Frequency response of uncertain systems with interval plants

Frequency domain properties of uncertain rational functions depending on two independent interval polynomials are investigated. Several simplifications for efficient computation of the envelope of the Nyquist plots of the uncertain family are reported. Stronger computational reduction results are given for the Bode magnitude and phase plots of an interval plant-controller family of transfer functions. These results allow for a clear understanding of the extremality properties of several frequency domain performance specifications commonly used in control system design     

Construction of bode envelopes using REP based range finding algorithms

The frequency domain analysis of systems is an important topic in control theory. Powerful graphical tools exist in classic control, such as the Nyquist plot, Bode plots, and Nichols chart. These methods have been widely used to evaluate the frequency domain behavior of system. A literature survey shows that various approaches are available for the computation of the frequency response of control systems under different types of parametric dependencies, such as affine, multi-linear, polynomial, etc. However, there is a lack of tools in the literature to construct the Bode envelopes for the general nonlinear type of parametric dependencies. In this paper, we address the problem of computation of the envelope of Bode frequency response of a non-rational transfer function with nonlinear parametric uncertainties varying over a box. We propose two techniques to compute the Bode envelopes:first, based on the natural interval extensions (NIE) combined with uniform subdivision and second, based on the existing Taylor model combined with subdivision strategy. We also propose the algorithms to further speed up both methods through extrapolation techniques.     

A Note on Robust Stability Analysis of Fractional Order Interval Systems by Minimum Argument Vertex and Edge Polynomials

By using power mapping (s=vm), stability analysis of fractional order polynomials was simplified to the stability analysis of expanded degree integer order polynomials in the first Riemann sheet. However, more investigation is needed for revealing properties of power mapping and demonstration of conformity of Hurwitz stability under power mapping of fractional order characteristic polynomials. Contributions of this study have two folds:Firstly, this paper demonstrates conservation of root argument and magnitude relations under power mapping of characteristic polynomials and thus substantiates validity of Hurwitz stability under power mapping of fractional order characteristic polynomials. This also ensures implications of edge theorem for fractional order interval systems. Secondly, in control engineering point of view, numerical robust stability analysis approaches based on the consideration of minimum argument roots of edge and vertex polynomials are presented. For the computer-aided design of fractional order interval control systems, the minimum argument root principle is applied for a finite set of edge and vertex polynomials, which are sampled from parametric uncertainty box. Several illustrative examples are presented to discuss effectiveness of these approaches.      

Computation of the frequency response of multilinear affine systems

Deals with the problem of computing the frequency response of an uncertain transfer function whose numerator and denominator polynomials are multiples of independent uncertain polynomials of the form P(s, q) = l/sub o/ (q) + l/sub 1/ (q) s + /spl middot//spl middot//spl middot/ + l/sub n/, (q) s/sup n/ whose coefficients depend linearly on q = [q/sub 1/, q/sub 2/, ..., q/sub q/]/sup T/ and the uncertainty box is Q = {q: q/sub i/ /spl epsiv/ [q/sub i/, q/sub i/], i = 1, 2,..., q}. Using the geometric structure of the value set of P(s, q), a powerful edge elimination procedure is proposed for computing the Bode, Nyquist, and Nichols envelopes of these uncertain systems. A numerical example is included to illustrate the benefit of the method presented.     

Generalized Bode envelopes and generalized Nyquist theorem for analysis of uncertain systems

This work is concerned with analysis of uncertain control systems, with regard to clustering of poles inside a simple symmetric bounded contour. We present and use the generalized Bode envelopes, generalized Mikhailov theorem, and the generalized Nyquist theorem for the simultaneous analysis of stability and quality of transient response of uncertain systems. Our results allow one to determine the number of open‐ and closed‐loop poles inside the given symmetric simply connected contour, for the entire uncertain family. The proposed method is computationally efficient and appropriate for solving complex control analysis problems, as evidenced in the examples. Copyright © 2010 John Wiley & Sons, Ltd.     

Fractional order and BICO disturbance observers for a run-of-mine ore milling circuit

Grinding mill circuits are hard to control due to poor plant models, large external disturbances, uncertainties from internal couplings, and process variables that are difficult to measure. This paper proposes a novel fractional order disturbance observer (FO-DOB) for a run-of-mine (ROM) ore milling circuit. A fractional order low pass filter (Q-filer) is used in the DOB to offer an additional degree of freedom in tuning for set-point tracking performance and disturbance rejection performance. Another disturbance observer is introduced in which a Bode ideal cut-off (BICO) filter is used for the Q-filter. A full non-linear plant model is used for evaluation of the performance gained over the ubiquitous PI controller. The simulation results show that the FO-DOB and BICO-DOB schemes are useful additional tools for ROM ore milling circuit control implementations.     

Solving fractional pantograph delay differential equations via fractional-order Boubaker polynomials

In the current study, we introduce fractional-order Boubaker polynomials related to the Boubaker polynomials to achieve the numerical result for pantograph differential equations of fractional order in any arbitrary interval. The features of these polynomials are exploited to construct the new fractional integration and pantograph operational matrices. Then these matrices and least square approximation method are used to reorganize the problem to a nonlinear equations system which can be resolved by means of the Newton’s iterative method. The brief discussion about errors of the used estimations is deliberated and, finally, some examples are included to demonstrate the validity and applicability of our method.

     

基于Nyquist图的光伏并网电能质量综合补偿方法

在单相光伏并网逆变控制中,针对谐振频率高于截止频率时Bode图及其稳定判据的使用受到限制等问题,提出一种基于 Nyquist图的谐波补偿方法。该方法将Nyquist曲线到临界稳定点（－1,j0）的最短距离作为系统稳定程度的表征量,通过对矢量比例积分控制器（VPI）设计合适的相角补偿使该距离在Nyquist图中达到最大,从而提高奇次谐波补偿次数,同时避免闭环异常峰,闭环系统的稳定性和抗干扰能力得到增强。改进后的VPI控制使系统保持较高的稳定性,参数变化对系统稳定性的影响可忽略不计,因而简化了参数设计过程。理论分析和仿真结果验证了基于Nyquist图进行改进后的VPI控制策略的有效性和可行性。     

Bode 理想传递函数在分数阶控制中的应用

本文将Bode 理想传递函数应用于分数阶控制器的设计和分数阶PID 控制器参数整定中．所得控制器 可以在满足系统要求的截止频率和相角裕度的前提下,使补偿后系统Bode 图的相频特性曲线在截止频率附近有一 个水平区域,即闭环系统对增益的变化具有鲁棒性．它不仅适合于分数阶对象,也适用于整数阶对象,并能够提高 系统的控制品质．仿真结果证明了上述方法的有效性．     

Kharitonov's theorem extension to interval polynomials which candrop in degree: a Nyquist approach

This paper studies the Hurwitz stability of interval polynomials which can drop in degree from a Nyquist point of view. These families make it possible to deal with system families taking into account different dynamic behaviors in the modeling of a plant. The behavior of the interval polynomials is studied, and it is proven that Kharitonov's theorem can be extended     

MATLAB环境下频率响应曲线的绘制方法

本文给出了MATLAB环境下线性系统的Bode图、Nyquist图、Nichols图的绘制方法,为线性控制系统的频域分析提供了一种简单有效的途径。     

An efficient approximate method for solving linear fractional Klein–Gordon equation based on the generalized Laguerre polynomials

In this paper, a new approximate formula of the fractional derivative is derived. The proposed formula is based on the generalized Laguerre polynomials. Global approximations to functions defined on a semi-infinite interval are constructed. The fractional derivatives are presented in terms of Caputo sense. Special attention is given to study the error and the convergence analysis of the proposed formula. A new spectral Laguerre collocation method is presented for solving linear fractional Klein–Gordon equation (LFKGE). The properties of Laguerre polynomials are utilized to reduce LFKGE to a system of ordinary differential equations, which solved using the finite difference method. Numerical results are provided to confirm the theoretical results and the efficiency of the proposed method.     

Control system design for parametric uncertainty

This paper introduces some recently developed frequency-domain design techniques that are effective in the design of control systems that are required to be robust under parametric uncertainty. We have extended the well-known classical control tools (i.e., Nyquist plot, Bode plot, and Nichols chart) developed for a fixed plant to the domain of families of plants where the uncertain parameter varies in intervals. Using this new family of plots, classical control design techniques can be used to design robust control systems. The technique is illustrated by examples.     

Controller Synthesis Free of Analytical Models: Three Term Controllers

The main focus of this paper is on direct data driven synthesis and design of controllers. We show that the complete set of stabilizing proportional–integral–derivative (PID) and first-order controllers for a finite dimensional linear time-invariant plant, possibly cascaded with a delay, can be calculated directly from the frequency response (Nyquist/Bode) data $P(jmath omega )$ for $omega in [0, infty )$ without the need of producing an identified analytical model. It is also shown that complete sets achieving guaranteed levels of performance measures such as gain margin, phase margin, and $H_{infty} $ norms can likewise be calculated directly from only Nyquist/Bode data. The solutions have important new features. For example it is not necessary to know the order of the plant or even the number of left half plane or right half plane poles or zeros. The solution also identifies, in the case of PID controllers an exact low frequency band over which the plant data must be known with accuracy and beyond which the plant information may be rough or approximate. These constitute important new guidelines for identification when the latter is to be used for control design.      

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     

Rotary flexible joint control by fractional order controllers

In this paper, a fractional order control law is proposed and implemented for the evaluation of trajectory tracking performance of a rotary flexible-joint system. A state feedback based fractional integral control scheme is used in this proposed method. In this scheme, state feedback is responsible for stabilizing the system. The compensator, in series with the fractional integrator leads to obtain a similar closed-loop transient response like Bode’s ideal transfer function. The effectiveness of the proposed controller in tracking and being robust against parameter uncertainties is demonstrated through simulation. In addition, to show the usefulness of the proposed control scheme, the fractional controller is compared to an integer state feedback control by simulation and through experimentation on the Quanser’s rotary flexible-joint system.     

A Comparative Approach for Time‐Delay Fractional Optimal Control Problems: Discrete Versus Continuous Chebyshev Polynomials

This paper aims to demonstrate the superiority of the discrete Chebyshev polynomials over the classical Chebyshev polynomials for solving time‐delay fractional optimal control problems (TDFOCPs). The discrete Chebyshev polynomials have been introduced and their properties are investigated thoroughly. Then, the fractional derivative of the state function in the dynamic constraint of TDFOCPs is approximated by these polynomials with unknown coefficients. The operational matrix of fractional integration together with the dynamical constraints is used to approximate the control function directly as a function of the state function. Finally, these approximations were put in the performance index and necessary conditions for optimality transform the under consideration TDFOCPs into an algabric system. A comparison has been made between the required CPU time and accuracy of the discrete and continuous Chebyshev polynomials methods. The obtained numerical results reveal that utilizing discrete Chebyshev polynomials is more efficient and less time‐consuming in comparison to the continuous Chebyshev polynomials.     

A novel root locus calibrated in real gain in the inverses plane

The design of feedback control systems is presented by means of a novel root locus method in the plane of the inverse of the Laplace's variable. The steady-state error constants are unified in a generalized static gain and denominated real gain, which coincides with the gain coefficient used in the Bode, Nyquist, and Nichols methods. The design of compensators becomes easier and straightforward for both steady-state and transient performance. Some illusory effects occurring in the conventional s-plane design are discussed and eliminated     

样条函数在系统频域分析中的应用——系统动态响应实用分析方法之二(对数频率特性法)

自动控制系统的动态性能的分析方法包括时域分析法,频域分析法,及稳定性判据.具体的分析法有微分方程,传递函数,方框图,伯德图,奈奎斯特图和根轨迹等.对数频率特性法是频域分析中应用较为广泛的一种.文章从自动控制系统的动态性能分析方法入手,着重探讨了样条函数在系统频域分析中的应用,并举例加以说明.     

样条函数在系统频域分析中的应用——系统动态响应实用分析方法之二（对数频率特性法）

自动控制系统的动态性能的分析方法包括时域分析法,频域分析法,及稳定性判据。具体的分析法有微分方程,传递函数,方框图,伯德图,奈奎斯特图和根轨迹等。对数频率特性法是频域分析中应用较为广泛的一种。文章从自动控制系统的动态性能分析方法入手,着重探讨了样条函数在系统频域分析中的应用,并举例加以说明。