paper provides an application of Fractional Model Predictive Control (FMPC) and fractional-order Proportional Integral controller (P I^λ) on a thermalsystem with time delay.The first controller is based on Grünwald-Letnikov’s method to predict the future dynamic behavior of the system. This methodconsists in replacing the non-integer derivation operator of the adopted system representation by a discrete approximation. Therefore, this controller isdeveloped on the basis of a fractional order model. However, the second controller is founded on an extended version of Hermite-Biehler theorem todetermine the complete set stabilizing P I^λ parameters Experiment results onto a time delay thermal system are given to illustrate the effectiveness of thedeveloped strategies.     

Fuzzy Adaptive Control of a Fractional Order Chaotic System With Unknown Control Gain Sign Using a Fractional Order Nussbaum Gain

In this paper we propose an improved fuzzy adaptive control strategy, for a class of nonlinear chaotic fractional order (SISO) systems with unknown control gain sign. The online control algorithm uses fuzzy logic sets for the identification of the fractional order chaotic system, whereas the lack of a priori knowledge on the control directions is solved by introducing a fractional order Nussbaum gain. Based on Lyapunov stability theorem, stability analysis is performed for the proposed control method for an acceptable synchronization error level. In this work, the Grünwald-Letnikov method is used for numerical approximation of the fractional order systems. A simulation example is given to illustrate the effectiveness of the proposed control scheme.      

垂直运动永磁直线同步电动机的建模与控制仿真

运用电动机统一理论及参照旋转电动机的数学建模方法并结合PMLSM的特点,建立了PMLSM的数学模型。通过对Matlab/Simulink下旋转同步电动机模型的分析,建立了基于Matlab/Simulink的通用PMLSM仿真模型,可以利用它方便地进行垂直运动的PMLSM控制系统的仿真和分析。仿真结果证明,该电动机模型能够较好地反映实际电动机的工作运行特性,对于实际控制系统的设计制造具有很好的参考价值。     

Type Number Based Steady‐State Error Analysis on Fractional Order Control Systems

In this paper, steady‐state error of fractional order control systems (FOCSs) is analyzed based on the concept of type number. Some practical definitions of the reference and the error for FOCSs with non‐unity feedback are first given. Based on these definitions, the type number concept is first introduced to FOCSs with non‐unity feedback. Then steady‐state error of various types of FOCSs are analyzed respectively via type number concept. It is also shown that FOCSs with type number more than 1 are not able to track particular reference without changing the sign of the tracking error.     

Robust Output Feedback Control for Fractional Order Nonlinear Systems with Time-varying Delays

Robust controller design problem is investigated for a class of fractional order nonlinear systems with time varying delays. Firstly, a reduced-order observer is designed. Then, an output feedback controller is designed. Both the designed observer and controller are independent of time delays. By choosing appropriate Lyapunov functions, we prove the designed controller can render the fractional order system asymptotically stable. A simulation example is given to verify the effectiveness of the proposed approach.      

Consensus Control of Fractional‐Order Systems Based on Delayed State Fractional Order Derivative

In this paper, the delayed state fractional order derivative (DSFOD) is introduced into the existing traditional consensus protocol aiming to improve the robustness of fractional‐order multi‐agent systems against communication time delay. Both of communication channels with time‐delay and without time‐delay cases are considered. Based on the frequency‐domain analysis and algebraic graph theory, it is shown that properly choosing the intensity of DSFOD can improve the robustness of fractional‐order multi‐agent systems against communication delay. Finally, a simulated example with simulations is presented to confirm the correctness and effectiveness of the theoretical results.     

复杂系统的分数阶内模控制器设计

针对高阶复杂系统提出一种分数阶内模控制器设计方法。利用微粒群算法(PSO)进行模型化简,基于内模控制(IMC)原理设计分数阶控制器,该控制器仅有一个可调参数,并根据鲁棒性能指标给出控制器参数整定的解析表达式。仿真结果表明,该方法可以使系统同时具有良好的目标值跟踪特性、扰动抑制特性以及克服参数变化的鲁棒性。     

基于分数阶滤波器的高泛化性神经网络建模

为了显著提高神经网络建模的泛化性,提出了激活函数是分数阶滤波器的神经网络.分数阶滤波器涵盖了Butterworth和Chebyshev滤波器的性能.滤波器集对样本信号进行频率分解,既提升了信息的致密性,也保证了遍历性,更有助于提高神经网络的泛化性.各滤波器参数由盲动粒子群优化算法寻优.神经网络解算时,既采用了线性回归求解神经网络输出层权重,又在有限频段上用线性传递函数模拟替代分数阶传递函数,这两种措施缩短了解算时间.仿真结果表明,线性系统的泛化性精度可达亿分之几,非线性系统可达万分之几,可以离线应用.     

分数阶PD控制器在时滞伺服系统中的研究

分数阶微积分理论的发展推动了分数阶微积分在各个领域的广泛应用,分数阶控制算法的研究也成为近几年的研究热点.目前分数阶控制理论的研究主要集中在理论分析,对于目前所研究的这类工程性较强的控制对象,成熟的研究成果较少.针对实际工程中含有时滞的伺服系统模型,将Flat phase法和系统稳定性裕度条件相结合,设计了分数阶PD控制器.此PD控制器结构简单,对参数变化稳定性好,控制精度高.为了对比仿真实验,文章还设计了整数阶PID控制器.仿真结果表明,分数阶控制器比传统整数阶PID控制器的控制精度更高,鲁棒性更好.     

State Feedback Control for a Class of Fractional Order Nonlinear Systems

Using the Lyapunov function method, this paper investigates the design of state feedback stabilization controllers for fractional order nonlinear systems in triangular form, and presents a number of new results. First, some new properties of Caputo fractional derivative are presented, and a sufficient condition of asymptotical stability for fractional order nonlinear systems is obtained based on the new properties. Then, by introducing appropriate transformations of coordinates, the problem of controller design is converted into the problem of finding some parameters, which can be certainly obtained by solving the Lyapunov equation and relevant matrix inequalities. Finally, based on the Lyapunov function method, state feedback stabilization controllers making the closed-loop system asymptotically stable are explicitly constructed. A simulation example is given to demonstrate the effectiveness of the proposed design procedure.      

Fractional Order Controller for Controlling Power System Dynamic Behavior

Several power system networks exhibit bifurcation, chaos and instability behavior for some specific values of initial conditions and parameters. Angle and voltage instability behavior of power system is prone to such specific values and parameter variation. This paper proposes fractional order proportional integral controller (FOPI) based state feedback for precise and robust control of such undesirable behavior. This paper proposes first ever use of FOPI for precise rotor angle control leading to instability in power system dynamic behavior. FOPI controller is applied on generator connected to IEEE‐14 bus benchmark model. The ripple frequency of turbine torque is chosen as one of the cause of instability behavior of power system, which has the potential possibility to push system behavior to chaos and instability mode. The proposed FOPI controller design will inhibit the dynamic behavior of power system to safe and stable bounds. Proposed strategy can be applied to other large power system models as well due to its simplicity in design philosophy. Several phase plane trajectories with and without FOPI controller are used to support the viewpoint.     

无刷直流电机的分数阶建模方法

传统的基于机理的整数阶无刷直流电机仿真模型和实际物理样机有较大差距.论文通过分析无刷直流电机的数学模型,在基于机理建模整数阶模型基础上引入分数阶微积分,将机理建模和数字建模相结合,提出一种新的无刷直流电机建模方法.首先根据物理样机建立基于机理的无刷直流电机仿真模型,然后采用基于输出误差的最小二乘法辨识出分数阶模型的参数...     

单相并网逆变电流分数阶PI控制策略研究

并网发电系统中,逆变器输出的并网电流易受到电网电压周期性扰动等非线性因素干扰,导致并网电流波形畸变。建立单相光伏并网逆变控制系统模型,设计一种逆变电流分数阶PI(PIλ或FO-PI)控制器,根据控制系统的频域特性,确立控制器的比例、积分系数及积分阶次与系统性能指标的关系;在整数阶PI控制器和分数阶PI控制器的分别作用下,对系统的动态和稳态性能、抗干扰性及并网逆变特性进行仿真对比,并应用快速傅里叶变换（FFT）对系统各自的逆变并网电流质量进行了分析。仿真结果表明,分数阶控制系统在满足各项稳态性能指标的同时,降低了并网电流谐波总畸变率（THD）,并提升了系统的动态性能和抗干扰能力。     

Constrained Fractional Variational Problems of Variable Order

Isoperimetric problems consist in minimizing or maximizing a cost functional subject to an integral constraint. In this work, we present two fractional isoperimetric problems where the Lagrangian depends on a combined Caputo derivative of variable fractional order and we present a new variational problem subject to a holonomic constraint. We establish necessary optimality conditions in order to determine the minimizers of the fractional problems. The terminal point in the cost integral, as well as the terminal state, are considered to be free, and we obtain corresponding natural boundary conditions.      

Optimum Design of Fractional Order Pid Controller Using Chaotic Firefly Algorithms for a Control CSTR System

A fractional‐order PID controller is a generalization of a standard PID controller using fractional calculus. Compared with the standard PID controller, two adjustable variables, “differential order” and “integral order”, are added to the PID controller. Fractional‐order PID is more flexible, has better responses, and the precise adjustment closed‐loop system stability region is larger than that of a classic PID controller. But the design and stability analysis is more complicated than for the PID controller. Therefore, the optimal setting of parameters is very important. A firefly algorithm in standard mode has only local optimization and accuracy is low. In order to fix this flaw an improved chaotic algorithm firefly is proposed for a design controller FOPID. To evaluate the performance of the proposed controller, it has been used in the control of a CSTR system with a variety of fitness functions. Simulations confirm the optimal performance of the proposed controller.     

基于模型降阶的分数阶系统动态性能研究

由于大多数分数阶系统阶次过高,使得系统的控制器设计变得非常困难,会造成系统控制精度变差且动态性能降低等不利因素,而模型降阶技术是解决这一问题的有效工具.首先简要介绍了H2范数模型降阶的一般方法,并提出一种新的降阶模型结构,可以使降阶后的模型扩展到分数阶并且更加精确地逼近各种高阶系统.仿真结果证明,采用改进型H2范数模型降阶方法不仅保持了原有分数阶模型系统的动态性能而且还提升了原有整数阶模型系统的动态性能.     

Adaptive Mittag‐Leffler Stabilization of a Class of Fractional Order Uncertain Nonlinear Systems

This paper deals with the stabilization of a class of commensurate fractional order uncertain nonlinear systems. The fractional order system concerned is of the strict‐feedback form with uncertain nonlinearity. An adaptive control scheme combined with fractional order update laws is proposed by extending classical backstepping control to fractional order backstepping scheme. The asymptotic stability of the closed‐loop system is guaranteed under the construction of fractional Lyapunov functions in the sense of generalized Mittag‐Leffler stability. The fractional order nonlinear system investigated can be stabilized asymptotically globally in presence of arbitrary uncertainty. Finally illustrative examples and numerical simulations are performed to verify the effectiveness of the proposed control scheme.