A review of time domain, frequency domain and stability analysis of linear complex-order systems

Jacob, Joel A.; Tare, Arti V.; Vyawahare, Vishwesh A.; Pande, Vijay N.

doi:10.1109/wiecon-ece.2016.8009110

Cited by 7 publications

(7 citation statements)

References 4 publications

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“…Here, the following definitions are introduced: q = u + iv , r = a + ib and p = c + id ( u > 0 and u , v , a , b , c , d ∈ R ). Due to the lack of sufficient time-response analysis tools for complex-order systems in the literature and therefore the use of approximation methods for complex orders, the values of u and v are usually considered in the range (0, 2) (Adams et al, 2010a, 2010b; Jacob et al, 2016). This deficiency currently limits work on higher-order complex-order systems.…”

Section: Preliminariesmentioning

confidence: 99%

“…Complex conjugate-order modelling studies are more popular for HIV infection with drug resistance during therapy (Ferna´ndez, 2020;Pinto and Carvalho, 2015;Sweilam et al, 2021). In Jacob et al (2016), the time/frequency domain and stability analyses of linear systems represented by differential equations with complex conjugate-order derivatives were investigated. The stability problem of the lateral vibrations of an elastic rod positioned on viscoelastic foundation of complex conjugate-order fractional derivative type was analysed in Atanackovic´et al (2015).…”

Section: Introductionmentioning

confidence: 99%

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Proportional–integral–derivative stabilization of complex conjugate-order systems

Cetintas

Hamamcı

2022

Transactions of the Institute of Measurement and Control

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Proportional–integral–derivative (PID) stabilization is an important control design strategy that provides the designer with all PID controller set which results control system stability absolutely. In this way, the designer has a wide range of freedom to obtain the controller that meets the desired criteria. This process is particularly advantageous in situations where it is difficult to clearly define the design criteria at the beginning of the design or to make a balanced decision among the design criteria. The main objective of this paper is to present for the first time a PID stabilization method for complex conjugate-order systems, which is a new type of system for the control community and has been little studied on. The method is based on obtaining of stability/instability regions using the D-decomposition method in the controller parameter space graphically. These regions are formed by stability boundaries that are defined as real root, infinite root and complex root boundaries. The stability of the regions is determined using generalized modified Mikhailov stability criterion that is a powerful stability tool of the system theory. The simulation results indicate that the presented stabilization method is effective and practically useful in the analysis and control of the complex conjugate-order systems.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Proportional–integral–derivative stabilization of complex conjugate-order systems

Cetintas

Hamamcı

2022

Transactions of the Institute of Measurement and Control

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“…This is due to the additional two parameters available for control in the fractional model structure. The complex order PI a+ jb D c+ jd controller is a logical extension of the fractional calculus based control as it includes two more parameters for control as compared to the FOPID controller [13]- [16].…”

Section: Introductionmentioning

confidence: 99%

“…In [21], the authors modeled HIV infection drug resistance using a complex order model. In [16], the authors reviewed time domain, stability and frequency domain in complex order modeling of various systems. In [22], the authors tuned complex order controller by numerical optimization for a time delay resonant plant.…”

Section: Introductionmentioning

confidence: 99%

Complex Order PIa+jbDc+jd Controller Design for a Fractional Order DC Motor System

Shah¹,

Sekhar²,

Iswanto³

et al. 2021

Adv. sci. technol. eng. syst. j.

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Fractional order controller DC motor Complex order controller Bode diagram Root locus Industry 4.0 implementation stipulates effective actuator control. In the present work, a complex order PI a+ jb D c+ jd (COPID) controller was designed for a fractional order model of a direct current (DC) motor system. For comparisons, the DC motor system model was also controlled using the conventional proportional integral (PI), proportional integral derivative (PID), proportional resonant (PR) and fractional order PID controllers (FOPID). Time domain results indicated that the PR controller performed exceedingly well for output signal responses, but fared poorly in case of control signal specifications. The PI controller responses suffered from high time domain characteristics for both control and output signals. The PID controller performed moderately in terms of time domain and peak overshoot metrics. The FOPID controller attained the best time domain characteristics, but was unable to effectively limit the control and output signal peak overshoots. It was only the COPID controller, that successfully minimised / eliminated peak overshoots in control and output signals (0.1 % and 0.0 % respectively). Moreover, the COPID controller was also successful in limiting the rise, peak and settling times. In addition, Bode diagram, root locus plot were obtained and system gain parameters were varied to confirm the robustness of the proposed COPID controller. Thus, COPID controller promises to be an effective solution towards accurate and robust actuator control in modern manufacturing.

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“…Differential equations of complex fractional order (which should be distinguished from complex differential equations) have been studied in the context of viscoelasticity, control systems, etc. Time domain, frequency domain and stability analysis of linear systems represented by differential equations with complex order derivative has been carried out [9]. It has found applications such as the design of controller for fractionalorder DC motor system [10], in PID controller and low-pass filter [11].…”

Section: Introductionmentioning

confidence: 99%

Stability and Dynamics of Complex Order Fractional Difference Equations

Bhalekar,

Gade,

Joshi

2021

Preprint

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We extend the definition of n-dimensional difference equations to complex order α ∈ C. We investigate the stability of linear systems defined by an n-dimensional matrix A and derive conditions for the stability of equilibrium points for linear systems. For the one-dimensional case where A = λ ∈ C, we find that the stability region, if any is enclosed by a boundary curve and we obtain a parametric equation for the same. Furthermore, we find that there is no stable region if this parametric curve is self-intersecting. Even for λ ∈ R, the solutions can be complex and dynamics in one-dimension is richer than the case for α ∈ R. These results can be extended to n-dimensions. For nonlinear systems, we observe that the stability of the linearized system determines the stability of the equilibrium point.

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A review of time domain, frequency domain and stability analysis of linear complex-order systems

Cited by 7 publications

References 4 publications

Proportional–integral–derivative stabilization of complex conjugate-order systems

Proportional–integral–derivative stabilization of complex conjugate-order systems

Complex Order PIa+jbDc+jd Controller Design for a Fractional Order DC Motor System

Stability and Dynamics of Complex Order Fractional Difference Equations

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