分数阶系统的分数阶PID控制器设计

对于一些复杂的实际系统,用分数阶微积分方程建模要比整数阶模型更简洁准确.分数阶微积分也为描述动态过程提供了一个很好的工具.对于分数阶模型需要提出相应的分数阶控制器来提高控制效果.本文针对分数阶受控对象,提出了一种分数阶PID控制器的设计方法.并用具体实例演示了对于分数阶系统模型,采用分数阶控制器比采用古典的PID控制器取得更好的效果.     

Extended Kalman filters for fractional‐order nonlinear continuous‐time systems containing unknown parameters with correlated colored noises

By using the Grünwald‐Letnikov (G‐L) difference method and the Tustin generating function method, this study presents extended Kalman filters to achieve satisfactory state estimation for fractional‐order nonlinear continuous‐time systems that containing some unknown parameters with the correlated fractional‐order colored noises. Based on the G‐L difference method and the Tustin generating function method, the difference equations corresponding to fractional‐order nonlinear continuous‐time systems are constructed respectively. The first‐order Taylor expansion is used to linearize the nonlinear functions in the estimated system, which provides the system model for extended Kalman filters. Using the augmented vector method, the unknown parameters are regarded as new state vectors, and the augmented difference equation is constructed. Based on the augmented difference equation, extended Kalman filters are designed to estimate the state of fractional‐order nonlinear systems with process noise as fractional‐order colored noise or measurement noise as fractional‐order colored noise. Meanwhile, the extended Kalman filters proposed in this paper can also estimate the unknown parameters effectively. Finally, the effectiveness of the proposed extended Kalman filters is validated in simulation with two examples.     

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.      

Nonlinear modeling of PEMFC based on fractional order subspace identification

Aiming at the multivariable, nonlinear and fractional‐order characteristics of proton exchange membrane fuel cell (PEMFC), this paper presents a nonlinear state space model based on a novel fractional Hammerstein model subspace identification theory. To reduce the complexity of modeling and choose the suitable input variables, canonical correlation analysis (CCA) method is used to select the most influential factors as the model input variables, and correlation analysis (CA) method is employed to remove the redundant input variables. To guarantee that the input‐output data are derivable at different fractional order, a Poisson moment function (PMF) is employed to construct the fractional order Hammerstein model with a six‐order polynomial as the front static nonlinear unit. To improve computing speed, a fractional differential short memory method (SMM) is proposed to reduce the computation cost of the identification algorithm. Meanwhile, a fuzzy genetic algorithm is adopted to acquire the best fractional order. Simulation results show that the fractional subspace identifying method can avoid fuel cell's internal complexity and PEMFC identification model can describe the working process of PEMFC accurately and quickly, which will provide an ideal control model for some advanced controller.     

Design of fractional order modeling based extended non-minimal state space MPC for temperature in an industrial electric heating furnace

In this paper, an improved approach of extended non-minimal state space (ENMSS) fractional order model predictive control (FMPC) is presented and tested on the temperature model of an industrial heating furnace. In the fractional order model predictive control algorithm, fractional order single-input single-output (SISO) system is discretized via fractional order Grünwald-Letnikov (GL) definition. The ENMSS fractional order model that contains the state variable and the fractional order output tracking error is formulated by choosing appropriate state variables. Meanwhile, the fractional order integral is introduced into the cost function and the GL definition is used to obtain the discrete form of the continuous cost function. Then the control signals are derived by minimizing the fractional order cost function. Lastly, the temperature process control of a heating furnace is illustrated to reflect the performance of the proposed FMPC method. Simulation results show the effectiveness of the proposed FMPC method.     

Characterization of Si‐BCB transmission line at millimeter‐wave frequency by compact fractional‐order equivalent circuit model

In this article, fractional‐order calculus is introduced to characterize Si‐BCB transmission line up to millimeter‐wave frequency region. Direct calculation method is employed to build this compact fractional‐order equivalent circuit model. Measured results have confirmed that the proposed fractional‐order model is capable of accurately describing the behavior of BCB‐Si T‐Line over a large range of frequencies up to 110 GHz.     

On the sliding‐mode control of fractional‐order nonlinear uncertain dynamics

This paper deals with applications of sliding‐mode‐based fractional control techniques to address tracking and stabilization control tasks for some classes of nonlinear uncertain fractional‐order systems. Both single‐input and multi‐input systems are considered. A second‐order sliding‐mode approach is taken, in suitable combination with PI‐based design, in the single‐input case, while the unit‐vector approach is the main tool of reference in the multi‐input case. Sliding manifolds containing fractional derivatives of the state variables are used in the present work. Constructive tuning conditions for the control parameters are derived by Lyapunov analysis, and the convergence properties of the proposed schemes are supported by simulation results. Copyright © 2015 John Wiley & Sons, Ltd.     

On fractional order composite model reference adaptive control

This paper presents a novel composite model reference adaptive control approach for a class of fractional order linear systems with unknown constant parameters. The method is extended from the model reference adaptive control. The parameter estimation error of our method depends on both the tracking error and the prediction error, whereas the existing method only depends on the tracking error, which makes our method has better transient performance in the sense of generating smooth system output. By the aid of the continuous frequency distributed model, stability of the proposed approach is established in the Lyapunov sense. Furthermore, the convergence property of the model parameters estimation is presented, on the premise that the closed-loop control system is stable. Finally, numerical simulation examples are given to demonstrate the effectiveness of the proposed schemes.     

Fractional Modeling and SOC Estimation of Lithium-ion Battery

This paper proposes a state of charge (SOC) estimator of Lithium-ion battery based on a fractional order impedance spectra model. Firstly, a battery fractional order impedance model is derived on the grounds of the characteristics of Warburg element and constant phase element (CPE) over a wide range of frequency domain. Secondly, a frequency fitting method and parameter identification algorithm based on output error are presented to identify parameters of the fractional order model of Lithium-ion battery. Finally, the fractional order Kalman filter approach is introduced to estimate the SOC of the lithium-ion battery based on the fractional order model. The simulation results show that the fractional-order model can ensure an acceptable accuracy of the SOC estimation, and the error of estimation reaches maximally up to 0.5% SOC.      

A Sufficient Condition for Checking the Attractiveness of a Sliding Manifold in Fractional Order Sliding Mode Control

Stability issues of fractional order sliding mode control laws are analyzed in this paper. For differentiation orders less than unity, it is shown that a stable reaching law in the fractional order case corresponds to a stable reaching law in the integer order case. The contribution of the current study is to explain the stability of the closed loop by the use of the Caputo and Riemann‐Liouville definitions of fractional order differentiation. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

网络控制系统分数阶PI~λD~μ控制器研究

针对带延时的分数阶的网络控制系统,采用分数阶的P I D控制器对系统进行仿真,并与不存在网络时延的系统进行对比。仿真结果表明,网络时延使系统敏感性增加,但分数阶控制器本身的鲁棒性,仍可保证系统具有良好的控制效果。     

一类分数阶超混沌系统的同步及其应用

应用分数阶微积分稳定性理论,提出了一类分数阶超混沌系统同步控制的新方法,通过在分数阶超混沌响应系统中设计两个控制器,实现了分数阶超混沌Chen系统之间的同步。并将该方案应用到保密通信中,利用混沌掩盖技术,实现了复杂非周期信号的安全传送,在接收端通过去掩盖,毫无失真地恢复了有用传送信号。数值仿真和理论分析结果的一致性表明了该方案的有效性和可行性。     

PDα‐type iterative learning control for fractional‐order linear continuous‐time switched systems

For a class of fractional‐order linear continuous‐time switched systems specified by an arbitrary switching rule, this paper proposes a PD^α‐type fractional‐order iterative learning control algorithm. For systems disturbed by bounded measurement noise, the robustness of PD^α‐type algorithm is first discussed in the iteration domain and the tracking performance is analyzed. Next, a sufficient condition for monotone convergence of the algorithm is studied when external noise is absent. The results of analysis and simulation illustrate the feasibility and effectiveness of the proposed control algorithm.     

基于最大灵敏度的加热系统分数阶内模控制

针对加热系统热传导过程模型不精确和系统参数不确定性问题, 提出一种新的基于最大灵敏度的分数阶内模控制方案. 采用分数阶模型描述加热系统可以提高精度, 而内模控制能够很好地处理系统参数不确定性问题. 利用最大灵敏度整定分数阶控制器参数, 并以此获得强鲁棒性控制系统. 数值结果验证了所提出的分数阶内模控制方案的有效性, 具有比整数阶内模控制方案更好的控制性能.

     

A composite fractional order active disturbance rejection control method for a class of uncertain systems with multiple disturbances

The existing active disturbance rejection control (ADRC) method may not provide sufficient disturbance rejection to multiple mismatched disturbances for the fractional order systems. In this paper, a composite disturbance rejection approach is developed for a class of fractional order uncertain systems, by synthesizing the fractional order ADRC (FOADRC) approach and a disturbance observer (DO)-based compensation scheme. Taking advantage of more disturbance information and a filter structure, an improved DO is developed to achieve precise estimation of disturbances in the presence of sensor noises. In addition, a state transformation is developed to convert the system into a simple integral chain model with only matched disturbances. Then a composite control law is designed to compensate the disturbances and provide satisfying dynamic performance. The efficiency of the proposed method is demonstrated by a numerical simulation and an actual servo control simulation, as well as the comparison with two kinds of the existing ADRC methods and the commonly used integral sliding mode control (I-SMC) method.     

分数阶控制器的数字实现及其特性

针对分数阶控制器数值计算和应用难的问题,研究了分数阶控制器的数字实现方法和控制特性．取Grunwald-Letnikov定义有限项,并直接离散化,得到有限记忆数字实现法;利用Tustin算子生成函数把分数阶微分由复频域变换到Z域,然后用连分式展开式CFE(continued fraction expansion)近似展开,可得到Tustin CFE数字实现法．两种方法的频域对比分析表明Tustin CFE法优于有限记忆法．采用设计的分数阶控制器的数字实现方法,对分数阶控制器和传统PID控制器的控制性能进行了对比实验分析．研究结果表明:分数阶控制器对非线性具有较强的控制能力．     

Sliding mode control approaches to the robust regulation of linear multivariable fractional‐order dynamics

Sliding mode control approaches are developed to stabilize a class of linear uncertain fractional‐order dynamics. After making a suitable transformation that simplifies the sliding manifold design, two sliding mode control schemes are presented. The first one is based on the conventional discontinuous first‐order sliding mode control technique. The second scheme is based on the chattering‐free second‐order sliding mode approach that leads to the same robust performance but using a continuous control action. Simple controller tuning formulas are constructively developed along the paper by Lyapunov analysis. The simulation results confirm the expected performance. Copyright © 2010 John Wiley & Sons, Ltd.     

航空发动机分数阶PID控制器的参数自整定方法

控制器作为航空发动机的大脑,是保障发动机正常运行的核心部件,随着对发动机控制器精度和时效性的要求越来越高,传统PID控制器的性能亟需进一步提升.本文提出了改进的分数阶PID离线和在线参数整定方法,应用于涡扇发动机推力的控制中.首先,利用Caputo分数阶微积分定义建立分数阶PID模型,实现时域上的数值计算;其次,基于对数正态分布提出了改进的布谷鸟算法,实现了分数阶PID离线参数整定;然后,结合RBF网络设计参数线上整定方法,解决了参数在线整定问题;最后将相关理论应用于发动机推力的控制中,结果表明,相比其他几种优化算法,改进的布谷鸟优化算法对分数阶PID控制参数整定效果最好;利用RBF神经网络对分数阶PID进行在线整定时控制效果稳定,且分数阶PID的控制效果优于传统的PID控制,能提高对推力的控制能力.     

基于分数阶滑模控制技术的永磁同步电机控制

针对传统整数阶滑模控制系统中存在的抖震问题,本文提出了分数阶滑模控制策略并应用到永磁同步电机的速度控制.传统滑模控制器中的开关函数由作用在切换流型或其整数阶导数面推广到其分数阶导数面,利用分数阶系统的特性,缓慢地传递系统的能量,有效地削减抖震.本文采用模糊逻辑推理算法,实现软开关切换增益的自整定.仿真和实验证明,本文提出的分数阶滑模控制系统不但能有效地削减抖震,而且能保持滑模控制器对系统参数变化和外部扰动的鲁棒性.     

Unscented Kalman filter for continuous‐time nonlinear fractional‐order systems with process and measurement noises

This study proposes the design of unscented Kalman filter for a continuous‐time nonlinear fractional‐order system involving the process noise and the measurement noise. The nonlinear fractional‐order system is discretized to get the difference equation. According to the unscented transformation, the design method of unscented Kalman filter for a continuous‐time nonlinear fractional‐order system is provided. Compared with the extended Kalman filter, the proposed method can obtain a more accurate estimation effect. For fractional‐order systems containing non‐differentiable nonlinear functions, the method proposed in this paper is still effective. The unknown parameters are also discussed by the augmented vector method to achieve the state estimation and parameter identification. Finally, two examples are offered to verify the effectiveness of the proposed unscented Kalman filter for nonlinear fractional‐order systems.