scholarly journals A New Stability Theory for Grünwald–Letnikov Inverse Model Control in the Multivariable LTI Fractional-Order Framework

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1322 ◽  
Author(s):  
Wojciech Przemysław Hunek ◽  
Łukasz Wach
Keyword(s):  
Fractional Order ◽  
Mimo Systems ◽  
Inverse Model ◽  
Integer Order ◽  
Transmission Zeros ◽  
Control Paradigm ◽  
Model Control ◽  
The Stability ◽  
Research Challenge ◽  
Stable Control

The new general theory dedicated to the stability for LTI MIMO, in particular nonsquare, fractional-order systems described by the Grünwald–Letnikov discrete-time state–space domain is presented in this paper. Such systems under inverse model control, principally MV/perfect control, represent a real research challenge due to an infinite number of solutions to the underlying inverse problem for nonsquare matrices. Therefore, the paper presents a new algorithm for fractional-order perfect control with corresponding stability formula involving recently given H- and σ -inverse of nonsquare matrices, up to now applied solely to the integer-order plants. On such foundation a new set of stability-related tools is introduced, among them the key role played by so-called control zeros. Control zeros constitute an extension of transmission zeros for nonsquare fractional-order LTI MIMO systems under inverse model control. Based on the sets of stable control zeros a minimum-phase behavior is specified because of the stability of newly defined perfect control law described in the non-integer-order framework. The whole theory is complemented by pole-free fractional-order perfect control paradigm, a special case of fractional-order perfect control strategy. A significant number of simulation examples confirm the correctness and research potential proposed in the paper methodology.

Improved Internal Model Control-Proportional-Integral- Derivative Fractional-Order Multiloop Controller Design for Non Integer Order Multivariable Systems

2018 ◽  
Vol 141 (1) ◽  
Author(s):  
Tassadit Chekari ◽  
Rachid Mansouri ◽  
Maamar Bettayeb
Keyword(s):  
Fractional Order ◽  
Degrees Of Freedom ◽  
Controller Design ◽  
Mimo Systems ◽  
Design Method ◽  
Internal Model Control ◽  
Tuning Parameter ◽  
Multivariable Systems ◽  
Integer Order ◽  
Model Control

This paper is aimed to propose a multiloop control scheme for fractional order multi-input multi-output (FO-MIMO) systems. It is an extension of the FO-multiloop controller design method developed for integer order multivariable systems to FO-MIMO ones. The interactions among the control loops are considered as disturbances and a two degrees-of-freedom (2DOF) paradigm is used to deal with the process outputs performance and the interactions reduction effect, separately. The proposed controller design method is simple, in relation with the desired closed-loop specifications and a tuning parameter. It presents an interest in controlling complex MIMO systems since fractional order models (FO-models) represent some real processes better than integer order ones and high order systems can be approximated by FO-models. Two examples are considered and compared with other existing methods to evaluate the proposed controller.

Robust fractional-order perfect control for non-full rank plants described in the Grünwald-Letnikov IMC framework

2021 ◽  
Vol 24 (4) ◽  
pp. 1257-1274
Author(s):  
Wojciech P. Hunek ◽  
Tomasz Feliks
Keyword(s):  
State Space ◽  
Fractional Order ◽  
Discrete Time ◽  
State Space Model ◽  
Numerical Algorithms ◽  
Full Rank ◽  
Computational Effort ◽  
Inverse Model ◽  
Control Paradigm ◽  
Model Control

Abstract The advanced analytical study in the field of fractional-order non-full rank inverse model control design is presented in the paper. Following the recent results in this matter it is certain, that the inverse model control-oriented perfect control law can be established for the non-full rank integer-order systems being under the discrete-time state-space reference with zero value. It is shown here, that the perfect control paradigm can be extended to cover the multivariable non-full rank plants governed by the more general Grünwald-Letnikov discrete-time state-space model. Indeed, the postulated approach significantly reduces both iterative and non-iterative computational effort, mainly derived from the approximation of the Moore-Penrose inverse of the non-full rank matrices to finally be inverted. A prevention provided by the new method excludes the detrimental matrix behavior in the form of singularity, often avoided due to the observed ill-conditioned sensitivity. Thus, the new defined robust fractional-order non-full rank instance of such control strategy, supported by the pole-free mechanism, gives rise to the introduction of the general unified non-full rank perfect control-originated theory. Numerical algorithms with simulation investigation clearly confirm the innovative peculiarities provided by the manuscript.

The Inverted Decoupling Based Fractional Order Two-Input-Two-Output IMC Controller

2017 ◽  
Author(s):  
Dazi Li ◽  
Xingyu He
Keyword(s):  
Frequency Domain ◽  
Fractional Order ◽  
Disturbance Rejection ◽  
Internal Model ◽  
Design Method ◽  
Internal Model Control ◽  
Order System ◽  
Single Input Single Output ◽  
Integer Order ◽  
Model Control

Many processes in the industry can be modeled as fractional order, research on the fractional order become more and more popular. Usually, controllers such as fractional order PID (FOPID) or fractional active disturbance rejection control (FADRC) are used to control single-input-single-output (SISO) fractional order system. However, when it comes to fractional order two-input-two-output (TITO) processes, few research focus on this. In this paper, a new design method for fractional order control based on multivariable non-internal model control with inverted decoupling is proposed to handle non-integer order two-input-two-output system. The controller proposed in this paper just has two parameters to tune compared with the five parameters of the FOPID controller, and the controller structure can be achieved by internal model control (IMC) method which means it is easy to implement. The parameters tuning method used in this paper is based on frequency domain strategy. Compared with integer order situation, fractional order method is more complex, because the calculation of the frequency domain characteristics is difficult. The controller proposed in this paper is robust to process gain variations, what’s more, it provides ideal performance for both set point-tracking and disturbance rejection. Numerical results are given to show the performance of the proposed controller.

scholarly journals CentrifugationSynchronization Control of Fractional Order Chaotic Systems Based on Memristor and Its Hardware Implementation

2021 ◽  
pp. 289-297
Author(s):  
Zhaohan zhang, Huiling Jin
Keyword(s):  
Fractional Order ◽  
Hardware Implementation ◽  
Chaotic Systems ◽  
Control Method ◽  
Sliding Mode ◽  
Sliding Mode Controller ◽  
Controlled System ◽  
Integer Order ◽  
The Stability ◽  
Dynamic Phenomena

This paper studies the synchronization control of fractional order chaotic systems based on memristor and its hardware implementation. This paper takes the complex dynamic phenomena of memristor turbidity system as the research background. Starting with the integer order memristor system, the fractional order form is derived based on the integer order turbid system, and its dynamics is deeply studied. At the same time, the turbidity phenomenon is applied to the watermark encryption algorithm, which effectively improves the confidentiality of the algorithm. Finally, in order to suppress the occurrence of turbidity, a fractional order sliding mode controller is proposed. In this paper, the sliding mode controller under the function switching control method is established, and the conditions for the parameters of the sliding mode controller are derived. Finally, the experimental results analyze the stability of the controlled system under different parameters, and give the corresponding time-domain waveform to verify the correctness of the theoretical analysis.

ANTICIPATING SYNCHRONIZATION OF INTEGER ORDER AND FRACTIONAL ORDER HYPER-CHAOTIC CHEN SYSTEM

2012 ◽  
Vol 26 (32) ◽  
pp. 1250211 ◽  
Author(s):  
PENGZHEN DONG ◽  
GANG SHANG ◽  
JIE LIU
Keyword(s):  
Fractional Order ◽  
Chaotic Systems ◽  
Lyapunov Stability Theory ◽  
Chen System ◽  
Integer Order ◽  
Research Fields ◽  
Long Time ◽  
Anticipation Time ◽  
Error System ◽  
The Stability

Such a problem, how to resolve the problem of long-term unpredictability of chaotic systems, has puzzled researchers in nonlinear research fields for a long time during the last decades. Recently, Voss et al. had proposed a new scheme to research the anticipating synchronization of integral-order nonlinear systems for arbitrary initial values and anticipation time. Can this anticipating synchronization be achieved with hyper-chaotic systems? In this paper, we discussed the application of anticipating synchronization in hyper-chaotic systems. Setting integer order and commensurate fractional order hyper-chaotic Chen systems as our research objects, we carry out the research on anticipating synchronization of above two systems based on analyzing the stability of the error system with the Krasovskill–Lyapunov stability theory. Simulation experiments show anticipating synchronization can be achieved in both integer order and fractional order hyper-chaotic Chen system for arbitrary initial value and arbitrary anticipation time.

D-Stability Based LMI Criteria of Stability and Stabilization for Fractional Order Systems

2015 ◽  
Author(s):  
XueFeng Zhang ◽  
YangQuan Chen
Keyword(s):  
Fractional Order ◽  
Feasible Solution ◽  
Order System ◽  
Linear Matrix ◽  
Fractional Order Systems ◽  
Matrix Inequalities ◽  
Integer Order ◽  
Stability And Stabilization ◽  
The Stability ◽  
Conditions Of Stability

This paper considers the stability and stabilization of fractional order systems (FOS) with the fractional order α: 0 < α < 1 case. The equivalence between stability of fractional order systems and D–stability of a matrix A in specific region is proven. The criteria of stability and stabilization of fractional order system are presented. The conditions are expressed in terms of linear matrix inequalities (LMIs) which can be easily calculated with standard feasible solution problem in MATLAB LMI toolbox. When α = 1, the results reduce to the conditions of stability and stabilization of integer order systems. Numerical examples are given to verify the effectiveness of the criteria. With the approach proposed in this paper, we can analyze and design fractional order systems in the same way as what we do to the integer order system state-space models.

scholarly journals Multistability Emergence through Fractional-Order-Derivatives in a PWL Multi-Scroll System

Electronics ◽  
2020 ◽  
Vol 9 (6) ◽  
pp. 880 ◽  
Author(s):  
José Luis Echenausía-Monroy ◽  
Guillermo Huerta-Cuellar ◽  
Rider Jaimes-Reátegui ◽  
Juan Hugo García-López ◽  
Vicente Aboites ◽  
...  
Keyword(s):  
Fractional Order ◽  
Fractional Integration ◽  
Basin Of Attraction ◽  
Parameter Variation ◽  
Order System ◽  
Bifurcation Parameter ◽  
Stable States ◽  
Integer Order ◽  
The Stability ◽  
Integration Order

In this paper, the emergence of multistable behavior through the use of fractional-order-derivatives in a Piece-Wise Linear (PWL) multi-scroll generator is presented. Using the integration-order as a bifurcation parameter, the stability in the system is modified in such a form that produces a basin of attraction segmentation, creating many stable states as scrolls are generated in the integer-order system. The results here presented reproduce the same phenomenon reported in systems with integer-order derivatives, where the multistable regimen is obtained through a parameter variation. The multistable behavior reported is also validated through electronic simulation. The presented results are not only applicable in engineering fields, but they also enrich the analysis and the understanding of the implications of using fractional integration orders, boosting the development of further and better studies.

Dynamical distributed controller for the synchronization problem of integer and fractional order partial differential equation systems

2021 ◽  
pp. 014233122110320
Author(s):  
Juan Pablo Flores-Flores ◽  
Rafael Martínez-Guerra
Keyword(s):  
Fractional Order ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Multiple Systems ◽  
Infinite Dimensional ◽  
Integer Order ◽  
Synchronization Problem ◽  
Distributed Controller ◽  
A Chain ◽  
The Stability

In this work we present a methodology to design dynamical distributed controllers for the synchronization problem of systems governed by partial differential equations of integer and fractional order. We consider fractional systems whose space derivatives are of integer order and time derivatives are of fractional commensurate order. The methodology is based on finding canonical forms by means of a change of variable, such that a distributed controller can be designed in a natural way in the form of a chain of integrators. To study the stability of the integer order closed loop system, we propose to use the spectral and semi-group theory for infinite dimensional Hilbert spaces. On the other hand, the stability criterion previously established by Matignon is used for the stability analysis of the fractional order case. Additionally, we tackle the synchronization problem of multiple systems, which is reduced to a problem of generalized multi-synchronization. To illustrate the effectiveness of the proposed methodology, examples and numerical results are given.

A novel IMC-FOF design for four wheel steering systems of distributed drive electric vehicles

2021 ◽  
pp. 095440702110314
Author(s):  
Yan Ti ◽  
Kangcheng Zheng ◽  
Wanzhong Zhao ◽  
Tinglun Song
Keyword(s):  
Fractional Order ◽  
Electric Vehicles ◽  
Internal Model ◽  
Rear Wheel ◽  
Internal Model Control ◽  
Steering Angle ◽  
Comparison Results ◽  
Model Control ◽  
Tracking Ability ◽  
Internal Model Controller

To improve handling and stability for distributed drive electric vehicles (DDEV), the study on four wheel steering (4WS) systems can improve the vehicle driving performance through enhancing the tracking capability to desired vehicle state. Most previous controllers are either a large amount of calculation, or requires a lot of experimental data, these are relatively time-consuming and laborious. According to the front and rear wheel steering angle of DDEV can be distributed independently, a novel controller named internal model controller with fractional-order filter (IMC-FOF) for 4WS systems is proposed and studied in this paper. The IMC-FOF is designed using the internal model control theory and compared with IMC and PID controller. The influence of time constant and fractional-order parameters which is optimized using quantum genetic algorithms (QGA) on tracking ability of vehicle state are also analyzed. Using a production vehicle as an example, the simulation is performed combining Matlab/Simulink and CarSim. The comparison results indicated that the proposed controller presents performance to distribute the front and rear wheel steering angle for ensuring better tracking capability to desired vehicle state, meanwhile it possesses strong robustness.

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