Fractional-order sinusoidal oscillators: Design procedure and practical examples

2008 ◽  
Vol 55 (7) ◽  
pp. 2051-2063 ◽  
Author(s):  
Ahmed Gomaa Radwan ◽  
Ahmed S. Elwakil ◽  
Ahmed M. Soliman
Keyword(s):  
Fractional Order ◽  
Design Procedure ◽  
Sinusoidal Oscillators

scholarly journals μ-Synthesis for Fractional-Order Robust Controllers

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 911
Author(s):  
Vlad Mihaly ◽  
Mircea Şuşcă ◽  
Dora Morar ◽  
Mihai Stănese ◽  
Petru Dobra
Keyword(s):  
Control Problem ◽  
Fractional Order ◽  
Design Procedure ◽  
High Order ◽  
Numerical Control ◽  
Performance Criteria ◽  
Iteration Algorithm ◽  
Optimization Approach ◽  
Cnc Machine ◽  
Bee Colony

The current article presents a design procedure for obtaining robust multiple-input and multiple-output (MIMO) fractional-order controllers using a μ-synthesis design procedure with D–K iteration. μ-synthesis uses the generalized Robust Control framework in order to find a controller which meets the stability and performance criteria for a family of plants. Because this control problem is NP-hard, it is usually solved using an approximation, the most common being the D–K iteration algorithm, but, this approximation leads to high-order controllers, which are not practically feasible. If a desired structure is imposed to the controller, the corresponding K step is a non-convex problem. The novelty of the paper consists in an artificial bee colony swarm optimization approach to compute the nearly optimal controller parameters. Further, a mixed-sensitivity μ-synthesis control problem is solved with the proposed approach for a two-axis Computer Numerical Control (CNC) machine benchmark problem. The resulting controller using the described algorithm manages to ensure, with mathematical guarantee, both robust stability and robust performance, while the high-order controller obtained with the classical μ-synthesis approach in MATLAB does not offer this.

From fractional-order to complex-order integrator loop gain: Robust control design and its stability analysis

2019 ◽  
Vol 41 (13) ◽  
pp. 3799-3807
Author(s):  
Mohammad Reza Rahmani ◽  
Ali Akbar Jalali
Keyword(s):  
Control System ◽  
Stability Analysis ◽  
Robust Control ◽  
Fractional Order ◽  
Design Procedure ◽  
Loop Control ◽  
Complex Order ◽  
Phase Slope ◽  
Nyquist Stability ◽  
Robust Control Design

Complex-order differintegral (COD) is the extended version of fractional-order one in which the differintegral order can be a complex number rather than a real number. In comparison with fractional-order differintegral (FOD), the distinguishing feature of the COD is that the phase slope of its Bode diagram is a function of imaginary part of the complex order of the COD. In this paper, by the use of this property of the COD, a robust control system is proposed. The design procedure and the realization of the proposed COD-based closed-loop control system are discussed. Since the phase of COD’s frequency response is a nonsymmetric function of frequency, stability analysis of the proposed control system is considered a problematic task. It is proven that for the stability of the control system, it is essential that the COD be applied in a limited frequency band that is derived by the use of the Nyquist stability criterion. Finally, some numerical examples are given to demonstrate the validity and superiority of the proposed complex-order control system.

Three Fractional-Order-Capacitors-Based Oscillators with Controllable Phase and Frequency

2017 ◽  
Vol 26 (10) ◽  
pp. 1750160 ◽  
Author(s):  
Lobna A. Said ◽  
Ahmed G. Radwan ◽  
Ahmed H. Madian ◽  
Ahmed M. Soliman
Keyword(s):  
Fractional Order ◽  
Degrees Of Freedom ◽  
Design Procedure ◽  
Building Blocks ◽  
State Matrix ◽  
Current Feedback ◽  
General Design ◽  
Op Amp ◽  
Special Cases ◽  
Current Feedback Operational Amplifier

This paper presents a generalization of six well-known quadrature third-order oscillators into the fractional-order domain. The generalization process involves replacement of three integer-order capacitors with fractional-order ones. The employment of fractional-order capacitors allows a complete tunability of oscillator frequency and phase. The presented oscillators are implemented with three active building blocks which are op-amp, current feedback operational amplifier (CFOA) and second generation current conveyor (CCII). The general state matrix, oscillation frequency and condition are deduced in terms of the fractional-order parameters. The extra degree of freedom provided by the fractional-order elements increases the design flexibility. Eight special cases including the integer case are illustrated with their numerical discussions. Three different phases are produced with fixed sum of [Formula: see text] which can be completely controlled by fractional-order elements. A general design procedure is introduced to design an oscillator with a specific phase and frequency. Two general design cases are discussed based on exploiting the degrees of freedom introduced by the fractional order to obtain the required design. Spice circuit simulations with experimental results for some special cases are presented to validate the theoretical findings.

scholarly journals μ-Synthesis FO-PID for Twin Rotor Aerodynamic System

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2504
Author(s):  
Vlad Mihaly ◽  
Mircea Şuşcă ◽  
Eva H. Dulf
Keyword(s):  
Fractional Order ◽  
Robust Stability ◽  
Optimization Problem ◽  
Order Approximation ◽  
Multiple Input Multiple Output ◽  
Design Procedure ◽  
Optimization Techniques ◽  
And Performance ◽  
Fixed Structure ◽  
Twin Rotor

μ-synthesis is a NP-hard optimization problem based on the generalized Robust Control framework which manages to find a controller which fulfills both robust stability and robust performance. In order to solve such problems, nonsmooth optimization techniques are employed to find nearly-optimal parameters values. However, the free parameters available for tuning must be involved only in classical arithmetic operations, which leads to a problem for the fractional-order operator or for its integer-order approximation, exponential operations being involved. The main goal of the current article consists of presenting a possibility to integrate a fixed-structure multiple-input-multiple-output (MIMO) fractional-order proportional-integral-derivative (FO-PID) controller in the μ-synthesis optimization problem. The solution consists in a possibility to find a set of tunable parameters isomorphic with the fractional-order such that the coefficients involved in the approximation of the fractional element, along with the formulation of a fixed-structure mixed-sensitivity loop shaping μ-synthesis control problem. The proposed design procedure is applied to a twin rotor aerodynamic system (TRAS) using both MATLAB numerical simulation and practical experiments on laboratory scale equipment. Moreover, a comparison with the unstructured μ-synthesis is performed, highlighting the advantages of the proposed solution: simpler form and guaranteed robust stability and performance.

Control of a class of fractional-order systems with mismatched disturbances via fractional-order sliding mode controller

2020 ◽  
Vol 42 (13) ◽  
pp. 2423-2439
Author(s):  
Shabnam Pashaei ◽  
Mohammad Ali Badamchizadeh
Keyword(s):  
Fractional Order ◽  
Control Method ◽  
Sliding Mode ◽  
Estimation Error ◽  
Design Procedure ◽  
Lyapunov Stability Theory ◽  
Fast Response ◽  
Sliding Mode Controller ◽  
Fractional Order Systems ◽  
Mismatched Disturbances

This paper presents a new fractional-order sliding mode controller (FOSMC) for disturbance rejection and stabilization of a class of fractional-order systems with mismatched disturbances. To design this control strategy, firstly, a fractional-order extended disturbance observer (FOEDO) is proposed to estimate the matched and mismatched disturbances and their derivatives. Then, according to the design procedure of the sliding mode controller and based on the designed FOEDO, a proper sliding mode surface is proposed. Subsequently, the proposed FOSMC is designed to guarantee that the system states reach the sliding surface and stay on it forever. The stability of the controlled fractional-order systems is proved via fractional-order Lyapunov stability theory. The numerical examples are used to illustrate the effectiveness of the proposed fractional-order controller. The simulation results of the proposed FOSMC are compared with the results of some other researchers’ works to show the superiority of the proposed control method. The new approach displays some attractive features such as fast response, the chattering reduction, robust stability, less disturbance estimation error, the mismatched disturbance, noise rejection, and better control performance.

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