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.      

Proportional stabilization and closed-loop identification of an unstable fractional order process

This paper deals with proportional stabilization and closed-loop step response identification of the fractional order counterparts of the unstable first order plus dead time (FOPDT) processes. At first, the necessary and sufficient condition for stabilizability of such processes by proportional controllers is found. Then, by assuming that a process of this kind has been stabilized by a proportional controller and the step response data of the closed-loop system is available, an algorithm is proposed for estimating the order and the parameters of an unstable fractional order model by using the mentioned data.     

State-space analysis and identification for a class of hysteretic systems

In this paper we present results on the twin subjects of system analysis and system identification for a class of state-space realizable dynamic systems under the influence of hysteresis. The class of systems in question consists of models in the form of a linear time-invariant dynamic system in series with a differential model of hysteresis. It will be demonstrated that under fairly light constraints on the differential model of hysteresis, it is possible to design a series of experiments leading towards the identification of the full state-space realization. The approach is tested successfully on a high-precision mechanical translation system affected by hysteresis.     

On-line identification of SISO linear time-invariant delay systems from output measurements

The paper presents a theoretical treatment of on-line identification issues and develops parameter identification algorithms for a class of single input single output (SISO) linear time delay systems (LTDS) with uncertain time-invariant parameters. The delays can also be unknown if their possible values belong to a set consisting of a finite number of known values. Since only output measurements are assumed to be available, a sliding mode-based observer of an underlying SISO LTDS with uncertain parameters is constructed and utilized to design an adaptive identifier of system parameters. The theory is supported by an application of the developed algorithms to transient fuel identification in a port fuel injected (PFI) internal combustion engine.     

Grey-box state-space identification of nonlinear mechanical vibrations

The present paper deals with the identification of nonlinear mechanical vibrations. A grey-box, or semi-physical, nonlinear state-space representation is introduced, expressing the nonlinear basis functions using a limited number of measured output variables. This representation assumes that the observed nonlinearities are localised in physical space, which is a generic case in mechanics. A two-step identification procedure is derived for the grey-box model parameters, integrating nonlinear subspace initialisation and weighted least-squares optimisation. The complete procedure is applied to an electrical circuit mimicking the behaviour of a single–input, single–output (SISO) nonlinear mechanical system and to a single–input, multiple–output (SIMO) geometrically nonlinear beam structure.     

分数阶线性系统的内部和外部稳定性研究

介绍了分数阶线性定常系统的状态方程描述和传递函数描述．运用拉普拉斯变换和留数定理，给出并证明了分数阶线性定常系统的内部和外部稳定性条件，并讨论了其相互关系．以一个粘弹性系统的实例验证了上述方法的正确性．     

分数阶系统时域子空间辨识

研究了分数阶系统的时域辨识问题,给出了一种新的分数阶系统时域子空间辨识算法.当分数阶微分阶次已知时,通过计算输入输出信号的分数阶微分,构造新的输入输出数据方程对系统的参数进行子空间辨识.当分数阶微分阶次未知时,通过代价函数将阶次辨识问题转化为参数寻优问题.采用Poisson滤波器有效避免了在计算分数阶微分时输入输出信号必须高阶可导的问题.通过分析给出了权矩阵的选取方式,提高了时域子空间辨识结果的精度.数值仿真结果表明了该算法的有效性.     

Subspace-based continuous-time identification of fractional order systems from non-uniformly sampled data

This work focuses on the identification of fractional commensurate order systems from non-uniformly sampled data. A novel scheme is proposed to solve such problem. In this scheme, the non-uniformly sampled data are first complemented by using fractional Laguerre generating functions. Then, the multivariable output error state space method is employed to identify the relevant system parameters. Moreover, an in-depth property analysis of the proposed scheme is provided. A numerical example is investigated to illustrate the effectiveness of the proposed method.     

Control of a grinding mill circuit using fractional order controllers

This paper presents the design and application of fractional single-input–single-output (SISO) controllers to a grinding mill circuit, which is a multiple-input–multiple-output (MIMO) process. Two kinds of controllers are presented: fractional order proportional-integral (FOPI) controllers, and a combination of FOPI and fractional order model reference adaptive controllers (FOMRAC). The parameters of the controller are tuned using off-line particle swarm optimization. In the presence of disturbances and process noise, the SISO fractional controllers achieve similar or better performance compared to linear model predictive control (LMPC).     

分数阶线性定常系统的稳定性及其判据

介绍了分数阶微分方程和分数阶系统 ,给出分数阶线性定常系统的传递函数描述和状态空间描述 .给出了分数阶线性定常系统的稳定性条件 ,并结合分数阶状态方程给出定理的证明 .直接从复分析中的辐角原理出发 ,推导出分数阶线性定常系统 2个有效的稳定性判据 :分数阶系统奈奎斯特判据和分数阶系统对数频率判据 .通过实例验证了其有效性     

Fractional variable order discrete-time systems,their solutions and properties

In this paper, fractional variable order discrete state-space systems based on different definitions of fractional variable order difference are investigated.The general solution of these systems is derived. Moreover, necessary and sufficient conditions for reachability and observability are given and proven. The sufficient conditions for controllability are proposed too.     

分数阶时滞系统的频域子空间辨识

研究了利用频率响应数据辨识分数阶时滞系统子空间模型的问题,给出了一种差分进化算法与频域子空间方法相结合的辨识算法.利用差分进化算法搜索最优分数微分阶次和时滞参数,而对于固定的分数微分阶次和时滞,则采用分数阶频域子空间辨识方法得到状态空间模型.通过仿真算例验证了该算法的有效性.     

Parameter identification of continuous-time systems using iterative learning control

This paper presents a system identification technique for continuous-time state-space system using the iterative learning control. The transfer function parameters are regarded as functions with respect to the state-space parameters which will be identified. The relationship between the state-space parameters and the response error is explicitly derived. An update law of the state-space parameters is proposed so as to improve the convergence speed. The effectiveness of the proposed identification technique is demonstrated by numerical examples.     

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.      

分数阶系统变阶次状态空间建模及稳定理论

结合分数阶微积分算子的叠加原理,提出了变阶次状态空间建模方法.将分数阶系统推广到状态空间领域,实现了最小阶状态空间转换,并可根据实际需要通过增加状态变量来提取某一阶次的输出信号.对于各阶次均小于1的变阶次状态空间实现的分数阶系统,提出了变阶次分数阶系统的稳定性判定定理.最后通过实例仿真验证了所提出方法的有效性.     

Fractional order [proportional derivative] controller for a class of fractional order systems

Recently, fractional order systems (FOS) have attracted more and more attention in various fields. But the control design techniques available for the FOS suffer from the lack of direct systematic approaches. In this paper, we focus on a given type of simple model of FOS. A fractional order [proportional derivative] (FO-[PD]) controller is proposed for this class of FOS, and a practical and systematic tuning procedure has been developed for the proposed FO-[PD] controller synthesis. The fairness issue in comparing with other controllers such as the traditional integer order PID (IO-PID) controller and the fractional order proportional derivative (FO-PD) controller has been addressed under the same number of design parameters and the same specifications. Fair comparisons of the three controllers (i.e., IO-PID, FO-PD and FO-[PD]) via the simulation tests illustrate that, the IO-PID controller designed may not always be stabilizing to achieve flat-phase specification while both FO-PD and FO-[PD] controllers designed are always stabilizing. Furthermore, the proposed FO-[PD] controller outperforms FO-PD controller for the class of fractional order systems.     

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.     

Control and Synchronization Of A Class Of Uncertain Fractional Order Chaotic Systems Via Adaptive Backstepping Control

This paper presents the stabilization and synchronization problem of a class of fractional order chaotic systems with unknown parameters. A systematic step by step approach is explained to derive control results using an adaptive backstepping strategy. The analytically obtained control structure, derived by blending a systematic backstepping procedure with Mittag‐Leffler stability results, helps in obtaining the stability of a strict feedback‐like class of uncertain fractional order chaotic systems. The results are further extended to achieve synchronization of these systems in master–slave configuration. Thereafter, the methodology has been applied to two example systems, that is, chaotic Chua's circuit and Genesio‐Tesi system, which belong to addressed class, in order to show the application of results. Numerical simulation given at the end confirms the efficacy of the scheme presented here.     

Model reduction for state-space symmetric systems

In this paper, the model reduction problem for state-space symmetric systems is investigated. First, it is shown that several model reduction methods, such as balanced truncation, balanced truncation which preserves the DC gain, optimal and suboptimal Hankel norm approximations, inherit the state-space symmetric property. Furthermore, for single input and single output (SISO) state-space symmetric systems, we prove that the H_∞ norm of its transfer functions can be calculated via two simple formulas. Moreover, the SISO state-space symmetric systems are equivalent to systems with zeros interlacing the poles (ZIP) under mild conditions.