A Numerical Method for a System of Fractional Differential-Algebraic Equations Based on Sliding Mode Control

Fractional calculus is widely used in engineering fields. In complex mechanical systems, multi-body dynamics can be modelled by fractional differential-algebraic equations when considering the fractional constitutive relations of some materials. In recent years, there have been a few works about the numerical method of the fractional differential-algebraic equations. However, most of the methods cannot be directly applied in the equations of dynamic systems. This paper presents a numerical algorithm of fractional differential-algebraic equations based on the theory of sliding mode control and the fractional calculus definition of Grünwald–Letnikov. The algebraic equation is considered as the sliding mode surface. The validity of the present method is verified by comparing with an example with exact solutions. The accuracy and efficiency of the present method are studied. It is found that the present method has very high accuracy and low time consumption. The effect of violation corrections on the accuracy is investigated for different time steps.

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Combined DAE and Sliding Mode Control Methods for Simulation of Constrained Mechanical Systems

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.1320450 ◽

2000 ◽

Vol 122 (4) ◽

pp. 691-698 ◽

Cited By ~ 6

Author(s):

M. D. Compere ◽

R. G. Longoria

Keyword(s):

Sliding Mode Control ◽

Numerical Method ◽

Hybrid Method ◽

Sliding Mode ◽

Initial Conditions ◽

Differential Algebraic Equations ◽

Control Law ◽

Stabilization Method ◽

Mode Control ◽

Hybrid Numerical Method

In dynamic analysis of constrained multibody systems (MBS), the computer simulation problem essentially reduces to finding a numerical solution to higher-index differential-algebraic equations (DAE). This paper presents a hybrid method composed of multi-input multi-output (MIMO), nonlinear, variable-structure control (VSC) theory and post-stabilization from DAE solution theory for the computer solution of constrained MBS equations. The primary contributions of this paper are: (1) explicit transformation of constrained MBS DAE into a general nonlinear MIMO control problem in canonical form; (2) development of a hybrid numerical method that incorporates benefits of both Sliding Mode Control (SMC) and DAE stabilization methods for the solution of index-2 or index-3 MBS DAE; (3) development of an acceleration-level stabilization method that draws from SMC’s boundary layer dynamics and the DAE literature’s post-stabilization; and (4) presentation of the hybrid numerical method as one way to eliminate chattering commonly found in simulation of SMC systems. The hybrid method presented can be used to simulate constrained MBS systems with either holonomic, nonholonomic, or both types of constraints. In addition, the initial conditions (ICs) may either be consistent or inconsistent. In this paper, MIMO SMC is used to find the control law that will provide two guarantees. First, if the constraints are initially not satisfied (i.e., for inconsistent ICs) the constraints will be driven to satisfaction within finite time using SMC’s stabilization method, urobust,i=−ηisgnsi. Second, once the constraints have been satisfied, the control law, ueq and hybrid stabilization techniques guarantee surface attractiveness and satisfaction for all time. For inconsistent ICs, Hermite-Birkhoff interpolants accurately locate when each surface reaches zero, indicating the transition time from SMC’s stabilization method to those in the DAE literature. [S0022-0434(00)02404-7]

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Sliding Modes for the Simulation of Mechanical and Electrical Systems Defined by Differential-Algebraic Equations

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4001904 ◽

2010 ◽

Vol 5 (4) ◽

Author(s):

Sachit Rao ◽

Vadim Utkin ◽

Martin Buss

Keyword(s):

Sliding Mode Control ◽

Sliding Mode ◽

Initial Conditions ◽

Differential Algebraic Equations ◽

Engineering Systems ◽

Sliding Modes ◽

Algebraic Equations ◽

Electric Networks ◽

Electrical Systems ◽

Mode Control

We offer a technique, motivated by feedback control and specifically sliding mode control, for the simulation of differential-algebraic equations (DAEs) that describe common engineering systems such as constrained multibody mechanical structures and electric networks. Our algorithm exploits the basic results from sliding mode control theory to establish a simulation environment that then requires only the most primitive of numerical solvers. We circumvent the most important requisite for the conventional simulation of DAEs: the calculation of a set of consistent initial conditions. Our algorithm, which relies on the enforcement and occurrence of sliding mode, will ensure that the algebraic equation is satisfied by the dynamic system even for inconsistent initial conditions and for all time thereafter.

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Dynamic Sliding Mode Control Based on Fractional Calculus Subject to Uncertain Delay Based Chaotic Pneumatic Robot

International Journal of Sensors Wireless Communications and Control ◽

10.2174/2210327909666190319142505 ◽

2020 ◽

Vol 10 (3) ◽

pp. 413-420 ◽

Cited By ~ 1

Author(s):

Sara Gholipour ◽

Heydar Toosian Shandiz ◽

Mobin Alizadeh ◽

Sara Minagar ◽

Javad Kazemitabar

Keyword(s):

Fractional Calculus ◽

Sliding Mode Control ◽

Control Method ◽

Sliding Mode ◽

Robot Manipulator ◽

Lyapunov Stability Theory ◽

Mode Control ◽

Control Scheme ◽

Dynamic Sliding Mode Control ◽

Dynamic Sliding Mode

Background & Objective: This paper considers the chattering problem of sliding mode control in the presence of delay in robot manipulator causing chaos in such electromechanical systems. Fractional calculus was used in order to produce a novel sliding mode to eliminate chatter. To realize the control of a class of chaotic systems in master-slave configuration, a novel fractional dynamic sliding mode control scheme is presented and examined on the delay based chaotic robot. Also, the stability of the closed-loop system is guaranteed by Lyapunov stability theory. Methods: A control scheme is proposed for reducing the chattering problem in finite time tracking and robust in presence of system matched disturbances. Results: Moreover, delayed robot motions are sorted out for qualitative and quantitative study. Finally, numerical simulations illustrate feasibility of the proposed control method. Conclusion: The control scheme is viable.

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Adaptive neural Sliding Mode Control for MEMS gyroscope using fractional calculus

2019 34rd Youth Academic Annual Conference of Chinese Association of Automation (YAC) ◽

10.1109/yac.2019.8787729 ◽

2019 ◽

Author(s):

Wang Huimin ◽

Hua Liang ◽

Guo Yunxiang ◽

Chen Hailong ◽

Lu Cheng

Keyword(s):

Fractional Calculus ◽

Sliding Mode Control ◽

Sliding Mode ◽

Mems Gyroscope ◽

Mode Control

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A New Result on Fractional Differential Inequality and Applications to Control of Dynamical Systems

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.4043025 ◽

2019 ◽

Vol 141 (9) ◽

Author(s):

Ngo Van Hoa ◽

Tran Minh Duc ◽

Ho Vu

Keyword(s):

Dynamical Systems ◽

Sliding Mode Control ◽

Fractional Order ◽

Dynamic Systems ◽

Differential Inequality ◽

Sliding Mode ◽

Control Law ◽

Asymptotically Stable ◽

Mode Control ◽

Fractional Differential

In this work, we establish a new estimate result for fractional differential inequality, and this inequality is used to derive a robust sliding mode control law for the fractional-order (FO) dynamic systems. The sliding mode control law is provided to make the states of the system asymptotically stable. Some examples are given to illustrate the results.

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Pseudo-State Sliding Mode Control of Fractional SISO Nonlinear Systems

Advances in Mathematical Physics ◽

10.1155/2013/918383 ◽

2013 ◽

Vol 2013 ◽

pp. 1-7 ◽

Cited By ~ 4

Author(s):

Bao Shi ◽

Jian Yuan ◽

Chao Dong

Keyword(s):

Nonlinear Systems ◽

Differential Equations ◽

Sliding Mode Control ◽

Fractional Differential Equations ◽

Chaotic Systems ◽

Sliding Mode ◽

Control Law ◽

Mode Control ◽

Fractional Differential ◽

Control Methodology

This paper deals with the problem of pseudo-state sliding mode control of fractional SISO nonlinear systems with model inaccuracies. Firstly, a stable fractional sliding mode surface is constructed based on the Routh-Hurwitz conditions for fractional differential equations. Secondly, a sliding mode control law is designed using the theory of Mittag-Leffler stability. Further, we utilize the control methodology to synchronize two fractional chaotic systems, which serves as an example of verifying the viability and effectiveness of the proposed technique.

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